Authors’ Note
With the exception of this note and small edits to the abstract, the entire document is machine written. We have taken care to supervise this writing to ensure that the proofs meet a minimum standard of readability and that previous literature is appropriately cited. However, the resulting document remains lacking in exposition and overall coherence. We are nonetheless releasing this manuscript on Hexagon to disseminate the result as quickly as possible, because we believe it may be of interest to the mathematics and physics communities. We welcome suggestions concerning mathematical corrections, omitted citations, and attribution of ideas.
P.L. was partially supported by NSF grant DMS-2450004.
Introduction
The Sherrington–Kirkpatrick model and the Parisi formula
Let ( g i j ) 1 ≤ i < j ≤ N (g_{ij})_{1\le i<j\le N} ( g ij ) 1 ≤ i < j ≤ N β > 0 \beta> 0 β > 0
H N ( σ ) = 1 N ∑ 1 ≤ i < j ≤ N g i j σ i σ j , σ ∈ { − 1 , 1 } N . H_N(\sigma)=\frac{1}{\sqrt{N}}\sum_{1\le i<j\le N}g_{ij}\sigma_i\sigma_j,\qquad\sigma\in\{-1,1\}^N. H N ( σ ) = N 1 1 ≤ i < j ≤ N ∑ g ij σ i σ j , σ ∈ { − 1 , 1 } N . Its logarithmic partition function and expected pressure are
F N , β = log ∑ σ ∈ { − 1 , 1 } N e β H N ( σ ) , p N , β = E F N , β N . F_{N,\beta}=\log\sum_{\sigma\in\{-1,1\}^N}e^{\beta H_N(\sigma)},\qquad p_{N,\beta}=\frac{\mathbb{E}F_{N,\beta}}{N}. F N , β = log σ ∈ { − 1 , 1 } N ∑ e β H N ( σ ) , p N , β = N E F N , β . For a probability measure μ \mu μ [ 0 , 1 ] [0,1] [ 0 , 1 ] α ( q ) = μ ( [ 0 , q ] ) \alpha(q)=\mu([0,q]) α ( q ) = μ ([ 0 , q ]) u α u_\alpha u α [ 0 , 1 ] × R [0,1]\times\mathbb{R} [ 0 , 1 ] × R P β \mathcal{P}_\beta P β P β P_\beta P β
∂ q u α = − β 2 2 ( ∂ x x u α + α ( q ) ( ∂ x u α ) 2 ) , u α ( 1 , x ) = log ( 2 cosh x ) , P β ( α ) = u α ( 0 , 0 ) − β 2 2 ∫ 0 1 q α ( q ) d q , P β = inf μ ∈ P ( [ 0 , 1 ] ) P β ( α ) , (1) \begin{aligned}
\partial_q u_\alpha&=-\frac{\beta^2}{2}\left(\partial_{xx}u_\alpha+\alpha(q)(\partial_xu_\alpha)^2\right),\qquad u_\alpha(1,x)=\log(2\cosh x),\\
\mathcal{P}_\beta(\alpha)&=u_\alpha(0,0)-\frac{\beta^2}{2}\int_0^1q\alpha(q)\,dq,\qquad P_\beta=\inf_{\mu\in\mathcal{P}([0,1])}\mathcal{P}_\beta(\alpha), \tag*{(1)}
\end{aligned} ∂ q u α P β ( α ) = − 2 β 2 ( ∂ xx u α + α ( q ) ( ∂ x u α ) 2 ) , u α ( 1 , x ) = log ( 2 cosh x ) , = u α ( 0 , 0 ) − 2 β 2 ∫ 0 1 q α ( q ) d q , P β = μ ∈ P ([ 0 , 1 ]) inf P β ( α ) , (1) where P ( [ 0 , 1 ] ) \mathcal{P}([0,1]) P ([ 0 , 1 ]) [ 0 , 1 ] [0,1] [ 0 , 1 ] μ \mu μ
Sherrington and Kirkpatrick introduced this model and studied it by a replica-symmetric calculation, which fails at low temperature [76]; a longer treatment, with numerical comparisons, is given in [45]. De Almeida and Thouless showed that the replica-symmetric saddle point becomes unstable [34]. Parisi proposed a hierarchical breaking of the replica symmetry in [62], and he developed a sequence of approximate solutions and the order parameter on [ 0 , 1 ] [0,1] [ 0 , 1 ]
The Parisi formula states that p N , β → P β p_{N,\beta}\to P_\beta p N , β → P β N → ∞ N\to\infty N → ∞ p N , β ≤ P β p_{N,\beta}\le P_\beta p N , β ≤ P β N N N μ β \mu_\beta μ β [10 ] , the Parisi measure, whose distribution function we denote by α β \alpha_\beta α β β > 1 \beta> 1 β > 1 [ 0 , q β ] [0,q_\beta] [ 0 , q β ] 0 < q β < 1 0 < q_\beta< 1 0 < q β < 1 β > 1 \beta> 1 β > 1
Finite-size corrections
In this paper, we study the rate of convergence in the Parisi formula at low temperature. By Guerra’s bound, the mean bias P β − p N , β P_\beta-p_{N,\beta} P β − p N , β N N N 1 / β 1/\beta 1/ β § 2.3 ).
At high temperature, the mean bias is small. For 0 < β < 1 0 < \beta< 1 0 < β < 1 N − 1 N^{-1} N − 1 O ( N − 1 ) O(N^{-1}) O ( N − 1 )
Below the critical temperature, the mean bias is expected to decay more slowly. In [4 Theorem 1.1, (1.5)] , we proved that 0 ≤ P β − p N , β ≤ C β N − 1 / 2 0 \le P_\beta-p_{N,\beta} \le C_\beta N^{-1/2} 0 ≤ P β − p N , β ≤ C β N − 1/2 β > 0 \beta> 0 β > 0 N ≥ 1 N \ge1 N ≥ 1 C C C β > 1 \beta> 1 β > 1 μ β \mu_\beta μ β C β N − 2 / 3 C_\beta N^{-2/3} C β N − 2/3 c β N − 11 / 12 c_\beta N^{-11/12} c β N − 11/12 N N N [4 Theorem 1.4] . In physics, the correction is expected to decay like N − ω N^{-\omega} N − ω ω < 1 \omega< 1 ω < 1 ω = 2 / 3 \omega= 2/3 ω = 2/3 ω = 3 / 4 \omega= 3/4 ω = 3/4 ω = 4 / 5 \omega= 4/5 ω = 4/5 § 2.4 . We improve the exponent of the upper bound from 2 / 3 2/3 2/3 7 / 10 7/10 7/10
Main result
Our main result is the following.
Theorem 1.1. For every β > 1 \beta> 1 β > 1 C β < ∞ C_\beta< \infty C β < ∞
P β − p N , β ≤ C β N − 7 / 10 ( N ≥ 2 ) . P_\beta-p_{N,\beta} \le C_\beta N^{-7/10} \qquad(N \ge2). P β − p N , β ≤ C β N − 7/10 ( N ≥ 2 ) . Since P β − p N , β ≥ 0 P_\beta-p_{N,\beta} \ge0 P β − p N , β ≥ 0 N N N E [ − F N , β / ( β N ) ] \mathbb{E}[-F_{N,\beta}/(\beta N)] E [ − F N , β / ( β N )] − P β / β -P_\beta/\beta − P β / β
0 ≤ E [ − F N , β β N ] − ( − P β β ) = P β − p N , β β ≤ C β β N − 7 / 10 ( N ≥ 2 ) . 0 \le\mathbb{E}\left[-\frac{F_{N,\beta}}{\beta N}\right]-\left(-\frac{P_\beta}{\beta}\right)=\frac{P_\beta-p_{N,\beta}}{\beta}\le\frac{C_\beta}{\beta}N^{-7/10}\qquad(N \ge2). 0 ≤ E [ − β N F N , β ] − ( − β P β ) = β P β − p N , β ≤ β C β N − 7/10 ( N ≥ 2 ) . For the expected logarithm of the partition function itself, the corresponding bound is
0 ≤ N P β − E F N , β ≤ C β N 3 / 10 . 0 \le N P_{\beta} - \mathbb{E}F_{N,\beta} \le C_{\beta}N^{3/10}. 0 ≤ N P β − E F N , β ≤ C β N 3/10 . The constant C β C_{\beta} C β β \beta β β ↓ 1 \beta\downarrow1 β ↓ 1 β → ∞ \beta\to\infty β → ∞
Together with the lower bound in [4 Theorem 1.4] , Theorem 1.1 implies that for every fixed β > 1 \beta> 1 β > 1
c β N − 11 / 12 ≤ P β − p N , β ≤ C β N − 7 / 10 . c_{\beta}N^{-11/12} \le P_{\beta} - p_{N,\beta} \le C_{\beta}N^{-7/10}. c β N − 11/12 ≤ P β − p N , β ≤ C β N − 7/10 . for all sufficiently large N N N 7 / 10 7/10 7/10 2 / 3 2/3 2/3 [4 Theorem 1.4] and the exponent 3 / 4 3/4 3/4 § 2.4 . In § 2.6 , we compare Theorem 1.1 with two conditional consequences of results stated in the OpenAI manuscripts [54, 56]. An uncentered comparison stated there would give an unspecified strict improvement over 2 / 3 2/3 2/3 P β − p N , β ≥ c β N − 5 / 6 P_{\beta} - p_{N,\beta} \ge c_{\beta}N^{-5/6} P β − p N , β ≥ c β N − 5/6
We now outline the proof; Section 2 gives a detailed overview. We follow the interpolation scheme of [4 ] , which is a quantitative form of the Guerra interpolation [40] that Talagrand used for the positive fractional moments of the partition function [80]. For a tilt 0 < λ ≤ λ 0 0 < \lambda\le\lambda_{0} 0 < λ ≤ λ 0 e N , β ( λ ) ≥ 0 e_{N,\beta}(\lambda) \ge0 e N , β ( λ ) ≥ 0 λ / β \lambda/\beta λ / β λ / β \lambda/\beta λ / β § 4.1 ). Imposing this mass raises the Parisi value by at most C β λ 5 C_{\beta}\lambda^{5} C β λ 5 [4 Lemma 7.2] . Using this bound and the convexity of Gaussian logarithmic moments [26, Theorem 1], as in the proof of [4 Theorem 7.8, Step 1] , we find
0 ≤ P β − p N , β β ≤ C β λ 5 + 2 e N , β ( λ ) . 0 \le\frac{P_{\beta} - p_{N,\beta}}{\beta} \le C_{\beta}\lambda^{5} + 2e_{N,\beta}(\lambda). 0 ≤ β P β − p N , β ≤ C β λ 5 + 2 e N , β ( λ ) . For sufficiently large N N N λ = N − 7 / 50 \lambda= N^{-7/50} λ = N − 7/50 e N , β ( N − 7 / 50 ) ≤ C β N − 7 / 10 e_{N,\beta}(N^{-7/50}) \le C_{\beta}N^{-7/10} e N , β ( N − 7/50 ) ≤ C β N − 7/10
Up to a factor of four, e N , β ( λ ) e_{N,\beta}(\lambda) e N , β ( λ ) ρ ∈ ( 0 , 1 ] \rho\in(0,1] ρ ∈ ( 0 , 1 ] D N ( ρ ) = β ( A + 2 J + V ) D_{N}(\rho) = \beta(A + 2J + V) D N ( ρ ) = β ( A + 2 J + V ) N N N A A A J J J § 4.1 and 6). The difficulty is V V V ρ \rho ρ ρ 0 \rho_{0} ρ 0 V V V V ≤ ( 1 − ρ ) ( 1 + ε N ) V + errors V \le(1-\rho)(1+\varepsilon_{N})V + \text{errors} V ≤ ( 1 − ρ ) ( 1 + ε N ) V + errors 1 + ε N 1+\varepsilon_{N} 1 + ε N ε N ≤ ρ / 4 \varepsilon_{N} \le\rho/4 ε N ≤ ρ /4 μ β \mu_{\beta} μ β ρ ≤ ρ 0 \rho\le\rho_{0} ρ ≤ ρ 0 V V V [4 Section 3] , refined in Section 5, and by a localization of the lowest quantiles of the minimizer (Section 8), which refines [4 Lemma 7.6] . The cutoff is ρ 0 = min { 1 , K β η 1 / 7 } \rho_0 = \min\{1, K_\beta\eta^{1/7}\} ρ 0 = min { 1 , K β η 1/7 } η = e N , β ( λ ) + C β λ 5 \eta= e_{N,\beta}(\lambda) + C_\beta\lambda^5 η = e N , β ( λ ) + C β λ 5
e N , β ( λ ) ≲ N − 2 / 3 ( e N , β ( λ ) + λ 5 ) 1 / 21 + smaller terms . e_{N,\beta}(\lambda) \lesssim N^{-2/3}\left(e_{N,\beta}(\lambda) + \lambda^5\right)^{1/21} + \text{smaller terms}. e N , β ( λ ) ≲ N − 2/3 ( e N , β ( λ ) + λ 5 ) 1/21 + smaller terms . If e N , β ( λ ) e_{N,\beta}(\lambda) e N , β ( λ ) λ 5 \lambda^5 λ 5 N − a N^{-a} N − a a = 2 / 3 + a / 21 a = 2/3 + a/21 a = 2/3 + a /21 a = 7 / 10 a = 7/10 a = 7/10
Relation to physical predictions
Expectations and early calculations. Below the critical temperature, physicists expect the finite-size correction to the free energy density to be nonanalytic, of order N − ω N^{-\omega} N − ω ω < 1 \omega< 1 ω < 1 ( log N ) / ( 12 N ) (\log N)/(12N) ( log N ) / ( 12 N ) [14 ] ), as well as slightly below criticality and along the de Almeida–Thouless line, including corrections to overlap distributions and to ultrametricity [70].
The exponent 2 / 3 2/3 2/3 At zero temperature, Bouchaud, Krzakala, and Martin found numerically, for Gaussian couplings, that the mean ground-state energy per spin approaches its limit like N − 2 / 3 N^{-2/3} N − 2/3 [19 Section VI B] . In a study of the SK model with binary couplings, Boettcher fits the corrections to the mean of the ground-state energy per spin and to the width of its distribution with exponents close to 2 / 3 2/3 2/3 3 / 4 3/4 3/4 [15 Section 3] . In a later study of diluted versions of the model, the fit for the undiluted model is again close to 2 / 3 2/3 2/3 [17 Table I] . At positive temperature, Billoire estimated the correction numerically for binary couplings, by integrating overlap observables along the interpolation of Guerra and Toninelli [41], and found data compatible with ω = 2 / 3 \omega= 2/3 ω = 2/3 [14 ] . Aspelmeier, Billoire, Marinari, and Moore derived ω = 2 / 3 \omega= 2/3 ω = 2/3 N N N N 1 / 6 N^{1/6} N 1/6 K K K K − 4 K^{-4} K − 4 [8 Section 2] .
Longitudinal fluctuations and the exponent 4 / 5 4/5 4/5 Parisi and Biscari obtained an earlier contribution of order N − 3 / 4 log N N^{-3/4}\log N N − 3/4 log N ω = 4 / 5 \omega= 4/5 ω = 4/5
The exponent 3/4. Franz, Parisi, and Ricci-Tersenghi [38] argue that ω = 2 / 3 \omega= 2/3 ω = 2/3 ω = 3 / 4 \omega= 3/4 ω = 3/4 ( N t ) − 1 / 2 (Nt)^{-1/2} ( N t ) − 1/2 t − 3 / 2 / N t^{-3/2}/N t − 3/2 / N z z z
Consequences of Theorem 1.1. Theorem 1.1 implies that P β − p N , β = O ( N − 7 / 10 ) P_\beta- p_{N,\beta} = O(N^{-7/10}) P β − p N , β = O ( N − 7/10 ) β > 1 \beta> 1 β > 1 o ( N − 2 / 3 ) o(N^{-2/3}) o ( N − 2/3 ) N − 2 / 3 N^{-2/3} N − 2/3 ω \omega ω β → ∞ \beta\to\infty β → ∞
The estimates on which our proof rests have a similar structure with two regimes, although they concern a different interpolation. If we ignore logarithms and the dependence on the tilt λ \lambda λ D N ( ρ ) D_N(\rho) D N ( ρ ) ( N ρ ) − 2 / 3 (N\rho)^{-2/3} ( N ρ ) − 2/3 ρ \rho ρ ( N ρ 2 ) − 1 (N\rho^2)^{-1} ( N ρ 2 ) − 1 ρ ≥ N − 1 / 4 \rho\ge N^{-1/4} ρ ≥ N − 1/4 N − 3 / 4 N^{-3/4} N − 3/4 ρ 0 ≥ N − 1 / 10 \rho_0 \ge N^{-1/10} ρ 0 ≥ N − 1/10 λ = N − 7 / 50 \lambda= N^{-7/50} λ = N − 7/50 7 / 10 7/10 7/10 3 / 4 3/4 3/4
Marginal stability also enters our proof, through a scalar statement: at the Parisi measure, the multiplier that dominates our scalar response operator equals one on supp ν β \operatorname{supp}\nu_\beta supp ν β
Mean, width, and large deviations. The exponents ω \omega ω n n n n n n lim N → ∞ ( N n ) − 1 log E e n F N , β \lim_{N\to\infty}(Nn)^{-1}\log\mathbb{E}e^{nF_{N,\beta}} lim N → ∞ ( N n ) − 1 log E e n F N , β P β P_\beta P β n 5 n^5 n 5 n ↓ 0 n\downarrow0 n ↓ 0 n > 0 n>0 n > 0 N − 1 log E e n ( F N , β − N P β ) N^{-1}\log\mathbb{E}e^{n(F_{N,\beta}-NP_\beta)} N − 1 log E e n ( F N , β − N P β ) n 6 n^6 n 6 6 / 5 6/5 6/5 F N , β / N F_{N,\beta}/N F N , β / N N − 5 / 6 N^{-5/6} N − 5/6 F N , β F_{N,\beta} F N , β N N N 6 / 5 6/5 6/5 N 2 N^2 N 2 12 / 5 12/5 12/5 N − 5 / 6 N^{-5/6} N − 5/6 n ↓ 0 n\downarrow0 n ↓ 0 N → ∞ N\to\infty N → ∞ N 1 / 3 N^{1/3} N 1/3 F N , β F_{N,\beta} F N , β [8 ] . Aspelmeier derived exact Gaussian interpolation identities relating the variance of the free energy to two- and four-replica overlaps between systems with correlated disorder [7 Eqs. (14) and (31)] . Combining these identities with replica estimates of the bond-chaos overlap distribution gives a predicted upper bound 1 / 4 1/4 1/4 [6 , 7 Section 4.3] . The identities are exact; this exponent estimate uses replica approximations and concerns the width. At zero temperature, the width of the distribution of the ground-state energy has been studied numerically in [15 , 16 ] [58]. For Gaussian couplings, Palassini finds data consistent with a mean correction exponent 2 / 3 2/3 2/3 5 / 6 5/6 5/6 3 / 4 3/4 3/4 3 / 4 3/4 3/4 5 / 6 5/6 5/6 [16 Section IV] . Monthus and Garel studied rare-disorder scenarios and the matching between typical and large-deviation tails for the ground-state energy of the SK model at zero temperature [53, Section III C–F]. Their alternative width–tail relations illustrate the additional assumptions involved in such matching.
Related work
The Parisi formula. Besides the proofs of Guerra and Talagrand, the rigorous theory of the Parisi formula includes the following results. Guerra and Toninelli proved by interpolation that the thermodynamic limit exists [41], and Aizenman, Sims, and Starr formulated an extended variational principle in terms of random overlap structures [2 ] . Panchenko proved that the Ghirlanda–Guerra identities [39] imply ultrametricity of overlaps [60, Theorem 1], and he proved the Parisi formula for mixed p p p
The Parisi measure. Toninelli proved that the replica-symmetric value is not optimal for β > 1 \beta> 1 β > 1 [10 ] , that zero belongs to its support, and that the largest point of its support is an atom for 1 < β ≤ 3 / 2 1 < \beta\le3/2 1 < β ≤ 3/2 [9 Theorems 1 and 4] . They also proved that its distribution function is smooth on [ a , b ) [a,b) [ a , b ) ( a , b ) (a,b) ( a , b ) 0 ≤ a < b ≤ 1 0 \le a < b \le1 0 ≤ a < b ≤ 1 [9 Theorem 2] . They expressed the Parisi functional through a stochastic control problem, which they used in their uniqueness proof, and they proved martingale identities for the derivatives of u α u_\alpha u α [10 Theorem 3 and Lemma 2] . An earlier Hamilton–Jacobi–Bellman representation appears in [20 Section 6.2] , and Jagannath and Tobasco developed a dynamic programming treatment of the functional [43, Lemma 18]. First-variation formulas along segments and optimality conditions on the support are given by Jagannath and Tobasco [44, Lemma 3.2 and Proposition 1.1] and by Chen [25, Theorem 2 and Proposition 1]. Lopatto proved, in addition to the interval support for every β > 1 \beta> 1 β > 1 μ β \mu_\beta μ β [ 0 , q β ] [0,q_\beta] [ 0 , q β ] q β q_\beta q β [11 ] . H.-B. Chen proved that q β → 1 q_\beta\to1 q β → 1 β → ∞ \beta\to\infty β → ∞ [ 0 , 1 ) [0,1) [ 0 , 1 ) [23 Theorems 1.2 and 1.3] . The full support at zero temperature is also stated in [55, Theorem 1.1]. We use the interval support at fixed β > 1 \beta> 1 β > 1 μ β \mu_\beta μ β
Fluctuations of the free energy. The fluctuations of F N , β F_{N,\beta} F N , β β \beta β 0 < β < 1 0 < \beta< 1 0 < β < 1 [1 ] , and Comets and Neveu studied this limit by stochastic calculus [30]. With a nonzero external field, Chen, Dey, and Panchenko proved that the variance is of order N N N β = 1 \beta= 1 β = 1 log 2 N \log^2 N log 2 N ( log N ) / 4 + C (\log N)/4 + C ( log N ) /4 + C log log log N \log\log\log N log log log N Var ( F N , 1 ) = ( log N ) / 6 + O ( 1 ) \operatorname{Var}(F_{N,1}) = (\log N)/6 + O(1) Var ( F N , 1 ) = ( log N ) /6 + O ( 1 ) β \beta β β = β N \beta= \beta_N β = β N N 1 / 3 ( 1 − β N ) → c ∈ ( 0 , ∞ ] N^{1/3}(1-\beta_N) \to c \in(0,\infty] N 1/3 ( 1 − β N ) → c ∈ ( 0 , ∞ ] β > 1 \beta> 1 β > 1 Var ( F N , β ) ≤ C β N / log N \operatorname{Var}(F_{N,\beta}) \le C_\beta N/\log N Var ( F N , β ) ≤ C β N / log N [22 Theorem 1.5] . In [4 Theorem 1.4] , we proved that Var ( F N , β ) \operatorname{Var}(F_{N,\beta}) Var ( F N , β ) N 4 / 15 N^{4/15} N 4/15 N 7 / 15 N^{7/15} N 7/15 N N N N 1 / 3 + o ( 1 ) N^{1/3+o(1)} N 1/3 + o ( 1 ) N − 1 / 3 Var ( F N , β ) N^{-1/3}\operatorname{Var}(F_{N,\beta}) N − 1/3 Var ( F N , β ) § 2.6 . A variance bound alone does not bound the mean bias from above, but a lower bound on the variance gives a lower bound on the mean bias through [4 Lemma 7.4] . For the spherical SK model, Baik and Lee proved that the free energy per spin has Gaussian fluctuations of order N − 1 N^{-1} N − 1 N − 2 / 3 N^{-2/3} N − 2/3 [12 Theorems 2.10 and 2.11] .
Gaussian-process comparisons. Chu and Raginsky use Gaussian-process geometry and information-theoretic tensorization to bound expected soft maxima [29]. Their application to the SK model gives a finite-volume comparison with the Parisi variational value up to universal multiplicative constants [29, Section 6, Theorem 10]. It uses a uniform spin prior and normalizes the Hamiltonian differently. Its conclusion is a multiplicative comparison; the present theorem controls the additive error to the limiting value.
Fractional moments. Talagrand computed the limits of normalized logarithms of positive moments of the partition function [80]. For moments of order 0 < s < 1 0 < s < 1 0 < s < 1 ( N s ) − 1 log E e s F N , β (Ns)^{-1}\log\mathbb{E}e^{sF_{N,\beta}} ( N s ) − 1 log E e s F N , β P β ( s ) P_\beta(s) P β ( s ) P β P_\beta P β μ \mu μ μ ( { 0 } ) ≥ s \mu(\{0\}) \ge s μ ({ 0 }) ≥ s [4 Lemma 7.2 and Proposition 7.3] , we proved that for fixed β > 1 \beta> 1 β > 1 P β ( s ) − P β P_\beta(s) - P_\beta P β ( s ) − P β s 5 s^5 s 5 s ↓ 0 s \downarrow0 s ↓ 0 § 2.4 , and Pei identified the leading coefficient of this increase [73, Theorem 1(a)]. The tilt λ = N − 7 / 50 \lambda= N^{-7/50} λ = N − 7/50
Origins of the methods. The interpolation of [4 Section 2] is a finite-dimensional, quantitative form of Guerra’s interpolation, with which he proved the upper bound in the Parisi formula [40]. For posterior estimates under its tilted laws, we combine the Brascamp–Lieb inequality [21 Theorem 4.1] , in its inverse-Hessian form [4 Proposition 2.2] , with a joint-source argument from the proof of [54, Lemma 2.3] (Section 4). Near the singular end of the interpolation, our estimates refine the quantile-window bound of [4 Lemma 3.3] . and the localization of lowest quantiles in [4 Lemma 7.6] . We close the final implicit inequality as in the proof of [4 Proposition 7.7] .
The cavity method originates in physics. An early precursor is the single-site description of Thouless, Anderson, and Palmer [83, Section 3, Eqs. (10)–(12)], and Mézard, Parisi, and Virasoro formulated the cavity construction for broken replica symmetry in [51]; see also the exposition in [52]. We use Talagrand’s rigorous version of the one-site argument [81, Section 1.6]. The expansion under hierarchical Gaussian laws relies on normalized replica trees and covariance insertions of the type developed in [54, Section 2]. In the scalar stability argument, the first variation of P β \mathcal{P}_{\beta} P β [9 Section 3, (19)] . We compute the second variation and compare it with a response multiplier, using convexity of P β \mathcal{P}_{\beta} P β [10 ] . This multiplier equals one on supp μ β \operatorname{supp}\mu_{\beta} supp μ β [9 Theorem 5] and [48, Lemma 2.6], and by the interval support [48, Theorem 1.1]. Above the support, we bound it by one using the curvature of the forward density and its heat evolution from [48, Proposition 4.1 and Lemma 2.4(ii)].
Our related work. This paper continues [4 ] , where we proved the quantitative Parisi formulas recalled in § 2.2 , the variance bounds above, and upper-tail estimates for F N , β F_{N,\beta} F N , β 6 / 5 6/5 6/5 [4 Theorem 1.5] . The interval-support theorem used in both papers was proved by the second author [48]. In [5 ] , we studied a different approximation of the SK pressure. On d d d L ≥ 3 L \ge3 L ≥ 3 1 / ( 2 d ) 1/(2d) 1/ ( 2 d ) 1 / d 1/d 1/ d d → ∞ d \to\infty d → ∞ β ≥ 0 \beta\ge0 β ≥ 0 h ∈ R h \in\mathbb{R} h ∈ R
Comparison with the OpenAI preprints
We obtained the exponent 7 / 10 7/10 7/10 The low-temperature Sherrington–Kirkpatrick fluctuation scale [54] and its companion The low-temperature Sherrington–Kirkpatrick free-energy limiting law [56], both dated September 24, 2026. We refer to their initial repository version.
Stated results. For every fixed β > 1 \beta> 1 β > 1 Var ( F N , β ) = N 1 / 3 + o ( 1 ) \operatorname{Var}(F_{N,\beta}) = N^{1/3+o(1)} Var ( F N , β ) = N 1/3 + o ( 1 ) N − 1 / 3 Var ( F N , β ) N^{-1/3}\sqrt{\operatorname{Var}(F_{N,\beta})} N − 1/3 Var ( F N , β ) F N , β F_{N,\beta} F N , β E F N , β EF_{N,\beta} E F N , β P β − p N , β P_\beta-p_{N,\beta} P β − p N , β
A conditional upper bound. An improvement of the exponent 2 / 3 2/3 2/3 T = β 2 T=\beta^2 T = β 2 x = N − 1 / 6 x=N^{-1/6} x = N − 1/6 c G > 0 c_G>0 c G > 0
0 ≤ F ∗ ( λ ) − f N ( b λ , h ∗ , λ ) ≤ x 4 + c G (2) 0 \le F_*(\lambda)-f_N(b_\lambda,h_{*,\lambda}) \le x^{4+c_G} \tag*{(2)} 0 ≤ F ∗ ( λ ) − f N ( b λ , h ∗ , λ ) ≤ x 4 + c G (2) for all sufficiently large N N N F ∗ F_* F ∗ f N f_N f N ( b λ , h ∗ , λ ) (b_\lambda,h_{*,\lambda}) ( b λ , h ∗ , λ ) λ \lambda λ λ = x 12 \lambda=x^{12} λ = x 12 λ = 0 \lambda=0 λ = 0 P β − log 2 − T / 4 P_\beta-\log2-T/4 P β − log 2 − T /4
∥ b λ − T ∥ ∞ + ∥ h ∗ , λ ∥ ∞ ≤ C β λ , ∣ F ∗ ( λ ) − ( P β − log 2 − T 4 ) ∣ ≤ C β λ . \lVert b_\lambda-T\rVert_\infty+\lVert h_{*,\lambda}\rVert_\infty\le C_\beta\lambda,\qquad\left|F_*(\lambda)-\left(P_\beta-\log2-\frac{T}{4}\right)\right|\le C_\beta\lambda. ∥ b λ − T ∥ ∞ + ∥ h ∗ , λ ∥ ∞ ≤ C β λ , F ∗ ( λ ) − ( P β − log 2 − 4 T ) ≤ C β λ . With the uniform spin prior and centered Gaussian completion of [56, Section 2.1], the unperturbed clocks satisfy f N ( T , 0 ) = p N , β − log 2 − T / 4 f_N(T,0)=p_{N,\beta}-\log2-T/4 f N ( T , 0 ) = p N , β − log 2 − T /4 [ 0 , 1 ] [0,1] [ 0 , 1 ]
∣ f N ( b λ , h ∗ , λ ) − f N ( T , 0 ) ∣ ≤ 1 4 ∥ b λ − T ∥ 1 + 1 2 ∥ h ∗ , λ ∥ 1 ≤ C β λ . \left|f_N(b_\lambda,h_{*,\lambda})-f_N(T,0)\right|\le\frac{1}{4}\lVert b_\lambda-T\rVert_1+\frac{1}{2}\lVert h_{*,\lambda}\rVert_1\le C_\beta\lambda. ∣ f N ( b λ , h ∗ , λ ) − f N ( T , 0 ) ∣ ≤ 4 1 ∥ b λ − T ∥ 1 + 2 1 ∥ h ∗ , λ ∥ 1 ≤ C β λ . We write P β − p N , β P_\beta-p_{N,\beta} P β − p N , β P β − log 2 − T / 4 − F ∗ ( λ ) P_\beta-\log2-T/4-F_*(\lambda) P β − log 2 − T /4 − F ∗ ( λ ) F ∗ ( λ ) − f N ( b λ , h ∗ , λ ) F_*(\lambda)-f_N(b_\lambda,h_{*,\lambda}) F ∗ ( λ ) − f N ( b λ , h ∗ , λ ) f N ( b λ , h ∗ , λ ) − f N ( T , 0 ) f_N(b_\lambda,h_{*,\lambda})-f_N(T,0) f N ( b λ , h ∗ , λ ) − f N ( T , 0 ) C β λ C_\beta\lambda C β λ x 4 + c G = N − 2 / 3 − c G / 6 x^{4+c_G}=N^{-2/3-c_G/6} x 4 + c G = N − 2/3 − c G /6 x 12 = N − 2 x^{12}=N^{-2} x 12 = N − 2
0 ≤ P β − p N , β ≤ x 4 + c G + C β x 12 ≤ C β N − 2 / 3 − δ β , δ β = c G / 6 > 0 , (3) 0\le P_\beta-p_{N,\beta}\le x^{4+c_G}+C_\beta x^{12}\le C_\beta N^{-2/3-\delta_\beta},\qquad\delta_\beta=c_G/6>0, \tag*{(3)} 0 ≤ P β − p N , β ≤ x 4 + c G + C β x 12 ≤ C β N − 2/3 − δ β , δ β = c G /6 > 0 , (3) after decreasing c G c_G c G c G < 8 c_G<8 c G < 8 N ≥ 2 N\ge2 N ≥ 2 C β C_\beta C β c G c_G c G c G ≥ 1 / 5 c_G\ge1/5 c G ≥ 1/5 7 / 10 7/10 7/10 x 5 + c x^{5+c} x 5 + c
Conditional lower bounds. In the other direction, the variance–mean-deficit inequality of [4 Lemma 7.4 and Remark 7.14, (7.68)] states that for every N ≥ 1 N \ge1 N ≥ 1
P β − p N , β ≥ c β ( Var ( F N , β ) + β 2 / 2 N ) 5 / 4 . P_\beta- p_{N,\beta} \ge c_\beta\left(\frac{\operatorname{Var}(F_{N,\beta})+\beta^2/2}{N}\right)^{5/4}. P β − p N , β ≥ c β ( N Var ( F N , β ) + β 2 /2 ) 5/4 . Inserting Var ( F N , β ) ≥ N 1 / 3 − 4 ε / 5 \operatorname{Var}(F_{N,\beta}) \ge N^{1/3-4\varepsilon/5} Var ( F N , β ) ≥ N 1/3 − 4 ε /5 ε > 0 \varepsilon> 0 ε > 0 P β − p N , β ≥ c β , ε N − 5 / 6 − ε P_\beta-p_{N,\beta} \ge c_{\beta,\varepsilon}N^{-5/6-\varepsilon} P β − p N , β ≥ c β , ε N − 5/6 − ε N N N N − 1 / 3 Var ( F N , β ) N^{-1/3}\operatorname{Var}(F_{N,\beta}) N − 1/3 Var ( F N , β )
P β − p N , β ≥ c β N − 5 / 6 for all sufficiently large N . (4) P_\beta-p_{N,\beta} \ge c_\beta N^{-5/6} \qquad\text{for all sufficiently large }N. \tag*{(4)} P β − p N , β ≥ c β N − 5/6 for all sufficiently large N . (4) For (4) , an eventual lower bound Var ( F N , β ) ≥ k β N 1 / 3 \operatorname{Var}(F_{N,\beta}) \ge k_\beta N^{1/3} Var ( F N , β ) ≥ k β N 1/3 k β > 0 k_\beta> 0 k β > 0 P β − p N , β P_\beta-p_{N,\beta} P β − p N , β N − a N^{-a} N − a 7 / 10 ≤ a ≤ 5 / 6 7/10 \le a \le5/6 7/10 ≤ a ≤ 5/6
Verification status. We have checked the statements quoted above and the conditional deductions from them. We have not independently reconstructed complete proofs of the global comparison and fluctuation estimates of [54, 56], and we do not assume their conclusions. The repository also contains Lean sources on the fluctuations of the SK model. Their main theorem on the fluctuation exponents, as we inspected it, takes as an explicit hypothesis a two-sided bound on the centered logarithmic Laplace transform of the completed free energy, PositiveLaplaceWindow; in the fluctuation-scale paper, this bound is the conclusion of [54, Proposition 7.5]. The conclusion of that theorem concerns the fluctuation exponents, not a positive variance prefactor or a limiting law.1
Proof techniques. The limiting-law paper contains a one-site cavity expansion whose structure is similar to that of Part IV. In [56, Lemma 2.2], one site is chosen by exchangeability and removed, and its Gaussian kernels are interpolated to those of an independent scalar model on one site; the linear cavity kernel is half of a connected four-spin correlation of the removed spin [56, (A8)]. In the proof of [56, Proposition 6.1], the resulting linear equation is inverted under a gap condition on the scalar response [56, Proposition 5.1, (H1)], and [56, Lemma 5.2, (H2)] writes the response on the additive sector as a susceptibility multiplier minus a positive semidefinite Hessian kernel. In our argument, the cavity inequality contracts because the scalar response operator is at most 1 + ε N 1+\varepsilon_N 1 + ε N ε N ≤ ρ / 4 \varepsilon_N \le\rho/4 ε N ≤ ρ /4 1 − ρ 1-\rho 1 − ρ
Organization and conventions
Section 2 contains an overview of the proof and a map of its dependencies. The proof itself occupies five parts.
Part I concerns the interpolation. Section 3 recalls the interpolation of [4 ] , its tilted law, and the decomposition of the defect, and reduces the mean bias to the fractional-moment error. Section 4 proves two Gaussian tools for tilted laws defined by arbitrary profiles, which we use in Parts III and IV.
Part II contains overlap estimates at a minimizing profile. Section 5 estimates moments of the self-overlap on quantile intervals, and Section 6 bounds the mean–covariance term.
Part III concerns the minimizing profile and the scalar response. Section 7 shows that the minimizing profile is close in L 1 L^1 L 1
Part IV contains the cavity argument. Section 10 sets up the cavity interpolation and its tree laws, Section 11 computes the derivative at the independent endpoint, and Section 12 closes the cavity inequality and bounds the defect.
In Part V, Section 13 integrates these estimates and proves Theorem 1.1 .
We rely on [4 ] for the interpolation identities and the scalar floor estimates, and on [48] and [9 ] for the structure and regularity of μ β \mu_\beta μ β [4 ] refer to its arXiv version 1, locators in [48] to its arXiv version 4, and locators in the OpenAI manuscripts to their initial repository version.
Throughout, β > 1 \beta> 1 β > 1 C β C_{\beta} C β β \beta β N N N ρ \rho ρ λ \lambda λ B ⪯ C B \preceq C B ⪯ C C − B C - B C − B log + x = log ( max { 1 , x } ) \log_{+} x = \log(\max\{1,x\}) log + x = log ( max { 1 , x }) x ≥ 0 x \ge0 x ≥ 0 α \alpha α a a a § 4.1 , are right-continuous. Atoms of measures are retained in Stieltjes integrals, and an interval ( q , 1 ] (q,1] ( q , 1 ] q q q
Overview of the proof
In this section, we explain the proof of Theorem 1.1, which occupies Sections 3 to 13. For each step, we state its main estimate in the form in which it is proved, give the precise reference, and explain briefly why the estimate holds and where it is used. Section 2.11 contains a map of the proof, and Section 2.10 discusses three conventions that recur in every part of it. Throughout, β > 1 \beta> 1 β > 1 X ≲ Y X \lesssim Y X ≲ Y X ≤ C β Y X \le C_{\beta}Y X ≤ C β Y C β C_{\beta} C β β \beta β N N N ρ \rho ρ λ \lambda λ
The structure of the argument
We follow the interpolation scheme of [4 ] ; what changes is how we estimate the self-overlap variance along the interpolation. The argument has four steps.
First, we reduce Theorem 1.1 to a bound on a fractional-moment error e N , β ( λ ) e_{N,\beta}(\lambda) e N , β ( λ ) λ / β \lambda/\beta λ / β λ / β \lambda/\beta λ / β β \beta β 2 e N , β ( λ ) 2e_{N,\beta}(\lambda) 2 e N , β ( λ ) λ 5 \lambda^5 λ 5 e N , β ( N − 7 / 50 ) ≲ N − 7 / 10 e_{N,\beta}(N^{-7/50}) \lesssim N^{-7/10} e N , β ( N − 7/50 ) ≲ N − 7/10 § 3.2 ).
Second, we write 4 e N , β ( λ ) 4e_{N,\beta}(\lambda) 4 e N , β ( λ ) ρ ∈ ( 0 , 1 ] \rho\in(0,1] ρ ∈ ( 0 , 1 ] D N ( ρ ) = β ( A + 2 J + V ) D_N(\rho) = \beta(A + 2J + V) D N ( ρ ) = β ( A + 2 J + V ) N N N § 3.3 ). The terms A A A J J J log 2 ( 2 + N ) / ( N ρ λ ) \log^2(2 + N)/(N\rho\lambda) log 2 ( 2 + N ) / ( N ρ λ ) § 3.4 ), and the difficulty is the self-overlap variance V V V
Third, we bound V V V ρ = 0 \rho= 0 ρ = 0 § 3.5 ). For ρ \rho ρ ρ 0 \rho_0 ρ 0 V V V § 3.6 ), which contracts if a scalar response operator at the minimizing profile is at most 1 + ρ / 4 1 + \rho/4 1 + ρ /4 § 3.7 ) and that the minimizing profile is close to a scalar trial profile (§ 3.8 ). The two regimes give the following bounds. For almost every ρ ∈ [ ρ 0 , 1 ] \rho\in[\rho_0,1] ρ ∈ [ ρ 0 , 1 ]
D N ( ρ ) ≲ log 2 ( 2 + N ) N λ ρ 2 (5) D_N(\rho) \lesssim\frac{\log^2(2 + N)}{N\lambda\rho^2} \tag*{(5)} D N ( ρ ) ≲ N λ ρ 2 log 2 ( 2 + N ) (5) by Corollary 12.5, and for almost every ρ ∈ ( 0 , 1 ] \rho\in(0,1] ρ ∈ ( 0 , 1 ]
D N ( ρ ) ≲ log 2 ( 2 + N ) N ρ λ + ( N ρ ) − 2 / 3 + ( N ρ ) − 2 / 3 λ − 1 / 3 ω ( ρ , η ) 2 / 3 (6) D_N(\rho) \lesssim\frac{\log^2(2+N)}{N\rho\lambda} + (N\rho)^{-2/3} + (N\rho)^{-2/3}\lambda^{-1/3}\omega(\rho,\eta)^{2/3} \tag*{(6)} D N ( ρ ) ≲ N ρ λ log 2 ( 2 + N ) + ( N ρ ) − 2/3 + ( N ρ ) − 2/3 λ − 1/3 ω ( ρ , η ) 2/3 (6) by Proposition 13.1. Here η = e N , β ( λ ) + L s λ 5 \eta=e_{N,\beta}(\lambda)+L_s\lambda^5 η = e N , β ( λ ) + L s λ 5 L s ≥ 1 L_s\geq1 L s ≥ 1 β \beta β ω ( ρ , η ) = min { 1 , ( η / ρ ) 1 / 6 } \omega(\rho,\eta)=\min\{1,(\eta/\rho)^{1/6}\} ω ( ρ , η ) = min { 1 , ( η / ρ ) 1/6 } ρ 0 = min { 1 , K η 1 / 7 } \rho_0=\min\{1,K\eta^{1/7}\} ρ 0 = min { 1 , K η 1/7 } K K K β \beta β ρ 0 \rho_0 ρ 0 (5) over [ ρ 0 , 1 ] [\rho_0,1] [ ρ 0 , 1 ] (6) over ( 0 , ρ 0 ] (0,\rho_0] ( 0 , ρ 0 ] N − 2 / 3 η 1 / 21 N^{-2/3}\eta^{1/21} N − 2/3 η 1/21 λ = N − 7 / 50 \lambda=N^{-7/50} λ = N − 7/50 e N , β ( λ ) ≲ N − 7 / 10 e_{N,\beta}(\lambda)\lesssim N^{-7/10} e N , β ( λ ) ≲ N − 7/10 § 3.9 ).
We take the first two steps from [4 ] , where the same scheme gives the rate N − 2 / 3 N^{-2/3} N − 2/3 [4 Theorem 1.4] , and our estimates near ρ = 0 \rho=0 ρ = 0 [4 Sections 3 and 7] . We use the cavity argument, and the stability and isolation estimates that it requires, only above the cutoff.
Reduction to the fractional-moment error
As in [4 Section 1.1] , we complete the Hamiltonian by an independent diagonal Gaussian term D N D_N D N N R 2 / 2 NR^2/2 N R 2 /2 R R R § 4.1.1 ). For 0 < λ < β 0<\lambda<\beta 0 < λ < β
e N , β ( λ ) = E β ( λ ) − ψ ( λ ) , ψ ( λ ) = 1 N λ log E r e f e λ ( F N , β / β + D N ) , e_{N,\beta}(\lambda)=E_\beta(\lambda)-\psi(\lambda),\qquad\psi(\lambda)=\frac{1}{N\lambda}\log\mathbb{E}_{\mathrm{ref}}e^{\lambda(F_{N,\beta}/\beta+D_N)}, e N , β ( λ ) = E β ( λ ) − ψ ( λ ) , ψ ( λ ) = N λ 1 log E ref e λ ( F N , β / β + D N ) , where E β ( λ ) = β − 1 P β ( λ / β ) E_\beta(\lambda)=\beta^{-1}P_\beta(\lambda/\beta) E β ( λ ) = β − 1 P β ( λ / β ) P β ( s ) P_\beta(s) P β ( s ) μ \mu μ [ 0 , 1 ] [0,1] [ 0 , 1 ] μ ( { 0 } ) ≥ s \mu(\{0\})\geq s μ ({ 0 }) ≥ s (14) . Talagrand showed that the normalized moments converge to this constrained value [80]; we need a rate.
The reduction is the midpoint argument of [4 Theorem 7.8, Step 1] . Since F N , β / β + D N F_{N,\beta}/\beta+D_N F N , β / β + D N ( g i j , z i ) (g_{ij},z_i) ( g ij , z i ) ψ \psi ψ [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) ψ ( 0 ) = p N , β / β \psi(0)=p_{N,\beta}/\beta ψ ( 0 ) = p N , β / β [4 Lemma 5.2] . Using ψ ( 0 ) ≥ 2 ψ ( λ ) − ψ ( 2 λ ) \psi(0)\geq2\psi(\lambda)-\psi(2\lambda) ψ ( 0 ) ≥ 2 ψ ( λ ) − ψ ( 2 λ ) e N , β ( 2 λ ) ≥ 0 e_{N,\beta}(2\lambda)\geq0 e N , β ( 2 λ ) ≥ 0
P β − p N , β β ≤ E β ( 0 ) − 2 E β ( λ ) + E β ( 2 λ ) + 2 e N , β ( λ ) . \frac{P_\beta-p_{N,\beta}}{\beta}\leq E_\beta(0)-2E_\beta(\lambda)+E_\beta(2\lambda)+2e_{N,\beta}(\lambda). β P β − p N , β ≤ E β ( 0 ) − 2 E β ( λ ) + E β ( 2 λ ) + 2 e N , β ( λ ) . The second difference of E β E_\beta E β [4 Lemma 7.2] , the Parisi profile γ β = β α β \gamma_\beta=\beta\alpha_\beta γ β = β α β 0 0 0 γ λ = max { λ , γ β } \gamma^\lambda=\max\{\lambda,\gamma_\beta\} γ λ = max { λ , γ β } λ \lambda λ γ β \gamma_\beta γ β L s λ 5 L_s\lambda^5 L s λ 5 (31) . Combining these facts, we arrive at the bound of Proposition 3.9,
0 ≤ P β − p N , β β ≤ 32 L s λ 5 + 2 e N , β ( λ ) ( 0 < λ ≤ λ 0 ) , 0\leq\frac{P_\beta-p_{N,\beta}}{\beta}\leq32L_s\lambda^5+2e_{N,\beta}(\lambda)\qquad(0<\lambda\leq\lambda_0), 0 ≤ β P β − p N , β ≤ 32 L s λ 5 + 2 e N , β ( λ ) ( 0 < λ ≤ λ 0 ) , with λ 0 > 0 \lambda_0 > 0 λ 0 > 0 β \beta β λ = N − 7 / 50 \lambda= N^{-7/50} λ = N − 7/50 N N N λ ≤ λ 0 \lambda\le\lambda_0 λ ≤ λ 0 λ 5 = N − 7 / 10 \lambda^5 = N^{-7/10} λ 5 = N − 7/10 λ ↓ 0 \lambda\downarrow0 λ ↓ 0 λ 5 \lambda^5 λ 5
The interpolation and the defect
The interpolation of [4 Section 2] is a finite-dimensional, quantitative version of Guerra’s interpolation [40], in the form that Talagrand used for positive fractional moments [80]. For 0 < ρ ≤ 1 0 < \rho\le1 0 < ρ ≤ 1 t = 1 − ρ t = 1 - \rho t = 1 − ρ c = β 2 ρ c = \beta^2\rho c = β 2 ρ M = N c M = Nc M = N c μ = s δ 0 + ν \mu= s\delta_0 + \nu μ = s δ 0 + ν [ 0 , 1 ] [0,1] [ 0 , 1 ] s = λ / β s = \lambda/\beta s = λ / β ν ≥ 0 \nu\ge0 ν ≥ 0 a ( q ) = μ ( [ 0 , q ] ) a(q) = \mu([0,q]) a ( q ) = μ ([ 0 , q ]) R N \mathbb{R}^N R N
∂ q U = − c 2 ( Δ U + a ( q ) ∣ ∇ U ∣ 2 ) , U ( 1 , x ; W ) = log ∑ σ ∈ { − 1 , 1 } N e β t σ T W / 2 + x ⋅ σ , \partial_q U = -\frac{c}{2}\left(\Delta U + a(q)|\nabla U|^2\right), \qquad U(1,x;W) = \log\sum_{\sigma\in\{-1,1\}^N} e^{\beta\sqrt{t}\,\sigma^{\mathsf T}W/2+x\cdot\sigma}, ∂ q U = − 2 c ( Δ U + a ( q ) ∣∇ U ∣ 2 ) , U ( 1 , x ; W ) = log σ ∈ { − 1 , 1 } N ∑ e β t σ T W /2 + x ⋅ σ , and we minimize the functional F t , s ( μ ) F_{t,s}(\mu) F t , s ( μ ) (16) over admissible profiles. At t = 0 t = 0 t = 0 N N N P β ( s ) P_\beta(s) P β ( s ) t = 1 t = 1 t = 1 (17) . The fractional-moment error is β − 1 \beta^{-1} β − 1
The tilted law. Each profile defines a joint law of the couplings, an auxiliary field X q ∈ R N X_q \in\mathbb{R}^N X q ∈ R N q ∈ [ 0 , 1 ] q \in[0,1] q ∈ [ 0 , 1 ] σ \sigma σ W W W e s U ( 0 , 0 ; W ) e^{sU(0,0;W)} e s U ( 0 , 0 ; W ) c a ( q ) ∇ U ( q , X q ; W ) ca(q)\nabla U(q,X_q;W) c a ( q ) ∇ U ( q , X q ; W ) c \sqrt{c} c σ \sigma σ X 1 X_1 X 1 § 4.1.3 ). We write G q \mathcal{G}_q G q q q q m q m_q m q C q C_q C q σ \sigma σ G q \mathcal{G}_q G q H q = ∇ 2 U ( q , X q ; W ) H_q = \nabla^2 U(q,X_q;W) H q = ∇ 2 U ( q , X q ; W ) S q = ∣ m q ∣ 2 / N S_q = |m_q|^2/N S q = ∣ m q ∣ 2 / N Proposition 3.1 , m q m_q m q
d m q = c H q d B q , dm_q = \sqrt{c}\,H_q\,dB_q, d m q = c H q d B q , and a ( q ) C q ⪯ H q ⪯ C q a(q)C_q \preceq H_q \preceq C_q a ( q ) C q ⪯ H q ⪯ C q μ \mu μ q q q supp ν \operatorname{supp}\nu supp ν E S q = q \mathbb{E}S_q = q E S q = q E Tr H q 2 ≤ N / c \mathbb{E}\operatorname{Tr} H_q^2 \le N/c E Tr H q 2 ≤ N / c
The defect. Let R q R_q R q G q \mathcal{G}_q G q Proposition 3.6 , the derivative of F t , s ( μ ) F_{t,s}(\mu) F t , s ( μ ) t t t − ( β 2 / 4 ) ∫ E ( R q − q ) 2 ν ( d q ) -(\beta^2/4)\int\mathbb{E}(R_q-q)^2\nu(dq) − ( β 2 /4 ) ∫ E ( R q − q ) 2 ν ( d q ) N R 2 / 2 NR^2/2 N R 2 /2 ν \nu ν s s s s δ 0 s\delta_0 s δ 0 R q − S q R_q-S_q R q − S q E ( R q − q ) 2 = a q t h + 2 j q + Var ( S q ) \mathbb{E}(R_q-q)^2 = a_q^{\mathrm{th}} + 2j_q + \operatorname{Var}(S_q) E ( R q − q ) 2 = a q th + 2 j q + Var ( S q ) Lemma 3.5 ). Integrating against ν \nu ν Corollary 3.7 , which states that at almost every ρ \rho ρ
D N ( ρ ) = − 4 β f N , s ′ ( 1 − ρ ) = β ( A + 2 J + V ) , 4 e N , β ( λ ) = ∫ 0 1 D N ( ρ ) d ρ , \mathcal{D}_N(\rho) = -\frac{4}{\beta}f'_{N,s}(1-\rho) = \beta(A+2J+V), \qquad4e_{N,\beta}(\lambda) = \int_0^1 \mathcal{D}_N(\rho)\,d\rho, D N ( ρ ) = − β 4 f N , s ′ ( 1 − ρ ) = β ( A + 2 J + V ) , 4 e N , β ( λ ) = ∫ 0 1 D N ( ρ ) d ρ , where f N , s ( t ) f_{N,s}(t) f N , s ( t ) F t , s \mathcal{F}_{t,s} F t , s
A = ∫ E Tr C q 2 N 2 ν ( d q ) , J = ∫ E [ m q T C q m q ] N 2 ν ( d q ) , V = ∫ Var ( S q ) ν ( d q ) . A=\int\frac{\mathbb{E}\operatorname{Tr} C_q^2}{N^2}\,\nu(\mathrm{d}q),\qquad J=\int\frac{\mathbb{E}[m_q^{\mathsf{T}}C_qm_q]}{N^2}\,\nu(\mathrm{d}q),\qquad V=\int\operatorname{Var}(S_q)\,\nu(\mathrm{d}q). A = ∫ N 2 E Tr C q 2 ν ( d q ) , J = ∫ N 2 E [ m q T C q m q ] ν ( d q ) , V = ∫ Var ( S q ) ν ( d q ) . The terms A A A J J J G q \mathcal{G}_q G q V V V q q q
The posterior variance bound. Let L q = ( σ ⋅ m q − ∣ m q ∣ 2 / 2 ) / N L_q=(\sigma\cdot m_q-\lvert m_q\rvert^2/2)/N L q = ( σ ⋅ m q − ∣ m q ∣ 2 /2 ) / N G q \mathcal{G}_q G q S q / 2 S_q/2 S q /2 k ∈ L 2 ( ν ) k\in L^2(\nu) k ∈ L 2 ( ν )
Var ( ∫ k ( q ) L q ν ( d q ) ) ≤ 1 M ∫ k ( q ) 2 a ( q ) ν ( d q ) , A ≤ 1 M s . \operatorname{Var}\left(\int k(q)L_q\,\nu(\mathrm{d}q)\right)\leq\frac{1}{M}\int\frac{k(q)^2}{a(q)}\,\nu(\mathrm{d}q),\qquad A\leq\frac{1}{Ms}. Var ( ∫ k ( q ) L q ν ( d q ) ) ≤ M 1 ∫ a ( q ) k ( q ) 2 ν ( d q ) , A ≤ M s 1 . With the terminal spin fixed, the Gaussian coordinates that generate W W W [21 ] , in the form of [4 Proposition 2.2] , and the contact identities. Note that the factor a ( q ) − 1 ≤ s − 1 a(q)^{-1}\leq s^{-1} a ( q ) − 1 ≤ s − 1 Q Q Q ν \nu ν ( 0 , 1 − s ) (0,1-s) ( 0 , 1 − s ) ν \nu ν a ( Q ( v ) ) ≥ s + v a(Q(v))\geq s+v a ( Q ( v )) ≥ s + v
The trial budget. In the energy normalization, we set γ = β a \gamma=\beta a γ = β a F ρ ( γ ) = β − 1 F 1 − ρ , s ( μ ) F_\rho(\gamma)=\beta^{-1}F_{1-\rho,s}(\mu) F ρ ( γ ) = β − 1 F 1 − ρ , s ( μ ) Lemma 3.8 , we have, for every ρ ∈ ( 0 , 1 ] \rho\in(0,1] ρ ∈ ( 0 , 1 ]
0 ≤ F ρ ( γ λ t r ) − min γ F ρ ( γ ) ≤ e N , β ( λ ) + L s λ 5 = η . 0\leq F_\rho(\gamma_\lambda^{\mathrm{tr}})-\min_\gamma F_\rho(\gamma)\leq e_{N,\beta}(\lambda)+L_s\lambda^5=\eta. 0 ≤ F ρ ( γ λ tr ) − γ min F ρ ( γ ) ≤ e N , β ( λ ) + L s λ 5 = η . Indeed, F t , s ( μ ) \mathcal{F}_{t,s}(\mu) F t , s ( μ ) t t t F ρ ( γ λ t r ) F_\rho(\gamma_\lambda^{\mathrm{tr}}) F ρ ( γ λ tr ) E β ( 0 ) + L s λ 5 E_\beta(0)+L_s\lambda^5 E β ( 0 ) + L s λ 5 min F ρ ≥ ψ ( λ ) = E β ( λ ) − e N , β ( λ ) \min F_\rho\geq\psi(\lambda)=E_\beta(\lambda)-e_{N,\beta}(\lambda) min F ρ ≥ ψ ( λ ) = E β ( λ ) − e N , β ( λ ) E β ( λ ) ≥ E β ( 0 ) E_\beta(\lambda)\geq E_\beta(0) E β ( λ ) ≥ E β ( 0 ) η \eta η ρ \rho ρ ρ 0 \rho_0 ρ 0
The thermal terms
At a contact level, by the covariance order, we have Tr H q 2 ≥ a ( q ) Tr ( H q C q ) ≥ a ( q ) 2 Tr C q 2 \operatorname{Tr}H_q^2\geq a(q)\operatorname{Tr}(H_qC_q)\geq a(q)^2\operatorname{Tr}C_q^2 Tr H q 2 ≥ a ( q ) Tr ( H q C q ) ≥ a ( q ) 2 Tr C q 2 E Tr H q 2 \mathbb{E}\operatorname{Tr}H_q^2 E Tr H q 2 E Tr C q 2 / N 2 ≤ ( M a ( q ) ) − 1 \mathbb{E}\operatorname{Tr}C_q^2/N^2\leq(Ma(q))^{-1} E Tr C q 2 / N 2 ≤ ( M a ( q ) ) − 1
A ≤ 1 M ∫ 0 1 − s d v a ( Q ( v ) ) 2 ≤ 1 M ∫ 0 1 − s d v ( s + v ) 2 ≤ 1 M s . A\leq\frac{1}{M}\int_0^{1-s}\frac{\mathrm{d}v}{a(Q(v))^2}\leq\frac{1}{M}\int_0^{1-s}\frac{\mathrm{d}v}{(s+v)^2}\leq\frac{1}{Ms}. A ≤ M 1 ∫ 0 1 − s a ( Q ( v ) ) 2 d v ≤ M 1 ∫ 0 1 − s ( s + v ) 2 d v ≤ M s 1 . The mean–covariance term needs more work, because j q = N − 2 E [ m q T C q m q ] j_q = N^{-2}\mathbb{E}[m_q^{T}C_qm_q] j q = N − 2 E [ m q T C q m q ] m q m_q m q q q q b b b G q ′ \mathcal{G}_{q'} G q ′ q ′ < q q' < q q ′ < q E [ b T C q b ] \mathbb{E}[b^{T}C_qb] E [ b T C q b ] E ∣ b ∣ 2 \mathbb{E}|b|^2 E ∣ b ∣ 2 c ∫ q ′ q a ( u ) 2 d u c\int_{q'}^q a(u)^2\,\mathrm{d}u c ∫ q ′ q a ( u ) 2 d u b = m q b = m_q b = m q m Q ( v ) m_{Q(v)} m Q ( v ) m Q ( w ) m_{Q(w)} m Q ( w ) w < v w < v w < v E S r \mathbb{E}S_r E S r q q q supp ν \operatorname{supp}\nu supp ν E ∣ m q − m r ∣ 2 \mathbb{E}|m_q-m_r|^2 E ∣ m q − m r ∣ 2 2 N ∣ q − r ∣ 2N|q-r| 2 N ∣ q − r ∣ Lemma 6.2 ). We control the increments by combining this bound with Lemma 6.1 in a dyadic decomposition (Lemma 6.3). The conditional covariance in the direction of the average is controlled by the posterior variance bound. With a quadratic weight ( s + w ) 2 (s+w)^2 ( s + w ) 2 ( s + v ) − 2 (s+v)^{-2} ( s + v ) − 2
j Q ( v ) ≤ 341 ( 1 + log + M ) 2 M ( s + v ) 2 , J ≤ 341 ( 1 + log + M ) 2 M s . j_{Q(v)} \le\frac{341(1+\log_{+} M)^2}{M(s+v)^2}, \qquad J \le\frac{341(1+\log_{+} M)^2}{Ms}. j Q ( v ) ≤ M ( s + v ) 2 341 ( 1 + log + M ) 2 , J ≤ M s 341 ( 1 + log + M ) 2 . (Proposition 6.4). Since M s = β N ρ λ Ms=\beta N\rho\lambda M s = β N ρ λ M ≤ β 2 N M\le\beta^2N M ≤ β 2 N
β ( A + 2 J ) ≤ 1 + 682 ( 1 + log + M ) 2 N ρ λ ≲ log 2 ( 2 + N ) N ρ λ . \beta(A+2J)\le\frac{1+682(1+\log_{+} M)^2}{N\rho\lambda}\lesssim\frac{\log^2(2+N)}{N\rho\lambda}. β ( A + 2 J ) ≤ N ρ λ 1 + 682 ( 1 + log + M ) 2 ≲ N ρ λ log 2 ( 2 + N ) . Integrated over ρ ∈ [ 1 / N , 1 ] \rho\in[1/N,1] ρ ∈ [ 1/ N , 1 ] log 3 ( 2 + N ) / ( N λ ) \log^3(2+N)/(N\lambda) log 3 ( 2 + N ) / ( N λ ) N − 43 / 50 log 3 ( 2 + N ) N^{-43/50}\log^3(2+N) N − 43/50 log 3 ( 2 + N ) λ = N − 7 / 50 \lambda=N^{-7/50} λ = N − 7/50 N − 7 / 10 N^{-7/10} N − 7/10 N N N
The self-overlap variance for small ρ \rho ρ
The posterior variance bound controls averages of L q L_q L q V V V S q S_q S q ν \nu ν [4 Lemma 3.3] , we compare S q S_q S q L p L_p L p ({xref}Section 5`).
Two comparisons. For p ≥ q p\ge q p ≥ q m m m S q ≤ 2 E [ L p ∣ G q ] S_q\le2\mathbb{E}[L_p\mid\mathcal{G}_q] S q ≤ 2 E [ L p ∣ G q ] p ≤ q p\le q p ≤ q E [ L p ∣ G q ] = S q / 2 − ∣ m q − m p ∣ 2 / ( 2 N ) \mathbb{E}[L_p\mid\mathcal{G}_q]=S_q/2-|m_q-m_p|^2/(2N) E [ L p ∣ G q ] = S q /2 − ∣ m q − m p ∣ 2 / ( 2 N ) q q q p ˉ \bar p p ˉ Lemma 5.1 that, at a minimizer, S q − q S_q-q S q − q q q q q q q p ˉ − q \bar p-q p ˉ − q L p L_p L p r r r a ≥ u a\ge u a ≥ u ( M u r ) − 1 (Mur)^{-1} ( M u r ) − 1 G q \mathcal{G}_q G q N − 1 ( σ − m q ) ⋅ m q N^{-1}(\sigma-m_q)\cdot m_q N − 1 ( σ − m q ) ⋅ m q L q L_q L q [4 Lemma 3.3] produces additional residual terms; we bound its variance j q j_q j q § 3.4 ).
Variance on a quantile interval. Let I ⊂ ( 0 , 1 − s ) I\subset(0,1-s) I ⊂ ( 0 , 1 − s ) h h h a ( Q ( v ) ) ≥ u a(Q(v))\ge u a ( Q ( v )) ≥ u v ∈ I v\in I v ∈ I ℓ \ell ℓ Q ( I ) Q(I) Q ( I ) ϵ = ( M u ) − 1 \epsilon=(Mu)^{-1} ϵ = ( M u ) − 1 τ \tau τ ϵ / τ \epsilon/\tau ϵ / τ τ \tau τ I I I Var ( S Q ( v ) ) > τ \operatorname{Var}(S_{Q(v)})>\tau Var ( S Q ( v ) ) > τ Q Q Q τ 1 / 2 \tau^{1/2} τ 1/2 Q Q Q ℓ \ell ℓ I I I ϵ ℓ τ − 3 / 2 \epsilon\ell\tau^{-3/2} ϵ ℓ τ − 3/2 ϵ / τ \epsilon/\tau ϵ / τ I I I min { h , ϵ ℓ τ − 3 / 2 } \min\{h,\epsilon\ell\tau^{-3/2}\} min { h , ϵ ℓ τ − 3/2 } τ \tau τ
∫ I Var ( S Q ( v ) ) d σ ≤ 128 [ min { h , ϵ ( 1 + log + ( h / ϵ ) ) } + ( ϵ ℓ ) 2 / 3 h 1 / 3 ] . (7) \int_I \operatorname{Var}(S_{Q(v)})\,\mathrm{d}\sigma\le128\left[\min\{h,\epsilon(1+\log_+(h/\epsilon))\}+(\epsilon\ell)^{2/3}h^{1/3}\right]. \tag*{(7)} ∫ I Var ( S Q ( v ) ) d σ ≤ 128 [ min { h , ϵ ( 1 + log + ( h / ϵ ))} + ( ϵ ℓ ) 2/3 h 1/3 ] . (7) The first term is at most ϵ \epsilon ϵ
Dyadic shells and root localization. In the proof of Proposition 13.1 , we cover ( 0 , 1 − s ) (0,1-s) ( 0 , 1 − s ) I u = { v : u ≤ s + v < 2 u } I_u=\{v:u\le s+v<2u\} I u = { v : u ≤ s + v < 2 u } u = 2 j s < 1 u=2^j s<1 u = 2 j s < 1 a ( Q ( v ) ) ≥ u a(Q(v))\ge u a ( Q ( v )) ≥ u h ≤ u h\le u h ≤ u ϵ = ( M u ) − 1 \epsilon=(Mu)^{-1} ϵ = ( M u ) − 1 ℓ ≤ 1 \ell\le1 ℓ ≤ 1 (7) is at most M − 2 / 3 u − 1 / 3 M^{-2/3}u^{-1/3} M − 2/3 u − 1/3 β M − 2 / 3 s − 1 / 3 = ( N ρ ) − 2 / 3 λ − 1 / 3 \beta M^{-2/3}s^{-1/3}=(N\rho)^{-2/3}\lambda^{-1/3} β M − 2/3 s − 1/3 = ( N ρ ) − 2/3 λ − 1/3
β V ≲ ( N ρ ) − 2 / 3 λ − 1 / 3 + log 2 ( 2 + N ) N ρ λ . \beta V\lesssim(N\rho)^{-2/3}\lambda^{-1/3}+\frac{\log^2(2+N)}{N\rho\lambda}. β V ≲ ( N ρ ) − 2/3 λ − 1/3 + N ρ λ log 2 ( 2 + N ) . The bound β V ≤ C ( N ρ ) − 2 / 3 λ − 1 / 3 \beta V\le C(N\rho)^{-2/3}\lambda^{-1/3} β V ≤ C ( N ρ ) − 2/3 λ − 1/3 [4 (4.2)] has the same form. At λ = N − 7 / 50 \lambda=N^{-7/50} λ = N − 7/50 λ − 1 / 3 = N 7 / 150 \lambda^{-1/3}=N^{7/150} λ − 1/3 = N 7/150 N N N § 3.9 . To remove it, we use the root localization of Corollary 8.4 , which states that at every minimizer and for every trial budget η \eta η
Q ( v ) ≤ C Q ( s + v + ω ( ρ , η ) ) , ω ( ρ , η ) = min { 1 , ( η / ρ ) 1 / 6 } . Q(v)\le C_Q(s+v+\omega(\rho,\eta)),\qquad\omega(\rho,\eta)=\min\{1,(\eta/\rho)^{1/6}\}. Q ( v ) ≤ C Q ( s + v + ω ( ρ , η )) , ω ( ρ , η ) = min { 1 , ( η / ρ ) 1/6 } . with C Q C_Q C Q β \beta β ℓ ( I u ) ≤ 2 C Q ( u + ω ) \ell(I_u)\le2C_Q(u+\omega) ℓ ( I u ) ≤ 2 C Q ( u + ω ) (7) is at most of order M − 2 / 3 ( u 1 / 3 + u − 1 / 3 ω 2 / 3 ) M^{-2/3}(u^{1/3}+u^{-1/3}\omega^{2/3}) M − 2/3 ( u 1/3 + u − 1/3 ω 2/3 ) M − 2 / 3 ( 1 + s − 1 / 3 ω 2 / 3 ) M^{-2/3}(1+s^{-1/3}\omega^{2/3}) M − 2/3 ( 1 + s − 1/3 ω 2/3 ) (6) , that is, (201) . Its leading term ( N ρ ) − 2 / 3 (N\rho)^{-2/3} ( N ρ ) − 2/3 λ \lambda λ ρ = 0 \rho=0 ρ = 0 ω 2 / 3 ≤ ( η / ρ ) 1 / 9 \omega^{2/3}\le(\eta/\rho)^{1/9} ω 2/3 ≤ ( η / ρ ) 1/9 ρ = 1 / N \rho=1/N ρ = 1/ N D N ( ρ ) ≤ C ( N ρ ) − 1 / 2 D_N(\rho)\le C(N\rho)^{-1/2} D N ( ρ ) ≤ C ( N ρ ) − 1/2 [4 Lemma 4.1] , recalled in (30) . Integrating, we arrive at (203) , which states that for 0 < ρ 0 ≤ 1 0<\rho_0\le1 0 < ρ 0 ≤ 1
∫ 0 ρ 0 D N ( ρ ) d ρ ≲ N − 1 + log 3 ( 2 + N ) N λ + N − 2 / 3 ρ 0 1 / 3 + N − 2 / 3 λ − 1 / 3 η 1 / 9 ρ 0 2 / 9 . (8) \int_0^{\rho_0}D_N(\rho)\,\mathrm{d}\rho\lesssim N^{-1}+\frac{\log^3(2+N)}{N\lambda}+N^{-2/3}\rho_0^{1/3}+N^{-2/3}\lambda^{-1/3}\eta^{1/9}\rho_0^{2/9}. \tag*{(8)} ∫ 0 ρ 0 D N ( ρ ) d ρ ≲ N − 1 + N λ log 3 ( 2 + N ) + N − 2/3 ρ 0 1/3 + N − 2/3 λ − 1/3 η 1/9 ρ 0 2/9 . (8) The barrier at 2 / 3 2/3 2/3 For ρ 0 = 1 \rho_0=1 ρ 0 = 1 (8) is of order N − 2 / 3 N^{-2/3} N − 2/3 [4 Proposition 7.7] , in which quantile estimates [4 Lemma 3.3] and a localization of low quantiles [4 Lemma 7.6] lead to e N , β ( λ ) ≲ N − 2 / 3 + N − 3 / 4 λ − 1 / 2 + log ( 2 + N ) / ( N λ ) e_{N,\beta}(\lambda)\lesssim N^{-2/3}+N^{-3/4}\lambda^{-1/2}+\log(2+N)/(N\lambda) e N , β ( λ ) ≲ N − 2/3 + N − 3/4 λ − 1/2 + log ( 2 + N ) / ( N λ ) [4 (7.43)] , and the tilt λ = N − 2 / 15 \lambda=N^{-2/15} λ = N − 2/15 N − 2 / 3 N^{-2/3} N − 2/3 [4 Theorem 1.4] . When ρ \rho ρ ( N ρ ) − 2 / 3 (N\rho)^{-2/3} ( N ρ ) − 2/3 (5) , as we discuss heuristically in § 2.4 . We therefore use (8) only below the cutoff, where it contributes N − 2 / 3 ρ 0 1 / 3 N^{-2/3}\rho_0^{1/3} N − 2/3 ρ 0 1/3
The cavity contraction
Let μ \mu μ ( ρ , s ) (\rho,s) ( ρ , s ) a a a d = β 2 ( 1 − t / N ) d=\beta^{2}(1-t/N) d = β 2 ( 1 − t / N ) J a , d s c J^{\mathrm{sc}}_{a,d} J a , d sc H ν \mathcal{H}_{\nu} H ν L 2 ( ν ) L^{2}(\nu) L 2 ( ν )
J a , d s c ⪯ ( 1 + ϵ N ) I on H ν , 0 ≤ ϵ N ≤ ρ 4 , (9) J^{\mathrm{sc}}_{a,d}\preceq(1+\epsilon_N)I\ \text{ on }\mathcal{H}_{\nu},\qquad0\leq\epsilon_N\leq\frac{\rho}{4}, \tag*{(9)} J a , d sc ⪯ ( 1 + ϵ N ) I on H ν , 0 ≤ ϵ N ≤ 4 ρ , (9) then
3 ρ 4 V ≤ C c a v [ N − 1 + A + J + H 3 ] , H 3 = ∫ ∣ E ∣ S q − q ∣ 3 ∣ ν ( d q ) , \frac{3\rho}{4}V\leq C_{\mathrm{cav}}\left[N^{-1}+A+J+H_{3}\right],\qquad H_{3}=\int\left|\mathbb{E}|S_q-q|^{3}\right|\nu(dq), 4 3 ρ V ≤ C cav [ N − 1 + A + J + H 3 ] , H 3 = ∫ E ∣ S q − q ∣ 3 ν ( d q ) , with C c a v C_{\mathrm{cav}} C cav β \beta β
Exchangeability. Let n = N − 1 n=N-1 n = N − 1 m N ( p ) m_N(p) m N ( p ) G p \mathcal{G}_p G p A p − A_p^{-} A p − n n n N S p = m N ( p ) 2 + n A p − NS_p=m_N(p)^2+nA_p^{-} N S p = m N ( p ) 2 + n A p − Cov ( S p , m N ( p ) 2 ) = Var ( S p ) \operatorname{Cov}(S_p,m_N(p)^2)=\operatorname{Var}(S_p) Cov ( S p , m N ( p ) 2 ) = Var ( S p )
Var ( S p ) = 1 N Var ( m N ( p ) 2 ) + n N Cov ( m N ( p ) 2 , A p − ) . \operatorname{Var}(S_p)=\frac{1}{N}\operatorname{Var}(m_N(p)^2)+\frac{n}{N}\operatorname{Cov}(m_N(p)^2,A_p^{-}). Var ( S p ) = N 1 Var ( m N ( p ) 2 ) + N n Cov ( m N ( p ) 2 , A p − ) . Up to an error of order N − 1 N^{-1} N − 1 V V V
The cavity interpolation. For u ∈ [ 0 , 1 ] u\in[0,1] u ∈ [ 0 , 1 ] α u / n \alpha u/n α u / n b ( u ) = d − α u b(u)=d-\alpha u b ( u ) = d − α u α = β 2 t n / N \alpha=\beta^{2}tn/N α = β 2 t n / N d = c + α d=c+\alpha d = c + α u = 1 u=1 u = 1 u = 0 u=0 u = 0 n n n d d d C p p ( u ) = Cov u ( m N ( p ) 2 , A p − ) C_{p p}(u)=\operatorname{Cov}_{u}(m_N(p)^2,A_p^{-}) C pp ( u ) = Cov u ( m N ( p ) 2 , A p − ) u = 0 u=0 u = 0
C p p ( 1 ) = C p p ′ ( 0 ) + ∫ 0 1 ( 1 − u ) C p p ′ ′ ( u ) d u . C_{p p}(1)=C'_{p p}(0)+\int_{0}^{1}(1-u)C''_{p p}(u)\,du. C pp ( 1 ) = C pp ′ ( 0 ) + ∫ 0 1 ( 1 − u ) C pp ′′ ( u ) d u . Note that the n n n u = 0 u=0 u = 0
Tree laws and the remainder. Derivatives in u u u § 9.1.2 ). Normalized replica trees of this kind were developed in [54, Section 2]. By Lemma 10.5, the derivative in u u u α / 2 \alpha/2 α /2 D a b = ϵ a ϵ b ( R a b − − q a b ) D_{ab}=\epsilon_a\epsilon_b(R_{ab}^{-}-q_{ab}) D ab = ϵ a ϵ b ( R ab − − q ab ) R a b − R_{ab}^{-} R ab − b ( u ) b(u) b ( u ) q a b q_{ab} q ab u u u K β , k K_{\beta,k} K β , k β \beta β k k k C p p ′ ′ C_{pp}'' C pp ′′ C β sup u M 3 ( u ) C_{\beta}\sup_u\mathcal{M}_3(u) C β sup u M 3 ( u ) M 3 ( u ) = ∫ E 2 , q , u ∣ R 12 − − q ∣ 3 ν ( d q ) \mathcal{M}_3(u)=\int\mathbb{E}_{2,q,u}|R_{12}^{-}-q|^3\,\nu(dq) M 3 ( u ) = ∫ E 2 , q , u ∣ R 12 − − q ∣ 3 ν ( d q ) K β , 2 [ 64 ( A + J ) + 16 H 3 + 32 N − 3 ] K_{\beta,2}[64(A+J)+16H_3+32N^{-3}] K β , 2 [ 64 ( A + J ) + 16 H 3 + 32 N − 3 ]
The linear term. Along the interpolation, the fields acting on the last spins of two leaves have covariance d q a b + α u ( R a b − − q a b ) d q_{ab}+\alpha u(R_{ab}^{-}-q_{ab}) d q ab + α u ( R ab − − q ab ) § 10.2.1 ). In the scalar law, if we replace the covariance d ( q ∧ r ) d(q\wedge r) d ( q ∧ r ) d T ( q ∧ r ) dT(q\wedge r) d T ( q ∧ r ) T ( q ) = q + θ φ ( q ) T(q)=q+\theta\varphi(q) T ( q ) = q + θ φ ( q ) p p p ( J a , d φ ) ( p ) (J_{a,d\varphi})(p) ( J a , d φ ) ( p ) ( α u / d ) ( A q − − q ) (\alpha u/d)(A_q^{-}-q) ( α u / d ) ( A q − − q ) C p p ′ ( 0 ) C_{pp}'(0) C pp ′ ( 0 ) ( α / d ) ( J a , d s c Σ − ) ( p , p ) (\alpha/d)(J_{a,d}^{\mathrm{sc}}\Sigma^{-})(p,p) ( α / d ) ( J a , d sc Σ − ) ( p , p ) Σ − ( p , q ) = Cov − ( A p − , A q − ) \Sigma^{-}(p,q)=\operatorname{Cov}^{-}(A_p^{-},A_q^{-}) Σ − ( p , q ) = Cov − ( A p − , A q − ) u = 0 u=0 u = 0
∫ C p p ′ ( 0 ) ν ( d p ) = α [ 1 d Tr ν ( J a , d s c Σ − ) − 1 2 P + ∫ T p ν ( d p ) ] , P ≥ 0. \int C_{pp}'(0)\,\nu(dp)=\alpha\left[\frac{1}{d}\operatorname{Tr}_{\nu}(J_{a,d}^{\mathrm{sc}}\Sigma^{-})-\frac{1}{2}\mathcal{P}+\int T_p\,\nu(dp)\right],\qquad\mathcal{P}\ge0. ∫ C pp ′ ( 0 ) ν ( d p ) = α [ d 1 Tr ν ( J a , d sc Σ − ) − 2 1 P + ∫ T p ν ( d p ) ] , P ≥ 0. The term − P / 2 -\mathcal{P}/2 − P /2 T p T_p T p A p − A_p^{-} A p − Var − ( A p − ) \operatorname{Var}^{-}(A_p^{-}) Var − ( A p − ) p p p (132) on the variance of ν \nu ν A − A^{-} A −
α ∣ ∫ T p ν ( d p ) ∣ ≤ 192 β 2 K β , 3 [ A + J + N − 2 ] . \alpha\left|\int T_p\,\nu(dp)\right|\le192\beta^2K_{\beta,3}\left[A+J+N^{-2}\right]. α ∫ T p ν ( d p ) ≤ 192 β 2 K β , 3 [ A + J + N − 2 ] . The factor 1 − ρ 1-\rho 1 − ρ Combining these identities, we obtain the trace identity of Lemma 12.2,
V = V s i t e N + n N [ α d Tr ν ( J a , d s c Σ − ) − α 2 P + α ∫ T p ν ( d p ) + R c a v ] . V=\frac{V^{\mathrm{site}}}{N}+\frac{n}{N}\left[\frac{\alpha}{d}\operatorname{Tr}_{\nu}(J_{a,d}^{\mathrm{sc}}\Sigma^{-})-\frac{\alpha}{2}\mathcal{P}+\alpha\int T_p\,\nu(dp)+\mathcal{R}_{\mathrm{cav}}\right]. V = N V site + N n [ d α Tr ν ( J a , d sc Σ − ) − 2 α P + α ∫ T p ν ( d p ) + R cav ] . where 0 ≤ V s i t e ≤ 1 / 4 0\le V^{\mathrm{site}}\le1/4 0 ≤ V site ≤ 1/4 R c a v \mathcal{R}_{\mathrm{cav}} R cav Σ − \Sigma^{-} Σ − H ν \mathcal{H}_{\nu} H ν V − = ∫ Var − ( A p − ) ν ( d p ) V^{-}=\int\operatorname{Var}^{-}(A_p^{-})\,\nu(dp) V − = ∫ Var − ( A p − ) ν ( d p ) (9) , the trace Tr ν ( J a , d s c Σ − ) \operatorname{Tr}_{\nu}(J_{a,d}^{\mathrm{sc}}\Sigma^{-}) Tr ν ( J a , d sc Σ − ) ( 1 + ε N ) V − (1+\varepsilon_N)V^- ( 1 + ε N ) V −
α d = t N − 1 N − t ≤ t = 1 − ρ (10) \frac{\alpha}{d}=t\frac{N-1}{N-t}\le t=1-\rho \tag*{(10)} d α = t N − t N − 1 ≤ t = 1 − ρ (10) Indeed, the field acting on the last spin has variance coefficient d d d u = 1 u=1 u = 1 c = β 2 ρ c=\beta^2\rho c = β 2 ρ α \alpha α V ≤ ( 1 − ρ ) ( 1 + ε N ) V − + errors V\le(1-\rho)(1+\varepsilon_N)V^-+\text{errors} V ≤ ( 1 − ρ ) ( 1 + ε N ) V − + errors
Comparison with coefficient one. It remains to compare V − V^- V − V V V V − ≤ K V + errors V^-\le KV+\text{errors} V − ≤ K V + errors K > 1 K>1 K > 1 V ≤ ( 1 − ρ ) K ( 1 + ε N ) V + errors V\le(1-\rho)K(1+\varepsilon_N)V+\text{errors} V ≤ ( 1 − ρ ) K ( 1 + ε N ) V + errors V V V ( 1 − ρ ) K ( 1 + ε N ) < 1 (1-\rho)K(1+\varepsilon_N)<1 ( 1 − ρ ) K ( 1 + ε N ) < 1 ρ ≤ 1 − 1 / K \rho\le1-1/K ρ ≤ 1 − 1/ K
V − ≤ V + C β sup 0 ≤ u ≤ 1 M 3 ( u ) + 1 4 n 2 . V^-\le V+C_\beta\sup_{0\le u\le1}\mathcal{M}_3(u)+\frac{1}{4n^2}. V − ≤ V + C β 0 ≤ u ≤ 1 sup M 3 ( u ) + 4 n 2 1 . For each p p p E u ( A p − − p ) 2 \mathbb{E}_u(A^-_p-p)^2 E u ( A p − − p ) 2 p p p u u u ν \nu ν u = 0 u=0 u = 0 u = 1 u=1 u = 1 C β sup u M 3 ( u ) C_\beta\sup_u\mathcal{M}_3(u) C β sup u M 3 ( u ) u = 0 u=0 u = 0 Var − ( A p − ) \operatorname{Var}^-(A^-_p) Var − ( A p − ) E − A p − = p \mathbb{E}^-A^-_p=p E − A p − = p u = 1 u=1 u = 1 E S p = p \mathbb{E}S_p=p E S p = p Var ( S p ) + 1 / ( 4 n 2 ) \operatorname{Var}(S_p)+1/(4n^2) Var ( S p ) + 1/ ( 4 n 2 )
Closing. As ( 1 − ρ ) ( 1 + ρ / 4 ) ≤ 1 − 3 ρ / 4 (1-\rho)(1+\rho/4)\le1-3\rho/4 ( 1 − ρ ) ( 1 + ρ /4 ) ≤ 1 − 3 ρ /4 V ≤ ( 1 − 3 ρ / 4 ) V + C β [ N − 1 + A + J + H 3 ] V\le(1-3\rho/4)V+C_\beta[N^{-1}+A+J+H_3] V ≤ ( 1 − 3 ρ /4 ) V + C β [ N − 1 + A + J + H 3 ] Proposition 12.4 . By Corollary 5.3, if M ≥ 1 M\ge1 M ≥ 1
H 3 ≤ z V + 425 M s log ( 2 M ) log 2 z ( 0 < z ≤ 1 / 4 ) . H_3\le zV+\frac{425}{M_s}\log(2M)\log\frac{2}{z}\qquad(0<z\le1/4). H 3 ≤ z V + M s 425 log ( 2 M ) log z 2 ( 0 < z ≤ 1/4 ) . We take z z z ρ \rho ρ z V zV z V 3 ρ / 4 3\rho/4 3 ρ /4 A A A J J J M s = β N ρ λ M_s=\beta N\rho\lambda M s = β N ρ λ Corollary 12.5 , which states that at every minimizer that satisfies (2.5), we have
β ( A + 2 J + V ) ≤ C β log 2 ( 2 + N ) N λ ρ 2 ( 1 / N ≤ ρ ≤ 1 ) . \beta(A+2J+V)\le C_\beta\frac{\log^2(2+N)}{N\lambda\rho^2}\qquad(1/N\le\rho\le1). β ( A + 2 J + V ) ≤ C β N λ ρ 2 log 2 ( 2 + N ) ( 1/ N ≤ ρ ≤ 1 ) . One factor ρ − 1 \rho^{-1} ρ − 1 M s M_s M s α / d \alpha/d α / d
Stability of the scalar response
At the independent endpoint, the last spin follows the scalar law with the profile a N a_N a N N N N d N = β 2 ( 1 − t / N ) d_N = \beta^2(1-t/N) d N = β 2 ( 1 − t / N ) (9) with an explicit ε N \varepsilon_N ε N
The response operator. For a profile a a a d ∈ [ β 2 / 2 , β 2 ] d \in[\beta^2/2,\beta^2] d ∈ [ β 2 /2 , β 2 ] u = u a , d u = u_{a,d} u = u a , d d d d Y Y Y m q = ∂ x u ( q , Y q ) m_q = \partial_x u(q,Y_q) m q = ∂ x u ( q , Y q ) h q = ∂ x x u ( q , Y q ) h_q = \partial_{xx}u(q,Y_q) h q = ∂ xx u ( q , Y q )
Γ a , d ( q ) = E m q 2 , k a , d ( q ) = d E h q 2 . \Gamma_{a,d}(q) = \mathbb{E}m_q^2,\qquad k_{a,d}(q) = d\mathbb{E}h_q^2. Γ a , d ( q ) = E m q 2 , k a , d ( q ) = d E h q 2 . so that Γ a , d ′ = k a , d \Gamma'_{a,d} = k_{a,d} Γ a , d ′ = k a , d k a , d k_{a,d} k a , d J a , d J_{a,d} J a , d Γ a , d \Gamma_{a,d} Γ a , d d ( q ∧ r ) d(q \wedge r) d ( q ∧ r ) d T θ ( q ∧ r ) dT_\theta(q \wedge r) d T θ ( q ∧ r ) T θ ( q ) = q + θ φ ( q ) T_\theta(q) = q + \theta\varphi(q) T θ ( q ) = q + θ φ ( q ) φ ( 0 ) = φ ( 1 ) = 0 \varphi(0) = \varphi(1) = 0 φ ( 0 ) = φ ( 1 ) = 0 p p p ( J a , d φ ) ( p ) (J_{a,d}\varphi)(p) ( J a , d φ ) ( p ) θ = 0 \theta= 0 θ = 0 J a , d s c J^{\mathrm{sc}}_{a,d} J a , d sc J a , d J_{a,d} J a , d H ν \mathcal{H}_\nu H ν L 2 ( ν ) L^2(\nu) L 2 ( ν ) ν \nu ν C β C_\beta C β
Domination by the multiplier. The following observation reduces the operator bound to a pointwise bound. In Lemma 9.7, we show that, for every profile, J a , d s c J^{\mathrm{sc}}_{a,d} J a , d sc
J a , d s c ⪯ M k a , d , ( M k a , d v ) ( q ) = k a , d ( q ) v ( q ) . J^{\mathrm{sc}}_{a,d} \preceq M_{k_{a,d}},\qquad(M_{k_{a,d}}v)(q) = k_{a,d}(q)v(q). J a , d sc ⪯ M k a , d , ( M k a , d v ) ( q ) = k a , d ( q ) v ( q ) . To prove it, we follow the scalar Parisi functional P d \mathcal{P}_d P d Π ( θ ) = P d ( a ∘ T θ − 1 ) \Pi(\theta) = \mathcal{P}_d(a \circ T_\theta^{-1}) Π ( θ ) = P d ( a ∘ T θ − 1 ) Π ′ ′ ( 0 ) = ( d / 2 ) ⟨ φ , ( I − J a , d ) φ ⟩ \Pi''(0) = (d/2)\langle\varphi,(I-J_{a,d})\varphi\rangle Π ′′ ( 0 ) = ( d /2 ) ⟨ φ , ( I − J a , d ) φ ⟩ (120) ; the first derivative is the first variation under transports of levels due to Talagrand [79] (see [9 ] , Section 3, (19)). Since the functional is convex along segments of distribution functions [10 ] , its increment along the transport is at least its first variation at a a a a ∘ T θ − 1 − a a \circ T_\theta^{-1} - a a ∘ T θ − 1 − a θ \theta θ ( 1 − k a , d ) φ (1-k_{a,d})\varphi ( 1 − k a , d ) φ ν \nu ν ⟨ φ , J a , d φ ⟩ ≤ ⟨ φ , k a , d φ ⟩ \langle\varphi,J_{a,d}\varphi\rangle\le\langle\varphi,k_{a,d}\varphi\rangle ⟨ φ , J a , d φ ⟩ ≤ ⟨ φ , k a , d φ ⟩ Π \Pi Π θ ↦ a ∘ T θ − 1 \theta\mapsto a \circ T_\theta^{-1} θ ↦ a ∘ T θ − 1 1 − k a , d 1-k_{a,d} 1 − k a , d sup q k a N , d N ( q ) ≤ 1 + ε N \sup_q k_{a_N,d_N}(q) \le1+\varepsilon_N sup q k a N , d N ( q ) ≤ 1 + ε N
The multiplier at the Parisi measure. In Lemma 9.8, we prove that
k α β , β 2 ( q ) = 1 ( 0 ≤ q ≤ q β ) , k α β , β 2 ( q ) ≤ e − β 2 ( q − q β ) / 2 ( q β ≤ q ≤ 1 ) . k_{\alpha_\beta,\beta^2}(q) = 1 \quad(0 \le q \le q_\beta),\qquad k_{\alpha_\beta,\beta^2}(q) \le e^{-\beta^2(q-q_\beta)/2} \quad(q_\beta\le q \le1). k α β , β 2 ( q ) = 1 ( 0 ≤ q ≤ q β ) , k α β , β 2 ( q ) ≤ e − β 2 ( q − q β ) /2 ( q β ≤ q ≤ 1 ) . On the support, we rely on an identity of Talagrand [79], in the form of [9 Theorem 5] and [48, Lemma 2.6], which states that Γ ( q ) = q \Gamma(q) = q Γ ( q ) = q supp μ β \operatorname{supp}\mu_\beta supp μ β [ 0 , q β ] [0,q_\beta] [ 0 , q β ] k = Γ ′ = 1 k = \Gamma' = 1 k = Γ ′ = 1 α β = 1 \alpha_\beta= 1 α β = 1 u u u z q z_q z q Y q Y_q Y q z q β z_{q_\beta} z q β z q β z_{q_\beta} z q β ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) z q z_q z q sech 3 \operatorname{sech}^3 sech 3 [ 0 , 1 ] [0,1] [ 0 , 1 ] q β q_\beta q β
Marginality. At the Parisi measure, the multiplier equals one on the whole support, and the domination gives no contraction there. The cavity inequality contracts because of the ratio α / d ≤ 1 − ρ \alpha/d \le1-\rho α / d ≤ 1 − ρ ρ \rho ρ ε N ≤ ρ / 4 \varepsilon_N \le\rho/4 ε N ≤ ρ /4 k = 1 k=1 k = 1 § 2.4 .
Transfer to the minimizer. By Lemma 9.3, the multiplier depends Lipschitz continuously on the profile and coefficient, in the sense that
sup 0 ≤ q ≤ 1 ∣ k a , d ( q ) − k a ~ , d ~ ( q ) ∣ ≤ C β ( ∥ a − a ~ ∥ L 1 ( 0 , 1 ) + ∣ d − d ~ ∣ ) . \sup_{0\le q\le1} \lvert k_{a,d}(q)-k_{\widetilde{a},\widetilde{d}}(q)\rvert\le C_\beta\left(\lVert a-\widetilde{a}\rVert_{L^1(0,1)}+\lvert d-\widetilde{d}\rvert\right). 0 ≤ q ≤ 1 sup ∣ k a , d ( q ) − k a , d ( q )∣ ≤ C β ( ∥ a − a ∥ L 1 ( 0 , 1 ) + ∣ d − d ∣ ) . We have ∣ d N − β 2 ∣ ≤ β 2 / N \lvert d_N-\beta^2\rvert\le\beta^2/N ∣ d N − β 2 ∣ ≤ β 2 / N ∥ γ λ t r − γ β ∥ L 1 ( 0 , 1 ) ≲ λ 2 \lVert\gamma_\lambda^{\mathrm{tr}}-\gamma_\beta\rVert_{L^1(0,1)}\lesssim\lambda^2 ∥ γ λ tr − γ β ∥ L 1 ( 0 , 1 ) ≲ λ 2 § 3.8 , we obtain Proposition 9.9, which states that on the space H ν N \mathcal{H}_{\nu_N} H ν N
J a N , d N s c ⪯ ( 1 + ε N ) I , ε N ≤ C s t [ ρ − 3 / 4 η 1 / 4 + λ 2 + N − 1 ] . J_{a_N,d_N}^{\mathrm{sc}}\preceq(1+\varepsilon_N)I,\qquad\varepsilon_N\le C_{\mathrm{st}}\left[\rho^{-3/4}\eta^{1/4}+\lambda^2+N^{-1}\right]. J a N , d N sc ⪯ ( 1 + ε N ) I , ε N ≤ C st [ ρ − 3/4 η 1/4 + λ 2 + N − 1 ] . with C s t C_{\mathrm{st}} C st β \beta β a N a_N a N
Localization of the minimizing profile
We need two bounds on the distance between a minimizer γ ∗ \gamma_* γ ∗ F ρ F_\rho F ρ γ t r \gamma^{\mathrm{tr}} γ tr η \eta η L 1 ( 0 , 1 ) L^1(0,1) L 1 ( 0 , 1 ) ρ \rho ρ
Convexity and response. From [4 Section 7.4 and Appendix A] , we import the following facts (Proposition 7.1). The functional F ρ F_\rho F ρ γ θ = ( 1 − θ ) γ 0 + θ γ 1 \gamma_\theta=(1-\theta)\gamma_0+\theta\gamma_1 γ θ = ( 1 − θ ) γ 0 + θ γ 1 f ( θ ) = F ρ ( γ θ ) f(\theta)=F_\rho(\gamma_\theta) f ( θ ) = F ρ ( γ θ ) ζ θ ( q ) = β 2 N − 1 E γ θ Tr H q 2 \zeta_\theta(q)=\beta^2N^{-1}\mathbb{E}_{\gamma_\theta}\operatorname{Tr}H_q^2 ζ θ ( q ) = β 2 N − 1 E γ θ Tr H q 2 ρ ζ θ \rho\zeta_\theta ρ ζ θ γ 1 − γ 0 \gamma_1-\gamma_0 γ 1 − γ 0
ρ ∣ ∫ 0 1 ( γ 1 − γ 0 ) ζ θ φ d q ∣ ≤ ( f ′ ′ ( θ ) ρ ) 1 / 2 [ γ 1 / 2 + 2 ( γ + Z ) 1 / 2 ] . \rho\left\lvert\int_0^1(\gamma_1-\gamma_0)\zeta_\theta\varphi\,dq\right\rvert\le\left(\frac{f''(\theta)}{\rho}\right)^{1/2}\left[\gamma^{1/2}+2(\gamma+Z)^{1/2}\right]. ρ ∫ 0 1 ( γ 1 − γ 0 ) ζ θ φ d q ≤ ( ρ f ′′ ( θ ) ) 1/2 [ γ 1/2 + 2 ( γ + Z ) 1/2 ] . (59) , for Lipschitz cutoffs φ \varphi φ Y Y Y φ \varphi φ Z Z Z ρ φ 2 ζ θ \rho\varphi^2\zeta_\theta ρ φ 2 ζ θ d γ θ d\gamma_\theta d γ θ ζ θ ≥ k β 2 \zeta_\theta\ge k_\beta^2 ζ θ ≥ k β 2 k β > 0 k_\beta> 0 k β > 0 (60) . The midpoint inequality (65) bounds the integral of min { θ , 1 − θ } f ′ ′ ( θ ) \min\{\theta,1-\theta\}f^{\prime\prime}(\theta) min { θ , 1 − θ } f ′′ ( θ ) ( 0 , 1 ) (0,1) ( 0 , 1 ) F ρ ( γ 0 ) + F ρ ( γ 1 ) − 2 min F ρ F_\rho(\gamma_0)+F_\rho(\gamma_1)-2\min F_\rho F ρ ( γ 0 ) + F ρ ( γ 1 ) − 2 min F ρ Lemma 7.4 ).
Isolation in L 1 L^1 L 1 A minimizer may have atoms, and Z Z Z d γ θ d\gamma_\theta d γ θ γ ∗ \gamma_* γ ∗ h h h Lemma 7.3 ). It creates a density at most β / h \beta/h β / h ρ β h 2 / 12 \rho\beta h^2/12 ρβ h 2 /12 [4 Lemma 7.5] . Along the segment from the regularized minimizer to γ t r \gamma^{\mathrm{tr}} γ tr γ t r \gamma^{\mathrm{tr}} γ tr q β q_\beta q β q β q_\beta q β f ′ ′ ≳ ρ 2 min { ρ D 4 , h D 2 } f^{\prime\prime}\gtrsim\rho^2\min\{\rho D^4,hD^2\} f ′′ ≳ ρ 2 min { ρ D 4 , h D 2 } D = ∥ γ ∗ − γ t r ∥ L 1 ( 0 , 1 ) D=\|\gamma_*-\gamma^{\mathrm{tr}}\|_{L^1(0,1)} D = ∥ γ ∗ − γ tr ∥ L 1 ( 0 , 1 ) (76) . For h h h ρ D 2 \rho D^2 ρ D 2 f ′ ′ ≳ ρ 3 D 4 f^{\prime\prime}\gtrsim\rho^3D^4 f ′′ ≳ ρ 3 D 4 ρ 3 D 4 \rho^3D^4 ρ 3 D 4 ρ 3 D 4 ≲ η \rho^3D^4\lesssim\eta ρ 3 D 4 ≲ η Proposition 7.6 ,
∥ γ ∗ − γ t r ∥ L 1 ( 0 , 1 ) ≤ C min { 1 , ρ − 3 / 4 η 1 / 4 } , \|\gamma_*-\gamma^{\mathrm{tr}}\|_{L^1(0,1)}\le C\min\{1,\rho^{-3/4}\eta^{1/4}\}, ∥ γ ∗ − γ tr ∥ L 1 ( 0 , 1 ) ≤ C min { 1 , ρ − 3/4 η 1/4 } , for every minimizer and every trial budget, with C C C β \beta β
Root localization. In the isolation argument, the lower bound ρ ∫ ζ θ φ ≥ ρ k β 2 ∫ φ \rho\int\zeta_\theta\varphi\ge\rho k_\beta^2\int\varphi ρ ∫ ζ θ φ ≥ ρ k β 2 ∫ φ ρ \rho ρ ρ = 0 \rho=0 ρ = 0 Lemma 8.1 . Suppose that two nearly minimal profiles have bounded densities on an interval I I I ℓ \ell ℓ I I I η / ρ \eta/\rho η / ρ ℓ \ell ℓ I I I F ρ F_\rho F ρ ℓ \ell ℓ ρ \rho ρ Q γ ∗ ( v ) Q_{\gamma_*}(v) Q γ ∗ ( v ) q = min { Q γ ∗ ( v ) , T 1 } q=\min\{Q_{\gamma_*}(v),T_1\} q = min { Q γ ∗ ( v ) , T 1 } T 1 T_1 T 1 Lemma 8.2 , which depends only on β \beta β Q γ ∗ ( v ) ≤ q / T 1 Q_{\gamma_*}(v)\le q/T_1 Q γ ∗ ( v ) ≤ q / T 1 q q q λ + v \lambda+v λ + v q q q ( q / 4 , 3 q / 4 ) (q/4,3q/4) ( q /4 , 3 q /4 ) k 0 / 2 k_0/2 k 0 /2 Lemma 8.2 ) and exceeds it by an amount of order q q q η / ρ ≳ min { q 6 , q 4 / ( m / h + C ) } \eta/\rho\gtrsim\min\{q^6,q^4/(m/h+C)\} η / ρ ≳ min { q 6 , q 4 / ( m / h + C )} m / h m/h m / h Proposition 8.3 , which states that
Q γ ∗ ( v ) ≤ C [ λ + v + min { 1 , ( η / ρ ) 1 / 6 } ] ( 0 < v < β − λ ) . Q_{\gamma_*}(v)\le C\left[\lambda+v+\min\left\{1,\left(\eta/\rho\right)^{1/6}\right\}\right]\qquad(0<v<\beta-\lambda). Q γ ∗ ( v ) ≤ C [ λ + v + min { 1 , ( η / ρ ) 1/6 } ] ( 0 < v < β − λ ) . together with its form (8.11) in the probability normalization. We follow the strategy of [4 Lemma 7.6] . Compared with [4, (7.30)}, the bound depends on η \eta η ρ \rho ρ η / ρ \eta/\rho η / ρ
Closing the implicit inequality
By Proposition 9.9, the stability assumption (9) holds as soon as C s t [ ρ − 3 / 4 η 1 / 4 + λ 2 + N − 1 ] ≤ ρ / 4 C_{\mathrm{st}}[\rho^{-3/4}\eta^{1/4}+\lambda^2+N^{-1}]\leq\rho/4 C st [ ρ − 3/4 η 1/4 + λ 2 + N − 1 ] ≤ ρ /4 η ≥ L s λ 5 ≥ λ 5 \eta\geq L_{\mathrm{s}}\lambda^5\geq\lambda^5 η ≥ L s λ 5 ≥ λ 5 η 1 / 7 ≥ λ 5 / 7 = N − 1 / 10 \eta^{1/7}\geq\lambda^{5/7}=N^{-1/10} η 1/7 ≥ λ 5/7 = N − 1/10 λ 2 \lambda^2 λ 2 N − 1 N^{-1} N − 1 ρ − 3 / 4 η 1 / 4 ≲ ρ \rho^{-3/4}\eta^{1/4}\lesssim\rho ρ − 3/4 η 1/4 ≲ ρ ρ ≳ η 1 / 7 \rho\gtrsim\eta^{1/7} ρ ≳ η 1/7 K = 12 C s t K=12C_{\mathrm{st}} K = 12 C st
ρ 0 = min { 1 , K η 1 / 7 } (11) \rho_0=\min\{1,K\eta^{1/7}\} \tag*{(11)} ρ 0 = min { 1 , K η 1/7 } (11) If ρ 0 < 1 \rho_0<1 ρ 0 < 1 ε N ≤ ρ / 4 \varepsilon_N\leq\rho/4 ε N ≤ ρ /4 ρ ∈ [ ρ 0 , 1 ] \rho\in[\rho_0,1] ρ ∈ [ ρ 0 , 1 ] (5) applies; if ρ 0 = 1 \rho_0=1 ρ 0 = 1 ρ 0 ≥ λ 5 / 7 \rho_0\geq\lambda^{5/7} ρ 0 ≥ λ 5/7
∫ ρ 0 1 D N ( ρ ) d ρ ≲ log 2 ( 2 + N ) N λ ρ 0 ≲ log 2 ( 2 + N ) N λ 12 / 7 = N − 19 / 25 log 2 ( 2 + N ) . \int_{\rho_0}^{1}\mathcal{D}_N(\rho)\,\mathrm{d}\rho\lesssim\frac{\log^2(2+N)}{N\lambda\rho_0}\lesssim\frac{\log^2(2+N)}{N\lambda^{12/7}}=N^{-19/25}\log^2(2+N). ∫ ρ 0 1 D N ( ρ ) d ρ ≲ N λ ρ 0 log 2 ( 2 + N ) ≲ N λ 12/7 log 2 ( 2 + N ) = N − 19/25 log 2 ( 2 + N ) . On ( 0 , ρ 0 ] (0,\rho_0] ( 0 , ρ 0 ] ρ 0 ≤ K η 1 / 7 \rho_0\leq K\eta^{1/7} ρ 0 ≤ K η 1/7 (8) , which gives ρ 0 1 / 3 ≲ η 1 / 21 \rho_0^{1/3}\lesssim\eta^{1/21} ρ 0 1/3 ≲ η 1/21 η 1 / 9 ρ 0 2 / 9 ≲ η 1 / 7 \eta^{1/9}\rho_0^{2/9}\lesssim\eta^{1/7} η 1/9 ρ 0 2/9 ≲ η 1/7 4 e N , β ( λ ) = ∫ 0 1 D N ( ρ ) d ρ 4e_{N,\beta}(\lambda)=\int_0^1\mathcal{D}_N(\rho)\,\mathrm{d}\rho 4 e N , β ( λ ) = ∫ 0 1 D N ( ρ ) d ρ (211) ,
e N , β ( λ ) ≲ N − 19 / 25 log 3 ( 2 + N ) + N − 2 / 3 η 1 / 21 + N − 2 / 3 λ − 1 / 3 η 1 / 7 . (12) e_{N,\beta}(\lambda)\lesssim N^{-19/25}\log^3(2+N)+N^{-2/3}\eta^{1/21}+N^{-2/3}\lambda^{-1/3}\eta^{1/7}. \tag*{(12)} e N , β ( λ ) ≲ N − 19/25 log 3 ( 2 + N ) + N − 2/3 η 1/21 + N − 2/3 λ − 1/3 η 1/7 . (12) But there is an apparent circularity here, since the cutoff ρ 0 \rho_0 ρ 0 η = e N , β ( λ ) + L s λ 5 \eta=e_{N,\beta}(\lambda)+L_{\mathrm{s}}\lambda^5 η = e N , β ( λ ) + L s λ 5 (12) only through powers of η \eta η r = N − 7 / 10 = λ 5 r=N^{-7/10}=\lambda^5 r = N − 7/10 = λ 5 x = η / r x=\eta/r x = η / r L s ≥ 1 L_{\mathrm{s}}\geq1 L s ≥ 1 (12) by r r r L s L_{\mathrm{s}} L s
x ≤ L s + C β [ N − 3 / 50 log 3 ( 2 + N ) + x 1 / 21 + N − 1 / 50 x 1 / 7 ] ≤ C β x 1 / 7 , (13) x\leq L_{\mathrm{s}}+C_\beta\left[N^{-3/50}\log^3(2+N)+x^{1/21}+N^{-1/50}x^{1/7}\right]\leq C_\beta x^{1/7}, \tag*{(13)} x ≤ L s + C β [ N − 3/50 log 3 ( 2 + N ) + x 1/21 + N − 1/50 x 1/7 ] ≤ C β x 1/7 , (13) which implies that x ≤ C β 7 / 6 x\leq C_\beta^{7/6} x ≤ C β 7/6 e N , β ( λ ) ≤ η = r x ≲ N − 7 / 10 e_{N,\beta}(\lambda)\leq\eta=rx\lesssim N^{-7/10} e N , β ( λ ) ≤ η = r x ≲ N − 7/10 [4 Proposition 7.7] closes an inequality in which the unknown error enters the trial budget in the same way. By Proposition 3.9, we conclude that P β − p N , β ≲ N − 7 / 10 P_\beta-p_{N,\beta}\lesssim N^{-7/10} P β − p N , β ≲ N − 7/10 N N N N N N
The exponent. If η \eta η N − a N^{-a} N − a N − 2 / 3 η 1 / 21 N^{-2/3}\eta^{1/21} N − 2/3 η 1/21 (12) is of the same order as η \eta η
a = 2 3 + a 21 , that is, a = 7 10 , a=\frac{2}{3}+\frac{a}{21},\qquad\text{that is,}\qquad a=\frac{7}{10}, a = 3 2 + 21 a , that is, a = 10 7 , and the cost λ 5 \lambda^5 λ 5 λ = N − 7 / 50 \lambda=N^{-7/50} λ = N − 7/50 Table 2.1 lists the terms of (2.6) after division by N − 7 / 10 N^{-7/10} N − 7/10 N − 1 / 50 N^{-1/50} N − 1/50 N − 3 / 50 N^{-3/50} N − 3/50 N − 4 / 25 N^{-4/25} N − 4/25 N − 3 / 10 N^{-3/10} N − 3/10 L s N − 7 / 10 ≤ η ≲ N − 7 / 10 L_sN^{-7/10}\le\eta\lesssim N^{-7/10} L s N − 7/10 ≤ η ≲ N − 7/10 N − 1 / 10 N^{-1/10} N − 1/10 N N N Table 2.2 collects the exponents of the proof and the statements in which they arise.
Table 2.1. The terms of the bound on e N , β ( λ ) e_{N,\beta}(\lambda) e N , β ( λ ) λ = N − 7 / 50 \lambda=N^{-7/50} λ = N − 7/50 x = η / N − 7 / 10 x=\eta/N^{-7/10} x = η / N − 7/10 (12) ; see § 12.1.2 .
Table 2.2. The exponents of the proof.
What would improve the exponent. The exponent 1 / 21 1/21 1/21 1 / 3 1/3 1/3 ρ 0 \rho_0 ρ 0 1 / 7 1/7 1/7 η \eta η log 2 ( 2 + N ) / ( N λ ρ 0 ) \log^{2}(2+N)/(N\lambda\rho_{0}) log 2 ( 2 + N ) / ( N λ ρ 0 ) Table 2.1 . If the cavity bound (5) held down to ρ \rho ρ N − 1 / 4 N^{-1/4} N − 1/4 N − 3 / 4 N^{-3/4} N − 3/4 § 2.4 .
Minimizers, arbitrary profiles, and atoms
Three conventions recur throughout the proof.
Which statements use minimality. Estimates that use the contact identities (22) or the trial budget hold at minimizers: the main results of Sections 5 to 8, Step 2 of the proof of Lemma 12.1, and everything built on them. The Gaussian tools of Section 4, the lemmas of Section 9 on the scalar law of an arbitrary profile, and the cavity calculus of Sections 10 and 11 hold for every deterministic profile (with ν ( { 1 } ) = 0 \nu(\{1\})=0 ν ({ 1 }) = 0 Lemma 9.8 concerns the Parisi measure μ β \mu_\beta μ β Proposition 9.9 concerns minimizing profiles. We need this generality, since at the independent endpoint, the n n n
Every minimizer. A minimizer need not be unique, and it may depend on N N N ρ \rho ρ λ \lambda λ (29) holds at every minimizer, and since our bounds on A A A J J J V V V ρ \rho ρ
Atoms. Minimizing profiles may have atoms in [ 0 , 1 ) [0,1) [ 0 , 1 ) ν \nu ν 0 0 0 ν ( { 1 } ) = 0 \nu(\{1\})=0 ν ({ 1 }) = 0 Proposition 3.1 . Distribution functions are right-continuous, so that a ( q ) a(q) a ( q ) q q q ( q , 1 ] (q,1] ( q , 1 ] ν \nu ν § 5.1.1 ). We handle scalar and tree laws through exact path likelihoods (Lemmas 9.2 and 10.3 ) and through the heat identity of Lemma 4.3 , in which profile measures are held fixed with their atoms. In Sections 7 and 8, we spread the free mass before using cutoffs.
A map of the proof
Table 2.3 lists, for each section, its main results, its main inputs, and the places where its results are used. The proof of Theorem 1.1 in § 12.1.2 begins with a table of its inputs and of the parameters at which they are used. In terms of these results, the proof has the following structure.
Table 2.3. The sections of the proof, their main results and inputs, and where the results are used.
Proposition 3.9 , Corollary 3.7 , and Lemma 3.8 reduce the theorem to a bound on the integral of D N D_N D N η \eta η ρ 0 \rho_0 ρ 0
On [ ρ 0 , 1 ] [\rho_{0},1] [ ρ 0 , 1 ] Corollary 12.5 bounds the defect, using Proposition 12.4 , Corollary 5.3, and Proposition 6.4 . The contraction of Proposition 12.4 combines Lemmas 10.8 , 11.2 , 11.3 , and 12.1 to 12.3 , and its stability hypothesis (194) is verified by Proposition 9.9 , which combines Lemmas 9.3 , 9.7, and 9.8 and Proposition 7.6 .
I The interpolation
The interpolation and the defect
In this section, we recall the interpolation of [4 Section 2] , which compares the fractional moments of the partition function with the constrained scalar Parisi problem. We also fix the normalizations, the tilted law, and the observables that we use in the rest of the paper. The later sections build on two results. The first is the defect identity (29) of Corollary 3.7 . It states that four times the fractional-moment error e N , β ( λ ) e_{N,\beta}(\lambda) e N , β ( λ ) D N ( ρ ) = β ( A + 2 J + V ) D_N(\rho)=\beta(A+2J+V) D N ( ρ ) = β ( A + 2 J + V ) Proposition 3.9 , where we show that ( P β − p N , β ) / β (P_\beta-p_{N,\beta})/\beta ( P β − p N , β ) / β 2 e N , β ( λ ) 2e_{N,\beta}(\lambda) 2 e N , β ( λ ) λ 5 \lambda^5 λ 5 Proposition 3.1 , the integrated variance bound in Proposition 3.3, the decomposition of the overlap error in Lemma 3.5 , the interpolation identity in Proposition 3.6 , and the trial budget in Lemma 3.8 .
Most of these facts are proved in [4 ] . We state each of them in full, with its location in [4 ] . Where the argument is short, we also sketch the proof, so that this section can be read without [4 ] at hand. Besides [4 ] , this section uses only the scalar Parisi functional (1.1) and the notation of § 2.1 , together with the support theorem [48, Theorem 1.1]. Its results are used throughout the paper. Propositions 3.1 and 3.3 and Lemma 3.5 are used in Sections 5, 6, 10, and § 10.2 ; the energy normalization and the trial budget in Sections 7 to 9; and (29) , (30) , and Proposition 3.9 in § 10.2 and 13. We treat the laws with anisotropic fields in Section 4, where we also prove the two Gaussian tools that we need for profiles that do not minimize the functional.
Throughout this section, we fix β > 1 \beta>1 β > 1 N ≥ 1 N\ge1 N ≥ 1 λ \lambda λ N N N § 4.1.3 to § 4.1.6 hold for every β > 0 \beta>0 β > 0 β > 1 \beta>1 β > 1 § 4.1.1 and for the cost of the scalar floor in § 4.1.7 .
The interpolation of [4 Section 2] is a quantitative form of the Guerra interpolation that Talagrand used for the positive fractional moments of the partition function, whose limits he determined in [80]; see [4 Section 1.2] . For fixed 0 < s < 1 0<s<1 0 < s < 1 s s s s s s log 2 \log2 log 2 N N N § 4.1.2 carries the same constraint, in the sense that its profiles have the prescribed mass s s s
Completion and the constrained scalar value
Let W W W N × N N \times N N × N W i j = g i j / N W_{ij}=g_{ij}/\sqrt{N} W ij = g ij / N i < j i<j i < j W i i = 2 / N z i W_{ii}=\sqrt{2/N}\,z_i W ii = 2/ N z i z 1 , … , z N z_1,\ldots,z_N z 1 , … , z N ( g i j ) (g_{ij}) ( g ij ) E r e f \mathbb{E}_{\mathrm{ref}} E ref ( g i j , z i ) (g_{ij},z_i) ( g ij , z i ) [4 Section 1.1] , we set
D N = 1 2 Tr W , F N , β c = log ∑ σ ∈ { − 1 , 1 } N e β σ T W σ / 2 . D_N=\frac{1}{2}\operatorname{Tr}W,\qquad F^c_{N,\beta}=\log\sum_{\sigma\in\{-1,1\}^N}e^{\beta\sigma^{T}W\sigma/2}. D N = 2 1 Tr W , F N , β c = log σ ∈ { − 1 , 1 } N ∑ e β σ T W σ /2 . and we recall that F N , β = log ∑ σ e β H N ( σ ) F_{N,\beta}=\log\sum_{\sigma}e^{\beta H_N(\sigma)} F N , β = log ∑ σ e β H N ( σ ) σ T W σ / 2 = H N ( σ ) + D N \sigma^{T}W\sigma/2=H_N(\sigma)+D_N σ T W σ /2 = H N ( σ ) + D N σ i 2 = 1 \sigma_i^2=1 σ i 2 = 1 F N , β c = F N , β + β D N F^c_{N,\beta}=F_{N,\beta}+\beta D_N F N , β c = F N , β + β D N D N = ( 2 N ) − 1 / 2 ∑ i z i D_N=(2N)^{-1/2}\sum_i z_i D N = ( 2 N ) − 1/2 ∑ i z i 1 / 2 1/2 1/2 F N , β F_{N,\beta} F N , β
The diagonal term completes the covariance of the Hamiltonian to an exact function of the overlap R ( σ , σ ′ ) = N − 1 σ ⋅ σ ′ R(\sigma,\sigma')=N^{-1}\sigma\cdot\sigma' R ( σ , σ ′ ) = N − 1 σ ⋅ σ ′ τ i = σ i σ i ′ \tau_i=\sigma_i\sigma'_i τ i = σ i σ i ′
E r e f [ H N ( σ ) H N ( σ ′ ) ] = 1 N ∑ i < j τ i τ j = 1 2 N [ ( ∑ i = 1 N τ i ) 2 − N ] = N 2 R ( σ , σ ′ ) 2 − 1 2 . \mathbb{E}_{\mathrm{ref}}\left[H_N(\sigma)H_N(\sigma')\right]=\frac{1}{N}\sum_{i<j}\tau_i\tau_j=\frac{1}{2N}\left[\left(\sum_{i=1}^{N}\tau_i\right)^2-N\right]=\frac{N}{2}R(\sigma,\sigma')^2-\frac{1}{2}. E ref [ H N ( σ ) H N ( σ ′ ) ] = N 1 i < j ∑ τ i τ j = 2 N 1 ( i = 1 ∑ N τ i ) 2 − N = 2 N R ( σ , σ ′ ) 2 − 2 1 . and adding Var ( D N ) = 1 / 2 \operatorname{Var}(D_N)=1/2 Var ( D N ) = 1/2 σ T W σ / 2 \sigma^{T}W\sigma/2 σ T W σ /2 N R ( σ , σ ′ ) 2 / 2 NR(\sigma,\sigma')^2/2 N R ( σ , σ ′ ) 2 /2 D N D_N D N F N , β F_{N,\beta} F N , β s > 0 s>0 s > 0 [4 (1.3)]
ϕ N , β c ( s ) = 1 N s log E r e f e s F N , β c = 1 N s log E r e f e s F N , β + β 2 s 4 N . \phi^c_{N,\beta}(s)=\frac{1}{Ns}\log\mathbb{E}_{\mathrm{ref}}e^{sF^c_{N,\beta}}=\frac{1}{Ns}\log\mathbb{E}_{\mathrm{ref}}e^{sF_{N,\beta}}+\frac{\beta^2s}{4N}. ϕ N , β c ( s ) = N s 1 log E ref e s F N , β c = N s 1 log E ref e s F N , β + 4 N β 2 s . where we used log E r e f e s β D N = β 2 s 2 / 4 \log\mathbb{E}_{\mathrm{ref}}e^{s\beta D_N}=\beta^2s^2/4 log E ref e s β D N = β 2 s 2 /4 s ↓ 0 s\downarrow0 s ↓ 0 ϕ N , β c ( s ) → p N , β \phi^c_{N,\beta}(s)\to p_{N,\beta} ϕ N , β c ( s ) → p N , β F N , β F_{N,\beta} F N , β
Recall the scalar Parisi functional P β \mathcal{P}_\beta P β [ 0 , 1 ] [0,1] [ 0 , 1 ] 0 ≤ s ≤ 1 0\leq s\leq1 0 ≤ s ≤ 1 0 ≤ λ ≤ β 0\leq\lambda\leq\beta 0 ≤ λ ≤ β
P β ( s ) = min μ ∈ P ( [ 0 , 1 ] ) : μ ( { 0 } ) ≥ s P β ( μ ( [ 0 , ⋅ ] ) ) , E β ( λ ) = β − 1 P β ( λ / β ) . \mathcal{P}_\beta(s)=\min_{\mu\in\mathcal{P}([0,1]):\,\mu(\{0\})\geq s}\mathcal{P}_\beta(\mu([0,\cdot])),\qquad E_\beta(\lambda)=\beta^{-1}\mathcal{P}_\beta(\lambda/\beta). P β ( s ) = μ ∈ P ([ 0 , 1 ]) : μ ({ 0 }) ≥ s min P β ( μ ([ 0 , ⋅ ])) , E β ( λ ) = β − 1 P β ( λ / β ) . The minimum is attained [4 Section 1.1] , and P β ( 0 ) = P β \mathcal{P}_\beta(0)=\mathcal{P}_\beta P β ( 0 ) = P β E β E_\beta E β [4 Section 7.1] , in which the order s s s λ = β s \lambda=\beta s λ = β s E β E_\beta E β s s s E β ( λ ) ≥ E β ( 0 ) E_\beta(\lambda)\ge E_\beta(0) E β ( λ ) ≥ E β ( 0 )
The unconstrained minimizer μ β \mu_\beta μ β [10 ] . We write α β ( q ) = μ β ( [ 0 , q ] ) \alpha_\beta(q)=\mu_\beta([0,q]) α β ( q ) = μ β ([ 0 , q ]) γ β = β α β \gamma_\beta=\beta\alpha_\beta γ β = β α β β > 1 \beta>1 β > 1 supp μ β = [ 0 , q β ] \operatorname{supp}\mu_\beta=[0,q_\beta] supp μ β = [ 0 , q β ] 0 < q β < 1 0<q_\beta<1 0 < q β < 1 μ β \mu_\beta μ β
For 0 < λ < β 0<\lambda<\beta 0 < λ < β s = λ / β s=\lambda/\beta s = λ / β fractional-moment error by
e N , β ( λ ) = P β ( s ) − Φ N , β c ( s ) β = E β ( λ ) − 1 N λ log E r e f e λ ( F N , β / β + D N ) . (14) e_{N,\beta}(\lambda)=\frac{P_\beta(s)-\Phi^c_{N,\beta}(s)}{\beta}=E_\beta(\lambda)-\frac{1}{N\lambda}\log\mathbb{E}_{\mathrm{ref}}e^{\lambda(F_{N,\beta}/\beta+D_N)}. \tag*{(14)} e N , β ( λ ) = β P β ( s ) − Φ N , β c ( s ) = E β ( λ ) − N λ 1 log E ref e λ ( F N , β / β + D N ) . (14) In the second expression, we used s F N , β c = λ ( F N , β / β + D N ) sF^c_{N,\beta}=\lambda(F_{N,\beta}/\beta+D_N) s F N , β c = λ ( F N , β / β + D N ) λ / ( 4 N ) \lambda/(4N) λ / ( 4 N ) β \beta β e N ( λ ) = e N , β ( λ ) e_N(\lambda)=e_{N,\beta}(\lambda) e N ( λ ) = e N , β ( λ ) s = λ / β ≤ 1 / 2 s=\lambda/\beta\le1/2 s = λ / β ≤ 1/2 [4 Section 2] . We will see in Corollary 3.7 that the error e N , β ( λ ) e_{N,\beta}(\lambda) e N , β ( λ )
The interpolating functional
For 0 ≤ ρ ≤ 1 0\le\rho\le1 0 ≤ ρ ≤ 1
t = 1 − ρ , c = β 2 ρ , M = N c . t=1-\rho,\qquad c=\beta^2\rho,\qquad M=Nc. t = 1 − ρ , c = β 2 ρ , M = N c . The interpolation reaches the fractional moment at ρ = 0 \rho=0 ρ = 0 ρ = 1 \rho=1 ρ = 1 ρ \rho ρ ρ ↓ 0 \rho\downarrow0 ρ ↓ 0 § 4.1.3 for 0 < ρ ≤ 1 0<\rho\le1 0 < ρ ≤ 1 c > 0 c>0 c > 0
An admissible profile is a deterministic probability measure μ = s δ 0 + ν \mu=s\delta_0+\nu μ = s δ 0 + ν [ 0 , 1 ] [0,1] [ 0 , 1 ] ν ≥ 0 \nu\ge0 ν ≥ 0 1 − s 1-s 1 − s 0 0 0 μ \mu μ μ ( { 0 } ) ≥ s \mu(\{0\})\ge s μ ({ 0 }) ≥ s s δ 0 s\delta_0 s δ 0 ν \nu ν μ \mu μ a ( q ) = μ ( [ 0 , q ] ) a(q)=\mu([0,q]) a ( q ) = μ ([ 0 , q ]) a ≥ s a\ge s a ≥ s a ( 1 ) = 1 a(1)=1 a ( 1 ) = 1 W W W U = U μ , t U=U_{\mu,t} U = U μ , t
∂ q U = − c 2 ( Δ U + a ( q ) ∣ ∇ U ∣ 2 ) , U ( 1 , x ; W ) = log ∑ σ ∈ { − 1 , 1 } N e β t σ T W / 2 + x ⋅ σ . (15) \partial_q U=-\frac{c}{2}\left(\Delta U+a(q)|\nabla U|^2\right),\qquad U(1,x;W)=\log\sum_{\sigma\in\{-1,1\}^N}e^{\beta\sqrt{t}\,\sigma^{T}W/2+x\cdot\sigma}. \tag*{(15)} ∂ q U = − 2 c ( Δ U + a ( q ) ∣∇ U ∣ 2 ) , U ( 1 , x ; W ) = log σ ∈ { − 1 , 1 } N ∑ e β t σ T W /2 + x ⋅ σ . (15) This is the Parisi equation in R N \mathbb{R}^N R N β t \beta\sqrt{t} β t x x x
F t , s ( μ ) = 1 N s log E r e f e s U ( 0 , 0 ; W ) − c 2 ∫ 0 1 q a ( q ) d q , f N , s ( t ) = min μ F t , s ( μ ) . (16) \mathcal{F}_{t,s}(\mu)=\frac{1}{Ns}\log\mathbb{E}_{\mathrm{ref}}e^{sU(0,0;W)}-\frac{c}{2}\int_0^1qa(q)\,dq,\qquad f_{N,s}(t)=\min_\mu\mathcal{F}_{t,s}(\mu). \tag*{(16)} F t , s ( μ ) = N s 1 log E ref e s U ( 0 , 0 ; W ) − 2 c ∫ 0 1 q a ( q ) d q , f N , s ( t ) = μ min F t , s ( μ ) . (16) By [4 Section 2.6 and Proposition 2.3] , the equation (15) has a solution for every admissible profile, and F t , s \mathcal{F}_{t,s} F t , s § 4.1.3 . For a general profile, it is the uniform limit of the solutions for step profiles whose cumulative functions converge in L 1 ( [ 0 , 1 ] ) L^1([0,1]) L 1 ([ 0 , 1 ]) U a U_a U a U a ~ U_{\tilde a} U a ~ a a a a ~ \tilde a a ~ [4 Section 2.6, (2.19)]
sup q , x , W ∣ U a ( q , x ; W ) − U a ~ ( q , x ; W ) ∣ ≤ c N 2 ∫ 0 1 ∣ a ( u ) − a ~ ( u ) ∣ d u . \sup_{q,x,W} \lvert U_a(q,x;W)-U_{\tilde a}(q,x;W)\rvert\le\frac{cN}{2}\int_0^1 \lvert a(u)-\tilde a(u)\rvert\,\mathrm{d}u. q , x , W sup ∣ U a ( q , x ; W ) − U a ~ ( q , x ; W )∣ ≤ 2 c N ∫ 0 1 ∣ a ( u ) − a ~ ( u )∣ d u . The admissible profiles form a weakly compact set, since they are the image of the compact set of measures of mass 1 − s 1-s 1 − s ν ↦ s δ 0 + ν \nu\mapsto s\delta_0+\nu ν ↦ s δ 0 + ν L 1 L^1 L 1 (16) is attained.
We note that minimizers are deterministic. The Gaussian expectation is taken inside the functional, which means that F t , s ( μ ) F_{t,s}(\mu) F t , s ( μ ) N N N ρ \rho ρ λ \lambda λ § 4.1.3 is constructed from a deterministic profile, and the estimates at a minimizer combine that law with the variational inequality that the same profile satisfies. A minimizer need not be unique; at each use, we say which minimizer is meant.
The prescribed root mass s δ 0 s\delta_0 s δ 0 s s s Proposition 3.6 , the outer logarithmic moment contributes a term that cancels the root mass exactly, so that only the free measure ν \nu ν
The two endpoints of the interpolation are explicit. At t = 0 t=0 t = 0 (15) is ∑ i log ( 2 cosh x i ) \sum_i \log(2\cosh x_i) ∑ i log ( 2 cosh x i ) c = β 2 c=\beta^2 c = β 2 N N N U ( q , x ; W ) = ∑ i u a ( q , x i ) U(q,x;W)=\sum_i u_a(q,x_i) U ( q , x ; W ) = ∑ i u a ( q , x i ) u a u_a u a α = a \alpha=a α = a U ( 0 , 0 ; W ) = N u a ( 0 , 0 ) U(0,0;W)=Nu_a(0,0) U ( 0 , 0 ; W ) = N u a ( 0 , 0 ) F 0 , s ( μ ) = P β ( a ) F_{0,s}(\mu)=P_\beta(a) F 0 , s ( μ ) = P β ( a ) t = 1 t=1 t = 1 c = 0 c=0 c = 0 U ( 0 , 0 ; W ) = U ( 1 , 0 ; W ) = F N , β c U(0,0;W)=U(1,0;W)=F^c_{N,\beta} U ( 0 , 0 ; W ) = U ( 1 , 0 ; W ) = F N , β c F 1 , s ( μ ) = ϕ N , β c ( s ) F_{1,s}(\mu)=\phi^c_{N,\beta}(s) F 1 , s ( μ ) = ϕ N , β c ( s ) [4 Proposition 2.3]
f N , s ( 0 ) = P β ( s ) , f N , s ( 1 ) = ϕ N , β c ( s ) . (17) f_{N,s}(0)=P_\beta(s),\qquad f_{N,s}(1)=\phi^c_{N,\beta}(s). \tag*{(17)} f N , s ( 0 ) = P β ( s ) , f N , s ( 1 ) = ϕ N , β c ( s ) . (17) For a fixed profile, the map t ↦ F t , s ( μ ) t\mapsto F_{t,s}(\mu) t ↦ F t , s ( μ ) Proposition 3.6 ). By (14) , the fractional-moment error is ( f N , s ( 0 ) − f N , s ( 1 ) ) / β (f_{N,s}(0)-f_{N,s}(1))/\beta ( f N , s ( 0 ) − f N , s ( 1 )) / β f N , s / β f_{N,s}/\beta f N , s / β § 4.1.6 is four times the rate of this decrease.
The energy normalization. We vary profiles in the energy normalization of [4 Section 7.1] . For an admissible profile μ \mu μ a a a
γ = β a , ν γ = β ν , F ρ ( γ ) = β − 1 F 1 − ρ , λ / β ( μ ) . \gamma=\beta a,\qquad\nu_\gamma=\beta\nu,\qquad F_\rho(\gamma)=\beta^{-1}F_{1-\rho,\lambda/\beta}(\mu). γ = β a , ν γ = β ν , F ρ ( γ ) = β − 1 F 1 − ρ , λ / β ( μ ) . Then d γ \mathrm{d}\gamma d γ β \beta β d γ ≥ λ δ 0 \mathrm{d}\gamma\ge\lambda\delta_0 d γ ≥ λ δ 0 γ \gamma γ min γ F ρ ( γ ) = β − 1 f N , s ( 1 − ρ ) \min_\gamma F_\rho(\gamma)=\beta^{-1}f_{N,s}(1-\rho) min γ F ρ ( γ ) = β − 1 f N , s ( 1 − ρ ) γ \gamma γ U ~ ( q , y ; W ) = β − 1 U ( q , β y ; W ) \widetilde{U}(q,y;W)=\beta^{-1}U(q,\beta y;W) U ( q , y ; W ) = β − 1 U ( q , β y ; W )
∂ q U ~ = − ρ 2 ( Δ U ~ + γ ( q ) ∣ ∇ U ~ ∣ 2 ) , \partial_q\widetilde{U}=-\frac{\rho}{2}\left(\Delta\widetilde{U}+\gamma(q)|\nabla\widetilde{U}|^2\right), ∂ q U = − 2 ρ ( Δ U + γ ( q ) ∣∇ U ∣ 2 ) , with terminal value β − 1 \beta^{-1} β − 1 β y \beta y β y
F ρ ( γ ) = 1 N λ log E r e f e λ U ~ ( 0 , 0 ; W ) − ρ 2 ∫ 0 1 q γ ( q ) d q . F_{\rho}(\gamma)=\frac{1}{N\lambda}\log\mathbb{E}_{\mathrm{ref}}e^{\lambda\widetilde{U}(0,0;W)}-\frac{\rho}{2}\int_0^1q\gamma(q)\,\mathrm{d}q. F ρ ( γ ) = N λ 1 log E ref e λ U ( 0 , 0 ; W ) − 2 ρ ∫ 0 1 q γ ( q ) d q . This is the functional of [4 (7.2)] with r = ρ r=\rho r = ρ W W W [4 Section 7.1] . Note that the rescaling leaves the gradient unchanged, ∇ U ~ ( q , y ; W ) = ∇ U ( q , β y ; W ) \nabla\widetilde{U}(q,y;W)=\nabla U(q,\beta y;W) ∇ U ( q , y ; W ) = ∇ U ( q , β y ; W ) β \beta β
We define the scalar trial profile by γ λ t r ( q ) = max { λ , γ β ( q ) } \gamma^{\mathrm{tr}}_{\lambda}(q)=\max\{\lambda,\gamma_{\beta}(q)\} γ λ tr ( q ) = max { λ , γ β ( q )} λ ≤ γ λ t r ≤ β \lambda\leq\gamma^{\mathrm{tr}}_{\lambda}\leq\beta λ ≤ γ λ tr ≤ β γ λ t r ( 1 ) = β \gamma^{\mathrm{tr}}_{\lambda}(1)=\beta γ λ tr ( 1 ) = β γ λ t r / β \gamma^{\mathrm{tr}}_{\lambda}/\beta γ λ tr / β λ \lambda λ γ t r = γ λ t r \gamma^{\mathrm{tr}}=\gamma^{\mathrm{tr}}_{\lambda} γ tr = γ λ tr § 4.1.7 .
The tilted joint law and overlap observables
Fix 0 < ρ ≤ 1 0<\rho\leq1 0 < ρ ≤ 1 μ \mu μ [4 Section 2] , under which all later expectations are taken. It is built in three stages.
The Gaussian law of W W W e s U ( 0 , 0 ; W ) / E r e f e s U ( 0 , 0 ; W ) e^{sU(0,0;W)}/\mathbb{E}_{\mathrm{ref}}e^{sU(0,0;W)} e s U ( 0 , 0 ; W ) / E ref e s U ( 0 , 0 ; W )
Given W W W
d X q = c a ( q ) ∇ U ( q , X q ; W ) d q + c d B q , X 0 = 0 , (18) \mathrm{d}X_q=c a(q)\nabla U(q,X_q;W)\,\mathrm{d}q+\sqrt{c}\,\mathrm{d}B_q,\qquad X_0=0, \tag*{(18)} d X q = c a ( q ) ∇ U ( q , X q ; W ) d q + c d B q , X 0 = 0 , (18) where B B B R N \mathbb{R}^N R N W W W
Given W W W X X X σ ∈ { − 1 , 1 } N \sigma\in\{-1,1\}^N σ ∈ { − 1 , 1 } N exp { β t σ T W σ / 2 + X 1 ⋅ σ } \exp\{\beta\sqrt{t}\sigma^{T}W\sigma/2+X_1\cdot\sigma\} exp { β t σ T W σ /2 + X 1 ⋅ σ }
We write P \mathbb{P} P E \mathbb{E} E G q = σ ( W , X r : 0 ≤ r ≤ q ) \mathcal{G}_q=\sigma(W,X_r:0\leq r\leq q) G q = σ ( W , X r : 0 ≤ r ≤ q ) q q q N N N x x x [4 Section 2.6.1] , and the equation has a unique strong solution. The tilt in (i) changes the law of W W W P \mathbb{P} P B B B W W W
For a step profile, the law has a discrete description, which explains the construction. The Gaussian Hopf–Cole recursion in the Parisi setting is described in [40, Section 3, (10)–(13)], and the finite-dimensional tilted law used here is the formulation of [4 Section 2.1] . Suppose that the profile has grid points 0 = q 0 < q 1 < ⋯ < q K = 1 0=q_0<q_1<\cdots<q_K=1 0 = q 0 < q 1 < ⋯ < q K = 1 a = a ℓ a=a_\ell a = a ℓ [ q ℓ , q ℓ + 1 ) [q_\ell,q_{\ell+1}) [ q ℓ , q ℓ + 1 )
U ( q ℓ , x ; W ) = 1 a ℓ log E g exp { a ℓ U ( q ℓ + 1 , x + c ( q ℓ + 1 − q ℓ ) g ; W ) } . U(q_\ell,x;W)=\frac{1}{a_\ell}\log\mathbb{E}_g\exp\left\{a_\ell U\left(q_{\ell+1},x+\sqrt{c(q_{\ell+1}-q_\ell)}\,g;W\right)\right\}. U ( q ℓ , x ; W ) = a ℓ 1 log E g exp { a ℓ U ( q ℓ + 1 , x + c ( q ℓ + 1 − q ℓ ) g ; W ) } . where g g g R N \mathbb{R}^{N} R N E g U ( q ℓ + 1 , x + c ( q ℓ + 1 − q ℓ ) g ; W ) \mathbb{E}_{g}U(q_{\ell+1},x+\sqrt{c(q_{\ell+1}-q_{\ell})}\,g;W) E g U ( q ℓ + 1 , x + c ( q ℓ + 1 − q ℓ ) g ; W ) a ℓ = 0 a_{\ell}=0 a ℓ = 0 W W W X q ℓ X_{q_{\ell}} X q ℓ X q ℓ + 1 − X q ℓ X_{q_{\ell+1}}-X_{q_{\ell}} X q ℓ + 1 − X q ℓ c ( q ℓ + 1 − q ℓ ) g \sqrt{c(q_{\ell+1}-q_{\ell})}\,g c ( q ℓ + 1 − q ℓ ) g exp { a ℓ ( U ( q ℓ + 1 , X q ℓ + 1 ; W ) − U ( q ℓ , X q ℓ ; W ) ) } \exp\{a_{\ell}(U(q_{\ell+1},X_{q_{\ell+1}};W)-U(q_{\ell},X_{q_{\ell}};W))\} exp { a ℓ ( U ( q ℓ + 1 , X q ℓ + 1 ; W ) − U ( q ℓ , X q ℓ ; W ))} s s s
Observables. We denote the conditional mean, the conditional covariance, and the spatial Hessian by
m q = E [ σ ∣ G q ] , C q = Cov ( σ ∣ G q ) , H q = ∇ x 2 U ( q , X q ; W ) . m_q=\mathbb{E}[\sigma\mid\mathcal{G}_q],\qquad C_q=\operatorname{Cov}(\sigma\mid\mathcal{G}_q),\qquad H_q=\nabla_x^2U(q,X_q;W). m q = E [ σ ∣ G q ] , C q = Cov ( σ ∣ G q ) , H q = ∇ x 2 U ( q , X q ; W ) . We will see in Proposition 3.1 that m q = ∇ U ( q , X q ; W ) m_q=\nabla U(q,X_q;W) m q = ∇ U ( q , X q ; W ) a ( q ) C q ⪯ H q ⪯ C q a(q)C_q\preceq H_q\preceq C_q a ( q ) C q ⪯ H q ⪯ C q
S q = ∣ m q ∣ 2 N , L q = σ ⋅ m q − ∣ m q ∣ 2 / 2 N , j q = E [ m q T C q m q ] N 2 , a q t h = E Tr C q 2 N 2 . S_q=\frac{|m_q|^2}{N},\qquad L_q=\frac{\sigma\cdot m_q-|m_q|^2/2}{N},\qquad j_q=\frac{\mathbb{E}[m_q^{T}C_qm_q]}{N^2},\qquad a_q^{\mathrm{th}}=\frac{\mathbb{E}\operatorname{Tr}C_q^2}{N^2}. S q = N ∣ m q ∣ 2 , L q = N σ ⋅ m q − ∣ m q ∣ 2 /2 , j q = N 2 E [ m q T C q m q ] , a q th = N 2 E Tr C q 2 . we use the superscript in a q t h a_q^{\mathrm{th}} a q th a a a q q q σ 1 \sigma^1 σ 1 σ 2 \sigma^2 σ 2 G q \mathcal{G}_q G q σ \sigma σ G q \mathcal{G}_q G q R q = R ( σ 1 , σ 2 ) R_q=R(\sigma^1,\sigma^2) R q = R ( σ 1 , σ 2 ) S q S_q S q R q R_q R q G q \mathcal{G}_q G q L q L_q L q S q / 2 S_q/2 S q /2 G q \mathcal{G}_q G q L q L_q L q 0 ≤ S q ≤ 1 0\leq S_q\leq1 0 ≤ S q ≤ 1 ∣ L q ∣ ≤ 3 / 2 |L_q|\leq3/2 ∣ L q ∣ ≤ 3/2
The processes m m m C C C H H H m q = ∇ U ( q , X q ; W ) m_q=\nabla U(q,X_q;W) m q = ∇ U ( q , X q ; W ) H q = ∇ 2 U ( q , X q ; W ) H_q=\nabla^2U(q,X_q;W) H q = ∇ 2 U ( q , X q ; W ) E [ σ σ T ∣ G q ] = Σ ( q , X q ; W ) \mathbb{E}[\sigma\sigma^T\mid\mathcal{G}_q]=\Sigma(q,X_q;W) E [ σ σ T ∣ G q ] = Σ ( q , X q ; W ) Σ \Sigma Σ c a ∇ U ca\nabla U c a ∇ U ∇ U \nabla U ∇ U ∇ 2 U \nabla^2U ∇ 2 U Σ \Sigma Σ ( q , x ) (q,x) ( q , x ) [4 Section 2.6.1] ; the path X X X S q S_q S q L q L_q L q ν \nu ν q ↦ j q q\mapsto j_q q ↦ j q a q t h a_q^{\mathrm{th}} a q th E S q \mathbb{E}S_q E S q Var ( S q ) \operatorname{Var}(S_q) Var ( S q ) E ∣ S q − q ∣ 3 \mathbb{E}|S_q-q|^3 E ∣ S q − q ∣ 3 [ 0 , 1 ] [0,1] [ 0 , 1 ] q ↦ E ( R q − q ) 2 q\mapsto\mathbb{E}(R_q-q)^2 q ↦ E ( R q − q ) 2 E [ R q ∣ G q ] = S q \mathbb{E}[R_q\mid\mathcal{G}_q]=S_q E [ R q ∣ G q ] = S q E [ R q 2 ∣ G q ] = N − 2 ∑ i , j Σ i j ( q , X q ; W ) 2 \mathbb{E}[R_q^2\mid\mathcal{G}_q]=N^{-2}\sum_{i,j}\Sigma_{ij}(q,X_q;W)^2 E [ R q 2 ∣ G q ] = N − 2 ∑ i , j Σ ij ( q , X q ; W ) 2
We also define the integrated quantities
A = ∫ a q t h ν ( d q ) , J = ∫ j q ν ( d q ) , A=\int a_q^{\mathrm{th}}\,\nu(dq),\qquad J=\int j_q\,\nu(dq), A = ∫ a q th ν ( d q ) , J = ∫ j q ν ( d q ) , V = ∫ Var ( S q ) ν ( d q ) , H 3 = ∫ E ∣ S q − q ∣ 3 ν ( d q ) . V=\int\operatorname{Var}(S_q)\,\nu(dq),\qquad H_3=\int\mathbb{E}|S_q-q|^3\,\nu(dq). V = ∫ Var ( S q ) ν ( d q ) , H 3 = ∫ E ∣ S q − q ∣ 3 ν ( d q ) . We note that they depend on N N N ρ \rho ρ λ \lambda λ A A A J J J V V V H 3 H_3 H 3 § 10.2 .
Quantiles and contact levels. The quantile function
Q ( v ) = inf { q ∈ [ 0 , 1 ] : ν ( [ 0 , q ] ) ≥ v } , 0 < v < 1 − s , (19) Q(v)=\inf\{q\in[0,1]:\nu([0,q])\ge v\},\qquad0<v<1-s, \tag*{(19)} Q ( v ) = inf { q ∈ [ 0 , 1 ] : ν ([ 0 , q ]) ≥ v } , 0 < v < 1 − s , (19) pushes Lebesgue measure on ( 0 , 1 − s ) (0,1-s) ( 0 , 1 − s ) ν \nu ν a ( Q ( v ) ) = s + ν ( [ 0 , Q ( v ) ] ) ≥ s + v a(Q(v))=s+\nu([0,Q(v)])\ge s+v a ( Q ( v )) = s + ν ([ 0 , Q ( v )]) ≥ s + v q ↦ ν ( [ 0 , q ] ) q\mapsto\nu([0,q]) q ↦ ν ([ 0 , q ]) ν \nu ν a a a ν \nu ν Q Q Q ( 0 , 1 − s ) (0,1-s) ( 0 , 1 − s ) ν \nu ν ν \nu ν [ 0 , 1 ] [0,1] [ 0 , 1 ] v v v ν \nu ν
If μ \mu μ F t , s \mathcal{F}_{t,s} F t , s supp ν \operatorname{supp}\nu supp ν contact level of μ \mu μ E S q \mathrm{ES}_q ES q q q q
Martingale and contact identities
In the next proposition, we collect the finite-dimensional martingale, first-variation, and contact identities of [4 Proposition 2.1] . For the scalar Parisi functional, the stochastic-control representation and the martingale identities are in [10 Theorem 3 and Lemma 2] , with a dynamic programming treatment in [43]. The first variation and the optimality conditions on the support are in [44, Lemma 3.2 and Proposition 1.1] and [25, Theorem 2 and Proposition 1].
Proposition 3.1 (Martingale, covariance order, and contact). Let 0 < ρ ≤ 1 0<\rho\le1 0 < ρ ≤ 1 μ = s δ 0 + ν \mu=s\delta_0+\nu μ = s δ 0 + ν q ∈ [ 0 , 1 ] q\in[0,1] q ∈ [ 0 , 1 ]
m q = ∇ U ( q , X q ; W ) , m 0 = 0 , d m q = c H q d B q , a ( q ) C q ⪯ H q ⪯ C q , (20) m_q=\nabla U(q,X_q;W),\qquad m_0=0,\qquad\mathrm{d}m_q=\sqrt{c}\,H_q\,\mathrm{d}B_q,\qquad a(q)C_q\preceq H_q\preceq C_q, \tag*{(20)} m q = ∇ U ( q , X q ; W ) , m 0 = 0 , d m q = c H q d B q , a ( q ) C q ⪯ H q ⪯ C q , (20) and q ↦ E S q q\mapsto\mathrm{ES}_q q ↦ ES q [ 0 , 1 ] [0,1] [ 0 , 1 ] c E Tr H q 2 / N c\,\mathbb{E}\operatorname{Tr}H_q^2/N c E Tr H q 2 / N ν ′ ≥ 0 \nu'\ge0 ν ′ ≥ 0 1 − s 1-s 1 − s θ ↦ F t , s ( s δ 0 + ( 1 − θ ) ν + θ ν ′ ) \theta\mapsto\mathcal{F}_{t,s}(s\delta_0+(1-\theta)\nu+\theta\nu') θ ↦ F t , s ( s δ 0 + ( 1 − θ ) ν + θ ν ′ ) θ = 0 \theta=0 θ = 0
D F t , s ( μ ) [ ν ′ − ν ] = c 2 ∫ 0 1 G ( x ) ( ν ′ − ν ) ( d x ) , G ( x ) = ∫ x 1 ( E S y − y ) d y . (21) D\mathcal{F}_{t,s}(\mu)[\nu'-\nu]=\frac{c}{2}\int_0^1G(x)(\nu'-\nu)(\mathrm{d}x),\qquad G(x)=\int_x^1(\mathrm{ES}_y-y)\,\mathrm{d}y. \tag*{(21)} D F t , s ( μ ) [ ν ′ − ν ] = 2 c ∫ 0 1 G ( x ) ( ν ′ − ν ) ( d x ) , G ( x ) = ∫ x 1 ( ES y − y ) d y . (21) If μ \mu μ F t , s \mathcal{F}_{t,s} F t , s supp ν \operatorname{supp}\nu supp ν G G G [ 0 , 1 ] [0,1] [ 0 , 1 ]
supp ν ⊂ [ 0 , 1 ) , E S q = q , E Tr H q 2 ≤ N c for every contact level q . (22) \operatorname{supp}\nu\subset[0,1),\qquad\mathrm{ES}_q=q,\qquad\mathbb{E}\operatorname{Tr}H_q^2\le\frac{N}{c}\qquad\text{for every contact level }q. \tag*{(22)} supp ν ⊂ [ 0 , 1 ) , ES q = q , E Tr H q 2 ≤ c N for every contact level q . (22) Remark 3.2. A contact level is a point of the support of the free measure ν \nu ν q = 0 q=0 q = 0 E S 0 = 0 \mathrm{ES}_0=0 ES 0 = 0 m 0 = 0 m_0=0 m 0 = 0 E Tr H 0 2 \mathbb{E}\operatorname{Tr}H_0^2 E Tr H 0 2 (22) holds when 0 ∈ supp ν 0\in\operatorname{supp}\nu 0 ∈ supp ν s δ 0 s\delta_0 s δ 0
Proof of Proposition 3.1. We recall the argument of [4 Proposition 2.1 and Section 2.6] . The proof has four steps. We first prove the martingale identities and the covariance order for step profiles. We then derive the first variation and deduce the support property and (3.9) at a minimizer. Finally, we extend these statements to general profiles.
Step 1: martingale identities for step profiles. Let μ \mu μ U U U x x x ( q , x , W ) (q,x,W) ( q , x , W ) N N N U U U q q q [4 Section 2.6.1] . Differentiating (3.2) in x x x [4 (2.5)]
d ∇ U ( q , X q ; W ) = c H q d B q , d H q = − c a ( q ) H q 2 d q + c ∇ H q d B q . \mathrm{d}\,\nabla U(q,X_q;W)=\sqrt{c}\,H_q\,\mathrm{d}B_q,\qquad\mathrm{d}H_q=-ca(q)H_q^2\,\mathrm{d}q+\sqrt{c}\,\nabla H_q\,\mathrm{d}B_q. d ∇ U ( q , X q ; W ) = c H q d B q , d H q = − c a ( q ) H q 2 d q + c ∇ H q d B q . where ∇ H q d B q \nabla H_q\,\mathrm{d}B_q ∇ H q d B q U U U ( q , X q ) (q,X_q) ( q , X q ) d B q \mathrm{d}B_q d B q q = 1 q=1 q = 1 ∇ U ( 1 , X 1 ; W ) \nabla U(1,X_1;W) ∇ U ( 1 , X 1 ; W ) ( W , X 1 ) (W,X_1) ( W , X 1 ) E [ σ ∣ G 1 ] \mathbb{E}[\sigma\mid\mathcal{G}_1] E [ σ ∣ G 1 ] ∇ U ( q , X q ; W ) = E [ σ ∣ G q ] = m q \nabla U(q,X_q;W)=\mathbb{E}[\sigma\mid\mathcal{G}_q]=m_q ∇ U ( q , X q ; W ) = E [ σ ∣ G q ] = m q q q q σ \sigma σ − σ -\sigma − σ U ( 1 , x ; W ) U(1,x;W) U ( 1 , x ; W ) U ( 1 , ⋅ ; W ) U(1,\cdot;W) U ( 1 , ⋅ ; W ) U ( q , ⋅ ; W ) U(q,\cdot;W) U ( q , ⋅ ; W ) m 0 = ∇ U ( 0 , 0 ; W ) = 0 m_0=\nabla U(0,0;W)=0 m 0 = ∇ U ( 0 , 0 ; W ) = 0
For the covariance order, we first note that C 1 = H 1 C_1=H_1 C 1 = H 1 m m m H H H
C q = E [ H 1 ∣ G q ] + c ∫ q 1 E [ H u 2 ∣ G q ] d u , H q = E [ H 1 ∣ G q ] + c ∫ q 1 a ( u ) E [ H u 2 ∣ G q ] d u . C_q=\mathbb{E}[H_1\mid\mathcal{G}_q]+c\int_q^1\mathbb{E}[H_u^2\mid\mathcal{G}_q]\,\mathrm{d}u,\qquad H_q=\mathbb{E}[H_1\mid\mathcal{G}_q]+c\int_q^1a(u)\mathbb{E}[H_u^2\mid\mathcal{G}_q]\,\mathrm{d}u. C q = E [ H 1 ∣ G q ] + c ∫ q 1 E [ H u 2 ∣ G q ] d u , H q = E [ H 1 ∣ G q ] + c ∫ q 1 a ( u ) E [ H u 2 ∣ G q ] d u . Observe that every matrix in these expressions is positive semidefinite, and that a ( q ) ≤ a ( u ) ≤ 1 a(q)\leq a(u)\leq1 a ( q ) ≤ a ( u ) ≤ 1 u ≥ q u\geq q u ≥ q
C q − H q = c ∫ q 1 ( 1 − a ( u ) ) E [ H u 2 ∣ G q ] d u ⪰ 0 C_q-H_q=c\int_q^1(1-a(u))\mathbb{E}[H_u^2\mid\mathcal{G}_q]\,\mathrm{d}u\succeq0 C q − H q = c ∫ q 1 ( 1 − a ( u )) E [ H u 2 ∣ G q ] d u ⪰ 0 and
H q − a ( q ) C q = ( 1 − a ( q ) ) E [ H 1 ∣ G q ] + c ∫ q 1 ( a ( u ) − a ( q ) ) E [ H u 2 ∣ G q ] d u ⪰ 0 , H_q-a(q)C_q=(1-a(q))\mathbb{E}[H_1\mid\mathcal{G}_q]+c\int_q^1(a(u)-a(q))\mathbb{E}[H_u^2\mid\mathcal{G}_q]\,\mathrm{d}u\succeq0, H q − a ( q ) C q = ( 1 − a ( q )) E [ H 1 ∣ G q ] + c ∫ q 1 ( a ( u ) − a ( q )) E [ H u 2 ∣ G q ] d u ⪰ 0 , which proves a ( q ) C q ⪯ H q ⪯ C q a(q)C_q\preceq H_q\preceq C_q a ( q ) C q ⪯ H q ⪯ C q [4 (2.6)] . Finally, Itô’s formula gives d ∣ m q ∣ 2 = 2 m q ⋅ d m q + c Tr H q 2 d q \mathrm{d}|m_q|^2=2m_q\cdot\mathrm{d}m_q+c\operatorname{Tr}H_q^2\,\mathrm{d}q d ∣ m q ∣ 2 = 2 m q ⋅ d m q + c Tr H q 2 d q q ↦ E S q q\mapsto\mathbb{E}S_q q ↦ E S q c E Tr H q 2 / N c\,\mathbb{E}\operatorname{Tr}H_q^2/N c E Tr H q 2 / N [4 (2.7)] , which is continuous in q q q
Step 2: the first variation. Let ν ′ ≥ 0 \nu' \ge0 ν ′ ≥ 0 1 − s 1-s 1 − s δ a ( q ) = ( ν ′ − ν ) ( [ 0 , q ] ) \delta a(q) = (\nu' - \nu)([0,q]) δ a ( q ) = ( ν ′ − ν ) ([ 0 , q ]) s δ 0 + ( 1 − θ ) ν + θ ν ′ s\delta_0 + (1-\theta)\nu+ \theta\nu' s δ 0 + ( 1 − θ ) ν + θ ν ′ a + θ δ a a + \theta\delta a a + θ δ a U ˙ \dot U U ˙ θ \theta θ θ = 0 \theta= 0 θ = 0
∂ q U ˙ + c 2 Δ U ˙ + c a ( q ) ∇ U ⋅ ∇ U ˙ = − c 2 δ a ( q ) ∣ ∇ U ∣ 2 , U ˙ ( 1 , ⋅ ; W ) = 0. \partial_q\dot U + \frac{c}{2}\Delta\dot U + ca(q)\nabla U \cdot\nabla\dot U = -\frac{c}{2}\delta a(q)|\nabla U|^2,\qquad\dot U(1,\cdot;W)=0. ∂ q U ˙ + 2 c Δ U ˙ + c a ( q ) ∇ U ⋅ ∇ U ˙ = − 2 c δ a ( q ) ∣∇ U ∣ 2 , U ˙ ( 1 , ⋅ ; W ) = 0. Its generator is the generator of (18) . By the Feynman–Kac formula, we have
U ˙ ( 0 , 0 ; W ) = c 2 E [ ∫ 0 1 δ a ( q ) ∣ m q ∣ 2 d q ∣ W ] . \dot U(0,0;W)=\frac{c}{2}\mathbb{E}\left[\int_0^1\delta a(q)|m_q|^2\,dq\mathrel{\Big|}W\right]. U ˙ ( 0 , 0 ; W ) = 2 c E [ ∫ 0 1 δ a ( q ) ∣ m q ∣ 2 d q W ] . When we differentiate the outer logarithmic moment in (16) , this identity is averaged under the tilted law of W W W − ( c / 2 ) ∫ 0 1 q δ a ( q ) d q -(c/2)\int_0^1q\delta a(q)\,dq − ( c /2 ) ∫ 0 1 q δ a ( q ) d q
D F t , s ( μ ) [ ν ′ − ν ] = c 2 ∫ 0 1 ( E S q − q ) δ a ( q ) d q . D\mathcal{F}_{t,s}(\mu)[\nu'-\nu]=\frac{c}{2}\int_0^1(\mathbb{E}S_q-q)\delta a(q)\,dq. D F t , s ( μ ) [ ν ′ − ν ] = 2 c ∫ 0 1 ( E S q − q ) δ a ( q ) d q . We write δ a ( q ) = ∫ 1 { x ≤ q } ( ν ′ − ν ) ( d x ) \delta a(q)=\int1\{x\le q\}(\nu'-\nu)(dx) δ a ( q ) = ∫ 1 { x ≤ q } ( ν ′ − ν ) ( d x ) ν ′ − ν \nu'-\nu ν ′ − ν (21) , which is [4 (2.8)] . We note that its derivation does not use minimality. In the energy normalization, the identity is stated for every admissible profile in [4 Section 7.4, (7.22)] .
Step 3: support and contact at a minimizer. Let μ \mu μ F t , s \mathcal{F}_{t,s} F t , s ν ′ ≥ 0 \nu' \ge0 ν ′ ≥ 0 1 − s 1-s 1 − s s δ 0 + ( 1 − θ ) ν + θ ν ′ s\delta_0+(1-\theta)\nu+\theta\nu' s δ 0 + ( 1 − θ ) ν + θ ν ′ 0 ≤ θ ≤ 1 0\le\theta\le1 0 ≤ θ ≤ 1 θ = 0 \theta=0 θ = 0 F t , s \mathcal{F}_{t,s} F t , s (21) is nonnegative. Taking ν ′ = ( 1 − s ) δ x \nu'=(1-s)\delta_x ν ′ = ( 1 − s ) δ x
( 1 − s ) G ( x ) ≥ ∫ G d ν for every x ∈ [ 0 , 1 ] . (1-s)G(x)\ge\int G\,d\nu\qquad\text{for every }x\in[0,1]. ( 1 − s ) G ( x ) ≥ ∫ G d ν for every x ∈ [ 0 , 1 ] . The function G G G [ 0 , 1 ] [0,1] [ 0 , 1 ] x x x G G G ∫ ( G − min [ 0 , 1 ] G ) d ν ≤ 0 \int(G-\min_{[0,1]}G)\,d\nu\le0 ∫ ( G − min [ 0 , 1 ] G ) d ν ≤ 0 G = min [ 0 , 1 ] G G=\min_{[0,1]}G G = min [ 0 , 1 ] G ν \nu ν G > min [ 0 , 1 ] G G>\min_{[0,1]}G G > min [ 0 , 1 ] G [ 0 , 1 ] [0,1] [ 0 , 1 ] ν \nu ν supp ν \operatorname{supp}\nu supp ν G G G
By Step 1, G G G
G ′ ( q ) = q − E S q , G ′ ′ ( q ) = 1 − c N E Tr H q 2 . G'(q)=q-\mathbb{E}S_q,\qquad G''(q)=1-\frac{c}{N}\mathbb{E}\operatorname{Tr}H_q^2. G ′ ( q ) = q − E S q , G ′′ ( q ) = 1 − N c E Tr H q 2 . If q ∈ supp ν ∩ ( 0 , 1 ) q\in\operatorname{supp}\nu\cap(0,1) q ∈ supp ν ∩ ( 0 , 1 ) q q q G G G G ′ ( q ) = 0 G'(q)=0 G ′ ( q ) = 0 G ′ ′ ( q ) ≥ 0 G''(q)\ge0 G ′′ ( q ) ≥ 0 (22) . At q = 0 q=0 q = 0 G ′ ( 0 ) = − E S 0 = 0 G'(0)=-\mathbb{E}S_0=0 G ′ ( 0 ) = − E S 0 = 0 m 0 = 0 m_0=0 m 0 = 0 0 ∈ supp ν 0\in\operatorname{supp}\nu 0 ∈ supp ν 0 0 0 G G G G ′ ′ ( 0 ) ≥ 0 G''(0)\ge0 G ′′ ( 0 ) ≥ 0 G ′ ′ < 0 G''<0 G ′′ < 0 0 0 0 G ( h ) < G ( 0 ) G(h)<G(0) G ( h ) < G ( 0 ) h > 0 h>0 h > 0 q = 1 q=1 q = 1 ∣ m 1 , i ∣ < 1 |m_{1,i}|<1 ∣ m 1 , i ∣ < 1 i i i G ′ ( 1 ) = 1 − E S 1 > 0 G'(1)=1-\mathbb{E}S_1>0 G ′ ( 1 ) = 1 − E S 1 > 0 G ( 1 − h ) < G ( 1 ) G(1-h)<G(1) G ( 1 − h ) < G ( 1 ) h > 0 h>0 h > 0 1 1 1 G G G supp ν ⊂ [ 0 , 1 ) \operatorname{supp}\nu\subset[0,1) supp ν ⊂ [ 0 , 1 ) (22) , which is [4 (2.9)] .
Step 4: general profiles. Let μ \mu μ [4 Section 2.6] , we approximate ν \nu ν ν j \nu_j ν j 0 0 0 1 1 1 μ j = s δ 0 + ν j \mu_j=s\delta_0+\nu_j μ j = s δ 0 + ν j μ j ⇒ μ \mu_j\Rightarrow\mu μ j ⇒ μ L 1 ( [ 0 , 1 ] ) L^1([0,1]) L 1 ([ 0 , 1 ]) a a a N N N L 1 L^1 L 1 § 4.1.2 to the solutions, and the same comparison argument to their differentiated equations. Using also the uniform bound on the derivatives and Grönwall’s inequality, we find that the solutions and their first and second spatial derivatives converge uniformly. We couple the laws for μ j \mu_j μ j μ \mu μ W W W B B B X X X m m m H H H μ j \mu_j μ j μ \mu μ q q q c a ∇ U ca\nabla U c a ∇ U C q C_q C q μ j \mu_j μ j μ \mu μ e − 2 s d j e^{-2sd_j} e − 2 s d j e 2 s d j e^{2sd_j} e 2 s d j d j = ( c N / 2 ) ∥ a j − a ∥ L 1 → 0 d_j=(cN/2)\lVert a_j-a\rVert_{L^1}\to0 d j = ( c N /2 ) ∥ a j − a ∥ L 1 → 0 L 1 L^1 L 1 N N N L p L^p L p (20) , the formula for the derivative of ES q \operatorname{ES}_q ES q q q q (21) to the limit [4 Section 2.6.2] . If μ \mu μ μ \mu μ μ \mu μ
The integrated variance bound and the overlap error
At a minimizer, we use the contact identities to turn the posterior variance bound of [4 Proposition 2.2] into bounds that depend only on the cumulative function a a a
Proposition 3.3 (Integrated variance and trace bounds). Let 0 < ρ ≤ 1 0<\rho\le1 0 < ρ ≤ 1 μ = s δ 0 + ν \mu=s\delta_0+\nu μ = s δ 0 + ν F t , s \mathcal{F}_{t,s} F t , s k ∈ L 2 ( ν ) k\in L^2(\nu) k ∈ L 2 ( ν )
Var ( ∫ k ( q ) L q ν ( d q ) ) ≤ 1 M ∫ k ( q ) 2 a ( q ) ν ( d q ) , A ≤ 1 M s ˙ . (23) \operatorname{Var}\left(\int k(q)L_q\,\nu(\mathrm{d}q)\right)\le\frac{1}{M}\int\frac{k(q)^2}{a(q)}\,\nu(\mathrm{d}q), \qquad A\le\frac{1}{M\dot{s}}. \tag*{(23)} Var ( ∫ k ( q ) L q ν ( d q ) ) ≤ M 1 ∫ a ( q ) k ( q ) 2 ν ( d q ) , A ≤ M s ˙ 1 . (23) The first bound is [4 Proposition 2.2, (2.16)] and the second is [4 Section 2.5, (2.18)] . We briefly recall the argument. We fix the terminal spin and apply the Brascamp–Lieb inequality to the law of the independent Gaussian coordinates that generate W W W W W W P \mathbb{P} P
Proof of Proposition 3.3. We first show that, for every admissible profile and every deterministic k ∈ L 2 ( ν ) k \in L^2(\nu) k ∈ L 2 ( ν )
Var ( ∫ k ( q ) L q ν ( d q ) ) ≤ 1 N 2 ∫ k ( q ) 2 E Tr ( H q C q ) ν ( d q ) . (24) \operatorname{Var}\left(\int k(q)L_q\,\nu(dq)\right) \le\frac{1}{N^2}\int k(q)^2 \mathbb{E}\operatorname{Tr}(H_qC_q)\,\nu(dq). \tag*{(24)} Var ( ∫ k ( q ) L q ν ( d q ) ) ≤ N 2 1 ∫ k ( q ) 2 E Tr ( H q C q ) ν ( d q ) . (24) which is [4 (2.15)] . Let μ \mu μ ε > 0 \varepsilon> 0 ε > 0 X 0 = ε z 0 X_0 = \sqrt{\varepsilon}z_0 X 0 = ε z 0 z 0 z_0 z 0 W W W z 0 z_0 z 0 Z Z Z Z Z Z e − Φ σ ( Z ) e^{-\Phi_\sigma(Z)} e − Φ σ ( Z ) Φ σ \Phi_\sigma Φ σ Q = I + ∫ A q ν ( d q ) ⪰ I Q = I + \int A_q\,\nu(dq) \succeq I Q = I + ∫ A q ν ( d q ) ⪰ I A q = ∇ Z 2 U ( q , X q ; W ) ⪰ 0 A_q = \nabla_Z^2 U(q,X_q;W) \succeq0 A q = ∇ Z 2 U ( q , X q ; W ) ⪰ 0 [4 (2.1)–(2.3)] . We derive this density for more general laws in the proof of Lemma 4.2. We work in these Gaussian reference coordinates, in which, at the grid points q ℓ q_\ell q ℓ
X q ℓ = ε z 0 + ∑ h = 1 ℓ c ( q h − q h − 1 ) z h ; X_{q_\ell} = \sqrt{\varepsilon}z_0 + \sum_{h=1}^{\ell}\sqrt{c(q_h-q_{h-1})}\,z_h; X q ℓ = ε z 0 + h = 1 ∑ ℓ c ( q h − q h − 1 ) z h ; the tilted law is obtained by reweighting these coordinates. In this representation, the field depends on z 0 z_0 z 0 ε z 0 \sqrt{\varepsilon}z_0 ε z 0 ∇ z 0 U ( q , X q ; W ) = ε m q \nabla_{z_0}U(q,X_q;W)=\sqrt{\varepsilon}m_q ∇ z 0 U ( q , X q ; W ) = ε m q A q A_q A q z 0 z_0 z 0 ε H q \varepsilon H_q ε H q v q v_q v q z 0 z_0 z 0 ( σ − m q ) / ε (\sigma-m_q)/\sqrt{\varepsilon} ( σ − m q ) / ε L q L_q L q
∇ Z L q = 1 N A q v q , v q T A q v q = ( σ − m q ) T H q ( σ − m q ) . \nabla_Z L_q = \frac{1}{N}A_qv_q,\qquad v_q^{T}A_qv_q=(\sigma-m_q)^{T}H_q(\sigma-m_q). ∇ Z L q = N 1 A q v q , v q T A q v q = ( σ − m q ) T H q ( σ − m q ) . For Y k = ∫ k L q d ν Y_k=\int kL_q\,d\nu Y k = ∫ k L q d ν b = ∫ k ( q ) A q v q ν ( d q ) b=\int k(q)A_qv_q\,\nu(dq) b = ∫ k ( q ) A q v q ν ( d q ) ∇ Z Y k = b / N \nabla_ZY_k=b/N ∇ Z Y k = b / N w w w A q A_q A q ν \nu ν ∫ A q d ν = Q − I ⪯ Q \int A_q\,d\nu=Q-I\preceq Q ∫ A q d ν = Q − I ⪯ Q
∣ w T b ∣ 2 ≤ ( ∫ w T A q w ν ( d q ) ) ∫ k ( q ) 2 v q T A q v q ν ( d q ) ≤ ( w T Q w ) ∫ k ( q ) 2 v q T A q v q ν ( d q ) . \lvert w^{T}b\rvert^2 \le\left(\int w^{T}A_qw\,\nu(dq)\right)\int k(q)^2v_q^{T}A_qv_q\,\nu(dq) \le(w^{T}Qw)\int k(q)^2v_q^{T}A_qv_q\,\nu(dq). ∣ w T b ∣ 2 ≤ ( ∫ w T A q w ν ( d q ) ) ∫ k ( q ) 2 v q T A q v q ν ( d q ) ≤ ( w T Qw ) ∫ k ( q ) 2 v q T A q v q ν ( d q ) . The choice w = Q − 1 b w=Q^{-1}b w = Q − 1 b b T Q − 1 b ≤ ∫ k 2 v q T A q v q d ν b^{T}Q^{-1}b\le\int k^2v_q^{T}A_qv_q\,d\nu b T Q − 1 b ≤ ∫ k 2 v q T A q v q d ν [21 ] , in the form [4 Proposition 2.2, (2.14)] , implies that the conditional variance of Y k Y_k Y k σ \sigma σ N − 2 E [ b T Q − 1 b ∣ σ ] N^{-2}\mathbb{E}[b^{T}Q^{-1}b\mid\sigma] N − 2 E [ b T Q − 1 b ∣ σ ] [4 Section 2.1] , the conditional law of Y k Y_k Y k σ \sigma σ σ \sigma σ Var ( Y k ) \operatorname{Var}(Y_k) Var ( Y k ) G q \mathcal{G}_q G q E [ ( σ − m q ) T H q ( σ − m q ) ∣ G q ] = Tr ( H q C q ) \mathbb{E}[(\sigma-m_q)^{T}H_q(\sigma-m_q)\mid\mathcal{G}_q]=\operatorname{Tr}(H_qC_q) E [( σ − m q ) T H q ( σ − m q ) ∣ G q ] = Tr ( H q C q ) ε ↓ 0 \varepsilon\downarrow0 ε ↓ 0 ∣ L q ∣ ≤ 3 / 2 \lvert L_q\rvert\le3/2 ∣ L q ∣ ≤ 3/2 Tr ( H q C q ) ≤ N 2 \operatorname{Tr}(H_qC_q)\le N^2 Tr ( H q C q ) ≤ N 2 k k k k ∈ L 2 ( ν ) k \in L^2(\nu) k ∈ L 2 ( ν ) [4 Section 2.6.2] .
Now suppose that μ \mu μ H q − a ( q ) C q ⪰ 0 H_q-a(q)C_q \succeq0 H q − a ( q ) C q ⪰ 0
Tr H q 2 = Tr ( H q ( H q − a ( q ) C q ) ) + a ( q ) Tr ( H q C q ) ≥ a ( q ) Tr ( H q C q ) , \operatorname{Tr} H_q^2=\operatorname{Tr}\left(H_q\left(H_q-a(q)C_q\right)\right)+a(q)\operatorname{Tr}(H_qC_q)\geq a(q)\operatorname{Tr}(H_qC_q), Tr H q 2 = Tr ( H q ( H q − a ( q ) C q ) ) + a ( q ) Tr ( H q C q ) ≥ a ( q ) Tr ( H q C q ) , and similarly
Tr ( H q C q ) = Tr ( ( H q − a ( q ) C q ) C q ) + a ( q ) Tr C q 2 ≥ a ( q ) Tr C q 2 . \operatorname{Tr}(H_qC_q)=\operatorname{Tr}\left(\left(H_q-a(q)C_q\right)C_q\right)+a(q)\operatorname{Tr} C_q^2\geq a(q)\operatorname{Tr} C_q^2. Tr ( H q C q ) = Tr ( ( H q − a ( q ) C q ) C q ) + a ( q ) Tr C q 2 ≥ a ( q ) Tr C q 2 . By (22) , E Tr H q 2 ≤ N / c \mathbb{E}\operatorname{Tr} H_q^2\leq N/c E Tr H q 2 ≤ N / c ν \nu ν a ( q ) ≥ s > 0 a(q)\geq s>0 a ( q ) ≥ s > 0 E Tr ( H q C q ) ≤ N / ( c a ( q ) ) \mathbb{E}\operatorname{Tr}(H_qC_q)\leq N/(ca(q)) E Tr ( H q C q ) ≤ N / ( c a ( q )) E Tr C q 2 ≤ N / ( c a ( q ) 2 ) \mathbb{E}\operatorname{Tr} C_q^2\leq N/(ca(q)^2) E Tr C q 2 ≤ N / ( c a ( q ) 2 ) ν \nu ν q q q (24) and using M = N c M=Nc M = N c (23) . For the second, the quantile function (19) and a ( Q ( v ) ) ≥ s + v a(Q(v))\geq s+v a ( Q ( v )) ≥ s + v
A ≤ 1 M ∫ ν ( d q ) a ( q ) 2 = 1 M ∫ 0 1 − s d v a ( Q ( v ) ) 2 ≤ 1 M ∫ 0 1 − s d v ( s + v ) 2 = 1 M ( 1 s − 1 ) ≤ 1 M s . A\leq\frac{1}{M}\int\frac{\nu(\mathrm{d}q)}{a(q)^2}=\frac{1}{M}\int_0^{1-s}\frac{\mathrm{d}v}{a(Q(v))^2}\leq\frac{1}{M}\int_0^{1-s}\frac{\mathrm{d}v}{(s+v)^2}=\frac{1}{M}\left(\frac{1}{s}-1\right)\leq\frac{1}{Ms}. A ≤ M 1 ∫ a ( q ) 2 ν ( d q ) = M 1 ∫ 0 1 − s a ( Q ( v ) ) 2 d v ≤ M 1 ∫ 0 1 − s ( s + v ) 2 d v = M 1 ( s 1 − 1 ) ≤ M s 1 . Remark 3.4. Since k k k (23) is arbitrary, the bound applies to sub-measures of ν \nu ν h h h 0 ≤ h ≤ 1 0\leq h\leq1 0 ≤ h ≤ 1 (23) with k h kh k h k k k h 2 ≤ h h^2\leq h h 2 ≤ h
Var ( ∫ k ( q ) h ( q ) L q ν ( d q ) ) ≤ 1 M ∫ k ( q ) 2 h ( q ) a ( q ) ν ( d q ) . \operatorname{Var}\left(\int k(q)h(q)L_q\,\nu(\mathrm{d}q)\right)\leq\frac{1}{M}\int\frac{k(q)^2h(q)}{a(q)}\,\nu(\mathrm{d}q). Var ( ∫ k ( q ) h ( q ) L q ν ( d q ) ) ≤ M 1 ∫ a ( q ) k ( q ) 2 h ( q ) ν ( d q ) . We now show that the overlap error splits into the three terms of the defect.
Lemma 3.5 (Decomposition of the overlap error). Let 0 < ρ ≤ 1 0<\rho\leq1 0 < ρ ≤ 1 q ∈ [ 0 , 1 ] q\in[0,1] q ∈ [ 0 , 1 ] Var ( L q ) = j q + Var ( S q ) / 4 \operatorname{Var}(L_q)=j_q+\operatorname{Var}(S_q)/4 Var ( L q ) = j q + Var ( S q ) /4 q q q
E ( R q − q ) 2 = a q t h + 2 j q + Var ( S q ) . (25) \mathbb{E}(R_q-q)^2=a_q^{\mathrm{th}}+2j_q+\operatorname{Var}(S_q). \tag*{(25)} E ( R q − q ) 2 = a q th + 2 j q + Var ( S q ) . (25) Proof. Both identities are displayed in [4 Section 2.5] before (2.17). Let ξ 1 = σ 1 − m q \xi^1=\sigma^1-m_q ξ 1 = σ 1 − m q ξ 2 = σ 2 − m q \xi^2=\sigma^2-m_q ξ 2 = σ 2 − m q G q \mathcal{G}_q G q C q C_q C q
R q = S q + 1 N ( m q ⋅ ξ 1 + m q ⋅ ξ 2 + ξ 1 ⋅ ξ 2 ) . R_q=S_q+\frac{1}{N}\left(m_q\cdot\xi^1+m_q\cdot\xi^2+\xi^1\cdot\xi^2\right). R q = S q + N 1 ( m q ⋅ ξ 1 + m q ⋅ ξ 2 + ξ 1 ⋅ ξ 2 ) . The three terms in parentheses have conditional mean zero, they are pairwise orthogonal given G q \mathcal{G}_q G q m q T C q m q m_q^{T}C_qm_q m q T C q m q m q T C q m q m_q^{T}C_qm_q m q T C q m q Tr C q 2 \operatorname{Tr} C_q^2 Tr C q 2 q q q
E ( R q − S q ) 2 = a q t h + 2 j q . (26) \mathbb{E}(R_q-S_q)^2=a_q^{\mathrm{th}}+2j_q. \tag*{(26)} E ( R q − S q ) 2 = a q th + 2 j q . (26) Since E [ R q − S q ∣ G q ] = 0 \mathbb{E}[R_q-S_q\mid\mathcal{G}_q]=0 E [ R q − S q ∣ G q ] = 0
E ( R q − q ) 2 = a q t h + 2 j q + E ( S q − q ) 2 = a q t h + 2 j q + Var ( S q ) + ( E S q − q ) 2 . \mathbb{E}(R_q-q)^2=a_q^{\mathrm{th}}+2j_q+\mathbb{E}(S_q-q)^2=a_q^{\mathrm{th}}+2j_q+\operatorname{Var}(S_q)+(\mathbb{E}S_q-q)^2. E ( R q − q ) 2 = a q th + 2 j q + E ( S q − q ) 2 = a q th + 2 j q + Var ( S q ) + ( E S q − q ) 2 . At a contact level of a minimizer, E S q = q \mathbb{E}S_q=q E S q = q (22) , which proves (25) . Similarly, L q = S q / 2 + N − 1 ( σ − m q ) ⋅ m q L_q=S_q/2+N^{-1}(\sigma-m_q)\cdot m_q L q = S q /2 + N − 1 ( σ − m q ) ⋅ m q N − 2 m q T C q m q N^{-2}m_q^{\mathsf T}C_qm_q N − 2 m q T C q m q G q \mathcal{G}_q G q Var ( L q ) = j q + Var ( S q ) / 4 \operatorname{Var}(L_q)=j_q+\operatorname{Var}(S_q)/4 Var ( L q ) = j q + Var ( S q ) /4
The optimized interpolation and the defect
We next recall from [4 Proposition 2.3] that the derivative of the interpolating functional in t t t
Proposition 3.6 (Interpolation identity). Let N ≥ 1 N\geq1 N ≥ 1 0 < s ≤ 1 / 2 0<s\leq1/2 0 < s ≤ 1/2
(a) For every admissible profile μ \mu μ t ↦ F t , s ( μ ) t\mapsto\mathcal{F}_{t,s}(\mu) t ↦ F t , s ( μ ) [ 0 , 1 ] [0,1] [ 0 , 1 ] ( 0 , 1 ) (0,1) ( 0 , 1 )
∂ t F t , s ( μ ) = − β 2 4 ∫ E ( R q − q ) 2 ν ( d q ) , (27) \partial_t\mathcal{F}_{t,s}(\mu)=-\frac{\beta^2}{4}\int\mathbb{E}(R_q-q)^2\,\nu(dq), \tag*{(27)} ∂ t F t , s ( μ ) = − 4 β 2 ∫ E ( R q − q ) 2 ν ( d q ) , (27) where the expectation is under the tilted law of μ \mu μ t t t [ 0 , 4 ( 1 − s ) ] [0,4(1-s)] [ 0 , 4 ( 1 − s )]
(b) The function f N , s f_{N,s} f N , s [ 0 , 1 ] [0,1] [ 0 , 1 ] t ∈ ( 0 , 1 ) t\in(0,1) t ∈ ( 0 , 1 ) f N , s f_{N,s} f N , s μ \mu μ F t , s \mathcal{F}_{t,s} F t , s f N , s ′ ( t ) = ∂ t F t , s ( μ ) f_{N,s}'(t)=\partial_t\mathcal{F}_{t,s}(\mu) f N , s ′ ( t ) = ∂ t F t , s ( μ )
Proof. Part (a) is [4 Proposition 2.3 and (2.12)] . The regularity in t t t [4 Section 2.6.3] . We sketch the computation for a step profile, which shows where the square comes from. The parameter t t t F t , s \mathcal{F}_{t,s} F t , s β t \beta\sqrt{t} β t c = β 2 ( 1 − t ) c=\beta^2(1-t) c = β 2 ( 1 − t ) (15) , and the penalty. We write B ( q ) = E R q 2 B(q)=\mathbb{E}R_q^2 B ( q ) = E R q 2 Γ ( q ) = E S q \Gamma(q)=\mathbb{E}S_q Γ ( q ) = E S q E R q = Γ ( q ) \mathbb{E}R_q=\Gamma(q) E R q = Γ ( q )
Matrix interaction. We write β σ T W σ / 2 = ∑ e g e v e ( σ ) \beta\sigma^{\mathsf T}W\sigma/2=\sum_e g_ev_e(\sigma) β σ T W σ /2 = ∑ e g e v e ( σ ) e e e i < j i<j i < j g e = g i j g_e=g_{ij} g e = g ij v e ( σ ) = β σ i σ j / N v_e(\sigma)=\beta\sigma_i\sigma_j/\sqrt{N} v e ( σ ) = β σ i σ j / N e = ( i , i ) e=(i,i) e = ( i , i ) g e = z i g_e=z_i g e = z i v e = β / 2 N v_e=\beta/\sqrt{2N} v e = β / 2 N § 4.1.1 , we have ∑ e v e ( σ ) v e ( σ ′ ) = β 2 N R ( σ , σ ′ ) 2 / 2 \sum_e v_e(\sigma)v_e(\sigma')=\beta^2NR(\sigma,\sigma')^2/2 ∑ e v e ( σ ) v e ( σ ′ ) = β 2 N R ( σ , σ ′ ) 2 /2 y e = t g e y_e=\sqrt{t}\,g_e y e = t g e t t t c c c t t t
1 2 N E ∑ e ( ∂ y e 2 U ( 0 , 0 ; W ) + s ( ∂ y e U ( 0 , 0 ; W ) ) 2 ) . \frac{1}{2N}\mathbb{E}\sum_e\left(\partial_{y_e}^2U(0,0;W)+s\bigl(\partial_{y_e}U(0,0;W)\bigr)^2\right). 2 N 1 E e ∑ ( ∂ y e 2 U ( 0 , 0 ; W ) + s ( ∂ y e U ( 0 , 0 ; W ) ) 2 ) . For a function b ( σ ) b(\sigma) b ( σ ) y b ( σ ) yb(\sigma) y b ( σ ) y y y y = 0 y=0 y = 0 [4 (2.10)]
∂ y U ( 0 , 0 ) = E [ b ( σ ) ∣ G 0 ] , ∂ y 2 U ( 0 , 0 ) = E [ b ( σ ) 2 ∣ G 0 ] − ∫ E [ E [ b ( σ ) ∣ G q ] 2 ∣ G 0 ] μ ( d q ) . \partial_yU(0,0)=\mathbb{E}[b(\sigma)\mid\mathcal{G}_0],\qquad
\partial_y^2U(0,0)=\mathbb{E}[b(\sigma)^2\mid\mathcal{G}_0]-\int\mathbb{E}\left[\mathbb{E}[b(\sigma)\mid\mathcal{G}_q]^2\mid\mathcal{G}_0\right]\mu(dq). ∂ y U ( 0 , 0 ) = E [ b ( σ ) ∣ G 0 ] , ∂ y 2 U ( 0 , 0 ) = E [ b ( σ ) 2 ∣ G 0 ] − ∫ E [ E [ b ( σ ) ∣ G q ] 2 ∣ G 0 ] μ ( d q ) . Applying this with b = v e b=v_e b = v e e e e E [ v e ( σ ) ∣ G q ] 2 = E [ v e ( σ 1 ) v e ( σ 2 ) ∣ G q ] \mathbb{E}[v_e(\sigma)\mid\mathcal{G}_q]^2=\mathbb{E}[v_e(\sigma^1)v_e(\sigma^2)\mid\mathcal{G}_q] E [ v e ( σ ) ∣ G q ] 2 = E [ v e ( σ 1 ) v e ( σ 2 ) ∣ G q ] μ = s δ 0 + ν \mu=s\delta_0+\nu μ = s δ 0 + ν
β 2 4 ( 1 − ∫ B ( q ) μ ( d q ) + s B ( 0 ) ) = β 2 4 ( 1 − ∫ B ( q ) ν ( d q ) ) \frac{\beta^2}{4}\left(1-\int B(q)\,\mu(dq)+sB(0)\right)=\frac{\beta^2}{4}\left(1-\int B(q)\,\nu(dq)\right) 4 β 2 ( 1 − ∫ B ( q ) μ ( d q ) + s B ( 0 ) ) = 4 β 2 ( 1 − ∫ B ( q ) ν ( d q ) ) [4 (2.11)] . The term s ( ∂ y e U ) 2 s(\partial_{y_e}U)^2 s ( ∂ y e U ) 2 s s s
Field. As in Step 2 of the proof of Proposition 3.1, the derivative of U U U c c c − ( Δ U + a ∣ ∇ U ∣ 2 ) / 2 -(\Delta U+a|\nabla U|^2)/2 − ( Δ U + a ∣∇ U ∣ 2 ) /2
∂ c ( 1 N s log E r e f e s U ( 0 , 0 ; W ) ) = 1 2 ∫ 0 1 ( h ( q ) + a ( q ) Γ ( q ) ) d q , h ( q ) = 1 N E Tr H q . \partial_c\left(\frac{1}{Ns}\log\mathbb{E}_{\mathrm{ref}}e^{sU(0,0;W)}\right)=\frac{1}{2}\int_0^1\left(h(q)+a(q)\Gamma(q)\right)dq,\qquad h(q)=\frac{1}{N}\mathbb{E}\operatorname{Tr}H_q. ∂ c ( N s 1 log E ref e s U ( 0 , 0 ; W ) ) = 2 1 ∫ 0 1 ( h ( q ) + a ( q ) Γ ( q ) ) d q , h ( q ) = N 1 E Tr H q . By the equation for H q H_q H q Γ \Gamma Γ h ′ = − a Γ ′ h'=-a\Gamma' h ′ = − a Γ ′ h ( 1 ) = 1 − Γ ( 1 ) h(1)=1-\Gamma(1) h ( 1 ) = 1 − Γ ( 1 ) H 1 = C 1 H_1=C_1 H 1 = C 1 Tr C 1 = N − ∣ m 1 ∣ 2 \operatorname{Tr}C_1=N-|m_1|^2 Tr C 1 = N − ∣ m 1 ∣ 2 h h h Γ \Gamma Γ a a a a ( 1 ) = 1 a(1)=1 a ( 1 ) = 1
h ( q ) + a ( q ) Γ ( q ) = 1 − ∫ ( q , 1 ] Γ ( r ) μ ( d r ) ( 0 ≤ q ≤ 1 ) . h(q)+a(q)\Gamma(q)=1-\int_{(q,1]}\Gamma(r)\,\mu(dr)\qquad(0\le q\le1). h ( q ) + a ( q ) Γ ( q ) = 1 − ∫ ( q , 1 ] Γ ( r ) μ ( d r ) ( 0 ≤ q ≤ 1 ) . The interval is open at q q q a ( q ) a(q) a ( q ) μ \mu μ q q q q q q d c / d t = − β 2 dc/dt=-\beta^2 d c / d t = − β 2
− β 2 2 ( 1 − ∫ q Γ ( q ) ν ( d q ) ) ; -\frac{\beta^2}{2}\left(1-\int q\Gamma(q)\,\nu(dq)\right); − 2 β 2 ( 1 − ∫ q Γ ( q ) ν ( d q ) ) ; here we may replace μ \mu μ ν \nu ν q = 0 q=0 q = 0
Penalty. By Fubini’s theorem, we have 2 ∫ 0 1 q a ( q ) d q = 1 − ∫ q 2 μ ( d q ) 2\int_0^1qa(q)\,dq=1-\int q^2\,\mu(dq) 2 ∫ 0 1 q a ( q ) d q = 1 − ∫ q 2 μ ( d q )
β 2 4 ( 1 − ∫ q 2 ν ( d q ) ) . \frac{\beta^2}{4}\left(1-\int q^2\,\nu(dq)\right). 4 β 2 ( 1 − ∫ q 2 ν ( d q ) ) . Adding the three contributions, in which the constants cancel, we obtain
∂ t F t , s ( μ ) = − β 2 4 ∫ ( B ( q ) − 2 q Γ ( q ) + q 2 ) ν ( d q ) . (28) \partial_t\mathcal{F}_{t,s}(\mu)=-\frac{\beta^2}{4}\int\left(B(q)-2q\Gamma(q)+q^2\right)\nu(dq). \tag*{(28)} ∂ t F t , s ( μ ) = − 4 β 2 ∫ ( B ( q ) − 2 q Γ ( q ) + q 2 ) ν ( d q ) . (28) which is (3.14). The integral lies in [ 0 , 4 ( 1 − s ) ] [0,4(1-s)] [ 0 , 4 ( 1 − s )] ∣ R q − q ∣ ≤ 2 |R_q-q|\le2 ∣ R q − q ∣ ≤ 2 ν \nu ν 1 − s 1-s 1 − s
For (b), the bound in (a) implies that the functions t ↦ F t , s ( μ ) t\mapsto\mathcal{F}_{t,s}(\mu) t ↦ F t , s ( μ ) β 2 ( 1 − s ) \beta^2(1-s) β 2 ( 1 − s ) f N , s f_{N,s} f N , s t ∈ ( 0 , 1 ) t\in(0,1) t ∈ ( 0 , 1 ) f N , s f_{N,s} f N , s μ \mu μ t t t h h h t + h ∈ [ 0 , 1 ] t+h\in[0,1] t + h ∈ [ 0 , 1 ] f N , s ( t + h ) ≤ F t + h , s ( μ ) f_{N,s}(t+h)\le\mathcal{F}_{t+h,s}(\mu) f N , s ( t + h ) ≤ F t + h , s ( μ ) f N , s ( t ) = F t , s ( μ ) f_{N,s}(t)=\mathcal{F}_{t,s}(\mu) f N , s ( t ) = F t , s ( μ ) h > 0 h>0 h > 0 h ↓ 0 h\downarrow0 h ↓ 0 f N , s ′ ( t ) ≤ ∂ t F t , s ( μ ) f'_{N,s}(t)\le\partial_t\mathcal{F}_{t,s}(\mu) f N , s ′ ( t ) ≤ ∂ t F t , s ( μ ) h < 0 h<0 h < 0 [4 (2.20)] .
In particular, t ↦ F t , s ( μ ) t \mapsto\mathcal{F}_{t,s}(\mu) t ↦ F t , s ( μ ) f N , s f_{N,s} f N , s defect
D N ( ρ ) = − 4 β f N , s ′ ( 1 − ρ ) D_N(\rho)=-\frac{4}{\beta}f'_{N,s}(1-\rho) D N ( ρ ) = − β 4 f N , s ′ ( 1 − ρ ) at every ρ ∈ ( 0 , 1 ) \rho\in(0,1) ρ ∈ ( 0 , 1 ) f N , s f_{N,s} f N , s 1 − ρ 1-\rho 1 − ρ f N , s f_{N,s} f N , s ρ \rho ρ D N D_N D N D N D_N D N ρ \rho ρ N N N β \beta β ρ \rho ρ λ \lambda λ
Corollary 3.7 (The defect identity). Let N ≥ 1 N \ge1 N ≥ 1 0 < λ ≤ β / 2 0 < \lambda\le\beta/2 0 < λ ≤ β /2 s = λ / β s=\lambda/\beta s = λ / β ρ ∈ ( 0 , 1 ) \rho\in(0,1) ρ ∈ ( 0 , 1 ) D N ( ρ ) D_N(\rho) D N ( ρ ) μ \mu μ F 1 − ρ , s \mathcal{F}_{1-\rho,s} F 1 − ρ , s
D N ( ρ ) = β ( A + 2 J + V ) , 4 e N , β ( λ ) = ∫ 0 1 D N ( ρ ) d ρ (29) D_N(\rho)=\beta(A+2J+V), \qquad4e_{N,\beta}(\lambda)=\int_0^1D_N(\rho)\,\mathrm{d}\rho \tag*{(29)} D N ( ρ ) = β ( A + 2 J + V ) , 4 e N , β ( λ ) = ∫ 0 1 D N ( ρ ) d ρ (29) where A A A J J J V V V μ \mu μ D N ≥ 0 D_N \ge0 D N ≥ 0 e N , β ( λ ) ≥ 0 e_{N,\beta}(\lambda) \ge0 e N , β ( λ ) ≥ 0
Proof. Let ρ \rho ρ μ \mu μ t = 1 − ρ t=1-\rho t = 1 − ρ Proposition 3.6 (b) and (3.14), we have D N ( ρ ) = β ∫ E ( R q − q ) 2 ν ( d q ) D_N(\rho)=\beta\int\mathbb{E}(R_q-q)^2\,\nu(\mathrm{d}q) D N ( ρ ) = β ∫ E ( R q − q ) 2 ν ( d q ) ν \nu ν μ \mu μ (25) holds. Integrating (25) against ν \nu ν f N , s f_{N,s} f N , s
∫ 0 1 D N ( ρ ) d ρ = − 4 β ∫ 0 1 f N , s ′ ( t ) d t = 4 β ( f N , s ( 0 ) − f N , s ( 1 ) ) = 4 β ( P β ( s ) − ϕ N , β c ( s ) ) . \int_0^1D_N(\rho)\,\mathrm{d}\rho=-\frac{4}{\beta}\int_0^1f'_{N,s}(t)\,\mathrm{d}t=\frac{4}{\beta}\bigl(f_{N,s}(0)-f_{N,s}(1)\bigr)=\frac{4}{\beta}\bigl(P_\beta(s)-\phi^c_{N,\beta}(s)\bigr). ∫ 0 1 D N ( ρ ) d ρ = − β 4 ∫ 0 1 f N , s ′ ( t ) d t = β 4 ( f N , s ( 0 ) − f N , s ( 1 ) ) = β 4 ( P β ( s ) − ϕ N , β c ( s ) ) . where we used (17) in the last step. By (14) , the right side is 4 e N , β ( λ ) 4e_{N,\beta}(\lambda) 4 e N , β ( λ ) [4 Proposition 2.3, (2.21)] .
The individual terms A A A J J J V V V (29) , their combination A + 2 J + V A+2J+V A + 2 J + V A A A J J J V V V D N ( ρ ) D_N(\rho) D N ( ρ ) ρ \rho ρ
Near ρ = 0 \rho=0 ρ = 0 N ≥ 1 N \ge1 N ≥ 1 0 < s ≤ 1 / 2 0<s\le1/2 0 < s ≤ 1/2 0 < ρ ≤ 1 0<\rho\le1 0 < ρ ≤ 1 μ \mu μ F 1 − ρ , s \mathcal{F}_{1-\rho,s} F 1 − ρ , s
∫ E ( R q − q ) 2 ν ( d q ) ≤ C min { 1 , M − 1 / 2 , M − 2 / 3 s − 1 / 3 } . \int\mathbb{E}(R_q-q)^2\,\nu(\mathrm{d}q)\le C\min\{1,M^{-1/2},M^{-2/3}s^{-1/3}\}. ∫ E ( R q − q ) 2 ν ( d q ) ≤ C min { 1 , M − 1/2 , M − 2/3 s − 1/3 } . with a numerical constant C C C [4 Lemma 4.1, (4.1)] , and the third comes from the proof of [4 Theorem 1.1] ; see [4 (4.2)] . Keeping the first two terms and using β M − 1 / 2 = ( N ρ ) − 1 / 2 \beta M^{-1/2}=(N\rho)^{-1/2} β M − 1/2 = ( N ρ ) − 1/2
∫ E ( R q − q ) 2 ν ( d q ) ≤ C min { 1 , M − 1 / 2 } , \int\mathbb{E}(R_q-q)^2\,\nu(\mathrm{d}q)\le C\min\{1,M^{-1/2}\}, ∫ E ( R q − q ) 2 ν ( d q ) ≤ C min { 1 , M − 1/2 } , 0 ≤ D N ( ρ ) ≤ C min { β , ( N ρ ) − 1 / 2 } ( 0 < ρ ≤ 1 ) (30) 0\le D_N(\rho)\le C\min\{\beta,(N\rho)^{-1/2}\}\qquad(0<\rho\le1) \tag*{(30)} 0 ≤ D N ( ρ ) ≤ C min { β , ( N ρ ) − 1/2 } ( 0 < ρ ≤ 1 ) (30) where the second bound holds at every ρ \rho ρ D N ( ρ ) D_N(\rho) D N ( ρ )
The scalar trial profile and the trial budget
The scalar trial profile γ λ t r = max { λ , γ β } \gamma_{\lambda}^{\mathrm{tr}}=\max\{\lambda,\gamma_{\beta}\} γ λ tr = max { λ , γ β } γ β \gamma_{\beta} γ β λ \lambda λ [4 Lemma 7.2] , γ β \gamma_{\beta} γ β 0 0 0 λ \lambda λ γ β \gamma_{\beta} γ β λ 5 \lambda^{5} λ 5 λ = β s \lambda=\beta s λ = β s [4 Lemma 7.2] . Precisely, by [4 Lemma 7.2, (7.3)] , applied with β 0 = β \beta_{0}=\beta β 0 = β β > 1 \beta>1 β > 1 λ ∗ > 0 \lambda_{*}>0 λ ∗ > 0 C ∗ < ∞ C_{*}<\infty C ∗ < ∞ β \beta β
β − 1 P β ( γ λ t r / β ) ≤ E β ( 0 ) + C ∗ λ 5 ( 0 < λ ≤ λ ∗ ) . \beta^{-1}\mathcal{P}_{\beta}(\gamma_{\lambda}^{\mathrm{tr}}/\beta)\le E_{\beta}(0)+C_{*}\lambda^{5}\qquad(0<\lambda\le\lambda_{*}). β − 1 P β ( γ λ tr / β ) ≤ E β ( 0 ) + C ∗ λ 5 ( 0 < λ ≤ λ ∗ ) . We set L s = max { 1 , C ∗ } L_{s}=\max\{1,C_{*}\} L s = max { 1 , C ∗ } λ 0 = min { 1 , λ ∗ / 2 , β / 4 } \lambda_{0}=\min\{1,\lambda_{*}/2,\beta/4\} λ 0 = min { 1 , λ ∗ /2 , β /4 } E β E_{\beta} E β γ λ t r / β \gamma_{\lambda}^{\mathrm{tr}}/\beta γ λ tr / β λ / β \lambda/\beta λ / β
0 ≤ E β ( λ ) − E β ( 0 ) ≤ β − 1 P β ( γ λ t r / β ) − E β ( 0 ) ≤ L s λ 5 ( 0 < λ ≤ 2 λ 0 ) . (31) 0\le E_{\beta}(\lambda)-E_{\beta}(0)\le\beta^{-1}\mathcal{P}_{\beta}(\gamma_{\lambda}^{\mathrm{tr}}/\beta)-E_{\beta}(0)\le L_{s}\lambda^{5}\qquad(0<\lambda\le2\lambda_{0}). \tag*{(31)} 0 ≤ E β ( λ ) − E β ( 0 ) ≤ β − 1 P β ( γ λ tr / β ) − E β ( 0 ) ≤ L s λ 5 ( 0 < λ ≤ 2 λ 0 ) . (31) The bound λ 0 ≤ λ ∗ / 2 \lambda_{0}\le\lambda_{*}/2 λ 0 ≤ λ ∗ /2 (31) at 2 λ 2\lambda 2 λ λ 0 ≤ β / 4 \lambda_{0}\le\beta/4 λ 0 ≤ β /4 2 λ / β ≤ 1 / 2 2\lambda/\beta\le1/2 2 λ / β ≤ 1/2 λ ≤ λ 0 \lambda\le\lambda_{0} λ ≤ λ 0 s = 2 λ / β s=2\lambda/\beta s = 2 λ / β λ 0 ≤ 1 \lambda_{0}\le1 λ 0 ≤ 1
Lemma 3.8 (Trial budget). For every N ≥ 1 N\ge1 N ≥ 1 0 < ρ ≤ 1 0<\rho\le1 0 < ρ ≤ 1 and 0 < λ ≤ λ 0 0<\lambda\le\lambda_{0} 0 < λ ≤ λ 0
0 ≤ F ρ ( γ λ t r ) − min γ F ρ ( γ ) ≤ e N , β ( λ ) + L s λ 5 . (32) 0\le F_{\rho}(\gamma_{\lambda}^{\mathrm{tr}})-\min_{\gamma}F_{\rho}(\gamma)\le e_{N,\beta}(\lambda)+L_{s}\lambda^{5}. \tag*{(32)} 0 ≤ F ρ ( γ λ tr ) − γ min F ρ ( γ ) ≤ e N , β ( λ ) + L s λ 5 . (32) Proof. The lower bound holds because γ λ t r \gamma_{\lambda}^{\mathrm{tr}} γ λ tr μ λ t r \mu_{\lambda}^{\mathrm{tr}} μ λ tr γ λ t r / β \gamma_{\lambda}^{\mathrm{tr}}/\beta γ λ tr / β t ↦ F t , s ( μ λ t r ) t\mapsto\mathcal{F}_{t,s}(\mu_{\lambda}^{\mathrm{tr}}) t ↦ F t , s ( μ λ tr ) F 0 , s ( μ λ t r ) = P β ( γ λ t r / β ) \mathcal{F}_{0,s}(\mu_{\lambda}^{\mathrm{tr}})=\mathcal{P}_{\beta}(\gamma_{\lambda}^{\mathrm{tr}}/\beta) F 0 , s ( μ λ tr ) = P β ( γ λ tr / β ) F ρ ( γ λ t r ) ≤ β − 1 P β ( γ λ t r / β ) F_{\rho}(\gamma_{\lambda}^{\mathrm{tr}})\le\beta^{-1}\mathcal{P}_{\beta}(\gamma_{\lambda}^{\mathrm{tr}}/\beta) F ρ ( γ λ tr ) ≤ β − 1 P β ( γ λ tr / β ) f N , s f_{N,s} f N , s (17) and (14) , we find
min γ F ρ ( γ ) = β − 1 f N , s ( 1 − ρ ) ≥ β − 1 f N , s ( 1 ) = β − 1 ϕ N , β c ( s ) = E β ( λ ) − e N , β ( λ ) . \min_{\gamma}F_{\rho}(\gamma)=\beta^{-1}f_{N,s}(1-\rho)\ge\beta^{-1}f_{N,s}(1)=\beta^{-1}\phi_{N,\beta}^{c}(s)=E_{\beta}(\lambda)-e_{N,\beta}(\lambda). γ min F ρ ( γ ) = β − 1 f N , s ( 1 − ρ ) ≥ β − 1 f N , s ( 1 ) = β − 1 ϕ N , β c ( s ) = E β ( λ ) − e N , β ( λ ) . Subtracting the two bounds, we obtain
F ρ ( γ λ t r ) − min γ F ρ ( γ ) ≤ e N , β ( λ ) + β − 1 P β ( γ λ t r / β ) − E β ( λ ) ≤ e N , β ( λ ) + L s λ 5 , F_{\rho}(\gamma_{\lambda}^{\mathrm{tr}})-\min_{\gamma}F_{\rho}(\gamma)\le e_{N,\beta}(\lambda)+\beta^{-1}\mathcal{P}_{\beta}(\gamma_{\lambda}^{\mathrm{tr}}/\beta)-E_{\beta}(\lambda)\le e_{N,\beta}(\lambda)+L_{s}\lambda^{5}, F ρ ( γ λ tr ) − γ min F ρ ( γ ) ≤ e N , β ( λ ) + β − 1 P β ( γ λ tr / β ) − E β ( λ ) ≤ e N , β ( λ ) + L s λ 5 , where in the last step we used (31) and E β ( λ ) ≥ E β ( 0 ) E_{\beta}(\lambda)\ge E_{\beta}(0) E β ( λ ) ≥ E β ( 0 ) [4 Proposition 7.7, (7.44)] .
We call a number η ≥ 0 \eta\ge0 η ≥ 0 (32) a trial budget . By Lemma 3.8, η = e N , β ( λ ) + L s λ 5 \eta=e_{N,\beta}(\lambda)+L_{s}\lambda^{5} η = e N , β ( λ ) + L s λ 5 ρ ∈ ( 0 , 1 ] \rho\in(0,1] ρ ∈ ( 0 , 1 ]
Reduction of the mean bias
We now reduce the mean bias to the fractional-moment error by the midpoint argument of Step 1 in the proof of [4 Theorem 7.8] . We define
ψ ( λ ) = 1 N λ log E r e f e λ ( F N , β C / β + D N ) ( λ > 0 ) , ψ ( 0 ) = p N , β β , \psi(\lambda)=\frac{1}{N\lambda}\log\mathbb{E}_{\mathrm{ref}}e^{\lambda(F_{N,\beta}^{C}/\beta+D_N)}
\quad(\lambda>0),\qquad\psi(0)=\frac{p_{N,\beta}}{\beta}, ψ ( λ ) = N λ 1 log E ref e λ ( F N , β C / β + D N ) ( λ > 0 ) , ψ ( 0 ) = β p N , β , so that ψ ( λ ) = E β ( λ ) − e N , β ( λ ) \psi(\lambda)=E_\beta(\lambda)-e_{N,\beta}(\lambda) ψ ( λ ) = E β ( λ ) − e N , β ( λ ) 0 < λ < β 0<\lambda<\beta 0 < λ < β N ( N + 1 ) / 2 N(N+1)/2 N ( N + 1 ) /2 ( g i j , z i ) (g_{ij},z_i) ( g ij , z i ) F N , β C / β F_{N,\beta}^{C}/\beta F N , β C / β β − 1 \beta^{-1} β − 1 N − 1 / 2 ⟨ σ i σ j ⟩ N^{-1/2}\langle\sigma_i\sigma_j\rangle N − 1/2 ⟨ σ i σ j ⟩ g i j g_{ij} g ij ⟨ ⋅ ⟩ \langle\cdot\rangle ⟨ ⋅ ⟩ F N , β C F_{N,\beta}^{C} F N , β C ( 2 N ) − 1 / 2 (2N)^{-1/2} ( 2 N ) − 1/2 z i z_i z i
1 N ⋅ N ( N − 1 ) 2 + N ⋅ 1 2 N = N 2 , \frac{1}{N}\cdot\frac{N(N-1)}{2}+N\cdot\frac{1}{2N}=\frac{N}{2}, N 1 ⋅ 2 N ( N − 1 ) + N ⋅ 2 N 1 = 2 N , and the function is N / 2 \sqrt{N}/2 N /2 [4 Lemma 5.2] , which is a special case of Chen’s convexity theorem [26, Theorem 1], the function λ ↦ λ − 1 log E r e f e λ F N , β C / β = N ψ ( λ ) \lambda\mapsto\lambda^{-1}\log\mathbb{E}_{\mathrm{ref}}e^{\lambda F_{N,\beta}^{C}/\beta}=N\psi(\lambda) λ ↦ λ − 1 log E ref e λ F N , β C / β = N ψ ( λ ) [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) E r e f F N , β C / β = N ψ ( 0 ) \mathbb{E}_{\mathrm{ref}}F_{N,\beta}^{C}/\beta=N\psi(0) E ref F N , β C / β = N ψ ( 0 ) E r e f D N = 0 \mathbb{E}_{\mathrm{ref}}D_N=0 E ref D N = 0 ψ \psi ψ [ 0 , ∞ ) [0,\infty) [ 0 , ∞ )
Proposition 3.9 (Mean bias). For every N ≥ 1 N\ge1 N ≥ 1 0 < λ ≤ λ 0 0<\lambda\le\lambda_0 0 < λ ≤ λ 0
0 ≤ P β − p N , β β ≤ E β ( 0 ) − 2 E β ( λ ) + E β ( 2 λ ) + 2 e N , β ( λ ) ≤ 32 L s λ 5 + 2 e N , β ( λ ) . (33) 0\le\frac{P_\beta-p_{N,\beta}}{\beta}\le E_\beta(0)-2E_\beta(\lambda)+E_\beta(2\lambda)+2e_{N,\beta}(\lambda)\le32L_s\lambda^5+2e_{N,\beta}(\lambda). \tag*{(33)} 0 ≤ β P β − p N , β ≤ E β ( 0 ) − 2 E β ( λ ) + E β ( 2 λ ) + 2 e N , β ( λ ) ≤ 32 L s λ 5 + 2 e N , β ( λ ) . (33) Proof. The lower bound is [4 Theorem 1.1, (1.5)] ; it is also Guerra’s bound [40]. Since ψ \psi ψ [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) ψ ( λ ) ≤ ( ψ ( 0 ) + ψ ( 2 λ ) ) / 2 \psi(\lambda)\le(\psi(0)+\psi(2\lambda))/2 ψ ( λ ) ≤ ( ψ ( 0 ) + ψ ( 2 λ )) /2 ψ ( 0 ) ≥ 2 ψ ( λ ) − ψ ( 2 λ ) \psi(0)\ge2\psi(\lambda)-\psi(2\lambda) ψ ( 0 ) ≥ 2 ψ ( λ ) − ψ ( 2 λ ) ψ ( λ ) = E β ( λ ) − e N , β ( λ ) \psi(\lambda)=E_\beta(\lambda)-e_{N,\beta}(\lambda) ψ ( λ ) = E β ( λ ) − e N , β ( λ ) λ ≤ λ 0 ≤ β / 4 \lambda\le\lambda_0\le\beta/4 λ ≤ λ 0 ≤ β /4 2 λ / β ≤ 1 / 2 2\lambda/\beta\le1/2 2 λ / β ≤ 1/2 2 λ 2\lambda 2 λ e N , β ( 2 λ ) ≥ 0 e_{N,\beta}(2\lambda)\ge0 e N , β ( 2 λ ) ≥ 0 ψ ( 2 λ ) ≤ E β ( 2 λ ) \psi(2\lambda)\le E_\beta(2\lambda) ψ ( 2 λ ) ≤ E β ( 2 λ ) E β ( 0 ) = P β / β E_\beta(0)=P_\beta/\beta E β ( 0 ) = P β / β
P β − p N , β β = E β ( 0 ) − ψ ( 0 ) ≤ E β ( 0 ) − 2 E β ( λ ) + 2 e N , β ( λ ) + E β ( 2 λ ) , \frac{P_\beta-p_{N,\beta}}{\beta}=E_\beta(0)-\psi(0)\le E_\beta(0)-2E_\beta(\lambda)+2e_{N,\beta}(\lambda)+E_\beta(2\lambda), β P β − p N , β = E β ( 0 ) − ψ ( 0 ) ≤ E β ( 0 ) − 2 E β ( λ ) + 2 e N , β ( λ ) + E β ( 2 λ ) , which is the middle inequality in (3.19). For the last inequality, we use E β ( λ ) ≥ E β ( 0 ) E_\beta(\lambda)\ge E_\beta(0) E β ( λ ) ≥ E β ( 0 ) 2 λ ≤ 2 λ 0 2\lambda\le2\lambda_0 2 λ ≤ 2 λ 0
E β ( 0 ) − 2 E β ( λ ) + E β ( 2 λ ) ≤ E β ( 2 λ ) − E β ( 0 ) ≤ L s ( 2 λ ) 5 = 32 L s λ 5 . E_\beta(0)-2E_\beta(\lambda)+E_\beta(2\lambda)\le E_\beta(2\lambda)-E_\beta(0)\le L_s(2\lambda)^5=32L_s\lambda^5. E β ( 0 ) − 2 E β ( λ ) + E β ( 2 λ ) ≤ E β ( 2 λ ) − E β ( 0 ) ≤ L s ( 2 λ ) 5 = 32 L s λ 5 . In Section 13, we take λ = N − 7 / 50 \lambda=N^{-7/50} λ = N − 7/50 λ 5 = N − 7 / 10 \lambda^5=N^{-7/10} λ 5 = N − 7/10 e N , β ( N − 7 / 50 ) ≤ C β N − 7 / 10 e_{N,\beta}(N^{-7/50})\le C_\beta N^{-7/10} e N , β ( N − 7/50 ) ≤ C β N − 7/10 N N N
The results of Section 3 concern the law of Section 3.3, and the sharper ones hold only at a minimizing profile. In Sections 9 to 11, we also need laws whose profile minimizes no functional, and laws whose auxiliary field has different variances in different coordinates. In this section, we introduce a class of generalized laws that contains these cases (Definition 4.1), and we prove two Gaussian tools for them.
The first tool is Lemma 4.2, which shows that the martingale identities and the covariance order hold for every generalized law. It also bounds the variance of averages over levels of the squared conditional mean on a set of coordinates. We note that this bound holds for every profile and that its proof does not use contact identities. We use it in Section 9 for the scalar law, which is the generalized law with N = 1 N=1 N = 1 d d d s = 0 s=0 s = 0 diag ( c I N , b ( u ) ) \operatorname{diag}(cI_N,b(u)) diag ( c I N , b ( u )) s = λ / β s=\lambda/\beta s = λ / β
The second tool is Lemma 4.3, a version of the Gaussian heat identity for normalized expectations. Here we take expectations of functionals of finitely many Gaussian variables and continuous Gaussian paths, and we hold the profile measure fixed. We use the lemma in Section 9 to differentiate the scalar law under transports of its levels, and in Sections 10 and 11 to differentiate the cavity laws in the cavity parameter. In this section, we use only the definitions of Section 3, the arguments of [4 Section 2] , and the proof of Proposition 3.1, which we adapt.
Note that some letters have a local meaning in this section. The letter ω \omega ω ω \omega ω M M M Q Q Q M = N c M=Nc M = N c M q M_q M q
Generalized laws and a posterior variance bound
In Section 10, we decouple one spin from the others. The remaining spins then evolve under a law whose profile need not minimize any functional and whose auxiliary field has unequal variances. The following class of laws contains both situations.
Definition 4.1 (Generalized law). Let Λ = diag ( ϑ 1 , … , ϑ N ) \Lambda=\operatorname{diag}(\vartheta_1,\ldots,\vartheta_N) Λ = diag ( ϑ 1 , … , ϑ N ) ϑ i > 0 \vartheta_i>0 ϑ i > 0 ( κ i j ) 1 ≤ i ≤ j ≤ N (\kappa_{ij})_{1\le i\le j\le N} ( κ ij ) 1 ≤ i ≤ j ≤ N Λ \Lambda Λ κ \kappa κ ω = ( ω i j ) 1 ≤ i ≤ j ≤ N \omega=(\omega_{ij})_{1\le i\le j\le N} ω = ( ω ij ) 1 ≤ i ≤ j ≤ N
independent standard Gaussian variables, and write E r e f \mathbb{E}_{\mathrm{ref}} E ref μ = s δ 0 + ν \mu=s\delta_{0}+\nu μ = s δ 0 + ν § 4.1.2 , so that 0 < s ≤ 1 / 2 0<s\le1/2 0 < s ≤ 1/2 s = 0 s=0 s = 0 μ = ν \mu=\nu μ = ν [ 0 , 1 ] [0,1] [ 0 , 1 ] a ( q ) = μ ( [ 0 , q ] ) a(q)=\mu([0,q]) a ( q ) = μ ([ 0 , q ]) U U U
∂ q U = − 1 2 ( Tr ( Λ ∇ 2 U ) + a ( q ) ∇ U T Λ ∇ U ) , U ( 1 , x ; ω ) = log ∑ σ ∈ { − 1 , 1 } N e H σ ( ω , x ) , (34) \partial_{q}U=-\frac{1}{2}\left(\operatorname{Tr}(\Lambda\nabla^{2}U)+a(q)\nabla U^{\mathsf T}\Lambda\nabla U\right),\qquad U(1,x;\omega)=\log\sum_{\sigma\in\{-1,1\}^{N}}e^{\mathcal{H}_{\sigma}(\omega,x)}, \tag*{(34)} ∂ q U = − 2 1 ( Tr ( Λ ∇ 2 U ) + a ( q ) ∇ U T Λ∇ U ) , U ( 1 , x ; ω ) = log σ ∈ { − 1 , 1 } N ∑ e H σ ( ω , x ) , (34) where H σ ( ω , x ) = ∑ i ≤ j κ i j ω i j σ i σ j + x ⋅ σ \mathcal{H}_{\sigma}(\omega,x)=\sum_{i\le j}\kappa_{ij}\omega_{ij}\sigma_{i}\sigma_{j}+x\cdot\sigma H σ ( ω , x ) = ∑ i ≤ j κ ij ω ij σ i σ j + x ⋅ σ generalized law of ( Λ , κ , μ ) (\Lambda,\kappa,\mu) ( Λ , κ , μ ) ( ω , X , σ ) (\omega,X,\sigma) ( ω , X , σ )
The Gaussian law of ω \omega ω e s U ( 0 , 0 ; ω ) / E r e f e s U ( 0 , 0 ; ω ) e^{sU(0,0;\omega)}/\mathbb{E}_{\mathrm{ref}}e^{sU(0,0;\omega)} e s U ( 0 , 0 ; ω ) / E ref e s U ( 0 , 0 ; ω )
Given ω \omega ω d X q = a ( q ) Λ ∇ U ( q , X q ; ω ) d q + Λ 1 / 2 d B q \mathrm{d}X_{q}=a(q)\Lambda\nabla U(q,X_{q};\omega)\,\mathrm{d}q+\Lambda^{1/2}\,\mathrm{d}B_{q} d X q = a ( q ) Λ∇ U ( q , X q ; ω ) d q + Λ 1/2 d B q X 0 = 0 X_{0}=0 X 0 = 0 B B B R N \mathbb{R}^{N} R N
Given ω \omega ω X X X σ \sigma σ e H σ ( ω , X 1 ) e^{\mathcal{H}_{\sigma}(\omega,X_{1})} e H σ ( ω , X 1 )
As in § 4.1.3 , we write P \mathbb{P} P E \mathbb{E} E G q = σ ( ω , X r : 0 ≤ r ≤ q ) \mathcal{G}_{q}=\sigma(\omega,X_{r}:0\le r\le q) G q = σ ( ω , X r : 0 ≤ r ≤ q ) m q = E [ σ ∣ G q ] m_{q}=\mathbb{E}[\sigma\mid\mathcal{G}_{q}] m q = E [ σ ∣ G q ] C q = Cov ( σ ∣ G q ) C_{q}=\operatorname{Cov}(\sigma\mid\mathcal{G}_{q}) C q = Cov ( σ ∣ G q ) H q = ∇ x 2 U ( q , X q ; ω ) H_{q}=\nabla_{x}^{2}U(q,X_{q};\omega) H q = ∇ x 2 U ( q , X q ; ω )
When s = 0 s=0 s = 0 § 4.1 still have s > 0 s>0 s > 0 (34) has a solution for every such profile, which is obtained from step profiles by the Hopf–Cole recursion and uniform approximation, as we show in Step 6 of the proof of Lemma 4.2. With Λ = c I \Lambda=cI Λ = c I ω i j = g i j \omega_{ij}=g_{ij} ω ij = g ij κ i j = β t / N \kappa_{ij}=\beta\sqrt{t/N} κ ij = β t / N i < j i<j i < j ω i i = z i \omega_{ii}=z_{i} ω ii = z i κ i i = β t / ( 2 N ) \kappa_{ii}=\beta\sqrt{t/(2N)} κ ii = β t / ( 2 N ) H σ ( ω , x ) = β t σ T W σ / 2 + x ⋅ σ \mathcal{H}_{\sigma}(\omega,x)=\beta\sqrt{t}\,\sigma^{\mathsf T}W\sigma/2+x\cdot\sigma H σ ( ω , x ) = β t σ T W σ /2 + x ⋅ σ § 4.1.3 . The scalar law of Section 9 is the generalized law with N = 1 N=1 N = 1 Λ = d \Lambda=d Λ = d κ = 0 \kappa=0 κ = 0 s = 0 s=0 s = 0 Λ = diag ( c I n , b ( u ) ) \Lambda=\operatorname{diag}(cI_{n},b(u)) Λ = diag ( c I n , b ( u ))
In the next lemma, we adapt the argument of the proof of [54, Lemma 2.3], which is also used in the proof of [56, Lemma 2.1]. That argument combines joint convexity in the Gaussian drivers and a spin source, the inverse-Hessian Brascamp–Lieb inequality, and a gauge symmetry that removes the conditioning on the terminal spin. We use it for averages of squared conditional means on selected coordinates, and for diagonal anisotropic laws.
Lemma 4.2 (Averaged self-overlap under an arbitrary profile). Let μ = s δ 0 + ν \mu=s\delta_{0}+\nu μ = s δ 0 + ν admissible profile , and consider either its joint law from § 4.1.3 , for some 0 < ρ ≤ 1 0<\rho\le1 0 < ρ ≤ 1 s = 0 s=0 s = 0 μ = ν \mu=\nu μ = ν m q = ∇ U ( q , X q ) = E [ σ ∣ G q ] m_{q}=\nabla U(q,X_{q})=\mathbb{E}[\sigma\mid\mathcal{G}_{q}] m q = ∇ U ( q , X q ) = E [ σ ∣ G q ] m 0 = 0 m_{0}=0 m 0 = 0 a ( q ) C q ⪯ H q ⪯ C q a(q)C_{q}\preceq H_{q}\preceq C_{q} a ( q ) C q ⪯ H q ⪯ C q q ∈ [ 0 , 1 ] q\in[0,1] q ∈ [ 0 , 1 ]
set of n ≥ 1 n \ge1 n ≥ 1 m ‾ q \overline{m}_q m q m q m_q m q H q − − H_q^{--} H q −− C q − − C_q^{--} C q −− H q H_q H q C q C_q C q k ∈ L 2 ( ν ) k \in L^2(\nu) k ∈ L 2 ( ν )
Var ( ∫ k ( q ) ∣ m ‾ q ∣ 2 n ν ( d q ) ) ≤ 4 n 2 ∫ k ( q ) 2 E [ ( m ‾ q ) T H q − − m ‾ q ] ν ( d q ) ≤ 4 n 2 ∫ k ( q ) 2 E [ ( m ‾ q ) T C q − − m ‾ q ] ν ( d q ) . (35) \begin{aligned}
\operatorname{Var}\left(\int k(q)\frac{|\overline{m}_q|^2}{n}\,\nu(\mathrm{d}q)\right) \le\frac{4}{n^2}\int k(q)^2\mathbb{E}\left[(\overline{m}_q)^{\mathsf{T}}H_q^{--}\overline{m}_q\right]\,\nu(\mathrm{d}q) \\
&\le\frac{4}{n^2}\int k(q)^2\mathbb{E}\left[(\overline{m}_q)^{\mathsf{T}}C_q^{--}\overline{m}_q\right]\,\nu(\mathrm{d}q). \tag*{(35)}
\end{aligned} Var ( ∫ k ( q ) n ∣ m q ∣ 2 ν ( d q ) ) ≤ n 2 4 ∫ k ( q ) 2 E [ ( m q ) T H q −− m q ] ν ( d q ) ≤ n 2 4 ∫ k ( q ) 2 E [ ( m q ) T C q −− m q ] ν ( d q ) . (35) Recall that the law of § 4.1.3 is a generalized law, so that it suffices to prove the lemma for generalized laws. The proof also gives the martingale differential
d m q = H q Λ 1 / 2 d B q ( 0 ≤ q ≤ 1 ) (36) \mathrm{d}m_q=H_q\Lambda^{1/2}\,\mathrm{d}B_q \qquad(0\le q\le1) \tag*{(36)} d m q = H q Λ 1/2 d B q ( 0 ≤ q ≤ 1 ) (36) for every generalized law, as we will see in Steps 1 and 6 below.
Proof of Lemma 4.2. The proof has six steps. We first prove the martingale identities and the covariance order for step profiles. Next, still for step profiles, we write the law in terms of independent Gaussian coordinates. We compare the curvature of the posterior density with the gradient of the squared conditional mean, and we apply the Brascamp–Lieb inequality with the terminal spin fixed. We then remove the conditioning on the spin by a gauge symmetry. Finally, we pass to general profiles and general k k k
Step 1: martingale and covariance order. Let μ \mu μ Proposition 3.1 , replacing c I cI c I Λ \Lambda Λ Tr ( Λ ∇ 2 ) / 2 + a ( q ) ( Λ ∇ U ) ⋅ ∇ \operatorname{Tr}(\Lambda\nabla^2)/2+a(q)(\Lambda\nabla U)\cdot\nabla Tr ( Λ ∇ 2 ) /2 + a ( q ) ( Λ∇ U ) ⋅ ∇ x x x
d ∇ U ( q , X q ; ω ) = H q Λ 1 / 2 d B q , d H q = − a ( q ) H q Λ H q d q + ∇ H q Λ 1 / 2 d B q , \mathrm{d}\nabla U(q,X_q;\omega)=H_q\Lambda^{1/2}\,\mathrm{d}B_q,\qquad\mathrm{d}H_q=-a(q)H_q\Lambda H_q\,\mathrm{d}q+\nabla H_q\Lambda^{1/2}\,\mathrm{d}B_q, d ∇ U ( q , X q ; ω ) = H q Λ 1/2 d B q , d H q = − a ( q ) H q Λ H q d q + ∇ H q Λ 1/2 d B q , where the last term is the contraction of the third spatial derivative of U U U Λ 1 / 2 d B q \Lambda^{1/2}\,\mathrm{d}B_q Λ 1/2 d B q N N N ∇ U ( 1 , X 1 ; ω ) \nabla U(1,X_1;\omega) ∇ U ( 1 , X 1 ; ω ) ∇ U ( q , X q ; ω ) = E [ σ ∣ G q ] = m q \nabla U(q,X_q;\omega)=\mathbb{E}[\sigma\mid\mathcal{G}_q]=m_q ∇ U ( q , X q ; ω ) = E [ σ ∣ G q ] = m q C 1 = H 1 C_1=H_1 C 1 = H 1 H H H
C q = E [ H 1 ∣ G q ] + ∫ q 1 E [ H u Λ H u ∣ G q ] d u , C_q=\mathbb{E}[H_1\mid\mathcal{G}_q]+\int_q^1\mathbb{E}[H_u\Lambda H_u\mid\mathcal{G}_q]\,\mathrm{d}u, C q = E [ H 1 ∣ G q ] + ∫ q 1 E [ H u Λ H u ∣ G q ] d u , H q = E [ H 1 ∣ G q ] + ∫ q 1 a ( u ) E [ H u Λ H u ∣ G q ] d u . H_q=\mathbb{E}[H_1\mid\mathcal{G}_q]+\int_q^1a(u)\mathbb{E}[H_u\Lambda H_u\mid\mathcal{G}_q]\,\mathrm{d}u. H q = E [ H 1 ∣ G q ] + ∫ q 1 a ( u ) E [ H u Λ H u ∣ G q ] d u . Note that the matrices H u Λ H u H_u\Lambda H_u H u Λ H u a ( q ) ≤ a ( u ) ≤ 1 a(q)\le a(u)\le1 a ( q ) ≤ a ( u ) ≤ 1 u ≥ q u\ge q u ≥ q Proposition 3.1 , we obtain a ( q ) C q ⪯ H q ⪯ C q a(q)C_q\preceq H_q\preceq C_q a ( q ) C q ⪯ H q ⪯ C q H − σ ( ω , x ) = H σ ( ω , − x ) \mathcal{H}_{-\sigma}(\omega,x)=\mathcal{H}_{\sigma}(\omega,-x) H − σ ( ω , x ) = H σ ( ω , − x ) σ \sigma σ − σ -\sigma − σ U ( 1 , ⋅ ; ω ) U(1,\cdot\,;\omega) U ( 1 , ⋅ ; ω ) U ( q , ⋅ ; ω ) U(q,\cdot\,;\omega) U ( q , ⋅ ; ω ) m 0 = 0 m_0=0 m 0 = 0
Step 2: Gaussian coordinates and the posterior density. Let 0 = q 0 < q 1 < ⋯ < q K = 1 0=q_0<q_1<\cdots<q_K=1 0 = q 0 < q 1 < ⋯ < q K = 1 a = a ℓ a=a_\ell a = a ℓ [ q ℓ , q ℓ + 1 ) [q_\ell,q_{\ell+1}) [ q ℓ , q ℓ + 1 ) a ℓ ≥ s ≥ 0 a_\ell\ge s\ge0 a ℓ ≥ s ≥ 0 ν \nu ν ν ( { q 0 } ) = a 0 − s \nu(\{q_0\})=a_0-s ν ({ q 0 }) = a 0 − s ν ( { q ℓ } ) = a ℓ − a ℓ − 1 \nu(\{q_\ell\})=a_\ell-a_{\ell-1} ν ({ q ℓ }) = a ℓ − a ℓ − 1 1 ≤ ℓ ≤ K − 1 1\le\ell\le K-1 1 ≤ ℓ ≤ K − 1 ν ( { q K } ) = 1 − a K − 1 \nu(\{q_K\})=1-a_{K-1} ν ({ q K }) = 1 − a K − 1 z 1 , … , z K z_1,\ldots,z_K z 1 , … , z K R N \mathbb{R}^N R N
X q ℓ = ∑ h = 1 ℓ ( q h − q h − 1 ) 1 / 2 Λ 1 / 2 z h ( 0 ≤ ℓ ≤ K ) . X_{q_\ell}=\sum_{h=1}^{\ell}(q_h-q_{h-1})^{1/2}\Lambda^{1/2}z_h \qquad(0\le\ell\le K). X q ℓ = h = 1 ∑ ℓ ( q h − q h − 1 ) 1/2 Λ 1/2 z h ( 0 ≤ ℓ ≤ K ) . and let Z Z Z ω \omega ω z 1 , … , z K z_1,\ldots,z_K z 1 , … , z K U q ( Z ) = U ( q , X q ; ω ) U_q(Z)=U(q,X_q;\omega) U q ( Z ) = U ( q , X q ; ω ) U q ℓ = a ℓ − 1 log E z ℓ + 1 e a ℓ U q ℓ + 1 U_{q_\ell}=a_\ell^{-1}\log\mathbb{E}_{z_{\ell+1}}e^{a_\ell U_{q_{\ell+1}}} U q ℓ = a ℓ − 1 log E z ℓ + 1 e a ℓ U q ℓ + 1 a ℓ > 0 a_\ell>0 a ℓ > 0 U q ℓ = E z ℓ + 1 U q ℓ + 1 U_{q_\ell}=\mathbb{E}_{z_{\ell+1}}U_{q_{\ell+1}} U q ℓ = E z ℓ + 1 U q ℓ + 1 a ℓ = 0 a_\ell=0 a ℓ = 0 [4 (2.1)–(2.3)] . We claim that, restricted to ω \omega ω ( ω , ( X q ℓ ) ℓ , σ ) (\omega,(X_{q_\ell})_\ell,\sigma) ( ω , ( X q ℓ ) ℓ , σ ) ( Z , σ ) (Z,\sigma) ( Z , σ )
exp { H σ ( ω , X 1 ) − ∫ U q ( Z ) ν ( d q ) } E r e f e s U ( 0 , 0 ; ω ) . \frac{\exp\left\{\mathcal{H}_\sigma(\omega,X_1)-\int U_q(Z)\,\nu(\mathrm{d}q)\right\}}{\mathbb{E}_{\mathrm{ref}}e^{sU(0,0;\omega)}}. E ref e s U ( 0 , 0 ; ω ) exp { H σ ( ω , X 1 ) − ∫ U q ( Z ) ν ( d q ) } . with respect to the standard Gaussian law of Z Z Z { − 1 , 1 } N \{-1,1\}^N { − 1 , 1 } N s U q 0 sU_{q_0} s U q 0 q ℓ q_\ell q ℓ q ℓ + 1 q_{\ell+1} q ℓ + 1 a ℓ ( U q ℓ + 1 − U q ℓ ) a_\ell(U_{q_{\ell+1}}-U_{q_\ell}) a ℓ ( U q ℓ + 1 − U q ℓ ) H σ ( ω , X 1 ) − U q K \mathcal{H}_\sigma(\omega,X_1)-U_{q_K} H σ ( ω , X 1 ) − U q K U q 0 , U q 1 , … , U q K U_{q_0},U_{q_1},\ldots,U_{q_K} U q 0 , U q 1 , … , U q K
s − a 0 , a 0 − a 1 , … , a K − 2 − a K − 1 , a K − 1 − 1. s-a_0,\qquad a_0-a_1,\qquad\ldots,\qquad a_{K-2}-a_{K-1},\qquad a_{K-1}-1. s − a 0 , a 0 − a 1 , … , a K − 2 − a K − 1 , a K − 1 − 1. which are the negatives of the masses of ν \nu ν s = 0 s=0 s = 0 σ \sigma σ Z Z Z
Φ σ ( Z ) = 1 2 ∣ Z ∣ 2 − H σ ( ω , X 1 ) + ∫ U q ( Z ) ν ( d q ) . \Phi_\sigma(Z)=\frac{1}{2}|Z|^2-\mathcal{H}_\sigma(\omega,X_1)+\int U_q(Z)\,\nu(\mathrm{d}q). Φ σ ( Z ) = 2 1 ∣ Z ∣ 2 − H σ ( ω , X 1 ) + ∫ U q ( Z ) ν ( d q ) . Note that H σ ( ω , X 1 ) \mathcal{H}_\sigma(\omega,X_1) H σ ( ω , X 1 ) Z Z Z U q U_q U q Z Z Z ( ω , x ) (\omega,x) ( ω , x ) X q X_q X q Z Z Z
∇ Z 2 Φ σ = Q = I + ∫ A q ν ( d q ) ⪰ I , A q = ∇ Z 2 U q ⪰ 0. \nabla_Z^2\Phi_\sigma=\mathcal{Q}=I+\int A_q\,\nu(\mathrm{d}q)\succeq I,\qquad A_q=\nabla_Z^2U_q\succeq0. ∇ Z 2 Φ σ = Q = I + ∫ A q ν ( d q ) ⪰ I , A q = ∇ Z 2 U q ⪰ 0. The matrix Q \mathcal{Q} Q σ \sigma σ
Step 3: joint source curvature. Let ι : R n → R N \iota:\mathbb{R}^{n}\to\mathbb{R}^{N} ι : R n → R N U q ( Z , y ) = U ( q , X q ( Z ) + ι y ; ω ) U_q(Z,y)=U(q,X_q(Z)+\iota y;\omega) U q ( Z , y ) = U ( q , X q ( Z ) + ιy ; ω ) y ∈ R n y\in\mathbb{R}^{n} y ∈ R n ( Z , y ) (Z,y) ( Z , y ) U ( q , x ; ω ) U(q,x;\omega) U ( q , x ; ω ) ( ω , x ) (\omega,x) ( ω , x ) ( Z , y ) ↦ ( ω , X q ( Z ) + ι y ) (Z,y)\mapsto(\omega,X_q(Z)+\iota y) ( Z , y ) ↦ ( ω , X q ( Z ) + ιy ) B q = ∇ Z ∇ y U q ( Z , 0 ) B_q=\nabla_Z\nabla_yU_q(Z,0) B q = ∇ Z ∇ y U q ( Z , 0 ) n n n ∇ y U q ( Z , 0 ) = m q − \nabla_yU_q(Z,0)=m_q^{-} ∇ y U q ( Z , 0 ) = m q − ∇ y 2 U q ( Z , 0 ) = H q − − \nabla_y^2U_q(Z,0)=H_q^{--} ∇ y 2 U q ( Z , 0 ) = H q −−
\begin{pmatrix}
A_q & B_q \\
B_q^{T} & H_q^{--}
\end{pmatrix}\succeq0,\qquad
\nabla_Z\lvert m_q^{-}\rvert^2=2B_qm_q^{-}. \tag*{(37)}\end{pmatrix}
For a vector w w w r ∈ R r\in\mathbb{R} r ∈ R ( r w , m q − ) (rw,m_q^{-}) ( r w , m q − ) r r r
∣ w T B q m q − ∣ 2 ≤ ( w T A q w ) ( m q − ) T H q − − m q − . \lvert w^{T}B_qm_q^{-}\rvert^2\leq(w^{T}A_qw)(m_q^{-})^{T}H_q^{--}m_q^{-}. ∣ w T B q m q − ∣ 2 ≤ ( w T A q w ) ( m q − ) T H q −− m q − . The same statements hold at q = 0 q=0 q = 0 m 0 − = 0 m_0^{-}=0 m 0 − = 0
Step 4: the Brascamp–Lieb inequality. Let Y = n − 1 ∫ k ( q ) ∣ m q − ∣ 2 ν ( d q ) Y=n^{-1}\int k(q)\lvert m_q^{-}\rvert^2\,\nu(\mathrm{d}q) Y = n − 1 ∫ k ( q ) ∣ m q − ∣ 2 ν ( d q ) b = ∫ k ( q ) B q m q − ν ( d q ) b=\int k(q)B_qm_q^{-}\,\nu(\mathrm{d}q) b = ∫ k ( q ) B q m q − ν ( d q ) ∇ Z Y = 2 b / n \nabla_ZY=2b/n ∇ Z Y = 2 b / n w w w ν \nu ν ∫ A q ν ( d q ) = Q − I ⪯ Q \int A_q\,\nu(\mathrm{d}q)=Q-I\preceq Q ∫ A q ν ( d q ) = Q − I ⪯ Q
∣ w T b ∣ 2 ≤ ( ∫ w T A q w ν ( d q ) ) ∫ k ( q ) 2 ( m q − ) T H q − − m q − ν ( d q ) ≤ ( w T Q w ) ∫ k ( q ) 2 ( m q − ) T H q − − m q − ν ( d q ) . \begin{aligned}
\lvert w^{T}b\rvert^2\leq\left(\int w^{T}A_qw\,\nu(\mathrm{d}q)\right)\int k(q)^2(m_q^{-})^{T}H_q^{--}m_q^{-}\,\nu(\mathrm{d}q) \\
&\leq(w^{T}Qw)\int k(q)^2(m_q^{-})^{T}H_q^{--}m_q^{-}\,\nu(\mathrm{d}q).
\end{aligned} ∣ w T b ∣ 2 ≤ ( ∫ w T A q w ν ( d q ) ) ∫ k ( q ) 2 ( m q − ) T H q −− m q − ν ( d q ) ≤ ( w T Qw ) ∫ k ( q ) 2 ( m q − ) T H q −− m q − ν ( d q ) . Taking w = Q − 1 b w=Q^{-1}b w = Q − 1 b b T Q − 1 b b^{T}Q^{-1}b b T Q − 1 b
b T Q − 1 b ≤ ∫ k ( q ) 2 ( m q − ) T H q − − m q − ν ( d q ) ; b^{T}Q^{-1}b\leq\int k(q)^2(m_q^{-})^{T}H_q^{--}m_q^{-}\,\nu(\mathrm{d}q); b T Q − 1 b ≤ ∫ k ( q ) 2 ( m q − ) T H q −− m q − ν ( d q ) ; if b T Q − 1 b = 0 b^{T}Q^{-1}b=0 b T Q − 1 b = 0 σ \sigma σ Φ σ \Phi_\sigma Φ σ Q Q Q I I I [21 ] , in the form [4 Proposition 2.2, (2.14)] , we have Var ( Y ∣ σ ) ≤ E [ ∇ Z Y T Q − 1 ∇ Z Y ∣ σ ] \operatorname{Var}(Y\mid\sigma)\leq\mathbb{E}[\nabla_ZY^{T}Q^{-1}\nabla_ZY\mid\sigma] Var ( Y ∣ σ ) ≤ E [ ∇ Z Y T Q − 1 ∇ Z Y ∣ σ ]
Var ( Y ∣ σ ) ≤ 4 n 2 E [ ∫ k ( q ) 2 ( m q − ) T H q − − m q − ν ( d q ) | σ ] . \operatorname{Var}(Y\mid\sigma)\leq\frac{4}{n^2}\mathbb{E}\left[\int k(q)^2(m_q^{-})^{T}H_q^{--}m_q^{-}\,\nu(\mathrm{d}q)\,\middle|\,\sigma\right]. Var ( Y ∣ σ ) ≤ n 2 4 E [ ∫ k ( q ) 2 ( m q − ) T H q −− m q − ν ( d q ) σ ] . Step 5: removing the conditioning. For τ ∈ { − 1 , 1 } N \tau\in\{-1,1\}^{N} τ ∈ { − 1 , 1 } N
σ i ↦ τ i σ i , ω i j ↦ τ i τ j ω i j , z h , i ↦ τ i z h , i . \sigma_i\mapsto\tau_i\sigma_i,\qquad\omega_{ij}\mapsto\tau_i\tau_j\omega_{ij},\qquad z_{h,i}\mapsto\tau_i z_{h,i}. σ i ↦ τ i σ i , ω ij ↦ τ i τ j ω ij , z h , i ↦ τ i z h , i . This map preserves the Gaussian reference law. Because Λ \Lambda Λ X q X_q X q τ \tau τ H τ σ H_{\tau\sigma} H τ σ ( ω , x ) (\omega,x) ( ω , x ) H σ ( ω , x ) \mathcal{H}_{\sigma}(\omega,x) H σ ( ω , x ) U q U_q U q m q , i m_{q,i} m q , i τ i m q , i \tau_i m_{q,i} τ i m q , i ( H q ) i j (H_q)_{ij} ( H q ) ij τ i τ j ( H q ) i j \tau_i\tau_j(H_q)_{ij} τ i τ j ( H q ) ij Y Y Y ( m q − ) T H q − m q − (m_q^-)^{\mathsf T}H_q^-m_q^- ( m q − ) T H q − m q − σ \sigma σ σ \sigma σ E [ Y ∣ σ ] \mathbb{E}[Y\mid\sigma] E [ Y ∣ σ ] Var ( Y ) = E Var ( Y ∣ σ ) \operatorname{Var}(Y)=\mathbb{E}\operatorname{Var}(Y\mid\sigma) Var ( Y ) = E Var ( Y ∣ σ ) (35) for step profiles. The second inequality follows from H q − − ⪯ C q − − H_q^{--}\preceq C_q^{--} H q −− ⪯ C q −− H q ⪯ C q H_q\preceq C_q H q ⪯ C q
Step 6: general profiles and general k k k We approximate μ \mu μ [4 Section 2.6] . That is, we push ν \nu ν 0 0 0 1 1 1 [4 Section 2.6] uses only that the spins are bounded, that spatial derivatives of fixed order are bounded uniformly over step profiles at fixed N N N L 1 L^1 L 1 Λ \Lambda Λ κ i j ω i j \kappa_{ij}\omega_{ij} κ ij ω ij U U U U ~ \widetilde{U} U a a a a ~ \widetilde{a} a φ = U − U ~ \varphi=U-\widetilde{U} φ = U − U
∂ q φ + 1 2 Tr ( Λ ∇ 2 φ ) + a 2 ( ∇ U + ∇ U ~ ) T Λ ∇ φ = − a − a ~ 2 ∇ U ~ T Λ ∇ U ~ , φ ( 1 , ⋅ ; ω ) = 0. \partial_q\varphi+\frac{1}{2}\operatorname{Tr}(\Lambda\nabla^2\varphi)+\frac{a}{2}(\nabla U+\nabla\widetilde{U})^{\mathsf T}\Lambda\nabla\varphi=-\frac{a-\widetilde{a}}{2}\nabla\widetilde{U}^{\mathsf T}\Lambda\nabla\widetilde{U},\qquad\varphi(1,\cdot\,;\omega)=0. ∂ q φ + 2 1 Tr ( Λ ∇ 2 φ ) + 2 a ( ∇ U + ∇ U ) T Λ∇ φ = − 2 a − a ∇ U T Λ∇ U , φ ( 1 , ⋅ ; ω ) = 0. Since each coordinate of ∇ U ~ \nabla\widetilde{U} ∇ U ∇ U ~ T Λ ∇ U ~ ≤ Tr Λ \nabla\widetilde{U}^{\mathsf T}\Lambda\nabla\widetilde{U}\leq\operatorname{Tr}\Lambda ∇ U T Λ∇ U ≤ Tr Λ φ \varphi φ
sup x , ω ∣ φ ( q , x ; ω ) ∣ ≤ Tr Λ 2 ∫ q 1 ∣ a ( u ) − a ~ ( u ) ∣ d u ≤ N 2 max i ϑ i ∥ a − a ~ ∥ L 1 ( [ 0 , 1 ] ) . \sup_{x,\omega}|\varphi(q,x;\omega)|\leq\frac{\operatorname{Tr}\Lambda}{2}\int_q^1|a(u)-\widetilde{a}(u)|\,\mathrm{d}u\leq\frac{N}{2}\max_i\vartheta_i\lVert a-\widetilde{a}\rVert_{L^1([0,1])}. x , ω sup ∣ φ ( q , x ; ω ) ∣ ≤ 2 Tr Λ ∫ q 1 ∣ a ( u ) − a ( u ) ∣ d u ≤ 2 N i max ϑ i ∥ a − a ∥ L 1 ([ 0 , 1 ]) . This replaces the constant c N / 2 cN/2 c N /2 [4 (2.19)] . It follows that the ratio of the outer densities for μ \mu μ ω \omega ω s = 0 s=0 s = 0
Under the common coupling of [4 Section 2.6.1] , the processes X X X m m m H H H C C C μ \mu μ q q q (36) , pass to the limit, where for (36) we use the Itô isometry and the bounded convergence of H H H k k k q ↦ k ( q ) ∣ m q − ∣ 2 q\mapsto k(q)|m_q^-|^2 q ↦ k ( q ) ∣ m q − ∣ 2 q ↦ k ( q ) 2 ( m q − ) T H q − m q − q\mapsto k(q)^2(m_q^-)^{\mathsf T}H_q^-m_q^- q ↦ k ( q ) 2 ( m q − ) T H q − m q − q q q ω \omega ω U ( 0 , 0 ; ω ) U(0,0;\omega) U ( 0 , 0 ; ω ) ω \omega ω k k k k ∈ L 2 ( ν ) k \in L^2(\nu) k ∈ L 2 ( ν ) ∣ m q − ∣ 2 ≤ n |m_q^-|^2 \le n ∣ m q − ∣ 2 ≤ n
∥ 1 n ∫ ( k 1 ( q ) − k 2 ( q ) ) ∣ m q − ∣ 2 ν ( d q ) ∥ L 2 ( P ) ≤ ν ( [ 0 , 1 ] ) 1 / 2 ∥ k 1 − k 2 ∥ L 2 ( ν ) , \left\|\frac{1}{n}\int(k_1(q)-k_2(q))|m_q^-|^2\nu(\mathrm{d}q)\right\|_{L^2(\mathbb{P})}\le\nu([0,1])^{1/2}\|k_1-k_2\|_{L^2(\nu)}, n 1 ∫ ( k 1 ( q ) − k 2 ( q )) ∣ m q − ∣ 2 ν ( d q ) L 2 ( P ) ≤ ν ([ 0 , 1 ] ) 1/2 ∥ k 1 − k 2 ∥ L 2 ( ν ) , which shows that the left side of (4.2) is continuous in k k k ( m q − ) T H q m q − ≤ ( m q − ) T C q − m q − ≤ n ∥ C q ∥ o p ≤ n N (m_q^-)^{T}H_qm_q^- \le(m_q^-)^{T}C_q^-m_q^- \le n\|C_q\|_{\mathrm{op}} \le nN ( m q − ) T H q m q − ≤ ( m q − ) T C q − m q − ≤ n ∥ C q ∥ op ≤ n N 4 n − 1 N ∥ k 1 − k 2 ∥ L 2 ( ν ) ∥ k 1 + k 2 ∥ L 2 ( ν ) 4n^{-1}N\|k_1-k_2\|_{L^2(\nu)}\|k_1+k_2\|_{L^2(\nu)} 4 n − 1 N ∥ k 1 − k 2 ∥ L 2 ( ν ) ∥ k 1 + k 2 ∥ L 2 ( ν ) L 2 ( ν ) L^2(\nu) L 2 ( ν ) □ \square □
Differentiating normalized Gaussian path expectations
In Sections 9 to 11, we differentiate normalized expectations under Gaussian field laws while keeping their profile measures fixed. The parameter changes the covariance of the Gaussian variables. In Section 9, it changes the clock of the field path, and in Sections 10 and 11, it changes the variance of a row of couplings and the clock of one field coordinate. The densities that we encounter are exponentials of integrals of smooth functions of the field against the profile measure, which may have atoms. We use the following version of the Gaussian heat identity, for finitely many Gaussian variables and continuous field paths. Its finite-dimensional input is the classical formula for the derivative of a Gaussian expectation with respect to the covariance. A general version, under hypotheses different from ours, is proved in [85, Theorem 1]. In the proof, we reduce to finite Gaussian vectors by Bernstein approximation of the paths, while we keep the profile measure, including its atoms, exact.
Setting. Let I ⊂ R I \subset\mathbb{R} I ⊂ R I ∘ I^\circ I ∘ z ∈ I z \in I z ∈ I E r e f \mathbb{E}_{\mathrm{ref}} E ref
(i) Finite coordinates G z f i n = ( v k ( z ) 1 / 2 ξ k ) 1 ≤ k ≤ n 0 G_z^{\mathrm{fin}}=(v_k(z)^{1/2}\xi_k)_{1\le k\le n_0} G z fin = ( v k ( z ) 1/2 ξ k ) 1 ≤ k ≤ n 0 ξ 1 , … , ξ n 0 \xi_1,\ldots,\xi_{n_0} ξ 1 , … , ξ n 0 v k : I → [ 0 , ∞ ) v_k\colon I\to[0,\infty) v k : I → [ 0 , ∞ )
(ii) Field paths on a fixed finite rooted tree. Each edge e e e [ 0 , 1 ] [0,1] [ 0 , 1 ] d d d e e e j ≤ d j\le d j ≤ d B e , j B_{e,j} B e , j T e , j , z : [ 0 , 1 ] → [ 0 , ∞ ) T_{e,j,z}\colon[0,1]\to[0,\infty) T e , j , z : [ 0 , 1 ] → [ 0 , ∞ ) T e , j , z ( 0 ) = 0 T_{e,j,z}(0)=0 T e , j , z ( 0 ) = 0 e e e
G z ( e , j , v ) = G z ( e ′ , j , 1 ) + B e , j ( T e , j , z ( v ) ) , 0 ≤ v ≤ 1. G_z(e,j,v)=G_z(e',j,1)+B_{e,j}(T_{e,j,z}(v)),\qquad0\le v\le1. G z ( e , j , v ) = G z ( e ′ , j , 1 ) + B e , j ( T e , j , z ( v )) , 0 ≤ v ≤ 1. where e ′ e' e ′ e e e e e e
(iii) A finite family ω \omega ω z z z σ \sigma σ
The variables ξ k \xi_k ξ k B e , j B_{e,j} B e , j I \mathcal{I} I 1 ≤ k ≤ n 0 1 \le k \le n_0 1 ≤ k ≤ n 0 ( e , j , v ) (e,j,v) ( e , j , v ) [ 0 , 1 ] [0,1] [ 0 , 1 ] G z = ( G z f i n , G z p a t h ) G_z = (G_z^{\mathrm{fin}},G_z^{\mathrm{path}}) G z = ( G z fin , G z path ) C \mathcal{C} C I \mathcal{I} I [ 0 , 1 ] [0,1] [ 0 , 1 ] ∥ G f i n ∥ + ∥ G p a t h ∥ ∞ \lVert G^{\mathrm{fin}}\rVert+ \lVert G^{\mathrm{path}}\rVert_\infty ∥ G fin ∥ + ∥ G path ∥ ∞ K z ( i , i ′ ) = E r e f [ G z ( i ) G z ( i ′ ) ] K_z(i,i') = \mathbb{E}_{\mathrm{ref}}[G_z(i)G_z(i')] K z ( i , i ′ ) = E ref [ G z ( i ) G z ( i ′ )] i , i ′ ∈ I i,i' \in\mathcal{I} i , i ′ ∈ I K z ( v , v ′ ) = T z ( min { v , v ′ } ) K_z(v,v') = T_z(\min\{v,v'\}) K z ( v , v ′ ) = T z ( min { v , v ′ })
Let ℓ z ( G ) = ℓ z ( G , ω , σ ) \ell_z(G) = \ell_z(G,\omega,\sigma) ℓ z ( G ) = ℓ z ( G , ω , σ ) Φ z ( G ) = Φ z ( G , ω , σ ) \Phi_z(G) = \Phi_z(G,\omega,\sigma) Φ z ( G ) = Φ z ( G , ω , σ ) z ∈ I z \in I z ∈ I G ∈ C G \in\mathcal{C} G ∈ C ω \omega ω σ \sigma σ ℓ ˙ z \dot{\ell}_z ℓ ˙ z Φ ˙ z \dot{\Phi}_z Φ ˙ z z z z ( G , ω , σ ) (G,\omega,\sigma) ( G , ω , σ ) K ˙ z \dot{K}_z K ˙ z K z K_z K z z z z A A A I 2 \mathcal{I}^2 I 2 B B B I 2 \mathcal{I}^2 I 2 ⟨ A , B ⟩ = ∬ A ( i , i ′ ) B ( d i , d i ′ ) \langle A,B\rangle= \iint A(i,i')B(\mathrm{d}i,\mathrm{d}i') ⟨ A , B ⟩ = ∬ A ( i , i ′ ) B ( d i , d i ′ ) S ⊗ P S \otimes P S ⊗ P I \mathcal{I} I
Lemma 4.3 (Normalized Gaussian path heat identity). In the setting above, assume the following.
(A1) Variances and clocks. Each v k v_k v k T e , j , z ( v ) T_{e,j,z}(v) T e , j , z ( v ) z ∈ I z \in I z ∈ I v v v z ∈ I ∘ z \in I^\circ z ∈ I ∘ z ∈ I ∘ z \in I^\circ z ∈ I ∘ v k v_k v k v v v v k ≡ 0 v_k \equiv0 v k ≡ 0
(A2) Covariance. The covariance K z ( i , i ′ ) K_z(i,i') K z ( i , i ′ ) z ∈ I ∘ z \in I^\circ z ∈ I ∘ ( z , i , i ′ ) ↦ K z ( i , i ′ ) (z,i,i') \mapsto K_z(i,i') ( z , i , i ′ ) ↦ K z ( i , i ′ ) ( z , i , i ′ ) ↦ K ˙ z ( i , i ′ ) (z,i,i') \mapsto\dot{K}_z(i,i') ( z , i , i ′ ) ↦ K ˙ z ( i , i ′ ) I × I 2 I \times\mathcal{I}^2 I × I 2 K ˙ z \dot{K}_z K ˙ z I I I
(A3) Functionals. For each σ \sigma σ ℓ z ( G , ω , σ ) \ell_z(G,\omega,\sigma) ℓ z ( G , ω , σ ) Φ z ( G , ω , σ ) \Phi_z(G,\omega,\sigma) Φ z ( G , ω , σ ) ℓ ˙ z ( G , ω , σ ) \dot{\ell}_z(G,\omega,\sigma) ℓ ˙ z ( G , ω , σ ) Φ ˙ z ( G , ω , σ ) \dot{\Phi}_z(G,\omega,\sigma) Φ ˙ z ( G , ω , σ ) ( z , G , ω ) ∈ I × C × R n 1 (z,G,\omega) \in I \times\mathcal{C} \times\mathbb{R}^{n_1} ( z , G , ω ) ∈ I × C × R n 1 n 1 n_1 n 1 ℓ ˙ z \dot{\ell}_z ℓ ˙ z Φ ˙ z \dot{\Phi}_z Φ ˙ z I I I Φ z \Phi_z Φ z Φ ˙ z \dot{\Phi}_z Φ ˙ z ℓ ˙ z \dot{\ell}_z ℓ ˙ z C < ∞ C < \infty C < ∞
∣ ℓ z ( G ) ∣ ≤ C ( 1 + ∣ ω ∣ + ∥ G f i n ∥ + ∥ G p a t h ∥ ∞ ) . |\ell_z(G)| \le C(1+|\omega|+\lVert G^{\mathrm{fin}}\rVert+\lVert G^{\mathrm{path}}\rVert_\infty). ∣ ℓ z ( G ) ∣ ≤ C ( 1 + ∣ ω ∣ + ∥ G fin ∥ + ∥ G path ∥ ∞ ) . (A4) Derivative measures. For fixed ( z , ω , σ ) (z,\omega,\sigma) ( z , ω , σ ) G ↦ ℓ z ( G ) G \mapsto\ell_z(G) G ↦ ℓ z ( G ) G ↦ Φ z ( G ) G \mapsto\Phi_z(G) G ↦ Φ z ( G ) C \mathcal{C} C
S = D ℓ z , M = D 2 ℓ z , P = D Φ z , Q = D 2 Φ z S = D\ell_z,\qquad M = D^2\ell_z,\qquad P = D\Phi_z,\qquad Q = D^2\Phi_z S = D ℓ z , M = D 2 ℓ z , P = D Φ z , Q = D 2 Φ z on I \mathcal{I} I I 2 \mathcal{I}^{2} I 2
D ℓ z ( G ) [ h ] = ∫ h d S , D 2 ℓ z ( G ) [ h , h ′ ] = ∬ h ( i ) h ′ ( i ′ ) M ( d i , d i ′ ) ( h , h ′ ∈ C ) . D\ell_z(G)[h] = \int h\,\mathrm{d}S,\qquad D^{2}\ell_z(G)[h,h'] = \iint h(i)h'(i')\,M(\mathrm{d}i,\mathrm{d}i')\qquad(h,h' \in\mathcal{C}). D ℓ z ( G ) [ h ] = ∫ h d S , D 2 ℓ z ( G ) [ h , h ′ ] = ∬ h ( i ) h ′ ( i ′ ) M ( d i , d i ′ ) ( h , h ′ ∈ C ) . For fixed ( ω , σ ) (\omega,\sigma) ( ω , σ ) ( z , G ) (z,G) ( z , G )
Let Z z = E r e f e ℓ z ( G z ) Z_z = \mathbb{E}_{\mathrm{ref}}e^{\ell_z(G_z)} Z z = E ref e ℓ z ( G z ) E r e f \mathbb{E}_{\mathrm{ref}} E ref σ \sigma σ E z Φ = Z z − 1 E r e f [ e ℓ z ( G z ) Φ ] \mathbb{E}_z\Phi= Z_z^{-1}\mathbb{E}_{\mathrm{ref}}[e^{\ell_z(G_z)}\Phi] E z Φ = Z z − 1 E ref [ e ℓ z ( G z ) Φ ] z ∈ I z \in I z ∈ I
d d z E z Φ z = E z Φ ˙ z + Cov z ( Φ z , ℓ ˙ z ) + 1 2 E z ⟨ K ˙ z , Q + S ⊗ P + P ⊗ S + ( Φ z − E z Φ z ) ( M + S ⊗ S ) ⟩ , (38) \frac{\mathrm{d}}{\mathrm{d}z}\mathbb{E}_z\Phi_z = \mathbb{E}_z\dot{\Phi}_z + \operatorname{Cov}_z(\Phi_z,\dot{\ell}_z)
+ \frac{1}{2}\mathbb{E}_z\left\langle\dot{K}_z,Q + S \otimes P + P \otimes S + (\Phi_z-\mathbb{E}_z\Phi_z)(M+S\otimes S)\right\rangle, \tag*{(38)} d z d E z Φ z = E z Φ ˙ z + Cov z ( Φ z , ℓ ˙ z ) + 2 1 E z ⟨ K ˙ z , Q + S ⊗ P + P ⊗ S + ( Φ z − E z Φ z ) ( M + S ⊗ S ) ⟩ , (38) where the functionals and measures are evaluated at G z G_z G z I I I
d d z log Z z = E z ℓ ˙ z + 1 2 E z ⟨ K ˙ z , M + S ⊗ S ⟩ . (39) \frac{\mathrm{d}}{\mathrm{d}z}\log Z_z = \mathbb{E}_z\dot{\ell}_z + \frac{1}{2}\mathbb{E}_z\langle\dot{K}_z,M+S\otimes S\rangle. \tag*{(39)} d z d log Z z = E z ℓ ˙ z + 2 1 E z ⟨ K ˙ z , M + S ⊗ S ⟩ . (39) Remark 4.4. We show in the proof that the right sides of (38) and (39) are continuous in z ∈ I z \in I z ∈ I z ↦ E z Φ z z \mapsto\mathbb{E}_z\Phi_z z ↦ E z Φ z z ↦ log Z z z \mapsto\log Z_z z ↦ log Z z I I I ϖ \varpi ϖ I I I ℓ z ( G ) = − ∫ V z ( i , G ( i ) ) ϖ ( d i ) \ell_z(G)=-\int V_z(i,G(i))\,\varpi(\mathrm{d}i) ℓ z ( G ) = − ∫ V z ( i , G ( i )) ϖ ( d i ) V z ( i , x ) V_z(i,x) V z ( i , x ) x x x ∂ x V z \partial_xV_z ∂ x V z ∂ x 2 V z \partial_x^2V_z ∂ x 2 V z ( z , i , x ) (z,i,x) ( z , i , x )
S = − ∂ x V z ( ⋅ , G ( ⋅ ) ) ϖ , M = − diag # ( ∂ x 2 V z ( ⋅ , G ( ⋅ ) ) ϖ ) . S=-\partial_xV_z(\cdot,G(\cdot))\,\varpi,\qquad M=-\operatorname{diag}_{\#}\bigl(\partial_x^2V_z(\cdot,G(\cdot))\,\varpi\bigr). S = − ∂ x V z ( ⋅ , G ( ⋅ )) ϖ , M = − diag # ( ∂ x 2 V z ( ⋅ , G ( ⋅ )) ϖ ) . where diag # \operatorname{diag}_{\#} diag # i ↦ ( i , i ) i\mapsto(i,i) i ↦ ( i , i ) G G G G ′ G' G ′ ϖ ( I ) sup i ∣ ∂ x V z ( i , G ( i ) ) − ∂ x V z ( i , G ′ ( i ) ) ∣ \varpi(I)\sup_i|\partial_xV_z(i,G(i))-\partial_xV_z(i,G'(i))| ϖ ( I ) sup i ∣ ∂ x V z ( i , G ( i )) − ∂ x V z ( i , G ′ ( i )) ∣ ∂ x 2 V z \partial_x^2V_z ∂ x 2 V z G G G ϖ \varpi ϖ M M M ⟨ K ˙ z , M ⟩ \langle\dot{K}_z,M\rangle ⟨ K ˙ z , M ⟩ K ˙ z \dot{K}_z K ˙ z
Proof of Lemma 4.3. The proof has seven steps. We first prove the identities for finite Gaussian vectors. We then approximate the paths by their Bernstein polynomials and couple the approximations on one probability space. Next, we show that the derivative measures and the covariance kernels converge, and we establish a common integrable envelope and a lower bound for the normalizers. Finally, we pass to the limit in the interior of I I I
Step 1: the finite-dimensional identity. We begin with a finite Gaussian vector. Let J J J Y z Y_z Y z z ∈ J z \in J z ∈ J R p \mathbb{R}^p R p Σ z \Sigma_z Σ z z z z Σ z \Sigma_z Σ z F z : R p → R F_z:\mathbb{R}^p \to\mathbb{R} F z : R p → R z z z y ∈ R p y \in\mathbb{R}^p y ∈ R p F z F_z F z ∂ z F z \partial_z F_z ∂ z F z ∇ F z \nabla F_z ∇ F z ∇ 2 F z \nabla^2 F_z ∇ 2 F z C e C ∣ y ∣ Ce^{C|y|} C e C ∣ y ∣ z z z
d d z E F z ( Y z ) = E ∂ z F z ( Y z ) + 1 2 E ⟨ Σ ˙ z , ∇ 2 F z ( Y z ) ⟩ . (40) \frac{\mathrm{d}}{\mathrm{d}z}\mathbb{E}F_z(Y_z)=\mathbb{E}\,\partial_zF_z(Y_z)+\frac{1}{2}\mathbb{E}\langle\dot{\Sigma}_z,\nabla^2F_z(Y_z)\rangle. \tag*{(40)} d z d E F z ( Y z ) = E ∂ z F z ( Y z ) + 2 1 E ⟨ Σ ˙ z , ∇ 2 F z ( Y z )⟩ . (40) Let ε > 0 \varepsilon>0 ε > 0 ζ \zeta ζ R p \mathbb{R}^p R p A z = ( Σ z + ε I ) 1 / 2 A_z=(\Sigma_z+\varepsilon I)^{1/2} A z = ( Σ z + ε I ) 1/2 A z A_z A z z z z E F z ( A z ζ ) \mathbb{E}F_z(A_z\zeta) E F z ( A z ζ ) E ∂ z F z ( A z ζ ) \mathbb{E}\,\partial_zF_z(A_z\zeta) E ∂ z F z ( A z ζ ) E [ ∇ F z ( A z ζ ) ⋅ A ˙ z ζ ] \mathbb{E}[\nabla F_z(A_z\zeta)\cdot\dot{A}_z\zeta] E [ ∇ F z ( A z ζ ) ⋅ A ˙ z ζ ] E [ ζ l g ( ζ ) ] = E [ ∂ l g ( ζ ) ] \mathbb{E}[\zeta_lg(\zeta)]=\mathbb{E}[\partial_lg(\zeta)] E [ ζ l g ( ζ )] = E [ ∂ l g ( ζ )] E Tr ( A ˙ z ∇ 2 F z ( A z ζ ) A z ) \mathbb{E}\operatorname{Tr}(\dot{A}_z\nabla^2F_z(A_z\zeta)A_z) E Tr ( A ˙ z ∇ 2 F z ( A z ζ ) A z ) ∇ 2 F z \nabla^2F_z ∇ 2 F z A ˙ z A z + A z A ˙ z = Σ ˙ z \dot{A}_zA_z+A_z\dot{A}_z=\dot{\Sigma}_z A ˙ z A z + A z A ˙ z = Σ ˙ z E ⟨ Σ ˙ z , ∇ 2 F z ( A z ζ ) ⟩ / 2 \mathbb{E}\langle\dot{\Sigma}_z,\nabla^2F_z(A_z\zeta)\rangle/2 E ⟨ Σ ˙ z , ∇ 2 F z ( A z ζ )⟩ /2 [ z 1 , z 2 ] ⊂ J [z_1,z_2]\subset J [ z 1 , z 2 ] ⊂ J ε ↓ 0 \varepsilon\downarrow0 ε ↓ 0 A z ζ A_z\zeta A z ζ Y z + ε ζ ′ Y_z+\sqrt{\varepsilon}\zeta' Y z + ε ζ ′ ζ ′ \zeta' ζ ′ Y z Y_z Y z (40) . Realizing Y z = Σ z 1 / 2 ζ Y_z=\Sigma_z^{1/2}\zeta Y z = Σ z 1/2 ζ z z z (40) is continuous in z z z (40) . This is the classical formula. Note that we never invert Σ z \Sigma_z Σ z
We will apply (40) to e ℓ z e^{\ell_z} e ℓ z e ℓ z Φ z e^{\ell_z}\Phi_z e ℓ z Φ z R p \mathbb{R}^p R p C \mathbb{C} C D e ℓ = e ℓ S D e^\ell=e^\ell S D e ℓ = e ℓ S D 2 e ℓ = e ℓ ( M + S ⊗ S ) D^2e^\ell=e^\ell(M+S\otimes S) D 2 e ℓ = e ℓ ( M + S ⊗ S )
D 2 ( e ℓ Φ ) = e ℓ [ Q + S ⊗ P + P ⊗ S + Φ ( M + S ⊗ S ) ] . D^2(e^\ell\Phi)=e^\ell\left[Q+S\otimes P+P\otimes S+\Phi(M+S\otimes S)\right]. D 2 ( e ℓ Φ ) = e ℓ [ Q + S ⊗ P + P ⊗ S + Φ ( M + S ⊗ S ) ] . Inserting these derivatives into (40) for the numerator E r e f [ e ℓ z Φ z ] \mathbb{E}_{\mathrm{ref}}[e^{\ell_z}\Phi_z] E ref [ e ℓ z Φ z ] E r e f e ℓ z \mathbb{E}_{\mathrm{ref}}e^{\ell_z} E ref e ℓ z Φ z − E z Φ z \Phi_z-\mathbb{E}_z\Phi_z Φ z − E z Φ z
Step 2: Bernstein approximation and kernel convergence. We use the Bernstein approximation introduced in [13 ] . Let m ≥ 1 m\geq1 m ≥ 1 v l = l / m v_l=l/m v l = l / m 0 ≤ l ≤ m 0\leq l\leq m 0 ≤ l ≤ m f f f [ 0 , 1 ] [0,1] [ 0 , 1 ]
Π m f ( v ) = ∑ l = 0 m b m , l ( v ) f ( v l ) , b m , l ( v ) = ( m l ) v l ( 1 − v ) m − l . \Pi_m f(v)=\sum_{l=0}^{m}b_{m,l}(v)f(v_l),\qquad b_{m,l}(v)=\binom{m}{l}v^l(1-v)^{m-l}. Π m f ( v ) = l = 0 ∑ m b m , l ( v ) f ( v l ) , b m , l ( v ) = ( l m ) v l ( 1 − v ) m − l . The operator Π m \Pi_m Π m Π m f ( 0 ) = f ( 0 ) \Pi_mf(0)=f(0) Π m f ( 0 ) = f ( 0 ) Π m f ( 1 ) = f ( 1 ) \Pi_mf(1)=f(1) Π m f ( 1 ) = f ( 1 ) ∥ Π m f ∥ ∞ ≤ ∥ f ∥ ∞ \|\Pi_mf\|_\infty\leq\|f\|_\infty ∥ Π m f ∥ ∞ ≤ ∥ f ∥ ∞ δ > 0 \delta>0 δ > 0
∥ Π m f − f ∥ ∞ ≤ ω f ( δ ) + ∥ f ∥ ∞ 2 m δ 2 . \|\Pi_mf-f\|_\infty\leq\omega_f(\delta)+\frac{\|f\|_\infty}{2m\delta^2}. ∥ Π m f − f ∥ ∞ ≤ ω f ( δ ) + 2 m δ 2 ∥ f ∥ ∞ . where ω f \omega_f ω f f f f m m m m / 4 m/4 m /4 Π m f → f \Pi_m f \to f Π m f → f f f f Π m ⊗ Π m \Pi_m \otimes\Pi_m Π m ⊗ Π m [ 0 , 1 ] 2 [0,1]^2 [ 0 , 1 ] 2 ω f ( δ ) + ∥ f ∥ ∞ / ( m δ 2 ) \omega_f(\delta) + \|f\|_\infty/(m\delta^2) ω f ( δ ) + ∥ f ∥ ∞ / ( m δ 2 )
On each edge and component, we let Π m \Pi_m Π m C \mathcal C C Π m \Pi_m Π m Π m G z \Pi_mG_z Π m G z Y z Y_z Y z G z ( e , j , v l ) G_z(e,j,v_l) G z ( e , j , v l ) Y z Y_z Y z Y z Y_z Y z K z K_z K z I ∘ I^\circ I ∘ K ˙ z \dot K_z K ˙ z ∥ Π m y ∥ C ≤ 2 ∣ y ∣ \|\Pi_m y\|_{\mathcal C} \le2|y| ∥ Π m y ∥ C ≤ 2∣ y ∣
We set
ℓ z ( m ) = ℓ z ( Π m G z ) , Φ z ( m ) = Φ z ( Π m G z ) , Z z ( m ) = E r e f e ℓ z ( m ) . \ell_z^{(m)}=\ell_z(\Pi_mG_z),\qquad\Phi_z^{(m)}=\Phi_z(\Pi_mG_z),\qquad Z_z^{(m)}=\mathbb{E}_{\mathrm{ref}}e^{\ell_z^{(m)}}. ℓ z ( m ) = ℓ z ( Π m G z ) , Φ z ( m ) = Φ z ( Π m G z ) , Z z ( m ) = E ref e ℓ z ( m ) . and we normalize the approximate moment E z ( m ) Φ z ( m ) = ( Z z ( m ) ) − 1 E r e f [ e ℓ z ( m ) Φ z ( m ) ] \mathbb{E}_z^{(m)}\Phi_z^{(m)}=(Z_z^{(m)})^{-1}\mathbb{E}_{\mathrm{ref}}[e^{\ell_z^{(m)}}\Phi_z^{(m)}] E z ( m ) Φ z ( m ) = ( Z z ( m ) ) − 1 E ref [ e ℓ z ( m ) Φ z ( m ) ] Z z ( m ) Z_z^{(m)} Z z ( m ) ℓ z ( Π m y ) \ell_z(\Pi_m y) ℓ z ( Π m y ) ( e , j , v l ) (e,j,v_l) ( e , j , v l ) ∫ b m , l d S \int b_{m,l}\,dS ∫ b m , l d S b m , l b_{m,l} b m , l e e e j j j M M M ( ω , σ ) (\omega,\sigma) ( ω , σ ) ω \omega ω σ \sigma σ z ∈ I ∘ z\in I^\circ z ∈ I ∘ E z \mathbb{E}_z E z E z ( m ) \mathbb{E}_z^{(m)} E z ( m ) Π m G z \Pi_mG_z Π m G z K ˙ z \dot K_z K ˙ z
K ˙ z ( m ) ( v , w ) = ∑ l , l ′ b m , l ( v ) b m , l ′ ( w ) K ˙ z ( v l , v l ′ ) \dot K_z^{(m)}(v,w)=\sum_{l,l'}b_{m,l}(v)b_{m,l'}(w)\dot K_z(v_l,v_{l'}) K ˙ z ( m ) ( v , w ) = l , l ′ ∑ b m , l ( v ) b m , l ′ ( w ) K ˙ z ( v l , v l ′ ) on each pair of edges and components. Blocks between finite coordinates are unchanged, and blocks between a finite coordinate k k k
K ˙ z ( m ) ( k , ( e , j , v ) ) = ∑ l b m , l ( v ) K ˙ z ( k , ( e , j , v l ) ) . \dot K_z^{(m)}(k,(e,j,v))=\sum_l b_{m,l}(v)\dot K_z(k,(e,j,v_l)). K ˙ z ( m ) ( k , ( e , j , v )) = l ∑ b m , l ( v ) K ˙ z ( k , ( e , j , v l )) . By (A2), the kernels K z K_z K z K ˙ z \dot K_z K ˙ z I × T 2 I\times\mathcal T^2 I × T 2 { K z } z ∈ I \{K_z\}_{z\in I} { K z } z ∈ I { K ˙ z } z ∈ I \{\dot K_z\}_{z\in I} { K ˙ z } z ∈ I
sup z ∈ I ∥ K ˙ z ( m ) − K ˙ z ∥ ∞ ⟶ 0 , sup z ∈ I ∥ K z ( m ) − K z ∥ ∞ ⟶ 0 , (41) \sup_{z\in I}\|\dot K_z^{(m)}-\dot K_z\|_\infty\longrightarrow0,\qquad
\sup_{z\in I}\|K_z^{(m)}-K_z\|_\infty\longrightarrow0, \tag*{(41)} z ∈ I sup ∥ K ˙ z ( m ) − K ˙ z ∥ ∞ ⟶ 0 , z ∈ I sup ∥ K z ( m ) − K z ∥ ∞ ⟶ 0 , (41) where K z ( m ) K_z^{(m)} K z ( m ) Π m G z \Pi_mG_z Π m G z
Step 3: coupling. In Step 1, we used only the law of Y z Y_z Y z z z z B e , j B_{e,j} B e , j v k ( z ) 1 / 2 ξ k v_k(z)^{1/2}\xi_k v k ( z ) 1/2 ξ k ξ k \xi_k ξ k ( z , v ) ↦ G z ( e , j , v ) (z,v) \mapsto G_z(e,j,v) ( z , v ) ↦ G z ( e , j , v ) I × [ 0 , 1 ] I \times[0,1] I × [ 0 , 1 ] { G z ( e , j , ⋅ ) } z ∈ I \{G_z(e,j,\cdot)\}_{z\in I} { G z ( e , j , ⋅ ) } z ∈ I sup z ∈ I ∥ Π m G z − G z ∥ C → 0 \sup_{z\in I}\|\Pi_mG_z-G_z\|_C \to0 sup z ∈ I ∥ Π m G z − G z ∥ C → 0 T ‾ \overline{T} T I × [ 0 , 1 ] I \times[0,1] I × [ 0 , 1 ] T ‾ = 0 \overline{T}=0 T = 0 v ‾ = max ( { 0 } ∪ { v k ( z ) : 1 ≤ k ≤ n 0 , z ∈ I } ) \overline{v}=\max(\{0\}\cup\{v_k(z):1\le k\le n_0,\ z\in I\}) v = max ({ 0 } ∪ { v k ( z ) : 1 ≤ k ≤ n 0 , z ∈ I }) v ‾ = 0 \overline{v}=0 v = 0 e e e e e e
sup z ∈ I ∥ Π m G z ∥ C ≤ sup z ∈ I ∥ G z ∥ C ≤ v ‾ 1 / 2 ∣ ξ ∣ + ∑ e , j sup 0 ≤ τ ≤ T ‾ ∣ B e , j ( τ ) ∣ . \sup_{z\in I}\|\Pi_mG_z\|_C\le\sup_{z\in I}\|G_z\|_C\le\overline{v}^{1/2}|\xi|+\sum_{e,j}\sup_{0\le\tau\le\overline{T}}|B_{e,j}(\tau)|. z ∈ I sup ∥ Π m G z ∥ C ≤ z ∈ I sup ∥ G z ∥ C ≤ v 1/2 ∣ ξ ∣ + e , j ∑ 0 ≤ τ ≤ T sup ∣ B e , j ( τ ) ∣. Step 4: convergence of the derivative measures. By (A4) and Step 3, we see that the measures S S S M M M P P P Q Q Q Π m G z \Pi_mG_z Π m G z G z G_z G z z ∈ I z\in I z ∈ I M ( m ) M^{(m)} M ( m ) M M M Π m G z \Pi_mG_z Π m G z
∣ ⟨ K ˙ z ( m ) , M ( m ) ⟩ − ⟨ K ˙ z , M ⟩ ∣ ≤ ∥ K ˙ z ( m ) − K ˙ z ∥ ∞ ∥ M ( m ) ∥ T V + ∥ K ˙ z ∥ ∞ ∥ M ( m ) − M ∥ T V . \left|\langle\dot{K}_z^{(m)},M^{(m)}\rangle-\langle\dot{K}_z,M\rangle\right|
\le\|\dot{K}_z^{(m)}-\dot{K}_z\|_\infty\|M^{(m)}\|_{\mathrm{TV}}+\|\dot{K}_z\|_\infty\|M^{(m)}-M\|_{\mathrm{TV}}. ⟨ K ˙ z ( m ) , M ( m ) ⟩ − ⟨ K ˙ z , M ⟩ ≤ ∥ K ˙ z ( m ) − K ˙ z ∥ ∞ ∥ M ( m ) ∥ TV + ∥ K ˙ z ∥ ∞ ∥ M ( m ) − M ∥ TV . and for product measures we use ∥ S ′ ⊗ P ′ − S ⊗ P ∥ T V ≤ ∥ S ′ − S ∥ T V ∥ P ′ ∥ T V + ∥ S ∥ T V ∥ P ′ − P ∥ T V \|S'\otimes P'-S\otimes P\|_{\mathrm{TV}}\le\|S'-S\|_{\mathrm{TV}}\|P'\|_{\mathrm{TV}}+\|S\|_{\mathrm{TV}}\|P'-P\|_{\mathrm{TV}} ∥ S ′ ⊗ P ′ − S ⊗ P ∥ TV ≤ ∥ S ′ − S ∥ TV ∥ P ′ ∥ TV + ∥ S ∥ TV ∥ P ′ − P ∥ TV K ˙ z ( m ) ( v , v ) → K ˙ z ( v , v ) \dot{K}_z^{(m)}(v,v)\to\dot{K}_z(v,v) K ˙ z ( m ) ( v , v ) → K ˙ z ( v , v )
Step 5: domination and the normalizers. By (A3) and Step 3, we have e ℓ z ( m ) ≤ W e^{\ell_z^{(m)}}\le W e ℓ z ( m ) ≤ W m m m z z z
W = exp { C ( 1 + ∣ ω ∣ + v ‾ 1 / 2 ∣ ξ ∣ + ∑ e , j sup 0 ≤ τ ≤ T ‾ ∣ B e , j ( τ ) ∣ ) } . W=\exp\left\{C\left(1+|\omega|+\overline{v}^{1/2}|\xi|+\sum_{e,j}\sup_{0\le\tau\le\overline{T}}|B_{e,j}(\tau)|\right)\right\}. W = exp { C ( 1 + ∣ ω ∣ + v 1/2 ∣ ξ ∣ + e , j ∑ 0 ≤ τ ≤ T sup ∣ B e , j ( τ ) ∣ ) } . and C C C W W W σ \sigma σ σ 0 \sigma_0 σ 0 ∣ ℓ z ∣ |\ell_z| ∣ ℓ z ∣
Z z ( m ) ≥ E r e f e ℓ z ( Π m G z , ω , σ 0 ) ≥ exp ( E r e f ℓ z ( Π m G z , ω , σ 0 ) ) ≥ e − C ′ , Z_z^{(m)}\ge\mathbb{E}_{\mathrm{ref}}e^{\ell_z(\Pi_mG_z,\omega,\sigma_0)}
\ge\exp\left(\mathbb{E}_{\mathrm{ref}}\ell_z(\Pi_mG_z,\omega,\sigma_0)\right)\ge e^{-C'}, Z z ( m ) ≥ E ref e ℓ z ( Π m G z , ω , σ 0 ) ≥ exp ( E ref ℓ z ( Π m G z , ω , σ 0 ) ) ≥ e − C ′ , where the expectations are over the Gaussian variables only, and C ′ C' C ′ m m m z z z
Step 6: passage to the limit in the interior. Let E m ( z ) = E z ( m ) Φ z ( m ) \mathcal{E}_m(z)=\mathbb{E}_z^{(m)}\Phi_z^{(m)} E m ( z ) = E z ( m ) Φ z ( m ) Ψ m ( z ) \Psi_m(z) Ψ m ( z ) [ z 1 , z 2 ] ⊂ I ∘ [z_1,z_2]\subset I^\circ [ z 1 , z 2 ] ⊂ I ∘
E m ( z 2 ) − E m ( z 1 ) = ∫ z 1 z 2 Ψ m ( z ) d z . \mathcal{E}_m(z_2)-\mathcal{E}_m(z_1)=\int_{z_1}^{z_2}\Psi_m(z)\,\mathrm{d}z. E m ( z 2 ) − E m ( z 1 ) = ∫ z 1 z 2 Ψ m ( z ) d z . By Steps 3–5 and dominated convergence, we see that E m ( z ) → E z Φ z \mathcal{E}_m(z)\to\mathbb{E}_z\Phi_z E m ( z ) → E z Φ z Ψ m ( z ) → Ψ ( z ) \Psi_m(z)\to\Psi(z) Ψ m ( z ) → Ψ ( z ) z ∈ I z\in I z ∈ I Ψ ( z ) \Psi(z) Ψ ( z ) (38) . Here we used the lower bound on the normalizers and the continuity of ℓ z \ell_z ℓ z Φ z \Phi_z Φ z ℓ ˙ z \dot{\ell}_z ℓ ˙ z Φ ˙ z \dot{\Phi}_z Φ ˙ z G G G Ψ m \Psi_m Ψ m m m m z z z z z z
E z 2 Φ z 2 − E z 1 Φ z 1 = ∫ z 1 z 2 Ψ ( z ) d z . \mathbb{E}_{z_2}\Phi_{z_2}-\mathbb{E}_{z_1}\Phi_{z_1}=\int_{z_1}^{z_2}\Psi(z)\,\mathrm{d}z. E z 2 Φ z 2 − E z 1 Φ z 1 = ∫ z 1 z 2 Ψ ( z ) d z . The same argument with z k → z z_k\to z z k → z m → ∞ m\to\infty m → ∞ Ψ \Psi Ψ z ↦ E z Φ z z\mapsto\mathbb{E}_z\Phi_z z ↦ E z Φ z I I I G z k → G z G_{z_k}\to G_z G z k → G z K ˙ z k → K ˙ z \dot{K}_{z_k}\to\dot{K}_z K ˙ z k → K ˙ z (38) holds on I ∘ I^\circ I ∘ Φ ≡ 1 \Phi\equiv1 Φ ≡ 1 log Z z ( m ) \log Z_z^{(m)} log Z z ( m ) (39) on I ∘ I^\circ I ∘
Step 7: endpoints. Let z − z_- z − I I I z − < z 1 < z 2 z_-<z_1<z_2 z − < z 1 < z 2 I ∘ I^\circ I ∘ z 1 ↓ z − z_1\downarrow z_- z 1 ↓ z − z ↦ E z Φ z z\mapsto\mathbb{E}_z\Phi_z z ↦ E z Φ z I I I Ψ \Psi Ψ I I I z − z_- z − K ˙ z \dot{K}_z K ˙ z E z 2 Φ z 2 − E z − Φ z − = ∫ z − z 2 Ψ ( z ) d z \mathbb{E}_{z_2}\Phi_{z_2}-\mathbb{E}_{z_-}\Phi_{z_-}=\int_{z_-}^{z_2}\Psi(z)\,\mathrm{d}z E z 2 Φ z 2 − E z − Φ z − = ∫ z − z 2 Ψ ( z ) d z Ψ \Psi Ψ z − z_- z − (39) in the same way.
Overlap estimates at a minimizer
Self-overlap moments on quantile intervals
In this section, we bound two moments of the self-overlap at a minimizing profile. The first is the integrated variance V = ∫ Var ( S q ) ν ( d q ) V = \int\operatorname{Var}(S_q)\,\nu(\mathrm{d}q) V = ∫ Var ( S q ) ν ( d q ) (29) . The second is the integrated cubic moment H 3 = ∫ E ∣ S q − q ∣ 3 ν ( d q ) H_3 = \int\mathbb{E}|S_q-q|^3\,\nu(\mathrm{d}q) H 3 = ∫ E ∣ S q − q ∣ 3 ν ( d q ) S q S_q S q L p L_p L p q q q (23) .
The section contains three results. In Lemma 5.1, we prove two comparisons: conditionally on G q \mathcal{G}_q G q S q S_q S q L p L_p L p p ≥ q p \ge q p ≥ q L p L_p L p p ≤ q p \le q p ≤ q Var ( S Q ( v ) ) \operatorname{Var}(S_{Q(v)}) Var ( S Q ( v ) ) Q Q Q H 3 H_3 H 3 z V zV z V V V V ( M s ) − 1 log ( 2 M ) log ( 2 / z ) (Ms)^{-1}\log(2M)\log(2/z) ( M s ) − 1 log ( 2 M ) log ( 2/ z )
We use the martingale and contact identities (20) and (22) of Proposition 3.1, the integrated variance bound (23) of Proposition 3.3, and the quantile function (19) . We use Proposition 5.2 in the proof of Proposition 13.1, which bounds the defect for small ρ \rho ρ
We refine the quantile-window argument of [4 Lemma 3.3] , which estimates integrals of Var ( L Q ( v ) ) \operatorname{Var}(L_{Q(v)}) Var ( L Q ( v ) ) Var ( L q ) = j q + Var ( S q ) / 4 \operatorname{Var}(L_q)=j_q+\operatorname{Var}(S_q)/4 Var ( L q ) = j q + Var ( S q ) /4 j q j_q j q N − 1 ( σ − m q ) ⋅ m q N^{-1}(\sigma-m_q)\cdot m_q N − 1 ( σ − m q ) ⋅ m q L q L_q L q Var ( L q ) \operatorname{Var}(L_q) Var ( L q ) [4 Lemma 3.3] it produces additional residual terms. Here we compare conditional expectations given G q \mathcal{G}_q G q j q j_q j q
Quantile windows and kernels
Standing assumptions. Throughout this section, we fix N ≥ 1 N \ge1 N ≥ 1 0 < ρ ≤ 1 0 < \rho\le1 0 < ρ ≤ 1 0 < s ≤ 1 / 2 0 < s \le1/2 0 < s ≤ 1/2 t = 1 − ρ ∈ [ 0 , 1 ) t=1-\rho\in[0,1) t = 1 − ρ ∈ [ 0 , 1 ) M = N c > 0 M=Nc>0 M = N c > 0 μ = s δ 0 + ν \mu=s\delta_0+\nu μ = s δ 0 + ν μ \mu μ F t , s F_{t,s} F t , s N N N β \beta β ρ \rho ρ s s s σ ∈ { − 1 , 1 } N \sigma\in\{-1,1\}^{N} σ ∈ { − 1 , 1 } N m q = E [ σ ∣ G q ] m_q = \mathbb{E}[\sigma\mid\mathcal{G}_q] m q = E [ σ ∣ G q ] ∣ m q ∣ 2 ≤ N |m_q|^2 \le N ∣ m q ∣ 2 ≤ N ∣ σ ⋅ m q ∣ ≤ N |\sigma\cdot m_q| \le N ∣ σ ⋅ m q ∣ ≤ N
0 ≤ S q ≤ 1 , ∣ L q ∣ ≤ 3 2 ( 0 ≤ q ≤ 1 ) . 0 \le S_q \le1, \qquad|L_q| \le\frac{3}{2} \qquad(0 \le q \le1). 0 ≤ S q ≤ 1 , ∣ L q ∣ ≤ 2 3 ( 0 ≤ q ≤ 1 ) . The observables S q S_q S q L q L_q L q q ↦ m q q \mapsto m_q q ↦ m q L 1 ( ν ) L^1(\nu) L 1 ( ν )
The variance bound. The estimates of this section rest on the first bound in (23) , and its factor a ( q ) − 1 a(q)^{-1} a ( q ) − 1 N − 2 ∫ k ( q ) 2 E Tr ( H q C q ) ν ( d q ) N^{-2}\int k(q)^2 \mathbb{E}\operatorname{Tr}(H_qC_q)\,\nu(\mathrm{d}q) N − 2 ∫ k ( q ) 2 E Tr ( H q C q ) ν ( d q ) [4 Proposition 2.2, (2.15)] and rests on the Brascamp–Lieb inequality for the posterior law of the Gaussian coordinates. The second is the inequality Tr ( H q C q ) ≤ a ( q ) − 1 Tr H q 2 \operatorname{Tr}(H_qC_q) \le a(q)^{-1}\operatorname{Tr}H_q^2 Tr ( H q C q ) ≤ a ( q ) − 1 Tr H q 2 H q − a ( q ) C q ⪰ 0 H_q-a(q)C_q\succeq0 H q − a ( q ) C q ⪰ 0 E Tr H q 2 ≤ N / c \mathbb{E}\operatorname{Tr}H_q^2\le N/c E Tr H q 2 ≤ N / c
Quantiles. Recall that the quantile function Q Q Q (19) pushes Lebesgue measure on ( 0 , 1 − s ) (0,1-s) ( 0 , 1 − s ) ν \nu ν F ( p ) = ν ( [ 0 , p ] ) F(p)=\nu([0,p]) F ( p ) = ν ([ 0 , p ]) F − ( p ) = ν ( [ 0 , p ) ) F_-(p)=\nu([0,p)) F − ( p ) = ν ([ 0 , p )) a ( p ) = s + F ( p ) a(p)=s+F(p) a ( p ) = s + F ( p ) (19) is attained because F F F F ( Q ( v ) ) ≥ v F(Q(v))\ge v F ( Q ( v )) ≥ v a ( Q ( v ) ) ≥ s + v a(Q(v))\ge s+v a ( Q ( v )) ≥ s + v
a ( q ) ≥ s ( 0 ≤ q ≤ 1 ) , Q ( v ) ∈ supp ν ( 0 < v < 1 − s ) . (42) a(q)\ge s \qquad(0\le q\le1), \qquad Q(v)\in\operatorname{supp}\nu\qquad(0<v<1-s). \tag*{(42)} a ( q ) ≥ s ( 0 ≤ q ≤ 1 ) , Q ( v ) ∈ supp ν ( 0 < v < 1 − s ) . (42) The first bound holds because μ ≥ s δ 0 \mu\ge s\delta_0 μ ≥ s δ 0 q ∗ = Q ( v ) q_* = Q(v) q ∗ = Q ( v ) ν ( [ 0 , q ∗ ] ) ≥ v > 0 \nu([0,q_*])\ge v>0 ν ([ 0 , q ∗ ]) ≥ v > 0 q ∗ = 0 q_*=0 q ∗ = 0 0 0 0 ν \nu ν supp ν \operatorname{supp}\nu supp ν q ∗ > 0 q_*>0 q ∗ > 0 ( q ∗ − r , q ∗ + r ) (q_*-r,q_*+r) ( q ∗ − r , q ∗ + r ) ν \nu ν 0 < r < q ∗ 0<r<q_* 0 < r < q ∗ ν ( [ 0 , q ∗ − r / 2 ] ) = ν ( [ 0 , q ∗ ] ) ≥ v \nu([0,q_*-r/2])=\nu([0,q_*])\ge v ν ([ 0 , q ∗ − r /2 ]) = ν ([ 0 , q ∗ ]) ≥ v Q ( v ) Q(v) Q ( v )
If μ \mu μ supp ν \operatorname{supp}\nu supp ν (42) and (22) , we then have E S Q ( v ) = Q ( v ) \mathbb{E}S_{Q(v)}=Q(v) E S Q ( v ) = Q ( v ) v ∈ ( 0 , 1 − s ) v\in(0,1-s) v ∈ ( 0 , 1 − s )
ξ v = S Q ( v ) − Q ( v ) ( 0 < v < 1 − s ) . \xi_v=S_{Q(v)}-Q(v) \qquad(0<v<1-s). ξ v = S Q ( v ) − Q ( v ) ( 0 < v < 1 − s ) . Note that ∣ ξ v ∣ ≤ 1 |\xi_v|\le1 ∣ ξ v ∣ ≤ 1 E ξ v = 0 \mathbb{E}\xi_v=0 E ξ v = 0 E ξ v 2 = Var ( S Q ( v ) ) \mathbb{E}\xi_v^2=\operatorname{Var}(S_{Q(v)}) E ξ v 2 = Var ( S Q ( v ) ) Q Q Q ν \nu ν
V = ∫ 0 1 − s E ξ v 2 d v , H 3 = ∫ 0 1 − s E ∣ ξ v ∣ 3 d v . V=\int_0^{1-s}\mathbb{E}\xi_v^2\,\mathrm{d}v,\qquad H_3=\int_0^{1-s}\mathbb{E}|\xi_v|^3\,\mathrm{d}v. V = ∫ 0 1 − s E ξ v 2 d v , H 3 = ∫ 0 1 − s E ∣ ξ v ∣ 3 d v . The integrands are Borel functions of v v v Q Q Q
Windows. Let I ⊂ ( 0 , 1 − s ) I \subset(0,1-s) I ⊂ ( 0 , 1 − s ) ∣ I ∣ > 0 |I|>0 ∣ I ∣ > 0 I I I Q Q Q ( 0 , 1 − s ) (0,1-s) ( 0 , 1 − s ) ν \nu ν [4 Section 3] , it has a density f I f_I f I ν \nu ν
0 ≤ f I ≤ 1 , ∫ f I d ν = ∣ I ∣ . 0 \le f_I \le1,\qquad\int f_I\,\mathrm{d}\nu=|I|. 0 ≤ f I ≤ 1 , ∫ f I d ν = ∣ I ∣. The density is explicit at an atom. If p p p ν \nu ν r p = ν ( { p } ) r_p=\nu(\{p\}) r p = ν ({ p }) { v : Q ( v ) = p } \{v:Q(v)=p\} { v : Q ( v ) = p } F − ( p ) F_-(p) F − ( p ) F ( p ) F(p) F ( p )
f I ( p ) = ∣ I ∩ ( F − ( p ) , F ( p ) ) ∣ r p . f_I(p)=\frac{\left|I\cap(F_-(p),F(p))\right|}{r_p}. f I ( p ) = r p ∣ I ∩ ( F − ( p ) , F ( p )) ∣ . In particular, a window may contain only part of the quantile interval of an atom; the atom then receives the corresponding fraction of its mass. We define the normalized density κ I = f I / ∣ I ∣ \kappa_I=f_I/|I| κ I = f I /∣ I ∣
0 ≤ κ I ≤ ∣ I ∣ − 1 , ∫ κ I d ν = 1 , ∫ φ κ I d ν = 1 ∣ I ∣ ∫ I φ ( Q ( v ) ) d v 0\le\kappa_I\le|I|^{-1},\qquad\int\kappa_I\,\mathrm{d}\nu=1,\qquad\int\varphi\kappa_I\,\mathrm{d}\nu=\frac{1}{|I|}\int_I\varphi(Q(v))\,\mathrm{d}v 0 ≤ κ I ≤ ∣ I ∣ − 1 , ∫ κ I d ν = 1 , ∫ φ κ I d ν = ∣ I ∣ 1 ∫ I φ ( Q ( v )) d v for every bounded Borel function φ \varphi φ I I I threshold u > 0 u>0 u > 0 a ( Q ( v ) ) ≥ u a(Q(v))\ge u a ( Q ( v )) ≥ u v ∈ I v\in I v ∈ I s s s
Kernels. More generally, we call a deterministic Borel function κ ≥ 0 \kappa\ge0 κ ≥ 0 [ 0 , 1 ] [0,1] [ 0 , 1 ] ∫ κ d ν = 1 \int\kappa\,\mathrm{d}\nu=1 ∫ κ d ν = 1 ∫ κ 2 d ν < ∞ \int\kappa^2\,\mathrm{d}\nu<\infty ∫ κ 2 d ν < ∞ kernel . For a kernel κ \kappa κ L p − p / 2 L_p-p/2 L p − p /2 μ \mu μ
p ˉ κ = ∫ p κ ( p ) ν ( d p ) , Λ κ = ∫ κ ( p ) ( L p − p 2 ) ν ( d p ) . \bar p_\kappa=\int p\kappa(p)\,\nu(\mathrm{d}p),\qquad\Lambda_\kappa=\int\kappa(p)\left(L_p-\frac{p}{2}\right)\nu(\mathrm{d}p). p ˉ κ = ∫ p κ ( p ) ν ( d p ) , Λ κ = ∫ κ ( p ) ( L p − 2 p ) ν ( d p ) . We write p ˉ I \bar p_I p ˉ I Λ I \Lambda_I Λ I κ = κ I \kappa=\kappa_I κ = κ I
p ˉ I = 1 ∣ I ∣ ∫ I Q ( v ) d v , Λ I = 1 ∣ I ∣ ∫ I ( L Q ( v ) − Q ( v ) 2 ) d v . \bar p_I=\frac{1}{|I|}\int_I Q(v)\,\mathrm{d}v,\qquad\Lambda_I=\frac{1}{|I|}\int_I\left(L_{Q(v)}-\frac{Q(v)}{2}\right)\,\mathrm{d}v. p ˉ I = ∣ I ∣ 1 ∫ I Q ( v ) d v , Λ I = ∣ I ∣ 1 ∫ I ( L Q ( v ) − 2 Q ( v ) ) d v . Fixed kernels. Each assertion of Lemma 5.1 below concerns one level q q q κ \kappa κ v v v v v v v v v v v v
Two-sided conditional comparison
For a later level p ≥ q p\ge q p ≥ q S q S_q S q 2 E [ L p ∣ G q ] 2\mathbb{E}[L_p\mid\mathcal{G}_q] 2 E [ L p ∣ G q ] p ≤ q p\le q p ≤ q L p L_p L p G q \mathcal{G}_q G q S q / 2 S_q/2 S q /2 ∣ m q − m p ∣ 2 / ( 2 N ) |m_q-m_p|^2/(2N) ∣ m q − m p ∣ 2 / ( 2 N ) p p p S q − q S_q-q S q − q Λ κ \Lambda_\kappa Λ κ G q \mathcal{G}_q G q Λ κ \Lambda_\kappa Λ κ
Lemma 5.1 (Two-sided conditional comparison). Let μ \mu μ q ∈ [ 0 , 1 ] q \in[0,1] q ∈ [ 0 , 1 ] κ \kappa κ
(a) If κ = 0 \kappa=0 κ = 0 ν \nu ν [ 0 , q ) [0,q) [ 0 , q )
S q − q ≤ ( p ˉ κ − q ) + 2 E [ Λ κ ∣ G q ] . S_q-q \le\left(\bar{p}_\kappa-q\right)+2\mathbb{E}[\Lambda_\kappa\mid\mathcal{G}_q]. S q − q ≤ ( p ˉ κ − q ) + 2 E [ Λ κ ∣ G q ] . (b) If κ = 0 \kappa=0 κ = 0 ν \nu ν ( q , 1 ] (q,1] ( q , 1 ]
S q − q = ( p ˉ κ − q ) + 2 E [ Λ κ ∣ G q ] + 1 N ∫ κ ( p ) ∣ m q − m p ∣ 2 ν ( d p ) ≥ ( p ˉ κ − q ) + 2 E [ Λ κ ∣ G q ] . \begin{aligned}
S_q-q
&=\left(\bar{p}_\kappa-q\right)+2\mathbb{E}[\Lambda_\kappa\mid\mathcal{G}_q]+\frac{1}{N}\int\kappa(p)|m_q-m_p|^2\,\nu(\mathrm{d}p)\\
&\ge\left(\bar{p}_\kappa-q\right)+2\mathbb{E}[\Lambda_\kappa\mid\mathcal{G}_q].
\end{aligned} S q − q = ( p ˉ κ − q ) + 2 E [ Λ κ ∣ G q ] + N 1 ∫ κ ( p ) ∣ m q − m p ∣ 2 ν ( d p ) ≥ ( p ˉ κ − q ) + 2 E [ Λ κ ∣ G q ] . If μ \mu μ F t , s F_{t,s} F t , s
(c) We have E Λ κ = 0 \mathbb{E}\Lambda_\kappa=0 E Λ κ = 0
E [ ( E [ Λ κ ∣ G q ] ) 2 ] ≤ Var ( Λ κ ) ≤ 1 M ∫ κ ( p ) 2 a ( p ) ν ( d p ) . \mathbb{E}\left[\left(\mathbb{E}[\Lambda_\kappa\mid\mathcal{G}_q]\right)^2\right]\le\operatorname{Var}(\Lambda_\kappa)\le\frac{1}{M}\int\frac{\kappa(p)^2}{a(p)}\,\nu(\mathrm{d}p). E [ ( E [ Λ κ ∣ G q ] ) 2 ] ≤ Var ( Λ κ ) ≤ M 1 ∫ a ( p ) κ ( p ) 2 ν ( d p ) . (d) Let ϑ > 0 \vartheta>0 ϑ > 0 κ \kappa κ p ˉ κ − q ≤ ϑ / 2 \bar{p}_\kappa-q\le\vartheta/2 p ˉ κ − q ≤ ϑ /2
E [ ( S q − q ) 2 1 { S q − q > ϑ } ] ≤ 16 Var ( Λ κ ) . \mathbb{E}\left[(S_q-q)^2\mathbf{1}_{\{S_q-q>\vartheta\}}\right]\le16\operatorname{Var}(\Lambda_\kappa). E [ ( S q − q ) 2 1 { S q − q > ϑ } ] ≤ 16 Var ( Λ κ ) . If κ \kappa κ q − p ˉ κ ≤ ϑ / 2 q-\bar{p}_\kappa\le\vartheta/2 q − p ˉ κ ≤ ϑ /2
E [ ( S q − q ) 2 1 { S q − q < − ϑ } ] ≤ 16 Var ( Λ κ ) . \mathbb{E}\left[(S_q-q)^2\mathbf{1}_{\{S_q-q<-\vartheta\}}\right]\le16\operatorname{Var}(\Lambda_\kappa). E [ ( S q − q ) 2 1 { S q − q < − ϑ } ] ≤ 16 Var ( Λ κ ) . (e) If I ⊂ ( 0 , 1 − s ) I\subset(0,1-s) I ⊂ ( 0 , 1 − s ) x > 0 x>0 x > 0 u u u Var ( Λ I ) ≤ 1 / ( M u x ) \operatorname{Var}(\Lambda_I)\le1/(Mux) Var ( Λ I ) ≤ 1/ ( M ux )
For every admissible profile and every interval I ⊂ ( 0 , 1 − s ) I\subset(0,1-s) I ⊂ ( 0 , 1 − s ) κ I \kappa_I κ I Q ≥ q Q\ge q Q ≥ q I I I Q ≤ q Q\le q Q ≤ q I I I
Proof. Parts (a) and (b) are statements about conditional expectations. We prove each first for a single level p p p κ ( p ) ν ( d p ) \kappa(p)\,\nu(\mathrm{d}p) κ ( p ) ν ( d p )
Λ ′ = ∫ κ ( p ) L p ν ( d p ) = Λ κ + p ˉ κ 2 . \Lambda'=\int\kappa(p)L_p\,\nu(\mathrm{d}p)=\Lambda_\kappa+\frac{\bar{p}_\kappa}{2}. Λ ′ = ∫ κ ( p ) L p ν ( d p ) = Λ κ + 2 p ˉ κ . Taking the conditional expectation given G p \mathcal{G}_p G p L p L_p L p m p = E [ σ ∣ G p ] m_p=\mathbb{E}[\sigma\mid\mathcal{G}_p] m p = E [ σ ∣ G p ] G p \mathcal{G}_p G p
E [ L p ∣ G p ] = 1 2 S p ( 0 ≤ p ≤ 1 ) . (43) \mathbb{E}[L_p\mid\mathcal{G}_p]=\frac{1}{2}S_p\qquad(0\le p\le1). \tag*{(43)} E [ L p ∣ G p ] = 2 1 S p ( 0 ≤ p ≤ 1 ) . (43) Part (a). Let p ≥ q p \ge q p ≥ q m m m (20) , and m q = E [ m p ∣ G q ] m_q = \mathbb{E}[m_p \mid\mathcal{G}_q] m q = E [ m p ∣ G q ] x ↦ ∣ x ∣ 2 x \mapsto|x|^2 x ↦ ∣ x ∣ 2 (43) , we have
S q ≤ E [ S p ∣ G q ] = 2 E [ L p ∣ G q ] . S_q \le\mathbb{E}[S_p \mid\mathcal{G}_q] = 2\mathbb{E}[L_p \mid\mathcal{G}_q]. S q ≤ E [ S p ∣ G q ] = 2 E [ L p ∣ G q ] . Let ζ ≥ 0 \zeta\ge0 ζ ≥ 0 G q \mathcal{G}_q G q E [ ζ S q ] ≤ 2 E [ ζ L p ] \mathbb{E}[\zeta S_q] \le2\mathbb{E}[\zeta L_p] E [ ζ S q ] ≤ 2 E [ ζ L p ] p ≥ q p \ge q p ≥ q κ ( p ) ν ( d p ) \kappa(p)\,\nu(dp) κ ( p ) ν ( d p ) [ 0 , q ) [0,q) [ 0 , q ) ∣ ζ L p ∣ |\zeta L_p| ∣ ζ L p ∣ E [ ζ S q ] ≤ 2 E [ ζ Λ ′ ] \mathbb{E}[\zeta S_q] \le2\mathbb{E}[\zeta\Lambda^\prime] E [ ζ S q ] ≤ 2 E [ ζ Λ ′ ] ζ \zeta ζ S q S_q S q G q \mathcal{G}_q G q S q ≤ 2 E [ Λ ′ ∣ G q ] S_q \le2\mathbb{E}[\Lambda^\prime\mid\mathcal{G}_q] S q ≤ 2 E [ Λ ′ ∣ G q ] q q q Λ ′ = Λ κ + p ˉ κ / 2 \Lambda^\prime= \Lambda_\kappa+ \bar p_\kappa/2 Λ ′ = Λ κ + p ˉ κ /2
Part (b). Let p ≤ q p \le q p ≤ q m p m_p m p m q m_q m q G q \mathcal{G}_q G q E [ σ ∣ G q ] = m q \mathbb{E}[\sigma\mid\mathcal{G}_q] = m_q E [ σ ∣ G q ] = m q ∣ m q − m p ∣ 2 |m_q-m_p|^2 ∣ m q − m p ∣ 2
E [ L p ∣ G q ] = m q ⋅ m p − ∣ m p ∣ 2 / 2 N = S q 2 − ∣ m q − m p ∣ 2 2 N . \mathbb{E}[L_p \mid\mathcal{G}_q] = \frac{m_q \cdot m_p - |m_p|^2/2}{N} = \frac{S_q}{2} - \frac{|m_q-m_p|^2}{2N}. E [ L p ∣ G q ] = N m q ⋅ m p − ∣ m p ∣ 2 /2 = 2 S q − 2 N ∣ m q − m p ∣ 2 . The right side is bounded and jointly measurable in p ∈ [ 0 , q ] p \in[0,q] p ∈ [ 0 , q ] κ ( p ) ν ( d p ) \kappa(p)\,\nu(dp) κ ( p ) ν ( d p ) [ 0 , q ] [0,q] [ 0 , q ] G q \mathcal{G}_q G q m m m G q \mathcal{G}_q G q
E [ Λ ′ ∣ G q ] = S q 2 − 1 2 N ∫ κ ( p ) ∣ m q − m p ∣ 2 ν ( d p ) . \mathbb{E}[\Lambda^\prime\mid\mathcal{G}_q] = \frac{S_q}{2} - \frac{1}{2N}\int\kappa(p)|m_q-m_p|^2\,\nu(dp). E [ Λ ′ ∣ G q ] = 2 S q − 2 N 1 ∫ κ ( p ) ∣ m q − m p ∣ 2 ν ( d p ) . Multiplying by two, subtracting q q q Λ ′ = Λ κ + p ˉ κ / 2 \Lambda^\prime= \Lambda_\kappa+ \bar p_\kappa/2 Λ ′ = Λ κ + p ˉ κ /2
Part (c). Let μ \mu μ (43) and (22) , we have E L p = E S p / 2 = p / 2 \mathbb{E}L_p = \mathbb{E}S_p/2 = p/2 E L p = E S p /2 = p /2 p ∈ supp ν p \in\operatorname{supp}\nu p ∈ supp ν ν \nu ν p p p E Λ κ = 0 \mathbb{E}\Lambda_\kappa= 0 E Λ κ = 0
E [ ( E [ Λ κ ∣ G q ] ) 2 ] ≤ E Λ κ 2 = Var ( Λ κ ) . \mathbb{E}\left[\left(\mathbb{E}[\Lambda_\kappa\mid\mathcal{G}_q]\right)^2\right] \le\mathbb{E}\Lambda_\kappa^2 = \operatorname{Var}(\Lambda_\kappa). E [ ( E [ Λ κ ∣ G q ] ) 2 ] ≤ E Λ κ 2 = Var ( Λ κ ) . As Λ κ \Lambda_\kappa Λ κ Λ ′ \Lambda^\prime Λ ′ (23) with k = κ k=\kappa k = κ
Part (d). Let ξ = S q − q \xi= S_q-q ξ = S q − q Y = E [ Λ κ ∣ G q ] Y = \mathbb{E}[\Lambda_\kappa\mid\mathcal{G}_q] Y = E [ Λ κ ∣ G q ] κ \kappa κ p ˉ κ − q ≤ ϑ / 2 \bar p_\kappa-q \le\vartheta/2 p ˉ κ − q ≤ ϑ /2 { ξ > ϑ } \{\xi>\vartheta\} { ξ > ϑ }
2 Y ≥ ξ − ( p ˉ κ − q ) ≥ ξ − ϑ 2 > ξ 2 > 0 , 2Y \ge\xi-(\bar p_\kappa-q) \ge\xi-\frac{\vartheta}{2} > \frac{\xi}{2} > 0, 2 Y ≥ ξ − ( p ˉ κ − q ) ≥ ξ − 2 ϑ > 2 ξ > 0 , so that ξ 2 < 16 Y 2 \xi^2 < 16Y^2 ξ 2 < 16 Y 2 κ \kappa κ q − p ˉ κ ≤ ϑ / 2 q-\bar p_\kappa\le\vartheta/2 q − p ˉ κ ≤ ϑ /2 { ξ < − ϑ } \{\xi<-\vartheta\} { ξ < − ϑ }
2 Y ≤ ξ + ( q − p ˉ κ ) ≤ ξ + ϑ 2 < ξ 2 < 0 , 2Y \le\xi+(q-\bar p_\kappa) \le\xi+\frac{\vartheta}{2} < \frac{\xi}{2} < 0, 2 Y ≤ ξ + ( q − p ˉ κ ) ≤ ξ + 2 ϑ < 2 ξ < 0 , and again ζ 2 < 16 Y 2 \zeta^2 < 16Y^2 ζ 2 < 16 Y 2 16 E Y 2 16\mathbb{E}Y^2 16 E Y 2 16 Var ( Λ κ ) 16\operatorname{Var}(\Lambda_\kappa) 16 Var ( Λ κ )
Part (e). The bound κ I ≤ 1 / x \kappa_I \le1/x κ I ≤ 1/ x κ I 2 ≤ κ I / x \kappa_I^2 \le\kappa_I/x κ I 2 ≤ κ I / x 1 / a 1/a 1/ a 1 / s 1/s 1/ s (42) . Using the identity for ∫ φ κ I d ν \int\varphi\kappa_I\,\mathrm{d}\nu ∫ φ κ I d ν φ = 1 / a \varphi= 1/a φ = 1/ a
∫ κ I ( p ) 2 a ( p ) ν ( d p ) ≤ 1 x ∫ κ I ( p ) a ( p ) ν ( d p ) = 1 x 2 ∫ I d v a ( Q ( v ) ) ≤ 1 u x . \int\frac{\kappa_I(p)^2}{a(p)}\,\nu(\mathrm{d}p) \le\frac{1}{x}\int\frac{\kappa_I(p)}{a(p)}\,\nu(\mathrm{d}p) = \frac{1}{x^2}\int_I \frac{\mathrm{d}v}{a(Q(v))} \le\frac{1}{ux}. ∫ a ( p ) κ I ( p ) 2 ν ( d p ) ≤ x 1 ∫ a ( p ) κ I ( p ) ν ( d p ) = x 2 1 ∫ I a ( Q ( v )) d v ≤ ux 1 . where we used the threshold in the last step. Inserting this bound into (c), we find that Var ( Λ I ) ≤ 1 / ( M u x ) \operatorname{Var}(\Lambda_I) \le1/(Mux) Var ( Λ I ) ≤ 1/ ( M ux ) [4 Section 3, (3.6)] , because Λ I \Lambda_I Λ I x − 1 ∫ I L Q ( v ) d v x^{-1}\int_I L_Q(v)\,\mathrm{d}v x − 1 ∫ I L Q ( v ) d v p ˉ I / 2 \bar p_I/2 p ˉ I /2
The hypotheses for κ I \kappa_I κ I Suppose that Q ≥ q Q \ge q Q ≥ q I I I
∫ 1 [ 0 , q ) κ I d ν = 1 ∣ I ∣ ∣ { v ∈ I : Q ( v ) < q } ∣ = 0 , \int\mathbf{1}_{[0,q)}\kappa_I\,\mathrm{d}\nu= \frac{1}{|I|}\left|\{v \in I: Q(v) < q\}\right| = 0, ∫ 1 [ 0 , q ) κ I d ν = ∣ I ∣ 1 ∣ { v ∈ I : Q ( v ) < q } ∣ = 0 , so κ I \kappa_I κ I Q ≤ q Q \le q Q ≤ q ( q , 1 ] (q,1] ( q , 1 ] [ 0 , q ) [0,q) [ 0 , q )
Variance on a quantile interval
We now use the comparisons of Lemma 5.1 to bound the integral of Var ( S Q ( v ) ) \operatorname{Var}(S_{Q(v)}) Var ( S Q ( v ) ) v v v r r r v v v ( M u r ) − 1 (Mur)^{-1} ( M u r ) − 1 r r r Q Q Q ( v − r , v + r ) (v-r,v+r) ( v − r , v + r ) Q Q Q R \mathbb{R} R 2 r 2r 2 r Q Q Q I ⊂ ( 0 , 1 − s ) I \subset(0,1-s) I ⊂ ( 0 , 1 − s )
ℓ ( I ) = ess sup v , w ∈ I ∣ Q ( v ) − Q ( w ) ∣ \ell(I) = \operatorname*{ess\,sup}_{v,w\in I}|Q(v)-Q(w)| ℓ ( I ) = v , w ∈ I ess sup ∣ Q ( v ) − Q ( w ) ∣ denote the diameter of its image.
Proposition 5.2 (Variance on a quantile interval). Let μ \mu μ F t , s F_{t,s} F t , s I ⊂ ( 0 , 1 − s ) I \subset(0,1-s) I ⊂ ( 0 , 1 − s ) h > 0 h > 0 h > 0 u > 0 u > 0 u > 0 ℓ = ℓ ( I ) \ell= \ell(I) ℓ = ℓ ( I ) ϵ = ( M u ) − 1 \epsilon= (Mu)^{-1} ϵ = ( M u ) − 1
∫ I Var ( S Q ( v ) ) d v ≤ 128 [ min { h , ϵ ( 1 + log + ( h / ϵ ) ) } + ( ϵ ℓ ) 2 / 3 h 1 / 3 ] . (44) \int_I \operatorname{Var}(S_{Q(v)})\,\mathrm{d}v \le128\left[\min\{h,\epsilon(1+\log_+(h/\epsilon))\}+(\epsilon\ell)^{2/3}h^{1/3}\right]. \tag*{(44)} ∫ I Var ( S Q ( v ) ) d v ≤ 128 [ min { h , ϵ ( 1 + log + ( h / ϵ ))} + ( ϵ ℓ ) 2/3 h 1/3 ] . (44) The constant is absolute, and I I I ν \nu ν
Proof. The proof has four steps. First, we record an identity for the increments of Q Q Q Var ( S Q ( v ) ) \operatorname{Var}(S_{Q(v)}) Var ( S Q ( v ) ) v v v r r r v v v τ ∈ ( 0 , 1 ) \tau\in(0,1) τ ∈ ( 0 , 1 ) r r r τ \tau τ τ \tau τ τ \tau τ
Step 1: increments of the quantile function. Discarding endpoints, which changes no integral below, we write I = ( v ‾ , v ‾ ) I=(\underline{v},\overline{v}) I = ( v , v ) h = v ‾ − v ‾ h=\overline{v}-\underline{v} h = v − v
χ ( v ) = E ξ v 2 = Var ( S Q ( v ) ) ∈ [ 0 , 1 ] ( v ∈ I ) . \chi(v)=\mathbb{E}\xi_v^2=\operatorname{Var}(S_{Q(v)})\in[0,1]\qquad(v\in I). χ ( v ) = E ξ v 2 = Var ( S Q ( v ) ) ∈ [ 0 , 1 ] ( v ∈ I ) . As Q Q Q I I I q − = lim v ↓ v ‾ Q ( v ) q_-=\lim_{v\downarrow\underline{v}}Q(v) q − = lim v ↓ v Q ( v ) q + = lim v ↑ v ‾ Q ( v ) q_+=\lim_{v\uparrow\overline{v}}Q(v) q + = lim v ↑ v Q ( v ) q + − q − = ℓ q_+-q_-=\ell q + − q − = ℓ Q Q Q I I I Q ~ \widetilde{Q} Q R \mathbb{R} R Q ~ = q − \widetilde{Q}=q_- Q = q − ( − ∞ , v ‾ ] (-\infty,\underline{v}] ( − ∞ , v ] Q ~ = q + \widetilde{Q}=q_+ Q = q + [ v ‾ , ∞ ) [\overline{v},\infty) [ v , ∞ ) d Q ~ \mathrm{d}\widetilde{Q} d Q Q ~ \widetilde{Q} Q q + − q − = ℓ q_+-q_-=\ell q + − q − = ℓ
∫ R ( Q ~ ( v + r ) − Q ~ ( v − r ) ) d v = 2 r ℓ ( r > 0 ) . (45) \int_{\mathbb{R}}\bigl(\widetilde{Q}(v+r)-\widetilde{Q}(v-r)\bigr)\,\mathrm{d}v=2r\ell\qquad(r>0). \tag*{(45)} ∫ R ( Q ( v + r ) − Q ( v − r ) ) d v = 2 r ℓ ( r > 0 ) . (45) Indeed, Q ~ \widetilde{Q} Q r > 0 r>0 r > 0 v v v r r r − r -r − r v − r v-r v − r v + r v+r v + r Q ~ ( v + r ) − Q ~ ( v − r ) \widetilde{Q}(v+r)-\widetilde{Q}(v-r) Q ( v + r ) − Q ( v − r ) d Q ~ \mathrm{d}\widetilde{Q} d Q [ v − r , v + r ] [v-r,v+r] [ v − r , v + r ]
∫ R d Q ~ ( [ v − r , v + r ] ) d v = ∫ R ∫ R 1 { ∣ x − v ∣ ≤ r } d v d Q ~ ( x ) = 2 r ℓ . \int_{\mathbb{R}}\mathrm{d}\widetilde{Q}([v-r,v+r])\,\mathrm{d}v
=\int_{\mathbb{R}}\int_{\mathbb{R}}\mathbf{1}_{\{|x-v|\le r\}}\,\mathrm{d}v\,\mathrm{d}\widetilde{Q}(x)=2r\ell. ∫ R d Q ([ v − r , v + r ]) d v = ∫ R ∫ R 1 { ∣ x − v ∣ ≤ r } d v d Q ( x ) = 2 r ℓ . In words, each unit of mass of d Q ~ \mathrm{d}\widetilde{Q} d Q v v v r r r 2 r 2r 2 r Q ~ \widetilde{Q} Q Q Q Q
Step 2: the two windows at a fixed radius. Let 0 < r < h / 2 0<r<h/2 0 < r < h /2 v ∈ ( v ‾ + r , v ‾ − r ) v\in(\underline{v}+r,\overline{v}-r) v ∈ ( v + r , v − r ) I − = ( v − r , v ) I_-=(v-r,v) I − = ( v − r , v ) I + = ( v , v + r ) I_+=(v,v+r) I + = ( v , v + r ) I I I r r r u u u
Δ r ( v ) = Q ( v + r ) − Q ( v − r ) ≥ 0 , Y ± = E [ Λ I ± ∣ G Q ( v ) ] . \Delta_r(v)=Q(v+r)-Q(v-r)\ge0,\qquad Y_\pm=\mathbb{E}[\Lambda_{I_\pm}\mid\mathcal{G}_{Q(v)}]. Δ r ( v ) = Q ( v + r ) − Q ( v − r ) ≥ 0 , Y ± = E [ Λ I ± ∣ G Q ( v ) ] . Note that Q ≥ Q ( v ) Q\ge Q(v) Q ≥ Q ( v ) I + I_+ I + Q ≤ Q ( v ) Q\le Q(v) Q ≤ Q ( v ) I − I_- I − Q Q Q κ I + \kappa_{I_+} κ I + κ I − \kappa_{I_-} κ I − q = Q ( v ) q=Q(v) q = Q ( v )
0 ≤ p ˉ I + − Q ( v ) ≤ Q ( v + r ) − Q ( v ) ≤ Δ r ( v ) , 0\le\bar{p}_{I_+}-Q(v)\le Q(v+r)-Q(v)\le\Delta_r(v), 0 ≤ p ˉ I + − Q ( v ) ≤ Q ( v + r ) − Q ( v ) ≤ Δ r ( v ) , 0 ≤ Q ( v ) − p ˉ I − ≤ Q ( v ) − Q ( v − r ) ≤ Δ r ( v ) . 0\le Q(v)-\bar{p}_{I_-}\le Q(v)-Q(v-r)\le\Delta_r(v). 0 ≤ Q ( v ) − p ˉ I − ≤ Q ( v ) − Q ( v − r ) ≤ Δ r ( v ) . Then Lemma 5.1(a) and (b) imply that
ξ v ≤ Δ r ( v ) + 2 Y + , − ξ v ≤ Δ r ( v ) − 2 Y − . \xi_v \le\Delta_r(v) + 2Y_+,\qquad-\xi_v \le\Delta_r(v) - 2Y_-. ξ v ≤ Δ r ( v ) + 2 Y + , − ξ v ≤ Δ r ( v ) − 2 Y − . On the event { ξ v > 0 } \{\xi_v > 0\} { ξ v > 0 } ξ v 2 ≤ ( Δ r ( v ) + 2 ∣ Y + ∣ ) 2 \xi_v^2 \le(\Delta_r(v) + 2|Y_+|)^2 ξ v 2 ≤ ( Δ r ( v ) + 2∣ Y + ∣ ) 2 2 Δ r ( v ) 2 + 8 Y + 2 2\Delta_r(v)^2 + 8Y_+^2 2 Δ r ( v ) 2 + 8 Y + 2 ( x + y ) 2 ≤ 2 x 2 + 2 y 2 (x+y)^2 \le2x^2 + 2y^2 ( x + y ) 2 ≤ 2 x 2 + 2 y 2 { ξ v < 0 } \{\xi_v < 0\} { ξ v < 0 } ξ v 2 ≤ 2 Δ r ( v ) 2 + 8 Y − 2 \xi_v^2 \le2\Delta_r(v)^2 + 8Y_-^2 ξ v 2 ≤ 2 Δ r ( v ) 2 + 8 Y − 2
ξ v 2 ≤ 2 Δ r ( v ) 2 + 8 Y + 2 + 8 Y − 2 . \xi_v^2 \le2\Delta_r(v)^2 + 8Y_+^2 + 8Y_-^2. ξ v 2 ≤ 2 Δ r ( v ) 2 + 8 Y + 2 + 8 Y − 2 . Lemma 5.1(c) and (e) imply that E Y ± 2 ≤ Var ( Λ I ± ) ≤ ( M u r ) − 1 = ϵ / r \mathbb{E}Y_\pm^2 \le\operatorname{Var}(\Lambda_{I_\pm}) \le(Mur)^{-1} = \epsilon/r E Y ± 2 ≤ Var ( Λ I ± ) ≤ ( M u r ) − 1 = ϵ / r
χ ( v ) ≤ 2 Δ r ( v ) 2 + 16 ϵ r for every v ∈ ( v ‾ + r , v ‾ − r ) . (46) \chi(v) \le2\Delta_r(v)^2 + \frac{16\epsilon}{r}\qquad\text{for every }v \in(\underline{v}+r,\overline{v}-r). \tag*{(46)} χ ( v ) ≤ 2 Δ r ( v ) 2 + r 16 ϵ for every v ∈ ( v + r , v − r ) . (46) Note that v ‾ \underline{v} v r r r Q ( v ) Q(v) Q ( v )
Step 3: the measure of the set where the variance is large. Fix τ ∈ ( 0 , 1 ) \tau\in(0,1) τ ∈ ( 0 , 1 ) r = 32 ϵ / τ r = 32\epsilon/\tau r = 32 ϵ / τ 16 ϵ / r = τ / 2 16\epsilon/r = \tau/2 16 ϵ / r = τ /2 r < h / 2 r < h/2 r < h /2 v ∈ ( v ‾ + r , v ‾ − r ) v \in(\underline{v}+r,\overline{v}-r) v ∈ ( v + r , v − r ) χ ( v ) > τ \chi(v) > \tau χ ( v ) > τ 2 Δ r ( v ) 2 > τ / 2 2\Delta_r(v)^2 > \tau/2 2 Δ r ( v ) 2 > τ /2 Δ r ( v ) > τ / 2 \Delta_r(v) > \sqrt{\tau}/2 Δ r ( v ) > τ /2 Δ r ( v ) = Q ~ ( v ‾ + r ) − Q ~ ( v ‾ − r ) \Delta_r(v) = \widetilde{Q}(\underline{v}+r) - \widetilde{Q}(\overline{v}-r) Δ r ( v ) = Q ( v + r ) − Q ( v − r ) χ ( v ) > τ \chi(v) > \tau χ ( v ) > τ
2 τ ∫ R ( Q ~ ( v + r ) − Q ~ ( v − r ) ) d v = 4 r ℓ τ = 128 ϵ ℓ τ 3 / 2 . \frac{2}{\sqrt{\tau}}\int_{\mathbb{R}}\bigl(\widetilde{Q}(v+r)-\widetilde{Q}(v-r)\bigr)\,\mathrm{d}v
=\frac{4r\ell}{\sqrt{\tau}}=\frac{128\epsilon\ell}{\tau^{3/2}}. τ 2 ∫ R ( Q ( v + r ) − Q ( v − r ) ) d v = τ 4 r ℓ = τ 3/2 128 ϵ ℓ . The two boundary strips ( v ‾ , v ‾ + r ] (\underline{v},\underline{v}+r] ( v , v + r ] [ v ‾ − r , v ‾ ) [\overline{v}-r,\overline{v}) [ v − r , v ) 2 r = 64 ϵ / τ 2r = 64\epsilon/\tau 2 r = 64 ϵ / τ r ≥ h / 2 r \ge h/2 r ≥ h /2 h ≤ 2 r = 64 ϵ / τ h \le2r = 64\epsilon/\tau h ≤ 2 r = 64 ϵ / τ 2 r 2r 2 r h h h
∣ { v ∈ I : χ ( v ) > τ } ∣ ≤ min { h , 64 ϵ τ + 128 ϵ ℓ τ 3 / 2 } ≤ min { h , 64 ϵ τ } + min { h , 128 ϵ ℓ τ 3 / 2 } . |\{v\in I:\chi(v)>\tau\}|
\le\min\left\{h,\frac{64\epsilon}{\tau}+\frac{128\epsilon\ell}{\tau^{3/2}}\right\}
\le\min\left\{h,\frac{64\epsilon}{\tau}\right\}
+\min\left\{h,\frac{128\epsilon\ell}{\tau^{3/2}}\right\}. ∣ { v ∈ I : χ ( v ) > τ } ∣ ≤ min { h , τ 64 ϵ + τ 3/2 128 ϵ ℓ } ≤ min { h , τ 64 ϵ } + min { h , τ 3/2 128 ϵ ℓ } . Step 4: integration over the threshold. By the layer-cake formula, and since 0 ≤ χ ≤ 1 0\le\chi\le1 0 ≤ χ ≤ 1
∫ I χ ( v ) d v = ∫ 0 1 ∣ { v ∈ I : χ ( v ) > τ } ∣ d τ . \int_I\chi(v)\,\mathrm{d}v
=\int_0^1|\{v\in I:\chi(v)>\tau\}|\,\mathrm{d}\tau. ∫ I χ ( v ) d v = ∫ 0 1 ∣ { v ∈ I : χ ( v ) > τ } ∣ d τ . We integrate the two terms of the last bound of Step 3 separately. For the first, we use min { h , 64 y } ≤ 64 min { h , y } \min\{h,64y\}\le64\min\{h,y\} min { h , 64 y } ≤ 64 min { h , y } y ≥ 0 y\ge0 y ≥ 0 ϵ ≥ h \epsilon\ge h ϵ ≥ h ϵ / τ ≥ h \epsilon/\tau\ge h ϵ / τ ≥ h τ ≤ 1 \tau\le1 τ ≤ 1 min { h , ϵ / τ } \min\{h,\epsilon/\tau\} min { h , ϵ / τ } ( 0 , 1 ) (0,1) ( 0 , 1 ) h h h ϵ < h \epsilon<h ϵ < h τ = ϵ / h \tau=\epsilon/h τ = ϵ / h ϵ + ϵ log ( h / ϵ ) \epsilon+\epsilon\log(h/\epsilon) ϵ + ϵ log ( h / ϵ ) h h h 1 + log y ≤ y 1+\log y\le y 1 + log y ≤ y y ≥ 1 y\ge1 y ≥ 1
∫ 0 1 min { h , 64 ϵ τ } d τ ≤ 64 min { h , ϵ ( 1 + log + ( h / ϵ ) ) } . \int_0^1\min\left\{h,\frac{64\epsilon}{\tau}\right\}\,\mathrm{d}\tau
\le64\min\{h,\epsilon(1+\log_+(h/\epsilon))\}. ∫ 0 1 min { h , τ 64 ϵ } d τ ≤ 64 min { h , ϵ ( 1 + log + ( h / ϵ ))} . For the second term, let B = 128 ϵ ℓ B = 128\epsilon\ell B = 128 ϵ ℓ ℓ = 0 \ell= 0 ℓ = 0 ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) τ 0 = ( B / h ) 2 / 3 \tau_0 = (B/h)^{2/3} τ 0 = ( B / h ) 2/3 B τ 0 − 3 / 2 = h B\tau_0^{-3/2} = h B τ 0 − 3/2 = h
∫ 0 ∞ min { h , B τ − 3 / 2 } d τ = h τ 0 + 2 B τ 0 − 1 / 2 = 3 B 2 / 3 h 1 / 3 = 3 ⋅ 128 2 / 3 ( ϵ ℓ ) 2 / 3 h 1 / 3 ≤ 128 ( ϵ ℓ ) 2 / 3 h 1 / 3 . \begin{aligned}
\int_0^\infty\min\{h,B\tau^{-3/2}\}\,\mathrm{d}\tau= h\tau_0 + 2B\tau_0^{-1/2} = 3B^{2/3}h^{1/3} \\
&= 3\cdot128^{2/3}(\epsilon\ell)^{2/3}h^{1/3} \le128(\epsilon\ell)^{2/3}h^{1/3}.
\end{aligned} ∫ 0 ∞ min { h , B τ − 3/2 } d τ = h τ 0 + 2 B τ 0 − 1/2 = 3 B 2/3 h 1/3 = 3 ⋅ 12 8 2/3 ( ϵ ℓ ) 2/3 h 1/3 ≤ 128 ( ϵ ℓ ) 2/3 h 1/3 . In the last inequality, we used that 3 ⋅ 128 2 / 3 ≤ 128 3\cdot128^{2/3}\le128 3 ⋅ 12 8 2/3 ≤ 128 27 ≤ 128 27\le128 27 ≤ 128 (44) .
A cubic-moment bound
The cavity argument produces third moments of S q − q S_q-q S q − q ξ v \xi_v ξ v z z z z V zV z V H 3 H_3 H 3 ϑ \vartheta ϑ ϑ / 4 \vartheta/4 ϑ /4 v v v Q ( v ) Q(v) Q ( v ) Q ( v ) + ϑ / 4 Q(v)+\vartheta/4 Q ( v ) + ϑ /4 v v v Q ( v ) Q(v) Q ( v ) Q ( v ) − ϑ / 4 Q(v)-\vartheta/4 Q ( v ) − ϑ /4 v v v ϑ \vartheta ϑ
Corollary 5.3 (Cubic self-overlap moment). Let μ \mu μ F t , s F_{t,s} F t , s M ≥ 1 M\ge1 M ≥ 1 0 < z ≤ 1 / 4 0<z\le1/4 0 < z ≤ 1/4
H 3 ≤ z V + 425 M s log ( 2 M ) log 2 z . (47) H_3 \le zV + \frac{425}{Ms}\log(2M)\log\frac{2}{z}. \tag*{(47)} H 3 ≤ z V + M s 425 log ( 2 M ) log z 2 . (47) The constant is absolute; atoms of ν \nu ν
Proof. The proof has three steps. We first reduce H 3 H_3 H 3 ξ v \xi_v ξ v
Step 1: reduction to tails. For ∣ y ∣ ≤ 1 |y|\le1 ∣ y ∣ ≤ 1 0 < z ≤ 1 / 4 0<z\le1/4 0 < z ≤ 1/4
∣ y ∣ ≤ z + ∫ z 1 1 { ∣ y ∣ > ϑ } d ϑ . |y| \le z + \int_z^1 \mathbf{1}_{\{|y|>\vartheta\}}\,\mathrm{d}\vartheta. ∣ y ∣ ≤ z + ∫ z 1 1 { ∣ y ∣ > ϑ } d ϑ . Indeed, the right side is at least z z z ∣ y ∣ > z |y|>z ∣ y ∣ > z ∣ y ∣ − z |y|-z ∣ y ∣ − z y 2 y^2 y 2
∣ y ∣ 3 ≤ z y 2 + ∫ z 1 y 2 1 { ∣ y ∣ > ϑ } d ϑ . |y|^3 \le zy^2 + \int_z^1 y^2\mathbf{1}_{\{|y|>\vartheta\}}\,\mathrm{d}\vartheta. ∣ y ∣ 3 ≤ z y 2 + ∫ z 1 y 2 1 { ∣ y ∣ > ϑ } d ϑ . We use this inequality with y = ξ v y=\xi_v y = ξ v v ∈ ( 0 , 1 − s ) v\in(0,1-s) v ∈ ( 0 , 1 − s )
T ( ϑ ) = ∫ 0 1 − s E [ ξ v 2 1 { ∣ ξ v ∣ > ϑ } ] d v ( 0 < ϑ ≤ 1 ) . T(\vartheta)=\int_0^{1-s}\mathbb{E}\left[\xi_v^2\mathbf{1}_{\{|\xi_v|>\vartheta\}}\right]\,\mathrm{d}v \qquad(0<\vartheta\le1). T ( ϑ ) = ∫ 0 1 − s E [ ξ v 2 1 { ∣ ξ v ∣ > ϑ } ] d v ( 0 < ϑ ≤ 1 ) . Using Tonelli’s theorem and the identities V = ∫ 0 1 − s E ξ v 2 d v V=\int_{0}^{1-s}\mathbb{E}\xi_v^2\,dv V = ∫ 0 1 − s E ξ v 2 d v H 3 = ∫ 0 1 − s E ∣ ξ v ∣ 3 d v H_3=\int_{0}^{1-s}\mathbb{E}|\xi_v|^3\,dv H 3 = ∫ 0 1 − s E ∣ ξ v ∣ 3 d v
H 3 ≤ z V + ∫ z 1 T ( ϑ ) d ϑ . (48) H_3\le zV+\int_z^1 T(\vartheta)\,d\vartheta. \tag*{(48)} H 3 ≤ z V + ∫ z 1 T ( ϑ ) d ϑ . (48) Step 2: the tail bound. Fix ϑ ∈ ( 0 , 1 ] \vartheta\in(0,1] ϑ ∈ ( 0 , 1 ] 0 ≤ k ≤ ⌊ 4 / ϑ ⌋ 0\le k\le\lfloor4/\vartheta\rfloor 0 ≤ k ≤ ⌊ 4/ ϑ ⌋ B k = [ k ϑ / 4 , ( k + 1 ) ϑ / 4 ) B_k=[k\vartheta/4,(k+1)\vartheta/4) B k = [ k ϑ /4 , ( k + 1 ) ϑ /4 ) [ 0 , 1 ] [0,1] [ 0 , 1 ] 4 / ϑ + 1 ≤ 5 / ϑ 4/\vartheta+1\le5/\vartheta 4/ ϑ + 1 ≤ 5/ ϑ
I k = { v ∈ ( 0 , 1 − s ) : Q ( v ) ∈ B k } I_k=\{v\in(0,1-s):Q(v)\in B_k\} I k = { v ∈ ( 0 , 1 − s ) : Q ( v ) ∈ B k } are intervals, because Q Q Q v k ≤ v ˉ k v_k\le\bar v_k v k ≤ v ˉ k I k I_k I k h k = v ˉ k − v k ≤ 1 h_k=\bar v_k-v_k\le1 h k = v ˉ k − v k ≤ 1 I k I_k I k ( 0 , 1 − s ) (0,1-s) ( 0 , 1 − s ) h k = 0 h_k=0 h k = 0 T ( ϑ ) T(\vartheta) T ( ϑ )
Fix k k k h k > 0 h_k>0 h k > 0 v ∈ ( v k , v ˉ k ) v\in(v_k,\bar v_k) v ∈ ( v k , v ˉ k ) ( v , v ˉ k ) (v,\bar v_k) ( v , v ˉ k ) I k I_k I k w w w Q ( v ) ≤ Q ( w ) < Q ( v ) + ϑ / 4 Q(v)\le Q(w)<Q(v)+\vartheta/4 Q ( v ) ≤ Q ( w ) < Q ( v ) + ϑ /4 Q ( w ) Q(w) Q ( w ) Q ( v ) Q(v) Q ( v ) ϑ / 4 \vartheta/4 ϑ /4 Q ( v ) Q(v) Q ( v ) Q ( v ) + ϑ / 4 Q(v)+\vartheta/4 Q ( v ) + ϑ /4 ( v k , v ) (v_k,v) ( v k , v ) Q ≤ Q ( v ) Q\le Q(v) Q ≤ Q ( v ) Q ( v ) − ϑ / 4 Q(v)-\vartheta/4 Q ( v ) − ϑ /4 s s s (42) . By the last assertion of Lemma 5.1 , their normalized densities satisfy the hypotheses of (a) and (b), respectively, with q = Q ( v ) q=Q(v) q = Q ( v ) ϑ / 4 ≤ ϑ / 2 \vartheta/4\le\vartheta/2 ϑ /4 ≤ ϑ /2 Lemma 5.1 (d) and (e) now imply that
E [ ξ v 2 1 { ξ v > ϑ } ] ≤ 16 M s ( v ˉ k − v ) , E [ ξ v 2 1 { ξ v < − ϑ } ] ≤ 16 M s ( v − v k ) . \mathbb{E}\left[\xi_v^2\mathbf{1}_{\{\xi_v>\vartheta\}}\right]\le\frac{16}{Ms(\bar v_k-v)},\qquad
\mathbb{E}\left[\xi_v^2\mathbf{1}_{\{\xi_v<-\vartheta\}}\right]\le\frac{16}{Ms(v-v_k)}. E [ ξ v 2 1 { ξ v > ϑ } ] ≤ M s ( v ˉ k − v ) 16 , E [ ξ v 2 1 { ξ v < − ϑ } ] ≤ M s ( v − v k ) 16 . Suppose that h k > 1 / M h_k>1/M h k > 1/ M ( v k , v ˉ k − 1 / M ) (v_k,\bar v_k-1/M) ( v k , v ˉ k − 1/ M ) ∣ ξ v ∣ ≤ 1 |\xi_v|\le1 ∣ ξ v ∣ ≤ 1 1 / M 1/M 1/ M
∫ I k E [ ξ v 2 1 { ξ v > ϑ } ] d v ≤ 16 M s ∫ v k v ˉ k − 1 / M d v v ˉ k − v + 1 M = 16 M s log ( M h k ) + 1 M ≤ 16 log ( 2 M ) M s + 1 M , \begin{aligned}
\int_{I_k}\mathbb{E}\left[\xi_v^2\mathbf{1}_{\{\xi_v>\vartheta\}}\right]\,dv
\le\frac{16}{Ms}\int_{v_k}^{\bar v_k-1/M}\frac{dv}{\bar v_k-v}+\frac{1}{M}
=\frac{16}{Ms}\log(Mh_k)+\frac{1}{M} \\
&\le\frac{16\log(2M)}{Ms}+\frac{1}{M},
\end{aligned} ∫ I k E [ ξ v 2 1 { ξ v > ϑ } ] d v ≤ M s 16 ∫ v k v ˉ k − 1/ M v ˉ k − v d v + M 1 = M s 16 log ( M h k ) + M 1 ≤ M s 16 log ( 2 M ) + M 1 , where we used h k ≤ 1 h_k\le1 h k ≤ 1 1 / M 1/M 1/ M I k I_k I k h k ≤ 1 / M h_k\le1/M h k ≤ 1/ M h k ≤ 1 / M h_k\le1/M h k ≤ 1/ M ∣ ξ v ∣ ≤ 1 |\xi_v|\le1 ∣ ξ v ∣ ≤ 1 M ≥ 1 M\ge1 M ≥ 1 s ≤ 1 / 2 s\le1/2 s ≤ 1/2
2 M ≤ 1 M s ≤ 2 log ( 2 M ) M s , \frac{2}{M}\le\frac{1}{Ms}\le\frac{2\log(2M)}{Ms}, M 2 ≤ M s 1 ≤ M s 2 log ( 2 M ) , where we used log 2 ≥ 1 / 2 \log2\ge1/2 log 2 ≥ 1/2 32 log ( 2 M ) / ( M s ) + 2 / M ≤ 34 log ( 2 M ) / ( M s ) 32\log(2M)/(Ms)+2/M\le34\log(2M)/(Ms) 32 log ( 2 M ) / ( M s ) + 2/ M ≤ 34 log ( 2 M ) / ( M s ) T ( ϑ ) T(\vartheta) T ( ϑ ) 5 / ϑ 5/\vartheta 5/ ϑ
T ( ϑ ) ≤ 5 ϑ ⋅ 34 log ( 2 M ) M s = 170 log ( 2 M ) M s ϑ ( 0 < ϑ ≤ 1 ) . T(\vartheta)\le\frac{5}{\vartheta}\cdot\frac{34\log(2M)}{Ms}
=\frac{170\log(2M)}{Ms\vartheta}\qquad(0<\vartheta\le1). T ( ϑ ) ≤ ϑ 5 ⋅ M s 34 log ( 2 M ) = M s ϑ 170 log ( 2 M ) ( 0 < ϑ ≤ 1 ) . Step 3: integration. The tail bound holds on the whole range ( z , 1 ] (z,1] ( z , 1 ] (48) . Inserting it, we obtain
H 3 ≤ z V + 170 log ( 2 M ) M s ∫ z 1 d ϑ ϑ = z V + 170 log ( 2 M ) M s log 1 z ≤ z V + 425 log ( 2 M ) M s log 2 z . H_3 \le zV + \frac{170\log(2M)}{Ms}\int_z^1\frac{\mathrm{d}\vartheta}{\vartheta}=zV+\frac{170\log(2M)}{Ms}\log\frac{1}{z}\le zV+\frac{425\log(2M)}{Ms}\log\frac{2}{z}. H 3 ≤ z V + M s 170 log ( 2 M ) ∫ z 1 ϑ d ϑ = z V + M s 170 log ( 2 M ) log z 1 ≤ z V + M s 425 log ( 2 M ) log z 2 . This proves (47) . In fact, we have proved the stronger bound with 170 log ( 1 / z ) 170\log(1/z) 170 log ( 1/ z ) 425 log ( 2 / z ) 425\log(2/z) 425 log ( 2/ z ) □ \square □
The mean–covariance term
In this section, we bound the mean–covariance term J = ∫ j q ν ( d q ) J = \int j_q\,\nu(\mathrm{d}q) J = ∫ j q ν ( d q ) j q = N − 2 E [ m q T C q m q ] j_q = N^{-2}\mathbb{E}[m_q^{\mathsf{T}}C_qm_q] j q = N − 2 E [ m q T C q m q ] Proposition 6.4 , states that J ≤ 341 ( 1 + log + M ) 2 / ( M s ) J \le341(1+\log_{+}M)^2/(Ms) J ≤ 341 ( 1 + log + M ) 2 / ( M s ) A ≤ 1 / ( M s ) A \le1/(Ms) A ≤ 1/ ( M s ) (23) , this controls the thermal part A + 2 J A+2J A + 2 J D N ( ρ ) = β ( A + 2 J + V ) D_N(\rho)=\beta(A+2J+V) D N ( ρ ) = β ( A + 2 J + V ) M s = β 2 N ρ ⋅ λ / β = β N ρ λ Ms=\beta^2N\rho\cdot\lambda/\beta=\beta N\rho\lambda M s = β 2 N ρ ⋅ λ / β = β N ρ λ
β ( A + 2 J ) ≤ 1 + 682 ( 1 + log + M ) 2 N ρ λ . \beta(A+2J)\le\frac{1+682(1+\log_{+}M)^2}{N\rho\lambda}. β ( A + 2 J ) ≤ N ρ λ 1 + 682 ( 1 + log + M ) 2 . We use Proposition 6.4 in this form in the proof of Corollary 12.5 , for those ρ \rho ρ (200) in the proof of Proposition 13.1 , for small ρ \rho ρ
The proof has four steps.
In Lemma 6.1, we show by a scalar Riccati inequality that conditional covariances decay in every direction that is measurable at an earlier level: if b b b G q ′ \mathcal{G}_{q'} G q ′ E [ b T C q b ] \mathbb{E}[b^{\mathsf{T}}C_qb] E [ b T C q b ] E ∣ b ∣ 2 \mathbb{E}|b|^2 E ∣ b ∣ 2 c ∫ q ′ q a ( u ) 2 d u c\int_{q'}^q a(u)^2\,\mathrm{d}u c ∫ q ′ q a ( u ) 2 d u
In Lemma 6.2 , we use first-order optimality to show that S ( r ) = E S r S(r)=\mathbb{E}S_r S ( r ) = E S r q q q supp ν \operatorname{supp}\nu supp ν r r r (50) below, this bounds E ∣ m q − m r ∣ 2 \mathbb{E}|m_q-m_r|^2 E ∣ m q − m r ∣ 2 2 N ∣ q − r ∣ 2N|q-r| 2 N ∣ q − r ∣
In Lemma 6.3, we combine these two estimates through a dyadic decomposition of m q − m p m_q-m_p m q − m p q ∈ supp ν q\in\operatorname{supp}\nu q ∈ supp ν p ≤ q p\le q p ≤ q R ( p , q ) = N − 2 E [ ( m q − m p ) T C q ( m q − m p ) ] R(p,q)=N^{-2}\mathbb{E}[(m_q-m_p)^{\mathsf{T}}C_q(m_q-m_p)] R ( p , q ) = N − 2 E [( m q − m p ) T C q ( m q − m p )] ( 1 + log + M ) 2 / ( M a ( ( p + q ) / 2 ) 2 ) (1+\log_{+}M)^2/(Ma((p+q)/2)^2) ( 1 + log + M ) 2 / ( M a (( p + q ) /2 ) 2 )
In Proposition 6.4 , we average earlier means m Q ( w ) m_{Q(w)} m Q ( w ) 0 < w < v 0<w<v 0 < w < v m 0 = 0 m_0=0 m 0 = 0 (23) , and its distance from m Q ( v ) m_{Q(v)} m Q ( v ) ( s + v ) − 2 (s+v)^{-2} ( s + v ) − 2 v ∈ ( 0 , 1 − s ) v\in(0,1-s) v ∈ ( 0 , 1 − s ) 1 / s 1/s 1/ s
We use Proposition 3.1 , namely the martingale equation and covariance order (20) , the first variation (21) , and the contact identities (22) ; Proposition 3.3; and facts about the quantile function from § 5.1.1 , including (42) . Lemmas 6.1 to 6.3 are used only in this section.
Standing assumptions
Throughout this section, we fix N ≥ 1 N \ge1 N ≥ 1 0 < ρ ≤ 1 0 < \rho\le1 0 < ρ ≤ 1 0 < s ≤ 1 / 2 0 < s \le1/2 0 < s ≤ 1/2 t = 1 − ρ ∈ [ 0 , 1 ) t = 1 - \rho\in[0,1) t = 1 − ρ ∈ [ 0 , 1 ) c > 0 c > 0 c > 0 M = N c > 0 M = Nc > 0 M = N c > 0 μ = s δ 0 + ν \mu= s\delta_0 + \nu μ = s δ 0 + ν μ \mu μ F t , s F_{t,s} F t , s
We write S ( q ) = E S q S(q) = \mathbb{E}S_q S ( q ) = E S q a ( q ) C q ⪯ H q ⪯ C q a(q)C_q \preceq H_q \preceq C_q a ( q ) C q ⪯ H q ⪯ C q 0 ⪯ H q ⪯ C q 0 \preceq H_q \preceq C_q 0 ⪯ H q ⪯ C q C q C_q C q 1 − m q , i 2 ≤ 1 1 - m_{q,i}^2 \le1 1 − m q , i 2 ≤ 1
∥ H q ∥ o p ≤ ∥ C q ∥ o p ≤ Tr C q ≤ N , 0 ≤ S ( q ) ≤ 1 , (49) \lVert H_q\rVert_{\mathrm{op}} \le\lVert C_q\rVert_{\mathrm{op}} \le\operatorname{Tr} C_q \le N,\qquad0 \le S(q) \le1, \tag*{(49)} ∥ H q ∥ op ≤ ∥ C q ∥ op ≤ Tr C q ≤ N , 0 ≤ S ( q ) ≤ 1 , (49) where ∥ ⋅ ∥ o p \lVert\cdot\rVert_{\mathrm{op}} ∥ ⋅ ∥ op m m m E [ m q ∣ G p ] = m p \mathbb{E}[m_q \mid\mathcal{G}_p] = m_p E [ m q ∣ G p ] = m p p ≤ q p \le q p ≤ q E ∣ m q ∣ 2 = E ∣ m p ∣ 2 + E ∣ m q − m p ∣ 2 \mathbb{E}|m_q|^2 = \mathbb{E}|m_p|^2 + \mathbb{E}|m_q - m_p|^2 E ∣ m q ∣ 2 = E ∣ m p ∣ 2 + E ∣ m q − m p ∣ 2 S S S
E ∣ m q − m p ∣ 2 = N ( S ( q ) − S ( p ) ) ( 0 ≤ p ≤ q ≤ 1 ) . (50) \mathbb{E}|m_q - m_p|^2 = N\bigl(S(q) - S(p)\bigr)\qquad(0 \le p \le q \le1). \tag*{(50)} E ∣ m q − m p ∣ 2 = N ( S ( q ) − S ( p ) ) ( 0 ≤ p ≤ q ≤ 1 ) . (50) For 0 ≤ p ≤ q ≤ 1 0 \le p \le q \le1 0 ≤ p ≤ q ≤ 1
R ( p , q ) = N − 2 E [ ( m q − m p ) T C q ( m q − m p ) ] , \mathcal{R}(p,q) = N^{-2}\mathbb{E}\bigl[(m_q - m_p)^{T}C_q(m_q - m_p)\bigr], R ( p , q ) = N − 2 E [ ( m q − m p ) T C q ( m q − m p ) ] , which is the analog of j q j_q j q m q − m p m_q - m_p m q − m p m q m_q m q R ( 0 , q ) = j q \mathcal{R}(0,q) = j_q R ( 0 , q ) = j q m 0 = 0 m_0 = 0 m 0 = 0
Covariance along an earlier measurable direction
Fix a direction that is measurable at some level. As more of the auxiliary field is revealed, the expected conditional variance of the spin in this direction can only decrease. The martingale equation determines the rate of decrease, and the covariance order bounds this rate from below by a multiple of the square of the variance itself. We integrate the resulting scalar Riccati inequality and obtain decay that is inversely proportional to the accumulated clock.
Lemma 6.1 (Directional covariance decay). Let μ = s δ 0 + ν \mu= s\delta_0 + \nu μ = s δ 0 + ν 0 ≤ q ′ < q ≤ 1 0 \le q' < q \le1 0 ≤ q ′ < q ≤ 1
I = c ∫ q ′ q a ( u ) 2 d u . \mathcal{I} = c\int_{q'}^q a(u)^2\,du. I = c ∫ q ′ q a ( u ) 2 d u . Then I ≥ c s 2 ( q − q ′ ) > 0 \mathcal{I} \ge cs^2(q-q') > 0 I ≥ c s 2 ( q − q ′ ) > 0 G q ′ \mathcal{G}_{q'} G q ′ b ∈ R N b \in\mathbb{R}^N b ∈ R N E ∣ b ∣ 2 < ∞ \mathbb{E}|b|^2 < \infty E ∣ b ∣ 2 < ∞
E [ b T C q b ] ≤ E ∣ b ∣ 2 I . \mathbb{E}\bigl[b^T C_q b\bigr] \le\frac{\mathbb{E}|b|^2}{\mathcal{I}}. E [ b T C q b ] ≤ I E ∣ b ∣ 2 . Proof. The lower bound for I \mathcal{I} I a ≥ s a \ge s a ≥ s b b b r ↦ E [ b T C r b ] r \mapsto\mathbb{E}[b^{T}C_{r}b] r ↦ E [ b T C r b ] [ q ′ , q ] [q',q] [ q ′ , q ]
Step 1: the rate of decrease. Suppose that b b b φ ( r ) = E [ b T C r b ] \varphi(r)=\mathbb{E}[b^{T}C_{r}b] φ ( r ) = E [ b T C r b ] r ∈ [ q ′ , q ] r\in[q',q] r ∈ [ q ′ , q ] r r r b b b G r \mathcal{G}_{r} G r b T C r b = E [ ( b ⋅ ( σ − m r ) ) 2 ∣ G r ] b^{T}C_{r}b=\mathbb{E}[(b\cdot(\sigma-m_{r}))^{2}\mid\mathcal{G}_{r}] b T C r b = E [( b ⋅ ( σ − m r ) ) 2 ∣ G r ] φ ( r ) = E ( b ⋅ ( σ − m r ) ) 2 \varphi(r)=\mathbb{E}(b\cdot(\sigma-m_{r}))^{2} φ ( r ) = E ( b ⋅ ( σ − m r ) ) 2
b ⋅ ( σ − m q ′ ) = b ⋅ ( σ − m r ) + b ⋅ ( m r − m q ′ ) . b\cdot(\sigma-m_{q'})=b\cdot(\sigma-m_{r})+b\cdot(m_{r}-m_{q'}). b ⋅ ( σ − m q ′ ) = b ⋅ ( σ − m r ) + b ⋅ ( m r − m q ′ ) . The second term is G r \mathcal{G}_{r} G r G r \mathcal{G}_{r} G r
φ ( q ′ ) = φ ( r ) + E ( b ⋅ ( m r − m q ′ ) ) 2 . \varphi(q')=\varphi(r)+\mathbb{E}\bigl(b\cdot(m_{r}-m_{q'})\bigr)^{2}. φ ( q ′ ) = φ ( r ) + E ( b ⋅ ( m r − m q ′ ) ) 2 . By (20) , and since b b b G q ′ \mathcal{G}_{q'} G q ′ b ⋅ ( m r − m q ′ ) = c ∫ q ′ r ( H u b ) ⋅ d B u b\cdot(m_{r}-m_{q'})=\sqrt{c}\int_{q'}^{r}(H_{u}b)\cdot\mathrm{d}B_{u} b ⋅ ( m r − m q ′ ) = c ∫ q ′ r ( H u b ) ⋅ d B u N ∣ b ∣ N|b| N ∣ b ∣ (49) . Itô’s isometry gives
φ ( r ) = φ ( q ′ ) − c ∫ q ′ r E ∣ H u b ∣ 2 d u ( q ′ ≤ r ≤ q ) . \varphi(r)=\varphi(q')-c\int_{q'}^{r}\mathbb{E}|H_{u}b|^{2}\,\mathrm{d}u\qquad(q'\le r\le q). φ ( r ) = φ ( q ′ ) − c ∫ q ′ r E ∣ H u b ∣ 2 d u ( q ′ ≤ r ≤ q ) . Then φ \varphi φ φ ′ ( r ) = − c E ∣ H r b ∣ 2 \varphi'(r)=-c\mathbb{E}|H_{r}b|^{2} φ ′ ( r ) = − c E ∣ H r b ∣ 2 r ∈ [ q ′ , q ] r\in[q',q] r ∈ [ q ′ , q ]
Step 2: the Riccati inequality. On the event { b ≠ 0 } \{b\ne0\} { b = 0 } 0 ≤ a ( r ) b T C r b ≤ b T H r b 0\le a(r)b^{T}C_{r}b\le b^{T}H_{r}b 0 ≤ a ( r ) b T C r b ≤ b T H r b
∣ H r b ∣ 2 ≥ ( b T H r b ) 2 ∣ b ∣ 2 ≥ a ( r ) 2 ( b T C r b ) 2 ∣ b ∣ 2 . |H_{r}b|^{2}\ge\frac{(b^{T}H_{r}b)^{2}}{|b|^{2}}\ge\frac{a(r)^{2}(b^{T}C_{r}b)^{2}}{|b|^{2}}. ∣ H r b ∣ 2 ≥ ∣ b ∣ 2 ( b T H r b ) 2 ≥ ∣ b ∣ 2 a ( r ) 2 ( b T C r b ) 2 . Let Z r = ∣ b ∣ − 1 b T C r b Z_{r}=|b|^{-1}b^{T}C_{r}b Z r = ∣ b ∣ − 1 b T C r b { b ≠ 0 } \{b\ne0\} { b = 0 } Z r = 0 Z_{r}=0 Z r = 0 E ∣ H r b ∣ 2 ≥ a ( r ) 2 E Z r 2 \mathbb{E}|H_{r}b|^{2}\ge a(r)^{2}\mathbb{E}Z_{r}^{2} E ∣ H r b ∣ 2 ≥ a ( r ) 2 E Z r 2 φ ( r ) = E [ Z r ∣ b ∣ ] \varphi(r)=\mathbb{E}[Z_{r}|b|] φ ( r ) = E [ Z r ∣ b ∣ ] φ ( r ) 2 ≤ E Z r 2 E ∣ b ∣ 2 \varphi(r)^{2}\le\mathbb{E}Z_{r}^{2}\mathbb{E}|b|^{2} φ ( r ) 2 ≤ E Z r 2 E ∣ b ∣ 2 E ∣ b ∣ 2 > 0 \mathbb{E}|b|^{2}>0 E ∣ b ∣ 2 > 0
φ ′ ( r ) ≤ − c a ( r ) 2 φ ( r ) 2 E ∣ b ∣ 2 for almost every r ∈ [ q ′ , q ] . \varphi'(r)\le-\frac{c\,a(r)^{2}\varphi(r)^{2}}{\mathbb{E}|b|^{2}}
\qquad\text{for almost every }r\in[q',q]. φ ′ ( r ) ≤ − E ∣ b ∣ 2 c a ( r ) 2 φ ( r ) 2 for almost every r ∈ [ q ′ , q ] . Step 3: integration. If E ∣ b ∣ 2 = 0 \mathbb{E}|b|^{2}=0 E ∣ b ∣ 2 = 0 φ ( q ) = 0 \varphi(q)=0 φ ( q ) = 0 φ ≥ φ ( q ) > 0 \varphi\ge\varphi(q)>0 φ ≥ φ ( q ) > 0 [ q ′ , q ] [q',q] [ q ′ , q ] φ \varphi φ 1 / φ 1/\varphi 1/ φ ( 1 / φ ) ′ = − φ ′ / φ 2 ≥ c a ( r ) 2 / E ∣ b ∣ 2 (1/\varphi)'=-\varphi'/\varphi^{2}\ge c\,a(r)^{2}/\mathbb{E}|b|^{2} ( 1/ φ ) ′ = − φ ′ / φ 2 ≥ c a ( r ) 2 / E ∣ b ∣ 2 [ q ′ , q ] [q',q] [ q ′ , q ]
1 φ ( q ) ≥ 1 φ ( q ′ ) + I E ∣ b ∣ 2 ≥ I E ∣ b ∣ 2 . \frac{1}{\varphi(q)}\ge\frac{1}{\varphi(q')}+\frac{\mathcal{I}}{\mathbb{E}|b|^{2}}\ge\frac{\mathcal{I}}{\mathbb{E}|b|^{2}}. φ ( q ) 1 ≥ φ ( q ′ ) 1 + E ∣ b ∣ 2 I ≥ E ∣ b ∣ 2 I . which is the claim for bounded b b b a a a
Step 4: general directions. For general b b b E ∣ b ∣ 2 < ∞ \mathbb{E}|b|^2 < \infty E ∣ b ∣ 2 < ∞ b n = b 1 { ∣ b ∣ ≤ n } b_n = b1_{\{|b|\le n\}} b n = b 1 { ∣ b ∣ ≤ n } G q ′ G_{q'} G q ′ b n T C q b n = ( b T C q b ) 1 { ∣ b ∣ ≤ n } b_n^{T}C_qb_n = (b^{T}C_qb)1_{\{|b|\le n\}} b n T C q b n = ( b T C q b ) 1 { ∣ b ∣ ≤ n } b T C q b b^{T}C_qb b T C q b n → ∞ n \to\infty n → ∞ C q ⪰ 0 C_q \succeq0 C q ⪰ 0 ∣ b n ∣ 2 |b_n|^2 ∣ b n ∣ 2 ∣ b ∣ 2 |b|^2 ∣ b ∣ 2 b n b_n b n n → ∞ n \to\infty n → ∞ b b b
Mean increments toward a support point
In the dyadic argument below, we need E ∣ m q − m r ∣ 2 \mathbb{E}|m_q-m_r|^2 E ∣ m q − m r ∣ 2 N ∣ q − r ∣ N|q-r| N ∣ q − r ∣ q q q supp ν \operatorname{supp}\nu supp ν r r r (50) , this is a Lipschitz bound for S S S q q q
G ( y ) = ∫ y 1 ( S ( u ) − u ) d u ( 0 ≤ y ≤ 1 ) G(y) = \int_y^1 (S(u)-u)\,\mathrm{d}u \qquad(0 \le y \le1) G ( y ) = ∫ y 1 ( S ( u ) − u ) d u ( 0 ≤ y ≤ 1 ) be the function in the first variation (21) . Proposition 3.1 states that if μ \mu μ supp ν \operatorname{supp}\nu supp ν G G G [ 0 , 1 ] [0,1] [ 0 , 1 ]
G ( y ) ≥ G ( q ) ( q ∈ supp ν , 0 ≤ y ≤ 1 ) . (51) G(y) \ge G(q) \qquad(q \in\operatorname{supp}\nu,\ 0 \le y \le1). \tag*{(51)} G ( y ) ≥ G ( q ) ( q ∈ supp ν , 0 ≤ y ≤ 1 ) . (51) We recall the argument from the proof of [4 Proposition 2.1] , because we will use (51) at points y y y supp ν \operatorname{supp}\nu supp ν y ∈ [ 0 , 1 ] y \in[0,1] y ∈ [ 0 , 1 ] s δ 0 + ( 1 − θ ) ν + θ ( 1 − s ) δ y s\delta_0 + (1-\theta)\nu+ \theta(1-s)\delta_y s δ 0 + ( 1 − θ ) ν + θ ( 1 − s ) δ y 0 ≤ θ ≤ 1 0 \le\theta\le1 0 ≤ θ ≤ 1 F t , s F_{t,s} F t , s θ = 0 \theta=0 θ = 0 (21) with ν ′ = ( 1 − s ) δ y \nu'=(1-s)\delta_y ν ′ = ( 1 − s ) δ y c / 2 c/2 c /2
( 1 − s ) G ( y ) ≥ ∫ G ( p ) ν ( d p ) ( 0 ≤ y ≤ 1 ) . (1-s)G(y) \ge\int G(p)\,\nu(\mathrm{d}p) \qquad(0 \le y \le1). ( 1 − s ) G ( y ) ≥ ∫ G ( p ) ν ( d p ) ( 0 ≤ y ≤ 1 ) . The function G G G y y y G G G ν \nu ν 1 − s 1-s 1 − s ∫ ( G − min G ) d ν ≤ 0 \int(G-\min G)\,\mathrm{d}\nu\le0 ∫ ( G − min G ) d ν ≤ 0 { G > min G } \{G>\min G\} { G > min G } ν \nu ν supp ν \operatorname{supp}\nu supp ν
Lemma 6.2 (Lipschitz bound toward the support). Let μ \mu μ F t , s F_{t,s} F t , s q ∈ supp ν q \in\operatorname{supp}\nu q ∈ supp ν r ∈ [ 0 , 1 ] r \in[0,1] r ∈ [ 0 , 1 ]
∣ S ( r ) − S ( q ) ∣ ≤ 2 ∣ r − q ∣ ; in particular, 0 ≤ S ( q ) − S ( r ) ≤ 2 ( q − r ) ( 0 ≤ r ≤ q ) . |S(r)-S(q)| \le2|r-q|;\qquad\text{in particular,}\qquad0 \le S(q)-S(r) \le2(q-r)\qquad(0 \le r \le q). ∣ S ( r ) − S ( q ) ∣ ≤ 2∣ r − q ∣ ; in particular, 0 ≤ S ( q ) − S ( r ) ≤ 2 ( q − r ) ( 0 ≤ r ≤ q ) . Proof. Let q ∈ supp ν q \in\operatorname{supp}\nu q ∈ supp ν S ( q ) = q S(q)=q S ( q ) = q (22) . Recall that S S S [ 0 , 1 ] [0,1] [ 0 , 1 ] G ( y 1 ) − G ( y 2 ) = ∫ y 1 y 2 ( S ( u ) − u ) d u G(y_1)-G(y_2)=\int_{y_1}^{y_2}(S(u)-u)\,\mathrm{d}u G ( y 1 ) − G ( y 2 ) = ∫ y 1 y 2 ( S ( u ) − u ) d u y 1 ≤ y 2 y_1 \le y_2 y 1 ≤ y 2 r ∈ [ 0 , 1 ] r \in[0,1] r ∈ [ 0 , 1 ] ξ = S ( r ) ∈ [ 0 , 1 ] \xi=S(r) \in[0,1] ξ = S ( r ) ∈ [ 0 , 1 ] G G G ξ \xi ξ q q q ξ \xi ξ supp ν \operatorname{supp}\nu supp ν (51) for every $y \in [0,1]`. The case r ≤ q r \le q r ≤ q By monotonicity, S ( q ) − S ( r ) ≥ 0 S(q)-S(r) \ge0 S ( q ) − S ( r ) ≥ 0 ξ ≥ r \xi\ge r ξ ≥ r S ( q ) − S ( r ) = q − ξ ≤ q − r S(q)-S(r)=q-\xi\le q-r S ( q ) − S ( r ) = q − ξ ≤ q − r ξ < r \xi<r ξ < r S ≤ S ( r ) = ξ S \le S(r)=\xi S ≤ S ( r ) = ξ [ ξ , r ] [\xi,r] [ ξ , r ] S ≤ S ( q ) = q S \le S(q)=q S ≤ S ( q ) = q [ r , q ] [r,q] [ r , q ] (51) with y = ξ y=\xi y = ξ
0 ≤ G ( ξ ) − G ( q ) = ∫ ξ r ( S ( u ) − u ) d u + ∫ r q ( S ( u ) − u ) d u ≤ ∫ ξ r ( ξ − u ) d u + ∫ r q ( q − u ) d u = ( q − r ) 2 − ( r − ξ ) 2 2 . 0 \le G(\xi)-G(q)=\int_{\xi}^{r}(S(u)-u)\,\mathrm{d}u+\int_{r}^{q}(S(u)-u)\,\mathrm{d}u
\le\int_{\xi}^{r}(\xi-u)\,\mathrm{d}u+\int_{r}^{q}(q-u)\,\mathrm{d}u=\frac{(q-r)^2-(r-\xi)^2}{2}. 0 ≤ G ( ξ ) − G ( q ) = ∫ ξ r ( S ( u ) − u ) d u + ∫ r q ( S ( u ) − u ) d u ≤ ∫ ξ r ( ξ − u ) d u + ∫ r q ( q − u ) d u = 2 ( q − r ) 2 − ( r − ξ ) 2 . Then r − ξ ≤ q − r r-\xi\le q-r r − ξ ≤ q − r S ( q ) − S ( r ) = ( q − r ) + ( r − ξ ) ≤ 2 ( q − r ) S(q)-S(r)=(q-r)+(r-\xi)\le2(q-r) S ( q ) − S ( r ) = ( q − r ) + ( r − ξ ) ≤ 2 ( q − r )
The case r > q r>q r > q By monotonicity, S ( r ) − S ( q ) ≥ 0 S(r)-S(q)\ge0 S ( r ) − S ( q ) ≥ 0 ξ ≤ r \xi\le r ξ ≤ r S ( r ) − S ( q ) = ξ − q ≤ r − q S(r)-S(q)=\xi-q\le r-q S ( r ) − S ( q ) = ξ − q ≤ r − q ξ > r \xi>r ξ > r S ≥ q S\ge q S ≥ q [ q , r ] [q,r] [ q , r ] S ≥ ξ S\ge\xi S ≥ ξ [ r , ξ ] [r,\xi] [ r , ξ ]
0 ≤ G ( ξ ) − G ( q ) = − ∫ q r ( S ( u ) − u ) d u − ∫ r ξ ( S ( u ) − u ) d u ≤ ∫ q r ( u − q ) d u + ∫ r ξ ( u − ξ ) d u = ( r − q ) 2 − ( ξ − r ) 2 2 . 0 \le G(\xi)-G(q)=-\int_{q}^{r}(S(u)-u)\,\mathrm{d}u-\int_{r}^{\xi}(S(u)-u)\,\mathrm{d}u
\le\int_{q}^{r}(u-q)\,\mathrm{d}u+\int_{r}^{\xi}(u-\xi)\,\mathrm{d}u=\frac{(r-q)^2-(\xi-r)^2}{2}. 0 ≤ G ( ξ ) − G ( q ) = − ∫ q r ( S ( u ) − u ) d u − ∫ r ξ ( S ( u ) − u ) d u ≤ ∫ q r ( u − q ) d u + ∫ r ξ ( u − ξ ) d u = 2 ( r − q ) 2 − ( ξ − r ) 2 . We obtain ξ − r ≤ r − q \xi-r\le r-q ξ − r ≤ r − q S ( r ) − S ( q ) = ( r − q ) + ( ξ − r ) ≤ 2 ( r − q ) S(r)-S(q)=(r-q)+(\xi-r)\le2(r-q) S ( r ) − S ( q ) = ( r − q ) + ( ξ − r ) ≤ 2 ( r − q )
Together, (50) and Lemma 6.2 imply that
E ∣ m q − m r ∣ 2 ≤ 2 N ( q − r ) ( q ∈ supp ν , 0 ≤ r ≤ q ) . \mathbb{E}|m_q-m_r|^2\le2N(q-r)\qquad(q\in\operatorname{supp}\nu,\ 0\le r\le q). E ∣ m q − m r ∣ 2 ≤ 2 N ( q − r ) ( q ∈ supp ν , 0 ≤ r ≤ q ) . A dyadic decomposition of the mean increment
Lemma 6.1 does not apply directly to the increment m q − m p m_q-m_p m q − m p q q q q q q q q q q q q C q ⪯ N I C_q\preceq NI C q ⪯ N I
Lemma 6.3 (Covariance of a mean increment). Let μ \mu μ F t , s \mathcal{F}_{t,s} F t , s q ∈ supp ν q\in\operatorname{supp}\nu q ∈ supp ν 0 ≤ p ≤ q 0\le p\le q 0 ≤ p ≤ q a ∗ = a ( ( p + q ) / 2 ) a_* = a((p+q)/2) a ∗ = a (( p + q ) /2 ) K ≥ 1 K\ge1 K ≥ 1
R ( p , q ) ≤ ( K + 1 ) [ 4 K M a ∗ 2 + 2 ( q − p ) 2 − K ] . (52) \mathcal{R}(p,q)\le(K+1)\left[\frac{4K}{Ma_*^2}+2(q-p)2^{-K}\right]. \tag*{(52)} R ( p , q ) ≤ ( K + 1 ) [ M a ∗ 2 4 K + 2 ( q − p ) 2 − K ] . (52) Moreover,
R ( p , q ) ≤ 56 ( 1 + log + ( M ( q − p ) ) ) 2 M a ∗ 2 . (53) \mathcal{R}(p,q)\le\frac{56(1+\log_+(M(q-p)))^2}{Ma_*^2}. \tag*{(53)} R ( p , q ) ≤ M a ∗ 2 56 ( 1 + log + ( M ( q − p )) ) 2 . (53) Proof. We may assume that p < q p<q p < q p = q p=q p = q m q − m p m_q-m_p m q − m p K + 1 K+1 K + 1 K K K K K K
For 0 ≤ k ≤ K 0\le k\le K 0 ≤ k ≤ K q k = q − ( q − p ) 2 − k q_k=q-(q-p)2^{-k} q k = q − ( q − p ) 2 − k q 0 = p q_0=p q 0 = p q 1 = ( p + q ) / 2 q_1=(p+q)/2 q 1 = ( p + q ) /2 q − q k = ( q − p ) 2 − k q-q_k=(q-p)2^{-k} q − q k = ( q − p ) 2 − k b k = m q k + 1 − m q k b_k=m_{q_{k+1}}-m_{q_k} b k = m q k + 1 − m q k 0 ≤ k < K 0\le k<K 0 ≤ k < K b K = m q − m q K b_K=m_q-m_{q_K} b K = m q − m q K m q − m p = ∑ k = 0 K b k m_q-m_p=\sum_{k=0}^{K}b_k m q − m p = ∑ k = 0 K b k C q C_q C q
( m q − m p ) T C q ( m q − m p ) ≤ ( K + 1 ) ∑ k = 0 K b k T C q b k . (m_q-m_p)^T C_q(m_q-m_p)\le(K+1)\sum_{k=0}^{K}b_k^T C_qb_k. ( m q − m p ) T C q ( m q − m p ) ≤ ( K + 1 ) k = 0 ∑ K b k T C q b k . The points q k q_k q k q ∈ supp ν q\in\operatorname{supp}\nu q ∈ supp ν
Step 1: the first K K K Let 0 ≤ k < K 0\le k<K 0 ≤ k < K b k b_k b k G q k + 1 \mathcal{G}_{q_{k+1}} G q k + 1 q k + 1 < q q_{k+1}<q q k + 1 < q a a a q k + 1 ≥ q 1 q_{k+1}\ge q_1 q k + 1 ≥ q 1 a ( u ) ≥ a ( q 1 ) = a ∗ a(u)\ge a(q_1)=a_* a ( u ) ≥ a ( q 1 ) = a ∗ u ≥ q k + 1 u\ge q_{k+1} u ≥ q k + 1 q ′ = q k + 1 q'=q_{k+1} q ′ = q k + 1
I = c ∫ q k + 1 q a ( u ) 2 d u ≥ c a ∗ 2 ( q − q k + 1 ) , \mathcal{I}=c\int_{q_{k+1}}^q a(u)^2\,du\ge ca_*^2(q-q_{k+1}), I = c ∫ q k + 1 q a ( u ) 2 d u ≥ c a ∗ 2 ( q − q k + 1 ) , and Lemma 6.1 implies that E [ b k T C q b k ] ≤ E ∣ b k ∣ 2 / ( c a ∗ 2 ( q − q k + 1 ) ) \mathbb{E}[b_k^T C_qb_k]\le\mathbb{E}|b_k|^2/(ca_*^2(q-q_{k+1})) E [ b k T C q b k ] ≤ E ∣ b k ∣ 2 / ( c a ∗ 2 ( q − q k + 1 )) S S S r = q k r=q_k r = q k
E ∣ b k ∣ 2 = N ( S ( q k + 1 ) − S ( q k ) ) ≤ N ( S ( q ) − S ( q k ) ) ≤ 2 N ( q − q k ) = 4 N ( q − q k + 1 ) . \mathbb{E}|b_k|^2=N\bigl(S(q_{k+1})-S(q_k)\bigr)\le N\bigl(S(q)-S(q_k)\bigr)\le2N(q-q_k)=4N(q-q_{k+1}). E ∣ b k ∣ 2 = N ( S ( q k + 1 ) − S ( q k ) ) ≤ N ( S ( q ) − S ( q k ) ) ≤ 2 N ( q − q k ) = 4 N ( q − q k + 1 ) . The last two bounds imply that
N − 2 E [ b k T C q b k ] ≤ 4 N c a ∗ 2 = 4 M a ∗ 2 ( 0 ≤ k < K ) . N^{-2}\mathbb{E}[b_k^T C_qb_k]\le\frac{4}{Nca_*^2}=\frac{4}{Ma_*^2}\qquad(0\le k<K). N − 2 E [ b k T C q b k ] ≤ N c a ∗ 2 4 = M a ∗ 2 4 ( 0 ≤ k < K ) . Step 2: the last piece. Using ∥ C q ∥ o p ≤ N \lVert C_q\rVert_{\mathrm{op}}\le N ∥ C q ∥ op ≤ N (49) , then (6.2) and Lemma 6.2 with r = q K r=q_K r = q K
N − 2 E [ b K T C q b K ] ≤ N − 1 E ∣ b K ∣ 2 = S ( q ) − S ( q K ) ≤ 2 ( q − q K ) = 2 ( q − p ) 2 − K . N^{-2}\mathbb{E}[b_K^T C_qb_K]\le N^{-1}\mathbb{E}|b_K|^2=S(q)-S(q_K)\le2(q-q_K)=2(q-p)2^{-K}. N − 2 E [ b K T C q b K ] ≤ N − 1 E ∣ b K ∣ 2 = S ( q ) − S ( q K ) ≤ 2 ( q − q K ) = 2 ( q − p ) 2 − K . We sum the bounds of Steps 1 and 2 over the K + 1 K+1 K + 1
Step 3: the choice of K K K Let z = M ( q − p ) z=M(q-p) z = M ( q − p ) θ = log + z \theta=\log_+z θ = log + z K = ⌈ log 2 ( 2 + z ) ⌉ K=\lceil\log_2(2+z)\rceil K = ⌈ log 2 ( 2 + z )⌉ K ≥ 1 K\ge1 K ≥ 1 2 K ≥ 2 + z > z 2^K\ge2+z>z 2 K ≥ 2 + z > z
2 ( q − p ) 2 − K = 2 z M 2 K ≤ 2 M ≤ 2 M a ∗ 2 , 2(q-p)2^{-K}=\frac{2z}{M2^K}\le\frac{2}{M}\le\frac{2}{Ma_*^2}, 2 ( q − p ) 2 − K = M 2 K 2 z ≤ M 2 ≤ M a ∗ 2 2 , where we used a ∗ ≤ 1 a_* \le1 a ∗ ≤ 1 K K K (52) implies that
R ( p , q ) ≤ ( K + 1 ) ( 4 K + 2 ) M a ∗ 2 . \mathcal{R}(p,q) \le\frac{(K+1)(4K+2)}{Ma_*^2}. R ( p , q ) ≤ M a ∗ 2 ( K + 1 ) ( 4 K + 2 ) . Since 2 + z ≤ 3 max { 1 , z } 2+z \le3\max\{1,z\} 2 + z ≤ 3 max { 1 , z } log 2 ( 2 + z ) ≤ log 2 3 + θ / log 2 < 2 + 2 θ \log_2(2+z) \le\log_2 3+\theta/\log2 < 2+2\theta log 2 ( 2 + z ) ≤ log 2 3 + θ / log 2 < 2 + 2 θ log 2 3 < 2 \log_2 3<2 log 2 3 < 2 1 / log 2 < 2 1/\log2<2 1/ log 2 < 2 K < 1 + log 2 ( 2 + z ) < 3 + 2 θ K<1+\log_2(2+z)<3+2\theta K < 1 + log 2 ( 2 + z ) < 3 + 2 θ
( K + 1 ) ( 4 K + 2 ) ≤ ( 4 + 2 θ ) ( 14 + 8 θ ) = 56 + 60 θ + 16 θ 2 ≤ 56 ( 1 + θ ) 2 . (K+1)(4K+2) \le(4+2\theta)(14+8\theta)=56+60\theta+16\theta^2 \le56(1+\theta)^2. ( K + 1 ) ( 4 K + 2 ) ≤ ( 4 + 2 θ ) ( 14 + 8 θ ) = 56 + 60 θ + 16 θ 2 ≤ 56 ( 1 + θ ) 2 . This proves (53) .
The two powers of the logarithm in (53) have different sources: there are K K K K + 1 K+1 K + 1
An integrated bound for j j j
With p = 0 p=0 p = 0 j q = R ( 0 , q ) j_q=\mathcal{R}(0,q) j q = R ( 0 , q ) ( 1 + log + M ) 2 / ( M a ( q / 2 ) 2 ) (1+\log_+ M)^2/(Ma(q/2)^2) ( 1 + log + M ) 2 / ( M a ( q /2 ) 2 ) ν \nu ν s + v s+v s + v q = Q ( v ) q=Q(v) q = Q ( v ) a ( q / 2 ) a(q/2) a ( q /2 ) s s s m Q ( w ) m_{Q(w)} m Q ( w ) 0 < w < v 0<w<v 0 < w < v ( s + w ) 2 (s+w)^2 ( s + w ) 2 m 0 = 0 m_0=0 m 0 = 0 (23) , and its distance from m Q ( v ) m_{Q(v)} m Q ( v ) ( s + v ) − 2 (s+v)^{-2} ( s + v ) − 2 v < s v<s v < s
Proposition 6.4 (Covariance in the mean direction). Let μ = s δ 0 + ν \mu=s\delta_0+\nu μ = s δ 0 + ν F t , s \mathcal{F}_{t,s} F t , s v ∈ ( 0 , 1 − s ) v\in(0,1-s) v ∈ ( 0 , 1 − s )
j Q ( v ) ≤ 341 ( 1 + log + M ) 2 M ( s + v ) 2 , (54) j_{Q(v)} \le\frac{341(1+\log_+ M)^2}{M(s+v)^2}, \tag*{(54)} j Q ( v ) ≤ M ( s + v ) 2 341 ( 1 + log + M ) 2 , (54) and
J = ∫ j q ν ( d q ) ≤ 341 ( 1 + log + M ) 2 M s . (55) J=\int j_q\,\nu(dq)\le\frac{341(1+\log_+ M)^2}{Ms}. \tag*{(55)} J = ∫ j q ν ( d q ) ≤ M s 341 ( 1 + log + M ) 2 . (55) The constant is absolute. The bounds hold for all N ≥ 1 N\ge1 N ≥ 1 c > 0 c>0 c > 0 0 < s ≤ 1 / 2 0<s\le1/2 0 < s ≤ 1/2
Proof. Fix v ∈ ( 0 , 1 − s ) v\in(0,1-s) v ∈ ( 0 , 1 − s ) q = Q ( v ) q=Q(v) q = Q ( v ) L = 1 + log + M \mathcal{L}=1+\log_+ M L = 1 + log + M (42) , we have q ∈ supp ν q\in\operatorname{supp}\nu q ∈ supp ν Q ( w ) ≤ q Q(w)\le q Q ( w ) ≤ q w < v w<v w < v Q Q Q q = 0 q=0 q = 0 m q = m 0 = 0 m_q=m_0=0 m q = m 0 = 0 j q = 0 j_q=0 j q = 0 q > 0 q>0 q > 0 m q m_q m q m ˉ \bar{m} m ˉ Δ ˉ \bar{\Delta} Δ ˉ (56) below. Second, we bound the conditional covariance in the direction of m ˉ \bar{m} m ˉ (23) , using the conditional Jensen inequality on quantile fibers. Third, we bound the remainder by Lemma 6.3. Fourth, we combine the bounds and integrate over v v v
Step 1: the decomposition. We use the deterministic weights
r 0 = s 3 ( s + v ) 3 , k v ( w ) = 3 ( s + v ) 2 ( s + v ) 3 ( 0 < w < v ) , r_0 = \frac{s^3}{(s+v)^3}, \qquad k_v(w) = \frac{3(s+v)^2}{(s+v)^3} \quad(0 < w < v), r 0 = ( s + v ) 3 s 3 , k v ( w ) = ( s + v ) 3 3 ( s + v ) 2 ( 0 < w < v ) , which satisfy
r 0 + ∫ 0 v k v ( w ) d w = s 3 + ( ( s + v ) 3 − s 3 ) ( s + v ) 3 = 1. r_0 + \int_0^v k_v(w)\,dw = \frac{s^3 + ((s+v)^3-s^3)}{(s+v)^3} = 1. r 0 + ∫ 0 v k v ( w ) d w = ( s + v ) 3 s 3 + (( s + v ) 3 − s 3 ) = 1. We use the probability measure r 0 δ 0 + k v ( w ) d w r_0\delta_0 + k_v(w)\,dw r 0 δ 0 + k v ( w ) d w [ 0 , v ) [0,v) [ 0 , v ) m 0 = 0 m_0=0 m 0 = 0 m Q ( w ) m_{Q(w)} m Q ( w ) 0 < w < v 0<w<v 0 < w < v
m ˉ = ∫ 0 v k v ( w ) m Q ( w ) d w , L ˉ = ∫ 0 v k v ( w ) L Q ( w ) d w , \bar m = \int_0^v k_v(w)m_{Q(w)}\,dw, \qquad\bar L = \int_0^v k_v(w)L_{Q(w)}\,dw, m ˉ = ∫ 0 v k v ( w ) m Q ( w ) d w , L ˉ = ∫ 0 v k v ( w ) L Q ( w ) d w , Δ ˉ = r 0 m q + ∫ 0 v k v ( w ) ( m q − m Q ( w ) ) d w . \bar\Delta= r_0m_q + \int_0^v k_v(w)(m_q-m_{Q(w)})\,dw. Δ ˉ = r 0 m q + ∫ 0 v k v ( w ) ( m q − m Q ( w ) ) d w . By the normalization of the weights, m q = m ˉ + Δ ˉ m_q=\bar m+\bar\Delta m q = m ˉ + Δ ˉ m ˉ \bar m m ˉ D = 2 − 1 ∫ 0 v k v ( w ) ∣ m Q ( w ) ∣ 2 d w D=2^{-1}\int_0^v k_v(w)|m_{Q(w)}|^2\,dw D = 2 − 1 ∫ 0 v k v ( w ) ∣ m Q ( w ) ∣ 2 d w G q \mathcal{G}_q G q Q ( w ) ≤ q Q(w)\le q Q ( w ) ≤ q w < v w<v w < v N L ˉ = σ ⋅ m ˉ − D N\bar L=\sigma\cdot\bar m-D N L ˉ = σ ⋅ m ˉ − D Var ( L ˉ ∣ G q ) = N − 2 m ˉ T C q m ˉ \operatorname{Var}(\bar L\mid\mathcal{G}_q)=N^{-2}\bar m^{T}C_q\bar m Var ( L ˉ ∣ G q ) = N − 2 m ˉ T C q m ˉ
N − 2 E [ m ˉ T C q m ˉ ] = E Var ( L ˉ ∣ G q ) ≤ Var ( L ˉ ) . N^{-2}\mathbb{E}[\bar m^{T}C_q\bar m]=\mathbb{E}\operatorname{Var}(\bar L\mid\mathcal{G}_q)\le\operatorname{Var}(\bar L). N − 2 E [ m ˉ T C q m ˉ ] = E Var ( L ˉ ∣ G q ) ≤ Var ( L ˉ ) . Next, we write Δ ˉ = r 0 ( m q − m 0 ) + ∫ 0 v k v ( w ) ( m q − m Q ( w ) ) d w \bar\Delta=r_0(m_q-m_0)+\int_0^v k_v(w)(m_q-m_{Q(w)})\,dw Δ ˉ = r 0 ( m q − m 0 ) + ∫ 0 v k v ( w ) ( m q − m Q ( w ) ) d w m q − m 0 m_q-m_0 m q − m 0 m q − m Q ( w ) m_q-m_{Q(w)} m q − m Q ( w ) C q C_q C q
Δ ˉ T C q Δ ˉ ≤ r 0 ( m q − m 0 ) T C q ( m q − m 0 ) + ∫ 0 v k v ( w ) ( m q − m Q ( w ) ) T C q ( m q − m Q ( w ) ) d w . \bar\Delta^{T}C_q\bar\Delta\le r_0(m_q-m_0)^{T}C_q(m_q-m_0)+\int_0^v k_v(w)(m_q-m_{Q(w)})^{T}C_q(m_q-m_{Q(w)})\,dw. Δ ˉ T C q Δ ˉ ≤ r 0 ( m q − m 0 ) T C q ( m q − m 0 ) + ∫ 0 v k v ( w ) ( m q − m Q ( w ) ) T C q ( m q − m Q ( w ) ) d w . The integrand is bounded by 4 N 2 4N^2 4 N 2 N 2 N^2 N 2
N − 2 E [ Δ ˉ T C q Δ ˉ ] ≤ r 0 R ( 0 , q ) + ∫ 0 v k v ( w ) R ( Q ( w ) , q ) d w . N^{-2}\mathbb{E}[\bar\Delta^{T}C_q\bar\Delta]\le r_0\mathcal{R}(0,q)+\int_0^v k_v(w)\mathcal{R}(Q(w),q)\,dw. N − 2 E [ Δ ˉ T C q Δ ˉ ] ≤ r 0 R ( 0 , q ) + ∫ 0 v k v ( w ) R ( Q ( w ) , q ) d w . Finally, since C q ⪰ 0 C_q\succeq0 C q ⪰ 0 m q T C q m q ≤ 2 m ˉ T C q m ˉ + 2 Δ ˉ T C q Δ ˉ m_q^{T}C_qm_q\le2\bar m^{T}C_q\bar m+2\bar\Delta^{T}C_q\bar\Delta m q T C q m q ≤ 2 m ˉ T C q m ˉ + 2 Δ ˉ T C q Δ ˉ
j q ≤ 2 Var ( L ˉ ) + 2 r 0 R ( 0 , q ) + 2 ∫ 0 v k v ( w ) R ( Q ( w ) , q ) d w . (56) j_q\le2\operatorname{Var}(\bar L)+2r_0\mathcal{R}(0,q)+2\int_0^v k_v(w)\mathcal{R}(Q(w),q)\,dw. \tag*{(56)} j q ≤ 2 Var ( L ˉ ) + 2 r 0 R ( 0 , q ) + 2 ∫ 0 v k v ( w ) R ( Q ( w ) , q ) d w . (56) The quantity R ( 0 , q ) = j q \mathcal{R}(0,q)=j_q R ( 0 , q ) = j q
Step 2: the variance of the weighted prefix. We extend k v k_v k v ( 0 , 1 − s ) (0,1-s) ( 0 , 1 − s ) k v ≤ 3 / ( s + v ) k_v \le3/(s+v) k v ≤ 3/ ( s + v ) k v ( w ) d w k_v(w)\,dw k v ( w ) d w Q Q Q 3 / ( s + v ) 3/(s+v) 3/ ( s + v ) ν \nu ν κ v \kappa_v κ v ν \nu ν 0 ≤ κ v ≤ 3 / ( s + v ) 0 \le\kappa_v \le3/(s+v) 0 ≤ κ v ≤ 3/ ( s + v ) κ v ∈ L 2 ( ν ) \kappa_v \in L^2(\nu) κ v ∈ L 2 ( ν ) κ v \kappa_v κ v Q Q Q φ \varphi φ
∫ 0 1 − s k v ( w ) φ ( Q ( w ) ) d w = ∫ κ v φ d ν = ∫ 0 1 − s κ v ( Q ( w ) ) φ ( Q ( w ) ) d w . \int_0^{1-s} k_v(w)\varphi(Q(w))\,dw = \int\kappa_v\varphi\,d\nu= \int_0^{1-s} \kappa_v(Q(w))\varphi(Q(w))\,dw. ∫ 0 1 − s k v ( w ) φ ( Q ( w )) d w = ∫ κ v φ d ν = ∫ 0 1 − s κ v ( Q ( w )) φ ( Q ( w )) d w . Applied for each outcome to the bounded function p ↦ L p p \mapsto L_p p ↦ L p L ˉ = ∫ κ v ( p ) L p ν ( d p ) \bar L = \int\kappa_v(p)L_p\,\nu(dp) L ˉ = ∫ κ v ( p ) L p ν ( d p ) ( 0 , 1 − s ) (0,1-s) ( 0 , 1 − s ) κ v ∘ Q \kappa_v\circ Q κ v ∘ Q k v k_v k v σ \sigma σ Q Q Q p p p ν \nu ν r p r_p r p
κ v ( p ) = 1 r p ∫ ( F − ( p ) , F ( p ) ) k v ( w ) d w . \kappa_v(p)=\frac{1}{r_p}\int_{(F_-(p),F(p))} k_v(w)\,dw. κ v ( p ) = r p 1 ∫ ( F − ( p ) , F ( p )) k v ( w ) d w . with the notation of Section 5.1. A prefix ( 0 , v ) (0,v) ( 0 , v ) k v k_v k v 1 / a ( Q ) 1/a(Q) 1/ a ( Q ) Q Q Q
∫ κ v ( p ) 2 a ( p ) ν ( d p ) = ∫ 0 1 − s κ v ( Q ( w ) ) 2 a ( Q ( w ) ) d w ≤ ∫ 0 1 − s k v ( w ) 2 a ( Q ( w ) ) d w = ∫ 0 v k v ( w ) 2 a ( Q ( w ) ) d w . \int\frac{\kappa_v(p)^2}{a(p)}\,\nu(dp)=\int_0^{1-s}\frac{\kappa_v(Q(w))^2}{a(Q(w))}\,dw\le\int_0^{1-s}\frac{k_v(w)^2}{a(Q(w))}\,dw=\int_0^v\frac{k_v(w)^2}{a(Q(w))}\,dw. ∫ a ( p ) κ v ( p ) 2 ν ( d p ) = ∫ 0 1 − s a ( Q ( w )) κ v ( Q ( w ) ) 2 d w ≤ ∫ 0 1 − s a ( Q ( w )) k v ( w ) 2 d w = ∫ 0 v a ( Q ( w )) k v ( w ) 2 d w . As μ \mu μ k = κ v k=\kappa_v k = κ v a ( Q ( w ) ) ≥ s + w a(Q(w))\ge s+w a ( Q ( w )) ≥ s + w
Var ( L ˉ ) ≤ 1 M ∫ 0 v k v ( w ) 2 s + w d w = 9 M ( s + v ) 6 ∫ 0 v ( s + w ) 3 d w = 9 4 M ( s + v ) 2 [ 1 − s 4 ( s + v ) 4 ] ≤ 9 4 M ( s + v ) 2 . \begin{aligned}
\operatorname{Var}(\bar L)\le\frac{1}{M}\int_0^v\frac{k_v(w)^2}{s+w}\,dw=\frac{9}{M(s+v)^6}\int_0^v(s+w)^3\,dw \\
&=\frac{9}{4M(s+v)^2}\left[1-\frac{s^4}{(s+v)^4}\right]\le\frac{9}{4M(s+v)^2}.
\end{aligned} Var ( L ˉ ) ≤ M 1 ∫ 0 v s + w k v ( w ) 2 d w = M ( s + v ) 6 9 ∫ 0 v ( s + w ) 3 d w = 4 M ( s + v ) 2 9 [ 1 − ( s + v ) 4 s 4 ] ≤ 4 M ( s + v ) 2 9 . Step 3: the remainder. Let 0 < w < v 0<w<v 0 < w < v p = Q ( w ) p=Q(w) p = Q ( w ) p ≤ q p\le q p ≤ q a a a a ( ( p + q ) / 2 ) ≥ a ( p ) ≥ s + w a((p+q)/2)\ge a(p)\ge s+w a (( p + q ) /2 ) ≥ a ( p ) ≥ s + w a ( q / 2 ) ≥ s a(q/2)\ge s a ( q /2 ) ≥ s log + ( M ( q − p ) ) ≤ log + M \log_+(M(q-p))\le\log_+M log + ( M ( q − p )) ≤ log + M q ∈ supp ν q\in\operatorname{supp}\nu q ∈ supp ν 0 0 0 Q ( w ) Q(w) Q ( w )
R ( 0 , q ) ≤ 56 L 2 M s 2 , R ( Q ( w ) , q ) ≤ 56 L 2 M ( s + w ) 2 ( 0 < w < v ) . \mathcal{R}(0,q)\le\frac{56\mathcal{L}^2}{Ms^2},\qquad\mathcal{R}(Q(w),q)\le\frac{56\mathcal{L}^2}{M(s+w)^2}\quad(0<w<v). R ( 0 , q ) ≤ M s 2 56 L 2 , R ( Q ( w ) , q ) ≤ M ( s + w ) 2 56 L 2 ( 0 < w < v ) . Using r 0 s − 2 = s / ( s + v ) 3 r_0s^{-2}=s/(s+v)^3 r 0 s − 2 = s / ( s + v ) 3 k v ( w ) ( s + w ) − 2 = 3 / ( s + v ) 3 k_v(w)(s+w)^{-2}=3/(s+v)^3 k v ( w ) ( s + w ) − 2 = 3/ ( s + v ) 3
r 0 R ( 0 , q ) + ∫ 0 v k v ( w ) R ( Q ( w ) , q ) d w ≤ 56 L 2 M ⋅ s + 3 v ( s + v ) 3 ≤ 168 L 2 M ( s + v ) 2 , r_0\mathcal{R}(0,q)+\int_0^v k_v(w)\mathcal{R}(Q(w),q)\,dw\le\frac{56\mathcal{L}^2}{M}\cdot\frac{s+3v}{(s+v)^3}\le\frac{168\mathcal{L}^2}{M(s+v)^2}, r 0 R ( 0 , q ) + ∫ 0 v k v ( w ) R ( Q ( w ) , q ) d w ≤ M 56 L 2 ⋅ ( s + v ) 3 s + 3 v ≤ M ( s + v ) 2 168 L 2 , where we used s + 3 v ≤ 3 ( s + v ) s+3v\le3(s+v) s + 3 v ≤ 3 ( s + v )
Step 4: conclusion. Inserting the bounds of Steps 2 and 3 into (56) and using L ≥ 1 \mathcal{L} \ge1 L ≥ 1
j Q ( v ) ≤ 9 2 M ( s + v ) 2 + 336 L 2 M ( s + v ) 2 ≤ ( 9 / 2 + 336 ) L 2 M ( s + v ) 2 ≤ 341 L 2 M ( s + v ) 2 . j_Q(v) \le\frac{9}{2M(s+v)^2}+\frac{336\mathcal{L}^2}{M(s+v)^2}\le\frac{(9/2+336)\mathcal{L}^2}{M(s+v)^2}\le\frac{341\mathcal{L}^2}{M(s+v)^2}. j Q ( v ) ≤ 2 M ( s + v ) 2 9 + M ( s + v ) 2 336 L 2 ≤ M ( s + v ) 2 ( 9/2 + 336 ) L 2 ≤ M ( s + v ) 2 341 L 2 . which is (54) . Finally, J = ∫ 0 1 − s j Q ( v ) d v J=\int_0^{1-s}j_Q(v)\,\mathrm{d}v J = ∫ 0 1 − s j Q ( v ) d v Q Q Q ( 0 , 1 − s ) (0,1-s) ( 0 , 1 − s ) ν \nu ν (55) follows from
∫ 0 1 − s d v ( s + v ) 2 = 1 s − 1 ≤ 1 s . \int_0^{1-s}\frac{\mathrm{d}v}{(s+v)^2}=\frac{1}{s}-1\le\frac{1}{s}. ∫ 0 1 − s ( s + v ) 2 d v = s 1 − 1 ≤ s 1 . This completes the proof. □ \square □
The minimizing profile and the scalar response
Isolation of the minimizing profile
In this section and the next, we study an arbitrary minimizer γ ∗ \gamma_* γ ∗ F ρ F_\rho F ρ § 4.1 . We compare γ ∗ \gamma_* γ ∗ γ λ t r = max { λ , γ β } \gamma_\lambda^{\mathrm{tr}}=\max\{\lambda,\gamma_\beta\} γ λ tr = max { λ , γ β } η \eta η F ρ ( γ λ t r ) − F ρ ( γ ∗ ) F_\rho(\gamma_\lambda^{\mathrm{tr}})-F_\rho(\gamma_*) F ρ ( γ λ tr ) − F ρ ( γ ∗ ) L 1 L^1 L 1 γ ∗ \gamma_* γ ∗ γ λ t r \gamma_\lambda^{\mathrm{tr}} γ λ tr C min { 1 , ρ − 3 / 4 η 1 / 4 } C\min\{1,\rho^{-3/4}\eta^{1/4}\} C min { 1 , ρ − 3/4 η 1/4 } C C C β \beta β γ ∗ \gamma_* γ ∗
We use the energy normalization, the tilted joint law, and the martingale identities of § 4.1 , in particular Proposition 3.1, and the trial budget of Lemma 3.8. From [4 ] , we import the convexity of F ρ F_\rho F ρ [4 Lemma 7.5] (Lemma 7.3(a)); and the linear growth of the scalar Parisi profile near the origin (61) . From [48], we use the structure of the scalar Parisi measure (62) . To prove Proposition 7.6, we first smooth the minimizer (Lemma 7.3(b)) and then join it to the trial profile by a segment. With cutoffs of one sign, built from one-sided averages of the superlevel sets of the difference of the two profiles, we turn the response inequality into lower bounds on the second derivative of F ρ F_\rho F ρ (65) then compares these lower bounds with the trial budget.
Proposition 7.1, Lemma 7.3(a), Lemma 7.4, the midpoint inequality (65) , and the facts (61) , (62) , and (63) are used again in Section 8. Section 9 uses (61) , (62) , (63) , and Proposition 7.6.
Energy normalization and variational facts
Throughout this section and Section 8, we fix 1 < β < ∞ 1<\beta<\infty 1 < β < ∞ N ≥ 1 N\ge1 N ≥ 1 0 < ρ ≤ 1 0<\rho\le1 0 < ρ ≤ 1 t = 1 − ρ t=1-\rho t = 1 − ρ 0 < λ ≤ λ 0 0<\lambda\le\lambda_0 0 < λ ≤ λ 0 λ 0 ≤ min { 1 , β / 4 } \lambda_0\le\min\{1,\beta/4\} λ 0 ≤ min { 1 , β /4 } (31) , and s = λ / β s=\lambda/\beta s = λ / β β \beta β N N N ρ \rho ρ λ \lambda λ
Admissible profiles and segments. We use the energy normalization of § 4.1 . An admissible profile is a function γ = β a \gamma=\beta a γ = β a a ( q ) = μ ( [ 0 , q ] ) a(q)=\mu([0,q]) a ( q ) = μ ([ 0 , q ]) μ = s δ 0 + ν \mu=s\delta_0+\nu μ = s δ 0 + ν § 4.1 . The function γ \gamma γ γ ( 1 ) = β \gamma(1)=\beta γ ( 1 ) = β γ ( 0 − ) = 0 \gamma(0-)=0 γ ( 0 − ) = 0 d γ d\gamma d γ β \beta β d γ ≥ λ δ 0 d\gamma\geq\lambda\delta_0 d γ ≥ λ δ 0
ν γ = d γ − λ δ 0 = β ν \nu_\gamma=d\gamma-\lambda\delta_0=\beta\nu ν γ = d γ − λ δ 0 = β ν the free measure of γ \gamma γ β − λ \beta-\lambda β − λ min F ρ \min F_\rho min F ρ F ρ F_\rho F ρ § 4.1 ). An admissible profile γ ∗ \gamma_* γ ∗ minimizer if F ρ ( γ ∗ ) = min F ρ F_\rho(\gamma_*)=\min F_\rho F ρ ( γ ∗ ) = min F ρ segment is a family γ θ = ( 1 − θ ) γ 0 + θ γ 1 \gamma_\theta=(1-\theta)\gamma_0+\theta\gamma_1 γ θ = ( 1 − θ ) γ 0 + θ γ 1 0 ≤ θ ≤ 1 0\leq\theta\leq1 0 ≤ θ ≤ 1 γ 0 , γ 1 \gamma_0,\gamma_1 γ 0 , γ 1 ν γ θ = ( 1 − θ ) ν γ 0 + θ ν γ 1 \nu_{\gamma_\theta}=(1-\theta)\nu_{\gamma_0}+\theta\nu_{\gamma_1} ν γ θ = ( 1 − θ ) ν γ 0 + θ ν γ 1 θ \theta θ
Self-overlap, trace, and curvature. For an admissible profile γ \gamma γ E γ \mathbb{E}_\gamma E γ § 4.1.3 with profile a = γ / β a=\gamma/\beta a = γ / β
S γ ( q ) = E γ S q , b γ ( q ) = β N E γ Tr H q , ζ γ ( q ) = β 2 N E γ Tr H q 2 . S_\gamma(q)=\mathbb{E}_\gamma S_q,\qquad b_\gamma(q)=\frac{\beta}{N}\mathbb{E}_\gamma\operatorname{Tr}H_q,\qquad\zeta_\gamma(q)=\frac{\beta^2}{N}\mathbb{E}_\gamma\operatorname{Tr}H_q^2. S γ ( q ) = E γ S q , b γ ( q ) = N β E γ Tr H q , ζ γ ( q ) = N β 2 E γ Tr H q 2 . By Proposition 3.1 , we have m 0 = 0 m_0=0 m 0 = 0 d m q = c H q d B q dm_q=\sqrt{c}H_q\,dB_q d m q = c H q d B q a ( q ) C q ⪯ H q ⪯ C q a(q)C_q\preceq H_q\preceq C_q a ( q ) C q ⪯ H q ⪯ C q q ↦ E γ S q q\mapsto\mathbb{E}_\gamma S_q q ↦ E γ S q ( c / N ) E γ Tr H q 2 (c/N)\mathbb{E}_\gamma\operatorname{Tr}H_q^2 ( c / N ) E γ Tr H q 2 c = β 2 ρ c=\beta^2\rho c = β 2 ρ ρ ζ γ ( q ) \rho\zeta_\gamma(q) ρ ζ γ ( q ) m q m_q m q [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] 0 ≤ S q ≤ 1 0\leq S_q\leq1 0 ≤ S q ≤ 1 Tr C q = ∑ i ( 1 − m q , i 2 ) ≤ N \operatorname{Tr}C_q=\sum_i(1-m_{q,i}^2)\leq N Tr C q = ∑ i ( 1 − m q , i 2 ) ≤ N 0 ⪯ H q ⪯ C q 0\preceq H_q\preceq C_q 0 ⪯ H q ⪯ C q 0 ≤ Tr H q ≤ Tr C q 0\leq\operatorname{Tr}H_q\leq\operatorname{Tr}C_q 0 ≤ Tr H q ≤ Tr C q
S γ ( 0 ) = 0 , 0 ≤ S γ ≤ 1 , S γ ′ = ρ ζ γ ≥ 0 , 0 ≤ b γ ≤ β . (57) S_\gamma(0)=0,\qquad0\leq S_\gamma\leq1,\qquad S_\gamma'=\rho\zeta_\gamma\geq0,\qquad0\leq b_\gamma\leq\beta. \tag*{(57)} S γ ( 0 ) = 0 , 0 ≤ S γ ≤ 1 , S γ ′ = ρ ζ γ ≥ 0 , 0 ≤ b γ ≤ β . (57) Note that S γ ( q ) = ρ ∫ 0 q ζ γ ( r ) d r S_\gamma(q)=\rho\int_0^q\zeta_\gamma(r)\,dr S γ ( q ) = ρ ∫ 0 q ζ γ ( r ) d r S γ S_\gamma S γ 0 ⪯ H q ⪯ C q 0\preceq H_q\preceq C_q 0 ⪯ H q ⪯ C q Tr H q 2 ≤ Tr C q 2 ≤ ( Tr C q ) 2 ≤ N 2 \operatorname{Tr}H_q^2\leq\operatorname{Tr}C_q^2\leq(\operatorname{Tr}C_q)^2\leq N^2 Tr H q 2 ≤ Tr C q 2 ≤ ( Tr C q ) 2 ≤ N 2 0 ≤ ζ γ ≤ β 2 N 0\leq\zeta_\gamma\leq\beta^2N 0 ≤ ζ γ ≤ β 2 N N N N S γ S_\gamma S γ b γ b_\gamma b γ ζ γ \zeta_\gamma ζ γ
The rescaled functional of [4 ] . The variational facts we need are proved in [4 ] , § 7.1 and § 8.2.2 and Appendix A, for the rescaled form of F ρ F_\rho F ρ § 4.1 , and we now check this identification in detail. Let γ \gamma γ U = U μ , t U=U_{\mu,t} U = U μ , t (15) with a = γ / β a=\gamma/\beta a = γ / β U ~ ( q , y ; W ) = β − 1 U ( q , β y ; W ) \widetilde U(q,y;W)=\beta^{-1}U(q,\beta y;W) U ( q , y ; W ) = β − 1 U ( q , β y ; W ) § 4.1 ; when we need to indicate the profile, we write U ~ γ \widetilde U_\gamma U γ § 4.1 that U ~ \widetilde U U ∂ q U ~ = − ( ρ / 2 ) ( Δ y U ~ + γ ( q ) ∣ ∇ y U ~ ∣ 2 ) \partial_q\widetilde U=-(\rho/2)(\Delta_y\widetilde U+\gamma(q)|\nabla_y\widetilde U|^2) ∂ q U = − ( ρ /2 ) ( Δ y U + γ ( q ) ∣ ∇ y U ∣ 2 ) ∇ y U ~ ( q , y ; W ) = ∇ x U ( q , β y ; W ) \nabla_y\widetilde U(q,y;W)=\nabla_xU(q,\beta y;W) ∇ y U ( q , y ; W ) = ∇ x U ( q , β y ; W ) ∇ y 2 U ~ ( q , y ; W ) = β ∇ x 2 U ( q , β y ; W ) \nabla_y^2\widetilde U(q,y;W)=\beta\nabla_x^2U(q,\beta y;W) ∇ y 2 U ( q , y ; W ) = β ∇ x 2 U ( q , β y ; W ) F ρ ( γ ) F_\rho(\gamma) F ρ ( γ ) ( N λ ) − 1 log E r e f e λ U ~ ( 0 , 0 ; W ) − ( ρ / 2 ) ∫ 0 1 q γ ( q ) d q (N\lambda)^{-1}\log\mathbb{E}_{\mathrm{ref}}e^{\lambda\widetilde U(0,0;W)}-(\rho/2)\int_0^1q\gamma(q)\,dq ( N λ ) − 1 log E ref e λ U ( 0 , 0 ; W ) − ( ρ /2 ) ∫ 0 1 q γ ( q ) d q (15) , the terminal value of U ~ \widetilde U U
U ~ ( 1 , y ; W ) = 1 β log ∑ σ ∈ { − 1 , 1 } N exp { β ( t σ T W σ 2 + y ⋅ σ ) } . \widetilde U(1,y;W)=\frac{1}{\beta}\log\sum_{\sigma\in\{-1,1\}^N}\exp\left\{\beta\left(\frac{\sqrt{t}\,\sigma^{T}W\sigma}{2}+y\cdot\sigma\right)\right\}. U ( 1 , y ; W ) = β 1 log σ ∈ { − 1 , 1 } N ∑ exp { β ( 2 t σ T W σ + y ⋅ σ ) } . Isolation of the minimizing profile
Let W o f f W^{\mathrm{off}} W off W W W σ ∈ { − 1 , 1 } N \sigma\in\{-1,1\}^{N} σ ∈ { − 1 , 1 } N σ T W σ / 2 = σ T W o f f σ / 2 + D N \sigma^{T}W\sigma/2=\sigma^{T}W^{\mathrm{off}}\sigma/2+D_N σ T W σ /2 = σ T W off σ /2 + D N D N = Tr W / 2 D_N=\operatorname{Tr}W/2 D N = Tr W /2 U ~ \widetilde U U U ~ ( q , y ; W ) = t D N + U ~ o f f ( q , y ) \widetilde U(q,y;W)=\sqrt{t}D_N+\widetilde U^{\mathrm{off}}(q,y) U ( q , y ; W ) = t D N + U off ( q , y ) U ~ o f f \widetilde U^{\mathrm{off}} U off W o f f W^{\mathrm{off}} W off W W W u u u [4 (7.2)] with r = ρ r=\rho r = ρ t W o f f \sqrt{t}W^{\mathrm{off}} t W off D N D_N D N 1 / 2 1/2 1/2 W o f f W^{\mathrm{off}} W off ( N λ ) − 1 log E r e f e λ t D N = λ t / ( 4 N ) (N\lambda)^{-1}\log\mathbb{E}_{\mathrm{ref}}e^{\lambda\sqrt{t}D_N}=\lambda t/(4N) ( N λ ) − 1 log E ref e λ t D N = λ t / ( 4 N ) F ρ F_\rho F ρ
F ρ ( γ ) = 1 N λ log E r e f e λ U ~ o f f ( 0 , 0 ) − ρ 2 ∫ 0 1 q γ ( q ) d q + λ t 4 N . F_\rho(\gamma)=\frac{1}{N\lambda}\log\mathbb{E}_{\mathrm{ref}}e^{\lambda\widetilde U^{\mathrm{off}}(0,0)}-\frac{\rho}{2}\int_0^1q\gamma(q)\,dq+\frac{\lambda t}{4N}. F ρ ( γ ) = N λ 1 log E ref e λ U off ( 0 , 0 ) − 2 ρ ∫ 0 1 q γ ( q ) d q + 4 N λ t . This is exactly the functional F t c ( γ ) F_t^c(\gamma) F t c ( γ ) [4 (7.2)] , which is written F r F_r F r [4 Section 7.4] ; the term λ t / ( 4 N ) \lambda t/(4N) λ t / ( 4 N ) γ \gamma γ
The joint laws also agree. The tilt e s U ( 0 , 0 ; W ) = e λ t D N e λ U ~ o f f ( 0 , 0 ) e^{sU(0,0;W)}=e^{\lambda\sqrt{t}D_N}e^{\lambda\widetilde U^{\mathrm{off}}(0,0)} e s U ( 0 , 0 ; W ) = e λ t D N e λ U off ( 0 , 0 ) D N D_N D N (18) , whose drift involves ∇ x U \nabla_xU ∇ x U W W W t W o f f \sqrt{t}W^{\mathrm{off}} t W off Q γ Q_\gamma Q γ [4 Appendix A] , and the rescaled field X q / β X_q/\beta X q / β
d ( X q / β ) = ρ γ ( q ) ∇ y U ~ ( q , X q / β ; W ) d q + ρ d B q , X 0 / β = 0. \mathrm{d}(X_q/\beta)=\rho\gamma(q)\nabla_y\widetilde U(q,X_q/\beta;W)\,\mathrm{d}q+\sqrt{\rho}\,\mathrm{d}B_q,\qquad X_0/\beta=0. d ( X q / β ) = ρ γ ( q ) ∇ y U ( q , X q / β ; W ) d q + ρ d B q , X 0 / β = 0. which is the field equation of [4 Section 7.1] . Under this law, the magnetization m q = ∇ x U ( q , X q ; W ) = ∇ y U ~ ( q , X q / β ; W ) m_q=\nabla_xU(q,X_q;W)=\nabla_y\widetilde U(q,X_q/\beta;W) m q = ∇ x U ( q , X q ; W ) = ∇ y U ( q , X q / β ; W ) ∇ y 2 U ~ ( q , X q / β ; W ) = β H q \nabla_y^2\widetilde U(q,X_q/\beta;W)=\beta H_q ∇ y 2 U ( q , X q / β ; W ) = β H q S γ S_\gamma S γ b γ b_\gamma b γ ζ γ \zeta_\gamma ζ γ S S S b b b ζ \zeta ζ [4 Sections 7.1 and 7.4] , and that b γ b_\gamma b γ ζ γ \zeta_\gamma ζ γ [4 Appendix A] . We use the results proved there in the following form.
Proposition 7.1 (Variational and response facts). Let γ \gamma γ G γ ( x ) = ∫ x 1 ( S γ ( q ) − q ) d q G_\gamma(x)=\int_x^1(S_\gamma(q)-q)\,dq G γ ( x ) = ∫ x 1 ( S γ ( q ) − q ) d q 0 ≤ x ≤ 1 0\le x\le1 0 ≤ x ≤ 1
(a) Let ν ′ ≥ 0 \nu'\ge0 ν ′ ≥ 0 [ 0 , 1 ] [0,1] [ 0 , 1 ] β − λ \beta-\lambda β − λ F ρ F_\rho F ρ γ \gamma γ ν ′ \nu' ν ′ γ \gamma γ
D + F ρ ( γ ) [ ν ′ − ν γ ] = ρ 2 ∫ G γ d ( ν ′ − ν γ ) . (58) D_+F_\rho(\gamma)[\nu'-\nu_\gamma]=\frac{\rho}{2}\int G_\gamma\,\mathrm{d}(\nu'-\nu_\gamma). \tag*{(58)} D + F ρ ( γ ) [ ν ′ − ν γ ] = 2 ρ ∫ G γ d ( ν ′ − ν γ ) . (58) The function G γ G_\gamma G γ G γ ′ ( x ) = x − S γ ( x ) G'_\gamma(x)=x-S_\gamma(x) G γ ′ ( x ) = x − S γ ( x ) G γ ′ ( 0 ) = 0 G'_\gamma(0)=0 G γ ′ ( 0 ) = 0 x ↦ G γ ′ ( x ) − x x\mapsto G'_\gamma(x)-x x ↦ G γ ′ ( x ) − x
(b) Let ( γ θ ) (\gamma_\theta) ( γ θ ) ψ = γ 1 − γ 0 \psi=\gamma_1-\gamma_0 ψ = γ 1 − γ 0 f ( θ ) = F ρ ( γ θ ) f(\theta)=F_\rho(\gamma_\theta) f ( θ ) = F ρ ( γ θ ) f f f [ 0 , 1 ] [0,1] [ 0 , 1 ] ( 0 , 1 ) (0,1) ( 0 , 1 ) θ ∈ ( 0 , 1 ) \theta\in(0,1) θ ∈ ( 0 , 1 ) b θ = b γ θ b_\theta=b_{\gamma_\theta} b θ = b γ θ ζ θ = ζ γ θ \zeta_\theta=\zeta_{\gamma_\theta} ζ θ = ζ γ θ φ \varphi φ ( 0 , 1 ) (0,1) ( 0 , 1 )
Y = ∫ 0 1 b θ ∣ φ ′ ∣ 2 d q , Z = ρ ∫ ( 0 , 1 ) φ 2 ζ θ d γ θ . Y=\int_0^1 b_\theta|\varphi'|^2\,dq,\qquad Z=\rho\int_{(0,1)}\varphi^2\zeta_\theta\,\mathrm{d}\gamma_\theta. Y = ∫ 0 1 b θ ∣ φ ′ ∣ 2 d q , Z = ρ ∫ ( 0 , 1 ) φ 2 ζ θ d γ θ . Then
ρ ∣ ∫ 0 1 ψ ζ γ φ d q ∣ ≤ ( f ′ ′ ( θ ) ρ ) 1 / 2 [ γ 1 / 2 + 2 ( γ + Z ) 1 / 2 ] . (59) \rho\left|\int_0^1 \psi\zeta_\gamma\varphi\,\mathrm{d}q\right| \le\left(\frac{f''(\theta)}{\rho}\right)^{1/2}\left[\gamma^{1/2}+2(\gamma+Z)^{1/2}\right]. \tag*{(59)} ρ ∫ 0 1 ψ ζ γ φ d q ≤ ( ρ f ′′ ( θ ) ) 1/2 [ γ 1/2 + 2 ( γ + Z ) 1/2 ] . (59) (c) There exists k β > 0 k_\beta>0 k β > 0 β \beta β
ζ γ ( q ) ≥ k β 2 ( 0 ≤ q < 1 ) . (60) \zeta_\gamma(q) \ge k_\beta^2 \qquad(0\le q<1). \tag*{(60)} ζ γ ( q ) ≥ k β 2 ( 0 ≤ q < 1 ) . (60) The integral Z Z Z ( 0 , 1 ) (0,1) ( 0 , 1 ) d γ θ \mathrm{d}\gamma_\theta d γ θ ( 0 , 1 ) (0,1) ( 0 , 1 )
Proof. We prove the three parts in order. By the discussion preceding the proposition, F ρ F_\rho F ρ F r F_r F r [4 Section 7.4] with r = ρ r=\rho r = ρ S γ S_\gamma S γ b γ b_\gamma b γ ζ γ \zeta_\gamma ζ γ
Part (a). The segment from γ \gamma γ ν ′ \nu' ν ′ ν γ \nu_\gamma ν γ ( 1 − θ ) ν γ + θ ν ′ (1-\theta)\nu_\gamma+\theta\nu' ( 1 − θ ) ν γ + θ ν ′ F r F_r F r [4 Section 7.4, (7.22)] , with G γ G_\gamma G γ β \beta β ν γ = β ν \nu_\gamma=\beta\nu ν γ = β ν
β − 1 ( c / 2 ) ∫ G d ( ν ′ / β − ν ) = ( ρ / 2 ) ∫ G d ( ν ′ − ν γ ) , \beta^{-1}(c/2)\int G\,\mathrm{d}(\nu'/\beta-\nu)=(\rho/2)\int G\,\mathrm{d}(\nu'-\nu_\gamma), β − 1 ( c /2 ) ∫ G d ( ν ′ / β − ν ) = ( ρ /2 ) ∫ G d ( ν ′ − ν γ ) , and the function G G G G γ G_\gamma G γ S γ S_\gamma S γ G γ G_\gamma G γ G γ ′ ( x ) = x − S γ ( x ) G'_\gamma(x)=x-S_\gamma(x) G γ ′ ( x ) = x − S γ ( x ) G γ ′ ( 0 ) = 0 G'_\gamma(0)=0 G γ ′ ( 0 ) = 0 G γ ′ ( x ) − x = − S γ ( x ) G'_\gamma(x)-x=-S_\gamma(x) G γ ′ ( x ) − x = − S γ ( x ) [4 (7.23)] .
Part (b). Applying (a) with γ = γ 0 \gamma=\gamma_0 γ = γ 0 ν ′ = ν γ 1 \nu'=\nu_{\gamma_1} ν ′ = ν γ 1 f f f [4 Section 7.4] states that
sup q , y ∣ U ~ γ o f f ( q , y ) − U ~ γ ′ o f f ( q , y ) ∣ ≤ 3 ρ N 2 ∥ γ − γ ′ ∥ L 1 ( 0 , 1 ) . \sup_{q,y}\left|\widetilde U^{\mathrm{off}}_\gamma(q,y)-\widetilde U^{\mathrm{off}}_{\gamma'}(q,y)\right| \le\frac{3\rho N}{2}\|\gamma-\gamma'\|_{L^1(0,1)}. q , y sup U γ off ( q , y ) − U γ ′ off ( q , y ) ≤ 2 3 ρN ∥ γ − γ ′ ∥ L 1 ( 0 , 1 ) . for admissible γ , γ ′ \gamma,\gamma' γ , γ ′ F ρ F_\rho F ρ
∣ F ρ ( γ ) − F ρ ( γ ′ ) ∣ ≤ 2 ρ ∥ γ − γ ′ ∥ L 1 ( 0 , 1 ) , |F_\rho(\gamma)-F_\rho(\gamma')|\le2\rho\|\gamma-\gamma'\|_{L^1(0,1)}, ∣ F ρ ( γ ) − F ρ ( γ ′ ) ∣ ≤ 2 ρ ∥ γ − γ ′ ∥ L 1 ( 0 , 1 ) , where the outer logarithmic moment contributes 3 ρ / 2 3\rho/2 3 ρ /2 ρ / 2 \rho/2 ρ /2 f f f [ 0 , 1 ] [0,1] [ 0 , 1 ] ∥ γ θ − γ θ ′ ∥ L 1 = ∣ θ − θ ′ ∣ ∥ γ 1 − γ 0 ∥ L 1 \|\gamma_\theta-\gamma_{\theta'}\|_{L^1}=|\theta-\theta'|\|\gamma_1-\gamma_0\|_{L^1} ∥ γ θ − γ θ ′ ∥ L 1 = ∣ θ − θ ′ ∣∥ γ 1 − γ 0 ∥ L 1
We now apply [4 Proposition A.1 and Lemma A.2] with r = ρ r=\rho r = ρ γ 0 , γ 1 \gamma_0,\gamma_1 γ 0 , γ 1 v = ψ v=\psi v = ψ N N N 0 ≤ t < 1 0\le t<1 0 ≤ t < 1 [4 Appendix A] (λ ≤ γ i ≤ β \lambda\le\gamma_i\le\beta λ ≤ γ i ≤ β [ 0 , 1 ) [0,1) [ 0 , 1 ) γ i ( 0 − ) = 0 \gamma_i(0-)=0 γ i ( 0 − ) = 0 γ i ( 1 ) = β \gamma_i(1)=\beta γ i ( 1 ) = β λ \lambda λ [4 Proposition A.1] allows their additional masses at 0 to differ. By [4 Proposition A.1, (A.1)] , f f f f ′ ′ ( θ ) f''(\theta) f ′′ ( θ ) ρ ∫ 0 1 e θ ( q ) d q \rho\int_0^1 e_\theta(q)\,\mathrm{d}q ρ ∫ 0 1 e θ ( q ) d q e θ ≥ 0 e_\theta\ge0 e θ ≥ 0
∫ 0 1 e θ ( q ) d q ≤ f ′ ′ ( θ ) ρ . \int_0^1 e_\theta(q)\,\mathrm{d}q\le\frac{f''(\theta)}{\rho}. ∫ 0 1 e θ ( q ) d q ≤ ρ f ′′ ( θ ) . The pairing estimate [4 (A.2)] , applied to φ \varphi φ − φ -\varphi − φ [4 (A.3)] give a nonnegative function J θ J_{\theta} J θ [4 Proposition A.1] , such that
ρ ∣ ∫ 0 1 ψ ζ θ φ d q ∣ ≤ ( ∫ 0 1 e θ ) 1 / 2 [ γ 1 / 2 + 2 ( ∫ 0 1 J θ φ 2 ) 1 / 2 ] , ∫ 0 1 J θ φ 2 ≤ Y + Z . \rho\left|\int_0^1 \psi_{\zeta_{\theta}}\varphi\,dq\right| \le\left(\int_0^1 e_{\theta}\right)^{1/2}\left[\gamma^{1/2}+2\left(\int_0^1 J_{\theta}\varphi^2\right)^{1/2}\right], \qquad\int_0^1 J_{\theta}\varphi^2 \le Y+Z. ρ ∫ 0 1 ψ ζ θ φ d q ≤ ( ∫ 0 1 e θ ) 1/2 [ γ 1/2 + 2 ( ∫ 0 1 J θ φ 2 ) 1/2 ] , ∫ 0 1 J θ φ 2 ≤ Y + Z . Combining the last three displays, we obtain (59) .
Part (c). We use [4 Lemma A.3, (A.5)] with the level in (A.5) equal to β \beta β β ≥ 1 \beta\ge1 β ≥ 1 max { 1 , β } = β \max\{1,\beta\}=\beta max { 1 , β } = β γ ( q ) ≤ β \gamma(q)\le\beta γ ( q ) ≤ β q < 1 q<1 q < 1 E γ [ β H q ] ⪰ k β I \mathbb{E}_{\gamma}[\beta H_q]\succeq k_{\beta}I E γ [ β H q ] ⪰ k β I
k β = e − 16 c ′ β − 2 β 2 2 ( 1 + c ′ ) , c ′ = π / 2 . k_{\beta}=\frac{e^{-16c'\beta-2\beta^2}}{2(1+c')}, \qquad c'=\sqrt{\pi/2}. k β = 2 ( 1 + c ′ ) e − 16 c ′ β − 2 β 2 , c ′ = π /2 . Every admissible profile satisfies γ ≤ β \gamma\le\beta γ ≤ β H ↦ Tr H 2 H\mapsto\operatorname{Tr}H^2 H ↦ Tr H 2 ζ γ ( q ) = N − 1 E γ Tr ( β H q ) 2 ≥ N − 1 Tr ( E γ β H q ) 2 ≥ k β 2 \zeta_{\gamma}(q)=N^{-1}\mathbb{E}_{\gamma}\operatorname{Tr}(\beta H_q)^2\ge N^{-1}\operatorname{Tr}(\mathbb{E}_{\gamma}\beta H_q)^2\ge k_{\beta}^2 ζ γ ( q ) = N − 1 E γ Tr ( β H q ) 2 ≥ N − 1 Tr ( E γ β H q ) 2 ≥ k β 2 (60) .
In [4 ] , these results are first proved for step profiles, and then extended to all admissible profiles by approximation at fixed N N N β \beta β [4 Section 7.4 and Appendix A.3] . This includes profiles with atoms, and the Stieltjes integral in Z Z Z d γ θ d\gamma_{\theta} d γ θ ( 0 , 1 ) (0,1) ( 0 , 1 )
Remark 7.2 (Atoms and the Stieltjes term). The term Z Z Z (59) . In the proof of [4 Lemma A.2] , the Fisher estimate (A.3) follows from the identity of measures d P = 2 J θ d q − ρ ζ θ d γ θ d\mathcal{P}=2J_{\theta}\,dq-\rho\zeta_{\theta}\,d\gamma_{\theta} d P = 2 J θ d q − ρ ζ θ d γ θ ( 0 , 1 ) (0,1) ( 0 , 1 ) [4 (A.19)] , where P ( q ) = N − 1 E θ Tr h ˙ \mathcal{P}(q)=N^{-1}\mathbb{E}_{\theta}\operatorname{Tr}\dot{h} P ( q ) = N − 1 E θ Tr h ˙ h ˙ \dot{h} h ˙ q q q U ~ γ θ \widetilde{U}_{\gamma_{\theta}} U γ θ d γ θ d\gamma_{\theta} d γ θ q q q P \mathcal{P} P − ρ ζ θ ( q ) d γ θ ( { q } ) -\rho\zeta_{\theta}(q)d\gamma_{\theta}(\{q\}) − ρ ζ θ ( q ) d γ θ ({ q }) d γ θ d\gamma_{\theta} d γ θ
The scalar reference profile
Let μ β \mu_{\beta} μ β α β = μ β ( [ 0 , ⋅ ] ) \alpha_{\beta}=\mu_{\beta}([0,\cdot]) α β = μ β ([ 0 , ⋅ ]) γ β = β α β \gamma_{\beta}=\beta\alpha_{\beta} γ β = β α β γ t r = max { λ , γ β } \gamma^{\mathrm{tr}}=\max\{\lambda,\gamma_{\beta}\} γ tr = max { λ , γ β } λ \lambda λ γ β \gamma_{\beta} γ β γ ↦ β − 1 P β ( γ / β ) \gamma\mapsto\beta^{-1}P_{\beta}(\gamma/\beta) γ ↦ β − 1 P β ( γ / β ) [4 Section 7.1] . By [4 Lemma 7.2] , applied with β 0 = β \beta_0=\beta β 0 = β T ∗ , k 0 > 0 T_*,k_0>0 T ∗ , k 0 > 0 C 0 < ∞ C_0<\infty C 0 < ∞ β \beta β γ β ( 0 ) = 0 \gamma_{\beta}(0)=0 γ β ( 0 ) = 0 γ β \gamma_{\beta} γ β [ 0 , T ∗ ] [0,T_*] [ 0 , T ∗ ]
k 0 q ≤ γ β ( q ) ≤ C 0 q , 0 ≤ γ β ′ ( q ) ≤ C 0 ( 0 ≤ q ≤ T ∗ ) . k_0q\le\gamma_{\beta}(q)\le C_0q,\qquad0\le\gamma_{\beta}'(q)\le C_0\qquad(0\le q\le T_*). k 0 q ≤ γ β ( q ) ≤ C 0 q , 0 ≤ γ β ′ ( q ) ≤ C 0 ( 0 ≤ q ≤ T ∗ ) . the second bound for almost every $q. We set T 0 = min { T ∗ , 1 / 4 , 2 / C 0 } T_0=\min\{T_*,1/4,2/C_0\} T 0 = min { T ∗ , 1/4 , 2/ C 0 } [ 0 , T 0 ] [0,T_0] [ 0 , T 0 ] γ β ≤ C 0 T 0 ≤ 2 \gamma_{\beta}\le C_0T_0\le2 γ β ≤ C 0 T 0 ≤ 2 λ ≤ 1 \lambda\le1 λ ≤ 1 γ t r ≤ 2 \gamma^{\mathrm{tr}}\le2 γ tr ≤ 2 λ \lambda λ γ β \gamma_{\beta} γ β [ 0 , T 0 ] [0,T_{0}] [ 0 , T 0 ] γ β ′ 1 { γ β > λ } ∈ [ 0 , C 0 ] \gamma_{\beta}'\mathbf{1}_{\{\gamma_{\beta}>\lambda\}}\in[0,C_{0}] γ β ′ 1 { γ β > λ } ∈ [ 0 , C 0 ]
γ β ( 0 ) = 0 , k 0 q ≤ γ β ( q ) , γ t r ( q ) ≤ 2 ( 0 ≤ q ≤ T 0 ) , (61) \gamma_{\beta}(0)=0,\qquad k_{0}q\le\gamma_{\beta}(q),\qquad\gamma^{\mathrm{tr}}(q)\le2\qquad(0\le q\le T_{0}), \tag*{(61)} γ β ( 0 ) = 0 , k 0 q ≤ γ β ( q ) , γ tr ( q ) ≤ 2 ( 0 ≤ q ≤ T 0 ) , (61) and the restriction of d γ t r d\gamma^{\mathrm{tr}} d γ tr ( 0 , T 0 ] (0,T_{0}] ( 0 , T 0 ] C 0 C_{0} C 0 0 0 0 d γ t r d\gamma^{\mathrm{tr}} d γ tr γ t r ( 0 ) = λ \gamma^{\mathrm{tr}}(0)=\lambda γ tr ( 0 ) = λ
By [48, Theorem 1.1, (1.4) and (1.5)], there exist q β ∈ ( 0 , 1 ) q_{\beta}\in(0,1) q β ∈ ( 0 , 1 ) c β ∈ ( 0 , 1 ) c_{\beta}\in(0,1) c β ∈ ( 0 , 1 ) ϱ β ∈ C ∞ ( [ 0 , q β ] ) \varrho_{\beta}\in C^{\infty}([0,q_{\beta}]) ϱ β ∈ C ∞ ([ 0 , q β ])
μ β ( d q ) = ϱ β ( q ) 1 [ 0 , q β ) ( q ) d q + c β δ q β ( d q ) , supp μ β = [ 0 , q β ] . (62) \mu_{\beta}(dq)=\varrho_{\beta}(q)\mathbf{1}_{[0,q_{\beta})}(q)\,dq+c_{\beta}\delta_{q_{\beta}}(dq),\qquad\operatorname{supp}\mu_{\beta}=[0,q_{\beta}]. \tag*{(62)} μ β ( d q ) = ϱ β ( q ) 1 [ 0 , q β ) ( q ) d q + c β δ q β ( d q ) , supp μ β = [ 0 , q β ] . (62) Here, as in [48], ϱ β ∈ C ∞ ( [ 0 , q β ] ) \varrho_{\beta}\in C^{\infty}([0,q_{\beta}]) ϱ β ∈ C ∞ ([ 0 , q β ]) ϱ β \varrho_{\beta} ϱ β ( 0 , q β ) (0,q_{\beta}) ( 0 , q β ) [ 0 , q β ] [0,q_{\beta}] [ 0 , q β ] μ β \mu_{\beta} μ β q β q_{\beta} q β α β = 1 \alpha_{\beta}=1 α β = 1 γ β = β \gamma_{\beta}=\beta γ β = β log cosh x \log\cosh x log cosh x log 2 \log2 log 2 μ β \mu_{\beta} μ β
Trial budgets
For a minimizer γ ∗ \gamma_{*} γ ∗ F ρ F_{\rho} F ρ trial budget is a number η ≥ 0 \eta\ge0 η ≥ 0
F ρ ( γ t r ) − F ρ ( γ ∗ ) ≤ η . (63) F_{\rho}(\gamma^{\mathrm{tr}})-F_{\rho}(\gamma_{*})\le\eta. \tag*{(63)} F ρ ( γ tr ) − F ρ ( γ ∗ ) ≤ η . (63) Since F ρ ( γ ∗ ) = min F ρ F_{\rho}(\gamma_{*})=\min F_{\rho} F ρ ( γ ∗ ) = min F ρ § 4.1 , measured in energy units. By (3.18), η = e N , β ( λ ) + L s λ 5 \eta=e_{N,\beta}(\lambda)+L_{s}\lambda^{5} η = e N , β ( λ ) + L s λ 5 ρ ∈ ( 0 , 1 ] \rho\in(0,1] ρ ∈ ( 0 , 1 ]
Smoothing and coercivity
A minimizer may have atoms, and the response inequality (59) involves the derivative measure d γ θ d\gamma_{\theta} d γ θ [4 Lemma 7.5] , and part (b) extends the reflection argument of that lemma to the upper endpoint 1. We then show that a difference of one sign between the endpoints of a segment forces curvature along the segment (Lemma 7.4).
Lemma 7.3 (Reflected regularization). Let γ \gamma γ h > 0 h>0 h > 0
(a) Let 0 < T < 1 0<T<1 0 < T < 1 h ≤ 1 − T h\le1-T h ≤ 1 − T m = ν γ ( [ 0 , T ) ) m=\nu_{\gamma}([0,T)) m = ν γ ([ 0 , T )) γ ˉ \bar{\gamma} γ ˉ d γ ˉ = λ δ 0 + ν ˉ d\bar{\gamma}=\lambda\delta_{0}+\bar{\nu} d γ ˉ = λ δ 0 + ν ˉ ν ˉ \bar{\nu} ν ˉ ν γ \nu_{\gamma} ν γ q < T q<T q < T [ q − h , q + h ] [q-h,q+h] [ q − h , q + h ] x ↦ ∣ x ∣ x\mapsto\lvert x\rvert x ↦ ∣ x ∣ γ ˉ \bar{\gamma} γ ˉ γ ˉ ( 0 ) = λ \bar{\gamma}(0)=\lambda γ ˉ ( 0 ) = λ d γ ˉ d\bar{\gamma} d γ ˉ ( 0 , T ) (0,T) ( 0 , T ) m / h m/h m / h F ρ ( γ ˉ ) − F ρ ( γ ) ≤ ρ m h 2 / 12 F_{\rho}(\bar{\gamma})-F_{\rho}(\gamma)\le\rho mh^{2}/12 F ρ ( γ ˉ ) − F ρ ( γ ) ≤ ρ m h 2 /12
(b) Let h ≤ 1 / 2 h \le1/2 h ≤ 1/2 π ( x ) = ∣ x ∣ \pi(x)=|x| π ( x ) = ∣ x ∣ x ≤ 1 x \le1 x ≤ 1 π ( x ) = 2 − x \pi(x)=2-x π ( x ) = 2 − x x > 1 x>1 x > 1 γ h \gamma^{h} γ h d γ h = λ δ 0 + ν h \mathrm{d}\gamma^{h}=\lambda\delta_{0}+\nu^{h} d γ h = λ δ 0 + ν h ν h \nu^{h} ν h ν γ \nu_{\gamma} ν γ q ∈ [ 0 , 1 ] q\in[0,1] q ∈ [ 0 , 1 ] [ q − h , q + h ] [q-h,q+h] [ q − h , q + h ] π \pi π γ h \gamma^{h} γ h γ h ( 0 ) = λ \gamma^{h}(0)=\lambda γ h ( 0 ) = λ β / h \beta/h β / h [ 0 , 1 ] [0,1] [ 0 , 1 ]
∥ γ h − γ ∥ L 1 ( 0 , 1 ) ≤ β h , F ρ ( γ h ) − F ρ ( γ ) ≤ ρ β h 2 12 . \lVert\gamma^{h}-\gamma\rVert_{L^{1}(0,1)}\le\beta h,\qquad F_{\rho}(\gamma^{h})-F_{\rho}(\gamma)\le\frac{\rho\beta h^{2}}{12}. ∥ γ h − γ ∥ L 1 ( 0 , 1 ) ≤ β h , F ρ ( γ h ) − F ρ ( γ ) ≤ 12 ρβ h 2 . Proof. Part (a) is [4 Lemma 7.5] , including the cost bound [4, (7.24)]; by Section 7.1`, the functional F r F_{r} F r F ρ F_{\rho} F ρ r = ρ r=\rho r = ρ [ 0 , T ) [0,T) [ 0 , T ) x ↦ ∣ x ∣ x\mapsto|x| x ↦ ∣ x ∣ h ≤ 1 − T h\le1-T h ≤ 1 − T [ 0 , T + h ) ⊂ [ 0 , 1 ] [0,T+h)\subset[0,1] [ 0 , T + h ) ⊂ [ 0 , 1 ] [ − h , T + h ] ⊂ [ − 1 , 1 ] [-h,T+h]\subset[-1,1] [ − h , T + h ] ⊂ [ − 1 , 1 ] 1 1 1 ( ρ / 2 ) ⋅ m ⋅ ( h 2 / 6 ) = ρ m h 2 / 12 (\rho/2)\cdot m\cdot(h^{2}/6)=\rho m h^{2}/12 ( ρ /2 ) ⋅ m ⋅ ( h 2 /6 ) = ρ m h 2 /12 m / h m/h m / h ( 0 , T ) (0,T) ( 0 , T ) [ T , 1 ] [T,1] [ T , 1 ] ( 0 , T ) (0,T) ( 0 , T )
For (b), we first verify the properties of γ h \gamma^{h} γ h q ∈ [ 0 , 1 ] q\in[0,1] q ∈ [ 0 , 1 ] κ q \kappa_{q} κ q [ q − h , q + h ] [q-h,q+h] [ q − h , q + h ] π \pi π γ h \gamma^{h} γ h ν h = ∫ κ q ν γ ( d q ) \nu^{h}=\int\kappa_{q}\,\nu_{\gamma}(\mathrm{d}q) ν h = ∫ κ q ν γ ( d q ) π \pi π [ − h , 1 + h ] [-h,1+h] [ − h , 1 + h ] [ 0 , 1 ] [0,1] [ 0 , 1 ] ± 1 \pm1 ± 1 x ∈ ( 0 , 1 ) x\in(0,1) x ∈ ( 0 , 1 ) [ q − h , q + h ] [q-h,q+h] [ q − h , q + h ] π \pi π x x x x x x − x -x − x 2 − x 2-x 2 − x κ q \kappa_{q} κ q
1 2 h ( 1 { ∣ x − q ∣ ≤ h } + 1 { x + q ≤ h } + 1 { x + q ≥ 2 − h } ) , x ∈ ( 0 , 1 ) . \frac{1}{2h}\left(\mathbf{1}_{\{|x-q|\le h\}}+\mathbf{1}_{\{x+q\le h\}}+\mathbf{1}_{\{x+q\ge2-h\}}\right),\qquad x\in(0,1). 2 h 1 ( 1 { ∣ x − q ∣ ≤ h } + 1 { x + q ≤ h } + 1 { x + q ≥ 2 − h } ) , x ∈ ( 0 , 1 ) . Since h ≤ 1 / 2 < 2 − h h\le1/2<2-h h ≤ 1/2 < 2 − h 1 / h 1/h 1/ h κ q \kappa_{q} κ q [ 0 , 1 ] [0,1] [ 0 , 1 ] ν γ \nu_{\gamma} ν γ ν h \nu^{h} ν h [ 0 , 1 ] [0,1] [ 0 , 1 ] β − λ \beta-\lambda β − λ ( β − λ ) / h ≤ β / h (\beta-\lambda)/h\le\beta/h ( β − λ ) / h ≤ β / h d γ h = λ δ 0 + ν h \mathrm{d}\gamma^{h}=\lambda\delta_{0}+\nu^{h} d γ h = λ δ 0 + ν h β \beta β γ h ( 0 ) = λ \gamma^{h}(0)=\lambda γ h ( 0 ) = λ γ h \gamma^{h} γ h
For the displacement bound, let Π \Pi Π [ 0 , 1 ] 2 [0,1]^{2} [ 0 , 1 ] 2
∫ F d Π = ∫ [ 0 , 1 ] 1 2 h ∫ − h h F ( q , π ( q + z ) ) d z ν γ ( d q ) \int F\,\mathrm{d}\Pi=\int_{[0,1]}\frac{1}{2h}\int_{-h}^{h}F(q,\pi(q+z))\,\mathrm{d}z\,\nu_{\gamma}(\mathrm{d}q) ∫ F d Π = ∫ [ 0 , 1 ] 2 h 1 ∫ − h h F ( q , π ( q + z )) d z ν γ ( d q ) for bounded Borel functions F F F ν γ \nu_{\gamma} ν γ ν h \nu^{h} ν h π \pi π 1 1 1 π ( q ) = q \pi(q)=q π ( q ) = q q ∈ [ 0 , 1 ] q\in[0,1] q ∈ [ 0 , 1 ] ∣ π ( q + z ) − q ∣ ≤ ∣ z ∣ ≤ h |\pi(q+z)-q|\le|z|\le h ∣ π ( q + z ) − q ∣ ≤ ∣ z ∣ ≤ h Π \Pi Π ( q , r ) (q,r) ( q , r ) ∣ r − q ∣ ≤ h |r-q|\le h ∣ r − q ∣ ≤ h d γ \mathrm{d}\gamma d γ d γ h \mathrm{d}\gamma^{h} d γ h x ∈ [ 0 , 1 ] x\in[0,1] x ∈ [ 0 , 1 ]
γ h ( x ) − γ ( x ) = ∫ ( 1 { r ≤ x } − 1 { q ≤ x } ) Π ( d q , d r ) . \gamma^{h}(x)-\gamma(x)=\int\left(\mathbf{1}_{\{r\le x\}}-\mathbf{1}_{\{q\le x\}}\right)\Pi(\mathrm{d}q,\mathrm{d}r). γ h ( x ) − γ ( x ) = ∫ ( 1 { r ≤ x } − 1 { q ≤ x } ) Π ( d q , d r ) . For q , r ∈ [ 0 , 1 ] q,r \in[0,1] q , r ∈ [ 0 , 1 ] ∫ 0 1 ∣ 1 { r ≤ x } − 1 { q ≤ x } ∣ d x = ∣ r − q ∣ \int_0^1 \lvert\mathbf{1}_{\{r\le x\}}-\mathbf{1}_{\{q\le x\}}\rvert\,\mathrm{d}x=\lvert r-q\rvert ∫ 0 1 ∣ 1 { r ≤ x } − 1 { q ≤ x } ∣ d x = ∣ r − q ∣
∥ γ h − γ ∥ L 1 ( 0 , 1 ) ≤ ∫ ∣ r − q ∣ Π ( d q , d r ) ≤ ( β − λ ) h ≤ β h . \lVert\gamma^h-\gamma\rVert_{L^1(0,1)}\le\int\lvert r-q\rvert\,\Pi(\mathrm{d}q,\mathrm{d}r)\le(\beta-\lambda)h\le\beta h. ∥ γ h − γ ∥ L 1 ( 0 , 1 ) ≤ ∫ ∣ r − q ∣ Π ( d q , d r ) ≤ ( β − λ ) h ≤ β h . To bound the cost, let G = G γ h G=G_{\gamma^h} G = G γ h S = S γ h S=S_{\gamma^h} S = S γ h H ( z ) = G ( π ( z ) ) − z 2 / 2 H(z)=G(\pi(z))-z^2/2 H ( z ) = G ( π ( z )) − z 2 /2 z ∈ [ − h , 1 + h ] z\in[-h,1+h] z ∈ [ − h , 1 + h ] G G G [ 0 , 1 ] [0,1] [ 0 , 1 ] G ′ ( x ) = x − S ( x ) G'(x)=x-S(x) G ′ ( x ) = x − S ( x ) π ( z ) = − z \pi(z)=-z π ( z ) = − z z < 0 z<0 z < 0 π ( z ) = 2 − z \pi(z)=2-z π ( z ) = 2 − z z > 1 z>1 z > 1 H H H 0 0 0 1 1 1
H ′ ( z ) = { S ( − z ) , − h < z < 0 , − S ( z ) , 0 < z < 1 , S ( 2 − z ) − 2 , 1 < z < 1 + h . H'(z)=
\begin{cases}
S(-z), & -h<z<0,\\
-S(z), & 0<z<1,\\
S(2-z)-2, & 1<z<1+h.
\end{cases} H ′ ( z ) = ⎩ ⎨ ⎧ S ( − z ) , − S ( z ) , S ( 2 − z ) − 2 , − h < z < 0 , 0 < z < 1 , 1 < z < 1 + h . Each piece is nonincreasing, because S S S (57) . The one-sided derivatives at 0 0 0 S ( 0 ) = 0 S(0)=0 S ( 0 ) = 0 1 1 1 − S ( 1 ) -S(1) − S ( 1 ) S ( 1 ) − 2 ≤ − S ( 1 ) S(1)-2\le-S(1) S ( 1 ) − 2 ≤ − S ( 1 ) S ( 1 ) ≤ 1 S(1)\le1 S ( 1 ) ≤ 1 H ′ H' H ′ H H H [ − h , 1 + h ] [-h,1+h] [ − h , 1 + h ] q ∈ [ 0 , 1 ] q\in[0,1] q ∈ [ 0 , 1 ] G ′ ( q ) − q G'(q)-q G ′ ( q ) − q H H H q q q q = 1 q=1 q = 1 G ′ ( 1 ) − 1 = − S ( 1 ) G'(1)-1=-S(1) G ′ ( 1 ) − 1 = − S ( 1 ) H H H H ( q + z ) ≤ H ( q ) + ( G ′ ( q ) − q ) z H(q+z)\le H(q)+(G'(q)-q)z H ( q + z ) ≤ H ( q ) + ( G ′ ( q ) − q ) z
G ( π ( q + z ) ) ≤ G ( q ) + G ′ ( q ) z + z 2 2 ( q ∈ [ 0 , 1 ] , ∣ z ∣ ≤ h ) . G(\pi(q+z))\le G(q)+G'(q)z+\frac{z^2}{2}\qquad(q\in[0,1],\ \lvert z\rvert\le h). G ( π ( q + z )) ≤ G ( q ) + G ′ ( q ) z + 2 z 2 ( q ∈ [ 0 , 1 ] , ∣ z ∣ ≤ h ) . We average over z z z [ − h , h ] [-h,h] [ − h , h ] z z z 0 0 0 z 2 z^2 z 2 h 2 / 3 h^2/3 h 2 /3
∫ G d κ q − G ( q ) ≤ h 2 6 ( q ∈ [ 0 , 1 ] ) . (64) \int G\,\mathrm{d}\kappa_q-G(q)\le\frac{h^2}{6}\qquad(q\in[0,1]). \tag*{(64)} ∫ G d κ q − G ( q ) ≤ 6 h 2 ( q ∈ [ 0 , 1 ]) . (64) By Proposition 7.1(a), F ρ F_\rho F ρ γ h \gamma^h γ h γ \gamma γ F ρ ( γ ) ≥ F ρ ( γ h ) + D + F ρ ( γ h ) [ ν γ − ν h ] F_\rho(\gamma)\ge F_\rho(\gamma^h)+D_+F_\rho(\gamma^h)[\nu_\gamma-\nu^h] F ρ ( γ ) ≥ F ρ ( γ h ) + D + F ρ ( γ h ) [ ν γ − ν h ] (58) at γ h \gamma^h γ h (64) , we conclude that
F ρ ( γ h ) − F ρ ( γ ) ≤ ρ 2 ∫ G d ( ν h − ν γ ) = ρ 2 ∫ [ ∫ G d κ q − G ( q ) ] ν γ ( d q ) ≤ ρ ( β − λ ) h 2 12 . F_\rho(\gamma^h)-F_\rho(\gamma)\le\frac{\rho}{2}\int G\,\mathrm{d}(\nu^h-\nu_\gamma)=\frac{\rho}{2}\int\left[\int G\,\mathrm{d}\kappa_q-G(q)\right]\nu_\gamma(\mathrm{d}q)\le\frac{\rho(\beta-\lambda)h^2}{12}. F ρ ( γ h ) − F ρ ( γ ) ≤ 2 ρ ∫ G d ( ν h − ν γ ) = 2 ρ ∫ [ ∫ G d κ q − G ( q ) ] ν γ ( d q ) ≤ 12 ρ ( β − λ ) h 2 . The midpoint inequality. We compare the second derivative along a segment with the values at its endpoints. Let f f f [ 0 , 1 ] [0,1] [ 0 , 1 ] ( 0 , 1 ) . (0,1). ( 0 , 1 ) . f ′ f' f ′ ( 0 , 1 ) (0,1) ( 0 , 1 ) f f f [ 0 , 1 ] [0,1] [ 0 , 1 ] [ ϵ , 1 − ϵ ] [\epsilon,1-\epsilon] [ ϵ , 1 − ϵ ] f ′ f' f ′ f f f 0 0 0 1 1 1 f ( y ) − f ( x ) = ∫ x y f ′ f(y)-f(x)=\int_x^y f' f ( y ) − f ( x ) = ∫ x y f ′ 0 ≤ x < y ≤ 1 0\le x<y\le1 0 ≤ x < y ≤ 1 g g g ( 0 , 1 ) (0,1) ( 0 , 1 ) g ( y ) − g ( x ) ≥ ∫ x y g ′ g(y)-g(x)\ge\int_x^y g' g ( y ) − g ( x ) ≥ ∫ x y g ′ 0 < x < y < 1 0<x<y<1 0 < x < y < 1 g = f ′ g=f' g = f ′ f ′ ′ ≥ 0 f''\ge0 f ′′ ≥ 0 ( 0 , 1 ) (0,1) ( 0 , 1 )
f ( 0 ) + f ( 1 ) − 2 f ( 1 2 ) = ∫ 0 1 / 2 ( f ′ ( x + 1 2 ) − f ′ ( x ) ) d x ≥ ∫ 0 1 / 2 ∫ x x + 1 / 2 f ′ ′ ( θ ) d θ d x . f(0)+f(1)-2f\left(\frac{1}{2}\right)=\int_0^{1/2}\left(f'\left(x+\frac{1}{2}\right)-f'(x)\right)\,\mathrm{d}x\ge\int_0^{1/2}\int_x^{x+1/2}f''(\theta)\,\mathrm{d}\theta\,\mathrm{d}x. f ( 0 ) + f ( 1 ) − 2 f ( 2 1 ) = ∫ 0 1/2 ( f ′ ( x + 2 1 ) − f ′ ( x ) ) d x ≥ ∫ 0 1/2 ∫ x x + 1/2 f ′′ ( θ ) d θ d x . For fixed θ ∈ ( 0 , 1 ) \theta\in(0,1) θ ∈ ( 0 , 1 ) x ∈ ( 0 , 1 / 2 ) x \in(0,1/2) x ∈ ( 0 , 1/2 ) x < θ < x + 1 / 2 x < \theta< x + 1/2 x < θ < x + 1/2 min { 1 / 2 , θ } − max { 0 , θ − 1 / 2 } = min { θ , 1 − θ } \min\{1/2,\theta\}-\max\{0,\theta-1/2\}=\min\{\theta,1-\theta\} min { 1/2 , θ } − max { 0 , θ − 1/2 } = min { θ , 1 − θ } ∫ 0 1 min { θ , 1 − θ } f ′ ′ ( θ ) d θ \int_0^1\min\{\theta,1-\theta\}f''(\theta)\,\mathrm{d}\theta ∫ 0 1 min { θ , 1 − θ } f ′′ ( θ ) d θ f ′ ′ ≥ 0 f''\ge0 f ′′ ≥ 0 ( 0 , 1 ) (0,1) ( 0 , 1 )
Now let γ ∗ \gamma_* γ ∗ ( γ θ ) (\gamma_\theta) ( γ θ ) f ( θ ) = F ρ ( γ θ ) f(\theta)=F_\rho(\gamma_\theta) f ( θ ) = F ρ ( γ θ ) f f f γ 1 / 2 \gamma_{1/2} γ 1/2 f ( 1 / 2 ) ≥ F ρ ( γ ∗ ) f(1/2)\ge F_\rho(\gamma_*) f ( 1/2 ) ≥ F ρ ( γ ∗ )
∫ Θ min { θ , 1 − θ } f ′ ′ ( θ ) d θ ≤ F ρ ( γ 0 ) + F ρ ( γ 1 ) − 2 F ρ ( γ ∗ ) (65) \int_\Theta\min\{\theta,1-\theta\}f''(\theta)\,\mathrm{d}\theta\le F_\rho(\gamma_0)+F_\rho(\gamma_1)-2F_\rho(\gamma_*) \tag*{(65)} ∫ Θ min { θ , 1 − θ } f ′′ ( θ ) d θ ≤ F ρ ( γ 0 ) + F ρ ( γ 1 ) − 2 F ρ ( γ ∗ ) (65) ( Θ ⊂ ( 0 , 1 ) (\Theta\subset(0,1) ( Θ ⊂ ( 0 , 1 )
The weight min { θ , 1 − θ } \min\{\theta,1-\theta\} min { θ , 1 − θ } 1 / 4 1/4 1/4 ( 0 , 1 ) (0,1) ( 0 , 1 ) 2 ∫ 1 / 4 1 / 2 θ d θ = 3 / 16 2\int_{1/4}^{1/2}\theta\,\mathrm{d}\theta=3/16 2 ∫ 1/4 1/2 θ d θ = 3/16 [ 1 / 4 , 3 / 4 ] [1/4,3/4] [ 1/4 , 3/4 ]
Lemma 7.4 (Coercivity from a difference of one sign). Let ( γ θ ) (\gamma_\theta) ( γ θ ) ψ = γ 1 − γ 0 \psi=\gamma_1-\gamma_0 ψ = γ 1 − γ 0 f ( θ ) = F ρ ( γ θ ) f(\theta)=F_\rho(\gamma_\theta) f ( θ ) = F ρ ( γ θ ) θ ∈ ( 0 , 1 ) \theta\in(0,1) θ ∈ ( 0 , 1 ) φ : [ 0 , 1 ] → [ 0 , 1 ] \varphi:[0,1]\to[0,1] φ : [ 0 , 1 ] → [ 0 , 1 ] φ ( 0 ) = φ ( 1 ) = 0 \varphi(0)=\varphi(1)=0 φ ( 0 ) = φ ( 1 ) = 0
Δ = ρ ∫ 0 1 ζ θ φ d q , Y 0 = ∫ 0 1 ∣ φ ′ ∣ 2 d q . \Delta=\rho\int_0^1\zeta_\theta\varphi\,\mathrm{d}q,\qquad Y_0=\int_0^1|\varphi'|^2\,\mathrm{d}q. Δ = ρ ∫ 0 1 ζ θ φ d q , Y 0 = ∫ 0 1 ∣ φ ′ ∣ 2 d q . Suppose that there exist A > 0 A>0 A > 0 d ≥ 0 d\ge0 d ≥ 0 σ ∈ { − 1 , 1 } \sigma\in\{-1,1\} σ ∈ { − 1 , 1 } σ ψ ≥ A / 2 \sigma\psi\ge A/2 σ ψ ≥ A /2 φ > 0 \varphi>0 φ > 0
ρ ∫ ( 0 , 1 ) φ 2 ζ θ d γ θ ≤ d Δ . (66) \rho\int_{(0,1)}\varphi^2\zeta_\theta\,\mathrm{d}\gamma_\theta\le d\Delta. \tag*{(66)} ρ ∫ ( 0 , 1 ) φ 2 ζ θ d γ θ ≤ d Δ. (66) The last condition holds if the restriction of d γ θ \mathrm{d}\gamma_\theta d γ θ { φ > 0 } \{\varphi>0\} { φ > 0 } d d d
f ′ ′ ( θ ) ≥ ρ A 2 72 min { Δ 2 β Y 0 , Δ d } , (67) f''(\theta)\ge\rho\frac{A^2}{72}\min\left\{\frac{\Delta^2}{\beta Y_0},\frac{\Delta}{d}\right\}, \tag*{(67)} f ′′ ( θ ) ≥ ρ 72 A 2 min { β Y 0 Δ 2 , d Δ } , (67) where Δ / 0 = + ∞ \Delta/0=+\infty Δ/0 = + ∞ Δ > 0 \Delta>0 Δ > 0 0 0 0 Δ = 0 \Delta=0 Δ = 0
For fixed Y 0 Y_0 Y 0 d d d (67) are nondecreasing in Δ \Delta Δ Δ \Delta Δ
Proof. We first check the sufficient condition for (66) . If d γ θ ≤ d d q \mathrm{d}\gamma_\theta\le d\,\mathrm{d}q d γ θ ≤ d d q { φ > 0 } \{\varphi>0\} { φ > 0 } 0 ≤ φ ≤ 1 0\le\varphi\le1 0 ≤ φ ≤ 1 ζ θ ≥ 0 \zeta_\theta\ge0 ζ θ ≥ 0 ρ ∫ ( 0 , 1 ) φ 2 ζ θ d γ θ ≤ d ρ ∫ 0 1 φ 2 ζ θ d q ≤ d Δ \rho\int_{(0,1)}\varphi^2\zeta_\theta\,\mathrm{d}\gamma_\theta\le d\rho\int_0^1\varphi^2\zeta_\theta\,\mathrm{d}q\le d\Delta ρ ∫ ( 0 , 1 ) φ 2 ζ θ d γ θ ≤ d ρ ∫ 0 1 φ 2 ζ θ d q ≤ d Δ
Next, we extend (59) to φ \varphi φ ( 0 , 1 ) (0,1) ( 0 , 1 ) 0 < ϵ < 1 / 4 0<\epsilon<1/4 0 < ϵ < 1/4 χ ϵ \chi_\epsilon χ ϵ ϵ \epsilon ϵ { 0 , 1 } \{0,1\} { 0 , 1 } 2 ϵ 2\epsilon 2 ϵ ∣ χ ϵ ′ ∣ ≤ 1 / ϵ |\chi_\epsilon'|\le1/\epsilon ∣ χ ϵ ′ ∣ ≤ 1/ ϵ ℓ φ \ell_\varphi ℓ φ φ \varphi φ φ ϵ = χ ϵ φ \varphi_\epsilon=\chi_\epsilon\varphi φ ϵ = χ ϵ φ φ ϵ \varphi_\epsilon φ ϵ ( 0 , 1 ) (0,1) ( 0 , 1 ) φ ( 0 ) = φ ( 1 ) = 0 \varphi(0)=\varphi(1)=0 φ ( 0 ) = φ ( 1 ) = 0 ∣ φ ( q ) ∣ ≤ ℓ φ min { q , 1 − q } |\varphi(q)|\le\ell_\varphi\min\{q,1-q\} ∣ φ ( q ) ∣ ≤ ℓ φ min { q , 1 − q } 2 ℓ φ ϵ 2\ell_{\varphi}\epsilon 2 ℓ φ ϵ χ ϵ ≠ 1 \chi_{\epsilon} \ne1 χ ϵ = 1 4 ϵ 4\epsilon 4 ϵ ∥ φ ϵ − φ ∥ ∞ ≤ 2 ℓ φ ϵ \|\varphi_{\epsilon}-\varphi\|_{\infty} \le2\ell_{\varphi}\epsilon ∥ φ ϵ − φ ∥ ∞ ≤ 2 ℓ φ ϵ φ ϵ ′ − φ ′ = χ ϵ ′ φ + ( χ ϵ − 1 ) φ ′ \varphi_{\epsilon}'-\varphi'=\chi_{\epsilon}'\varphi+(\chi_{\epsilon}-1)\varphi' φ ϵ ′ − φ ′ = χ ϵ ′ φ + ( χ ϵ − 1 ) φ ′ 3 ℓ φ 3\ell_{\varphi} 3 ℓ φ ∥ φ ϵ ′ − φ ′ ∥ L 2 2 ≤ 36 ℓ φ 2 ϵ \|\varphi_{\epsilon}'-\varphi'\|_{L^2}^2 \le36\ell_{\varphi}^2\epsilon ∥ φ ϵ ′ − φ ′ ∥ L 2 2 ≤ 36 ℓ φ 2 ϵ N N N θ \theta θ ψ \psi ψ b θ b_{\theta} b θ ζ θ \zeta_{\theta} ζ θ β \beta β β \beta β β 2 N \beta^2N β 2 N [ λ , β ] [\lambda,\beta] [ λ , β ] (57) , and the third because 0 ⪯ H q ⪯ C q 0 \preceq H_q \preceq C_q 0 ⪯ H q ⪯ C q Tr C q ≤ N \operatorname{Tr} C_q \le N Tr C q ≤ N d γ θ \mathrm{d}\gamma_{\theta} d γ θ φ ϵ \varphi_{\epsilon} φ ϵ L 2 L^2 L 2 φ ϵ ′ \varphi_{\epsilon}' φ ϵ ′ (59) and Y Y Y ϵ → 0 \epsilon\to0 ϵ → 0 φ \varphi φ Z Z Z φ ϵ 2 → φ 2 \varphi_{\epsilon}^2 \to\varphi^2 φ ϵ 2 → φ 2 d γ θ \mathrm{d}\gamma_{\theta} d γ θ (59) holds for φ \varphi φ
If Δ = 0 \Delta=0 Δ = 0 f f f Y 0 > 0 Y_0>0 Y 0 > 0 φ \varphi φ 0 0 0 Y ≤ β Y 0 Y \le\beta Y_0 Y ≤ β Y 0 (57) and Z ≤ d Δ Z \le d\Delta Z ≤ d Δ (66) . As ζ θ \zeta_{\theta} ζ θ φ \varphi φ σ ρ ∫ 0 1 ψ ζ θ φ d q ≥ A Δ / 2 \sigma\rho\int_0^1\psi\zeta_{\theta}\varphi\,\mathrm{d}q \ge A\Delta/2 σ ρ ∫ 0 1 ψ ζ θ φ d q ≥ A Δ/2 (59) , we find
A 2 Δ ≤ ( f ′ ′ ( θ ) ρ ) 1 / 2 [ Y 1 / 2 + 2 ( Y + Z ) 1 / 2 ] ≤ 3 ( f ′ ′ ( θ ) ρ ) 1 / 2 ( Y + Z ) 1 / 2 . \frac{A}{2}\Delta\le\left(\frac{f''(\theta)}{\rho}\right)^{1/2}\left[Y^{1/2}+2(Y+Z)^{1/2}\right] \le3\left(\frac{f''(\theta)}{\rho}\right)^{1/2}(Y+Z)^{1/2}. 2 A Δ ≤ ( ρ f ′′ ( θ ) ) 1/2 [ Y 1/2 + 2 ( Y + Z ) 1/2 ] ≤ 3 ( ρ f ′′ ( θ ) ) 1/2 ( Y + Z ) 1/2 . The left side is positive, which implies that Y + Z > 0 Y+Z>0 Y + Z > 0 f ′ ′ ( θ ) ≥ ρ A 2 Δ 2 / ( 36 ( Y + Z ) ) f''(\theta)\ge\rho A^2\Delta^2/(36(Y+Z)) f ′′ ( θ ) ≥ ρ A 2 Δ 2 / ( 36 ( Y + Z )) Y + Z ≤ β Y 0 + d Δ ≤ 2 max { β Y 0 , d Δ } Y+Z\le\beta Y_0+d\Delta\le2\max\{\beta Y_0,d\Delta\} Y + Z ≤ β Y 0 + d Δ ≤ 2 max { β Y 0 , d Δ } (67) .
A density bound for the scalar measure
By (62) , the density ϱ β \varrho_{\beta} ϱ β μ β \mu_{\beta} μ β [ 0 , q β ) [0,q_{\beta}) [ 0 , q β ) [ 0 , q β ] [0,q_{\beta}] [ 0 , q β ] α β \alpha_{\beta} α β [9 Theorem 2(ii)] . For the SK model, this identity is [48, Lemma 6.2, (6.1)], and the smoothness of the resulting quotient up to q β q_{\beta} q β
Lemma 7.5 (Density bound). There exists L β < ∞ L_{\beta}<\infty L β < ∞ such that
α β ( y ) − α β ( x ) ≤ L β ( y − x ) ( 0 ≤ x < y < q β ) . (68) \alpha_{\beta}(y)-\alpha_{\beta}(x)\le L_{\beta}(y-x)\qquad(0\le x<y<q_{\beta}). \tag*{(68)} α β ( y ) − α β ( x ) ≤ L β ( y − x ) ( 0 ≤ x < y < q β ) . (68) With L = max { 1 , β L β } L=\max\{1,\beta L_{\beta}\} L = max { 1 , β L β } the trial profile satisfies γ t r ( 0 ) = λ \gamma^{\mathrm{tr}}(0)=\lambda γ tr ( 0 ) = λ γ t r = β \gamma^{\mathrm{tr}}=\beta γ tr = β on [ q β , 1 ] [q_{\beta},1] [ q β , 1 ] and ∣ γ t r ( x ) − γ t r ( y ) ∣ ≤ L ∣ x − y ∣ |\gamma^{\mathrm{tr}}(x)-\gamma^{\mathrm{tr}}(y)|\le L|x-y| ∣ γ tr ( x ) − γ tr ( y ) ∣ ≤ L ∣ x − y ∣ for x , y ∈ [ 0 , q β ) x,y\in[0,q_{\beta}) x , y ∈ [ 0 , q β ) In particular, the restriction of d γ t r \mathrm{d}\gamma^{\mathrm{tr}} d γ tr to ( 0 , 1 ] ∖ { q β } (0,1]\setminus\{q_{\beta}\} ( 0 , 1 ] ∖ { q β } is absolutely continuous, with density at most L L L on ( 0 , q β ) (0,q_{\beta}) ( 0 , q β ) and 0 0 0 on ( q β , 1 ] (q_{\beta},1] ( q β , 1 ]
The bound (68) is stated for y < q β y<q_{\beta} y < q β μ β \mu_{\beta} μ β q β q_{\beta} q β α ˉ \bar{\alpha} α ˉ α β \alpha_{\beta} α β [ 0 , q β ) [0,q_{\beta}) [ 0 , q β ) q β q_{\beta} q β μ β ( [ 0 , q β ) ) = 1 − c β \mu_{\beta}([0,q_{\beta}))=1-c_{\beta} μ β ([ 0 , q β )) = 1 − c β α β ( q β ) = 1 \alpha_{\beta}(q_{\beta})=1 α β ( q β ) = 1
Proof. We write α β \alpha_\beta α β γ t r \gamma^{\mathrm{tr}} γ tr
Let u u u α = α β \alpha=\alpha_\beta α = α β log cosh x \log\cosh x log cosh x log ( 2 cosh x ) \log(2\cosh x) log ( 2 cosh x ) log 2 \log2 log 2 w w w ζ \zeta ζ
d ζ r = β 2 α β ( r ) u x ( r , ζ r ) d r + β d w r , ζ 0 = 0 , \mathrm{d}\zeta_r=\beta^2\alpha_\beta(r)u_x(r,\zeta_r)\,\mathrm{d}r+\beta\,\mathrm{d}w_r,\qquad\zeta_0=0, d ζ r = β 2 α β ( r ) u x ( r , ζ r ) d r + β d w r , ζ 0 = 0 , which is the optimal diffusion [48, (2.4)]. We write C ^ r = u x x ( r , ζ r ) \widehat{C}_r=u_{xx}(r,\zeta_r) C r = u xx ( r , ζ r ) D ^ r = u x x x ( r , ζ r ) \widehat{D}_r=u_{xxx}(r,\zeta_r) D r = u xxx ( r , ζ r ) A ^ r = u x x x x ( r , ζ r ) \widehat{A}_r=u_{xxxx}(r,\zeta_r) A r = u xxxx ( r , ζ r )
Γ ( r ) = E u x ( r , ζ r ) 2 , Λ ( r ) = E D ^ r 2 , Ξ ( r ) = E C ^ r 3 ( 0 ≤ r ≤ 1 ) . \Gamma(r)=\mathbb{E}u_x(r,\zeta_r)^2,\qquad\Lambda(r)=\mathbb{E}\widehat{D}_r^2,\qquad\Xi(r)=\mathbb{E}\widehat{C}_r^3\qquad(0\le r\le1). Γ ( r ) = E u x ( r , ζ r ) 2 , Λ ( r ) = E D r 2 , Ξ ( r ) = E C r 3 ( 0 ≤ r ≤ 1 ) . We use three facts from [48, Section 2].
[48, Proposition 2.1], which combines [9 Propositions 1, 2, and 5] : every spatial derivative of u u u [ 0 , 1 ] × R [0,1]\times\mathbb{R} [ 0 , 1 ] × R β \beta β ∣ u x ∣ ≤ 1 |u_x|\le1 ∣ u x ∣ ≤ 1 0 < u x x ≤ 1 0<u_{xx}\le1 0 < u xx ≤ 1
[48, Lemma 2.6, (2.19)]: Γ ( r ) = r \Gamma(r)=r Γ ( r ) = r r ∈ supp μ β = [ 0 , q β ] r\in\operatorname{supp}\mu_\beta=[0,q_\beta] r ∈ supp μ β = [ 0 , q β ]
[48, Lemma 2.5, (2.13) and (2.14)]: the function Γ \Gamma Γ
Γ ′ ( r ) = β 2 E C ^ r 2 , \Gamma'(r)=\beta^2\mathbb{E}\widehat{C}_r^2, Γ ′ ( r ) = β 2 E C r 2 , and
Γ ′ ( y ) − Γ ′ ( x ) = β 4 ∫ x y ( Λ ( r ) − 2 α β ( r ) Ξ ( r ) ) d r ( 0 ≤ x < y ≤ 1 ) . \Gamma'(y)-\Gamma'(x)=\beta^4\int_x^y\left(\Lambda(r)-2\alpha_\beta(r)\Xi(r)\right)\,\mathrm{d}r\qquad(0\le x<y\le1). Γ ′ ( y ) − Γ ′ ( x ) = β 4 ∫ x y ( Λ ( r ) − 2 α β ( r ) Ξ ( r ) ) d r ( 0 ≤ x < y ≤ 1 ) . Further, by the same lemma, every polynomial moment of the spatial derivatives along ζ \zeta ζ [ 0 , 1 ] [0,1] [ 0 , 1 ]
Let K = max j = 3 , 4 sup [ 0 , 1 ] × R ∣ ∂ x j u ∣ K=\max_{j=3,4}\sup_{[0,1]\times\mathbb{R}}|\partial_x^j u| K = max j = 3 , 4 sup [ 0 , 1 ] × R ∣ ∂ x j u ∣ β \beta β
We begin with the moments Λ \Lambda Λ Ξ \Xi Ξ j = 2 , 3 j=2,3 j = 2 , 3
d C ^ r = − β 2 α β C ^ r 2 d r + β D ^ r d w r \mathrm{d}\widehat{C}_r=-\beta^2\alpha_\beta\widehat{C}_r^2\,\mathrm{d}r+\beta\widehat{D}_r\,\mathrm{d}w_r d C r = − β 2 α β C r 2 d r + β D r d w r and
d D ^ r = − 3 β 2 α β C ^ r D ^ r d r + β A ^ r d w r , \mathrm{d}\widehat{D}_r=-3\beta^2\alpha_\beta\widehat{C}_r\widehat{D}_r\,\mathrm{d}r+\beta\widehat{A}_r\,\mathrm{d}w_r, d D r = − 3 β 2 α β C r D r d r + β A r d w r , and Itô’s formula yields
d ( D ^ r 2 ) = β 2 ( A ^ r 2 − 6 α β C ^ r D ^ r 2 ) d r + 2 β D ^ r A ^ r d w r \mathrm{d}\left(\widehat{D}_r^2\right)=\beta^2\left(\widehat{A}_r^2-6\alpha_\beta\widehat{C}_r\widehat{D}_r^2\right)\,\mathrm{d}r+2\beta\widehat{D}_r\widehat{A}_r\,\mathrm{d}w_r d ( D r 2 ) = β 2 ( A r 2 − 6 α β C r D r 2 ) d r + 2 β D r A r d w r d ( C ^ r 3 ) = 3 β 2 ( C ^ r D ^ r 2 − α β C ^ r 4 ) d r + 3 β C ^ r 2 D ^ r d w r . \mathrm{d}\left(\widehat{C}_r^3\right)=3\beta^2\left(\widehat{C}_r\widehat{D}_r^2-\alpha_\beta\widehat{C}_r^4\right)\,\mathrm{d}r+3\beta\widehat{C}_r^2\widehat{D}_r\,\mathrm{d}w_r. d ( C r 3 ) = 3 β 2 ( C r D r 2 − α β C r 4 ) d r + 3 β C r 2 D r d w r . The stochastic integrals are martingales, since their integrands are bounded by (i). For the measure μ β \mu_\beta μ β f = u x x x 2 f=u_{xxx}^2 f = u xxx 2 f = u x x 3 f=u_{xx}^3 f = u xx 3 f 0 , f 1 f_0,f_1 f 0 , f 1
Λ ( y ) − Λ ( x ) = β 2 ∫ x y E [ A ^ r 2 − 6 α β ( r ) C ^ r D ^ r 2 ] d r , \Lambda(y)-\Lambda(x)=\beta^2\int_x^y\mathbb{E}\left[\widehat{A}_r^2-6\alpha_\beta(r)\widehat{C}_r\widehat{D}_r^2\right]\,\mathrm{d}r, Λ ( y ) − Λ ( x ) = β 2 ∫ x y E [ A r 2 − 6 α β ( r ) C r D r 2 ] d r , Ξ ( y ) − Ξ ( x ) = 3 β 2 ∫ x y E [ C ^ r D ^ r 2 − α β ( r ) C ^ r 4 ] d r ( 0 ≤ x < y ≤ 1 ) . \Xi(y)-\Xi(x)=3\beta^2\int_x^y\mathbb{E}\left[\widehat{C}_r\widehat{D}_r^2-\alpha_\beta(r)\widehat{C}_r^4\right]\,\mathrm{d}r\qquad(0\le x<y\le1). Ξ ( y ) − Ξ ( x ) = 3 β 2 ∫ x y E [ C r D r 2 − α β ( r ) C r 4 ] d r ( 0 ≤ x < y ≤ 1 ) . Since 0 ≤ α β ≤ 1 0 \le\alpha_\beta\le1 0 ≤ α β ≤ 1 0 < C ^ ≤ 1 0 < \widehat{C} \le1 0 < C ≤ 1 ∣ D ^ ∣ , ∣ A ^ ∣ ≤ K |\widehat{D}|,|\widehat{A}| \le K ∣ D ∣ , ∣ A ∣ ≤ K 7 K 2 7K^2 7 K 2 K 2 + 1 K^2+1 K 2 + 1 Λ \Lambda Λ Ξ \Xi Ξ [ 0 , 1 ] [0,1] [ 0 , 1 ] 7 β 2 K 2 7\beta^2K^2 7 β 2 K 2 3 β 2 ( K 2 + 1 ) 3\beta^2(K^2+1) 3 β 2 ( K 2 + 1 ) 0 ≤ Λ ≤ K 2 0 \le\Lambda\le K^2 0 ≤ Λ ≤ K 2 Ξ \Xi Ξ C ^ r > 0 \widehat{C}_r > 0 C r > 0
Ξ 0 = min 0 ≤ r ≤ 1 Ξ ( r ) > 0 , \Xi_0=\min_{0\le r\le1}\Xi(r)>0, Ξ 0 = 0 ≤ r ≤ 1 min Ξ ( r ) > 0 , which depends only on β \beta β μ β \mu_\beta μ β Ξ \Xi Ξ u x x > 0 u_{xx}>0 u xx > 0
The quotient α ‾ = Λ / ( 2 Ξ ) \overline{\alpha}=\Lambda/(2\Xi) α = Λ/ ( 2Ξ ) [ 0 , 1 ] [0,1] [ 0 , 1 ]
L β = 7 β 2 K 2 2 Ξ 0 + 3 β 2 K 2 ( K 2 + 1 ) 2 Ξ 0 2 . L_\beta=\frac{7\beta^2K^2}{2\Xi_0}+\frac{3\beta^2K^2(K^2+1)}{2\Xi_0^2}. L β = 2 Ξ 0 7 β 2 K 2 + 2 Ξ 0 2 3 β 2 K 2 ( K 2 + 1 ) . For x , y ∈ [ 0 , 1 ] x,y\in[0,1] x , y ∈ [ 0 , 1 ] Λ \Lambda Λ Ξ \Xi Ξ Λ ≤ K 2 \Lambda\le K^2 Λ ≤ K 2 Ξ ≥ Ξ 0 \Xi\ge\Xi_0 Ξ ≥ Ξ 0
∣ α ‾ ( y ) − α ‾ ( x ) ∣ ≤ ∣ Λ ( y ) − Λ ( x ) ∣ 2 Ξ ( y ) + Λ ( x ) ∣ Ξ ( x ) − Ξ ( y ) ∣ 2 Ξ ( x ) Ξ ( y ) ≤ L β ∣ y − x ∣ . \left|\overline{\alpha}(y)-\overline{\alpha}(x)\right|
\le\frac{|\Lambda(y)-\Lambda(x)|}{2\Xi(y)}
+\Lambda(x)\frac{|\Xi(x)-\Xi(y)|}{2\Xi(x)\Xi(y)}
\le L_\beta|y-x|. ∣ α ( y ) − α ( x ) ∣ ≤ 2Ξ ( y ) ∣Λ ( y ) − Λ ( x ) ∣ + Λ ( x ) 2Ξ ( x ) Ξ ( y ) ∣Ξ ( x ) − Ξ ( y ) ∣ ≤ L β ∣ y − x ∣. Next, we identify α β \alpha_\beta α β α ‾ \overline{\alpha} α [ 0 , q β ) [0,q_\beta) [ 0 , q β ) Γ ( r ) = r \Gamma(r)=r Γ ( r ) = r [ 0 , q β ] [0,q_\beta] [ 0 , q β ] Γ \Gamma Γ Γ ′ = 1 \Gamma'=1 Γ ′ = 1 ( 0 , q β ) (0,q_\beta) ( 0 , q β )
∫ x y ( Λ − 2 α β Ξ ) d r = 0 \int_x^y\left(\Lambda-2\alpha_\beta\Xi\right)\,\mathrm{d}r=0 ∫ x y ( Λ − 2 α β Ξ ) d r = 0 for 0 < x < y < q β 0<x<y<q_\beta 0 < x < y < q β Λ − 2 α β Ξ = 0 \Lambda-2\alpha_\beta\Xi=0 Λ − 2 α β Ξ = 0 α β = α ‾ \alpha_\beta=\overline{\alpha} α β = α ( 0 , q β ) (0,q_\beta) ( 0 , q β ) r ∈ [ 0 , q β ) r\in[0,q_\beta) r ∈ [ 0 , q β ) r n ↓ r r_n\downarrow r r n ↓ r ( r , q β ) (r,q_\beta) ( r , q β ) α β ( r n ) = α ‾ ( r n ) \alpha_\beta(r_n)=\overline{\alpha}(r_n) α β ( r n ) = α ( r n ) n → ∞ n\to\infty n → ∞ α β ( r ) \alpha_\beta(r) α β ( r ) α β \alpha_\beta α β α ‾ ( r ) \overline{\alpha}(r) α ( r ) α β = α ‾ \alpha_\beta=\overline{\alpha} α β = α [ 0 , q β ) [0,q_\beta) [ 0 , q β ) α ‾ \overline{\alpha} α
For the consequences, we have γ t r ( 0 ) = max { λ , 0 } = λ \gamma^{\mathrm{tr}}(0)=\max\{\lambda,0\}=\lambda γ tr ( 0 ) = max { λ , 0 } = λ γ t r = max { λ , β } = β \gamma^{\mathrm{tr}}=\max\{\lambda,\beta\}=\beta γ tr = max { λ , β } = β [ q β , 1 ] [q_\beta,1] [ q β , 1 ] [ 0 , q β ) [0,q_\beta) [ 0 , q β ) γ β = β α β \gamma_\beta=\beta\alpha_\beta γ β = β α β β L β \beta L_\beta β L β λ \lambda λ L ≥ β L β L\ge\beta L_\beta L ≥ β L β γ t r \gamma^{\mathrm{tr}} γ tr ( 0 , q β ) (0,q_\beta) ( 0 , q β ) ( q β , 1 ] (q_\beta,1] ( q β , 1 ] γ t r \gamma^{\mathrm{tr}} γ tr
Distance from the scalar trial profile
We now compare a minimizer with γ t r \gamma^{\mathrm{tr}} γ tr L 1 ( 0 , 1 ) L^1(0,1) L 1 ( 0 , 1 ) γ t r \gamma^{\mathrm{tr}} γ tr ( 0 , 1 ) (0,1) ( 0 , 1 ) γ t r \gamma^{\mathrm{tr}} γ tr q β q_\beta q β
Proposition 7.6 (Isolation against the trial). There exists C = C ( β ) < ∞ C=C(\beta)<\infty C = C ( β ) < ∞ such that, for all N ≥ 1 N\ge1 N ≥ 1 0 < ρ ≤ 1 0<\rho\le1 0 < ρ ≤ 1 0 < λ ≤ λ 0 0<\lambda\le\lambda_0 0 < λ ≤ λ 0 every minimizer γ ∗ \gamma_* γ ∗ of F ρ F_\rho F ρ and every trial budget η \eta η for γ ∗ \gamma_* γ ∗
∥ γ ∗ − γ t r ∥ L 1 ( 0 , 1 ) ≤ C min { 1 , ρ − 3 / 4 η 1 / 4 } . (69) \lVert\gamma_*-\gamma^{\mathrm{tr}}\rVert_{L^1(0,1)}\le C\min\left\{1,\rho^{-3/4}\eta^{1/4}\right\}. \tag*{(69)} ∥ γ ∗ − γ tr ∥ L 1 ( 0 , 1 ) ≤ C min { 1 , ρ − 3/4 η 1/4 } . (69) The constant depends on β \beta β β \beta β k β k_\beta k β L L L
Proof. The proof has seven steps. We first fix a smoothing width h h h γ ∗ \gamma_* γ ∗ γ ∗ h \gamma_*^h γ ∗ h ρ β h 2 / 12 \rho\beta h^2/12 ρβ h 2 /12 γ ∗ h \gamma_*^h γ ∗ h γ t r \gamma^{\mathrm{tr}} γ tr ψ = γ t r − γ ∗ h \psi=\gamma^{\mathrm{tr}}-\gamma_*^h ψ = γ tr − γ ∗ h L L L q β q_\beta q β q β q_\beta q β A > 0 A>0 A > 0 ψ \psi ψ q β q_\beta q β q β q_\beta q β q β q_\beta q β γ t r \gamma^{\mathrm{tr}} γ tr f ′ ′ ( θ ) f''(\theta) f ′′ ( θ ) f ′ ′ ( θ ) f''(\theta) f ′′ ( θ ) L 1 L^1 L 1 h h h ρ \rho ρ
Step 1: Setup and constraints on h h h Let D = ∥ γ ∗ − γ t r ∥ L 1 ( 0 , 1 ) D=\lVert\gamma_*-\gamma^{\mathrm{tr}}\rVert_{L^1(0,1)} D = ∥ γ ∗ − γ tr ∥ L 1 ( 0 , 1 ) D = 0 D=0 D = 0 D > 0 D>0 D > 0 [ λ , β ] [\lambda,\beta] [ λ , β ] D ≤ β D\le\beta D ≤ β L ≥ 1 L\ge1 L ≥ 1 h h h
0 < h ≤ 1 2 , h ≤ D 2 β , h ≤ β L . (70) 0<h\le\frac{1}{2},\qquad h\le\frac{D}{2\beta},\qquad h\le\frac{\beta}{L}. \tag*{(70)} 0 < h ≤ 2 1 , h ≤ 2 β D , h ≤ L β . (70) The final choice, in Step 7, will be a small multiple of ρ D 2 \rho D^2 ρ D 2
Step 2: The regularized segment. Let γ ∗ h \gamma_*^h γ ∗ h γ ∗ \gamma_* γ ∗
γ θ = ( 1 − θ ) γ ∗ h + θ γ t r , ψ = γ t r − γ ∗ h , f ( θ ) = F ρ ( γ θ ) ( 0 ≤ θ ≤ 1 ) . \gamma_\theta=(1-\theta)\gamma_*^h+\theta\gamma^{\mathrm{tr}},\qquad\psi=\gamma^{\mathrm{tr}}-\gamma_*^h,\qquad f(\theta)=F_\rho(\gamma_\theta)\qquad(0\le\theta\le1). γ θ = ( 1 − θ ) γ ∗ h + θ γ tr , ψ = γ tr − γ ∗ h , f ( θ ) = F ρ ( γ θ ) ( 0 ≤ θ ≤ 1 ) . The regularization is defined because h ≤ 1 / 2 h\le1/2 h ≤ 1/2 ( γ θ ) (\gamma_\theta) ( γ θ ) h ≤ D / ( 2 β ) h\le D/(2\beta) h ≤ D / ( 2 β )
D ‾ = ∥ ψ ∥ L 1 ( 0 , 1 ) ≥ D − ∥ γ ∗ h − γ ∗ ∥ L 1 ( 0 , 1 ) ≥ D − β h ≥ D 2 . \overline{D}=\lVert\psi\rVert_{L^1(0,1)}\ge D-\lVert\gamma_*^h-\gamma_*\rVert_{L^1(0,1)}\ge D-\beta h\ge\frac{D}{2}. D = ∥ ψ ∥ L 1 ( 0 , 1 ) ≥ D − ∥ γ ∗ h − γ ∗ ∥ L 1 ( 0 , 1 ) ≥ D − β h ≥ 2 D . For θ ∈ ( 0 , 1 ) \theta\in(0,1) θ ∈ ( 0 , 1 ) d γ θ \mathrm{d}\gamma_\theta d γ θ ( 1 − θ ) d γ ∗ h + θ d γ t r (1-\theta)\,\mathrm{d}\gamma_*^h+\theta\,\mathrm{d}\gamma^{\mathrm{tr}} ( 1 − θ ) d γ ∗ h + θ d γ tr ( 0 , 1 ) ∖ { q β } (0,1)\setminus\{q_\beta\} ( 0 , 1 ) ∖ { q β } d γ ∗ h \mathrm{d}\gamma_*^h d γ ∗ h β / h \beta/h β / h d γ t r \mathrm{d}\gamma^{\mathrm{tr}} d γ tr L L L L ≤ β / h L\le\beta/h L ≤ β / h d γ θ \mathrm{d}\gamma_\theta d γ θ ( 0 , 1 ) ∖ { q β } (0,1)\setminus\{q_\beta\} ( 0 , 1 ) ∖ { q β } β / h + L ≤ 2 β / h \beta/h+L\le2\beta/h β / h + L ≤ 2 β / h d γ θ \mathrm{d}\gamma_\theta d γ θ ( 0 , 1 ) (0,1) ( 0 , 1 ) d γ t r \mathrm{d}\gamma^{\mathrm{tr}} d γ tr q β q_\beta q β ζ θ ≥ k β 2 \zeta_\theta\ge k_\beta^2 ζ θ ≥ k β 2 Step 3: One-sided bounds on ψ \psi ψ By Lemma 7.3(b) and Lemma 7.5, we have γ ∗ h ( 0 ) = γ t r ( 0 ) = λ \gamma_*^h(0)=\gamma^{\mathrm{tr}}(0)=\lambda γ ∗ h ( 0 ) = γ tr ( 0 ) = λ 0 ≤ z < x < q β 0\le z<x<q_\beta 0 ≤ z < x < q β γ t r ( x ) − γ t r ( z ) ≤ L ( x − z ) \gamma^{\mathrm{tr}}(x)-\gamma^{\mathrm{tr}}(z)\le L(x-z) γ tr ( x ) − γ tr ( z ) ≤ L ( x − z ) γ ∗ h ( x ) − γ ∗ h ( z ) ≥ 0 \gamma_*^h(x)-\gamma_*^h(z)\ge0 γ ∗ h ( x ) − γ ∗ h ( z ) ≥ 0 γ ∗ h \gamma_*^h γ ∗ h
ψ ( 0 ) = 0 , ψ ( x ) − ψ ( z ) ≤ L ( x − z ) , ψ ( x ) ≤ L x ( 0 ≤ z < x < q β ) . (71) \psi(0)=0,\qquad\psi(x)-\psi(z)\le L(x-z),\qquad\psi(x)\le Lx\qquad(0\le z<x<q_\beta). \tag*{(71)} ψ ( 0 ) = 0 , ψ ( x ) − ψ ( z ) ≤ L ( x − z ) , ψ ( x ) ≤ Lx ( 0 ≤ z < x < q β ) . (71) On [ q β , 1 ] [q_\beta,1] [ q β , 1 ] ψ = β − γ ∗ h \psi=\beta-\gamma_*^h ψ = β − γ ∗ h γ ∗ h \gamma_*^h γ ∗ h [ 0 , 1 ] [0,1] [ 0 , 1 ] [ λ , β ] [\lambda,\beta] [ λ , β ] ∣ ψ ∣ ≤ β |\psi|\le\beta ∣ ψ ∣ ≤ β
Step 4: Cutoffs for the superlevel sets. We fix θ ∈ ( 0 , 1 ) \theta\in(0,1) θ ∈ ( 0 , 1 ) A > 0 A>0 A > 0 r = A / ( 4 L ) r=A/(4L) r = A / ( 4 L )
U A = { x ∈ ( 0 , q β ) : ψ ( x ) > A } , V A i = { x ∈ ( 0 , q β − r ) : ψ ( x ) < − A } , V A b = { x ∈ [ max { 0 , q β − r } , q β ) : ψ ( x ) < − A } , W A = { x ∈ [ q β , 1 ] : ψ ( x ) > A } . \begin{aligned}
U_A&=\{x\in(0,q_\beta):\psi(x)>A\},\\
V_A^i&=\{x\in(0,q_\beta-r):\psi(x)<-A\},\\
V_A^b&=\{x\in[\max\{0,q_\beta-r\},q_\beta):\psi(x)<-A\},\\
W_A&=\{x\in[q_\beta,1]:\psi(x)>A\}.
\end{aligned} U A V A i V A b W A = { x ∈ ( 0 , q β ) : ψ ( x ) > A } , = { x ∈ ( 0 , q β − r ) : ψ ( x ) < − A } , = { x ∈ [ max { 0 , q β − r } , q β ) : ψ ( x ) < − A } , = { x ∈ [ q β , 1 ] : ψ ( x ) > A } . where an interval of nonpositive length is empty. These four sets are disjoint, and since ψ ( 0 ) = 0 \psi(0)=0 ψ ( 0 ) = 0 ψ ≥ 0 \psi\ge0 ψ ≥ 0 [ q β , 1 ] [q_\beta,1] [ q β , 1 ] { x ∈ [ 0 , 1 ] : ∣ ψ ( x ) ∣ > A } \{x\in[0,1]:|\psi(x)|>A\} { x ∈ [ 0 , 1 ] : ∣ ψ ( x ) ∣ > A } U A U_A U A V A i V_A^i V A i V A b V_A^b V A b W A W_A W A
For a Borel set E ⊂ ( 0 , 1 ) E\subset(0,1) E ⊂ ( 0 , 1 ) R \mathbb{R} R
A + 1 E ( x ) = 1 r ∫ 0 r 1 E ( x + u ) d u , A − 1 E ( x ) = 1 r ∫ 0 r 1 E ( x − u ) d u ( x ∈ R ) . \mathcal{A}_{+}\mathbf{1}_E(x)=\frac{1}{r}\int_0^r\mathbf{1}_E(x+u)\,\mathrm{d}u,\qquad
\mathcal{A}_{-}\mathbf{1}_E(x)=\frac{1}{r}\int_0^r\mathbf{1}_E(x-u)\,\mathrm{d}u\qquad(x\in\mathbb{R}). A + 1 E ( x ) = r 1 ∫ 0 r 1 E ( x + u ) d u , A − 1 E ( x ) = r 1 ∫ 0 r 1 E ( x − u ) d u ( x ∈ R ) . With Φ E ( x ) = ∣ E ∩ ( − ∞ , x ] ∣ \Phi_E(x)=|E\cap(-\infty,x]| Φ E ( x ) = ∣ E ∩ ( − ∞ , x ] ∣ A + 1 E ( x ) = ( Φ E ( x + r ) − Φ E ( x ) ) / r \mathcal{A}_{+}\mathbf{1}_E(x)=(\Phi_E(x+r)-\Phi_E(x))/r A + 1 E ( x ) = ( Φ E ( x + r ) − Φ E ( x )) / r A − 1 E ( x ) = ( Φ E ( x ) − Φ E ( x − r ) ) / r \mathcal{A}_{-}\mathbf{1}_E(x)=(\Phi_E(x)-\Phi_E(x-r))/r A − 1 E ( x ) = ( Φ E ( x ) − Φ E ( x − r )) / r Φ E \Phi_E Φ E 1 E \mathbf{1}_E 1 E A ± 1 E \mathcal{A}_{\pm}\mathbf{1}_E A ± 1 E ( 1 E ( x + r ) − 1 E ( x ) ) / r (\mathbf{1}_E(x+r)-\mathbf{1}_E(x))/r ( 1 E ( x + r ) − 1 E ( x )) / r ( 1 E ( x ) − 1 E ( x − r ) ) / r (\mathbf{1}_E(x)-\mathbf{1}_E(x-r))/r ( 1 E ( x ) − 1 E ( x − r )) / r x x x ∣ ( E − r ) △ E ∣ ≤ 2 ∣ E ∣ |(E-r)\mathbin{\triangle}E|\le2|E| ∣ ( E − r ) △ E ∣ ≤ 2∣ E ∣
0 ≤ A ± 1 E ≤ 1 , ∫ R A ± 1 E = ∣ E ∣ , ∫ R ∣ ∂ x A ± 1 E ∣ 2 ≤ 2 ∣ E ∣ r 2 . (72) 0\le\mathcal{A}_{\pm}\mathbf{1}_E\le1,\qquad
\int_{\mathbb{R}}\mathcal{A}_{\pm}\mathbf{1}_E=|E|,\qquad
\int_{\mathbb{R}}|\partial_x\mathcal{A}_{\pm}\mathbf{1}_E|^2\le\frac{2|E|}{r^2}. \tag*{(72)} 0 ≤ A ± 1 E ≤ 1 , ∫ R A ± 1 E = ∣ E ∣ , ∫ R ∣ ∂ x A ± 1 E ∣ 2 ≤ r 2 2∣ E ∣ . (72) For E = U A E=U_A E = U A φ = A + 1 E \varphi=\mathcal{A}_{+}\mathbf{1}_E φ = A + 1 E φ ( x ) > 0 \varphi(x)>0 φ ( x ) > 0 y ∈ U A y\in U_A y ∈ U A x ≤ y ≤ x + r x\le y\le x+r x ≤ y ≤ x + r A < ψ ( y ) ≤ L y A<\psi(y)\le Ly A < ψ ( y ) ≤ L y y > A / L = 4 r y>A/L=4r y > A / L = 4 r x ≥ y − r > 3 r > 0 x\ge y-r>3r>0 x ≥ y − r > 3 r > 0 x ≤ y < q β x\le y<q_\beta x ≤ y < q β ψ ( x ) ≥ ψ ( y ) − L ( y − x ) > A − L r = 3 A / 4 \psi(x)\ge\psi(y)-L(y-x)>A-Lr=3A/4 ψ ( x ) ≥ ψ ( y ) − L ( y − x ) > A − L r = 3 A /4 { φ > 0 } ⊂ ( 0 , q β ) \{\varphi>0\}\subset(0,q_\beta) { φ > 0 } ⊂ ( 0 , q β ) ψ > 3 A / 4 \psi>3A/4 ψ > 3 A /4
For E = V A i E=V_A^i E = V A i φ = A − 1 E \varphi=\mathcal{A}_{-}\mathbf{1}_E φ = A − 1 E φ ( x ) > 0 \varphi(x)>0 φ ( x ) > 0 y ∈ V A i y\in V_A^i y ∈ V A i y ≤ x ≤ y + r y\le x\le y+r y ≤ x ≤ y + r y + r < q β y+r<q_\beta y + r < q β V A i V_A^i V A i 0 < x < q β 0<x<q_\beta 0 < x < q β ψ ( x ) ≤ ψ ( y ) + L ( x − y ) < − A + L r = − 3 A / 4 \psi(x)\le\psi(y)+L(x-y)<-A+Lr=-3A/4 ψ ( x ) ≤ ψ ( y ) + L ( x − y ) < − A + L r = − 3 A /4 { φ > 0 } ⊂ ( 0 , q β ) \{\varphi>0\}\subset(0,q_\beta) { φ > 0 } ⊂ ( 0 , q β ) ψ < − 3 A / 4 \psi<-3A/4 ψ < − 3 A /4
In both cases, φ \varphi φ [ 0 , 1 ] [0,1] [ 0 , 1 ] [ 0 , 1 ] [0,1] [ 0 , 1 ] 0 0 0 [ q β , 1 ] [q_{\beta},1] [ q β , 1 ] [ 0 , 1 ] [0,1] [ 0 , 1 ] ∣ E ∣ |E| ∣ E ∣ [ 0 , 1 ] [0,1] [ 0 , 1 ] 2 ∣ E ∣ / r 2 2|E|/r^2 2∣ E ∣/ r 2 { φ > 0 } \{\varphi>0\} { φ > 0 } q β q_{\beta} q β d γ θ \mathrm{d}\gamma_{\theta} d γ θ d = 2 β / h d=2\beta/h d = 2 β / h ∣ E ∣ > 0 |E|>0 ∣ E ∣ > 0 φ \varphi φ σ = 1 \sigma=1 σ = 1 U A U_A U A σ = − 1 \sigma=-1 σ = − 1 V A i V_A^i V A i
Δ = ρ ∫ 0 1 ζ θ φ d q ≥ ρ k β 2 ∣ E ∣ , γ 0 ≤ 2 ∣ E ∣ r 2 , d = 2 β h . \Delta=\rho\int_0^1\zeta_{\theta}\varphi\,\mathrm{d}q\geq\rho k_{\beta}^{2}|E|,\qquad\gamma_0\leq\frac{2|E|}{r^2},\qquad d=\frac{2\beta}{h}. Δ = ρ ∫ 0 1 ζ θ φ d q ≥ ρ k β 2 ∣ E ∣ , γ 0 ≤ r 2 2∣ E ∣ , d = h 2 β . by (7.4) and (7.16). Inserting these bounds into (7.11) and using r = A / ( 4 L ) r=A/(4L) r = A / ( 4 L )
f ′ ′ ( θ ) ≥ ρ 2 A 2 ∣ E ∣ 72 min { ρ k β 4 A 2 32 β L 2 , k β 2 h 2 β } ( E = U A , V A i ) . (73) f''(\theta)\geq\frac{\rho^2A^2|E|}{72}\min\left\{\frac{\rho k_{\beta}^{4}A^2}{32\beta L^2},\frac{k_{\beta}^{2}h}{2\beta}\right\}\qquad(E=U_A,V_A^i). \tag*{(73)} f ′′ ( θ ) ≥ 72 ρ 2 A 2 ∣ E ∣ min { 32 β L 2 ρ k β 4 A 2 , 2 β k β 2 h } ( E = U A , V A i ) . (73) When ∣ E ∣ = 0 |E|=0 ∣ E ∣ = 0
Step 5: The boundary sets. The set V A b V_A^b V A b r r r q β q_{\beta} q β 1 V A b \mathbf{1}_{V_A^b} 1 V A b r r r q β q_{\beta} q β ( x 0 , x 0 + z ) (x_0,x_0+z) ( x 0 , x 0 + z ) z > 0 z>0 z > 0
T x 0 , z ( x ) = ( 1 − 4 ∣ x − x 0 − z / 2 ∣ z ) + . T_{x_0,z}(x)=\left(1-\frac{4|x-x_0-z/2|}{z}\right)_+. T x 0 , z ( x ) = ( 1 − z 4∣ x − x 0 − z /2∣ ) + . This function takes values in [ 0 , 1 ] [0,1] [ 0 , 1 ] ( x 0 + z / 4 , x 0 + 3 z / 4 ) (x_0+z/4,x_0+3z/4) ( x 0 + z /4 , x 0 + 3 z /4 )
∫ R T x 0 , z = z 4 , ∫ R ∣ T x 0 , z ′ ∣ 2 = 8 z . \int_{\mathbb{R}}T_{x_0,z}=\frac{z}{4},\qquad\int_{\mathbb{R}}|T'_{x_0,z}|^2=\frac{8}{z}. ∫ R T x 0 , z = 4 z , ∫ R ∣ T x 0 , z ′ ∣ 2 = z 8 . Suppose that V A b V_A^b V A b x A = inf V A b x_A=\inf V_A^b x A = inf V A b l = q β − x A l=q_{\beta}-x_A l = q β − x A 0 < l ≤ r 0<l\leq r 0 < l ≤ r ∣ V A b ∣ ≤ l |V_A^b|\leq l ∣ V A b ∣ ≤ l z ∈ ( x A , q β ) z\in(x_A,q_{\beta}) z ∈ ( x A , q β ) x ∈ V A b x\in V_A^b x ∈ V A b x < z x<z x < z ψ ( z ) ≤ ψ ( x ) + L ( z − x ) < − A + L r = − 3 A / 4 \psi(z)\leq\psi(x)+L(z-x)<-A+Lr=-3A/4 ψ ( z ) ≤ ψ ( x ) + L ( z − x ) < − A + L r = − 3 A /4 φ = T x A , l \varphi=T_{x_A,l} φ = T x A , l ( x A , q β ) (x_A,q_{\beta}) ( x A , q β ) ( 0 , q β ) (0,q_{\beta}) ( 0 , q β ) ψ < − 3 A / 4 \psi<-3A/4 ψ < − 3 A /4 d γ θ \mathrm{d}\gamma_{\theta} d γ θ 2 β / h 2\beta/h 2 β / h σ = − 1 \sigma=-1 σ = − 1 Δ ≥ ρ k β 2 l / 4 \Delta\geq\rho k_{\beta}^{2}l/4 Δ ≥ ρ k β 2 l /4 γ 0 = 8 / l \gamma_0=8/l γ 0 = 8/ l d = 2 β / h d=2\beta/h d = 2 β / h
f ′ ′ ( θ ) ≥ ρ 2 A 2 l 72 min { ρ k β 4 l 2 128 β , k β 2 h 8 β } . (74) f''(\theta)\geq\frac{\rho^2A^2l}{72}\min\left\{\frac{\rho k_{\beta}^{4}l^2}{128\beta},\frac{k_{\beta}^{2}h}{8\beta}\right\}. \tag*{(74)} f ′′ ( θ ) ≥ 72 ρ 2 A 2 l min { 128 β ρ k β 4 l 2 , 8 β k β 2 h } . (74) Indeed, Δ 2 / ( β γ 0 ) ≥ ρ 2 k β 4 l 3 / ( 128 β ) \Delta^2/(\beta\gamma_0)\geq\rho^2k_{\beta}^{4}l^3/(128\beta) Δ 2 / ( β γ 0 ) ≥ ρ 2 k β 4 l 3 / ( 128 β ) Δ / d ≥ ρ k β 2 l h / ( 8 β ) \Delta/d\geq\rho k_{\beta}^{2}lh/(8\beta) Δ/ d ≥ ρ k β 2 l h / ( 8 β )
By Step 3, the function ψ \psi ψ [ q β , 1 ] [q_{\beta},1] [ q β , 1 ] W A W_A W A [ q β , q β + w ) [q_{\beta},q_{\beta}+w) [ q β , q β + w ) w ≤ 1 w\leq1 w ≤ 1 w > 0 w>0 w > 0 φ = T q β , w \varphi=T_{q_{\beta},w} φ = T q β , w ( q β , 1 ) (q_{\beta},1) ( q β , 1 ) ψ > A \psi>A ψ > A d γ t r \mathrm{d}\gamma^{\mathrm{tr}} d γ tr d γ θ \mathrm{d}\gamma_{\theta} d γ θ 2 β / h 2\beta/h 2 β / h σ = 1 \sigma=1 σ = 1 w w w l l l w = 0 w=0 w = 0
Step 6: Layer-cake integration. Let
c 1 = 1 72 min { k β 4 32 β L 2 , k β 4 128 β 2 , k β 2 8 β } , K = f ′ ′ ( θ ) c 1 ρ 2 . c_1 = \frac{1}{72}\min\left\{\frac{k_\beta^4}{32\beta L^2},\frac{k_\beta^4}{128\beta^2},\frac{k_\beta^2}{8\beta}\right\}, \qquad K = \frac{f''(\theta)}{c_1\rho^2}. c 1 = 72 1 min { 32 β L 2 k β 4 , 128 β 2 k β 4 , 8 β k β 2 } , K = c 1 ρ 2 f ′′ ( θ ) . Since min { a x , b y } ≥ min { a , b } min { x , y } \min\{ax,by\} \ge\min\{a,b\}\min\{x,y\} min { a x , b y } ≥ min { a , b } min { x , y } (73) and (74) imply that
A 2 ∣ E ∣ min { ρ A 2 , h } ≤ K ( E = U A , V A i ) , A 2 z min { ρ z 2 , h } ≤ K ( z = l , w ) . A^2|E|\min\{\rho A^2,h\}\le K \quad(E=U_A,V_A^i), \qquad A^2z\min\{\rho z^2,h\}\le K \quad(z=l,w). A 2 ∣ E ∣ min { ρ A 2 , h } ≤ K ( E = U A , V A i ) , A 2 z min { ρ z 2 , h } ≤ K ( z = l , w ) . For E = U A E=U_A E = U A E = V A i E=V_A^i E = V A i ∣ E ∣ ≤ K / ( ρ A 4 ) |E|\le K/(\rho A^4) ∣ E ∣ ≤ K / ( ρ A 4 ) ∣ E ∣ ≤ K / ( h A 2 ) |E|\le K/(hA^2) ∣ E ∣ ≤ K / ( h A 2 ) ∣ E ∣ ≤ 1 |E|\le1 ∣ E ∣ ≤ 1
∣ E ∣ ≤ min { 1 , K / ( ρ A 4 ) } + min { 1 , K / ( h A 2 ) } . |E|\le\min\{1,K/(\rho A^4)\}+\min\{1,K/(hA^2)\}. ∣ E ∣ ≤ min { 1 , K / ( ρ A 4 )} + min { 1 , K / ( h A 2 )} . For the boundary interval, one of l ≤ ( K / ( ρ A 2 ) ) 1 / 3 l\le(K/(\rho A^2))^{1/3} l ≤ ( K / ( ρ A 2 ) ) 1/3 l ≤ K / ( h A 2 ) l\le K/(hA^2) l ≤ K / ( h A 2 ) ∣ V A b ∣ ≤ l ≤ r = A / ( 4 L ) |V_A^b|\le l\le r=A/(4L) ∣ V A b ∣ ≤ l ≤ r = A / ( 4 L )
∣ V A b ∣ ≤ ( K ρ A 2 ) 1 / 3 + min { A 4 L , K h A 2 } . |V_A^b|\le\left(\frac{K}{\rho A^2}\right)^{1/3}+\min\left\{\frac{A}{4L},\frac{K}{hA^2}\right\}. ∣ V A b ∣ ≤ ( ρ A 2 K ) 1/3 + min { 4 L A , h A 2 K } . The same reasoning with w ≤ 1 w\le1 w ≤ 1
∣ W A ∣ ≤ ( K ρ A 2 ) 1 / 3 + min { 1 , K h A 2 } . |W_A|\le\left(\frac{K}{\rho A^2}\right)^{1/3}+\min\left\{1,\frac{K}{hA^2}\right\}. ∣ W A ∣ ≤ ( ρ A 2 K ) 1/3 + min { 1 , h A 2 K } . All four sets are empty for A ≥ β A\ge\beta A ≥ β ∣ ψ ∣ ≤ β |\psi|\le\beta ∣ ψ ∣ ≤ β
∫ 0 ∞ min { 1 , x A 4 } d A = 4 3 x 1 / 4 , ∫ 0 ∞ min { 1 , x A 2 } d A = 2 x 1 / 2 , \int_0^\infty\min\left\{1,\frac{x}{A^4}\right\}\,dA=\frac{4}{3}x^{1/4}, \qquad\int_0^\infty\min\left\{1,\frac{x}{A^2}\right\}\,dA=2x^{1/2}, ∫ 0 ∞ min { 1 , A 4 x } d A = 3 4 x 1/4 , ∫ 0 ∞ min { 1 , A 2 x } d A = 2 x 1/2 , ∫ 0 ∞ min { A 4 L , x A 2 } d A = C L x 2 / 3 , ∫ 0 β ( x A 2 ) 1 / 3 d A = 3 β 1 / 3 x 1 / 3 , \int_0^\infty\min\left\{\frac{A}{4L},\frac{x}{A^2}\right\}\,dA=C_Lx^{2/3}, \qquad\int_0^\beta\left(\frac{x}{A^2}\right)^{1/3}\,dA=3\beta^{1/3}x^{1/3}, ∫ 0 ∞ min { 4 L A , A 2 x } d A = C L x 2/3 , ∫ 0 β ( A 2 x ) 1/3 d A = 3 β 1/3 x 1/3 , where C L = 3 / ( 2 ( 4 L ) 1 / 3 ) C_L=3/(2(4L)^{1/3}) C L = 3/ ( 2 ( 4 L ) 1/3 ) x ≥ 0 x\ge0 x ≥ 0 x = 0 x=0 x = 0 x > 0 x>0 x > 0 A 0 = ( 4 L x ) 1 / 3 A_0=(4Lx)^{1/3} A 0 = ( 4 Lx ) 1/3 A 0 2 / ( 8 L ) + x / A 0 = 3 x / ( 2 A 0 ) A_0^2/(8L)+x/A_0=3x/(2A_0) A 0 2 / ( 8 L ) + x / A 0 = 3 x / ( 2 A 0 ) D ˉ = ∫ 0 β ∣ { ∣ ψ ∣ > A } ∣ d A \bar D=\int_0^\beta|\{|\psi|>A\}|\,dA D ˉ = ∫ 0 β ∣ { ∣ ψ ∣ > A } ∣ d A 0 < A < β 0<A<\beta 0 < A < β
D ˉ ≤ 8 3 ( K ρ ) 1 / 4 + 6 ( K h ) 1 / 2 + 6 β 1 / 3 ( K ρ ) 1 / 3 + C L ( K h ) 2 / 3 . (75) \bar D\le\frac{8}{3}\left(\frac{K}{\rho}\right)^{1/4}+6\left(\frac{K}{h}\right)^{1/2}+6\beta^{1/3}\left(\frac{K}{\rho}\right)^{1/3}+C_L\left(\frac{K}{h}\right)^{2/3}. \tag*{(75)} D ˉ ≤ 3 8 ( ρ K ) 1/4 + 6 ( h K ) 1/2 + 6 β 1/3 ( ρ K ) 1/3 + C L ( h K ) 2/3 . (75) Let C w = max { 6 β 1 / 3 , C L } C_w=\max\{6\beta^{1/3},C_L\} C w = max { 6 β 1/3 , C L } (75) is at most C w C_w C w β > 1 \beta>1 β > 1 K ≥ c 2 min { ρ D 4 , h D 2 } K\ge c_2\min\{\rho D^4,hD^2\} K ≥ c 2 min { ρ D 4 , h D 2 } c 2 = min { ( 8 C w ) − 4 , β − 4 } c_2=\min\{(8C_w)^{-4},\beta^{-4}\} c 2 = min {( 8 C w ) − 4 , β − 4 } K ≤ ρ K\le\rho K ≤ ρ K ≤ h K\le h K ≤ h ( K / ρ ) 1 / 3 ≤ ( K / ρ ) 1 / 4 (K/\rho)^{1/3}\le(K/\rho)^{1/4} ( K / ρ ) 1/3 ≤ ( K / ρ ) 1/4 ( K / h ) 2 / 3 ≤ ( K / h ) 1 / 2 (K/h)^{2/3}\le(K/h)^{1/2} ( K / h ) 2/3 ≤ ( K / h ) 1/2 (75) gives D ˉ ≤ 2 C w [ ( K / ρ ) 1 / 4 + ( K / h ) 1 / 2 ] \bar{D} \le2C_w[(K/\rho)^{1/4} + (K/h)^{1/2}] D ˉ ≤ 2 C w [( K / ρ ) 1/4 + ( K / h ) 1/2 ] D ˉ ≥ D / 2 \bar{D} \ge D/2 D ˉ ≥ D /2 D / ( 8 C w ) D/(8C_w) D / ( 8 C w ) K ≥ ρ D 4 / ( 8 C w ) 4 K \ge\rho D^4/(8C_w)^4 K ≥ ρ D 4 / ( 8 C w ) 4 K ≥ h D 2 / ( 8 C w ) 2 K \ge hD^2/(8C_w)^2 K ≥ h D 2 / ( 8 C w ) 2 8 C w ≥ 1 8C_w \ge1 8 C w ≥ 1 K ≥ ( 8 C w ) − 4 min { ρ D 4 , h D 2 } K \ge(8C_w)^{-4}\min\{\rho D^4,hD^2\} K ≥ ( 8 C w ) − 4 min { ρ D 4 , h D 2 } K > ρ K > \rho K > ρ D ≤ β D \le\beta D ≤ β min { ρ D 4 , h D 2 } ≤ ρ β 4 < β 4 K \min\{\rho D^4,hD^2\} \le\rho\beta^4 < \beta^4K min { ρ D 4 , h D 2 } ≤ ρ β 4 < β 4 K K > h K > h K > h min { ρ D 4 , h D 2 } ≤ h β 2 < β 4 K \min\{\rho D^4,hD^2\} \le h\beta^2 < \beta^4K min { ρ D 4 , h D 2 } ≤ h β 2 < β 4 K β > 1 \beta> 1 β > 1 θ ∈ ( 0 , 1 ) \theta\in(0,1) θ ∈ ( 0 , 1 )
f ′ ′ ( θ ) ≥ c 3 ρ 2 min { ρ D 4 , h D 2 } ( 0 < θ < 1 ) , c 3 = c 1 c 2 . (76) f''(\theta) \ge c_3\rho^2\min\{\rho D^4,hD^2\} \qquad(0 < \theta< 1), \qquad c_3 = c_1c_2. \tag*{(76)} f ′′ ( θ ) ≥ c 3 ρ 2 min { ρ D 4 , h D 2 } ( 0 < θ < 1 ) , c 3 = c 1 c 2 . (76) Step 7: Choice of h h h We now take h = c 4 ρ D 2 h = c_4\rho D^2 h = c 4 ρ D 2
c 4 = min { 1 , 1 2 β 2 , 1 β L , 3 c 3 2 β } . c_4 = \min\left\{1,\frac{1}{2\beta^2},\frac{1}{\beta L},\frac{3c_3}{2\beta}\right\}. c 4 = min { 1 , 2 β 2 1 , β L 1 , 2 β 3 c 3 } . Since D ≤ β D \le\beta D ≤ β ρ ≤ 1 \rho\le1 ρ ≤ 1 h ≤ c 4 β 2 ≤ 1 / 2 h \le c_4\beta^2 \le1/2 h ≤ c 4 β 2 ≤ 1/2 h ≤ c 4 β D ≤ D / ( 2 β ) h \le c_4\beta D \le D/(2\beta) h ≤ c 4 β D ≤ D / ( 2 β ) h ≤ c 4 β 2 ≤ β / L h \le c_4\beta^2 \le\beta/L h ≤ c 4 β 2 ≤ β / L (70) . Because c 4 ≤ 1 c_4 \le1 c 4 ≤ 1 (76) equals c 4 ρ D 4 c_4\rho D^4 c 4 ρ D 4 (76) gives f ′ ′ ≥ c 3 c 4 ρ 3 D 4 f'' \ge c_3c_4\rho^3D^4 f ′′ ≥ c 3 c 4 ρ 3 D 4 ( 0 , 1 ) (0,1) ( 0 , 1 ) (65) with Θ = ( 0 , 1 ) \Theta= (0,1) Θ = ( 0 , 1 ) 1 / 4 1/4 1/4 (63) , it yields
c 3 c 4 4 ρ 3 D 4 ≤ F ρ ( γ h ∗ ) + F ρ ( γ t r ) − 2 F ρ ( γ ∗ ) ≤ ρ β h 2 12 + η ≤ c 3 c 4 8 ρ 3 D 4 + η . \frac{c_3c_4}{4}\rho^3D^4 \le F_\rho(\gamma_h^*) + F_\rho(\gamma^{\mathrm{tr}}) - 2F_\rho(\gamma^*) \le\frac{\rho\beta h^2}{12} + \eta\le\frac{c_3c_4}{8}\rho^3D^4 + \eta. 4 c 3 c 4 ρ 3 D 4 ≤ F ρ ( γ h ∗ ) + F ρ ( γ tr ) − 2 F ρ ( γ ∗ ) ≤ 12 ρβ h 2 + η ≤ 8 c 3 c 4 ρ 3 D 4 + η . In the last inequality, we used that ρ β h 2 / 12 = ( β c 4 2 / 12 ) ρ 3 D 4 \rho\beta h^2/12 = (\beta c_4^2/12)\rho^3D^4 ρβ h 2 /12 = ( β c 4 2 /12 ) ρ 3 D 4 c 4 ≤ 3 c 3 / ( 2 β ) c_4 \le3c_3/(2\beta) c 4 ≤ 3 c 3 / ( 2 β ) η ≥ ( c 3 c 4 / 8 ) ρ 3 D 4 \eta\ge(c_3c_4/8)\rho^3D^4 η ≥ ( c 3 c 4 /8 ) ρ 3 D 4 D ≤ ( 8 / ( c 3 c 4 ) ) 1 / 4 ρ − 3 / 4 η 1 / 4 D \le(8/(c_3c_4))^{1/4}\rho^{-3/4}\eta^{1/4} D ≤ ( 8/ ( c 3 c 4 ) ) 1/4 ρ − 3/4 η 1/4 D ≤ β D \le\beta D ≤ β (69) with C = max { β , ( 8 / ( c 3 c 4 ) ) 1 / 4 } C = \max\{\beta,(8/(c_3c_4))^{1/4}\} C = max { β , ( 8/ ( c 3 c 4 ) ) 1/4 } c 1 , c 2 , c 3 , c 4 c_1,c_2,c_3,c_4 c 1 , c 2 , c 3 , c 4 β \beta β k β k_\beta k β L L L
Localization of the lowest quantiles
In this section, we show that the lowest quantiles of the free mass of every minimizer of F ρ F_\rho F ρ μ = s δ 0 + ν \mu= s\delta_0 + \nu μ = s δ 0 + ν η \eta η Q ( v ) Q(v) Q ( v ) ν \nu ν v v v C Q ( s + v + ω ( ρ , η ) ) C_Q(s+v+\omega(\rho,\eta)) C Q ( s + v + ω ( ρ , η )) ω ( ρ , η ) = min { 1 , ( η / ρ ) 1 / 6 } \omega(\rho,\eta)=\min\{1,(\eta/\rho)^{1/6}\} ω ( ρ , η ) = min { 1 , ( η / ρ ) 1/6 } C Q C_Q C Q β \beta β ρ = 0 \rho=0 ρ = 0
We keep the standing assumptions and the notation of Section 7.1. From Section 7, we use the convexity and the first variation of F ρ F_\rho F ρ μ β \mu_\beta μ β
The main tool is a local deficit certificate (Lemma 8.1). It combines a variation that moves mass of a profile toward the middle of an interval with the response inequality (7.3). If the variation violates its room or sign condition, this failure gives a lower bound on η / ρ \eta/\rho η / ρ [4 Lemma 7.6] . We regularize only the free mass below the selected quantile. Then we compare the result with the scalar trial profile on an interval where the latter has a positive density (Lemma 8.2). The certificate gives the dependence on η / ρ \eta/\rho η / ρ
A local deficit certificate
In the proof of Proposition 7.6, we used the uniform lower bound (7.4) on ζ \zeta ζ Δ = ρ ∫ ζ φ ≥ ρ k β 2 ∫ φ \Delta=\rho\int\zeta\varphi\geq\rho k_\beta^2\int\varphi Δ = ρ ∫ ζ φ ≥ ρ k β 2 ∫ φ Δ \Delta Δ φ \varphi φ ρ \rho ρ
A local deficit certificate
Lemma 8.1 (Local deficit certificate). There exists c c e r t = c c e r t ( β ) > 0 c_{\mathrm{cert}} = c_{\mathrm{cert}}(\beta) > 0 c cert = c cert ( β ) > 0 with the following property. Let I ⊂ ( 0 , 1 ) I \subset(0,1) I ⊂ ( 0 , 1 ) ℓ \ell ℓ γ 0 , γ 1 \gamma_0,\gamma_1 γ 0 , γ 1 η > 0 \eta> 0 η > 0 F ρ ( γ i ) ≤ min F ρ + η F_\rho(\gamma_i) \le\min F_\rho+ \eta F ρ ( γ i ) ≤ min F ρ + η i = 0 , 1 i = 0,1 i = 0 , 1 k > 0 k > 0 k > 0 d 0 , d 1 ≥ 0 d_0,d_1 \ge0 d 0 , d 1 ≥ 0 A > 0 A > 0 A > 0 I I I
d γ 0 ≤ d 0 d x , k d x ≤ d γ 1 ≤ d 1 d x , γ 1 − γ 0 ≥ A . \mathrm{d}\gamma_0 \le d_0\,\mathrm{d}x,\qquad k\,\mathrm{d}x \le\mathrm{d}\gamma_1 \le d_1\,\mathrm{d}x,\qquad\gamma_1-\gamma_0 \ge A. d γ 0 ≤ d 0 d x , k d x ≤ d γ 1 ≤ d 1 d x , γ 1 − γ 0 ≥ A . Then
η ρ ≥ c c e r t min { k ℓ 3 , A ℓ 2 , A 2 ℓ 4 , A 2 ℓ 2 d 0 + d 1 + k } . (77) \frac{\eta}{\rho} \ge c_{\mathrm{cert}}\min\left\{k\ell^3,A\ell^2,A^2\ell^4,\frac{A^2\ell^2}{d_0+d_1+k}\right\}. \tag*{(77)} ρ η ≥ c cert min { k ℓ 3 , A ℓ 2 , A 2 ℓ 4 , d 0 + d 1 + k A 2 ℓ 2 } . (77) Proof. The proof has four steps. In Step 1, we fix a bump function φ \varphi φ I I I τ \tau τ τ \tau τ (77) holds. Otherwise, we perturb the segment from γ 0 \gamma_0 γ 0 γ 1 \gamma_1 γ 1 I I I F ρ F_\rho F ρ ℓ \ell ℓ Lemma 7.4 implies a lower bound on the curvature along the perturbed segment. Comparing this curvature with the excesses at its endpoints, we obtain the last two alternatives (Step 4). Throughout, we write F min = min F ρ F_{\min}=\min F_\rho F m i n = min F ρ I = ( x 0 , x 0 + ℓ ) I=(x_0,x_0+\ell) I = ( x 0 , x 0 + ℓ )
Step 1: The bump and the room condition. Let B ( y ) = 16 y 2 ( 1 − y ) 2 B(y)=16y^2(1-y)^2 B ( y ) = 16 y 2 ( 1 − y ) 2 y ∈ [ 0 , 1 ] y\in[0,1] y ∈ [ 0 , 1 ] φ ( x ) = B ( ( x − x 0 ) / ℓ ) \varphi(x)=B((x-x_0)/\ell) φ ( x ) = B (( x − x 0 ) / ℓ ) x ∈ I x\in I x ∈ I φ ( x ) = 0 \varphi(x)=0 φ ( x ) = 0 x ∉ I x\notin I x ∈ / I
B ′ ( y ) = 32 y ( 1 − y ) ( 1 − 2 y ) , B ′ ′ ( y ) = 32 − 192 y + 192 y 2 , B'(y)=32y(1-y)(1-2y),\qquad B''(y)=32-192y+192y^2, B ′ ( y ) = 32 y ( 1 − y ) ( 1 − 2 y ) , B ′′ ( y ) = 32 − 192 y + 192 y 2 , and B B B B ′ B' B ′ ∫ 0 1 B = 8 / 15 \int_0^1 B=8/15 ∫ 0 1 B = 8/15 max ∣ B ′ ∣ = 16 3 / 9 < 4 \max|B'|=16\sqrt{3}/9<4 max ∣ B ′ ∣ = 16 3 /9 < 4 max ∣ B ′ ′ ∣ = 32 \max|B''|=32 max ∣ B ′′ ∣ = 32 ∫ 0 1 ∣ B ′ ∣ 2 = 512 / 105 \int_0^1|B'|^2=512/105 ∫ 0 1 ∣ B ′ ∣ 2 = 512/105 B B B φ \varphi φ R \mathbb{R} R [ 0 , 1 ] [0,1] [ 0 , 1 ] φ ′ \varphi' φ ′ φ \varphi φ φ ′ \varphi' φ ′ I I I
∫ φ = 8 ℓ 15 , ∥ φ ′ ∥ ∞ ≤ 4 ℓ , ∥ φ ′ ′ ∥ ∞ ≤ 32 ℓ 2 , ∫ ∣ φ ′ ∣ 2 = 512 105 ℓ , (78) \int\varphi=\frac{8\ell}{15},\qquad\|\varphi'\|_\infty\le\frac{4}{\ell},\qquad\|\varphi''\|_\infty\le\frac{32}{\ell^2},\qquad\int|\varphi'|^2=\frac{512}{105\ell}, \tag*{(78)} ∫ φ = 15 8 ℓ , ∥ φ ′ ∥ ∞ ≤ ℓ 4 , ∥ φ ′′ ∥ ∞ ≤ ℓ 2 32 , ∫ ∣ φ ′ ∣ 2 = 105 ℓ 512 , (78) where φ ′ ′ \varphi'' φ ′′ I I I φ ′ \varphi' φ ′ φ ′ ′ \varphi'' φ ′′ I I I ∫ φ ′ ′ = 0 \int\varphi''=0 ∫ φ ′′ = 0
We define τ = 30 η / ( ρ ℓ ) \tau=30\eta/(\rho\ell) τ = 30 η / ( ρ ℓ ) 8 η / ( ρ τ ) = 4 ℓ / 15 8\eta/(\rho\tau)=4\ell/15 8 η / ( ρ τ ) = 4 ℓ /15 ∫ φ \int\varphi ∫ φ
τ ≤ min { k ℓ 2 64 , A ℓ 8 } (79) \tau\le\min\left\{\frac{k\ell^2}{64},\frac{A\ell}{8}\right\} \tag*{(79)} τ ≤ min { 64 k ℓ 2 , 8 A ℓ } (79) fails. Then
η ρ = ℓ τ 30 > min { k ℓ 3 1920 , A ℓ 2 240 } ≥ 1 1920 min { k ℓ 3 , A ℓ 2 } , \frac{\eta}{\rho}=\frac{\ell\tau}{30}>\min\left\{\frac{k\ell^3}{1920},\frac{A\ell^2}{240}\right\}\ge\frac{1}{1920}\min\{k\ell^3,A\ell^2\}, ρ η = 30 ℓ τ > min { 1920 k ℓ 3 , 240 A ℓ 2 } ≥ 1920 1 min { k ℓ 3 , A ℓ 2 } , and (8.1) holds whenever c c e r t ≤ 1 / 1920 c_{\mathrm{cert}}\le1/1920 c cert ≤ 1/1920
Step 2: The perturbed segment. For 0 ≤ θ ≤ 1 0 \le\theta\le1 0 ≤ θ ≤ 1
γ θ = ( 1 − θ ) γ 0 + θ γ 1 , γ ~ θ = γ θ − θ τ φ ′ . \gamma_\theta= (1-\theta)\gamma_0+\theta\gamma_1,\qquad\widetilde{\gamma}_\theta=\gamma_\theta-\theta\tau\varphi'. γ θ = ( 1 − θ ) γ 0 + θ γ 1 , γ θ = γ θ − θ τ φ ′ . We claim that γ θ − u φ ′ \gamma_\theta-u\varphi' γ θ − u φ ′ 0 ≤ θ ≤ 1 0\le\theta\le1 0 ≤ θ ≤ 1 0 ≤ u ≤ θ τ 0\le u\le\theta\tau 0 ≤ u ≤ θ τ φ ′ \varphi' φ ′ I I I I ‾ ⊂ ( 0 , 1 ) \overline{I}\subset(0,1) I ⊂ ( 0 , 1 ) γ θ − u φ ′ \gamma_\theta-u\varphi' γ θ − u φ ′ γ θ \gamma_\theta γ θ 0 0 0 1 1 1 d γ θ − u φ ′ ′ d x d\gamma_\theta-u\varphi''\,dx d γ θ − u φ ′′ d x I I I d γ θ d\gamma_\theta d γ θ I I I d γ θ ≥ θ k d x d\gamma_\theta\ge\theta k\,dx d γ θ ≥ θ k d x (78) and (79) , we have
d ( γ θ − u φ ′ ) ≥ ( θ k − 32 u ℓ 2 ) d x ≥ ( θ k − 32 θ τ ℓ 2 ) d x ≥ θ k 2 d x . d(\gamma_\theta-u\varphi')\ge\left(\theta k-\frac{32u}{\ell^2}\right)dx\ge\left(\theta k-\frac{32\theta\tau}{\ell^2}\right)dx\ge\frac{\theta k}{2}\,dx. d ( γ θ − u φ ′ ) ≥ ( θ k − ℓ 2 32 u ) d x ≥ ( θ k − ℓ 2 32 θ τ ) d x ≥ 2 θ k d x . The signed measure φ ′ ′ d x \varphi''\,dx φ ′′ d x 0 0 0 γ θ − u φ ′ \gamma_\theta-u\varphi' γ θ − u φ ′ β \beta β 0 0 0 d γ θ d\gamma_\theta d γ θ γ θ − u φ ′ \gamma_\theta-u\varphi' γ θ − u φ ′ γ ~ θ = ( 1 − θ ) γ 0 + θ ( γ 1 − τ φ ′ ) \widetilde{\gamma}_\theta=(1-\theta)\gamma_0+\theta(\gamma_1-\tau\varphi') γ θ = ( 1 − θ ) γ 0 + θ ( γ 1 − τ φ ′ ) γ 1 − τ φ ′ \gamma_1-\tau\varphi' γ 1 − τ φ ′ θ = 1 \theta=1 θ = 1 u = τ u=\tau u = τ ( γ ~ θ ) (\widetilde{\gamma}_\theta) ( γ θ ) γ ~ 0 = γ 0 \widetilde{\gamma}_0=\gamma_0 γ 0 = γ 0 γ ~ 1 = γ 1 − τ φ ′ \widetilde{\gamma}_1=\gamma_1-\tau\varphi' γ 1 = γ 1 − τ φ ′
Step 3: The response integrated against the bump. Fix θ ∈ ( 0 , 1 ] \theta\in(0,1] θ ∈ ( 0 , 1 ] g θ ( u ) = F ρ ( γ θ − u φ ′ ) g_\theta(u)=F_\rho(\gamma_\theta-u\varphi') g θ ( u ) = F ρ ( γ θ − u φ ′ ) 0 ≤ u ≤ θ τ 0\le u\le\theta\tau 0 ≤ u ≤ θ τ γ θ − u φ ′ \gamma_\theta-u\varphi' γ θ − u φ ′ 0 ≤ u ≤ θ τ 0\le u\le\theta\tau 0 ≤ u ≤ θ τ g θ g_\theta g θ Proposition 7.1 . We compute its one-sided derivatives. Let 0 ≤ u ≤ θ τ 0\le u\le\theta\tau 0 ≤ u ≤ θ τ γ ′ = γ θ − u φ ′ \gamma'=\gamma_\theta-u\varphi' γ ′ = γ θ − u φ ′ G = G γ ′ G=G_{\gamma'} G = G γ ′ S = S γ ′ S=S_{\gamma'} S = S γ ′ u u u γ ′ \gamma' γ ′ − φ ′ ′ d x -\varphi''\,dx − φ ′′ d x u u u + φ ′ ′ d x +\varphi''\,dx + φ ′′ d x (58) , the right derivative of g θ g_\theta g θ u < θ τ u<\theta\tau u < θ τ u > 0 u>0 u > 0 − ( ρ / 2 ) ∫ G φ ′ ′ d x -(\rho/2)\int G\varphi''\,dx − ( ρ /2 ) ∫ G φ ′′ d x φ ′ \varphi' φ ′ I I I G ′ ( x ) = x − S ( x ) G'(x)=x-S(x) G ′ ( x ) = x − S ( x ) ∫ G φ ′ ′ d x = − ∫ ( x − S ) φ ′ d x \int G\varphi''\,dx=-\int(x-S)\varphi'\,dx ∫ G φ ′′ d x = − ∫ ( x − S ) φ ′ d x φ \varphi φ I I I S S S (57) ; integrating by parts once more, we find
− ∫ ( x − S ) φ ′ d x = ∫ φ d ( x − S ) = 8 ℓ 15 − ∫ φ d S . -\int(x-S)\varphi'\,dx=\int\varphi\,d(x-S)=\frac{8\ell}{15}-\int\varphi\,dS. − ∫ ( x − S ) φ ′ d x = ∫ φ d ( x − S ) = 15 8 ℓ − ∫ φ d S . We obtain
g θ ′ ( u ) = ρ 2 ( ∫ φ d S γ θ − u φ ′ − 8 ℓ 15 ) (80) g_\theta'(u)=\frac{\rho}{2}\left(\int\varphi\,dS_{\gamma_\theta-u\varphi'}-\frac{8\ell}{15}\right) \tag*{(80)} g θ ′ ( u ) = 2 ρ ( ∫ φ d S γ θ − u φ ′ − 15 8 ℓ ) (80) for the right derivative at 0 ≤ u < θ τ 0\le u<\theta\tau 0 ≤ u < θ τ 0 < u ≤ θ τ 0<u\le\theta\tau 0 < u ≤ θ τ F ρ F_\rho F ρ ( γ θ ) (\gamma_\theta) ( γ θ ) g θ ( 0 ) = F ρ ( γ θ ) ≤ ( 1 − θ ) F ρ ( γ 0 ) + θ F ρ ( γ 1 ) ≤ F min + η g_\theta(0)=F_\rho(\gamma_\theta)\le(1-\theta)F_\rho(\gamma_0)+\theta F_\rho(\gamma_1)\le F_{\min}+\eta g θ ( 0 ) = F ρ ( γ θ ) ≤ ( 1 − θ ) F ρ ( γ 0 ) + θ F ρ ( γ 1 ) ≤ F m i n + η g θ ( θ τ ) ≥ F min g_\theta(\theta\tau)\ge F_{\min} g θ ( θ τ ) ≥ F m i n
g θ ′ ( θ τ ) ≥ g θ ( θ τ ) − g θ ( 0 ) θ τ ≥ − η θ τ . g_\theta'(\theta\tau)\ge\frac{g_\theta(\theta\tau)-g_\theta(0)}{\theta\tau}\ge-\frac{\eta}{\theta\tau}. g θ ′ ( θ τ ) ≥ θ τ g θ ( θ τ ) − g θ ( 0 ) ≥ − θ τ η . Inserting (80) at u = θ τ u=\theta\tau u = θ τ γ θ − θ τ φ ′ = γ ~ θ \gamma_\theta-\theta\tau\varphi'=\widetilde{\gamma}_\theta γ θ − θ τ φ ′ = γ θ 1 / 4 ≤ θ ≤ 3 / 4 1/4\le\theta\le3/4 1/4 ≤ θ ≤ 3/4
Δ θ = ∫ φ d S γ ~ θ ≥ 8 ℓ 15 − 2 η ρ θ τ ≥ 8 ℓ 15 − 8 η ρ τ = 4 ℓ 15 . \Delta_\theta=\int\varphi\,\mathrm{d}S_{\widetilde{\gamma}_\theta}\ge\frac{8\ell}{15}-\frac{2\eta}{\rho\theta\tau}\ge\frac{8\ell}{15}-\frac{8\eta}{\rho\tau}=\frac{4\ell}{15}. Δ θ = ∫ φ d S γ θ ≥ 15 8 ℓ − ρθ τ 2 η ≥ 15 8 ℓ − ρ τ 8 η = 15 4 ℓ . By (57) , we have d S γ ~ θ = ρ γ ~ θ d x \mathrm{d}S_{\widetilde{\gamma}_\theta}=\rho\widetilde{\gamma}_\theta\,\mathrm{d}x d S γ θ = ρ γ θ d x Δ θ = ρ ∫ γ ~ θ φ d x \Delta_\theta=\rho\int\widetilde{\gamma}_\theta\varphi\,\mathrm{d}x Δ θ = ρ ∫ γ θ φ d x Δ \Delta Δ ( γ ~ θ ) (\widetilde{\gamma}_\theta) ( γ θ ) φ \varphi φ
Step 4: Curvature and the endpoint excesses. By (78) and (79) , the direction of the segment ( γ ~ θ ) (\widetilde{\gamma}_\theta) ( γ θ )
γ ~ 1 − γ ~ 0 = γ 1 − γ 0 − τ φ ′ ≥ A − 4 τ ℓ ≥ A 2 on I = { φ > 0 } . \widetilde{\gamma}_1-\widetilde{\gamma}_0=\gamma_1-\gamma_0-\tau\varphi'\ge A-\frac{4\tau}{\ell}\ge\frac{A}{2}\quad\text{on }I=\{\varphi>0\}. γ 1 − γ 0 = γ 1 − γ 0 − τ φ ′ ≥ A − ℓ 4 τ ≥ 2 A on I = { φ > 0 } . On I I I γ ~ θ \widetilde{\gamma}_\theta γ θ ( ( 1 − θ ) d 0 + θ d 1 + 32 τ / ℓ 2 ) d x ≤ d ˉ d x ((1-\theta)d_0+\theta d_1+32\tau/\ell^2)\,\mathrm{d}x\le\bar d\,\mathrm{d}x (( 1 − θ ) d 0 + θ d 1 + 32 τ / ℓ 2 ) d x ≤ d ˉ d x
d ˉ = d 0 + d 1 + 32 τ ℓ 2 ≤ d 0 + d 1 + k 2 ; \bar d=d_0+d_1+\frac{32\tau}{\ell^2}\le d_0+d_1+\frac{k}{2}; d ˉ = d 0 + d 1 + ℓ 2 32 τ ≤ d 0 + d 1 + 2 k ; note that d ˉ ≥ d 1 ≥ k > 0 \bar d\ge d_1\ge k>0 d ˉ ≥ d 1 ≥ k > 0 f ~ ( θ ) = F ρ ( γ ~ θ ) \widetilde f(\theta)=F_\rho(\widetilde{\gamma}_\theta) f ( θ ) = F ρ ( γ θ ) ( γ ~ θ ) (\widetilde{\gamma}_\theta) ( γ θ ) φ \varphi φ σ = 1 \sigma=1 σ = 1 A A A d = d ˉ d=\bar d d = d ˉ Δ θ ≥ 4 ℓ / 15 \Delta_\theta\ge4\ell/15 Δ θ ≥ 4 ℓ /15 Y 0 = 512 / ( 105 ℓ ) Y_0=512/(105\ell) Y 0 = 512/ ( 105 ℓ ) (78) , we obtain, for 1 / 4 ≤ θ ≤ 3 / 4 1/4\le\theta\le3/4 1/4 ≤ θ ≤ 3/4
f ~ ′ ′ ( θ ) ≥ ρ A 2 72 min { ( 4 ℓ 15 ) 2 105 ℓ 512 β ′ , 4 ℓ 15 d ˉ } = ρ A 2 72 min { 7 ℓ 3 480 β ′ , 4 ℓ 15 d ˉ } . \widetilde f''(\theta)\ge\frac{\rho A^2}{72}\min\left\{\left(\frac{4\ell}{15}\right)^2\frac{105\ell}{512\beta'},\frac{4\ell}{15\bar d}\right\}=\frac{\rho A^2}{72}\min\left\{\frac{7\ell^3}{480\beta'},\frac{4\ell}{15\bar d}\right\}. f ′′ ( θ ) ≥ 72 ρ A 2 min { ( 15 4 ℓ ) 2 512 β ′ 105 ℓ , 15 d ˉ 4 ℓ } = 72 ρ A 2 min { 480 β ′ 7 ℓ 3 , 15 d ˉ 4 ℓ } . Because β > 1 \beta>1 β > 1 4 / 15 ≥ 7 / ( 480 β ) 4/15\ge7/(480\beta) 4/15 ≥ 7/ ( 480 β )
d 2 d θ 2 F ρ ( γ ~ θ ) ≥ 7 34560 β ρ A 2 min { ℓ 3 , ℓ d ˉ } ( 1 / 4 ≤ θ ≤ 3 / 4 ) . (81) \frac{\mathrm{d}^2}{\mathrm{d}\theta^2}F_\rho(\widetilde{\gamma}_\theta)\ge\frac{7}{34560\beta}\rho A^2\min\left\{\ell^3,\frac{\ell}{\bar d}\right\}\qquad(1/4\le\theta\le3/4). \tag*{(81)} d θ 2 d 2 F ρ ( γ θ ) ≥ 34560 β 7 ρ A 2 min { ℓ 3 , d ˉ ℓ } ( 1/4 ≤ θ ≤ 3/4 ) . (81) It remains to bound the excesses at the endpoints of the perturbed segment. At θ = 0 \theta=0 θ = 0 f ~ ( 0 ) = F ρ ( γ 0 ) ≤ F min + η \widetilde f(0)=F_\rho(\gamma_0)\le F_{\min}+\eta f ( 0 ) = F ρ ( γ 0 ) ≤ F m i n + η θ = 1 \theta=1 θ = 1 f ~ ( 1 ) = g 1 ( τ ) \widetilde f(1)=g_1(\tau) f ( 1 ) = g 1 ( τ ) g 1 g_1 g 1 g 1 ( τ ) − g 1 ( 0 ) ≤ τ g 1 ′ ( τ ) g_1(\tau)-g_1(0)\le\tau g_1'(\tau) g 1 ( τ ) − g 1 ( 0 ) ≤ τ g 1 ′ ( τ ) (57) , the function S γ ~ 1 S_{\widetilde{\gamma}_1} S γ 1 [ 0 , 1 ] [0,1] [ 0 , 1 ] 0 ≤ φ ≤ 1 0\le\varphi\le1 0 ≤ φ ≤ 1 0 ≤ ∫ φ d S γ ~ 1 ≤ 1 0\le\int\varphi\,\mathrm{d}S_{\widetilde{\gamma}_1}\le1 0 ≤ ∫ φ d S γ 1 ≤ 1 (80) gives g 1 ′ ( τ ) ≤ ρ / 2 g_1'(\tau)\le\rho/2 g 1 ′ ( τ ) ≤ ρ /2 f ~ ( 1 ) ≤ F ρ ( γ 1 ) + ρ τ / 2 ≤ F min + η + 15 η / ℓ \widetilde f(1)\le F_\rho(\gamma_1)+\rho\tau/2\le F_{\min}+\eta+15\eta/\ell f ( 1 ) ≤ F ρ ( γ 1 ) + ρ τ /2 ≤ F m i n + η + 15 η / ℓ ℓ ≤ 1 \ell\le1 ℓ ≤ 1
f ~ ( 0 ) + f ~ ( 1 ) − 2 F min ≤ 2 η + 15 η ℓ ≤ 17 η ℓ . \widetilde f(0)+\widetilde f(1)-2F_{\min}\le2\eta+\frac{15\eta}{\ell}\le\frac{17\eta}{\ell}. f ( 0 ) + f ( 1 ) − 2 F m i n ≤ 2 η + ℓ 15 η ≤ ℓ 17 η . We now use the midpoint inequality (65) for the segment ( γ ~ θ ) (\widetilde{\gamma}_\theta) ( γ θ ) Θ = [ 1 / 4 , 3 / 4 ] \Theta=[1/4,3/4] Θ = [ 1/4 , 3/4 ] 3 / 16 3/16 3/16 (81) , it yields
3 16 ⋅ 7 34560 β ρ A 2 min { ℓ 3 , ℓ d ˉ } ≤ 17 η ℓ , \frac{3}{16}\cdot\frac{7}{34560\beta}\rho A^2\min\left\{\ell^3,\frac{\ell}{\bar d}\right\}\le\frac{17\eta}{\ell}, 16 3 ⋅ 34560 β 7 ρ A 2 min { ℓ 3 , d ˉ ℓ } ≤ ℓ 17 η , that is,
η ρ ≥ 7 3133440 β A 2 min { ℓ 4 , ℓ 2 d ˉ } ≥ 7 3133440 β min { A 2 ℓ 4 , A 2 ℓ 2 d 0 + d 1 + k } . \frac{\eta}{\rho} \ge\frac{7}{3133440\beta} A^2 \min\left\{\ell^4,\frac{\ell^2}{\bar d}\right\} \ge\frac{7}{3133440\beta}\min\left\{A^2\ell^4,\frac{A^2\ell^2}{d_0+d_1+k}\right\}. ρ η ≥ 3133440 β 7 A 2 min { ℓ 4 , d ˉ ℓ 2 } ≥ 3133440 β 7 min { A 2 ℓ 4 , d 0 + d 1 + k A 2 ℓ 2 } . where we used d ˉ ≤ d 0 + d 1 + k \bar d \le d_0+d_1+k d ˉ ≤ d 0 + d 1 + k (77) with c c e r t = 7 / ( 3133440 β ) c_{\mathrm{cert}}=7/(3133440\beta) c cert = 7/ ( 3133440 β ) 1 / 1920 1/1920 1/1920 β > 1 \beta>1 β > 1
Root localization
For an admissible profile γ \gamma γ
Q γ ( v ) = inf { q ∈ [ 0 , 1 ] : ν γ ( [ 0 , q ] ) ≥ v } , 0 < v < β − λ . Q_\gamma(v)=\inf\{q\in[0,1]:\nu_\gamma([0,q])\ge v\},\qquad0<v<\beta-\lambda. Q γ ( v ) = inf { q ∈ [ 0 , 1 ] : ν γ ([ 0 , q ]) ≥ v } , 0 < v < β − λ . If a quantile of a minimizer is large compared with the prescribed mass plus its level, then the minimizer is small on an interval where the scalar trial profile has a positive density. We control the density of the partially regularized minimizer there, and then compare the two profiles on that interval using Lemma 8.1. We follow the strategy of the proof of [4 Lemma 7.6] , which regularizes only the free mass below the selected quantile and compares the result with the scalar trial profile. We combine this strategy with Lemma 8.1, which gives the dependence on η / ρ \eta/\rho η / ρ (83) . The positive density of the trial profile near the origin comes from the next lemma.
Lemma 8.2 (Trial density near the root). There exists T 1 = T 1 ( β ) ∈ ( 0 , T 0 ] T_1=T_1(\beta)\in(0,T_0] T 1 = T 1 ( β ) ∈ ( 0 , T 0 ]
d γ β ≥ k 0 2 d x as measures on [ 0 , T 1 ] . (82) \mathrm{d}\gamma_\beta\ge\frac{k_0}{2}\,\mathrm{d}x\quad\text{as measures on }[0,T_1]. \tag*{(82)} d γ β ≥ 2 k 0 d x as measures on [ 0 , T 1 ] . (82) Proof. Recall from (62) that μ β \mu_\beta μ β [ 0 , q β ) [0,q_\beta) [ 0 , q β ) ϱ β \varrho_\beta ϱ β [ 0 , q β ] [0,q_\beta] [ 0 , q β ] γ β ( x ) = β ∫ 0 x ϱ β \gamma_\beta(x)=\beta\int_0^x\varrho_\beta γ β ( x ) = β ∫ 0 x ϱ β 0 ≤ x < q β 0\le x<q_\beta 0 ≤ x < q β γ β ( x ) / x → β ϱ β ( 0 ) \gamma_\beta(x)/x\to\beta\varrho_\beta(0) γ β ( x ) / x → β ϱ β ( 0 ) x ↓ 0 x\downarrow0 x ↓ 0 (61) , we have γ β ( x ) / x ≥ k 0 \gamma_\beta(x)/x\ge k_0 γ β ( x ) / x ≥ k 0 0 < x ≤ T 0 0<x\le T_0 0 < x ≤ T 0 β ϱ β ( 0 ) ≥ k 0 \beta\varrho_\beta(0)\ge k_0 β ϱ β ( 0 ) ≥ k 0 ϱ β \varrho_\beta ϱ β 0 0 0 T 1 ∈ ( 0 , min { T 0 , q β } ) T_1\in(0,\min\{T_0,q_\beta\}) T 1 ∈ ( 0 , min { T 0 , q β }) β ϱ β ≥ k 0 / 2 \beta\varrho_\beta\ge k_0/2 β ϱ β ≥ k 0 /2 [ 0 , T 1 ] [0,T_1] [ 0 , T 1 ] d γ β \mathrm{d}\gamma_\beta d γ β 0 0 0 γ β ( 0 ) = 0 \gamma_\beta(0)=0 γ β ( 0 ) = 0 [ 0 , T 1 ] [0,T_1] [ 0 , T 1 ] β ϱ β d x \beta\varrho_\beta\,\mathrm{d}x β ϱ β d x (82) . The scalar Parisi measure is unique [10 Corollary 1] , and q β q_\beta q β ϱ β \varrho_\beta ϱ β β \beta β k 0 k_0 k 0 T 0 T_0 T 0 β \beta β T 1 T_1 T 1 β \beta β
The width T 1 T_1 T 1 ϱ β \varrho_\beta ϱ β 0 0 0
Proposition 8.3 (Root localization). There exists C = C ( β ) < ∞ C=C(\beta)<\infty C = C ( β ) < ∞ N ≥ 1 N\ge1 N ≥ 1 0 < ρ ≤ 1 0<\rho\le1 0 < ρ ≤ 1 0 < λ ≤ λ 0 0<\lambda\le\lambda_0 0 < λ ≤ λ 0 γ ∗ \gamma_* γ ∗ F ρ F_\rho F ρ η \eta η γ ∗ \gamma_* γ ∗
Q γ ∗ ( v ) ≤ C [ λ + v + min { 1 , ( η ρ ) 1 / 6 } ] ( 0 < v < β − λ ) . (83) Q_{\gamma_*}(v)\le C\left[\lambda+v+\min\left\{1,\left(\frac{\eta}{\rho}\right)^{1/6}\right\}\right]\qquad(0<v<\beta-\lambda). \tag*{(83)} Q γ ∗ ( v ) ≤ C [ λ + v + min { 1 , ( ρ η ) 1/6 } ] ( 0 < v < β − λ ) . (83) The constant depends on β \beta β β \beta β k 0 k_0 k 0 C 0 C_0 C 0 T 1 T_1 T 1
Proof. The proof has five steps. After reducing to 0 < η < ρ 0 < \eta< \rho 0 < η < ρ q = min { Q γ ∗ ( v ) , T 1 } q = \min\{Q_{\gamma_*}(v), T_1\} q = min { Q γ ∗ ( v ) , T 1 } q q q λ + v \lambda+ v λ + v γ ∗ \gamma_* γ ∗ q q q γ t r \gamma^{\mathrm{tr}} γ tr ( q / 4 , 3 q / 4 ) (q/4,3q/4) ( q /4 , 3 q /4 ) Lemma 8.1 (Step 4). The resulting inequality bounds q q q η / ρ \eta/\rho η / ρ
Step 1: Reductions. Fix v ∈ ( 0 , β − λ ) v \in(0,\beta-\lambda) v ∈ ( 0 , β − λ ) v ˉ = λ + v \bar v = \lambda+ v v ˉ = λ + v C C C η ≥ ρ \eta\ge\rho η ≥ ρ Q γ ∗ ( v ) ≤ 1 = min { 1 , ( η / ρ ) 1 / 6 } Q_{\gamma_*}(v) \le1 = \min\{1,(\eta/\rho)^{1/6}\} Q γ ∗ ( v ) ≤ 1 = min { 1 , ( η / ρ ) 1/6 } η = 0 \eta= 0 η = 0 η ′ ∈ ( 0 , ρ ) \eta' \in(0,\rho) η ′ ∈ ( 0 , ρ ) γ ∗ \gamma_* γ ∗ η ′ \eta' η ′ η ′ ↓ 0 \eta' \downarrow0 η ′ ↓ 0 0 < η < ρ 0 < \eta< \rho 0 < η < ρ ε = η / ρ ∈ ( 0 , 1 ) \varepsilon= \eta/\rho\in(0,1) ε = η / ρ ∈ ( 0 , 1 )
We set q = min { Q γ ∗ ( v ) , T 1 } q = \min\{Q_{\gamma_*}(v),T_1\} q = min { Q γ ∗ ( v ) , T 1 } Q γ ∗ ( v ) ≤ q / T 1 Q_{\gamma_*}(v) \le q/T_1 Q γ ∗ ( v ) ≤ q / T 1 Q γ ∗ ( v ) ≤ T 1 Q_{\gamma_*}(v) \le T_1 Q γ ∗ ( v ) ≤ T 1 q = Q γ ∗ ( v ) q = Q_{\gamma_*}(v) q = Q γ ∗ ( v ) T 1 ≤ 1 T_1 \le1 T 1 ≤ 1 q = T 1 q = T_1 q = T 1 Q γ ∗ ( v ) ≤ 1 = q / T 1 Q_{\gamma_*}(v) \le1 = q/T_1 Q γ ∗ ( v ) ≤ 1 = q / T 1 q q q q ≤ 16 v ˉ / k 0 q \le16\bar v/k_0 q ≤ 16 v ˉ / k 0 Q γ ∗ ( v ) ≤ 16 v ˉ / ( k 0 T 1 ) Q_{\gamma_*}(v) \le16\bar v/(k_0T_1) Q γ ∗ ( v ) ≤ 16 v ˉ / ( k 0 T 1 )
q > 16 v ˉ k 0 . (84) q > \frac{16\bar v}{k_0}. \tag*{(84)} q > k 0 16 v ˉ . (84) Step 2: Regularization below q q q Let m = ν γ ∗ ( [ 0 , q ) ) m = \nu_{\gamma_*}([0,q)) m = ν γ ∗ ([ 0 , q )) x < q x < q x < q x < Q γ ∗ ( v ) x < Q_{\gamma_*}(v) x < Q γ ∗ ( v ) ν γ ∗ ( [ 0 , x ] ) < v \nu_{\gamma_*}([0,x]) < v ν γ ∗ ([ 0 , x ]) < v [ 0 , q ) [0,q) [ 0 , q ) [ 0 , x ] [0,x] [ 0 , x ] x < q x < q x < q m ≤ v m \le v m ≤ v ν γ ∗ \nu_{\gamma_*} ν γ ∗ q q q m m m
We take
h = min { q 8 , ε m } if m > 0 , h = q 8 if m = 0. h = \min\left\{\frac{q}{8},\sqrt{\frac{\varepsilon}{m}}\right\}\quad\text{if }m>0,\qquad h = \frac{q}{8}\quad\text{if }m=0. h = min { 8 q , m ε } if m > 0 , h = 8 q if m = 0. Then h ≤ q / 8 h \le q/8 h ≤ q /8 m h 2 ≤ ε mh^2 \le\varepsilon m h 2 ≤ ε h ≤ 1 − q h \le1-q h ≤ 1 − q q ≤ T 1 ≤ T 0 ≤ 1 / 4 q \le T_1 \le T_0 \le1/4 q ≤ T 1 ≤ T 0 ≤ 1/4 γ ˉ \bar\gamma γ ˉ T = q T=q T = q F ρ ( γ ˉ ) − F ρ ( γ ∗ ) ≤ ρ m h 2 / 12 ≤ η / 12 F_\rho(\bar\gamma)-F_\rho(\gamma_*) \le\rho mh^2/12 \le\eta/12 F ρ ( γ ˉ ) − F ρ ( γ ∗ ) ≤ ρ m h 2 /12 ≤ η /12 d γ ˉ d\bar\gamma d γ ˉ ( 0 , q ) (0,q) ( 0 , q ) m / h m/h m / h 0 ≤ x < q 0 \le x < q 0 ≤ x < q γ ˉ \bar\gamma γ ˉ [ 0 , x ] [0,x] [ 0 , x ]
γ ˉ ( x ) ≤ λ + m ≤ v ˉ < k 0 q 16 ( 0 ≤ x < q ) . \bar\gamma(x) \le\lambda+ m \le\bar v < \frac{k_0q}{16}\qquad(0 \le x < q). γ ˉ ( x ) ≤ λ + m ≤ v ˉ < 16 k 0 q ( 0 ≤ x < q ) . Step 3: The middle interval. Let I = ( q / 4 , 3 q / 4 ) I = (q/4,3q/4) I = ( q /4 , 3 q /4 ) x ∈ I x \in I x ∈ I γ β ( x ) ≥ k 0 x > k 0 q / 4 > 4 v ˉ ≥ λ \gamma_\beta(x) \ge k_0x > k_0q/4 > 4\bar v \ge\lambda γ β ( x ) ≥ k 0 x > k 0 q /4 > 4 v ˉ ≥ λ γ t r = γ β \gamma^{\mathrm{tr}} = \gamma_\beta γ tr = γ β I I I I ⊂ ( 0 , T 1 ] I \subset(0,T_1] I ⊂ ( 0 , T 1 ] Lemma 8.2 , and the density bound for d γ t r d\gamma^{\mathrm{tr}} d γ tr ( 0 , T 0 ] (0,T_0] ( 0 , T 0 ]
k 0 2 d x ≤ d γ t r ≤ C 0 d x on I . \frac{k_0}{2}\,dx \le d\gamma^{\mathrm{tr}} \le C_0\,dx\quad\text{on }I. 2 k 0 d x ≤ d γ tr ≤ C 0 d x on I . By Step 2, we also have d γ ˉ ≤ ( m / h ) d x \mathrm{d}\bar{\gamma} \le(m/h)\,\mathrm{d}x d γ ˉ ≤ ( m / h ) d x I I I
γ t r − γ ˉ > k 0 q 4 − k 0 q 16 ≥ k 0 q 8 on I . \gamma^{\mathrm{tr}}-\bar{\gamma} > \frac{k_0q}{4}-\frac{k_0q}{16} \ge\frac{k_0q}{8}\quad\text{on }I. γ tr − γ ˉ > 4 k 0 q − 16 k 0 q ≥ 8 k 0 q on I . Step 4: The certificate. We apply Lemma 8.1 with the interval I = ( q / 4 , 3 q / 4 ) ⊂ ( 0 , 1 ) I=(q/4,3q/4)\subset(0,1) I = ( q /4 , 3 q /4 ) ⊂ ( 0 , 1 ) γ 0 = γ ˉ \gamma_0=\bar{\gamma} γ 0 = γ ˉ γ 1 = γ t r \gamma_1=\gamma^{\mathrm{tr}} γ 1 = γ tr
ℓ = q 2 , A = k 0 q 8 , k = k 0 2 , d 0 = m h , d 1 = C 0 . \ell=\frac{q}{2},\qquad A=\frac{k_0q}{8},\qquad k=\frac{k_0}{2},\qquad d_0=\frac{m}{h},\qquad d_1=C_0. ℓ = 2 q , A = 8 k 0 q , k = 2 k 0 , d 0 = h m , d 1 = C 0 . Both endpoints have excess at most η \eta η min F ρ = F ρ ( γ ∗ ) \min F_\rho=F_\rho(\gamma_\ast) min F ρ = F ρ ( γ ∗ ) γ ˉ \bar{\gamma} γ ˉ γ t r \gamma^{\mathrm{tr}} γ tr
k ℓ 3 = k 0 q 3 16 , A ℓ 2 = k 0 q 3 32 , A 2 ℓ 4 = k 0 2 q 6 1024 , k\ell^3=\frac{k_0q^3}{16},\qquad A\ell^2=\frac{k_0q^3}{32},\qquad A^2\ell^4=\frac{k_0^2q^6}{1024}, k ℓ 3 = 16 k 0 q 3 , A ℓ 2 = 32 k 0 q 3 , A 2 ℓ 4 = 1024 k 0 2 q 6 , A 2 ℓ 2 d 0 + d 1 + k = k 0 2 q 4 256 ( m / h + C 0 + k 0 / 2 ) . \frac{A^2\ell^2}{d_0+d_1+k}=\frac{k_0^2q^4}{256(m/h+C_0+k_0/2)}. d 0 + d 1 + k A 2 ℓ 2 = 256 ( m / h + C 0 + k 0 /2 ) k 0 2 q 4 . Since q ≤ 1 q\le1 q ≤ 1 k 0 q 3 / 32 k_0q^3/32 k 0 q 3 /32 c ′ = c c e r t min { k 0 / 32 , k 0 2 / 1024 } c'=c_{\mathrm{cert}}\min\{k_0/32,k_0^2/1024\} c ′ = c cert min { k 0 /32 , k 0 2 /1024 }
ε ≥ c ′ min { q 6 , q 4 m / h + C 0 + k 0 / 2 } . (85) \varepsilon\ge c'\min\left\{q^6,\frac{q^4}{m/h+C_0+k_0/2}\right\}. \tag*{(85)} ε ≥ c ′ min { q 6 , m / h + C 0 + k 0 /2 q 4 } . (85) Step 5: Conclusion. If the minimum in (8.9) is q 6 q^6 q 6 q ≤ ( ε / c ′ ) 1 / 6 q\le(\varepsilon/c')^{1/6} q ≤ ( ε / c ′ ) 1/6 q 4 ≤ ε ( m / h + C 0 + k 0 / 2 ) / c ′ q^4\le\varepsilon(m/h+C_0+k_0/2)/c' q 4 ≤ ε ( m / h + C 0 + k 0 /2 ) / c ′ h h h
m h = max { 8 m q , ε − 1 / 2 m 3 / 2 } ≤ 8 m q + ε − 1 / 2 m 3 / 2 \frac{m}{h}=\max\left\{\frac{8m}{q},\varepsilon^{-1/2}m^{3/2}\right\}\le\frac{8m}{q}+\varepsilon^{-1/2}m^{3/2} h m = max { q 8 m , ε − 1/2 m 3/2 } ≤ q 8 m + ε − 1/2 m 3/2 if m > 0 m>0 m > 0 m / h = 0 m/h=0 m / h = 0 m = 0 m=0 m = 0 m ≤ v ≤ v ˉ m\le v\le\bar{v} m ≤ v ≤ v ˉ 8 m / q < k 0 / 2 8m/q<k_0/2 8 m / q < k 0 /2
q 4 ≤ ( C 0 + k 0 ) ε + ε 1 / 2 m 3 / 2 c ′ , q ≤ ( c ′ ) − 1 / 4 ( ( C 0 + k 0 ) 1 / 4 ε 1 / 4 + ε 1 / 8 m 3 / 8 ) . q^4\le\frac{(C_0+k_0)\varepsilon+\varepsilon^{1/2}m^{3/2}}{c'},\qquad q\le(c')^{-1/4}\left((C_0+k_0)^{1/4}\varepsilon^{1/4}+\varepsilon^{1/8}m^{3/8}\right). q 4 ≤ c ′ ( C 0 + k 0 ) ε + ε 1/2 m 3/2 , q ≤ ( c ′ ) − 1/4 ( ( C 0 + k 0 ) 1/4 ε 1/4 + ε 1/8 m 3/8 ) . where we used ( x + y ) 1 / 4 ≤ x 1 / 4 + y 1 / 4 (x+y)^{1/4}\le x^{1/4}+y^{1/4} ( x + y ) 1/4 ≤ x 1/4 + y 1/4 x , y ≥ 0 x,y\ge0 x , y ≥ 0 8 / 5 8/5 8/5 8 / 3 8/3 8/3
ε 1 / 8 m 3 / 8 ≤ 5 8 ε 1 / 5 + 3 8 m ≤ ε 1 / 6 + m , \varepsilon^{1/8}m^{3/8}\le\frac{5}{8}\varepsilon^{1/5}+\frac{3}{8}m\le\varepsilon^{1/6}+m, ε 1/8 m 3/8 ≤ 8 5 ε 1/5 + 8 3 m ≤ ε 1/6 + m , where in the last inequality we used that 0 < ε < 1 0<\varepsilon<1 0 < ε < 1 ε 1 / 4 ≤ ε 1 / 6 \varepsilon^{1/4}\le\varepsilon^{1/6} ε 1/4 ≤ ε 1/6 m ≤ v ˉ m\le\bar{v} m ≤ v ˉ q ≤ ( c ′ ) − 1 / 4 ( ( C 0 + k 0 ) 1 / 4 + 1 ) ( v ˉ + ε 1 / 6 ) q\le(c')^{-1/4}((C_0+k_0)^{1/4}+1)(\bar{v}+\varepsilon^{1/6}) q ≤ ( c ′ ) − 1/4 (( C 0 + k 0 ) 1/4 + 1 ) ( v ˉ + ε 1/6 )
q ≤ C 1 ( v ˉ + ε 1 / 6 ) , C 1 = max { 16 k 0 , ( c ′ ) − 1 / 6 , ( c ′ ) − 1 / 4 ( ( C 0 + k 0 ) 1 / 4 + 1 ) } . q\le C_1(\bar{v}+\varepsilon^{1/6}),\qquad C_1=\max\left\{\frac{16}{k_0},(c')^{-1/6},(c')^{-1/4}((C_0+k_0)^{1/4}+1)\right\}. q ≤ C 1 ( v ˉ + ε 1/6 ) , C 1 = max { k 0 16 , ( c ′ ) − 1/6 , ( c ′ ) − 1/4 (( C 0 + k 0 ) 1/4 + 1 ) } . Recalling that Q γ ∗ ( v ) ≤ q / T 1 Q_{\gamma_\ast}(v)\le q/T_1 Q γ ∗ ( v ) ≤ q / T 1 ε 1 / 6 = min { 1 , ( η / ρ ) 1 / 6 } \varepsilon^{1/6}=\min\{1,(\eta/\rho)^{1/6}\} ε 1/6 = min { 1 , ( η / ρ ) 1/6 } η < ρ \eta<\rho η < ρ C = max { 1 , C 1 / T 1 } C=\max\{1,C_1/T_1\} C = max { 1 , C 1 / T 1 } β \beta β c c e r t c_{\mathrm{cert}} c cert k 0 k_0 k 0 C 0 C_0 C 0 T 1 T_1 T 1
We comment on the exponent 1 / 6 1/6 1/6 (83) . In the setting of the proof above, the variation in the proof of Lemma 8.1 moves mass toward the middle of I I I ρ τ / 2 \rho\tau/2 ρ τ /2 η / q \eta/q η / q η / q \eta/q η / q 4 ℓ / 15 = 2 q / 15 4\ell/15=2q/15 4 ℓ /15 = 2 q /15 (81) , the second derivative along the perturbed segment is bounded below by a multiple of ρ min { q 5 , q 3 / d ˉ } \rho\min\{q^5,q^3/\bar d\} ρ min { q 5 , q 3 / d ˉ } d ˉ \bar d d ˉ η / ρ ≳ β min { q 6 , q 4 / d ˉ } \eta/\rho\gtrsim_\beta\min\{q^6,q^4/\bar d\} η / ρ ≳ β min { q 6 , q 4 / d ˉ } (85) . Its first alternative gives the exponent 1 / 6 1/6 1/6 h h h m / h m/h m / h v v v
Corollary 8.4 (Quantile bound in probability normalization). Let N ≥ 1 N\ge1 N ≥ 1 0 < ρ ≤ 1 0<\rho\le1 0 < ρ ≤ 1 0 < λ ≤ λ 0 0<\lambda\le\lambda_0 0 < λ ≤ λ 0 s = λ / β s=\lambda/\beta s = λ / β μ = s δ 0 + ν \mu=s\delta_0+\nu μ = s δ 0 + ν F 1 − ρ , s F_{1-\rho,s} F 1 − ρ , s (16) , so that γ ∗ = β μ ( [ 0 , ⋅ ] ) \gamma_*=\beta\mu([0,\cdot]) γ ∗ = β μ ([ 0 , ⋅ ]) F ρ F_\rho F ρ η ≥ 0 \eta\ge0 η ≥ 0 γ ∗ \gamma_* γ ∗
Q ( v ) = inf { q ∈ [ 0 , 1 ] : ν ( [ 0 , q ] ) ≥ v } ( 0 < v < 1 − s ) Q(v)=\inf\{q\in[0,1]:\nu([0,q])\ge v\}\qquad(0<v<1-s) Q ( v ) = inf { q ∈ [ 0 , 1 ] : ν ([ 0 , q ]) ≥ v } ( 0 < v < 1 − s ) be the quantile function (19) of ν \nu ν
ω ( ρ , η ) = min { 1 , ( η / ρ ) 1 / 6 } . (86) \omega(\rho,\eta)=\min\{1,(\eta/\rho)^{1/6}\}. \tag*{(86)} ω ( ρ , η ) = min { 1 , ( η / ρ ) 1/6 } . (86) Then, with C Q = max { 1 , β C } C_Q=\max\{1,\beta C\} C Q = max { 1 , β C } C C C Proposition 8.3 ,
Q ( v ) ≤ C Q ( s + v + ω ( ρ , η ) ) ( 0 < v < 1 − s ) . (87) Q(v)\le C_Q(s+v+\omega(\rho,\eta))\qquad(0<v<1-s). \tag*{(87)} Q ( v ) ≤ C Q ( s + v + ω ( ρ , η )) ( 0 < v < 1 − s ) . (87) Proof. The free measure of γ ∗ \gamma_* γ ∗ ν γ ∗ = β ν \nu_{\gamma_*}=\beta\nu ν γ ∗ = β ν ν γ ∗ ( [ 0 , q ] ) ≥ β v \nu_{\gamma_*}([0,q])\ge\beta v ν γ ∗ ([ 0 , q ]) ≥ β v ν ( [ 0 , q ] ) ≥ v \nu([0,q])\ge v ν ([ 0 , q ]) ≥ v Q ( v ) = Q γ ∗ ( β v ) Q(v)=Q_{\gamma_*}(\beta v) Q ( v ) = Q γ ∗ ( β v ) 0 < β v < β − λ 0<\beta v<\beta-\lambda 0 < β v < β − λ 0 < v < 1 − s 0<v<1-s 0 < v < 1 − s Proposition 8.3 and λ = β s \lambda=\beta s λ = β s
Q ( v ) ≤ C [ β ( s + v ) + ω ( ρ , η ) ] ≤ β C ( s + v + ω ( ρ , η ) ) , Q(v)\le C\left[\beta(s+v)+\omega(\rho,\eta)\right]\le\beta C(s+v+\omega(\rho,\eta)), Q ( v ) ≤ C [ β ( s + v ) + ω ( ρ , η ) ] ≤ β C ( s + v + ω ( ρ , η )) , where we used β > 1 \beta>1 β > 1 (87) , because β C ≤ C Q \beta C\le C_Q β C ≤ C Q
Scalar response and stability
In this section, we study the scalar Parisi equation of a single spin under an arbitrary profile and the scalar response operator J a , d s c J^{\mathrm{sc}}_{a,d} J a , d sc § 8.5.1 with coefficient d = β 2 ( 1 − t / N ) d = \beta^2(1-t/N) d = β 2 ( 1 − t / N ) J a , d s c J^{\mathrm{sc}}_{a,d} J a , d sc § 10.2 needs the order bound J a , d s c ⪯ ( 1 + ε N ) I J^{\mathrm{sc}}_{a,d} \preceq(1+\varepsilon_N)I J a , d sc ⪯ ( 1 + ε N ) I ε N ≤ ρ / 4 \varepsilon_N \leq\rho/4 ε N ≤ ρ /4 ε N ≤ C s t [ ρ − 3 / 4 η 1 / 4 + λ 2 + N − 1 ] \varepsilon_N \leq\mathrm{Cst}[\rho^{-3/4}\eta^{1/4}+\lambda^2+N^{-1}] ε N ≤ Cst [ ρ − 3/4 η 1/4 + λ 2 + N − 1 ] η \eta η Proposition 9.9 ).
The argument has four steps.
We show in Lemma 9.6 that, for every profile, J a , d J_{a,d} J a , d Γ a , d \Gamma_{a,d} Γ a , d q ↦ q + θ φ ( q ) q \mapsto q+\theta\varphi(q) q ↦ q + θ φ ( q )
The scalar Parisi functional is convex. We compare its second derivative along a transport of levels with the lower bound given by convexity and the first variation, and we find that J a , d s c J^{\mathrm{sc}}_{a,d} J a , d sc k a , d ( q ) = d E 0 h q 2 k_{a,d}(q)=d\,\mathbb{E}^{0}h_q^2 k a , d ( q ) = d E 0 h q 2
At the scalar Parisi measure, we have k = 1 k=1 k = 1 supp μ β = [ 0 , q β ] \operatorname{supp}\mu_\beta=[0,q_\beta] supp μ β = [ 0 , q β ] k ( q ) ≤ e − β 2 ( q − q β ) / 2 k(q)\leq e^{-\beta^2(q-q_\beta)/2} k ( q ) ≤ e − β 2 ( q − q β ) /2 Lemma 9.8 ). On the support, this follows from an identity of Talagrand. Above the support, we use the log-concavity of a transformed density of the scalar diffusion and its heat evolution, both from [48].
The multiplier depends Lipschitz continuously on the profile in L 1 L^1 L 1 d d d Lemma 9.3 ). By Proposition 7.6 , a finite-dimensional minimizer is close in L 1 L^1 L 1 β α β \beta\alpha_\beta β α β Proposition 9.9 .
The bound is marginal: at the Parisi measure, k = 1 k=1 k = 1 § 10.2 comes instead from the factor t = 1 − ρ t=1-\rho t = 1 − ρ ε N ≤ ρ / 4 \varepsilon_N\leq\rho/4 ε N ≤ ρ /4
Sommers and Dupont gave an earlier variational formulation of the coupled equations for the local magnetization and the distribution of frozen fields [77, Section 2, Eqs. (1)–(7), and Eq. (33)]; see also [32, Section III, (14)–(18), (22)–(23)]. Those equations use the physical field and the order-parameter function q ( x ) q(x) q ( x ) q q q
We use the following results from earlier sections: the generalized law and Lemma 4.2 of Section 4 for the basic properties of the scalar law, Lemma 4.3 for the differentiation in Lemma 9.6, and, from Section 7, the trial budget (7.7), the root growth (7.5), the description (7.6) of μ β \mu_\beta μ β
Standing assumptions and notation. Throughout this section, β > 1 \beta> 1 β > 1 d ∈ [ β 2 / 2 , β 2 ] d \in[\beta^2/2,\beta^2] d ∈ [ β 2 /2 , β 2 ] profile is the distribution function a ( q ) = μ ( [ 0 , q ] ) a(q) = \mu([0,q]) a ( q ) = μ ([ 0 , q ]) μ \mu μ [ 0 , 1 ] [0,1] [ 0 , 1 ] a ( 1 ) = 1 a(1) = 1 a ( 1 ) = 1 μ = s δ 0 + ν \mu= s\delta_0 + \nu μ = s δ 0 + ν s ≥ 0 s \ge0 s ≥ 0 ν ≥ 0 \nu\ge0 ν ≥ 0 a a a ν \nu ν step profile is a profile with finitely many values. Integrals over [ 0 , q ] [0,q] [ 0 , q ] q q q ( q , 1 ] (q,1] ( q , 1 ] C β C_\beta C β β \beta β d ∈ [ β 2 / 2 , β 2 ] d \in[\beta^2/2,\beta^2] d ∈ [ β 2 /2 , β 2 ] s s s
The scalar law
The scalar Parisi equation and its diffusion are
∂ q u = − d 2 ( ∂ x x u + a ( q ) ( ∂ x u ) 2 ) , u ( 1 , x ) = log ( 2 cosh x ) , d Y q = d a ( q ) ∂ x u ( q , Y q ) d q + d d B q , Y 0 = 0 , (88) \begin{aligned}
\partial_q u &= -\frac{d}{2}\left(\partial_{xx}u+a(q)(\partial_xu)^2\right), \qquad u(1,x)=\log(2\cosh x),\\
dY_q &= d\,a(q)\partial_xu(q,Y_q)\,dq+\sqrt{d}\,dB_q,\qquad Y_0=0, \tag*{(88)}
\end{aligned} ∂ q u d Y q = − 2 d ( ∂ xx u + a ( q ) ( ∂ x u ) 2 ) , u ( 1 , x ) = log ( 2 cosh x ) , = d a ( q ) ∂ x u ( q , Y q ) d q + d d B q , Y 0 = 0 , (88) where B B B u = u a , d u=u_{a,d} u = u a , d x x x u j = ∂ x j u u_j=\partial_x^j u u j = ∂ x j u a = α β a=\alpha_\beta a = α β d = β 2 d=\beta^2 d = β 2 u ( 0 , 0 ) u(0,0) u ( 0 , 0 )
Solutions and regularity. For a step profile, (9.1) has a classical solution on each interval where a a a u u u μ \mu μ β 2 \beta^{2} β 2 d d d u ( q , ⋅ ) u(q,\cdot) u ( q , ⋅ ) [ 0 , 1 ] × R [0,1]\times\mathbb{R} [ 0 , 1 ] × R log cosh x \log\cosh x log cosh x log 2 \log2 log 2 u u u log 2 \log2 log 2 a a a a a a
u ( q , x ) − u ( r , x ) = d 2 ∫ q r ( u x x + a u x 2 ) ( v , x ) d v , 0 ≤ q ≤ r ≤ 1 , x ∈ R . (89) u(q,x)-u(r,x)=\frac{d}{2}\int_q^r\left(u_{xx}+a\,u_x^2\right)(v,x)\,\mathrm{d}v,\qquad0\leq q\leq r\leq1,\ x\in\mathbb{R}. \tag*{(89)} u ( q , x ) − u ( r , x ) = 2 d ∫ q r ( u xx + a u x 2 ) ( v , x ) d v , 0 ≤ q ≤ r ≤ 1 , x ∈ R . (89) In particular, u u u q q q μ \mu μ q q q (88) and every other parabolic equation in this section in this integrated form, including the linearized equations derived below, and we never differentiate a a a
Linear backward equations. Several proofs below use linear equations of the form
∂ q w + c ( q ) 2 w x x + b w x + V w = − F , (90) \partial_q w+\frac{c(q)}{2}w_{xx}+b\,w_x+Vw=-F, \tag*{(90)} ∂ q w + 2 c ( q ) w xx + b w x + V w = − F , (90) where c c c [ β 2 / 4 , 2 β 2 ] [\beta^2/4,2\beta^2] [ β 2 /4 , 2 β 2 ] b b b V V V F F F ( q , x ) (q,x) ( q , x ) q q q (90) only for functions w w w q q q (89) . Fix ( q , x ) (q,x) ( q , x ) Z Z Z d Z r = b ( r , Z r ) d r + c ( r ) d B r \mathrm{d}Z_r=b(r,Z_r)\,\mathrm{d}r+\sqrt{c(r)}\,\mathrm{d}B_r d Z r = b ( r , Z r ) d r + c ( r ) d B r r ≥ q r\geq q r ≥ q Z q = x Z_q=x Z q = x r ↦ e ∫ q r V ( v , Z v ) d v w ( r , Z r ) r\mapsto e^{\int_q^r V(v,Z_v)\,\mathrm{d}v}w(r,Z_r) r ↦ e ∫ q r V ( v , Z v ) d v w ( r , Z r ) w w w q q q w w w
w ( q , x ) = E [ e ∫ q 1 V ( v , Z v ) d v w ( 1 , Z 1 ) + ∫ q 1 e ∫ q r V ( v , Z v ) d v F ( r , Z r ) d r ] . (91) w(q,x)=\mathbb{E}\left[e^{\int_q^1 V(v,Z_v)\,\mathrm{d}v}w(1,Z_1)+\int_q^1 e^{\int_q^r V(v,Z_v)\,\mathrm{d}v}F(r,Z_r)\,\mathrm{d}r\right]. \tag*{(91)} w ( q , x ) = E [ e ∫ q 1 V ( v , Z v ) d v w ( 1 , Z 1 ) + ∫ q 1 e ∫ q r V ( v , Z v ) d v F ( r , Z r ) d r ] . (91) In particular, if w ( 1 , ⋅ ) = 0 w(1,\cdot)=0 w ( 1 , ⋅ ) = 0 sup x ∣ w ( q , x ) ∣ ≤ e ∥ V ∥ ∞ ∫ q 1 sup x ∣ F ( r , x ) ∣ d r \sup_x|w(q,x)|\leq e^{\|V\|_\infty}\int_q^1\sup_x|F(r,x)|\,\mathrm{d}r sup x ∣ w ( q , x ) ∣ ≤ e ∥ V ∥ ∞ ∫ q 1 sup x ∣ F ( r , x ) ∣ d r
Notation. We write P 0 \mathbb{P}^0 P 0 E 0 \mathbb{E}^0 E 0 Y Y Y ϵ ∈ { − 1 , 1 } \epsilon\in\{-1,1\} ϵ ∈ { − 1 , 1 } Y 1 Y_1 Y 1
P 0 ( ϵ = ± 1 ∣ Y 1 ) = e ± Y 1 2 cosh Y 1 = e ± Y 1 − u ( 1 , Y 1 ) . \mathbb{P}^0(\epsilon=\pm1\mid Y_1)=\frac{e^{\pm Y_1}}{2\cosh Y_1}=e^{\pm Y_1-u(1,Y_1)}. P 0 ( ϵ = ± 1 ∣ Y 1 ) = 2 cosh Y 1 e ± Y 1 = e ± Y 1 − u ( 1 , Y 1 ) . Let G q 0 \mathcal{G}_q^0 G q 0 Y Y Y G q 0 \mathcal{G}_q^0 G q 0
m q = u x ( q , Y q ) , h q = u x x ( q , Y q ) , Γ a , d ( q ) = E 0 m q 2 , k a , d ( q ) = d E 0 h q 2 . m_q=u_x(q,Y_q),\qquad h_q=u_{xx}(q,Y_q),\qquad\Gamma_{a,d}(q)=\mathbb{E}^0m_q^2,\qquad k_{a,d}(q)=d\,\mathbb{E}^0h_q^2. m q = u x ( q , Y q ) , h q = u xx ( q , Y q ) , Γ a , d ( q ) = E 0 m q 2 , k a , d ( q ) = d E 0 h q 2 . We call k a , d k_{a,d} k a , d scalar multiplier . For 0 ≤ r ≤ ℓ ≤ 1 0 \le r \le\ell\le1 0 ≤ r ≤ ℓ ≤ 1 P r , ℓ 0 ψ ( y ) = E 0 [ ψ ( Y ℓ ) ∣ Y r = y ] P^0_{r,\ell}\psi(y) = \mathbb{E}^0[\psi(Y_\ell) \mid Y_r = y] P r , ℓ 0 ψ ( y ) = E 0 [ ψ ( Y ℓ ) ∣ Y r = y ] Y Y Y
f p = m p 2 , f ˉ p = f p − E 0 f p , g r ℓ ( y ) = P r , ℓ 0 ( u x ( ℓ , ⋅ ) 2 ) ( y ) , f_p = m_p^2,\qquad\bar f_p = f_p - \mathbb{E}^0 f_p,\qquad g^\ell_r(y) = P^0_{r,\ell}(u_x(\ell,\cdot)^2)(y), f p = m p 2 , f ˉ p = f p − E 0 f p , g r ℓ ( y ) = P r , ℓ 0 ( u x ( ℓ , ⋅ ) 2 ) ( y ) , so that g r ℓ ( Y r ) = E 0 [ m ℓ 2 ∣ G r 0 ] g^\ell_r(Y_r) = \mathbb{E}^0[m_\ell^2 \mid\mathcal{G}^0_r] g r ℓ ( Y r ) = E 0 [ m ℓ 2 ∣ G r 0 ] y y y r r r Y r , y Y^{r,y} Y r , y E r , y \mathbb{E}_{r,y} E r , y
Lemma 9.1 (Basic properties of the scalar law). Let a a a d ∈ [ β 2 / 2 , β 2 ] d \in[\beta^2/2,\beta^2] d ∈ [ β 2 /2 , β 2 ]
(a) On [ 0 , 1 ] × R [0,1] \times\mathbb{R} [ 0 , 1 ] × R
∣ ∂ x u ∣ ≤ 1 , 0 ≤ ∂ x x u ≤ 1 − ( ∂ x u ) 2 ≤ 1. (92) |\partial_x u| \le1,\qquad0 \le\partial_{xx}u \le1-(\partial_xu)^2 \le1. \tag*{(92)} ∣ ∂ x u ∣ ≤ 1 , 0 ≤ ∂ xx u ≤ 1 − ( ∂ x u ) 2 ≤ 1. (92) (b) For every j ≥ 1 j \ge1 j ≥ 1 C j ( β ) C_j(\beta) C j ( β ) β \beta β j j j ∣ u j ∣ ≤ C j ( β ) |u_j| \le C_j(\beta) ∣ u j ∣ ≤ C j ( β ) [ 0 , 1 ] × R [0,1] \times\mathbb{R} [ 0 , 1 ] × R sup x ∣ u x x ( q , x ) − u x x ( r , x ) ∣ ≤ C β ∣ q − r ∣ \sup_x|u_{xx}(q,x)-u_{xx}(r,x)| \le C_\beta|q-r| sup x ∣ u xx ( q , x ) − u xx ( r , x ) ∣ ≤ C β ∣ q − r ∣ q , r ∈ [ 0 , 1 ] q,r \in[0,1] q , r ∈ [ 0 , 1 ]
(c) We have
m q = E 0 [ ϵ ∣ G q 0 ] , m 0 = 0 , d m q = d h q d B q , Γ a , d ′ ( q ) = k a , d ( q ) . (93) m_q = \mathbb{E}^0[\epsilon\mid\mathcal{G}^0_q],\qquad m_0 = 0,\qquad\mathrm{d}m_q = \sqrt{d}\,h_q\,\mathrm{d}B_q,\qquad\Gamma'_{a,d}(q)=k_{a,d}(q). \tag*{(93)} m q = E 0 [ ϵ ∣ G q 0 ] , m 0 = 0 , d m q = d h q d B q , Γ a , d ′ ( q ) = k a , d ( q ) . (93) The function k a , d k_{a,d} k a , d [ 0 , 1 ] [0,1] [ 0 , 1 ] Γ a , d \Gamma_{a,d} Γ a , d [ 0 , 1 ] [0,1] [ 0 , 1 ]
Proof. We first identify the scalar law as a generalized law. We consider the generalized law of Definition 4.1 with N = 1 N=1 N = 1 Λ = d \Lambda=d Λ = d s = 0 s=0 s = 0 μ \mu μ ν \nu ν (34) is then log ∑ σ = ± 1 e x σ = log ( 2 cosh x ) \log\sum_{\sigma=\pm1}e^{x\sigma} = \log(2\cosh x) log ∑ σ = ± 1 e x σ = log ( 2 cosh x ) ( Y , ϵ ) (Y,\epsilon) ( Y , ϵ ) a a a C q = 1 − m q 2 C_q=1-m_q^2 C q = 1 − m q 2 H q = h q H_q=h_q H q = h q Lemma 4.2 implies that m q = E 0 [ ϵ ∣ G q 0 ] m_q=\mathbb{E}^0[\epsilon\mid\mathcal{G}^0_q] m q = E 0 [ ϵ ∣ G q 0 ] m 0 = 0 m_0=0 m 0 = 0 q ∈ [ 0 , 1 ] q\in[0,1] q ∈ [ 0 , 1 ]
a ( q ) ( 1 − m q 2 ) ≤ h q ≤ 1 − m q 2 . (94) a(q)(1-m_q^2) \le h_q \le1-m_q^2. \tag*{(94)} a ( q ) ( 1 − m q 2 ) ≤ h q ≤ 1 − m q 2 . (94) Differentiating (89) in x x x u x u_x u x ∂ q u x = − ( d / 2 ) ( u x x x + 2 a u x u x x ) \partial_q u_x=-(d/2)(u_{xxx}+2a\,u_xu_{xx}) ∂ q u x = − ( d /2 ) ( u xxx + 2 a u x u xx ) Y Y Y d m q = d h q d B q \mathrm{d}m_q=\sqrt{d}\,h_q\,\mathrm{d}B_q d m q = d h q d B q N = 1 N=1 N = 1 (36) .
Proof of (a). Fix q ∈ ( 0 , 1 ] q\in(0,1] q ∈ ( 0 , 1 ] Y Y Y d d d Y q Y_q Y q d q dq d q
u x x ( q , ⋅ ) − a ( q ) ( 1 − u x ( q , ⋅ ) 2 ) , 1 − u x ( q , ⋅ ) 2 − u x x ( q , ⋅ ) , 1 − u x ( q , ⋅ ) 2 u_{xx}(q,\cdot)-a(q)(1-u_x(q,\cdot)^2),\qquad1-u_x(q,\cdot)^2-u_{xx}(q,\cdot),\qquad1-u_x(q,\cdot)^2 u xx ( q , ⋅ ) − a ( q ) ( 1 − u x ( q , ⋅ ) 2 ) , 1 − u x ( q , ⋅ ) 2 − u xx ( q , ⋅ ) , 1 − u x ( q , ⋅ ) 2 are continuous, and by (9.7) and ∣ m q ∣ ≤ 1 |m_q| \le1 ∣ m q ∣ ≤ 1 Y q Y_q Y q x x x q > 0 q>0 q > 0 a ( q ) ( 1 − u x 2 ) ≥ 0 a(q)(1-u_x^2) \ge0 a ( q ) ( 1 − u x 2 ) ≥ 0 q ↓ 0 q \downarrow0 q ↓ 0 u x u_x u x u x x u_{xx} u xx q = 0 q=0 q = 0 a a a u x x ≥ a ( q ) ( 1 − u x 2 ) u_{xx} \ge a(q)(1-u_x^2) u xx ≥ a ( q ) ( 1 − u x 2 )
Proof of (b). We differentiate the equation and use the Feynman–Kac representation, following the method of [43, Lemma 16]. By (9.5), we have ∣ u 1 ∣ ≤ 1 |u_1| \le1 ∣ u 1 ∣ ≤ 1 ∣ u 2 ∣ ≤ 1 |u_2| \le1 ∣ u 2 ∣ ≤ 1 j ≥ 3 j \ge3 j ≥ 3
∂ x j ( u x 2 ) = 2 u 1 u j + 1 + 2 j u 2 u j + ∑ i = 2 j − 2 ( j i ) u i + 1 u j − i + 1 . \partial_x^j(u_x^2) = 2u_1u_{j+1} + 2ju_2u_j + \sum_{i=2}^{j-2} \binom{j}{i} u_{i+1}u_{j-i+1}. ∂ x j ( u x 2 ) = 2 u 1 u j + 1 + 2 j u 2 u j + i = 2 ∑ j − 2 ( i j ) u i + 1 u j − i + 1 . with an empty sum for j = 3 j=3 j = 3 j j j x x x u j u_j u j
c = d , b = d a u 1 , V = j d a u 2 , F = d a 2 ∑ i = 2 j − 2 ( j i ) u i + 1 u j − i + 1 . c=d,\qquad b=da u_1,\qquad V=jda u_2,\qquad F=\frac{da}{2}\sum_{i=2}^{j-2}\binom{j}{i}u_{i+1}u_{j-i+1}. c = d , b = d a u 1 , V = j d a u 2 , F = 2 d a i = 2 ∑ j − 2 ( i j ) u i + 1 u j − i + 1 . and terminal value ∂ x j log ( 2 cosh x ) \partial_x^j\log(2\cosh x) ∂ x j log ( 2 cosh x ) u u u Z = Y Z=Y Z = Y V V V [ 0 , j β 2 ] [0,j\beta^2] [ 0 , j β 2 ]
∥ u j ∥ ∞ ≤ e j β 2 ( ∥ ∂ x j log ( 2 cosh x ) ∥ ∞ + β 2 2 ∑ i = 2 j − 2 ( j i ) ∥ u i + 1 ∥ ∞ ∥ u j − i + 1 ∥ ∞ ) . \|u_j\|_\infty\le e^{j\beta^2}\left(\|\partial_x^j\log(2\cosh x)\|_\infty+ \frac{\beta^2}{2}\sum_{i=2}^{j-2}\binom{j}{i}\|u_{i+1}\|_\infty\|u_{j-i+1}\|_\infty\right). ∥ u j ∥ ∞ ≤ e j β 2 ( ∥ ∂ x j log ( 2 cosh x ) ∥ ∞ + 2 β 2 i = 2 ∑ j − 2 ( i j ) ∥ u i + 1 ∥ ∞ ∥ u j − i + 1 ∥ ∞ ) . Every derivative on the right has order between 3 and j − 1 j-1 j − 1 j j j ∥ u j ∥ ∞ ≤ C j ( β ) \|u_j\|_\infty\le C_j(\beta) ∥ u j ∥ ∞ ≤ C j ( β ) j ≥ 1 j \ge1 j ≥ 1 d ∈ [ β 2 / 2 , β 2 ] d \in[\beta^2/2,\beta^2] d ∈ [ β 2 /2 , β 2 ] x x x
u x x ( q , x ) − u x x ( r , x ) = d 2 ∫ q r [ u 4 + 2 a ( u 2 2 + u 1 u 3 ) ] ( v , x ) d v . u_{xx}(q,x)-u_{xx}(r,x)=\frac{d}{2}\int_q^r\left[u_4+2a\left(u_2^2+u_1u_3\right)\right](v,x)\,dv. u xx ( q , x ) − u xx ( r , x ) = 2 d ∫ q r [ u 4 + 2 a ( u 2 2 + u 1 u 3 ) ] ( v , x ) d v . and the integrand is bounded by a constant depending only on β \beta β
Proof of (c). The first three identities in (9.6) were shown above. By Itô’s isometry and m 0 = 0 m_0=0 m 0 = 0
Γ a , d ( q ) = E 0 m q 2 = d ∫ 0 q E 0 h r 2 d r = ∫ 0 q k a , d ( r ) d r . \Gamma_{a,d}(q)=\mathbb{E}^0m_q^2=d\int_0^q\mathbb{E}^0h_r^2\,dr=\int_0^q k_{a,d}(r)\,dr. Γ a , d ( q ) = E 0 m q 2 = d ∫ 0 q E 0 h r 2 d r = ∫ 0 q k a , d ( r ) d r . By (b), we have ∣ h q − h r ∣ ≤ C β ∣ q − r ∣ + C 3 ( β ) ∣ Y q − Y r ∣ |h_q-h_r| \le C_\beta|q-r|+C_3(\beta)|Y_q-Y_r| ∣ h q − h r ∣ ≤ C β ∣ q − r ∣ + C 3 ( β ) ∣ Y q − Y r ∣ Y Y Y ∣ h ∣ ≤ 1 |h| \le1 ∣ h ∣ ≤ 1 k a , d k_{a,d} k a , d Γ a , d \Gamma_{a,d} Γ a , d [ 0 , 1 ] [0,1] [ 0 , 1 ] Γ a , d ′ = k a , d \Gamma'_{a,d}=k_{a,d} Γ a , d ′ = k a , d
We also use the exact path likelihood of the scalar law, for every profile, including profiles with atoms. It is the change of measure of [44, Lemma 8.5 and Corollary 8.7]; we include the derivation in order to record the conventions at atoms. It extends the likelihood used for step profiles in the proof of Lemma 4.2 to every profile, without approximation.
Lemma 9.2 (Exact path likelihood). Let a a a d ∈ [ β 2 / 2 , β 2 ] d \in[\beta^2/2,\beta^2] d ∈ [ β 2 /2 , β 2 ] X q = d B q X_q=\sqrt{d}B_q X q = d B q B B B q ∈ [ 0 , 1 ] q \in[0,1] q ∈ [ 0 , 1 ] ( Y r ) 0 ≤ r ≤ q (Y_r)_{0\le r\le q} ( Y r ) 0 ≤ r ≤ q
exp { a ( q ) u ( q , X q ) − ∫ [ 0 , q ] u ( r , X r ) μ ( d r ) } (95) \exp\left\{a(q)u(q,X_q)-\int_{[0,q]}u(r,X_r)\,\mu(\mathrm{d}r)\right\} \tag*{(95)} exp { a ( q ) u ( q , X q ) − ∫ [ 0 , q ] u ( r , X r ) μ ( d r ) } (95) with respect to the law of ( X r ) 0 ≤ r ≤ q (X_r)_{0\le r\le q} ( X r ) 0 ≤ r ≤ q ( Y , ϵ ) (Y,\epsilon) ( Y , ϵ )
exp { ϵ X 1 − s u ( 0 , 0 ) − ∫ [ 0 , 1 ] u ( q , X q ) ν ( d q ) } (96) \exp\left\{\epsilon X_1-su(0,0)-\int_{[0,1]}u(q,X_q)\,\nu(\mathrm{d}q)\right\} \tag*{(96)} exp { ϵ X 1 − s u ( 0 , 0 ) − ∫ [ 0 , 1 ] u ( q , X q ) ν ( d q ) } (96) with respect to the product of the law of X X X { − 1 , 1 } \{-1,1\} { − 1 , 1 } d d d c ( q ) > 0 c(q)>0 c ( q ) > 0 X q = ∫ 0 q c ( r ) d B r X_q=\int_0^q\sqrt{c(r)}\,\mathrm{d}B_r X q = ∫ 0 q c ( r ) d B r
Proof. We write Υ r = u ( r , X r ) \Upsilon_r=u(r,X_r) Υ r = u ( r , X r ) d a u x d\,a\,u_x d a u x Y Y Y
exp { d 2 ∫ 0 q a ( r ) 2 u x ( r , X r ) 2 d r } ≤ e β 2 / 2 , \exp\left\{\frac{d}{2}\int_0^q a(r)^2u_x(r,X_r)^2\,\mathrm{d}r\right\}\le e^{\beta^2/2}, exp { 2 d ∫ 0 q a ( r ) 2 u x ( r , X r ) 2 d r } ≤ e β 2 /2 , which is Novikov’s condition. By Girsanov’s theorem, the density of the law of ( Y r ) r ≤ q (Y_r)_{r\le q} ( Y r ) r ≤ q ( X r ) r ≤ q (X_r)_{r\le q} ( X r ) r ≤ q
exp { ∫ 0 q a ( r ) u x ( r , X r ) d X r − d 2 ∫ 0 q a ( r ) 2 u x ( r , X r ) 2 d r } . \exp\left\{\int_0^q a(r)u_x(r,X_r)\,\mathrm{d}X_r-\frac{d}{2}\int_0^q a(r)^2u_x(r,X_r)^2\,\mathrm{d}r\right\}. exp { ∫ 0 q a ( r ) u x ( r , X r ) d X r − 2 d ∫ 0 q a ( r ) 2 u x ( r , X r ) 2 d r } . Under the reference law, using Itô’s formula and (9.2), we have
d Υ r = u x ( r , X r ) d X r − d 2 a ( r ) u x ( r , X r ) 2 d r . \mathrm{d}\Upsilon_r=u_x(r,X_r)\,\mathrm{d}X_r-\frac{d}{2}a(r)u_x(r,X_r)^2\,\mathrm{d}r. d Υ r = u x ( r , X r ) d X r − 2 d a ( r ) u x ( r , X r ) 2 d r . The exponent therefore equals ∫ 0 q a ( r ) d Υ r \int_0^q a(r)\,\mathrm{d}\Upsilon_r ∫ 0 q a ( r ) d Υ r a a a d a = μ \mathrm{d}a=\mu d a = μ ( 0 , 1 ] (0,1] ( 0 , 1 ] Υ \Upsilon Υ
a ( q ) Υ q − a ( 0 ) Υ 0 = ∫ ( 0 , q ] a ( r − ) d Υ r + ∫ ( 0 , q ] Υ r μ ( d r ) . a(q)\Upsilon_q-a(0)\Upsilon_0=\int_{(0,q]}a(r-)\,\mathrm{d}\Upsilon_r+\int_{(0,q]}\Upsilon_r\,\mu(\mathrm{d}r). a ( q ) Υ q − a ( 0 ) Υ 0 = ∫ ( 0 , q ] a ( r − ) d Υ r + ∫ ( 0 , q ] Υ r μ ( d r ) . The functions a ( r − ) a(r-) a ( r − ) a ( r ) a(r) a ( r ) μ \mu μ d Υ r \mathrm{d}\Upsilon_r d Υ r a ( 0 ) = μ ( { 0 } ) a(0)=\mu(\{0\}) a ( 0 ) = μ ({ 0 }) a ( 0 ) Υ 0 = ∫ { 0 } u ( r , X r ) μ ( d r ) a(0)\Upsilon_0=\int_{\{0\}}u(r,X_r)\,\mu(\mathrm{d}r) a ( 0 ) Υ 0 = ∫ { 0 } u ( r , X r ) μ ( d r )
∫ 0 q a ( r ) d Υ r = a ( q ) u ( q , X q ) − ∫ [ 0 , q ] u ( r , X r ) μ ( d r ) . \int_0^q a(r)\,\mathrm{d}\Upsilon_r=a(q)u(q,X_q)-\int_{[0,q]}u(r,X_r)\,\mu(\mathrm{d}r). ∫ 0 q a ( r ) d Υ r = a ( q ) u ( q , X q ) − ∫ [ 0 , q ] u ( r , X r ) μ ( d r ) . This proves (95) . At an atom of μ \mu μ q q q q = 0 q=0 q = 0
Given the path, the terminal spin has probability e ϵ Y 1 − u ( 1 , Y 1 ) e^{\epsilon Y_1-u(1,Y_1)} e ϵ Y 1 − u ( 1 , Y 1 ) (95) at q = 1 q=1 q = 1 a ( 1 ) = 1 a(1)=1 a ( 1 ) = 1 μ = s δ 0 + ν \mu=s\delta_0+\nu μ = s δ 0 + ν (96) . No further normalization is needed, since both factors are probability densities.
For a coefficient c ( q ) c(q) c ( q ) c + ( q ) = c ( q ) c^+(q)=c(q) c + ( q ) = c ( q ) c ( q ) > 0 c(q)>0 c ( q ) > 0 c + ( q ) = 1 c^+(q)=1 c + ( q ) = 1 c c c X X X C ( q ) = ∫ 0 q c ( r ) d r C(q)=\int_0^q c(r)\,\mathrm{d}r C ( q ) = ∫ 0 q c ( r ) d r τ = C ( q ) / C ( 1 ) \tau=C(q)/C(1) τ = C ( q ) / C ( 1 ) c ( q ) c(q) c ( q ) (88) with coefficient C ( 1 ) C(1) C ( 1 ) μ \mu μ q ↦ C ( q ) / C ( 1 ) q\mapsto C(q)/C(1) q ↦ C ( q ) / C ( 1 ) d τ = c ( q ) d q / C ( 1 ) \mathrm{d}\tau=c(q)\,\mathrm{d}q/C(1) d τ = c ( q ) d q / C ( 1 ) c > 0 c>0 c > 0 q ↦ C ( q ) / C ( 1 ) q\mapsto C(q)/C(1) q ↦ C ( q ) / C ( 1 ) [ 0 , 1 ] [0,1] [ 0 , 1 ] c c c C ( 1 ) C(1) C ( 1 ) ∣ u x ∣ ≤ 1 |u_x|\le1 ∣ u x ∣ ≤ 1 d r \mathrm{d}r d r c ( r ) d r c(r)\,\mathrm{d}r c ( r ) d r β 2 \beta^2 β 2 c c c
Continuity and kernel bounds
The proof of the next lemma follows the comparison argument of [43], Lemma 14, which bounds the differences of the solutions and of their first spatial derivatives by the L 1 L^1 L 1 d d d
Lemma 9.3 (Continuity of the scalar multiplier). There exists C β < ∞ C_\beta<\infty C β < ∞ a , a ~ a,\tilde a a , a ~ d , d ~ ∈ [ β 2 / 2 , β 2 ] d,\tilde d\in[\beta^2/2,\beta^2] d , d ~ ∈ [ β 2 /2 , β 2 ]
sup 0 ≤ q ≤ 1 ∣ k a , d ( q ) − k a ~ , d ~ ( q ) ∣ ≤ C β ( ∥ a − a ~ ∥ L 1 ( 0 , 1 ) + ∣ d − d ~ ∣ ) . (97) \sup_{0\le q\le1}\left|k_{a,d}(q)-k_{\tilde a,\tilde d}(q)\right|\le C_\beta\left(\lVert a-\tilde a\rVert_{L^1(0,1)}+|d-\tilde d|\right). \tag*{(97)} 0 ≤ q ≤ 1 sup k a , d ( q ) − k a ~ , d ~ ( q ) ≤ C β ( ∥ a − a ~ ∥ L 1 ( 0 , 1 ) + ∣ d − d ~ ∣ ) . (97) Proof. Let u u u u ~ \tilde u u ~ (88) for the two parameter pairs, and let Y Y Y Y ~ \tilde Y Y ~ u ^ = u − u ~ \hat u=u-\tilde u u ^ = u − u ~ Δ = ∥ a − a ~ ∥ L 1 ( 0 , 1 ) + ∣ d − d ~ ∣ \Delta=\lVert a-\tilde a\rVert_{L^1(0,1)}+|d-\tilde d| Δ = ∥ a − a ~ ∥ L 1 ( 0 , 1 ) + ∣ d − d ~ ∣ u ^ x \hat u_x u ^ x u ^ x x \hat u_{xx} u ^ xx C β Δ C_\beta\Delta C β Δ
Step 1: the difference of the solutions. Subtracting the two equations and writing u x 2 − u ~ x 2 = ( u x + u ~ x ) u ^ x u_x^2-\widetilde{u}_x^2=(u_x+\widetilde{u}_x)\widehat{u}_x u x 2 − u x 2 = ( u x + u x ) u x u ^ \widehat{u} u
c = d , b = ω , V = 0 , F = Ξ , u ^ ( 1 , ⋅ ) = 0 , c=d,\qquad b=\omega,\qquad V=0,\qquad F=\Xi,\qquad\widehat{u}(1,\cdot)=0, c = d , b = ω , V = 0 , F = Ξ , u ( 1 , ⋅ ) = 0 , where
ω = d a ( q ) 2 ( u x + u ~ x ) , Ξ = d − d ~ 2 u ^ x x + δ ( q ) u ~ x 2 , δ ( q ) = d a ( q ) − d ~ a ~ ( q ) 2 . \omega=\frac{d\,a(q)}{2}(u_x+\widetilde{u}_x),\qquad\Xi=\frac{d-\widetilde{d}}{2}\widehat{u}_{xx}+\delta(q)\widetilde{u}_x^2,\qquad\delta(q)=\frac{d\,a(q)-\widetilde{d}\,\widetilde{a}(q)}{2}. ω = 2 d a ( q ) ( u x + u x ) , Ξ = 2 d − d u xx + δ ( q ) u x 2 , δ ( q ) = 2 d a ( q ) − d a ( q ) . Using Lemma 9.1(a) and (b), we see that 0 ≤ ω x ≤ β 2 0\leq\omega_x\leq\beta^2 0 ≤ ω x ≤ β 2 ∣ ω x x ∣ ≤ C β |\omega_{xx}|\leq C\beta ∣ ω xx ∣ ≤ C β
Ξ x = d − d ~ 2 u ~ 3 + 2 δ u ~ 1 u ~ 2 , Ξ x x = d − d ~ 2 u ~ 4 + 2 δ ( u ~ 2 2 + u ~ 1 u ~ 3 ) , \Xi_x=\frac{d-\widetilde{d}}{2}\widetilde{u}_3+2\delta\,\widetilde{u}_1\widetilde{u}_2,\qquad\Xi_{xx}=\frac{d-\widetilde{d}}{2}\widetilde{u}_4+2\delta(\widetilde{u}_2^2+\widetilde{u}_1\widetilde{u}_3), Ξ x = 2 d − d u 3 + 2 δ u 1 u 2 , Ξ xx = 2 d − d u 4 + 2 δ ( u 2 2 + u 1 u 3 ) , and 2 ∣ δ ( q ) ∣ ≤ ∣ d − d ~ ∣ + β 2 ∣ a ( q ) − a ~ ( q ) ∣ 2|\delta(q)|\leq|d-\widetilde{d}|+\beta^2|a(q)-\widetilde{a}(q)| 2∣ δ ( q ) ∣ ≤ ∣ d − d ∣ + β 2 ∣ a ( q ) − a ( q ) ∣
∫ 0 1 sup x ( ∣ Ξ x ∣ + ∣ Ξ x x ∣ ) ( q , x ) d q ≤ C β ∫ 0 1 ( ∣ d − d ~ ∣ + ∣ a ( q ) − a ~ ( q ) ∣ ) d q ≤ C β Δ . \int_0^1\sup_x\bigl(|\Xi_x|+|\Xi_{xx}|\bigr)(q,x)\,dq\leq C_\beta\int_0^1\bigl(|d-\widetilde{d}|+|a(q)-\widetilde{a}(q)|\bigr)\,dq\leq C_\beta\Delta. ∫ 0 1 x sup ( ∣ Ξ x ∣ + ∣ Ξ xx ∣ ) ( q , x ) d q ≤ C β ∫ 0 1 ( ∣ d − d ∣ + ∣ a ( q ) − a ( q ) ∣ ) d q ≤ C β Δ. Differentiating the equation for u ^ \widehat{u} u x x x w 1 = u ^ x w_1=\widehat{u}_x w 1 = u x w 2 = u ^ x x w_2=\widehat{u}_{xx} w 2 = u xx c = d c=d c = d b = ω b=\omega b = ω
V = ω x , F = Ξ x and V = 2 ω x , F = Ξ x x + ω x x w 1 , V=\omega_x,\qquad F=\Xi_x\qquad\text{and}\qquad V=2\omega_x,\qquad F=\Xi_{xx}+\omega_{xx}w_1, V = ω x , F = Ξ x and V = 2 ω x , F = Ξ xx + ω xx w 1 , respectively. Then (9.4) implies that
∥ w 1 ∥ ∞ ≤ e β 2 ∫ 0 1 ∥ Ξ x ( q , ⋅ ) ∥ ∞ d q ≤ C β Δ , \|w_1\|_\infty\leq e^{\beta^2}\int_0^1\|\Xi_x(q,\cdot)\|_\infty\,dq\leq C_\beta\Delta, ∥ w 1 ∥ ∞ ≤ e β 2 ∫ 0 1 ∥ Ξ x ( q , ⋅ ) ∥ ∞ d q ≤ C β Δ , ∥ w 2 ∥ ∞ ≤ e 2 β 2 ∫ 0 1 ( ∥ Ξ x x ( q , ⋅ ) ∥ ∞ + ∥ ω x x ∥ ∞ ∥ w 1 ∥ ∞ ) d q ≤ C β Δ . \|w_2\|_\infty\leq e^{2\beta^2}\int_0^1\bigl(\|\Xi_{xx}(q,\cdot)\|_\infty+\|\omega_{xx}\|_\infty\|w_1\|_\infty\bigr)\,dq\leq C_\beta\Delta. ∥ w 2 ∥ ∞ ≤ e 2 β 2 ∫ 0 1 ( ∥ Ξ xx ( q , ⋅ ) ∥ ∞ + ∥ ω xx ∥ ∞ ∥ w 1 ∥ ∞ ) d q ≤ C β Δ. This shows that
sup q , x ( ∣ u x − u ~ x ∣ + ∣ u x x − u ~ x x ∣ ) ≤ C β Δ . (98) \sup_{q,x}\bigl(|u_x-\widetilde{u}_x|+|u_{xx}-\widetilde{u}_{xx}|\bigr)\leq C_\beta\Delta. \tag*{(98)} q , x sup ( ∣ u x − u x ∣ + ∣ u xx − u xx ∣ ) ≤ C β Δ. (98) Step 2: coupling the diffusions. We drive Y Y Y Y ~ \widetilde{Y} Y D ( q ) = sup r ≤ q ∣ Y r − Y ~ r ∣ D(q)=\sup_{r\leq q}|Y_r-\widetilde{Y}_r| D ( q ) = sup r ≤ q ∣ Y r − Y r ∣ d a ( r ) u x ( r , ⋅ ) d\,a(r)u_x(r,\cdot) d a ( r ) u x ( r , ⋅ ) β 2 \beta^2 β 2 d ~ a ~ ( r ) u ~ x ( r , ⋅ ) \widetilde{d}\,\widetilde{a}(r)\widetilde{u}_x(r,\cdot) d a ( r ) u x ( r , ⋅ ) ∣ d − d ~ ∣ + β 2 ∣ a ( r ) − a ~ ( r ) ∣ + β 2 C β Δ |d-\widetilde{d}|+\beta^2|a(r)-\widetilde{a}(r)|+\beta^2C_\beta\Delta ∣ d − d ∣ + β 2 ∣ a ( r ) − a ( r ) ∣ + β 2 C β Δ Y Y Y Y ~ \widetilde{Y} Y
D ( q ) ≤ β 2 ∫ 0 q D ( r ) d r + C β Δ + ∣ d − d ~ ∣ sup r ≤ 1 ∣ B r ∣ . D(q)\leq\beta^2\int_0^qD(r)\,dr+C_\beta\Delta+|\sqrt{d}-\sqrt{\widetilde{d}}|\sup_{r\leq1}|B_r|. D ( q ) ≤ β 2 ∫ 0 q D ( r ) d r + C β Δ + ∣ d − d ∣ r ≤ 1 sup ∣ B r ∣. Since d , d ~ ≥ β 2 / 2 d,\widetilde{d}\geq\beta^2/2 d , d ≥ β 2 /2 ∣ d − d ~ ∣ ≤ ∣ d − d ~ ∣ / ( 2 β ) |\sqrt{d}-\sqrt{\widetilde{d}}|\leq|d-\widetilde{d}|/(\sqrt{2}\,\beta) ∣ d − d ∣ ≤ ∣ d − d ∣/ ( 2 β ) E sup r ≤ 1 ∣ B r ∣ ≤ 2 \mathbb{E}\sup_{r\leq1}|B_r|\leq2 E sup r ≤ 1 ∣ B r ∣ ≤ 2
E sup q ≤ 1 ∣ Y q − Y ~ q ∣ ≤ C β ( Δ + ∣ d − d ~ ∣ E sup q ≤ 1 ∣ B q ∣ ) ≤ C β Δ . (99) \mathbb{E}\sup_{q\leq1}|Y_q-\widetilde{Y}_q|\leq C_\beta\bigl(\Delta+|\sqrt{d}-\sqrt{\widetilde{d}}|\,\mathbb{E}\sup_{q\leq1}|B_q|\bigr)\leq C_\beta\Delta. \tag*{(99)} E q ≤ 1 sup ∣ Y q − Y q ∣ ≤ C β ( Δ + ∣ d − d ∣ E q ≤ 1 sup ∣ B q ∣ ) ≤ C β Δ. (99) The same argument, applied to the two diffusions started from a common point y y y r r r ( r , y ) (r,y) ( r , y )
Step 3: the multipliers. We write h = u x x ( q , Y q ) h=u_{xx}(q,Y_q) h = u xx ( q , Y q ) h ~ = u ~ x x ( q , Y ~ q ) \widetilde{h}=\widetilde{u}_{xx}(q,\widetilde{Y}_q) h = u xx ( q , Y q ) ∣ h 2 − h ~ 2 ∣ ≤ 2 ∣ h − h ~ ∣ |h^2-\widetilde{h}^2|\le2|h-\widetilde{h}| ∣ h 2 − h 2 ∣ ≤ 2∣ h − h ∣ 0 ≤ h , h ~ ≤ 1 0\le h,\widetilde{h}\le1 0 ≤ h , h ≤ 1
∣ k a , d ( q ) − k a ~ , d ~ ( q ) ∣ ≤ ∣ d − d ~ ∣ + 2 β 2 E ∣ h − h ~ ∣ ≤ ∣ d − d ~ ∣ + 2 β 2 ( ∥ u x x − u ~ x x ∥ ∞ + ∥ u 3 ∥ ∞ E ∣ Y q − Y ~ q ∣ ) . \begin{aligned}
|k_{a,d}(q)-k_{\widetilde{a},\widetilde{d}}(q)|\le|d-\widetilde{d}|+2\beta^2\mathbb{E}|h-\widetilde{h}| \\
&\le|d-\widetilde{d}|+2\beta^2\left(\|u_{xx}-\widetilde{u}_{xx}\|_{\infty}+\|u_3\|_{\infty}\mathbb{E}|Y_q-\widetilde{Y}_q|\right).
\end{aligned} ∣ k a , d ( q ) − k a , d ( q ) ∣ ≤ ∣ d − d ∣ + 2 β 2 E ∣ h − h ∣ ≤ ∣ d − d ∣ + 2 β 2 ( ∥ u xx − u xx ∥ ∞ + ∥ u 3 ∥ ∞ E ∣ Y q − Y q ∣ ) . Inserting (98) , (99) , and the bound ∥ u 3 ∥ ∞ ≤ C 3 ( β ) \|u_3\|_{\infty}\le C_3(\beta) ∥ u 3 ∥ ∞ ≤ C 3 ( β ) (97) .
The next lemma collects the bounds on the kernels of the response operator that the cavity estimates of Section 11 use. Each bound retains a factor Γ \Gamma Γ
Lemma 9.4 (Scalar kernels and their level weight). For every profile a a a and d ∈ [ β 2 / 2 , β 2 ] d\in[\beta^2/2,\beta^2] d ∈ [ β 2 /2 , β 2 ] write Γ = Γ a , d \Gamma=\Gamma_{a,d} Γ = Γ a , d Then Γ \Gamma Γ is nondecreasing and
0 ≤ Γ ( p ) ≤ min { 1 , d p } ≤ min { 1 , β 2 p } . (100) 0\le\Gamma(p)\le\min\{1,dp\}\le\min\{1,\beta^2p\}. \tag*{(100)} 0 ≤ Γ ( p ) ≤ min { 1 , d p } ≤ min { 1 , β 2 p } . (100) For all p , r , ℓ ∈ [ 0 , 1 ] p,r,\ell\in[0,1] p , r , ℓ ∈ [ 0 , 1 ]
∣ E 0 [ m p h p m r ] ∣ ≤ Γ ( p ) Γ ( r ) , ∣ E 0 [ f ‾ p ϵ m r ] ∣ ≤ Γ ( p ) Γ ( r ) . (101) |\mathbb{E}^0[m_ph_pm_r]|\le\sqrt{\Gamma(p)\Gamma(r)},\qquad|\mathbb{E}^0[\overline{f}_p\epsilon m_r]|\le\sqrt{\Gamma(p)\Gamma(r)}. \tag*{(101)} ∣ E 0 [ m p h p m r ] ∣ ≤ Γ ( p ) Γ ( r ) , ∣ E 0 [ f p ϵ m r ] ∣ ≤ Γ ( p ) Γ ( r ) . (101) ∣ E 0 [ f ‾ p m r m ℓ ] ∣ ≤ Γ ( p ) Γ ( r ∧ ℓ ) , ∣ E 0 [ f ‾ p m p m r ] ∣ ≤ Γ ( p ) Γ ( p ∧ r ) . (102) |\mathbb{E}^0[\overline{f}_pm_rm_\ell]|\le\sqrt{\Gamma(p)\Gamma(r\wedge\ell)},\qquad|\mathbb{E}^0[\overline{f}_pm_pm_r]|\le\sqrt{\Gamma(p)\Gamma(p\wedge r)}. \tag*{(102)} ∣ E 0 [ f p m r m ℓ ] ∣ ≤ Γ ( p ) Γ ( r ∧ ℓ ) , ∣ E 0 [ f p m p m r ] ∣ ≤ Γ ( p ) Γ ( p ∧ r ) . (102) 0 ≤ Var 0 ( f p ) ≤ Γ ( p ) , 0 ≤ E 0 [ m p 2 h p ] ≤ Γ ( p ) . (103) 0\le\operatorname{Var}^0(f_p)\le\Gamma(p),\qquad0\le\mathbb{E}^0[m_p^2h_p]\le\Gamma(p). \tag*{(103)} 0 ≤ Var 0 ( f p ) ≤ Γ ( p ) , 0 ≤ E 0 [ m p 2 h p ] ≤ Γ ( p ) . (103) For ℓ ≥ p \ell\ge p ℓ ≥ p
∣ Cov 0 ( f p , g p ℓ ( Y p ) ) ∣ ≤ Γ ( p ) Γ ( ℓ ) . (104) |\operatorname{Cov}^0(f_p,g_p^\ell(Y_p))|\le\sqrt{\Gamma(p)\Gamma(\ell)}. \tag*{(104)} ∣ Cov 0 ( f p , g p ℓ ( Y p )) ∣ ≤ Γ ( p ) Γ ( ℓ ) . (104) Proof. By Lemma 9.1 (c) and (92) , we have Γ ( p ) = d ∫ 0 p E 0 h r 2 d r \Gamma(p)=d\int_0^p\mathbb{E}^0h_r^2\,\mathrm{d}r Γ ( p ) = d ∫ 0 p E 0 h r 2 d r 0 ≤ h r ≤ 1 0\le h_r\le1 0 ≤ h r ≤ 1 Γ \Gamma Γ 0 ≤ Γ ( p ) ≤ d p 0\le\Gamma(p)\le dp 0 ≤ Γ ( p ) ≤ d p Γ ≤ 1 \Gamma\le1 Γ ≤ 1 ∣ m p ∣ ≤ 1 |m_p|\le1 ∣ m p ∣ ≤ 1 (100) . Using ∣ m p ∣ ≤ 1 |m_p|\le1 ∣ m p ∣ ≤ 1
Var 0 ( f p ) = E 0 f p 2 ≤ E 0 m p 4 ≤ E 0 m p 2 = Γ ( p ) . \operatorname{Var}^0(f_p)=\mathbb{E}^0f_p^2\le\mathbb{E}^0m_p^4\le\mathbb{E}^0m_p^2=\Gamma(p). Var 0 ( f p ) = E 0 f p 2 ≤ E 0 m p 4 ≤ E 0 m p 2 = Γ ( p ) . The bounds below follow from the Cauchy–Schwarz inequality, together with ∣ h p ∣ ≤ 1 |h_p|\le1 ∣ h p ∣ ≤ 1 ∣ ϵ ∣ = 1 |\epsilon|=1 ∣ ϵ ∣ = 1 ∣ m ∣ ≤ 1 |m|\le1 ∣ m ∣ ≤ 1
∣ E 0 [ m p h p m r ] ∣ 2 ≤ E 0 [ m p 2 h p 2 ] E 0 m r 2 ≤ Γ ( p ) Γ ( r ) , |\mathbb{E}^0[m_ph_pm_r]|^2\le\mathbb{E}^0[m_p^2h_p^2]\mathbb{E}^0m_r^2\le\Gamma(p)\Gamma(r), ∣ E 0 [ m p h p m r ] ∣ 2 ≤ E 0 [ m p 2 h p 2 ] E 0 m r 2 ≤ Γ ( p ) Γ ( r ) , ∣ E 0 [ f ‾ p ϵ m r ] ∣ 2 ≤ E 0 f ‾ p 2 E 0 m r 2 ≤ Γ ( p ) Γ ( r ) , |\mathbb{E}^0[\overline{f}_p\epsilon m_r]|^2\le\mathbb{E}^0\overline{f}_p^2\mathbb{E}^0m_r^2\le\Gamma(p)\Gamma(r), ∣ E 0 [ f p ϵ m r ] ∣ 2 ≤ E 0 f p 2 E 0 m r 2 ≤ Γ ( p ) Γ ( r ) , ∣ E 0 [ f ‾ p m r m ℓ ] ∣ 2 ≤ E 0 f ‾ p 2 E 0 [ m r 2 m ℓ 2 ] ≤ Γ ( p ) Γ ( r ∧ ℓ ) . |\mathbb{E}^0[\overline{f}_pm_rm_\ell]|^2\le\mathbb{E}^0\overline{f}_p^2\mathbb{E}^0[m_r^2m_\ell^2]\le\Gamma(p)\Gamma(r\wedge\ell). ∣ E 0 [ f p m r m ℓ ] ∣ 2 ≤ E 0 f p 2 E 0 [ m r 2 m ℓ 2 ] ≤ Γ ( p ) Γ ( r ∧ ℓ ) . In the last line, we used m r 2 m ℓ 2 ≤ m r ∧ ℓ 2 m_r^2m_\ell^2\le m_{r\wedge\ell}^2 m r 2 m ℓ 2 ≤ m r ∧ ℓ 2 (101) and the first
bound in (102) ; the second bound in (102) is the case ℓ = p \ell=p ℓ = p (103) follow from the estimate on Var 0 ( f p ) \operatorname{Var}^{0}(f_p) Var 0 ( f p ) 0 ≤ h p ≤ 1 0 \le h_p \le1 0 ≤ h p ≤ 1 0 ≤ E 0 [ m p 2 h p ] ≤ E 0 m p 2 0 \le\mathbb{E}^{0}[m_p^2h_p] \le\mathbb{E}^{0}m_p^2 0 ≤ E 0 [ m p 2 h p ] ≤ E 0 m p 2 ℓ ≥ p \ell\ge p ℓ ≥ p 0 ≤ g p ℓ ≤ 1 0 \le g_p^\ell\le1 0 ≤ g p ℓ ≤ 1 E 0 g p ℓ ( Y p ) = E 0 m ℓ 2 = Γ ( ℓ ) \mathbb{E}^{0}g_p^\ell(Y_p)=\mathbb{E}^{0}m_\ell^2=\Gamma(\ell) E 0 g p ℓ ( Y p ) = E 0 m ℓ 2 = Γ ( ℓ )
Var 0 ( g p ℓ ( Y p ) ) ≤ E 0 ( g p ℓ ( Y p ) ) 2 ≤ E 0 g p ℓ ( Y p ) = Γ ( ℓ ) , \operatorname{Var}^{0}\left(g_p^\ell(Y_p)\right) \le\mathbb{E}^{0}\left(g_p^\ell(Y_p)\right)^2 \le\mathbb{E}^{0}g_p^\ell(Y_p)=\Gamma(\ell), Var 0 ( g p ℓ ( Y p ) ) ≤ E 0 ( g p ℓ ( Y p ) ) 2 ≤ E 0 g p ℓ ( Y p ) = Γ ( ℓ ) , and (104) follows from the Cauchy–Schwarz inequality and the bound on Var 0 ( f p ) \operatorname{Var}^{0}(f_p) Var 0 ( f p )
The response operator
For every profile, the operator defined below is the response of Γ \Gamma Γ [10 ] . In § 8.5.4 , this shows that the scalar response operator is dominated by multiplication by k a , d k_{a,d} k a , d scalar transport space
H ν = { v ∈ L 2 ( ν ) : v ( 0 ) = 0 if ν ( { 0 } ) > 0 , v ( 1 ) = 0 if ν ( { 1 } ) > 0 } . \mathcal{H}_{\nu}=\left\{v\in L^2(\nu):v(0)=0\text{ if }\nu(\{0\})>0,\ v(1)=0\text{ if }\nu(\{1\})>0\right\}. H ν = { v ∈ L 2 ( ν ) : v ( 0 ) = 0 if ν ({ 0 }) > 0 , v ( 1 ) = 0 if ν ({ 1 }) > 0 } . This is a closed subspace of L 2 ( ν ) L^2(\nu) L 2 ( ν ) 1 ( 0 , 1 ) \mathbf{1}_{(0,1)} 1 ( 0 , 1 )
The operator involves the derivative of g r ℓ g_r^\ell g r ℓ Y Y Y Y ℓ r , y Y_\ell^{r,y} Y ℓ r , y y y y r r r y y y
ξ ~ ℓ = exp { d ∫ r ℓ a ( v ) u x x ( v , Y v r , y ) d v } , \widetilde{\xi}_{\ell}=\exp\left\{d\int_r^\ell a(v)u_{xx}\left(v,Y_v^{r,y}\right)\,dv\right\}, ξ ℓ = exp { d ∫ r ℓ a ( v ) u xx ( v , Y v r , y ) d v } , which lies in [ 1 , e β 2 ] [1,e^{\beta^2}] [ 1 , e β 2 ] g r ℓ ( y ) = E [ u x ( ℓ , Y ℓ r , y ) 2 ] g_r^\ell(y)=\mathbb{E}[u_x(\ell,Y_\ell^{r,y})^2] g r ℓ ( y ) = E [ u x ( ℓ , Y ℓ r , y ) 2 ]
∂ y g r ℓ ( y ) = E [ 2 u x ( ℓ , Y ℓ r , y ) u x x ( ℓ , Y ℓ r , y ) ξ ~ ℓ ] , ∣ ∂ y g r ℓ ∣ ≤ 2 e β 2 . (105) \partial_y g_r^\ell(y)=\mathbb{E}\left[2u_x\left(\ell,Y_\ell^{r,y}\right)u_{xx}\left(\ell,Y_\ell^{r,y}\right)\widetilde{\xi}_{\ell}\right],\qquad\left|\partial_y g_r^\ell\right|\le2e^{\beta^2}. \tag*{(105)} ∂ y g r ℓ ( y ) = E [ 2 u x ( ℓ , Y ℓ r , y ) u xx ( ℓ , Y ℓ r , y ) ξ ℓ ] , ∂ y g r ℓ ≤ 2 e β 2 . (105) Definition 9.5 (Response operator). Let a a a μ = s δ 0 + ν \mu=s\delta_0+\nu μ = s δ 0 + ν d ∈ [ β 2 / 2 , β 2 ] d\in[\beta^2/2,\beta^2] d ∈ [ β 2 /2 , β 2 ] φ \varphi φ [ 0 , 1 ] [0,1] [ 0 , 1 ] p ∈ [ 0 , 1 ] p\in[0,1] p ∈ [ 0 , 1 ]
1 d ( J a , d φ ) ( p ) = E 0 [ h p 2 + 2 ( 1 − a ( p ) ) m p 2 h p ] φ ( p ) − ∫ ( p , 1 ] E 0 [ m p ∂ y g p ℓ ( Y p ) ] φ ( ℓ ) ν ( d ℓ ) − 2 ∫ E 0 [ m p h p m r ] φ ( p ∧ r ) ν ( d r ) − ∫ E 0 [ f ‾ p m r ] φ ( r ) ν ( d r ) + 1 2 ∫ a ( r ) E 0 [ f ‾ p m r 2 ] φ ( r ) ν ( d r ) + 1 2 ∬ E 0 [ f ‾ p m r m ℓ ] φ ( r ∧ ℓ ) ν ( d r ) ν ( d ℓ ) + 1 2 ∫ ν ( d r ) ∫ ( r , 1 ] E 0 [ f ‾ p m r g r ℓ ( Y r ) ] φ ( ℓ ) ν ( d ℓ ) . (106) \begin{aligned}
\frac{1}{d}(J_{a,d}\varphi)(p)
={}&\mathbb{E}^{0}\left[h_p^2+2(1-a(p))m_p^2h_p\right]\varphi(p)
-\int_{(p,1]}\mathbb{E}^{0}\left[m_p\partial_y g_p^\ell(Y_p)\right]\varphi(\ell)\,\nu(d\ell)\\
&-2\int\mathbb{E}^{0}[m_ph_pm_r]\varphi(p\wedge r)\,\nu(dr)
-\int\mathbb{E}^{0}[\overline{f}_pm_r]\varphi(r)\,\nu(dr)\\
&+\frac{1}{2}\int a(r)\mathbb{E}^{0}[\overline{f}_pm_r^2]\varphi(r)\,\nu(dr)
+\frac{1}{2}\iint\mathbb{E}^{0}[\overline{f}_pm_rm_\ell]\varphi(r\wedge\ell)\,\nu(dr)\nu(d\ell)\\
&+\frac{1}{2}\int\nu(dr)\int_{(r,1]}\mathbb{E}^{0}[\overline{f}_pm_rg_r^\ell(Y_r)]\varphi(\ell)\,\nu(d\ell). \tag*{(106)}
\end{aligned} d 1 ( J a , d φ ) ( p ) = E 0 [ h p 2 + 2 ( 1 − a ( p )) m p 2 h p ] φ ( p ) − ∫ ( p , 1 ] E 0 [ m p ∂ y g p ℓ ( Y p ) ] φ ( ℓ ) ν ( d ℓ ) − 2 ∫ E 0 [ m p h p m r ] φ ( p ∧ r ) ν ( d r ) − ∫ E 0 [ f p m r ] φ ( r ) ν ( d r ) + 2 1 ∫ a ( r ) E 0 [ f p m r 2 ] φ ( r ) ν ( d r ) + 2 1 ∬ E 0 [ f p m r m ℓ ] φ ( r ∧ ℓ ) ν ( d r ) ν ( d ℓ ) + 2 1 ∫ ν ( d r ) ∫ ( r , 1 ] E 0 [ f p m r g r ℓ ( Y r )] φ ( ℓ ) ν ( d ℓ ) . (106) Integrals without a domain are over [ 0 , 1 ] [0,1] [ 0 , 1 ] J a , d v J_{a,d}v J a , d v v ∈ L 2 ( ν ) v \in L^2(\nu) v ∈ L 2 ( ν ) scalar response operator J a , d s c J^{\mathrm{sc}}_{a,d} J a , d sc J a , d J_{a,d} J a , d H ν \mathcal{H}_\nu H ν J a , d s c v = 1 ( 0 , 1 ) J a , d v J^{\mathrm{sc}}_{a,d}v=\mathbf{1}_{(0,1)}J_{a,d}v J a , d sc v = 1 ( 0 , 1 ) J a , d v v ∈ H ν v \in\mathcal{H}_\nu v ∈ H ν
At an atom of ν \nu ν p p p a ( p ) a(p) a ( p ) ( p , 1 ] (p,1] ( p , 1 ] v ∈ L 2 ( ν ) v \in L^2(\nu) v ∈ L 2 ( ν ) v v v v ( p ) v(p) v ( p ) ν \nu ν p ∧ r ∈ { p , r } p \wedge r \in\{p,r\} p ∧ r ∈ { p , r } r ∧ ℓ ∈ { r , ℓ } r \wedge\ell\in\{r,\ell\} r ∧ ℓ ∈ { r , ℓ } J a , d v J_{a,d}v J a , d v L 2 ( ν ) L^2(\nu) L 2 ( ν )
Kernel form and boundedness. We now rewrite (106) . For r > p r>p r > p G p 0 \mathcal{G}_p^0 G p 0 (93) , we have E 0 [ m p h p m r ] = E 0 [ m p 2 h p ] \mathbb{E}^0[m_p h_p m_r]=\mathbb{E}^0[m_p^2h_p] E 0 [ m p h p m r ] = E 0 [ m p 2 h p ] ν ( ( p , 1 ] ) = 1 − a ( p ) \nu((p,1])=1-a(p) ν (( p , 1 ]) = 1 − a ( p ) (106) at r = p r=p r = p ( p , 1 ] (p,1] ( p , 1 ] − 2 ( 1 − a ( p ) ) E 0 [ m p 2 h p ] φ ( p ) -2(1-a(p))\mathbb{E}^0[m_p^2h_p]\varphi(p) − 2 ( 1 − a ( p )) E 0 [ m p 2 h p ] φ ( p ) φ ( r ∧ ℓ ) \varphi(r\wedge\ell) φ ( r ∧ ℓ ) r < ℓ r<\ell r < ℓ ℓ < r \ell<r ℓ < r r = ℓ r=\ell r = ℓ ν \nu ν
( J a , d v ) ( p ) = k a , d ( p ) v ( p ) + d ∫ K a , d ( p , r ) v ( r ) ν ( d r ) , (107) (J_{a,d}v)(p)=k_{a,d}(p)v(p)+d\int\mathcal{K}_{a,d}(p,r)v(r)\,\nu(\mathrm{d}r), \tag*{(107)} ( J a , d v ) ( p ) = k a , d ( p ) v ( p ) + d ∫ K a , d ( p , r ) v ( r ) ν ( d r ) , (107) for every bounded Borel v v v
K a , d ( p , r ) = − 1 { r > p } E 0 [ m p ∂ y g p r ( Y p ) ] − 2 1 { r ≤ p } E 0 [ m p h p m r ] − E 0 [ f ˉ p ϵ m r ] + 1 2 a ( r ) E 0 [ f ˉ p m r 2 ] + ∫ ( r , 1 ] E 0 [ f ˉ p m r m ℓ ] ν ( d ℓ ) + 1 2 ν ( { r } ) E 0 [ f ˉ p m r 2 ] + 1 2 ∫ [ 0 , r ) E 0 [ f ˉ p g ℓ r ( Y ℓ ) ] ν ( d ℓ ) . (108) \begin{aligned}
\mathcal{K}_{a,d}(p,r)={}&-1_{\{r>p\}}\mathbb{E}^0[m_p\partial_y g_p^r(Y_p)]-2\,1_{\{r\le p\}}\mathbb{E}^0[m_ph_pm_r]-\mathbb{E}^0[\bar f_p\epsilon m_r]\\
&+\frac{1}{2}a(r)\mathbb{E}^0[\bar f_pm_r^2]+\int_{(r,1]}\mathbb{E}^0[\bar f_pm_rm_\ell]\,\nu(\mathrm{d}\ell)+\frac{1}{2}\nu(\{r\})\mathbb{E}^0[\bar f_pm_r^2]\\
&+\frac{1}{2}\int_{[0,r)}\mathbb{E}^0[\bar f_pg_\ell^r(Y_\ell)]\,\nu(\mathrm{d}\ell). \tag*{(108)}
\end{aligned} K a , d ( p , r ) = − 1 { r > p } E 0 [ m p ∂ y g p r ( Y p )] − 2 1 { r ≤ p } E 0 [ m p h p m r ] − E 0 [ f ˉ p ϵ m r ] + 2 1 a ( r ) E 0 [ f ˉ p m r 2 ] + ∫ ( r , 1 ] E 0 [ f ˉ p m r m ℓ ] ν ( d ℓ ) + 2 1 ν ({ r }) E 0 [ f ˉ p m r 2 ] + 2 1 ∫ [ 0 , r ) E 0 [ f ˉ p g ℓ r ( Y ℓ )] ν ( d ℓ ) . (108) The term with ν ( { r } ) \nu(\{r\}) ν ({ r }) K a , d \mathcal{K}_{a,d} K a , d 2 e β 2 2e^{\beta^2} 2 e β 2 (105) ; since ∣ f ˉ p ∣ ≤ 1 |\bar f_p|\le1 ∣ f ˉ p ∣ ≤ 1 ∣ m ∣ ≤ 1 |m|\le1 ∣ m ∣ ≤ 1 0 ≤ g ≤ 1 0\le g\le1 0 ≤ g ≤ 1 1 1 1 1 / 2 1/2 1/2 ν ( [ 0 , 1 ] ) ≤ 1 \nu([0,1])\le1 ν ([ 0 , 1 ]) ≤ 1 ∣ K a , d ∣ ≤ 2 e β 2 + 3 |\mathcal{K}_{a,d}|\le2e^{\beta^2}+3 ∣ K a , d ∣ ≤ 2 e β 2 + 3 ∥ ∫ K a , d ( ⋅ , r ) v ( r ) ν ( d r ) ∥ L 2 ( ν ) ≤ ( 2 e β 2 + 3 ) ν ( [ 0 , 1 ] ) ∥ v ∥ L 2 ( ν ) \left\|\int\mathcal{K}_{a,d}(\cdot,r)v(r)\,\nu(\mathrm{d}r)\right\|_{L^2(\nu)}\le(2e^{\beta^2}+3)\nu([0,1])\|v\|_{L^2(\nu)} ∫ K a , d ( ⋅ , r ) v ( r ) ν ( d r ) L 2 ( ν ) ≤ ( 2 e β 2 + 3 ) ν ([ 0 , 1 ]) ∥ v ∥ L 2 ( ν ) (107) extends to v ∈ L 2 ( ν ) v\in L^2(\nu) v ∈ L 2 ( ν ) (106) . Using 0 ≤ k a , d ≤ d ≤ β 2 0\le k_{a,d}\le d\le\beta^2 0 ≤ k a , d ≤ d ≤ β 2 ∥ J a , d ∥ L 2 ( ν ) → L 2 ( ν ) ≤ β 2 ( 4 + 2 e β 2 ) \|J_{a,d}\|_{L^2(\nu)\to L^2(\nu)}\le\beta^2(4+2e^{\beta^2}) ∥ J a , d ∥ L 2 ( ν ) → L 2 ( ν ) ≤ β 2 ( 4 + 2 e β 2 )
∥ J a , d s c ∥ H ν → H ν ≤ C β . (109) \|J^{\mathrm{sc}}_{a,d}\|_{\mathcal{H}_\nu\to\mathcal{H}_\nu}\le C_\beta. \tag*{(109)} ∥ J a , d sc ∥ H ν → H ν ≤ C β . (109) Transports of levels. Let φ \varphi φ [ 0 , 1 ] [0,1] [ 0 , 1 ] φ ( 0 ) = φ ( 1 ) = 0 \varphi(0)=\varphi(1)=0 φ ( 0 ) = φ ( 1 ) = 0 T θ ( q ) = q + θ φ ( q ) T_\theta(q)=q+\theta\varphi(q) T θ ( q ) = q + θ φ ( q ) θ 0 ∈ ( 0 , 1 ] \theta_0\in(0,1] θ 0 ∈ ( 0 , 1 ] θ 0 ∥ φ ′ ∥ ∞ ≤ 1 / 2 \theta_0\|\varphi'\|_\infty\le1/2 θ 0 ∥ φ ′ ∥ ∞ ≤ 1/2 ∣ θ ∣ ≤ θ 0 |\theta|\le\theta_0 ∣ θ ∣ ≤ θ 0 T θ T_\theta T θ [ 0 , 1 ] [0,1] [ 0 , 1 ] T θ ′ ∈ [ 1 / 2 , 3 / 2 ] T_\theta'\in[1/2,3/2] T θ ′ ∈ [ 1/2 , 3/2 ] d ( q ∧ r ) d(q \wedge r) d ( q ∧ r ) X X X (96) by d T θ ( q ∧ r ) dT_\theta(q \wedge r) d T θ ( q ∧ r ) ν \nu ν d d d (88) by
c θ ( q ) = d T θ ′ ( q ) = d ( 1 + θ φ ′ ( q ) ) ∈ [ d 2 , 3 d 2 ] ⊂ [ β 2 4 , 2 β 2 ] , c_\theta(q)=dT_\theta'(q)=d\left(1+\theta\varphi'(q)\right)\in\left[\frac{d}{2},\frac{3d}{2}\right]\subset\left[\frac{\beta^2}{4},2\beta^2\right], c θ ( q ) = d T θ ′ ( q ) = d ( 1 + θ φ ′ ( q ) ) ∈ [ 2 d , 2 3 d ] ⊂ [ 4 β 2 , 2 β 2 ] , and keep a a a u θ u_\theta u θ Y θ Y^\theta Y θ u θ , x = ∂ x u θ u_{\theta,x}=\partial_xu_\theta u θ , x = ∂ x u θ
Γ ~ θ ( p ) = E [ ( ∂ x u θ ( p , Y p θ ) ) 2 ] . \widetilde{\Gamma}_\theta(p)=\mathbb{E}\left[(\partial_xu_\theta(p,Y^\theta_p))^2\right]. Γ θ ( p ) = E [ ( ∂ x u θ ( p , Y p θ ) ) 2 ] . for the resulting mean square magnetization. Let a θ = a ∘ T θ − 1 a_\theta=a\circ T_\theta^{-1} a θ = a ∘ T θ − 1 T θ # μ T_{\theta\#}\mu T θ # μ x ′ = T θ ( q ) x'=T_\theta(q) x ′ = T θ ( q ) u θ u_\theta u θ (88) with coefficient d d d a θ a_\theta a θ Y θ Y^\theta Y θ a θ a_\theta a θ
u θ ( q , x ) = u a θ , d ( T θ ( q ) , x ) , Γ ~ θ ( p ) = Γ a θ , d ( T θ ( p ) ) , (110) u_\theta(q,x)=u_{a_\theta,d}(T_\theta(q),x),\qquad\widetilde{\Gamma}_\theta(p)=\Gamma_{a_\theta,d}(T_\theta(p)), \tag*{(110)} u θ ( q , x ) = u a θ , d ( T θ ( q ) , x ) , Γ θ ( p ) = Γ a θ , d ( T θ ( p )) , (110) that is, ( u θ , Y θ ) (u_\theta,Y^\theta) ( u θ , Y θ ) T θ # μ T_{\theta\#}\mu T θ # μ d d d Γ ~ 0 = Γ a , d \widetilde{\Gamma}_0=\Gamma_{a,d} Γ 0 = Γ a , d u θ u_\theta u θ ∣ θ ∣ ≤ θ 0 |\theta|\leq\theta_0 ∣ θ ∣ ≤ θ 0
Lemma 9.6 (Response identity). Let a a a d ∈ [ β 2 / 2 , β 2 ] d\in[\beta^2/2,\beta^2] d ∈ [ β 2 /2 , β 2 ] φ \varphi φ Γ ~ θ \widetilde{\Gamma}_\theta Γ θ p ∈ [ 0 , 1 ] p\in[0,1] p ∈ [ 0 , 1 ] θ ↦ Γ ~ θ ( p ) \theta\mapsto\widetilde{\Gamma}_\theta(p) θ ↦ Γ θ ( p ) θ = 0 \theta=0 θ = 0
d d θ Γ ~ θ ( p ) ∣ θ = 0 = ( J a , d φ ) ( p ) . \left.\frac{\mathrm{d}}{\mathrm{d}\theta}\widetilde{\Gamma}_\theta(p)\right|_{\theta=0}=(J_{a,d}\varphi)(p). d θ d Γ θ ( p ) θ = 0 = ( J a , d φ ) ( p ) . Proof. The proof has four steps. In Step 1, we show that u θ u_\theta u θ θ \theta θ a a a u θ u_\theta u θ (115) . In Step 3, we write Γ ~ θ ( p ) \widetilde{\Gamma}_\theta(p) Γ θ ( p ) (106) . Throughout the proof, ∣ θ ∣ ≤ θ 0 |\theta|\leq\theta_0 ∣ θ ∣ ≤ θ 0 β \beta β φ \varphi φ
Step 1: differentiability in the parameter. Fix θ \theta θ δ ≠ 0 \delta\neq0 δ = 0 v = u θ + δ v=u_{\theta+\delta} v = u θ + δ u = u θ u=u_\theta u = u θ D δ = ( v − u ) / δ D_\delta=(v-u)/\delta D δ = ( v − u ) / δ c θ + δ − c θ = δ d φ ′ c_{\theta+\delta}-c_\theta=\delta d\varphi' c θ + δ − c θ = δ d φ ′ δ \delta δ
∂ q D δ + c θ 2 ( D δ ) x x + c θ a 2 ( v x + u x ) ( D δ ) x = − d φ ′ 2 ( v x x + a v x 2 ) , D δ ( 1 , ⋅ ) = 0. (111) \partial_qD_\delta+\frac{c_\theta}{2}(D_\delta)_{xx}+\frac{c_\theta a}{2}(v_x+u_x)(D_\delta)_x=-\frac{d\varphi'}{2}(v_{xx}+av_x^2),\qquad D_\delta(1,\cdot)=0. \tag*{(111)} ∂ q D δ + 2 c θ ( D δ ) xx + 2 c θ a ( v x + u x ) ( D δ ) x = − 2 d φ ′ ( v xx + a v x 2 ) , D δ ( 1 , ⋅ ) = 0. (111) This is (90) with c = c θ c=c_\theta c = c θ V = 0 V=0 V = 0 δ \delta δ θ \theta θ (111) j j j x x x ∂ x j D δ \partial_x^j D_\delta ∂ x j D δ j j j D δ D_\delta D δ j j j (111) . By (91) and induction on j j j sup q , x ∣ ∂ x j D δ ∣ ≤ C j \sup_{q,x}|\partial_x^jD_\delta|\le C_j sup q , x ∣ ∂ x j D δ ∣ ≤ C j j ≥ 0 j\ge0 j ≥ 0 δ \delta δ θ \theta θ v − u = δ D δ v-u=\delta D_\delta v − u = δ D δ O ( ∣ δ ∣ ) O(|\delta|) O ( ∣ δ ∣ )
Next, we write the drift in (111) as c θ a u x + ( c θ a / 2 ) ( v x − u x ) c_\theta a u_x+(c_\theta a/2)(v_x-u_x) c θ a u x + ( c θ a /2 ) ( v x − u x ) δ \delta δ δ ′ \delta' δ ′ D δ − D δ ′ D_\delta-D_{\delta'} D δ − D δ ′ (90) with drift c θ a u x c_\theta a u_x c θ a u x V = 0 V=0 V = 0
d φ ′ 2 [ ( v − v ′ ) x x + a ( v x 2 − v x ′ 2 ) ] + c θ a 2 [ ( v x − u x ) ( D δ ) x − ( v x ′ − u x ) ( D δ ′ ) x ] , \frac{d\varphi'}{2}\left[(v-v')_{xx}+a(v_x^2-v_x'^2)\right]+\frac{c_\theta a}{2}\left[(v_x-u_x)(D_\delta)_x-(v_x'-u_x)(D_{\delta'})_x\right], 2 d φ ′ [ ( v − v ′ ) xx + a ( v x 2 − v x ′2 ) ] + 2 c θ a [ ( v x − u x ) ( D δ ) x − ( v x ′ − u x ) ( D δ ′ ) x ] , where v ′ = u θ + δ ′ v'=u_{\theta+\delta'} v ′ = u θ + δ ′ C j ( ∣ δ ∣ + ∣ δ ′ ∣ ) C_j(|\delta|+|\delta'|) C j ( ∣ δ ∣ + ∣ δ ′ ∣ ) sup q , x ∣ ∂ x j ( D δ − D δ ′ ) ∣ ≤ C j ( ∣ δ ∣ + ∣ δ ′ ∣ ) \sup_{q,x}|\partial_x^j(D_\delta-D_{\delta'})|\le C_j(|\delta|+|\delta'|) sup q , x ∣ ∂ x j ( D δ − D δ ′ ) ∣ ≤ C j ( ∣ δ ∣ + ∣ δ ′ ∣ ) δ → 0 \delta\to0 δ → 0 D δ D_\delta D δ u ˙ θ \dot{u}_\theta u ˙ θ (111) , we find that u ˙ θ \dot{u}_\theta u ˙ θ
∂ q u ˙ θ + c θ 2 ( u ˙ θ ) x x + c θ a u θ , x ( u ˙ θ ) x = − d φ ′ 2 ( u θ , x x + a u θ , x 2 ) , u ˙ θ ( 1 , ⋅ ) = 0. (112) \partial_q\dot{u}_\theta+\frac{c_\theta}{2}(\dot{u}_\theta)_{xx}+c_\theta a u_{\theta,x}(\dot{u}_\theta)_x=-\frac{d\varphi'}{2}(u_{\theta,xx}+a u_{\theta,x}^2),\qquad\dot{u}_\theta(1,\cdot)=0. \tag*{(112)} ∂ q u ˙ θ + 2 c θ ( u ˙ θ ) xx + c θ a u θ , x ( u ˙ θ ) x = − 2 d φ ′ ( u θ , xx + a u θ , x 2 ) , u ˙ θ ( 1 , ⋅ ) = 0. (112) We conclude that u θ u_\theta u θ θ \theta θ ∂ θ ∂ x j u θ = ∂ x j u ˙ θ \partial_\theta\partial_x^j u_\theta=\partial_x^j\dot{u}_\theta ∂ θ ∂ x j u θ = ∂ x j u ˙ θ ( q , x ) (q,x) ( q , x ) θ \theta θ θ \theta θ b θ = c θ a u θ , x b_\theta=c_\theta a u_{\theta,x} b θ = c θ a u θ , x F θ = ( d φ ′ / 2 ) ( u θ , x x + a u θ , x 2 ) F_\theta=(d\varphi'/2)(u_{\theta,xx}+a u_{\theta,x}^2) F θ = ( d φ ′ /2 ) ( u θ , xx + a u θ , x 2 ) R = u ˙ θ − u ˙ θ ′ R=\dot{u}_\theta-\dot{u}_{\theta'} R = u ˙ θ − u ˙ θ ′ (112) at the two parameters, we obtain
( ∂ q + c θ 2 ∂ x x + b θ ∂ x ) R = − ( F θ − F θ ′ ) − c θ − c θ ′ 2 ( u ˙ θ ′ ) x x − ( b θ − b θ ′ ) ( u ˙ θ ′ ) x , R ( 1 , ⋅ ) = 0. \left(\partial_q+\frac{c_\theta}{2}\partial_{xx}+b_\theta\partial_x\right)R=-(F_\theta-F_{\theta'})-\frac{c_\theta-c_{\theta'}}{2}(\dot{u}_{\theta'})_{xx}-(b_\theta-b_{\theta'})(\dot{u}_{\theta'})_x,\qquad R(1,\cdot)=0. ( ∂ q + 2 c θ ∂ xx + b θ ∂ x ) R = − ( F θ − F θ ′ ) − 2 c θ − c θ ′ ( u ˙ θ ′ ) xx − ( b θ − b θ ′ ) ( u ˙ θ ′ ) x , R ( 1 , ⋅ ) = 0. Since c θ − c θ ′ = ( θ − θ ′ ) d φ ′ c_\theta-c_{\theta'}=(\theta-\theta')d\varphi' c θ − c θ ′ = ( θ − θ ′ ) d φ ′ u θ − u θ ′ u_\theta-u_{\theta'} u θ − u θ ′ O ( ∣ θ − θ ′ ∣ ) O(|\theta-\theta'|) O ( ∣ θ − θ ′ ∣ ) ∂ θ ∂ x j u θ \partial_\theta\partial_x^j u_\theta ∂ θ ∂ x j u θ O ( ∣ θ − θ ′ ∣ ) O(|\theta-\theta'|) O ( ∣ θ − θ ′ ∣ ) sup q , x ∣ ∂ x j R ∣ ≤ C j ∣ θ − θ ′ ∣ \sup_{q,x}|\partial_x^jR|\le C_j|\theta-\theta'| sup q , x ∣ ∂ x j R ∣ ≤ C j ∣ θ − θ ′ ∣ ∂ x j u ˙ θ \partial_x^j\dot{u}_\theta ∂ x j u ˙ θ θ \theta θ ( q , x ) (q,x) ( q , x ) q ↦ q + θ 1 φ 1 ( q ) + θ 2 φ 2 ( q ) q\mapsto q+\theta_1\varphi_1(q)+\theta_2\varphi_2(q) q ↦ q + θ 1 φ 1 ( q ) + θ 2 φ 2 ( q )
Step 2: the variation of the solution. We write u ˙ = u ˙ 0 \dot{u}=\dot{u}_0 u ˙ = u ˙ 0 θ = 0 \theta=0 θ = 0 (112) and (91) , with Z = Y Z=Y Z = Y
u ˙ ( q , y ) = d 2 ∫ q 1 φ ′ ( r ) Ψ q ( r ; y ) d r , Ψ q ( r ; y ) = E q , y [ h r + a ( r ) m r 2 ] , (113) \dot{u}(q,y)=\frac{d}{2}\int_q^1\varphi'(r)\Psi_q(r;y)\,dr,\qquad\Psi_q(r;y)=\mathbb{E}_{q,y}[h_r+a(r)m_r^2], \tag*{(113)} u ˙ ( q , y ) = 2 d ∫ q 1 φ ′ ( r ) Ψ q ( r ; y ) d r , Ψ q ( r ; y ) = E q , y [ h r + a ( r ) m r 2 ] , (113) where E q , y \mathbb{E}_{q,y} E q , y y y y q q q Ψ q ( r ; y ) \Psi_q(r;y) Ψ q ( r ; y ) Y Y Y (89) twice in x x x u x x ( r , Y r ) u_{xx}(r,Y_r) u xx ( r , Y r )
d h r = − d a ( r ) h r 2 d r + d u x x x ( r , Y r ) d B r , dh_r=-d\,a(r)h_r^2\,dr+\sqrt{d}\,u_{xxx}(r,Y_r)\,dB_r, d h r = − d a ( r ) h r 2 d r + d u xxx ( r , Y r ) d B r , and by (9.6), d ( m r 2 ) = d h r 2 d r + 2 d m r h r d B r d(m_r^2)=d h_r^2\,dr+2\sqrt{d}\,m_rh_r\,dB_r d ( m r 2 ) = d h r 2 d r + 2 d m r h r d B r m r 2 m_r^2 m r 2 a a a d ( a ( r ) m r 2 ) = a ( r − ) d ( m r 2 ) + m r 2 μ ( d r ) d(a(r)m_r^2)=a(r-)\,d(m_r^2)+m_r^2\,\mu(dr) d ( a ( r ) m r 2 ) = a ( r − ) d ( m r 2 ) + m r 2 μ ( d r ) ( 0 , 1 ] (0,1] ( 0 , 1 ] a ( r − ) = a ( r ) a(r-)=a(r) a ( r − ) = a ( r ) r r r ∓ d a ( r ) h r 2 d r \mp d a(r)h_r^2\,dr ∓ d a ( r ) h r 2 d r
d ( h r + a ( r ) m r 2 ) = d [ u x x x ( r , Y r ) + 2 a ( r ) m r h r ] d B r + m r 2 μ ( d r ) . d\bigl(h_r+a(r)m_r^2\bigr)=\sqrt{d}\bigl[u_{xxx}(r,Y_r)+2a(r)m_rh_r\bigr]\,dB_r+m_r^2\,\mu(dr). d ( h r + a ( r ) m r 2 ) = d [ u xxx ( r , Y r ) + 2 a ( r ) m r h r ] d B r + m r 2 μ ( d r ) . Taking conditional expectations given Y q = y Y_q=y Y q = y μ = ν \mu=\nu μ = ν ( q , r ] ⊂ ( 0 , 1 ] (q,r]\subset(0,1] ( q , r ] ⊂ ( 0 , 1 ]
Ψ q ( r ; y ) − Ψ q ( q ; y ) = ∫ ( q , r ] ξ q v ( y ) ν ( d v ) , q ≤ r ≤ 1 , (114) \Psi_q(r;y)-\Psi_q(q;y)=\int_{(q,r]}\xi_q^v(y)\,\nu(dv),\qquad q\le r\le1, \tag*{(114)} Ψ q ( r ; y ) − Ψ q ( q ; y ) = ∫ ( q , r ] ξ q v ( y ) ν ( d v ) , q ≤ r ≤ 1 , (114) where Ψ q ( q ; y ) = u x x ( q , y ) + a ( q ) u x ( q , y ) 2 \Psi_q(q;y)=u_{xx}(q,y)+a(q)u_x(q,y)^2 Ψ q ( q ; y ) = u xx ( q , y ) + a ( q ) u x ( q , y ) 2
We insert (114) into (113) and exchange the order of integration over q < v ≤ r ≤ 1 q<v\le r\le1 q < v ≤ r ≤ 1 ∫ q 1 ϕ ′ ( r ) d r = − ϕ ( q ) \int_q^1\phi'(r)\,dr=-\phi(q) ∫ q 1 ϕ ′ ( r ) d r = − ϕ ( q ) ∫ v 1 ϕ ′ ( r ) d r = − ϕ ( v ) \int_v^1\phi'(r)\,dr=-\phi(v) ∫ v 1 ϕ ′ ( r ) d r = − ϕ ( v ) ϕ ( 1 ) = 0 \phi(1)=0 ϕ ( 1 ) = 0
u ˙ ( q , y ) = − d 2 [ ϕ ( q ) ( u x x ( q , y ) + a ( q ) u x ( q , y ) 2 ) + ∫ ( q , 1 ] ϕ ( r ) ξ q r ( y ) ν ( d r ) ] . (115) \dot{u}(q,y)=-\frac{d}{2}\left[\phi(q)\bigl(u_{xx}(q,y)+a(q)u_x(q,y)^2\bigr)+\int_{(q,1]}\phi(r)\xi_q^r(y)\,\nu(dr)\right]. \tag*{(115)} u ˙ ( q , y ) = − 2 d [ ϕ ( q ) ( u xx ( q , y ) + a ( q ) u x ( q , y ) 2 ) + ∫ ( q , 1 ] ϕ ( r ) ξ q r ( y ) ν ( d r ) ] . (115) An atom at q q q a ( q ) a(q) a ( q ) ( q , 1 ] (q,1] ( q , 1 ] 1 1 1 ϕ ( 1 ) \phi(1) ϕ ( 1 )
The same computation holds at every base point θ \theta θ Y θ Y^\theta Y θ Y Y Y d d d c θ ( r ) c_\theta(r) c θ ( r ) ∓ c θ ( r ) a ( r ) h r 2 d r \mp c_\theta(r)a(r)h_r^2\,dr ∓ c θ ( r ) a ( r ) h r 2 d r (112) still carries the factor d ϕ ′ d\phi' d ϕ ′ c θ c_\theta c θ θ \theta θ ( q , y ) = ( 0 , 0 ) (q,y)=(0,0) ( q , y ) = ( 0 , 0 ) ϕ ( 0 ) = 0 \phi(0)=0 ϕ ( 0 ) = 0 E [ ∂ x u θ ( r , Y 0 θ ) 2 ∣ Y 0 θ = 0 ] = Γ ~ θ ( r ) \mathbb{E}[\partial_xu_\theta(r,Y_0^\theta)^2\mid Y_0^\theta=0]=\widetilde{\Gamma}_\theta(r) E [ ∂ x u θ ( r , Y 0 θ ) 2 ∣ Y 0 θ = 0 ] = Γ θ ( r )
d d θ u θ ( 0 , 0 ) = − d 2 ∫ ϕ ( r ) Γ ~ θ ( r ) ν ( d r ) , ∣ θ ∣ < θ 0 . (116) \frac{d}{d\theta}u_\theta(0,0)=-\frac{d}{2}\int\phi(r)\widetilde{\Gamma}_\theta(r)\,\nu(dr),\qquad|\theta|<\theta_0. \tag*{(116)} d θ d u θ ( 0 , 0 ) = − 2 d ∫ ϕ ( r ) Γ θ ( r ) ν ( d r ) , ∣ θ ∣ < θ 0 . (116) Step 3: Gaussian differentiation at the fixed profile. Let B B B X q θ = B d T θ ( q ) X_q^\theta=B_{dT_\theta(q)} X q θ = B d T θ ( q ) K θ ( q , r ) = d T θ ( q ∧ r ) K_\theta(q,r)=dT_\theta(q\wedge r) K θ ( q , r ) = d T θ ( q ∧ r ) ∫ 0 q c θ ( r ) d B r \int_0^q\sqrt{c_\theta(r)}\,dB_r ∫ 0 q c θ ( r ) d B r c θ c_\theta c θ ( Y θ , ϵ ) (Y^\theta,\epsilon) ( Y θ , ϵ ) u θ u_\theta u θ u u u X θ X^\theta X θ e − s u θ ( 0 , 0 ) e^{-s u_\theta(0,0)} e − s u θ ( 0 , 0 )
ℓ θ ( X , ϵ ) = ϵ X 1 − ∫ u θ ( r , X r ) ν ( d r ) , Φ θ ( X ) = u θ , x ( p , X p ) 2 , \ell_\theta(X,\epsilon)=\epsilon X_1-\int u_\theta(r,X_r)\,\nu(dr),\qquad\Phi_\theta(X)=u_{\theta,x}(p,X_p)^2, ℓ θ ( X , ϵ ) = ϵ X 1 − ∫ u θ ( r , X r ) ν ( d r ) , Φ θ ( X ) = u θ , x ( p , X p ) 2 , we have Γ ~ θ ( p ) = E θ Φ θ \widetilde{\Gamma}_\theta(p)=\mathbb{E}_\theta\Phi_\theta Γ θ ( p ) = E θ Φ θ E θ \mathbb{E}_\theta E θ W θ = e ℓ θ W_\theta=e^{\ell_\theta} W θ = e ℓ θ Lemma 4.3 . At θ = 0 \theta=0 θ = 0 X X X E 0 \mathbb{E}_0 E 0 Y Y Y § 8.5.1 , we write m r = u x ( r , X r ) m_r=u_x(r,X_r) m r = u x ( r , X r ) h r = u x x ( r , X r ) h_r=u_{xx}(r,X_r) h r = u xx ( r , X r ) u 3 , r = u x x x ( r , X r ) u_{3,r}=u_{xxx}(r,X_r) u 3 , r = u xxx ( r , X r )
We check that the assumptions (A1)–(A4) of Lemma 4.3 hold, with z = θ z=\theta z = θ I = [ − θ 0 / 2 , θ 0 / 2 ] I=[-\theta_0/2,\theta_0/2] I = [ − θ 0 /2 , θ 0 /2 ] d T θ ( q ) dT_\theta(q) d T θ ( q ) ( q , θ ) (q,\theta) ( q , θ ) θ \theta θ q q q ϵ \epsilon ϵ K θ K_\theta K θ K ˙ θ ( q , r ) = d φ ( q ∧ r ) \dot K_\theta(q,r)=d\varphi(q\wedge r) K ˙ θ ( q , r ) = d φ ( q ∧ r ) ( θ , q , r ) (\theta,q,r) ( θ , q , r ) Lemma 4.3 , where M M M Q Q Q ℓ θ \ell_\theta ℓ θ Φ θ \Phi_\theta Φ θ
S = ϵ δ 1 − u θ , x ( r , X r ) ν ( d r ) , M = − ι # ( u θ , x x ( r , X r ) ν ( d r ) ) , S=\epsilon\delta_1-u_{\theta,x}(r,X_r)\,\nu(dr),\qquad M=-\iota_{\#}\bigl(u_{\theta,xx}(r,X_r)\,\nu(dr)\bigr), S = ϵ δ 1 − u θ , x ( r , X r ) ν ( d r ) , M = − ι # ( u θ , xx ( r , X r ) ν ( d r ) ) , P = 2 u θ , x u θ , x x ( p , X p ) δ p , Q = 2 ( u θ , x x 2 + u θ , x u θ , x x x ) ( p , X p ) δ ( p , p ) . P=2u_{\theta,x}u_{\theta,xx}(p,X_p)\,\delta_p,\qquad Q=2\bigl(u_{\theta,xx}^{2}+u_{\theta,x}u_{\theta,xxx}\bigr)(p,X_p)\,\delta_{(p,p)}. P = 2 u θ , x u θ , xx ( p , X p ) δ p , Q = 2 ( u θ , xx 2 + u θ , x u θ , xxx ) ( p , X p ) δ ( p , p ) . where ι ( r ) = ( r , r ) \iota(r)=(r,r) ι ( r ) = ( r , r ) M M M ( r , r ) (r,r) ( r , r ) r r r ν \nu ν Lemma 9.1 (b), we see that these measures have total variation bounded uniformly in θ \theta θ X X X ( θ , X ) (\theta,X) ( θ , X ) X X X X ↦ ℓ θ ( X , ϵ ) X\mapsto\ell_\theta(X,\epsilon) X ↦ ℓ θ ( X , ϵ ) X ↦ Φ θ ( X ) X\mapsto\Phi_\theta(X) X ↦ Φ θ ( X ) u θ u_\theta u θ Φ θ \Phi_\theta Φ θ [ 0 , 1 ] [0,1] [ 0 , 1 ]
Φ ˙ θ = 2 u θ , x ( p , X p ) ∂ x u ˙ θ ( p , X p ) , ℓ ˙ θ = − ∫ u ˙ θ ( r , X r ) ν ( d r ) \dot\Phi_\theta=2u_{\theta,x}(p,X_p)\,\partial_x\dot u_\theta(p,X_p),\qquad\dot\ell_\theta=-\int\dot u_\theta(r,X_r)\,\nu(dr) Φ ˙ θ = 2 u θ , x ( p , X p ) ∂ x u ˙ θ ( p , X p ) , ℓ ˙ θ = − ∫ u ˙ θ ( r , X r ) ν ( d r ) are bounded and jointly continuous in ( θ , X ) (\theta,X) ( θ , X ) ∣ u θ , x ∣ ≤ 1 |u_{\theta,x}|\le1 ∣ u θ , x ∣ ≤ 1 u θ ( r , 0 ) u_\theta(r,0) u θ ( r , 0 ) ∣ ℓ θ ∣ ≤ C + 2 sup r ∣ X r ∣ |\ell_\theta|\le C+2\sup_r|X_r| ∣ ℓ θ ∣ ≤ C + 2 sup r ∣ X r ∣ Lemma 4.3 applies to E θ Φ θ \mathbb{E}_\theta\Phi_\theta E θ Φ θ (38) is continuous in θ \theta θ θ ↦ Γ ~ θ ( p ) \theta\mapsto\widetilde{\Gamma}_\theta(p) θ ↦ Γ θ ( p ) ( − θ 0 / 2 , θ 0 / 2 ) (-\theta_0/2,\theta_0/2) ( − θ 0 /2 , θ 0 /2 ) β \beta β φ \varphi φ p ∈ [ 0 , 1 ] p\in[0,1] p ∈ [ 0 , 1 ] θ \theta θ
Step 4: the terms of the identity. We evaluate the right side of (38) at θ = 0 \theta=0 θ = 0 d d d R R R [ 0 , 1 ] 2 [0,1]^2 [ 0 , 1 ] 2 d − 1 ⟨ K ˙ 0 , R ⟩ = ∬ φ ( q ∧ r ) R ( d q , d r ) d^{-1}\langle\dot K_0,R\rangle=\iint\varphi(q\wedge r)R(dq,dr) d − 1 ⟨ K ˙ 0 , R ⟩ = ∬ φ ( q ∧ r ) R ( d q , d r ) E 0 \mathbb{E}_0 E 0 Φ 0 − E 0 Φ 0 = f ˉ p \Phi_0-\mathbb{E}_0\Phi_0=\bar f_p Φ 0 − E 0 Φ 0 = f ˉ p
The second derivative of the observable contributes
I 1 = 1 2 d E 0 ⟨ K ˙ 0 , Q ⟩ = φ ( p ) E 0 [ h p 2 + m p u 3 , p ] . I_1=\frac{1}{2d}\mathbb{E}_0\langle\dot K_0,Q\rangle=\varphi(p)\mathbb{E}^0\left[h_p^2+m_pu_{3,p}\right]. I 1 = 2 d 1 E 0 ⟨ K ˙ 0 , Q ⟩ = φ ( p ) E 0 [ h p 2 + m p u 3 , p ] . The explicit derivative of the observable is Φ ˙ 0 = 2 m p ∂ y u ˙ ( p , X p ) \dot\Phi_0=2m_p\partial_y\dot u(p,X_p) Φ ˙ 0 = 2 m p ∂ y u ˙ ( p , X p ) (115) in y y y (105) , we find
I 2 = 1 d E 0 Φ ˙ 0 = − φ ( p ) E 0 [ m p u 3 , p + 2 a ( p ) m p 2 h p ] − ∫ ( p , 1 ] φ ( ℓ ) E 0 [ m p ∂ y g p 0 ( Y p ) ] ν ( d ℓ ) . I_2=\frac{1}{d}\mathbb{E}_0\dot\Phi_0=-\varphi(p)\mathbb{E}^0\left[m_pu_{3,p}+2a(p)m_p^2h_p\right]-\int_{(p,1]}\varphi(\ell)\mathbb{E}^0\left[m_p\partial_y g_p^0(Y_p)\right]\nu(d\ell). I 2 = d 1 E 0 Φ ˙ 0 = − φ ( p ) E 0 [ m p u 3 , p + 2 a ( p ) m p 2 h p ] − ∫ ( p , 1 ] φ ( ℓ ) E 0 [ m p ∂ y g p 0 ( Y p ) ] ν ( d ℓ ) . Since K ˙ 0 \dot{K}_0 K ˙ 0 K ˙ 0 ( 1 , p ) = d ϕ ( p ) \dot{K}_0(1,p)=d\phi(p) K ˙ 0 ( 1 , p ) = d ϕ ( p )
I 3 = 1 2 d E 0 ⟨ K ˙ 0 , S ⊗ P + P ⊗ S ⟩ = 1 d E 0 ⟨ K ˙ 0 , S ⊗ P ⟩ = 2 ϕ ( p ) E 0 [ m p h p ϵ ] − 2 ∫ ϕ ( p ∧ r ) E 0 [ m p h p m r ] ν ( d r ) . \begin{aligned}
I_3=\frac{1}{2d}\mathbb{E}_0\langle\dot{K}_0,S\otimes P+P\otimes S\rangle=\frac{1}{d}\mathbb{E}_0\langle\dot{K}_0,S\otimes P\rangle\\
&=2\phi(p)\mathbb{E}^0[m_ph_p\epsilon]-2\int\phi(p\wedge r)\mathbb{E}^0[m_ph_pm_r]\,\nu(\mathrm{d}r).
\end{aligned} I 3 = 2 d 1 E 0 ⟨ K ˙ 0 , S ⊗ P + P ⊗ S ⟩ = d 1 E 0 ⟨ K ˙ 0 , S ⊗ P ⟩ = 2 ϕ ( p ) E 0 [ m p h p ϵ ] − 2 ∫ ϕ ( p ∧ r ) E 0 [ m p h p m r ] ν ( d r ) . The product of the scores contributes the following; the terminal–terminal term vanishes, since ϕ ( 1 ) = 0 \phi(1)=0 ϕ ( 1 ) = 0
I 4 = 1 2 d E 0 [ f ˉ p ⟨ K ˙ 0 , S ⊗ S ⟩ ] = − ∫ ϕ ( r ) E 0 [ f ˉ p ϵ m r ] ν ( d r ) + 1 2 ∬ ϕ ( r ∧ ℓ ) E 0 [ f ˉ p m r m ℓ ] ν ( d r ) ν ( d ℓ ) . \begin{aligned}
I_4=\frac{1}{2d}\mathbb{E}_0[\bar{f}_p\langle\dot{K}_0,S\otimes S\rangle] \\
&=-\int\phi(r)\mathbb{E}^0[\bar{f}_p\epsilon m_r]\,\nu(\mathrm{d}r)+\frac{1}{2}\iint\phi(r\wedge\ell)\mathbb{E}^0[\bar{f}_pm_rm_\ell]\,\nu(\mathrm{d}r)\nu(\mathrm{d}\ell).
\end{aligned} I 4 = 2 d 1 E 0 [ f ˉ p ⟨ K ˙ 0 , S ⊗ S ⟩] = − ∫ ϕ ( r ) E 0 [ f ˉ p ϵ m r ] ν ( d r ) + 2 1 ∬ ϕ ( r ∧ ℓ ) E 0 [ f ˉ p m r m ℓ ] ν ( d r ) ν ( d ℓ ) . The Hessian of the logarithmic weight contributes
I 5 = 1 2 d E 0 [ f ˉ p ⟨ K 0 , M ⟩ ] = − 1 2 ∫ ϕ ( r ) E 0 [ f ˉ p h r ] ν ( d r ) . I_5=\frac{1}{2d}\mathbb{E}_0[\bar{f}_p\langle K_0,M\rangle]=-\frac{1}{2}\int\phi(r)\mathbb{E}^0[\bar{f}_ph_r]\,\nu(\mathrm{d}r). I 5 = 2 d 1 E 0 [ f ˉ p ⟨ K 0 , M ⟩] = − 2 1 ∫ ϕ ( r ) E 0 [ f ˉ p h r ] ν ( d r ) . The explicit derivative of the weight is ℓ ˙ 0 = − ∫ u ˙ ( r , X r ) ν ( d r ) \dot{\ell}_0=-\int\dot{u}(r,X_r)\,\nu(\mathrm{d}r) ℓ ˙ 0 = − ∫ u ˙ ( r , X r ) ν ( d r ) (115) , we find
I 6 = 1 d Cov 0 ( Φ 0 , ℓ ˙ 0 ) = 1 2 ∫ ϕ ( r ) E 0 [ f ˉ p ( h r + a ( r ) m r 2 ) ] ν ( d r ) + 1 2 ∫ ν ( d r ) ∫ ( r , 1 ] ϕ ( ℓ ) E 0 [ f ˉ p g r ℓ ( Y r ) ] ν ( d ℓ ) . \begin{aligned}
I_6=\frac{1}{d}\operatorname{Cov}_0(\Phi_0,\dot{\ell}_0)=\frac{1}{2}\int\phi(r)\mathbb{E}^0[\bar{f}_p(h_r+a(r)m_r^2)]\,\nu(\mathrm{d}r) \\
&+\frac{1}{2}\int\nu(\mathrm{d}r)\int_{(r,1]}\phi(\ell)\mathbb{E}^0[\bar{f}_pg_r^\ell(Y_r)]\,\nu(\mathrm{d}\ell).
\end{aligned} I 6 = d 1 Cov 0 ( Φ 0 , ℓ ˙ 0 ) = 2 1 ∫ ϕ ( r ) E 0 [ f ˉ p ( h r + a ( r ) m r 2 )] ν ( d r ) + 2 1 ∫ ν ( d r ) ∫ ( r , 1 ] ϕ ( ℓ ) E 0 [ f ˉ p g r ℓ ( Y r )] ν ( d ℓ ) . By (38) , we have
1 d d d θ Γ ~ θ ( p ) ∣ θ = 0 = I 1 + ⋯ + I 6 . \left.\frac{1}{d}\frac{\mathrm{d}}{\mathrm{d}\theta}\widetilde{\Gamma}_\theta(p)\right|_{\theta=0}=I_1+\cdots+I_6. d 1 d θ d Γ θ ( p ) θ = 0 = I 1 + ⋯ + I 6 . Conditioning on G p 0 \mathcal{G}_p^0 G p 0 (93) , we have E 0 [ m p h p ϵ ] = E 0 [ m p 2 h p ] \mathbb{E}^0[m_ph_p\epsilon]=\mathbb{E}^0[m_p^2h_p] E 0 [ m p h p ϵ ] = E 0 [ m p 2 h p ] I 1 , … , I 6 I_1,\ldots,I_6 I 1 , … , I 6 i i i i i i (106) .
Table 1.
In summary, the terms with m p μ 3 , p m_p\mu_{3,p} m p μ 3 , p h r h_r h r E 0 [ h p 2 − 2 a ( p ) m p 2 h p + 2 m p 2 h p ] = E 0 [ h p 2 + 2 ( 1 − a ( p ) ) m p 2 h p ] \mathbb{E}^0[h_p^2-2a(p)m_p^2h_p+2m_p^2h_p]=\mathbb{E}^0[h_p^2+2(1-a(p))m_p^2h_p] E 0 [ h p 2 − 2 a ( p ) m p 2 h p + 2 m p 2 h p ] = E 0 [ h p 2 + 2 ( 1 − a ( p )) m p 2 h p ] I 1 + ⋯ + I 6 I_1+\cdots+I_6 I 1 + ⋯ + I 6 (106) , including its signs, its factors 1 / 2 1/2 1/2
Domination by the scalar multiplier
We now show that the scalar response operator is dominated by its multiplier. We use the scalar Parisi functional with coefficient d d d
P d ( a ) = u a , d ( 0 , 0 ) − d 2 ∫ 0 1 q a ( q ) d q , \mathcal{P}_d(a)=u_{a,d}(0,0)-\frac{d}{2}\int_0^1q\,a(q)\,\mathrm{d}q, P d ( a ) = u a , d ( 0 , 0 ) − 2 d ∫ 0 1 q a ( q ) d q , defined on the convex set of profiles. For d = β 2 d=\beta^2 d = β 2 P β \mathcal{P}_\beta P β
Lemma 9.7 (Scalar multiplier domination). For every profile a a a d ∈ [ β 2 / 2 , β 2 ] d\in[\beta^2/2,\beta^2] d ∈ [ β 2 /2 , β 2 ] J a , d s c J^{\mathrm{sc}}_{a,d} J a , d sc H ν \mathcal{H}_\nu H ν C β C_\beta C β
J a , d s c ⪯ M k a , d , ( M k a , d v ) ( q ) = k a , d ( q ) v ( q ) . (117) J^{\mathrm{sc}}_{a,d}\preceq M_{k_{a,d}},\qquad(M_{k_{a,d}}v)(q)=k_{a,d}(q)v(q). \tag*{(117)} J a , d sc ⪯ M k a , d , ( M k a , d v ) ( q ) = k a , d ( q ) v ( q ) . (117) Proof. The bound on the norm is (109) . The proof of the rest has five steps. We first recall that P d \mathcal{P}_d P d P d \mathcal{P}_d P d a a a (117) for smooth directions. Two-parameter transports then give the symmetry, and a density argument extends both conclusions to H ν \mathcal{H}_\nu H ν a a a Γ = Γ a , d \Gamma=\Gamma_{a,d} Γ = Γ a , d k = k a , d k=k_{a,d} k = k a , d ⟨ ⋅ , ⋅ ⟩ \langle\cdot,\cdot\rangle ⟨ ⋅ , ⋅ ⟩ L 2 ( ν ) L^2(\nu) L 2 ( ν )
Step 1: convexity and the first variation. We first note that P d \mathcal{P}_d P d [10 Theorem 2] , applied with ξ ( q ) = d q 2 / 2 \xi(q)=dq^2/2 ξ ( q ) = d q 2 /2 ( 0 , 0 ) (0,0) ( 0 , 0 ) log cosh x \log\cosh x log cosh x u a , d ( 0 , 0 ) u_{a,d}(0,0) u a , d ( 0 , 0 ) log 2 \log2 log 2 P d \mathcal{P}_d P d a a a
The first variation of P d \mathcal{P}_d P d [4 (2.8)] (see also [44, Lemma 3.2] and [25, Theorem 2]): for profiles a a a a ~ \widetilde a a
lim ϑ ↓ 0 P d ( a + ϑ ( a ~ − a ) ) − P d ( a ) ϑ = d 2 ∫ 0 1 ( Γ ( q ) − q ) ( a ~ ( q ) − a ( q ) ) d q . (118) \lim_{\vartheta\downarrow0}\frac{\mathcal{P}_d(a+\vartheta(\widetilde a-a))-\mathcal{P}_d(a)}{\vartheta}
=\frac{d}{2}\int_0^1(\Gamma(q)-q)(\widetilde a(q)-a(q))\,\mathrm{d}q. \tag*{(118)} ϑ ↓ 0 lim ϑ P d ( a + ϑ ( a − a )) − P d ( a ) = 2 d ∫ 0 1 ( Γ ( q ) − q ) ( a ( q ) − a ( q )) d q . (118) The statement in [4 ] assumes a positive root mass. We note that the argument of Step 1 of the proof of Lemma 9.6, with the coefficient fixed and the profile varied, gives a direct proof of (118) for all profiles. The difference quotients of u a + ϑ ( a ~ − a ) , d u_{a+\vartheta(\tilde a-a),d} u a + ϑ ( a ~ − a ) , d u ˙ \dot u u ˙
∂ q u ˙ + d 2 u ˙ x x + d a u x u ˙ x = − d 2 ( a ~ − a ) u x 2 , u ˙ ( 1 , ⋅ ) = 0 , \partial_q\dot u+\frac{d}{2}\dot u_{xx}+da u_x\dot u_x=-\frac{d}{2}(\tilde a-a)u_x^2,\qquad\dot u(1,\cdot)=0, ∂ q u ˙ + 2 d u ˙ xx + d a u x u ˙ x = − 2 d ( a ~ − a ) u x 2 , u ˙ ( 1 , ⋅ ) = 0 , and (91) gives
u ˙ ( 0 , 0 ) = d 2 ∫ 0 1 ( a ~ ( q ) − a ( q ) ) E 0 m q 2 d q = d 2 ∫ 0 1 ( a ~ ( q ) − a ( q ) ) Γ ( q ) d q . \dot u(0,0)=\frac{d}{2}\int_0^1(\tilde a(q)-a(q))\mathbb{E}^0m_q^2\,dq=\frac{d}{2}\int_0^1(\tilde a(q)-a(q))\Gamma(q)\,dq. u ˙ ( 0 , 0 ) = 2 d ∫ 0 1 ( a ~ ( q ) − a ( q )) E 0 m q 2 d q = 2 d ∫ 0 1 ( a ~ ( q ) − a ( q )) Γ ( q ) d q . The penalty contributes − ( d / 2 ) ∫ 0 1 q ( a ~ ( q ) − a ( q ) ) d q -(d/2)\int_0^1q(\tilde a(q)-a(q))\,dq − ( d /2 ) ∫ 0 1 q ( a ~ ( q ) − a ( q )) d q P d \mathcal{P}_d P d (118) is nondecreasing in ϑ ∈ ( 0 , 1 ] \vartheta\in(0,1] ϑ ∈ ( 0 , 1 ] ϑ = 1 \vartheta=1 ϑ = 1 ϑ ↓ 0 \vartheta\downarrow0 ϑ ↓ 0
P d ( a ~ ) − P d ( a ) ≥ d 2 ∫ 0 1 ( Γ ( q ) − q ) ( a ~ ( q ) − a ( q ) ) d q . (119) \mathcal{P}_d(\tilde a)-\mathcal{P}_d(a)\geq\frac{d}{2}\int_0^1(\Gamma(q)-q)(\tilde a(q)-a(q))\,dq. \tag*{(119)} P d ( a ~ ) − P d ( a ) ≥ 2 d ∫ 0 1 ( Γ ( q ) − q ) ( a ~ ( q ) − a ( q )) d q . (119) Step 2: the second derivative along a transport. Let φ \varphi φ T θ T_\theta T θ θ 0 \theta_0 θ 0 a θ = a ∘ T θ − 1 a_\theta=a\circ T_\theta^{-1} a θ = a ∘ T θ − 1 § 8.5.3 , and let Π ( θ ) = P d ( a θ ) \Pi(\theta)=\mathcal{P}_d(a_\theta) Π ( θ ) = P d ( a θ ) ∣ θ ∣ < θ 0 / 2 |\theta|<\theta_0/2 ∣ θ ∣ < θ 0 /2 (110) at q = 0 q=0 q = 0 u a θ , d ( 0 , 0 ) = u θ ( 0 , 0 ) u_{a_\theta,d}(0,0)=u_\theta(0,0) u a θ , d ( 0 , 0 ) = u θ ( 0 , 0 )
∫ 0 1 x a θ ( x ) d x = ∫ ( ∫ T θ ( q ) 1 x d x ) μ ( d q ) = 1 2 − 1 2 ∫ T θ ( q ) 2 μ ( d q ) , \int_0^1x a_\theta(x)\,dx=\int\left(\int_{T_\theta(q)}^1x\,dx\right)\mu(dq)=\frac{1}{2}-\frac{1}{2}\int T_\theta(q)^2\,\mu(dq), ∫ 0 1 x a θ ( x ) d x = ∫ ( ∫ T θ ( q ) 1 x d x ) μ ( d q ) = 2 1 − 2 1 ∫ T θ ( q ) 2 μ ( d q ) , which gives
Π ( θ ) = u θ ( 0 , 0 ) − d 4 + d 4 ∫ T θ 2 d μ . \Pi(\theta)=u_\theta(0,0)-\frac{d}{4}+\frac{d}{4}\int T_\theta^2\,d\mu. Π ( θ ) = u θ ( 0 , 0 ) − 4 d + 4 d ∫ T θ 2 d μ . The derivative of the last term is ( d / 2 ) ∫ T θ φ d μ (d/2)\int T_\theta\varphi\,d\mu ( d /2 ) ∫ T θ φ d μ s δ 0 s\delta_0 s δ 0 φ ( 0 ) = 0 \varphi(0)=0 φ ( 0 ) = 0 (116) , we obtain
Π ′ ( θ ) = d 2 ∫ φ ( T θ − Γ ~ θ ) d ν , Π ′ ′ ( 0 ) = d 2 ⟨ φ , ( I − J a , d ) φ ⟩ . (120) \Pi'(\theta)=\frac{d}{2}\int\varphi(T_\theta-\widetilde{\Gamma}_\theta)\,d\nu,\qquad\Pi''(0)=\frac{d}{2}\langle\varphi,(I-J_{a,d})\varphi\rangle. \tag*{(120)} Π ′ ( θ ) = 2 d ∫ φ ( T θ − Γ θ ) d ν , Π ′′ ( 0 ) = 2 d ⟨ φ , ( I − J a , d ) φ ⟩ . (120) For the second identity, we used Lemma 9.6 and differentiated under the integral over ν \nu ν ∂ θ Γ ~ θ ( p ) \partial_\theta\widetilde{\Gamma}_\theta(p) ∂ θ Γ θ ( p ) θ \theta θ p p p θ \theta θ Π ′ ′ \Pi'' Π ′′ θ = 0 \theta=0 θ = 0 (120) is the first variation along transports in [9 Section 3, (19)] , where it is credited to [79, Lemma 3.7, Eq. (3.34)].
Step 3: the lower bound from convexity. Let G ( y ) = ∫ y 1 ( Γ ( q ) − q ) d q G(y)=\int_y^1(\Gamma(q)-q)\,dq G ( y ) = ∫ y 1 ( Γ ( q ) − q ) d q a θ ( q ) = μ ( { y : T θ ( y ) ≤ q } ) a_\theta(q)=\mu(\{y:T_\theta(y)\leq q\}) a θ ( q ) = μ ({ y : T θ ( y ) ≤ q }) ∫ 0 1 ( Γ ( q ) − q ) a θ ( q ) d q = ∫ G ( T θ y ) μ ( d y ) \int_0^1(\Gamma(q)-q)a_\theta(q)\,dq=\int G(T_\theta y)\,\mu(dy) ∫ 0 1 ( Γ ( q ) − q ) a θ ( q ) d q = ∫ G ( T θ y ) μ ( d y ) (119) with a ~ = a θ \tilde a=a_\theta a ~ = a θ
Π ( θ ) − Π ( 0 ) ≥ d 2 ∫ ( G ( T θ y ) − G ( y ) ) μ ( d y ) = d 2 ∫ ( G ( T θ y ) − G ( y ) ) ν ( d y ) , \Pi(\theta)-\Pi(0)\geq\frac{d}{2}\int(G(T_\theta y)-G(y))\,\mu(dy)=\frac{d}{2}\int(G(T_\theta y)-G(y))\,\nu(dy), Π ( θ ) − Π ( 0 ) ≥ 2 d ∫ ( G ( T θ y ) − G ( y )) μ ( d y ) = 2 d ∫ ( G ( T θ y ) − G ( y )) ν ( d y ) , where the atom s δ 0 s\delta_0 s δ 0 T θ ( 0 ) = 0 T_\theta(0)=0 T θ ( 0 ) = 0 G G G [ 0 , 1 ] [0,1] [ 0 , 1 ]
G ′ ( y ) = y − Γ ( y ) , G ′ ′ ( y ) = 1 − k ( y ) . G'(y)=y-\Gamma(y),\qquad G''(y)=1-k(y). G ′ ( y ) = y − Γ ( y ) , G ′′ ( y ) = 1 − k ( y ) . The function G ′ ′ G'' G ′′ φ \varphi φ
G ( T θ y ) − G ( y ) = θ φ ( y ) G ′ ( y ) + θ 2 2 φ ( y ) 2 ( 1 − k ( y ) ) + o ( θ 2 ) , G(T_\theta y)-G(y)=\theta\varphi(y)G'(y)+\frac{\theta^2}{2}\varphi(y)^2(1-k(y))+o(\theta^2), G ( T θ y ) − G ( y ) = θ φ ( y ) G ′ ( y ) + 2 θ 2 φ ( y ) 2 ( 1 − k ( y )) + o ( θ 2 ) , uniformly in y ∈ [ 0 , 1 ] y\in[0,1] y ∈ [ 0 , 1 ] θ = 0 \theta=0 θ = 0 Π ′ ( 0 ) = ( d / 2 ) ∫ φ G ′ d ν \Pi'(0)=(d/2)\int\varphi G'\,d\nu Π ′ ( 0 ) = ( d /2 ) ∫ φ G ′ d ν ν \nu ν
Π ( θ ) − Π ( 0 ) − θ Π ′ ( 0 ) ≥ θ 2 2 ⋅ d 2 ∫ ( 1 − k ) φ 2 d ν + o ( θ 2 ) . \Pi(\theta)-\Pi(0)-\theta\Pi'(0)\ge\frac{\theta^2}{2}\cdot\frac{d}{2}\int(1-k)\varphi^2\,d\nu+o(\theta^2). Π ( θ ) − Π ( 0 ) − θ Π ′ ( 0 ) ≥ 2 θ 2 ⋅ 2 d ∫ ( 1 − k ) φ 2 d ν + o ( θ 2 ) . On the other hand, Π \Pi Π 0 0 0 ( θ 2 / 2 ) Π ′ ′ ( 0 ) + o ( θ 2 ) (\theta^2/2)\Pi''(0)+o(\theta^2) ( θ 2 /2 ) Π ′′ ( 0 ) + o ( θ 2 ) θ 2 / 2 \theta^2/2 θ 2 /2 θ → 0 \theta\to0 θ → 0
Π ′ ′ ( 0 ) ≥ d 2 ∫ ( 1 − k ) φ 2 d ν . \Pi''(0)\ge\frac{d}{2}\int(1-k)\varphi^2\,d\nu. Π ′′ ( 0 ) ≥ 2 d ∫ ( 1 − k ) φ 2 d ν . Comparing this with (9.33), we conclude that
⟨ φ , J a , d φ ⟩ ≤ ⟨ φ , k φ ⟩ . (121) \langle\varphi,J_{a,d}\varphi\rangle\le\langle\varphi,k\varphi\rangle. \tag*{(121)} ⟨ φ , J a , d φ ⟩ ≤ ⟨ φ , k φ ⟩ . (121) Note that Π \Pi Π θ ↦ a θ \theta\mapsto a_\theta θ ↦ a θ 1 − k = G ′ ′ 1-k=G'' 1 − k = G ′′
Step 4: symmetry. We now take two functions φ 1 \varphi_1 φ 1 φ 2 \varphi_2 φ 2 T ( q ) = q + θ 1 φ 1 ( q ) + θ 2 φ 2 ( q ) T(q)=q+\theta_1\varphi_1(q)+\theta_2\varphi_2(q) T ( q ) = q + θ 1 φ 1 ( q ) + θ 2 φ 2 ( q ) Γ ~ θ 1 , θ 2 \widetilde{\Gamma}_{\theta_1,\theta_2} Γ θ 1 , θ 2 Π ( θ 1 , θ 2 ) \Pi(\theta_1,\theta_2) Π ( θ 1 , θ 2 ) ∣ θ 1 ∣ , ∣ θ 2 ∣ |\theta_1|,|\theta_2| ∣ θ 1 ∣ , ∣ θ 2 ∣ Γ ~ θ 1 , θ 2 ( p ) \widetilde{\Gamma}_{\theta_1,\theta_2}(p) Γ θ 1 , θ 2 ( p ) p p p d φ i ′ d\varphi_i' d φ i ′
∂ θ i Π = d 2 ∫ φ i ( T − Γ ~ θ 1 , θ 2 ) d ν , i = 1 , 2. \partial_{\theta_i}\Pi=\frac{d}{2}\int\varphi_i(T-\widetilde{\Gamma}_{\theta_1,\theta_2})\,d\nu,\qquad i=1,2. ∂ θ i Π = 2 d ∫ φ i ( T − Γ θ 1 , θ 2 ) d ν , i = 1 , 2. It follows that Π \Pi Π φ 2 \varphi_2 φ 2 θ 1 = 0 \theta_1=0 θ 1 = 0
∂ θ 2 ∂ θ 1 Π ( 0 , 0 ) = d 2 ⟨ φ 1 , ( I − J a , d ) φ 2 ⟩ . \partial_{\theta_2}\partial_{\theta_1}\Pi(0,0)=\frac{d}{2}\langle\varphi_1,(I-J_{a,d})\varphi_2\rangle. ∂ θ 2 ∂ θ 1 Π ( 0 , 0 ) = 2 d ⟨ φ 1 , ( I − J a , d ) φ 2 ⟩ . By the equality of mixed partial derivatives, we conclude that this bilinear form is symmetric, that is, ⟨ φ 1 , J a , d φ 2 ⟩ = ⟨ φ 2 , J a , d φ 1 ⟩ \langle\varphi_1,J_{a,d}\varphi_2\rangle=\langle\varphi_2,J_{a,d}\varphi_1\rangle ⟨ φ 1 , J a , d φ 2 ⟩ = ⟨ φ 2 , J a , d φ 1 ⟩
Step 5: extension to the transport space. In Steps 3 and 4, we proved the domination inequality and the symmetry for restrictions to [ 0 , 1 ] [0,1] [ 0 , 1 ] H ν \mathcal{H}_{\nu} H ν v ∈ H ν v \in\mathcal{H}_{\nu} v ∈ H ν ψ n \psi_n ψ n ∥ ψ n − v ∥ → 0 \lVert\psi_n-v\rVert\to0 ∥ ψ n − v ∥ → 0 L 2 ( ν ) L^2(\nu) L 2 ( ν ) n ≥ 3 n \ge3 n ≥ 3 χ n \chi_n χ n [ 0 , 1 ] [0,1] [ 0 , 1 ] 0 0 0 1 1 1 [ 1 / n , 1 − 1 / n ] [1/n,1-1/n] [ 1/ n , 1 − 1/ n ] ∥ χ n ψ n − v ∥ ≤ ∥ ψ n − v ∥ + ∥ ( 1 − χ n ) v ∥ \lVert\chi_n\psi_n-v\rVert\le\lVert\psi_n-v\rVert+\lVert(1-\chi_n)v\rVert ∥ χ n ψ n − v ∥ ≤ ∥ ψ n − v ∥ + ∥( 1 − χ n ) v ∥ ∥ ( 1 − χ n ) v ∥ → ∥ 1 { 0 , 1 } v ∥ = 0 \lVert(1-\chi_n)v\rVert\to\lVert\mathbf{1}_{\{0,1\}}v\rVert=0 ∥( 1 − χ n ) v ∥ → ∥ 1 { 0 , 1 } v ∥ = 0 v v v ν \nu ν { 0 , 1 } \{0,1\} { 0 , 1 } f f f f ( 0 ) = f ( 1 ) = 0 f(0)=f(1)=0 f ( 0 ) = f ( 1 ) = 0 P P P ∣ P − f ∣ ≤ τ \lvert P-f\rvert\le\tau ∣ P − f ∣ ≤ τ [ 0 , 1 ] [0,1] [ 0 , 1 ] P ( x ) − P ( 0 ) ( 1 − x ) − P ( 1 ) x P(x)-P(0)(1-x)-P(1)x P ( x ) − P ( 0 ) ( 1 − x ) − P ( 1 ) x 3 τ 3\tau 3 τ f f f f f f v ∈ H ν v \in\mathcal{H}_{\nu} v ∈ H ν ⟨ v , J α , d s c v ⟩ = ⟨ v , J α , d v ⟩ \langle v,J^{\mathrm{sc}}_{\alpha,d}v\rangle=\langle v,J_{\alpha,d}v\rangle ⟨ v , J α , d sc v ⟩ = ⟨ v , J α , d v ⟩ (121) and of the symmetry identity are continuous on L 2 ( ν ) L^2(\nu) L 2 ( ν ) J α , d J_{\alpha,d} J α , d L 2 ( ν ) L^2(\nu) L 2 ( ν ) (109) ) and 0 ≤ k ≤ β 2 0 \le k \le\beta^2 0 ≤ k ≤ β 2 H ν \mathcal{H}_{\nu} H ν k k k H ν \mathcal{H}_{\nu} H ν (117) follows.
The reference multiplier
A finite-dimensional minimizer may place mass outside the scalar support. For this reason, we control the multiplier at the scalar Parisi measure on all of [ 0 , 1 ] [0,1] [ 0 , 1 ] (62) that supp μ β = [ 0 , q β ] \operatorname{supp}\mu_{\beta}=[0,q_{\beta}] supp μ β = [ 0 , q β ] 0 < q β < 1 0<q_{\beta}<1 0 < q β < 1 μ β \mu_{\beta} μ β q β q_{\beta} q β α β = 1 \alpha_{\beta}=1 α β = 1 [ q β , 1 ] [q_{\beta},1] [ q β , 1 ]
Lemma 9.8 (Reference multiplier on the whole interval). At the scalar Parisi measure,
k α β , β 2 ( q ) = 1 ( 0 ≤ q ≤ q β ) , k α β , β 2 ( q ) ≤ e − β 2 ( q − q β ) / 2 ( q β ≤ q ≤ 1 ) . (122) k_{\alpha_{\beta},\beta^2}(q)=1 \qquad(0\le q\le q_{\beta}),\qquad k_{\alpha_{\beta},\beta^2}(q)\le e^{-\beta^2(q-q_{\beta})/2}\qquad(q_{\beta}\le q\le1). \tag*{(122)} k α β , β 2 ( q ) = 1 ( 0 ≤ q ≤ q β ) , k α β , β 2 ( q ) ≤ e − β 2 ( q − q β ) /2 ( q β ≤ q ≤ 1 ) . (122) In particular, sup q ∈ [ 0 , 1 ] k α β , β 2 ( q ) ≤ 1 \sup_{q\in[0,1]}k_{\alpha_{\beta},\beta^2}(q)\le1 sup q ∈ [ 0 , 1 ] k α β , β 2 ( q ) ≤ 1
Proof. We abbreviate d = β 2 d=\beta^2 d = β 2 a = α β a=\alpha_{\beta} a = α β k = k α β , d k=k_{\alpha_{\beta},d} k = k α β , d Γ = Γ α β , d \Gamma=\Gamma_{\alpha_{\beta},d} Γ = Γ α β , d u u u Y Y Y k = 1 k=1 k = 1 Y q β Y_{q_{\beta}} Y q β q β q_{\beta} q β k k k
Step 1: the support. By an identity of Talagrand [79], in the form stated in [9 Theorem 5] and [48, Lemma 2.6] (see also [44, Proposition 1.1] and [25, Proposition 1]), we have Γ ( q ) = q \Gamma(q)=q Γ ( q ) = q q ∈ supp μ β = [ 0 , q β ] q\in\operatorname{supp}\mu_{\beta}=[0,q_{\beta}] q ∈ supp μ β = [ 0 , q β ] Γ \Gamma Γ u x u_x u x log 2 \log2 log 2 Γ ′ = k \Gamma'=k Γ ′ = k k = 1 k=1 k = 1 ( 0 , q β ) (0,q_{\beta}) ( 0 , q β ) k k k k = 1 k=1 k = 1 [ 0 , q β ] [0,q_{\beta}] [ 0 , q β ]
Step 2: log-concavity of the transformed density. For q ∈ ( 0 , 1 ] q \in(0,1] q ∈ ( 0 , 1 ] π q \pi_q π q Y q Y_q Y q
z q ( x ) = π q ( x ) e − a ( q ) u ( q , x ) , V q ( x ) = − log z q ( x ) = a ( q ) u ( q , x ) − log π q ( x ) . z_q(x) = \pi_q(x)e^{-a(q)u(q,x)}, \qquad V_q(x) = -\log z_q(x) = a(q)u(q,x) - \log\pi_q(x). z q ( x ) = π q ( x ) e − a ( q ) u ( q , x ) , V q ( x ) = − log z q ( x ) = a ( q ) u ( q , x ) − log π q ( x ) . By (95) , z q ( x ) d x z_q(x)\,dx z q ( x ) d x X q X_q X q exp { − ∫ [ 0 , q ] u ( r , X r ) μ β ( d r ) } \exp\left\{-\int_{[0,q]} u(r,X_r)\,\mu_\beta(dr)\right\} exp { − ∫ [ 0 , q ] u ( r , X r ) μ β ( d r ) } [ 0 , 1 ] [0,1] [ 0 , 1 ] π q \pi_q π q a ( q ) u L ( q , x ) − log π q ( x ) a(q)u^{L}(q,x)-\log\pi_q(x) a ( q ) u L ( q , x ) − log π q ( x ) 1 / ( d q ) 1/(dq) 1/ ( d q ) u L = u − log 2 u^{L}=u-\log2 u L = u − log 2 a ( q ) a(q) a ( q ) a ( q ) log 2 a(q)\log2 a ( q ) log 2
V q ′ ′ ( x ) ≥ 1 d q > 0 , V q ( − x ) = V q ( x ) , x ∈ R . V_q''(x) \ge\frac{1}{dq} > 0, \qquad V_q(-x) = V_q(x), \qquad x \in\mathbb{R}. V q ′′ ( x ) ≥ d q 1 > 0 , V q ( − x ) = V q ( x ) , x ∈ R . Note that V q ′ ( 0 ) = 0 V_q'(0)=0 V q ′ ( 0 ) = 0 V q ′ ( x ) ≥ x / ( d q ) > 0 V_q'(x)\ge x/(dq)>0 V q ′ ( x ) ≥ x / ( d q ) > 0 x > 0 x>0 x > 0 z q z_q z q ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) q = q β q=q_\beta q = q β a ( q β ) = 1 a(q_\beta)=1 a ( q β ) = 1 μ β \mu_\beta μ β q β q_\beta q β z q β = π q β e − u ( q β , ⋅ ) z_{q_\beta}=\pi_{q_\beta}e^{-u(q_\beta,\cdot)} z q β = π q β e − u ( q β , ⋅ )
Step 3: heat evolution above the support. On [ q β , 1 ] [q_\beta,1] [ q β , 1 ] a = 1 a=1 a = 1 (88) is
u ( q , x ) = log ( 2 cosh x ) + d ( 1 − q ) 2 , q β ≤ q ≤ 1. u(q,x) = \log(2\cosh x) + \frac{d(1-q)}{2}, \qquad q_\beta\le q \le1. u ( q , x ) = log ( 2 cosh x ) + 2 d ( 1 − q ) , q β ≤ q ≤ 1. by [48, Lemma 2.4(i)]; this formula can also be checked by direct substitution, since u x = tanh x u_x=\tanh x u x = tanh x u x x = sech 2 x u_{xx}=\operatorname{sech}^{2}x u xx = sech 2 x u x x + u x 2 = 1 u_{xx}+u_x^2=1 u xx + u x 2 = 1 ∂ q u = − d / 2 \partial_q u=-d/2 ∂ q u = − d /2 G v G_v G v v v v m = 1 m=1 m = 1 [ q β , 1 ) [q_\beta,1) [ q β , 1 ) q β < q ≤ 1 q_\beta<q\le1 q β < q ≤ 1
π q ( x ) = e u ( q , x ) ∫ G d ( q − q β ) ( x − y ) e − u ( q β , y ) π q β ( y ) d y ; \pi_q(x)=e^{u(q,x)}\int G_{d(q-q_\beta)}(x-y)e^{-u(q_\beta,y)}\pi_{q_\beta}(y)\,dy; π q ( x ) = e u ( q , x ) ∫ G d ( q − q β ) ( x − y ) e − u ( q β , y ) π q β ( y ) d y ; the constants log 2 \log2 log 2 u ( q , x ) u(q,x) u ( q , x ) u ( q β , y ) u(q_\beta,y) u ( q β , y )
z q = G d ( q − q β ) ∗ z q β , q β < q ≤ 1. (123) z_q=G_{d(q-q_\beta)}*z_{q_\beta}, \qquad q_\beta<q\le1. \tag*{(123)} z q = G d ( q − q β ) ∗ z q β , q β < q ≤ 1. (123) Equivalently, z q z_q z q ∂ q z q = ( d / 2 ) ∂ x x z q \partial_q z_q=(d/2)\partial_{xx}z_q ∂ q z q = ( d /2 ) ∂ xx z q π q = e u z q \pi_q=e^u z_q π q = e u z q Y Y Y
∂ q z q + ( ∂ q u ) z q = e − u ( d 2 ∂ x x π q − d ∂ x ( u x π q ) ) = d 2 ∂ x x z q − d 2 ( u x x + u x 2 ) z q ; \partial_q z_q+(\partial_q u)z_q=e^{-u}\left(\frac{d}{2}\partial_{xx}\pi_q-d\partial_x(u_x\pi_q)\right)=\frac{d}{2}\partial_{xx}z_q-\frac{d}{2}(u_{xx}+u_x^2)z_q; ∂ q z q + ( ∂ q u ) z q = e − u ( 2 d ∂ xx π q − d ∂ x ( u x π q ) ) = 2 d ∂ xx z q − 2 d ( u xx + u x 2 ) z q ; the zeroth-order terms cancel because u x x + u x 2 = 1 u_{xx}+u_x^2=1 u xx + u x 2 = 1 ∂ q u = − d / 2 \partial_q u=-d/2 ∂ q u = − d /2
On the plateau, we have u x x = sech 2 u_{xx}=\operatorname{sech}^{2} u xx = sech 2 π q = e u ( q , ⋅ ) z q = 2 cosh ( ⋅ ) e d ( 1 − q ) / 2 z q \pi_q=e^{u(q,\cdot)}z_q=2\cosh(\cdot)e^{d(1-q)/2}z_q π q = e u ( q , ⋅ ) z q = 2 cosh ( ⋅ ) e d ( 1 − q ) /2 z q
k ( q ) = d ∫ R π q ( x ) sech 4 x d x = 2 d e d ( 1 − q ) / 2 ∫ R z q ( x ) sech 3 x d x , q β ≤ q ≤ 1. (124) k(q)=d\int_{\mathbb{R}}\pi_q(x)\operatorname{sech}^{4}x\,\mathrm{d}x=2d e^{d(1-q)/2}\int_{\mathbb{R}}z_q(x)\operatorname{sech}^{3}x\,\mathrm{d}x,\qquad q_\beta\le q\le1. \tag*{(124)} k ( q ) = d ∫ R π q ( x ) sech 4 x d x = 2 d e d ( 1 − q ) /2 ∫ R z q ( x ) sech 3 x d x , q β ≤ q ≤ 1. (124) Step 4: the rearrangement inequality. Let f f f g g g [ 0 , ∞ ) [0,\infty) [ 0 , ∞ )
∫ f ( x − y ) g ( x ) d x ≤ ∫ f ( x ) g ( x ) d x , y ∈ R . (125) \int f(x-y)g(x)\,\mathrm{d}x\le\int f(x)g(x)\,\mathrm{d}x,\qquad y\in\mathbb{R}. \tag*{(125)} ∫ f ( x − y ) g ( x ) d x ≤ ∫ f ( x ) g ( x ) d x , y ∈ R . (125) This is the centered rearrangement inequality in one dimension, which also follows from Anderson’s inequality [3 Theorem 1] by the layer-cake representation. We include a proof based on intersections of intervals. By the layer-cake representation and Tonelli’s theorem,
∫ f ( x − y ) g ( x ) d x = ∫ 0 ∞ ∫ 0 ∞ ∣ ( { f > t } + y ) ∩ { g > t ′ } ∣ d t d t ′ , \int f(x-y)g(x)\,\mathrm{d}x=\int_0^\infty\int_0^\infty\left|(\{f>t\}+y)\cap\{g>t'\}\right|\,\mathrm{d}t\,\mathrm{d}t', ∫ f ( x − y ) g ( x ) d x = ∫ 0 ∞ ∫ 0 ∞ ∣ ({ f > t } + y ) ∩ { g > t ′ } ∣ d t d t ′ , where ∣ ⋅ ∣ |\cdot| ∣ ⋅ ∣ { f > t } \{f>t\} { f > t } { g > t ′ } \{g>t'\} { g > t ′ } I I I I ′ I' I ′ ∣ ( I + y ) ∩ I ′ ∣ ≤ min { ∣ I ∣ , ∣ I ′ ∣ } = ∣ I ∩ I ′ ∣ |(I+y)\cap I'|\le\min\{|I|,|I'|\}=|I\cap I'| ∣ ( I + y ) ∩ I ′ ∣ ≤ min { ∣ I ∣ , ∣ I ′ ∣ } = ∣ I ∩ I ′ ∣ (125) follows.
We use (125) with f = z q β f=z_{q_\beta} f = z q β g = sech 3 g=\operatorname{sech}^{3} g = sech 3 sech ≤ 1 \operatorname{sech}\le1 sech ≤ 1 ∫ z q β = E [ e − u ( q β , Y q β ) ] ≤ 1 / 2 \int z_{q_\beta}=\mathbb{E}[e^{-u(q_\beta,Y_{q_\beta})}]\le1/2 ∫ z q β = E [ e − u ( q β , Y q β ) ] ≤ 1/2 (123) and Tonelli’s theorem, for q β < q ≤ 1 q_\beta<q\le1 q β < q ≤ 1
∫ z q ( x ) sech 3 x d x = ∫ G d ( q − q β ) ( y ) ( ∫ z q β ( x − y ) sech 3 x d x ) d y ≤ ∫ z q β ( x ) sech 3 x d x . \int z_q(x)\operatorname{sech}^{3}x\,\mathrm{d}x=\int G_{d(q-q_\beta)}(y)\left(\int z_{q_\beta}(x-y)\operatorname{sech}^{3}x\,\mathrm{d}x\right)\mathrm{d}y\le\int z_{q_\beta}(x)\operatorname{sech}^{3}x\,\mathrm{d}x. ∫ z q ( x ) sech 3 x d x = ∫ G d ( q − q β ) ( y ) ( ∫ z q β ( x − y ) sech 3 x d x ) d y ≤ ∫ z q β ( x ) sech 3 x d x . Inserting this into (124) , and using (124) at q = q β q=q_\beta q = q β k ( q β ) = 1 k(q_\beta)=1 k ( q β ) = 1
k ( q ) ≤ 2 d e d ( 1 − q ) / 2 ∫ z q β ( x ) sech 3 x d x = e − d ( q − q β ) / 2 k ( q β ) = e − d ( q − q β ) / 2 , q β ≤ q ≤ 1. k(q)\le2d e^{d(1-q)/2}\int z_{q_\beta}(x)\operatorname{sech}^{3}x\,\mathrm{d}x=e^{-d(q-q_\beta)/2}k(q_\beta)=e^{-d(q-q_\beta)/2},\qquad q_\beta\le q\le1. k ( q ) ≤ 2 d e d ( 1 − q ) /2 ∫ z q β ( x ) sech 3 x d x = e − d ( q − q β ) /2 k ( q β ) = e − d ( q − q β ) /2 , q β ≤ q ≤ 1. Together with Step 1, this proves (122) .
Stability at a minimizing profile
The isolation estimate of Proposition 7.6 compares a minimizer with the scalar trial profile γ λ t r = max { λ , γ β } \gamma_\lambda^{\mathrm{tr}}=\max\{\lambda,\gamma_\beta\} γ λ tr = max { λ , γ β } γ β = β α β \gamma_\beta=\beta\alpha_\beta γ β = β α β γ β \gamma_\beta γ β (61) that γ β ( q ) ≥ k 0 q \gamma_\beta(q)\ge k_0q γ β ( q ) ≥ k 0 q 0 ≤ q ≤ T 0 0\le q\le T_0 0 ≤ q ≤ T 0 k 0 , T 0 > 0 k_0,T_0>0 k 0 , T 0 > 0 β \beta β λ > 0 \lambda>0 λ > 0
∣ γ λ t r − γ β ∣ L 1 ( 0 , 1 ) ≤ C β λ 2 . (126) \left\lvert\gamma_\lambda^{\mathrm{tr}}-\gamma_\beta\right\rvert_{L^1(0,1)}\le C_\beta\lambda^2. \tag*{(126)} γ λ tr − γ β L 1 ( 0 , 1 ) ≤ C β λ 2 . (126) Indeed, γ λ t r − γ β = ( λ − γ β ) + \gamma_{\lambda}^{\mathrm{tr}}-\gamma_{\beta}=(\lambda-\gamma_{\beta})_{+} γ λ tr − γ β = ( λ − γ β ) + λ ≤ k 0 T 0 \lambda\le k_{0}T_{0} λ ≤ k 0 T 0 q ≥ T 0 q\ge T_{0} q ≥ T 0 γ β \gamma_{\beta} γ β γ β ( q ) ≥ γ β ( T 0 ) ≥ k 0 T 0 ≥ λ \gamma_{\beta}(q)\ge\gamma_{\beta}(T_{0})\ge k_{0}T_{0}\ge\lambda γ β ( q ) ≥ γ β ( T 0 ) ≥ k 0 T 0 ≥ λ
∥ γ λ t r − γ β ∥ L 1 ( 0 , 1 ) = ∫ 0 T 0 ( λ − γ β ( q ) ) + d q ≤ ∫ 0 λ / k 0 ( λ − k 0 q ) d q = λ 2 2 k 0 2 . \lVert\gamma_{\lambda}^{\mathrm{tr}}-\gamma_{\beta}\rVert_{L^{1}(0,1)}=\int_{0}^{T_{0}}(\lambda-\gamma_{\beta}(q))_{+}\,\mathrm{d}q\le\int_{0}^{\lambda/k_{0}}(\lambda-k_{0}q)\,\mathrm{d}q=\frac{\lambda^{2}}{2k_{0}^{2}}. ∥ γ λ tr − γ β ∥ L 1 ( 0 , 1 ) = ∫ 0 T 0 ( λ − γ β ( q ) ) + d q ≤ ∫ 0 λ / k 0 ( λ − k 0 q ) d q = 2 k 0 2 λ 2 . If λ > k 0 T 0 \lambda>k_{0}T_{0} λ > k 0 T 0 λ \lambda λ λ ≤ λ 2 / ( k 0 T 0 ) \lambda\le\lambda^{2}/(k_{0}T_{0}) λ ≤ λ 2 / ( k 0 T 0 ) T 0 ≤ 1 / 4 < 2 T_{0}\le1/4<2 T 0 ≤ 1/4 < 2 (126) with C β = 1 / ( k 0 T 0 ) C_{\beta}=1/(k_{0}T_{0}) C β = 1/ ( k 0 T 0 )
Proposition 9.9 (Scalar stability at the minimizing profile). There exists a constant C s t ≥ 1 C_{\mathrm{st}}\ge1 C st ≥ 1 β \beta β Let N ≥ 2 N\ge2 N ≥ 2 0 < ρ ≤ 1 0<\rho\le1 0 < ρ ≤ 1 t = 1 − ρ t=1-\rho t = 1 − ρ 0 < λ ≤ λ 0 0<\lambda\le\lambda_{0} 0 < λ ≤ λ 0 s = λ / β s=\lambda/\beta s = λ / β μ N = s δ 0 + ν N \mu_{N}=s\delta_{0}+\nu_{N} μ N = s δ 0 + ν N F t , s F_{t,s} F t , s (16) , let a N ( q ) = μ N ( [ 0 , q ] ) a_{N}(q)=\mu_{N}([0,q]) a N ( q ) = μ N ([ 0 , q ]) η \eta η β a N \beta a_{N} β a N F ρ F_{\rho} F ρ (63) . Let d N = β 2 ( 1 − t / N ) d_{N}=\beta^{2}(1-t/N) d N = β 2 ( 1 − t / N )
∥ a N − α β ∥ L 1 ( 0 , 1 ) ≤ min { 1 , C s t [ ρ − 3 / 4 η 1 / 4 + λ 2 ] } . (127) \lVert a_{N}-\alpha_{\beta}\rVert_{L^{1}(0,1)}\le\min\left\{1,C_{\mathrm{st}}\left[\rho^{-3/4}\eta^{1/4}+\lambda^{2}\right]\right\}. \tag*{(127)} ∥ a N − α β ∥ L 1 ( 0 , 1 ) ≤ min { 1 , C st [ ρ − 3/4 η 1/4 + λ 2 ] } . (127) Further, let ε N = max { 0 , sup q ∈ [ 0 , 1 ] k a N , d N ( q ) − 1 } \varepsilon_{N}=\max\{0,\sup_{q\in[0,1]}k_{a_{N},d_{N}}(q)-1\} ε N = max { 0 , sup q ∈ [ 0 , 1 ] k a N , d N ( q ) − 1 } H ν N \mathcal{H}_{\nu_{N}} H ν N μ N \mu_{N} μ N
J a N , d N s c ⪯ ( 1 + ε N ) I , ε N ≤ C s t [ ρ − 3 / 4 η 1 / 4 + λ 2 + N − 1 ] . (128) J^{\mathrm{sc}}_{a_{N},d_{N}}\preceq(1+\varepsilon_{N})I,\qquad\varepsilon_{N}\le C_{\mathrm{st}}\left[\rho^{-3/4}\eta^{1/4}+\lambda^{2}+N^{-1}\right]. \tag*{(128)} J a N , d N sc ⪯ ( 1 + ε N ) I , ε N ≤ C st [ ρ − 3/4 η 1/4 + λ 2 + N − 1 ] . (128) Proof. We use the energy normalization of Section 3, in which γ = β a \gamma=\beta a γ = β a F ρ ( γ ) = β − 1 F t , s ( μ ) F_{\rho}(\gamma)=\beta^{-1}F_{t,s}(\mu) F ρ ( γ ) = β − 1 F t , s ( μ ) μ ↦ β a \mu\mapsto\beta a μ ↦ β a β − 1 \beta^{-1} β − 1 γ N = β a N \gamma_{N}=\beta a_{N} γ N = β a N F ρ F_{\rho} F ρ η \eta η (69) implies that ∥ γ N − γ λ t r ∥ L 1 ≤ C ρ − 3 / 4 η 1 / 4 \lVert\gamma_{N}-\gamma_{\lambda}^{\mathrm{tr}}\rVert_{L^{1}}\le C\rho^{-3/4}\eta^{1/4} ∥ γ N − γ λ tr ∥ L 1 ≤ C ρ − 3/4 η 1/4 C C C β \beta β (126) , we have
∥ a N − α β ∥ L 1 ≤ β − 1 ( ∥ γ N − γ λ t r ∥ L 1 + ∥ γ λ t r − γ β ∥ L 1 ) ≤ C β [ ρ − 3 / 4 η 1 / 4 + λ 2 ] . \lVert a_{N}-\alpha_{\beta}\rVert_{L^{1}}\le\beta^{-1}\left(\lVert\gamma_{N}-\gamma_{\lambda}^{\mathrm{tr}}\rVert_{L^{1}}+\lVert\gamma_{\lambda}^{\mathrm{tr}}-\gamma_{\beta}\rVert_{L^{1}}\right)\le C_{\beta}\left[\rho^{-3/4}\eta^{1/4}+\lambda^{2}\right]. ∥ a N − α β ∥ L 1 ≤ β − 1 ( ∥ γ N − γ λ tr ∥ L 1 + ∥ γ λ tr − γ β ∥ L 1 ) ≤ C β [ ρ − 3/4 η 1/4 + λ 2 ] . The distance is also at most one, since both functions take values in [ 0 , 1 ] [0,1] [ 0 , 1 ] (127) .
Since N ≥ 2 N\ge2 N ≥ 2 0 ≤ t < 1 0\le t<1 0 ≤ t < 1 d N ∈ [ β 2 / 2 , β 2 ] d_{N}\in[\beta^{2}/2,\beta^{2}] d N ∈ [ β 2 /2 , β 2 ] ∣ d N − β 2 ∣ = β 2 t / N ≤ β 2 / N \lvert d_{N}-\beta^{2}\rvert=\beta^{2}t/N\le\beta^{2}/N ∣ d N − β 2 ∣ = β 2 t / N ≤ β 2 / N ( a , d ) = ( a N , d N ) (a,d)=(a_{N},d_{N}) ( a , d ) = ( a N , d N ) ( a ~ , d ~ ) = ( α β , β 2 ) (\widetilde a,\widetilde d)=(\alpha_{\beta},\beta^{2}) ( a , d ) = ( α β , β 2 ) q ∈ [ 0 , 1 ] q\in[0,1] q ∈ [ 0 , 1 ] q q q supp μ β \operatorname{supp}\mu_{\beta} supp μ β
k a N , d N ( q ) ≤ k α β , β 2 ( q ) + C β ( ∥ a N − α β ∥ L 1 + N − 1 ) ≤ 1 + C β ( ∥ a N − α β ∥ L 1 + N − 1 ) . k_{a_{N},d_{N}}(q)\le k_{\alpha_{\beta},\beta^{2}}(q)+C_{\beta}\left(\lVert a_{N}-\alpha_{\beta}\rVert_{L^{1}}+N^{-1}\right)\le1+C_{\beta}\left(\lVert a_{N}-\alpha_{\beta}\rVert_{L^{1}}+N^{-1}\right). k a N , d N ( q ) ≤ k α β , β 2 ( q ) + C β ( ∥ a N − α β ∥ L 1 + N − 1 ) ≤ 1 + C β ( ∥ a N − α β ∥ L 1 + N − 1 ) . Together with (127) , this yields the bound on ε N \varepsilon_{N} ε N (128) . Finally, since k a N , d N ≤ 1 + ε N k_{a_{N},d_{N}}\le1+\varepsilon_{N} k a N , d N ≤ 1 + ε N a N a_{N} a N ν = ν N \nu=\nu_{N} ν = ν N J a N , d N s c ⪯ ( 1 + ε N ) I J^{\mathrm{sc}}_{a_{N},d_{N}}\preceq(1+\varepsilon_{N})I J a N , d N sc ⪯ ( 1 + ε N ) I H ν N \mathcal{H}_{\nu_{N}} H ν N C s t ≥ 1 C_{\mathrm{st}}\ge1 C st ≥ 1 β \beta β □ \square □
The cavity contraction
The cavity interpolation and its tree laws
This section and the next two prove Proposition 12.4 . Suppose that at a minimizing profile, the scalar response operator J a , d s c J^{\mathrm{sc}}_{a,d} J a , d sc J a , d s c ⪯ ( 1 + ε N ) I J^{\mathrm{sc}}_{a,d} \preceq(1+\varepsilon_N)I J a , d sc ⪯ ( 1 + ε N ) I ε N ≤ ρ / 4 \varepsilon_N \le\rho/4 ε N ≤ ρ /4 V = ∫ Var ( S q ) ν ( d q ) V = \int\operatorname{Var}(S_q)\,\nu(\mathrm{d}q) V = ∫ Var ( S q ) ν ( d q ) ( 3 ρ / 4 ) V ≤ C c a v [ N − 1 + A + J + H 3 ] (3\rho/4)V \le C_{\mathrm{cav}}[N^{-1}+A+J+H_3] ( 3 ρ /4 ) V ≤ C cav [ N − 1 + A + J + H 3 ] C c a v C_{\mathrm{cav}} C cav β \beta β Corollary 12.5 , we deduce from it the bound on the defect D N ( ρ ) D_N(\rho) D N ( ρ )
The argument is a cavity interpolation on one site. We single out the last spin and move the variance of its couplings to the other n = N − 1 n=N-1 n = N − 1 V V V n / N n/N n / N N − 1 N^{-1} N − 1 Lemma 11.2 ). We also bound the errors by A A A J J J Lemma 11.3 ). In § 10.2 , we compare the variance of this self-overlap at the independent endpoint with V V V V V V Lemma 12.2 ), bound the cubic moment that controls the remainder by A A A J J J H 3 H_3 H 3 Lemma 12.3 ), and close the resulting inequality.
In this section, we set up the interpolation and the calculus that the next two sections use. In § 9.1.1 , we define the cavity law P u \mathbb{P}_u P u 0 ≤ u ≤ 1 0 \le u \le1 0 ≤ u ≤ 1 § 9.1.2 , we represent products of conditional expectations by laws of finitely many descendants on marked trees. We show that the solution of the Parisi equation depends smoothly on the incident row and on the variance of the last field (Lemma 10.2 ). We compute the density of the tree laws (Lemma 10.3 ) and the responses (10.11) and (10.13) to these two parameters (Lemma 10.4). From them, we derive an exact formula for the derivative of a tree moment in u u u Lemma 10.5 ) and a comparison of tree moments at different values of u u u § 9.5.1 , we bound integrated products of three overlap defects by a cubic moment M 3 ( u ) \mathcal{M}_3(u) M 3 ( u ) Lemma 10.7 ), and deduce a bound on the Taylor remainder (Lemma 10.8 ).
We use the generalized law of Section 4, with its variance bound (Lemma 4.2 ) and its heat identity (Lemma 4.3 ). We also use the scalar law (9.1) with Lemma 9.4 , and the differentiation and approximation arguments of [4 Sections 2.1, 2.3, and 2.6] . The results of this section are used as follows. The identity (10.11) and the density (10.9) with one leaf enter the proof of Lemma 11.2 . The bound (10.4) and Corollary 10.6 enter the proofs of Lemmas 11.3 and 12.3 . The bound of Lemma 10.7 enters the proof of Lemma 12.1, and Lemma 10.8 enters the proofs of Lemma 12.2 and Proposition 12.4 .
The cavity method originates in the physics literature [51, 52]. We use its rigorous form on one site, the framework of differentiation, comparison, and contraction in [81, Section 1.6] (see Lemmas 1.6.3–1.6.4 and Propositions 1.6.5–1.6.6 there). An expansion on one site for hierarchical Gaussian laws appears in [56, Lemma 2.2]. There, a site is chosen by exchangeability and deleted, and its Gaussian kernels are interpolated to those of an independent scalar model on one site. The linear cavity kernel is half of a connected correlation of four spins of the removed site [56, (A8)], and a linear equation with this kernel is inverted under a gap condition on the scalar response in [56, Proposition 5.1]. The tree laws of § 9.1.2 and their calculus are adapted from [54, Section 2]; we give the precise references there.
Standing assumptions. Throughout Part IV, we fix N ≥ 2 N \ge2 N ≥ 2 n = N − 1 n = N - 1 n = N − 1 0 < ρ ≤ 1 0 < \rho\le1 0 < ρ ≤ 1 t = 1 − ρ t = 1 - \rho t = 1 − ρ c = β 2 ρ c = \beta^{2}\rho c = β 2 ρ 0 < s = λ / β ≤ 1 / 2 0 < s = \lambda/\beta\le1/2 0 < s = λ / β ≤ 1/2 μ = s δ 0 + ν \mu= s\delta_{0} + \nu μ = s δ 0 + ν ν \nu ν [ 0 , 1 ] [0,1] [ 0 , 1 ] n n n n n n C C C C β C_{\beta} C β β \beta β N N N ρ \rho ρ s s s u u u u u u U 0 U^{0} U 0
Removing the incident row
We write σ = ( τ , ϵ ) ∈ { − 1 , 1 } n × { − 1 , 1 } \sigma= (\tau,\epsilon) \in\{-1,1\}^{n} \times\{-1,1\} σ = ( τ , ϵ ) ∈ { − 1 , 1 } n × { − 1 , 1 } ϵ = σ N \epsilon= \sigma_{N} ϵ = σ N g i N g_{iN} g i N i ≤ n i \le n i ≤ n ϵ \epsilon ϵ τ \tau τ incident row . Let u ∈ [ 0 , 1 ] u \in[0,1] u ∈ [ 0 , 1 ] t t t
α = β 2 t n N , d = c + α = β 2 ( 1 − t N ) , b ( u ) = d − α u , 0 ≤ u ≤ 1. (129) \alpha= \beta^{2}t\frac{n}{N}, \qquad d = c + \alpha= \beta^{2}\left(1-\frac{t}{N}\right), \qquad b(u) = d - \alpha u, \qquad0 \le u \le1. \tag*{(129)} α = β 2 t N n , d = c + α = β 2 ( 1 − N t ) , b ( u ) = d − α u , 0 ≤ u ≤ 1. (129) The second identity holds because c + α = β 2 ( ρ + t − t / N ) = β 2 ( 1 − t / N ) c + \alpha= \beta^{2}(\rho+ t - t/N) = \beta^{2}(1 - t/N) c + α = β 2 ( ρ + t − t / N ) = β 2 ( 1 − t / N ) N ≥ 2 N \ge2 N ≥ 2 d ∈ [ β 2 / 2 , β 2 ] d \in[\beta^{2}/2,\beta^{2}] d ∈ [ β 2 /2 , β 2 ] b ( u ) ≥ b ( 1 ) = c > 0 b(u) \ge b(1) = c > 0 b ( u ) ≥ b ( 1 ) = c > 0
For x ∈ R n x \in\mathbb{R}^n x ∈ R n y ∈ R y \in\mathbb{R} y ∈ R
E ( σ ; x , y ) = E − ( τ ) + α u n ϵ ∑ i = 1 n g i N τ i + x ⋅ τ + y ϵ , \mathcal{E}(\sigma;x,y)=\mathcal{E}^{-}(\tau)+\sqrt{\frac{\alpha u}{n}}\,\epsilon\sum_{i=1}^{n}g_iN\tau_i+x\cdot\tau+y\epsilon, E ( σ ; x , y ) = E − ( τ ) + n α u ϵ i = 1 ∑ n g i N τ i + x ⋅ τ + y ϵ , E − ( τ ) = β t N ∑ 1 ≤ i < j ≤ n g i j τ i τ j , \mathcal{E}^{-}(\tau)=\frac{\beta\sqrt{t}}{\sqrt{N}}\sum_{1\leq i<j\leq n}g_{ij}\tau_i\tau_j, E − ( τ ) = N β t 1 ≤ i < j ≤ n ∑ g ij τ i τ j , where the couplings are independent standard Gaussian variables under the reference law. Let U ( q , x , y ) U(q,x,y) U ( q , x , y )
∂ q U = − 1 2 { c ( Δ x U + a ( q ) ∣ ∇ x U ∣ 2 ) + b ( u ) ( ∂ y y U + a ( q ) ( ∂ y U ) 2 ) } , \partial_q U=-\frac{1}{2}\left\{c\left(\Delta_x U+a(q)|\nabla_x U|^2\right)+b(u)\left(\partial_{yy}U+a(q)(\partial_y U)^2\right)\right\}, ∂ q U = − 2 1 { c ( Δ x U + a ( q ) ∣ ∇ x U ∣ 2 ) + b ( u ) ( ∂ y y U + a ( q ) ( ∂ y U ) 2 ) } , U ( 1 , x , y ) = log ∑ σ e E ( σ ; x , y ) . (130) U(1,x,y)=\log\sum_{\sigma}e^{\mathcal{E}(\sigma;x,y)}. \tag*{(130)} U ( 1 , x , y ) = log σ ∑ e E ( σ ; x , y ) . (130) At u = 1 u=1 u = 1 α / n = β 2 t / N \alpha/n=\beta^2t/N α / n = β 2 t / N b ( 1 ) = c b(1)=c b ( 1 ) = c u u u 0 0 0 α u / n \alpha u/n α u / n b ( 0 ) = d = c + α b(0)=d=c+\alpha b ( 0 ) = d = c + α
The completion of Section 3 adds to the terminal exponent the Gaussian term G = β t ∑ i ≤ N z i / 2 N G=\beta\sqrt{t}\sum_{i\leq N}z_i/\sqrt{2N} G = β t ∑ i ≤ N z i / 2 N U U U U + G U+G U + G e s G e^{sG} e s G G G G g i j g_{ij} g ij z i z_i z i
We construct the law P u \mathbb{P}_u P u E u \mathbb{E}_u E u e s U ( 0 , 0 , 0 ) / Z s ( u ) e^{sU(0,0,0)}/Z_s(u) e s U ( 0 , 0 , 0 ) / Z s ( u ) Z s ( u ) = E r e f e s U ( 0 , 0 , 0 ) Z_s(u)=\mathbb{E}_{\mathrm{ref}}e^{sU(0,0,0)} Z s ( u ) = E ref e s U ( 0 , 0 , 0 ) ( X q , Y q ) ∈ R n × R (X_q,Y_q)\in\mathbb{R}^n\times\mathbb{R} ( X q , Y q ) ∈ R n × R
d X q = c a ( q ) ∇ x U ( q , X q , Y q ) d q + c d B q , X 0 = 0 , dX_q=c\,a(q)\nabla_xU(q,X_q,Y_q)\,dq+\sqrt{c}\,dB_q,\qquad X_0=0, d X q = c a ( q ) ∇ x U ( q , X q , Y q ) d q + c d B q , X 0 = 0 , d Y q = b ( u ) a ( q ) ∂ y U ( q , X q , Y q ) d q + b ( u ) d B q ′ , Y 0 = 0 , dY_q=b(u)\,a(q)\partial_yU(q,X_q,Y_q)\,dq+\sqrt{b(u)}\,dB'_q,\qquad Y_0=0, d Y q = b ( u ) a ( q ) ∂ y U ( q , X q , Y q ) d q + b ( u ) d B q ′ , Y 0 = 0 , where ( B , B ′ ) (B,B') ( B , B ′ ) R n + 1 \mathbb{R}^{n+1} R n + 1 diag ( c I n , b ( u ) ) \operatorname{diag}(cI_n,b(u)) diag ( c I n , b ( u )) a ( q ) a(q) a ( q ) ∇ U ( q , X q , Y q ) \nabla U(q,X_q,Y_q) ∇ U ( q , X q , Y q ) σ \sigma σ e E ( σ ; X 1 , Y 1 ) e^{\mathcal{E}(\sigma;X_1,Y_1)} e E ( σ ; X 1 , Y 1 ) G q \mathcal{G}_q G q σ \sigma σ q q q
The two endpoints. At u = 1 u=1 u = 1 P 1 \mathbb{P}_1 P 1 u = 0 u=0 u = 0 U ( 1 , x , y ) = U − ( 1 , x ) + log ( 2 cosh y ) U(1,x,y)=U^{-}(1,x)+\log(2\cosh y) U ( 1 , x , y ) = U − ( 1 , x ) + log ( 2 cosh y ) U − ( 1 , x ) = log ∑ τ e E − ( τ ) + x ⋅ τ U^{-}(1,x)=\log\sum_{\tau}e^{\mathcal{E}^{-}(\tau)+x\cdot\tau} U − ( 1 , x ) = log ∑ τ e E − ( τ ) + x ⋅ τ U − U^{-} U − n n n E − \mathcal{E}^{-} E − c c c μ \mu μ U 0 U^0 U 0 d d d μ \mu μ
We claim that U = U − + U 0 U=U^{-}+U^{0} U = U − + U 0 u = 0 u=0 u = 0 (130) factors, because the increments of x x x y y y b ( 0 ) = d b(0)=d b ( 0 ) = d [4 (2.19)] to each of the three equations, we extend the identity to every profile through the approximation of [4 Section 2.6] . The outer normalizer factors as Z s ( 0 ) = e s U 0 ( 0 , 0 ) E r e f e s U − ( 0 , 0 ) Z_s(0)=e^{sU^{0}(0,0)}\mathbb{E}_{\mathrm{ref}}e^{sU^{-}(0,0)} Z s ( 0 ) = e s U 0 ( 0 , 0 ) E ref e s U − ( 0 , 0 ) e s U 0 ( 0 , 0 ) e^{sU^{0}(0,0)} e s U 0 ( 0 , 0 ) X X X ∇ x U − \nabla_xU^{-} ∇ x U − Y Y Y ∂ y U 0 \partial_yU^{0} ∂ y U 0 e ϵ − ( τ ) + X 1 ⋅ τ − U − ( 1 , X 1 ) e^{\epsilon^{-(\tau)+X_1\cdot\tau-U^{-}(1,X_1)}} e ϵ − ( τ ) + X 1 ⋅ τ − U − ( 1 , X 1 ) e ϵ Y 1 / ( 2 cosh Y 1 ) e^{\epsilon Y_1}/(2\cosh Y_1) e ϵ Y 1 / ( 2 cosh Y 1 ) P 0 \mathbb{P}_0 P 0 n n n E − \mathbb{E}^{-} E − d d d μ \mu μ E 0 \mathbb{E}^{0} E 0 d ∈ [ β 2 / 2 , β 2 ] d\in[\beta^2/2,\beta^2] d ∈ [ β 2 /2 , β 2 ] m p m_p m p h p h_p h p f p = m p 2 f_p=m_p^2 f p = m p 2 f ˉ p = f p − E 0 f p \bar f_p=f_p-\mathbb{E}^{0}f_p f ˉ p = f p − E 0 f p g p ℓ g_p^\ell g p ℓ Γ ( p ) = Γ a , d ( p ) = E 0 m p 2 \Gamma(p)=\Gamma_{a,d}(p)=\mathbb{E}^{0}m_p^2 Γ ( p ) = Γ a , d ( p ) = E 0 m p 2
The remaining spins. For the first n n n
M q = E u [ τ ∣ G q ] , A q − = ∣ M q ∣ 2 n , C q − = Cov u ( τ ∣ G q ) , H q − = ∇ x 2 U ( q , X q , Y q ) . M_q=\mathbb{E}_u[\tau\mid\mathcal{G}_q],\qquad A_q^{-}=\frac{|M_q|^2}{n},\qquad C_q^{-}=\operatorname{Cov}_u(\tau\mid\mathcal{G}_q),\qquad H_q^{-}=\nabla_x^2U(q,X_q,Y_q). M q = E u [ τ ∣ G q ] , A q − = n ∣ M q ∣ 2 , C q − = Cov u ( τ ∣ G q ) , H q − = ∇ x 2 U ( q , X q , Y q ) . We let m N ( q ) = E u [ ϵ ∣ G q ] m_N(q)=\mathbb{E}_u[\epsilon\mid\mathcal{G}_q] m N ( q ) = E u [ ϵ ∣ G q ]
a q − ( u ) = E u Tr ( ( C q − ) 2 ) n 2 , j q − ( u ) = E u [ M q T C q − M q ] n 2 . (131) a_q^{-}(u)=\frac{\mathbb{E}_u\operatorname{Tr}((C_q^{-})^2)}{n^2},\qquad j_q^{-}(u)=\frac{\mathbb{E}_u[M_q^{T}C_q^{-}M_q]}{n^2}. \tag*{(131)} a q − ( u ) = n 2 E u Tr (( C q − ) 2 ) , j q − ( u ) = n 2 E u [ M q T C q − M q ] . (131) The rest self-overlap A q − A_q^{-} A q − A A A
We note that P u \mathbb{P}_u P u N N N Λ = diag ( c I n , b ( u ) ) \Lambda=\operatorname{diag}(cI_n,b(u)) Λ = diag ( c I n , b ( u )) β t / N \beta\sqrt{t/N} β t / N i < j ≤ n i<j\le n i < j ≤ n ( α u / n ) 1 / 2 (\alpha u/n)^{1/2} ( α u / n ) 1/2 ( i , N ) (i,N) ( i , N ) 0 0 0 Λ \Lambda Λ c > 0 c>0 c > 0 b ( u ) ≥ c b(u)\ge c b ( u ) ≥ c ( α u / n ) 1 / 2 (\alpha u/n)^{1/2} ( α u / n ) 1/2 u = 0 u=0 u = 0 n n n M q = ∇ x U ( q , X q , Y q ) M_q=\nabla_xU(q,X_q,Y_q) M q = ∇ x U ( q , X q , Y q ) m N ( q ) = ∂ y U ( q , X q , Y q ) m_N(q)=\partial_yU(q,X_q,Y_q) m N ( q ) = ∂ y U ( q , X q , Y q ) a ( q ) C q ⪯ H q ⪯ C q a(q)C_q\preceq H_q\preceq C_q a ( q ) C q ⪯ H q ⪯ C q a ( q ) C q − ⪯ H q − ⪯ C q − a(q)C_q^{-}\preceq H_q^{-}\preceq C_q^{-} a ( q ) C q − ⪯ H q − ⪯ C q − 0 ⪯ H q − ⪯ C q − 0\preceq H_q^{-}\preceq C_q^{-} 0 ⪯ H q − ⪯ C q − k = ψ k=\psi k = ψ u ∈ [ 0 , 1 ] u\in[0,1] u ∈ [ 0 , 1 ] ψ ∈ L 2 ( ν ) \psi\in L^2(\nu) ψ ∈ L 2 ( ν )
Var u ( ∫ ψ ( q ) A q − ν ( d q ) ) ≤ 4 ∫ ψ ( q ) 2 j q − ( u ) ν ( d q ) . (132) \operatorname{Var}_u\left(\int\psi(q)A_q^{-}\,\nu(dq)\right)\le4\int\psi(q)^2j_q^{-}(u)\,\nu(dq). \tag*{(132)} Var u ( ∫ ψ ( q ) A q − ν ( d q ) ) ≤ 4 ∫ ψ ( q ) 2 j q − ( u ) ν ( d q ) . (132) Laws of finitely many descendants
Derivatives in u u u , {xrefext}(4.25)–(4.26)]. Normalized finite replica trees of the present kind were developed in {citep}[54, Section 2], together with the differentiation of their moments by attaching new leaves and the resulting Taylor and comparison bounds. They are also used in [56, Sections 2.1–2.3]. In {citep}[54]`, the covariance differentiation formula is Proposition 2.5, the attachment rules and their uniform variation bounds are Lemma 2.6, the bounds on the levels at which new pairs split are Lemma 2.8, and the Taylor and comparison bounds are Lemma 2.9. We derive below the versions that we need for the cavity law, with its root tilt and an arbitrary profile, including its atoms.
Marked trees
Definition 10.1 (Marked trees and tree laws). A marked tree T \mathcal{T} T k ≥ 1 k \ge1 k ≥ 1 v v v q v ∈ [ 0 , 1 ] q_v \in[0,1] q v ∈ [ 0 , 1 ] 0 0 0 ( 0 , 1 ) (0,1) ( 0 , 1 ) 1 1 1 k k k a = 1 , … , k a=1,\ldots,k a = 1 , … , k
Given u ∈ [ 0 , 1 ] u \in[0,1] u ∈ [ 0 , 1 ] tree law P T , u \mathbb{P}_{\mathcal{T},u} P T , u E T , u \mathbb{E}_{\mathcal{T},u} E T , u P u \mathbb{P}_u P u σ a = ( τ a , ϵ a ) \sigma^a=(\tau^a,\epsilon_a) σ a = ( τ a , ϵ a ) σ a \sigma^a σ a e E ( σ ; X 1 a , Y 1 a ) e^{\mathcal{E}(\sigma;X_1^a,Y_1^a)} e E ( σ ; X 1 a , Y 1 a ) ( X 1 a , Y 1 a ) (X_1^a,Y_1^a) ( X 1 a , Y 1 a ) a a a E k , q , u \mathbb{E}_{k,q,u} E k , q , u k k k q q q
The field is defined at every point v v v T \mathcal{T} T ( X v , Y v ) (X_v,Y_v) ( X v , Y v ) q v q_v q v v v v k ≥ 2 k \ge2 k ≥ 2 0 < q < 1 0<q<1 0 < q < 1 k k k q q q q q q k k k k k k k ≥ 2 k \ge2 k ≥ 2 q = 0 q=0 q = 0 k k k k ≥ 2 k \ge2 k ≥ 2 q = 1 q=1 q = 1 k k k k = 1 k=1 k = 1 q q q
Given the couplings and the field at a vertex, the continuations along different outgoing edges are conditionally independent, and their laws are normalized. Integrating out a leaf yields the tree law of the remaining leaves. We call this the deletion property . In particular, every leaf has law P u \mathbb{P}_u P u G q \mathcal{G}_q G q q q q a ≠ b a \ne b a = b q a b q_{ab} q ab q a b = 1 q_{ab}=1 q ab = 1 rest overlap , their overlap defect , and the product of the overlap defect with their last spins by
R a b − = τ a ⋅ τ b n , Δ a b = R a b − − q a b , D a b = ϵ a ϵ b Δ a b . (133) R_{ab}^{-}=\frac{\tau^{a}\cdot\tau^{b}}{n},\qquad\Delta_{ab}=R_{ab}^{-}-q_{ab},\qquad D_{ab}=\epsilon_a\epsilon_b\Delta_{ab}. \tag*{(133)} R ab − = n τ a ⋅ τ b , Δ ab = R ab − − q ab , D ab = ϵ a ϵ b Δ ab . (133) Since ∣ R a b − ∣ ≤ 1 |R_{ab}^{-}|\le1 ∣ R ab − ∣ ≤ 1 ∣ D a b ∣ = ∣ Δ a b ∣ ≤ 2 |D_{ab}|=|\Delta_{ab}|\le2 ∣ D ab ∣ = ∣ Δ ab ∣ ≤ 2 E 4 , p , u \mathsf{E}_{4,p,u} E 4 , p , u G p \mathcal{G}_p G p σ \sigma σ G p \mathcal{G}_p G p E u [ m N ( p ) 2 A p − ] = E 4 , p , u [ ϵ 1 ϵ 2 R 34 − ] \mathbb{E}_u[m_N(p)^2A_p^{-}]=\mathbb{E}_{4,p,u}[\epsilon_1\epsilon_2R_{34}^{-}] E u [ m N ( p ) 2 A p − ] = E 4 , p , u [ ϵ 1 ϵ 2 R 34 − ]
Regularity in the row and in the last-field variance
To differentiate tree moments, we temporarily replace the row by a deterministic coupling ϵ ∑ i θ i τ i \epsilon\sum_i\theta_i\tau_i ϵ ∑ i θ i τ i θ ∈ R n \theta\in\mathbb{R}^n θ ∈ R n b ( u ) b(u) b ( u ) b > 0 b>0 b > 0 Λ b = diag ( c I n , b ) \Lambda_b=\operatorname{diag}(cI_n,b) Λ b = diag ( c I n , b ) g i j g_{ij} g ij i < j ≤ n i<j\le n i < j ≤ n P u \mathbb{P}_u P u θ = ( α u / n ) 1 / 2 ( g 1 N , … , g n N ) \theta=(\alpha u/n)^{1/2}(g_{1N},\ldots,g_{nN}) θ = ( α u / n ) 1/2 ( g 1 N , … , g n N ) b = b ( u ) b=b(u) b = b ( u )
Lemma 10.2 (Regularity in the parameters). Let μ \mu μ U = U ( q , x , y ) U=U(q,x,y) U = U ( q , x , y ) (130) with the row replaced by ϵ ∑ i θ i τ i \epsilon\sum_i\theta_i\tau_i ϵ ∑ i θ i τ i b ( u ) b(u) b ( u ) b > 0 b>0 b > 0 U U U ( θ , x , y ) (\theta,x,y) ( θ , x , y ) N N N θ \theta θ b b b ( q , x , y ) (q,x,y) ( q , x , y ) U U U ( b , q , θ , x , y ) ∈ ( 0 , ∞ ) × [ 0 , 1 ] × R n × R n × R (b,q,\theta,x,y)\in(0,\infty)\times[0,1]\times\mathbb{R}^n\times\mathbb{R}^n\times\mathbb{R} ( b , q , θ , x , y ) ∈ ( 0 , ∞ ) × [ 0 , 1 ] × R n × R n × R ∇ \nabla ∇ ( x , y ) (x,y) ( x , y ) 0 ≤ r ≤ h ≤ 1 0\le r\le h\le1 0 ≤ r ≤ h ≤ 1
U ( r , x , y ) − U ( h , x , y ) = 1 2 ∫ r h [ Tr ( Λ b ∇ 2 U ) + a ( q ) ∇ U T Λ b ∇ U ] ( q , x , y ) d q . (134) U(r,x,y)-U(h,x,y)=\frac{1}{2}\int_r^h\left[\operatorname{Tr}(\Lambda_b\nabla^2U)+a(q)\nabla U^{\mathsf T}\Lambda_b\nabla U\right](q,x,y)\,\mathrm{d}q. \tag*{(134)} U ( r , x , y ) − U ( h , x , y ) = 2 1 ∫ r h [ Tr ( Λ b ∇ 2 U ) + a ( q ) ∇ U T Λ b ∇ U ] ( q , x , y ) d q . (134) Proof. The proof has three steps. We first prove the bounds for step profiles, using the density of the future of the field. We then pass to arbitrary profiles with the approximation of [4 Section 2.6] , and finally we prove the continuity in q q q b b b
Step 1: step profiles. Let μ \mu μ (130) do not depend on ( x , y ) (x,y) ( x , y ) ( x ′ , y ′ ) (x',y') ( x ′ , y ′ ) x ′ ⋅ τ + y ′ ϵ x'\cdot\tau+y'\epsilon x ′ ⋅ τ + y ′ ϵ ( θ , x , y ) (\theta,x,y) ( θ , x , y ) z = 0 z=0 z = 0 z j z_j z j
E z ( σ ; x , y ) = E ( σ ; x , y ) + ∑ j z j η j ( σ ) , \mathcal{E}_z(\sigma;x,y)=\mathcal{E}(\sigma;x,y)+\sum_j z_j\eta_j(\sigma), E z ( σ ; x , y ) = E ( σ ; x , y ) + j ∑ z j η j ( σ ) , where the features η j \eta_j η j τ i ϵ \tau_i\epsilon τ i ϵ τ i \tau_i τ i i ≤ n i\le n i ≤ n ϵ \epsilon ϵ ∣ η j ∣ ≤ 1 |\eta_j|\le1 ∣ η j ∣ ≤ 1 U z U_z U z q q q q q q ( x , y ) (x,y) ( x , y ) q q q [ q ℓ , q ℓ + 1 ) [q_\ell,q_{\ell+1}) [ q ℓ , q ℓ + 1 ) a = a ℓ > 0 a=a_\ell>0 a = a ℓ > 0 e a ℓ U z ( q ℓ ) = E e a ℓ U z ( q ℓ + 1 ) e^{a_\ell U_z(q_\ell)}=\mathbb{E}e^{a_\ell U_z(q_{\ell+1})} e a ℓ U z ( q ℓ ) = E e a ℓ U z ( q ℓ + 1 ) ( q ℓ + 1 − q ℓ ) Λ b (q_{\ell+1}-q_\ell)\Lambda_b ( q ℓ + 1 − q ℓ ) Λ b e a ℓ ( U z ( q ℓ + 1 ) − U z ( q ℓ ) ) e^{a_\ell(U_z(q_{\ell+1})-U_z(q_\ell))} e a ℓ ( U z ( q ℓ + 1 ) − U z ( q ℓ )) a ℓ = 0 a_\ell=0 a ℓ = 0 e E z ( σ ) − U z ( 1 ) e^{\mathcal{E}_z(\sigma)-U_z(1)} e E z ( σ ) − U z ( 1 ) [4 Section 2.1, (2.1)] . The coefficient of U z U_z U z q ℓ > q q_\ell>q q ℓ > q a ℓ − 1 − a ℓ = − ν ( { q ℓ } ) a_{\ell-1}-a_\ell=-\nu(\{q_\ell\}) a ℓ − 1 − a ℓ = − ν ({ q ℓ }) 1 1 1 a ( 1 − ) − 1 = − ν ( { 1 } ) a(1-)-1=-\nu(\{1\}) a ( 1 − ) − 1 = − ν ({ 1 }) q q q
L z = exp { E z ( σ ; Z 1 ) − a ( q ) U z ( q , Z q ) − ∫ ( q , 1 ] U z ( r , Z r ) ν ( d r ) } (135) L_z=\exp\left\{\mathcal{E}_z(\sigma;Z_1)-a(q)U_z(q,Z_q)-\int_{(q,1]}U_z(r,Z_r)\,\nu(\mathrm{d}r)\right\} \tag*{(135)} L z = exp { E z ( σ ; Z 1 ) − a ( q ) U z ( q , Z q ) − ∫ ( q , 1 ] U z ( r , Z r ) ν ( d r ) } (135) with respect to driftless Gaussian increments of covariance Λ b \Lambda_b Λ b σ \sigma σ Z r Z_r Z r r r r
We write E z \mathbb{E}_z E z ℓ z = log L z \ell_z=\log L_z ℓ z = log L z a ℓ > 0 a_\ell>0 a ℓ > 0 e a ℓ U z ( q ℓ ) = E e a ℓ U z ( q ℓ + 1 ) e^{a_\ell U_z(q_\ell)}=\mathbb{E}e^{a_\ell U_z(q_{\ell+1})} e a ℓ U z ( q ℓ ) = E e a ℓ U z ( q ℓ + 1 ) ∂ z j U z ( q ℓ ) \partial_{z_j}U_z(q_\ell) ∂ z j U z ( q ℓ ) ∂ z j U z ( q ℓ + 1 ) \partial_{z_j}U_z(q_{\ell+1}) ∂ z j U z ( q ℓ + 1 ) a ℓ = 0 a_\ell=0 a ℓ = 0 1 1 1 ∂ z j U z ( 1 ) \partial_{z_j}U_z(1) ∂ z j U z ( 1 ) η j \eta_j η j
∂ z j U z ( q ) = E z η j , so that ∣ ∂ z j U z ∣ ≤ 1. (136) \partial_{z_j}U_z(q)=\mathbb{E}_z\eta_j,\qquad\text{so that}\qquad\lvert\partial_{z_j}U_z\rvert\le1. \tag*{(136)} ∂ z j U z ( q ) = E z η j , so that ∣ ∂ z j U z ∣ ≤ 1. (136) Since a ( q ) + ν ( ( q , 1 ] ) = 1 a(q)+\nu((q,1])=1 a ( q ) + ν (( q , 1 ]) = 1 ∣ ∂ z j ℓ z ∣ ≤ 2 \lvert\partial_{z_j}\ell_z\rvert\le2 ∣ ∂ z j ℓ z ∣ ≤ 2 U z U_z U z m ′ ≥ 2 m'\ge2 m ′ ≥ 2 C m ′ C_{m'} C m ′ ℓ z \ell_z ℓ z m ′ m' m ′ C m ′ C_{m'} C m ′ z z z j , j 1 , … , j m j,j_1,\ldots,j_m j , j 1 , … , j m
∂ z j 1 ⋯ ∂ z j m ∂ z j U z ( q ) = E z [ η j ∑ π ∏ B ∈ π ∂ B ℓ z ] . \partial_{z_{j_1}}\cdots\partial_{z_{j_m}}\partial_{z_j}U_z(q)=\mathbb{E}_z\left[\eta_j\sum_{\pi}\prod_{B\in\pi}\partial_B\ell_z\right]. ∂ z j 1 ⋯ ∂ z j m ∂ z j U z ( q ) = E z [ η j π ∑ B ∈ π ∏ ∂ B ℓ z ] . where π \pi π { 1 , … , m } \{1,\ldots,m\} { 1 , … , m } ∂ B \partial_B ∂ B j i j_i j i i ∈ B i\in B i ∈ B m m m m + 1 m+1 m + 1 z j z_j z j m m m 2 n + 1 2n+1 2 n + 1
Step 2: arbitrary profiles. As in [4 Section 2.6.2] , we approximate μ \mu μ μ m = s δ 0 + ν m \mu_m=s\delta_0+\nu_m μ m = s δ 0 + ν m μ \mu μ a m a_m a m a a a L 1 ( 0 , 1 ) L^1(0,1) L 1 ( 0 , 1 ) U m U_m U m [4 (2.19)] , with c I N c_{IN} c I N Λ b \Lambda_b Λ b
sup ∣ U m − U ∣ ≤ Tr Λ b 2 ∥ a m − a ∥ L 1 ( 0 , 1 ) , \sup\lvert U_m-U\rvert\le\frac{\operatorname{Tr}\Lambda_b}{2}\lVert a_m-a\rVert_{L^1(0,1)}, sup ∣ U m − U ∣ ≤ 2 Tr Λ b ∥ a m − a ∥ L 1 ( 0 , 1 ) , where the supremum is over q , x , y q,x,y q , x , y θ \theta θ f f f h > 0 h>0 h > 0 ∥ f ′ ∥ ∞ ≤ 2 ∥ f ∥ ∞ / h + ( h / 2 ) ∥ f ′ ′ ∥ ∞ \lVert f'\rVert_\infty\le2\lVert f\rVert_\infty/h+(h/2)\lVert f''\rVert_\infty ∥ f ′ ∥ ∞ ≤ 2 ∥ f ∥ ∞ / h + ( h /2 ) ∥ f ′′ ∥ ∞ U m − U m ′ U_m-U_{m'} U m − U m ′ U m U_m U m ( θ , x , y ) (\theta,x,y) ( θ , x , y ) U U U ( θ , x , y ) (\theta,x,y) ( θ , x , y ) U m U_m U m U m U_m U m (134) with a m a_m a m a a a a m → a a_m \to a a m → a L 1 L^1 L 1
Step 3: continuity. For a step profile, the derivatives of U U U ( θ , x , y ) (\theta,x,y) ( θ , x , y ) (130) . By Step 1, their right sides are bounded by constants depending only on N N N c c c b b b κ \kappa κ ( θ , x , y ) (\theta,x,y) ( θ , x , y ) κ = 0 \kappa=0 κ = 0
sup ∣ ∂ q D κ U m ∣ ≤ C N , ∣ κ ∣ ( n c + b ) \sup|\partial_q D^\kappa U_m| \le C_{N,|\kappa|}(nc+b) sup ∣ ∂ q D κ U m ∣ ≤ C N , ∣ κ ∣ ( n c + b ) on each grid interval, where C N , ∣ κ ∣ C_{N,|\kappa|} C N , ∣ κ ∣ N N N ∣ κ ∣ |\kappa| ∣ κ ∣ q q q [ 0 , 1 ] [0,1] [ 0 , 1 ] U U U ( θ , x , y ) (\theta,x,y) ( θ , x , y ) q q q N N N n c + b nc+b n c + b b b b U b U_b U b U b ′ U_{b'} U b ′ b , b ′ > 0 b,b'>0 b , b ′ > 0 U b − U b ′ U_b-U_{b'} U b − U b ′ q = 1 q=1 q = 1 Λ b ′ \Lambda_{b'} Λ b ′ 1 2 ( b − b ′ ) [ ∂ y y U b + a ( ∂ y U b ) 2 ] \frac{1}{2}(b-b')[\partial_{yy}U_b+a(\partial_yU_b)^2] 2 1 ( b − b ′ ) [ ∂ y y U b + a ( ∂ y U b ) 2 ] ∣ ∂ y y U b ∣ ≤ 2 |\partial_{yy}U_b|\le2 ∣ ∂ y y U b ∣ ≤ 2 ∣ ∂ y U b ∣ ≤ 1 |\partial_yU_b|\le1 ∣ ∂ y U b ∣ ≤ 1 sup ∣ U b − U b ′ ∣ ≤ 3 2 ∣ b − b ′ ∣ \sup|U_b-U_{b'}|\le\frac{3}{2}|b-b'| sup ∣ U b − U b ′ ∣ ≤ 2 3 ∣ b − b ′ ∣ μ \mu μ ( θ , x , y ) (\theta,x,y) ( θ , x , y ) b b b ( θ , x , y ) (\theta,x,y) ( θ , x , y ) q q q
The density of a tree law
Let T \mathcal{T} T v v v l v l_v l v q v < 1 q_v<1 q v < 1 v v v v v v ω \omega ω T \mathcal{T} T r < h r<h r < h ν \nu ν ( r , h ) (r,h) ( r , h ) v v v
ω v = ω ( { v } ) = ν ( { q v } ) + ( l v − 1 ) a ( q v ) . \omega_v=\omega(\{v\})=\nu(\{q_v\})+(l_v-1)a(q_v). ω v = ω ({ v }) = ν ({ q v }) + ( l v − 1 ) a ( q v ) . For example, the root carries ν ( { 0 } ) + ( l − 1 ) a ( 0 ) \nu(\{0\})+(l-1)a(0) ν ({ 0 }) + ( l − 1 ) a ( 0 ) l l l l l l ν ( { 1 } ) + l − 1 \nu(\{1\})+l-1 ν ({ 1 }) + l − 1 ∑ v ω v F ( v ) \sum_v\omega_vF(v) ∑ v ω v F ( v ) ∫ F d ω \int F\,d\omega ∫ F d ω driftless Gaussian field on T \mathcal{T} T Λ \Lambda Λ
Lemma 10.3 (Density of a tree law). Let μ \mu μ u ∈ [ 0 , 1 ] u \in[0,1] u ∈ [ 0 , 1 ] T \mathcal{T} T k k k T \mathcal{T} T T \mathcal{T} T P T , u \mathbb{P}_{\mathcal{T},u} P T , u Z s ( u ) − 1 D T Z_s(u)^{-1}\mathcal{D}_{\mathcal{T}} Z s ( u ) − 1 D T
D T = exp { ∑ a = 1 k E ( σ a ; X 1 a , Y 1 a ) − ∫ T U ( q v , X v , Y v ) ω ( d v ) } , ω ≥ 0 , ω ( T ) = k − s . (137) \mathcal{D}_{\mathcal{T}} = \exp\left\{\sum_{a=1}^{k}\mathcal{E}(\sigma^a;X_1^a,Y_1^a)-\int_{\mathcal{T}}U(q_v,X_v,Y_v)\,\omega(\mathrm{d}v)\right\},\qquad\omega\ge0,\qquad\omega(\mathcal{T})=k-s. \tag*{(137)} D T = exp { a = 1 ∑ k E ( σ a ; X 1 a , Y 1 a ) − ∫ T U ( q v , X v , Y v ) ω ( d v ) } , ω ≥ 0 , ω ( T ) = k − s . (137) For fixed couplings, the conditional density of the field and the spins is e − s U ( 0 , 0 , 0 ) D T e^{-sU(0,0,0)}\mathcal{D}_{\mathcal{T}} e − s U ( 0 , 0 , 0 ) D T
Proof. The proof has three steps: we compute the likelihood of one edge, collect the coefficients of U U U ω \omega ω U U U ( x , y ) (x,y) ( x , y ) a ( q ) Λ ∇ U a(q)\Lambda\nabla U a ( q ) Λ∇ U a ( q ) 2 ∣ Λ 1 / 2 ∇ U ∣ 2 ≤ n c + b ( u ) a(q)^2\lvert\Lambda^{1/2}\nabla U\rvert^2\le nc+b(u) a ( q ) 2 ∣ Λ 1/2 ∇ U ∣ 2 ≤ n c + b ( u )
Step 1: one edge. Let Z Z Z r < h r<h r < h d Z q = Λ 1 / 2 d B q \mathrm{d}Z_q=\Lambda^{1/2}\,\mathrm{d}B_q d Z q = Λ 1/2 d B q Z r Z_r Z r
exp { ∫ r h a ( q ) ∇ U ( q , Z q ) ⋅ d Z q − 1 2 ∫ r h a ( q ) 2 ∣ Λ 1 / 2 ∇ U ( q , Z q ) ∣ 2 d q } \exp\left\{\int_r^h a(q)\nabla U(q,Z_q)\cdot\mathrm{d}Z_q-\frac{1}{2}\int_r^h a(q)^2\lvert\Lambda^{1/2}\nabla U(q,Z_q)\rvert^2\,\mathrm{d}q\right\} exp { ∫ r h a ( q ) ∇ U ( q , Z q ) ⋅ d Z q − 2 1 ∫ r h a ( q ) 2 ∣ Λ 1/2 ∇ U ( q , Z q ) ∣ 2 d q } with respect to the law of Z Z Z U U U ( x , y ) (x,y) ( x , y ) U U U q q q U U U q q q
d U ( q , Z q ) = ∇ U ( q , Z q ) ⋅ d Z q − a ( q ) 2 ∣ Λ 1 / 2 ∇ U ( q , Z q ) ∣ 2 d q . \mathrm{d}U(q,Z_q)=\nabla U(q,Z_q)\cdot\mathrm{d}Z_q-\frac{a(q)}{2}\lvert\Lambda^{1/2}\nabla U(q,Z_q)\rvert^2\,\mathrm{d}q. d U ( q , Z q ) = ∇ U ( q , Z q ) ⋅ d Z q − 2 a ( q ) ∣ Λ 1/2 ∇ U ( q , Z q ) ∣ 2 d q . In the last identity, we used that the time derivative of U U U q q q ∫ r h a ( q ) d U ( q , Z q ) \int_r^h a(q)\,\mathrm{d}U(q,Z_q) ∫ r h a ( q ) d U ( q , Z q ) U q = U ( q , Z q ) U_q=U(q,Z_q) U q = U ( q , Z q ) a a a ν \nu ν ( r , h ] (r,h] ( r , h ] a a a U q U_q U q ∫ r h a d U q = a ( h ) U h − a ( r ) U r − ∫ ( r , h ] U q ν ( d q ) \int_r^h a\,\mathrm{d}U_q=a(h)U_h-a(r)U_r-\int_{(r,h]}U_q\,\nu(\mathrm{d}q) ∫ r h a d U q = a ( h ) U h − a ( r ) U r − ∫ ( r , h ] U q ν ( d q ) a ( h ) − ν ( { h } ) = a ( h − ) a(h)-\nu(\{h\})=a(h-) a ( h ) − ν ({ h }) = a ( h − )
exp { a ( h − ) U ( h , X h , Y h ) − a ( r ) U ( r , X r , Y r ) − ∫ ( r , h ) U ( q , X q , Y q ) ν ( d q ) } (138) \exp\left\{a(h-)U(h,X_h,Y_h)-a(r)U(r,X_r,Y_r)-\int_{(r,h)}U(q,X_q,Y_q)\,\nu(\mathrm{d}q)\right\} \tag*{(138)} exp { a ( h − ) U ( h , X h , Y h ) − a ( r ) U ( r , X r , Y r ) − ∫ ( r , h ) U ( q , X q , Y q ) ν ( d q ) } (138) where we wrote the field as ( X , Y ) (X,Y) ( X , Y ) ν \nu ν r r r h h h
Step 2: the vertices. We multiply the root tilt e s U ( 0 , 0 , 0 ) / Z s ( u ) e^{sU(0,0,0)}/Z_s(u) e s U ( 0 , 0 , 0 ) / Z s ( u ) (138) , and the Gibbs probabilities e ε ( σ a X 1 a , γ 1 a ) − U ( 1 , X 1 a , γ 1 a ) e^{\varepsilon(\sigma^a X_1^a,\gamma_1^a)-U(1,X_1^a,\gamma_1^a)} e ε ( σ a X 1 a , γ 1 a ) − U ( 1 , X 1 a , γ 1 a ) − ∫ U d ν -\int U\,\mathrm{d}\nu − ∫ U d ν v v v 0 < q v < 1 0<q_v<1 0 < q v < 1 a ( q v − ) U ( v ) a(q_{v-})U(v) a ( q v − ) U ( v ) l v l_v l v − l v a ( q v ) U ( v ) -l_v a(q_v)U(v) − l v a ( q v ) U ( v ) U ( v ) U(v) U ( v )
a ( q v − ) − l v a ( q v ) = − [ ν ( { q v } ) + ( l v − 1 ) a ( q v ) ] = − ω v . a(q_{v-})-l_v a(q_v)=-\left[\nu(\{q_v\})+(l_v-1)a(q_v)\right]=-\omega_v. a ( q v − ) − l v a ( q v ) = − [ ν ({ q v }) + ( l v − 1 ) a ( q v ) ] = − ω v . At the root, the tilt and the l l l s − l a ( 0 ) = − [ ν ( { 0 } ) + ( l − 1 ) a ( 0 ) ] s-la(0)=-[\nu(\{0\})+(l-1)a(0)] s − l a ( 0 ) = − [ ν ({ 0 }) + ( l − 1 ) a ( 0 )] a ( 0 ) = s + ν ( { 0 } ) a(0)=s+\nu(\{0\}) a ( 0 ) = s + ν ({ 0 }) l l l l l l a ( 1 − ) − l = − [ ν ( { 1 } ) + l − 1 ] a(1-)-l=-[\nu(\{1\})+l-1] a ( 1 − ) − l = − [ ν ({ 1 }) + l − 1 ] a ( 1 ) = 1 a(1)=1 a ( 1 ) = 1 ω ≥ 0 \omega\geq0 ω ≥ 0 Z s ( u ) − 1 D T Z_s(u)^{-1}\mathcal{D}_{T} Z s ( u ) − 1 D T D T \mathcal{D}_{T} D T (137) . The edge likelihoods and the Gibbs probabilities are normalized. Integrating D T \mathcal{D}_{T} D T e s U ( 0 , 0 , 0 ) e^{sU(0,0,0)} e s U ( 0 , 0 , 0 ) Z s ( u ) Z_s(u) Z s ( u ) Z s ( u ) Z_s(u) Z s ( u )
Step 3: the mass. For a single leaf, ω \omega ω ν \nu ν ω ( T ) = ν ( [ 0 , 1 ] ) = 1 − s \omega(T)=\nu([0,1])=1-s ω ( T ) = ν ([ 0 , 1 ]) = 1 − s l − 1 ≥ 1 l-1\geq1 l − 1 ≥ 1 q q q 1 1 1 l − 1 l-1 l − 1 ( l − 1 ) a ( q ) (l-1)a(q) ( l − 1 ) a ( q ) ( l − 1 ) ν ( ( q , 1 ] ) (l-1)\nu((q,1]) ( l − 1 ) ν (( q , 1 ]) a ( q ) + ν ( ( q , 1 ] ) = 1 a(q)+\nu((q,1])=1 a ( q ) + ν (( q , 1 ]) = 1 ω \omega ω l − 1 l-1 l − 1 ω ( T ) = 1 − s + ( k − 1 ) = k − s \omega(T)=1-s+(k-1)=k-s ω ( T ) = 1 − s + ( k − 1 ) = k − s
We also see from the density (137) that an extended tree, obtained by attaching new leaves to T T T P T , u \mathbb{P}_{T,u} P T , u ε ∑ i θ i τ i \varepsilon\sum_i\theta_i\tau_i ε ∑ i θ i τ i b ( u ) b(u) b ( u ) b > 0 b>0 b > 0 Lemma 10.2 . We then take the density with respect to the Gaussian law of the couplings g i j g_{ij} g ij i < j ≤ n i<j\leq n i < j ≤ n e s U ( 0 , 0 , 0 ) e^{sU(0,0,0)} e s U ( 0 , 0 , 0 )
Responses to the row and to the last-field variance
Let the field ( X , Y ) (X,Y) ( X , Y ) σ \sigma σ Lemma 10.2 , and let G q \mathcal{G}_q G q q q q
π i ( q ) = E [ τ i ε ∣ G q ] , m N ( q ) = E [ ε ∣ G q ] , \pi_i(q)=\mathbb{E}[\tau_i\varepsilon\mid\mathcal{G}_q],\qquad m_N(q)=\mathbb{E}[\varepsilon\mid\mathcal{G}_q], π i ( q ) = E [ τ i ε ∣ G q ] , m N ( q ) = E [ ε ∣ G q ] , and we evaluate the derivatives of U U U ( q , X q , Y q ) (q,X_q,Y_q) ( q , X q , Y q ) ∂ θ i U ( q ) \partial_{\theta_i}U(q) ∂ θ i U ( q )
Lemma 10.4 (Parameter responses). Let μ \mu μ U U U π i \pi_i π i m N m_N m N θ ∈ R n \theta\in\mathbb{R}^n θ ∈ R n b > 0 b>0 b > 0 q ∈ [ 0 , 1 ] q\in[0,1] q ∈ [ 0 , 1 ] i ≤ n i\leq n i ≤ n
∂ θ i U ( q ) = π i ( q ) , ∂ θ i 2 U ( q ) = 1 − a ( q ) π i ( q ) 2 − ∫ ( q , 1 ] E [ π i ( r ) 2 ∣ G q ] ν ( d r ) . (139) \partial_{\theta_i}U(q)=\pi_i(q),\qquad
\partial_{\theta_i}^2U(q)=1-a(q)\pi_i(q)^2-\int_{(q,1]}\mathbb{E}[\pi_i(r)^2\mid\mathcal{G}_q]\,\nu(\mathrm{d}r). \tag*{(139)} ∂ θ i U ( q ) = π i ( q ) , ∂ θ i 2 U ( q ) = 1 − a ( q ) π i ( q ) 2 − ∫ ( q , 1 ] E [ π i ( r ) 2 ∣ G q ] ν ( d r ) . (139) ∂ y U ( q ) = m N ( q ) , ∂ y y U ( q ) = 1 − a ( q ) m N ( q ) 2 − ∫ ( q , 1 ] E [ m N ( r ) 2 ∣ G q ] ν ( d r ) . (140) \partial_y U(q) = m_N(q), \qquad\partial_{yy} U(q) = 1-a(q)m_N(q)^2-\int_{(q,1]} \mathbb{E}[m_N(r)^2\mid\mathcal{G}_q]\,\nu(\mathrm{d}r). \tag*{(140)} ∂ y U ( q ) = m N ( q ) , ∂ y y U ( q ) = 1 − a ( q ) m N ( q ) 2 − ∫ ( q , 1 ] E [ m N ( r ) 2 ∣ G q ] ν ( d r ) . (140) Further, the derivative of U U U b b b q q q ( b , θ , x , y ) (b,\theta,x,y) ( b , θ , x , y )
∂ b U ( q ) = 1 2 [ ( 1 − q ) − ∫ ( q , 1 ] ( r − q ) E [ m N ( r ) 2 ∣ G q ] ν ( d r ) ] . (141) \partial_b U(q)=\frac{1}{2}\left[(1-q)-\int_{(q,1]}(r-q)\mathbb{E}[m_N(r)^2\mid\mathcal{G}_q]\,\nu(\mathrm{d}r)\right]. \tag*{(141)} ∂ b U ( q ) = 2 1 [ ( 1 − q ) − ∫ ( q , 1 ] ( r − q ) E [ m N ( r ) 2 ∣ G q ] ν ( d r ) ] . (141) In particular, 0 ≤ ∂ b U ( q ) ≤ ( 1 − q ) / 2 0\leq\partial_b U(q)\leq(1-q)/2 0 ≤ ∂ b U ( q ) ≤ ( 1 − q ) /2
For a single row, (139) and (140) are the conditional form of [4 (2.10)] . The identity (141) is the conditional, one-coordinate form of the field-variance computation following [4 (2.11)] . Under P u \mathbb{P}_u P u G q \mathcal{G}_q G q θ = ( α u / n ) 1 / 2 ( g 1 N , … , g n N ) \theta=(\alpha u/n)^{1/2}(g_{1N},\ldots,g_{nN}) θ = ( α u / n ) 1/2 ( g 1 N , … , g n N ) b = b ( u ) b=b(u) b = b ( u )
Proof. The proof has four steps. We prove (139) and (140) for step profiles with the density (135) , and we pass to arbitrary profiles with the exact density of Lemma 10.3. We then treat the derivative in b b b
Step 1: step profiles. Let μ \mu μ η \eta η τ i ϵ \tau_i\epsilon τ i ϵ ϵ \epsilon ϵ z z z (136) , we have ∂ z U ( r ) = E [ η ∣ G r ] \partial_zU(r)=\mathbb{E}[\eta\mid\mathcal{G}_r] ∂ z U ( r ) = E [ η ∣ G r ] r r r ϖ ( r ) \varpi(r) ϖ ( r ) (136) once more under the density (135) , we find
∂ z 2 U ( q ) = E [ η ∂ z ℓ z ∣ G q ] = E [ η 2 ∣ G q ] − a ( q ) ϖ ( q ) 2 − ∫ ( q , 1 ] E [ η ∂ z U ( r ) ∣ G q ] ν ( d r ) . \partial_z^2U(q)=\mathbb{E}[\eta\,\partial_z\ell_z\mid\mathcal{G}_q]=\mathbb{E}[\eta^2\mid\mathcal{G}_q]-a(q)\varpi(q)^2-\int_{(q,1]}\mathbb{E}[\eta\,\partial_zU(r)\mid\mathcal{G}_q]\,\nu(\mathrm{d}r). ∂ z 2 U ( q ) = E [ η ∂ z ℓ z ∣ G q ] = E [ η 2 ∣ G q ] − a ( q ) ϖ ( q ) 2 − ∫ ( q , 1 ] E [ η ∂ z U ( r ) ∣ G q ] ν ( d r ) . For r ≥ q r\geq q r ≥ q E [ η ∂ z U ( r ) ∣ G q ] = E [ ϖ ( r ) E [ η ∣ G r ] ∣ G q ] = E [ ϖ ( r ) 2 ∣ G q ] \mathbb{E}[\eta\,\partial_zU(r)\mid\mathcal{G}_q]=\mathbb{E}[\varpi(r)\mathbb{E}[\eta\mid\mathcal{G}_r]\mid\mathcal{G}_q]=\mathbb{E}[\varpi(r)^2\mid\mathcal{G}_q] E [ η ∂ z U ( r ) ∣ G q ] = E [ ϖ ( r ) E [ η ∣ G r ] ∣ G q ] = E [ ϖ ( r ) 2 ∣ G q ] η 2 = 1 \eta^2=1 η 2 = 1 (139) for η = τ i ϵ \eta=\tau_i\epsilon η = τ i ϵ (140) for η = ϵ \eta=\epsilon η = ϵ μ ( [ 0 , q ] ) = a ( q ) \mu([0,q])=a(q) μ ([ 0 , q ]) = a ( q ) μ = ν \mu=\nu μ = ν ( q , 1 ] (q,1] ( q , 1 ]
∂ z 2 U ( q ) = 1 − ∫ [ 0 , 1 ] E [ ϖ ( q ∨ r ) 2 ∣ G q ] μ ( d r ) . (142) \partial_z^2U(q)=1-\int_{[0,1]}\mathbb{E}[\varpi(q\vee r)^2\mid\mathcal{G}_q]\,\mu(\mathrm{d}r). \tag*{(142)} ∂ z 2 U ( q ) = 1 − ∫ [ 0 , 1 ] E [ ϖ ( q ∨ r ) 2 ∣ G q ] μ ( d r ) . (142) Step 2: arbitrary profiles. Let μ m ⇒ μ \mu_m\Rightarrow\mu μ m ⇒ μ q q q 1 1 1 (135) with z = 0 z=0 z = 0 q q q E ( σ ; Z 1 ) − ∫ [ 0 , 1 ] U ( q ∨ r , Z q ∨ r ) μ ( d r ) \mathcal{E}(\sigma;Z_1)-\int_{[0,1]}U(q\vee r,Z_{q\vee r})\,\mu(\mathrm{d}r) E ( σ ; Z 1 ) − ∫ [ 0 , 1 ] U ( q ∨ r , Z q ∨ r ) μ ( d r ) r r r U m → U U_m\to U U m → U μ m ⇒ μ \mu_m\Rightarrow\mu μ m ⇒ μ μ m \mu_m μ m μ \mu μ L m L_m L m μ m \mu_m μ m
log L m = E ( σ ; Z 1 ) − U m ( 1 , Z 1 ) + ∫ [ 0 , 1 ] [ U m ( 1 , Z 1 ) − U m ( q ∨ r , Z q ∨ r ) ] μ m ( d r ) , \log L_m=\mathcal{E}(\sigma;Z_1)-U_m(1,Z_1)+\int_{[0,1]}\left[U_m(1,Z_1)-U_m(q\vee r,Z_{q\vee r})\right]\,\mu_m(\mathrm{d}r), log L m = E ( σ ; Z 1 ) − U m ( 1 , Z 1 ) + ∫ [ 0 , 1 ] [ U m ( 1 , Z 1 ) − U m ( q ∨ r , Z q ∨ r ) ] μ m ( d r ) , since μ m ( [ 0 , 1 ] ) = 1 \mu_m([0,1])=1 μ m ([ 0 , 1 ]) = 1 U m ( 1 , ⋅ ) U_m(1,\cdot) U m ( 1 , ⋅ ) e E ( σ ; ⋅ ) e^{E(\sigma;\cdot)} e E ( σ ; ⋅ ) Lemma 10.2 , and the Lipschitz bound in q q q
log L m ≤ C N ( n c + b + sup q ≤ r ≤ 1 ∣ Z r ∣ ) . \log L_m \le C_N\left(nc+b+\sup_{q\le r\le1}|Z_r|\right). log L m ≤ C N ( n c + b + q ≤ r ≤ 1 sup ∣ Z r ∣ ) . where C N C_N C N N N N μ \mu μ L 1 L^1 L 1 (136) converges by Lemma 10.2 , so ∂ z U ( r ) = E [ η ∣ G r ] \partial_z U(r)=\mathbb{E}[\eta\mid\mathcal{G}_r] ∂ z U ( r ) = E [ η ∣ G r ] μ \mu μ r ↦ E [ ϖ ( q ∨ r ) 2 ∣ G q ] r\mapsto\mathbb{E}[\varpi(q\vee r)^2\mid\mathcal{G}_q] r ↦ E [ ϖ ( q ∨ r ) 2 ∣ G q ] (142) . Its value for μ m \mu_m μ m μ \mu μ L 1 L^1 L 1 r r r r r r Lemma 10.2 . Together with the weak convergence of μ m \mu_m μ m (142) , and hence (139) and (140) , for μ \mu μ ν \nu ν q q q a ( q ) a(q) a ( q ) 1 1 1
Step 3: the derivative in b b b Let μ \mu μ q ℓ ≤ q < q ℓ + 1 q_\ell\le q<q_{\ell+1} q ℓ ≤ q < q ℓ + 1 a = a ℓ a=a_\ell a = a ℓ [ q ℓ , q ℓ + 1 ) [q_\ell,q_{\ell+1}) [ q ℓ , q ℓ + 1 ) a ℓ > 0 a_\ell>0 a ℓ > 0 ϕ = e a ℓ U \phi=e^{a_\ell U} ϕ = e a ℓ U ϕ \phi ϕ ∂ q ϕ + 1 2 Tr ( Λ b ∇ 2 ϕ ) = 0 \partial_q\phi+\frac{1}{2}\operatorname{Tr}(\Lambda_b\nabla^2\phi)=0 ∂ q ϕ + 2 1 Tr ( Λ b ∇ 2 ϕ ) = 0 ϕ ( q ) \phi(q) ϕ ( q ) ϕ ( q ℓ + 1 ) \phi(q_{\ell+1}) ϕ ( q ℓ + 1 ) ( q ℓ + 1 − q ) Λ b (q_{\ell+1}-q)\Lambda_b ( q ℓ + 1 − q ) Λ b b b b a ℓ ϕ ( q ) a_\ell\phi(q) a ℓ ϕ ( q )
∂ b U ( q ) = E [ ∂ b U ( q ℓ + 1 ) ∣ G q ] + q ℓ + 1 − q 2 E [ ( ∂ y y U + a ℓ ( ∂ y U ) 2 ) ( q ℓ + 1 ) ∣ G q ] . \partial_b U(q)=\mathbb{E}[\partial_b U(q_{\ell+1})\mid\mathcal{G}_q]+\frac{q_{\ell+1}-q}{2}\mathbb{E}\left[(\partial_{yy}U+a_\ell(\partial_yU)^2)(q_{\ell+1})\mid\mathcal{G}_q\right]. ∂ b U ( q ) = E [ ∂ b U ( q ℓ + 1 ) ∣ G q ] + 2 q ℓ + 1 − q E [ ( ∂ y y U + a ℓ ( ∂ y U ) 2 ) ( q ℓ + 1 ) ∣ G q ] . The function ∂ y y ϕ = a ℓ ( ∂ y y U + a ℓ ( ∂ y U ) 2 ) ϕ \partial_{yy}\phi=a_\ell(\partial_{yy}U+a_\ell(\partial_yU)^2)\phi ∂ y y ϕ = a ℓ ( ∂ y y U + a ℓ ( ∂ y U ) 2 ) ϕ q q q r ≤ q ℓ + 1 r\le q_{\ell+1} r ≤ q ℓ + 1 ϕ ( r ) / ϕ ( q ) \phi(r)/\phi(q) ϕ ( r ) / ϕ ( q ) E [ ( ∂ y y U + a ℓ ( ∂ y U ) 2 ) ( r ) ∣ G q ] \mathbb{E}[(\partial_{yy}U+a_\ell(\partial_yU)^2)(r)\mid\mathcal{G}_q] E [( ∂ y y U + a ℓ ( ∂ y U ) 2 ) ( r ) ∣ G q ] r ∈ [ q , q ℓ + 1 ] r\in[q,q_{\ell+1}] r ∈ [ q , q ℓ + 1 ] 1 2 ∫ q q ℓ + 1 E [ ( ∂ y y U + a ( ∂ y U ) 2 ) ( r ) ∣ G q ] d r \frac{1}{2}\int_q^{q_{\ell+1}}\mathbb{E}[(\partial_{yy}U+a(\partial_yU)^2)(r)\mid\mathcal{G}_q]\,dr 2 1 ∫ q q ℓ + 1 E [( ∂ y y U + a ( ∂ y U ) 2 ) ( r ) ∣ G q ] d r a ℓ = 0 a_\ell=0 a ℓ = 0 ϕ = U \phi=U ϕ = U ∂ b U ( 1 ) = 0 \partial_bU(1)=0 ∂ b U ( 1 ) = 0
∂ b U ( q ) = 1 2 ∫ q 1 E [ ∂ y y U ( r ) + a ( r ) m N ( r ) 2 ∣ G q ] d r . (143) \partial_bU(q)=\frac{1}{2}\int_q^1\mathbb{E}\left[\partial_{yy}U(r)+a(r)m_N(r)^2\mid\mathcal{G}_q\right]\,dr. \tag*{(143)} ∂ b U ( q ) = 2 1 ∫ q 1 E [ ∂ y y U ( r ) + a ( r ) m N ( r ) 2 ∣ G q ] d r . (143) where we used that ∂ y U ( r ) = m N ( r ) \partial_yU(r)=m_N(r) ∂ y U ( r ) = m N ( r )
Step 4: the derivative in b b b By (140) , the integrand in (143) lies in [ 0 , 1 ] [0,1] [ 0 , 1 ] R ( b ) R(b) R ( b ) (143) for μ \mu μ μ m \mu_m μ m (143) converges to R ( b ) R(b) R ( b ) b b b a m → a a_m\to a a m → a L 1 ( 0 , 1 ) L^1(0,1) L 1 ( 0 , 1 ) 1 / 2 1/2 1/2 U m ( b 2 ) − U m ( b 1 ) = ∫ b 1 b 2 ∂ b U m d b U_m(b_2)-U_m(b_1)=\int_{b_1}^{b_2}\partial_bU_m\,\mathrm{d}b U m ( b 2 ) − U m ( b 1 ) = ∫ b 1 b 2 ∂ b U m d b U ( b 2 ) − U ( b 1 ) = ∫ b 1 b 2 R ( b ) d b U(b_2)-U(b_1)=\int_{b_1}^{b_2}R(b)\,\mathrm{d}b U ( b 2 ) − U ( b 1 ) = ∫ b 1 b 2 R ( b ) d b b \sqrt{b} b q q q R R R ( b , θ , x , y ) (b,\theta,x,y) ( b , θ , x , y ) ∂ b U \partial_bU ∂ b U R R R μ \mu μ r r r
∂ b U ( q ) = 1 2 ∫ q 1 ( 1 − ∫ ( r , 1 ] E [ m N ( ℓ ) 2 ∣ G q ] ν ( d ℓ ) ) d r . \partial_b U(q)=\frac{1}{2}\int_q^1\left(1-\int_{(r,1]}\mathbb{E}\left[m_N(\ell)^2\mid\mathcal{G}_q\right]\nu(\mathrm{d}\ell)\right)\mathrm{d}r. ∂ b U ( q ) = 2 1 ∫ q 1 ( 1 − ∫ ( r , 1 ] E [ m N ( ℓ ) 2 ∣ G q ] ν ( d ℓ ) ) d r . By Fubini’s theorem, we have ∫ q 1 d r ∫ ( r , 1 ] F ( ℓ ) ν ( d ℓ ) = ∫ ( q , 1 ] ( ℓ − q ) F ( ℓ ) ν ( d ℓ ) \int_q^1\mathrm{d}r\int_{(r,1]}F(\ell)\nu(\mathrm{d}\ell)=\int_{(q,1]}(\ell-q)F(\ell)\nu(\mathrm{d}\ell) ∫ q 1 d r ∫ ( r , 1 ] F ( ℓ ) ν ( d ℓ ) = ∫ ( q , 1 ] ( ℓ − q ) F ( ℓ ) ν ( d ℓ ) F ≥ 0 F\geq0 F ≥ 0 0 0 0 ( 1 − q ) ν ( ( q , 1 ] ) ≤ 1 − q (1-q)\nu((q,1])\leq1-q ( 1 − q ) ν (( q , 1 ]) ≤ 1 − q 0 ≤ ∂ b U ( q ) ≤ ( 1 − q ) / 2 0\leq\partial_bU(q)\leq(1-q)/2 0 ≤ ∂ b U ( q ) ≤ ( 1 − q ) /2
Differentiation in the interpolation parameter
For a point v v v T \mathcal{T} T v ^ \hat{v} v ^ v ˇ \check{v} v ˇ v v v q v q_v q v v v v v v v v v v v v v v v v r > q v r>q_v r > q v r ^ \hat{r} r ^ r ˇ \check{r} r ˇ v v v r r r T \mathcal{T} T v v v v v v F F F T \mathcal{T} T G , G ′ G,G' G , G ′
E [ F G ( σ v ^ ) ] = E T , u [ F E [ G ( σ ) ∣ G q v ] ( v ) ] , \mathbb{E}\left[F G(\sigma^{\hat{v}})\right]=\mathbb{E}_{\mathcal{T},u}\left[F\mathbb{E}\left[G(\sigma)\mid\mathcal{G}_{q_v}\right](v)\right], E [ F G ( σ v ^ ) ] = E T , u [ F E [ G ( σ ) ∣ G q v ] ( v ) ] , E [ F G ( σ r ^ ) G ′ ( σ r ˇ ) ] = E T , u [ F E [ E [ G ( σ ) ∣ G r ] E [ G ′ ( σ ) ∣ G r ] ∣ G q v ] ( v ) ] . (144) \mathbb{E}\left[F G(\sigma^{\hat{r}})G'(\sigma^{\check{r}})\right]=\mathbb{E}_{\mathcal{T},u}\left[F\mathbb{E}\left[\mathbb{E}\left[G(\sigma)\mid\mathcal{G}_r\right]\mathbb{E}\left[G'(\sigma)\mid\mathcal{G}_r\right]\mid\mathcal{G}_{q_v}\right](v)\right]. \tag*{(144)} E [ F G ( σ r ^ ) G ′ ( σ r ˇ ) ] = E T , u [ F E [ E [ G ( σ ) ∣ G r ] E [ G ′ ( σ ) ∣ G r ] ∣ G q v ] ( v ) ] . (144) Here the left sides are taken under the extended trees. A conditional expectation under P u \mathbb{P}_u P u G q v \mathcal{G}_{q_v} G q v q v q_v q v ( v ) (v) ( v ) v v v v ^ \hat{v} v ^ w ˇ \check{w} w ˇ v v v w w w v v v w w w
Let φ a b \varphi_{ab} φ ab F F F T \mathcal{T} T
S φ ( F ) = ∑ a ≠ b E ( F ϵ a ϵ b φ a b ) − 2 ∑ a , v ω v E ( F ϵ a ϵ v ^ φ a v ^ ) + ∑ v , w ω v ω w E ( F ϵ v ^ ϵ w ˇ φ v ^ w ˇ ) + ∑ v ω v a ( q v ) E ( F ϵ v ^ ϵ v ˇ φ v ^ v ˇ ) + ∑ v ω v ∫ ( q v , 1 ] E ( F ϵ r ^ ϵ r ˇ φ r ^ r ˇ ) ν ( d r ) . \mathcal{S}_{\varphi}(F)=\sum_{a\ne b}\mathbb{E}(F\epsilon_a\epsilon_b\varphi_{ab})-2\sum_{a,v}\omega_v\mathbb{E}(F\epsilon_a\epsilon_{\hat{v}}\varphi_{a\hat{v}})+\sum_{v,w}\omega_v\omega_w\mathbb{E}(F\epsilon_{\hat{v}}\epsilon_{\check{w}}\varphi_{\hat{v}\check{w}})
+\sum_v\omega_v a(q_v)\mathbb{E}(F\epsilon_{\hat{v}}\epsilon_{\check{v}}\varphi_{\hat{v}\check{v}})+\sum_v\omega_v\int_{(q_v,1]}\mathbb{E}(F\epsilon_{\hat{r}}\epsilon_{\check{r}}\varphi_{\hat{r}\check{r}})\nu(\mathrm{d}r). S φ ( F ) = a = b ∑ E ( F ϵ a ϵ b φ ab ) − 2 a , v ∑ ω v E ( F ϵ a ϵ v ^ φ a v ^ ) + v , w ∑ ω v ω w E ( F ϵ v ^ ϵ w ˇ φ v ^ w ˇ ) + v ∑ ω v a ( q v ) E ( F ϵ v ^ ϵ v ˇ φ v ^ v ˇ ) + v ∑ ω v ∫ ( q v , 1 ] E ( F ϵ r ^ ϵ r ˇ φ r ^ r ˇ ) ν ( d r ) . The sums run over leaves a , b a,b a , b T T T v , w v,w v , w T T T ω \omega ω T T T φ a b = R a b − \varphi_{ab}=R_{ab}^{-} φ ab = R ab − φ a b = q a b \varphi_{ab}=q_{ab} φ ab = q ab φ a b = Δ a b \varphi_{ab}=\Delta_{ab} φ ab = Δ ab S Δ = S R − − S q \mathcal{S}_{\Delta}=\mathcal{S}_{R^{-}}-\mathcal{S}_{q} S Δ = S R − − S q ϵ a ϵ b Δ a b = D a b \epsilon_a\epsilon_b\Delta_{ab}=D_{ab} ϵ a ϵ b Δ ab = D ab q a b q_{ab} q ab
Table 10.5.
Lemma 10.5 (Differentiation under the tree law). Let μ \mu μ T T T k k k F F F u ↦ E T , u F u\mapsto\mathbb{E}_{T,u}F u ↦ E T , u F C 2 ( [ 0 , 1 ] ) C^2([0,1]) C 2 ([ 0 , 1 ])
d d u E T , u F = α 2 [ ∑ a ≠ b E ( F D a b ) − 2 ∑ a , v ω v E ( F D a v ^ ) + ∑ v , w ω v ω w E ( F D v ^ w ˇ ) + ∑ v ω v a ( q v ) E ( F D v ^ v ˇ ) + ∑ v ω v ∫ ( q v , 1 ] E ( F D r ^ r ˇ ) ν ( d r ) + s E T , u F ∫ E 2 , r , u D 12 ν ( d r ) ] . (145) \frac{\mathrm{d}}{\mathrm{d}u}\mathbb{E}_{T,u}F=\frac{\alpha}{2}\left[\sum_{a\ne b}\mathbb{E}(FD_{ab})-2\sum_{a,v}\omega_v\mathbb{E}(FD_{a\hat{v}})+\sum_{v,w}\omega_v\omega_w\mathbb{E}(FD_{\hat{v}\check{w}})
+\sum_v\omega_v a(q_v)\mathbb{E}(FD_{\hat{v}\check{v}})+\sum_v\omega_v\int_{(q_v,1]}\mathbb{E}(FD_{\hat{r}\check{r}})\nu(\mathrm{d}r)
+s\mathbb{E}_{T,u}F\int\mathbb{E}_{2,r,u}D_{12}\nu(\mathrm{d}r)\right]. \tag*{(145)} d u d E T , u F = 2 α a = b ∑ E ( F D ab ) − 2 a , v ∑ ω v E ( F D a v ^ ) + v , w ∑ ω v ω w E ( F D v ^ w ˇ ) + v ∑ ω v a ( q v ) E ( F D v ^ v ˇ ) + v ∑ ω v ∫ ( q v , 1 ] E ( F D r ^ r ˇ ) ν ( d r ) + s E T , u F ∫ E 2 , r , u D 12 ν ( d r ) . (145) The sums run over leaves a , b a,b a , b v , w v,w v , w T T T ω \omega ω 4 k 2 + 2 k + 1 4k^2+2k+1 4 k 2 + 2 k + 1
In the notation above, (145) reads d d u E T , u F = α 2 [ S Δ ( F ) + s E T , u F ∫ E 2 , r , u D 12 ν ( d r ) ] \frac{\mathrm{d}}{\mathrm{d}u}\mathbb{E}_{T,u}F=\frac{\alpha}{2}\left[\mathcal{S}_{\Delta}(F)+s\mathbb{E}_{T,u}F\int\mathbb{E}_{2,r,u}D_{12}\nu(\mathrm{d}r)\right] d u d E T , u F = 2 α [ S Δ ( F ) + s E T , u F ∫ E 2 , r , u D 12 ν ( d r ) ]
Proof. If α = 0 \alpha=0 α = 0 u u u α > 0 \alpha>0 α > 0 Lemma 4.3 applies, and we compute the contributions of the row and of the last field to the derivative of the unnormalized moment. We then compute the derivative of the normalizer. Finally, we bound the coefficients and prove the C 2 C^2 C 2
Step 1: the heat identity. By Lemma 10.3 , we have E T , u F = Z s ( u ) − 1 E r e f [ F D T ] \mathbb{E}_{T,u}F=Z_s(u)^{-1}\mathbb{E}_{\mathrm{ref}}[FD_T] E T , u F = Z s ( u ) − 1 E ref [ F D T ] E r e f \mathbb{E}_{\mathrm{ref}} E ref θ = ( α u / n ) 1 / 2 ( g i N ) i ≤ n \theta=(\alpha u/n)^{1/2}(g_{iN})_{i\le n} θ = ( α u / n ) 1/2 ( g i N ) i ≤ n Lemma 4.3 on I = [ 0 , 1 ] I=[0,1] I = [ 0 , 1 ] θ \theta θ α u / n \alpha u/n α u / n
n + 1 n+1 n + 1 T \mathcal{T} T e e e r e < h e r_e<h_e r e < h e v ∈ [ 0 , 1 ] v\in[0,1] v ∈ [ 0 , 1 ]
T e , i , u ( v ) = c ( h e − r e ) v ( i ≤ n ) , T e , N , u ( v ) = b ( u ) ( h e − r e ) v . T_{e,i,u}(v)=c\left(h_e-r_e\right)v\quad(i\le n),\qquad T_{e,N,u}(v)=b(u)\left(h_e-r_e\right)v. T e , i , u ( v ) = c ( h e − r e ) v ( i ≤ n ) , T e , N , u ( v ) = b ( u ) ( h e − r e ) v . The couplings g i j g_{ij} g ij i < j ≤ n i<j\le n i < j ≤ n u u u g i j g_{ij} g ij ℓ = log D T \ell=\log\mathcal{D}_{\mathcal{T}} ℓ = log D T Φ = F \Phi=F Φ = F C \mathcal{C} C ℓ \ell ℓ e 0 e_0 e 0 e 0 e_0 e 0 (137) on the paths of the driftless field. The derivative measures below, including the atom of ω \omega ω
(A1): the variances α u / n \alpha u/n α u / n u > 0 u>0 u > 0 u u u v v v c > 0 c>0 c > 0 b ( u ) ≥ c b(u)\ge c b ( u ) ≥ c
(A2): the covariance of the last component at two field points v , w v,w v , w b ( u ) h ( v , w ) b(u)h(v,w) b ( u ) h ( v , w ) h ( v , w ) h(v,w) h ( v , w ) v v v w w w c h ( v , w ) ch(v,w) c h ( v , w ) K K K u u u K ˙ = ( α / n ) I n \dot{K}=(\alpha/n)I_n K ˙ = ( α / n ) I n K ˙ = − α h \dot{K}=-\alpha h K ˙ = − α h K ˙ = 0 \dot{K}=0 K ˙ = 0
(A3): Φ = F \Phi=F Φ = F Φ ˙ = 0 \dot{\Phi}=0 Φ ˙ = 0 ℓ \ell ℓ b ( u ) b(u) b ( u ) U U U ℓ ˙ = − b ′ ( u ) ∑ v ω v ∂ b U ( v ) = α ∑ v ω v ∂ b U ( v ) \dot{\ell}=-b'(u)\sum_v\omega_v\partial_bU(v)=\alpha\sum_v\omega_v\partial_bU(v) ℓ ˙ = − b ′ ( u ) ∑ v ω v ∂ b U ( v ) = α ∑ v ω v ∂ b U ( v ) q v q_v q v ω \omega ω ω \omega ω ℓ ˙ \dot{\ell} ℓ ˙ ( u , G ) (u,G) ( u , G ) α k / 2 \alpha k/2 α k /2 g i j g_{ij} g ij β t / N τ i τ j \beta\sqrt{t/N}\,\tau_i\tau_j β t / N τ i τ j U U U ∂ b U \partial_bU ∂ b U ℓ \ell ℓ ℓ ˙ \dot{\ell} ℓ ˙ ( u , G , g ) (u,G,g) ( u , G , g ) U U U g i j g_{ij} g ij i < j ≤ n i<j\le n i < j ≤ n β t / N τ i τ j \beta\sqrt{t/N}\,\tau_i\tau_j β t / N τ i τ j q q q U U U ∣ U ( q , 0 , 0 ) ∣ ≤ N log 2 + 3 2 ( n c + b ( u ) ) |U(q,0,0)|\le N\log2+\frac{3}{2}(nc+b(u)) ∣ U ( q , 0 , 0 ) ∣ ≤ N log 2 + 2 3 ( n c + b ( u )) E \mathcal{E} E ∣ ℓ ∣ ≤ C ( 1 + ∣ g ∣ + ∣ θ ∣ + ∥ G p a t h ∥ ∞ ) |\ell|\le C(1+|g|+|\theta|+\|G^{\mathrm{path}}\|_\infty) ∣ ℓ ∣ ≤ C ( 1 + ∣ g ∣ + ∣ θ ∣ + ∥ G path ∥ ∞ ) C C C N N N k k k β \beta β g g g g i j g_{ij} g ij i < j ≤ n i<j\le n i < j ≤ n
(A4): the score of ℓ \ell ℓ τ i a ϵ a \tau_i^a\epsilon_a τ i a ϵ a τ a \tau^a τ a ϵ a \epsilon_a ϵ a a a a − ∇ U ( v ) ω ( d v ) -\nabla U(v)\,\omega(\mathrm{d}v) − ∇ U ( v ) ω ( d v ) ∇ \nabla ∇ E \mathcal{E} E ℓ \ell ℓ − ∇ 2 U ( v ) ω ( d v ) -\nabla^2 U(v)\,\omega(\mathrm{d}v) − ∇ 2 U ( v ) ω ( d v ) v v v − ∂ θ i θ j U ( v ) ω ( d v ) -\partial_{\theta_i\theta_j}U(v)\,\omega(\mathrm{d}v) − ∂ θ i θ j U ( v ) ω ( d v ) − ∂ θ i y U ( v ) ω ( d v ) -\partial_{\theta_i y}U(v)\,\omega(\mathrm{d}v) − ∂ θ i y U ( v ) ω ( d v ) − ∂ θ i x j U ( v ) ω ( d v ) -\partial_{\theta_i x_j}U(v)\,\omega(\mathrm{d}v) − ∂ θ i x j U ( v ) ω ( d v ) − ∂ y y U ( v ) ω ( d v ) -\partial_{yy}U(v)\,\omega(\mathrm{d}v) − ∂ y y U ( v ) ω ( d v ) v v v U U U ( b , q , θ , x , y ) (b,q,\theta,x,y) ( b , q , θ , x , y ) ( q v , X v , Y v ) (q_v,X_v,Y_v) ( q v , X v , Y v ) G G G ( u , G ) (u,G) ( u , G ) C N k C_N k C N k Φ \Phi Φ
The constants in these hypotheses depend on N N N k k k N N N Φ = F \Phi=F Φ = F u u u
d d u E T , u F = ∂ u E r e f [ F D T ] Z s ( u ) − E T , u F d d u log Z s ( u ) , (146) \frac{\mathrm{d}}{\mathrm{d}u}\mathbb{E}_{T,u}F=\frac{\partial_u\mathbb{E}_{\mathrm{ref}}[F\mathcal{D}_T]}{Z_s(u)}-\mathbb{E}_{T,u}F\frac{\mathrm{d}}{\mathrm{d}u}\log Z_s(u), \tag*{(146)} d u d E T , u F = Z s ( u ) ∂ u E ref [ F D T ] − E T , u F d u d log Z s ( u ) , (146) where the derivative of the unnormalized moment is
∂ u E r e f [ F D T ] Z s ( u ) = E T , u [ F ( ℓ ˙ + α 2 n ∑ i = 1 n ( ( ∂ θ i ℓ ) 2 + ∂ θ i 2 ℓ ) − α 2 ⟨ h , S y ⊗ S y + M y y ⟩ ) ] . \frac{\partial_u\mathbb{E}_{\mathrm{ref}}[F\mathcal{D}_T]}{Z_s(u)}
=
\mathbb{E}_{T,u}\left[
F\left(\dot{\ell}+\frac{\alpha}{2n}\sum_{i=1}^{n}\left((\partial_{\theta_i}\ell)^2+\partial_{\theta_i}^2\ell\right)-\frac{\alpha}{2}\left\langle h,S_y\otimes S_y+M_{yy}\right\rangle\right)
\right]. Z s ( u ) ∂ u E ref [ F D T ] = E T , u [ F ( ℓ ˙ + 2 n α i = 1 ∑ n ( ( ∂ θ i ℓ ) 2 + ∂ θ i 2 ℓ ) − 2 α ⟨ h , S y ⊗ S y + M y y ⟩ ) ] . Here S y S_y S y M y y M_{yy} M y y ℓ \ell ℓ
Step 2: the row. By (10.9) and (10.11), we have
∂ θ i ℓ = ∑ a τ i a ϵ a − ∑ v ω v π i ( v ) , ∂ θ i 2 ℓ = − ∑ v ω v ∂ θ i 2 U ( v ) . \partial_{\theta_i}\ell=\sum_a\tau_i^a\epsilon_a-\sum_v\omega_v\pi_i(v),\qquad
\partial_{\theta_i}^2\ell=-\sum_v\omega_v\partial_{\theta_i}^2U(v). ∂ θ i ℓ = a ∑ τ i a ϵ a − v ∑ ω v π i ( v ) , ∂ θ i 2 ℓ = − v ∑ ω v ∂ θ i 2 U ( v ) . where π i ( v ) \pi_i(v) π i ( v ) ∂ θ i 2 U ( v ) \partial_{\theta_i}^2U(v) ∂ θ i 2 U ( v ) v v v ( τ i a ϵ a ) 2 = 1 (\tau_i^a\epsilon_a)^2=1 ( τ i a ϵ a ) 2 = 1 k k k 1 1 1 − ω ( T ) -\omega(\mathcal{T}) − ω ( T ) s s s ( ∂ θ i ℓ ) 2 + ∂ θ i 2 ℓ (\partial_{\theta_i}\ell)^2+\partial_{\theta_i}^2\ell ( ∂ θ i ℓ ) 2 + ∂ θ i 2 ℓ
∑ a ≠ b τ i a ϵ a τ i b ϵ b − 2 ∑ a , v ω v τ i a ϵ a π i ( v ) + ∑ v , w ω v ω w π i ( v ) π i ( w ) + ∑ v ω v a ( q v ) π i ( v ) 2 + ∑ v ω v ∫ ( q v , 1 ] E [ π i ( r ) 2 ∣ G q v ] ( v ) ν ( d r ) . \begin{aligned}
&\sum_{a\ne b}\tau_i^a\epsilon_a\tau_i^b\epsilon_b
-2\sum_{a,v}\omega_v\tau_i^a\epsilon_a\pi_i(v)
+\sum_{v,w}\omega_v\omega_w\pi_i(v)\pi_i(w)\\
&\quad+\sum_v\omega_v a(q_v)\pi_i(v)^2
+\sum_v\omega_v\int_{(q_v,1]}\mathbb{E}\left[\pi_i(r)^2\mid\mathcal{G}_{q_v}\right](v)\,\nu(\mathrm{d}r).
\end{aligned} a = b ∑ τ i a ϵ a τ i b ϵ b − 2 a , v ∑ ω v τ i a ϵ a π i ( v ) + v , w ∑ ω v ω w π i ( v ) π i ( w ) + v ∑ ω v a ( q v ) π i ( v ) 2 + v ∑ ω v ∫ ( q v , 1 ] E [ π i ( r ) 2 ∣ G q v ] ( v ) ν ( d r ) . By (10.16), inside E T , u [ F ⋅ ] \mathbb{E}_{T,u}[F\cdot] E T , u [ F ⋅ ] π i ( v ) \pi_i(v) π i ( v ) τ i v ^ ϵ v ^ \tau_i^{\hat v}\epsilon_{\hat v} τ i v ^ ϵ v ^ π i ( v ) π i ( w ) \pi_i(v)\pi_i(w) π i ( v ) π i ( w ) v ^ \hat v v ^ w ˇ \check w w ˇ π i ( v ) 2 \pi_i(v)^2 π i ( v ) 2 v ^ \hat v v ^ v ˇ \check v v ˇ E [ π i ( r ) 2 ∣ G q v ] ( v ) \mathbb{E}[\pi_i(r)^2\mid\mathcal{G}_{q_v}](v) E [ π i ( r ) 2 ∣ G q v ] ( v ) r ^ \hat r r ^ r ˇ \check r r ˇ i i i τ i a τ i b \tau_i^a\tau_i^b τ i a τ i b R a b − R^-_{ab} R ab −
α 2 n ∑ i = 1 n E T , u [ F ( ( ∂ θ i ℓ ) 2 + ∂ θ i 2 ℓ ) ] = α 2 [ s E T , u F + S R − ( F ) ] . (147) \frac{\alpha}{2n}\sum_{i=1}^{n}\mathbb{E}_{T,u}\left[F\left((\partial_{\theta_i}\ell)^2+\partial_{\theta_i}^2\ell\right)\right]
=
\frac{\alpha}{2}\left[s\mathbb{E}_{T,u}F+\mathcal{S}_{R^-}(F)\right]. \tag*{(147)} 2 n α i = 1 ∑ n E T , u [ F ( ( ∂ θ i ℓ ) 2 + ∂ θ i 2 ℓ ) ] = 2 α [ s E T , u F + S R − ( F ) ] . (147) Step 3: the last field. By (137) and (140) , the score and Hessian measures of ℓ \ell ℓ
S y = ∑ a ϵ a δ a − m N ( v ) ω ( d v ) , M y y = − diag # ( ∂ y y U ( v ) ω ( d v ) ) . S_y=\sum_a\epsilon_a\delta_a-m_N(v)\,\omega(\mathrm{d}v),\qquad M_{yy}=-\operatorname{diag}_{\#}\bigl(\partial_{yy}U(v)\,\omega(\mathrm{d}v)\bigr). S y = a ∑ ϵ a δ a − m N ( v ) ω ( d v ) , M y y = − diag # ( ∂ y y U ( v ) ω ( d v ) ) . where δ a \delta_a δ a a a a diag # \operatorname{diag}_{\#} diag # v ↦ ( v , v ) v\mapsto(v,v) v ↦ ( v , v ) ω \omega ω h h h h ( a , a ) = 1 h(a,a)=1 h ( a , a ) = 1 h ( a , b ) = q a b h(a,b)=q_{ab} h ( a , b ) = q ab h ( a , v ) = q a v ^ h(a,v)=q_{a\widehat{v}} h ( a , v ) = q a v h ( v , w ) = q v ^ w ^ h(v,w)=q_{\widehat{v}\widehat{w}} h ( v , w ) = q v w h ( v , v ) = q v h(v,v)=q_v h ( v , v ) = q v ⟨ h , S y ⊗ S y ⟩ \langle h,S_y\otimes S_y\rangle ⟨ h , S y ⊗ S y ⟩ m N ( v ) m_N(v) m N ( v ) ϵ v ^ \epsilon_{\widehat{v}} ϵ v (144) , we find
− α 2 E T , u [ F ⟨ h , S y ⊗ S y ⟩ ] = − α 2 [ k E T , u F + ∑ a ≠ b q a b E ( F ϵ a ϵ b ) − 2 ∑ a , v ω v q a v ^ E ( F ϵ a ϵ v ^ ) + ∑ v , w ω v ω w q v ^ w ^ E ( F ϵ v ^ ϵ w ^ ) ] . -\frac{\alpha}{2}\mathbb{E}_{T,u}[F\langle h,S_y\otimes S_y\rangle]=-\frac{\alpha}{2}\left[k\mathbb{E}_{T,u}F+\sum_{a\ne b}q_{ab}\mathbb{E}(F\epsilon_a\epsilon_b)-2\sum_{a,v}\omega_vq_{a\widehat{v}}\mathbb{E}(F\epsilon_a\epsilon_{\widehat{v}})+\sum_{v,w}\omega_v\omega_wq_{\widehat{v}\widehat{w}}\mathbb{E}(F\epsilon_{\widehat{v}}\epsilon_{\widehat{w}})\right]. − 2 α E T , u [ F ⟨ h , S y ⊗ S y ⟩] = − 2 α k E T , u F + a = b ∑ q ab E ( F ϵ a ϵ b ) − 2 a , v ∑ ω v q a v E ( F ϵ a ϵ v ) + v , w ∑ ω v ω w q v w E ( F ϵ v ϵ w ) . The bracket consists of the constant k E T , u F k\mathbb{E}_{T,u}F k E T , u F S q ( F ) \mathcal{S}_q(F) S q ( F ) − α 2 ⟨ h , M y y ⟩ = α 2 ∑ v ω v q v ∂ y y U ( v ) -\frac{\alpha}{2}\langle h,M_{yy}\rangle=\frac{\alpha}{2}\sum_v\omega_vq_v\partial_{yy}U(v) − 2 α ⟨ h , M y y ⟩ = 2 α ∑ v ω v q v ∂ y y U ( v ) ℓ ˙ = α ∑ v ω v ∂ b U ( v ) \dot{\ell}=\alpha\sum_v\omega_v\partial_bU(v) ℓ ˙ = α ∑ v ω v ∂ b U ( v ) α 2 ∑ v ω v [ q v ∂ y y U ( v ) + 2 ∂ b U ( v ) ] \frac{\alpha}{2}\sum_v\omega_v[q_v\partial_{yy}U(v)+2\partial_bU(v)] 2 α ∑ v ω v [ q v ∂ y y U ( v ) + 2 ∂ b U ( v )] (140) and (141) , we have
q v ∂ y y U ( v ) + 2 ∂ b U ( v ) = 1 − q v a ( q v ) m N ( v ) 2 − ∫ ( q v , 1 ] r E [ m N ( r ) 2 ∣ G q v ] ( v ) ν ( d r ) . q_v\partial_{yy}U(v)+2\partial_bU(v)=1-q_{va}(q_v)m_N(v)^2-\int_{(q_v,1]}r\mathbb{E}\left[m_N(r)^2\mid\mathcal{G}_{q_v}\right](v)\,\nu(\mathrm{d}r). q v ∂ y y U ( v ) + 2 ∂ b U ( v ) = 1 − q v a ( q v ) m N ( v ) 2 − ∫ ( q v , 1 ] r E [ m N ( r ) 2 ∣ G q v ] ( v ) ν ( d r ) . In the integral, we combined q v ∫ ( q v , 1 ] q_v\int_{(q_v,1]} q v ∫ ( q v , 1 ] (140) with ∫ ( q v , 1 ] ( r − q v ) \int_{(q_v,1]}(r-q_v) ∫ ( q v , 1 ] ( r − q v ) (141) . After the replacements (144) , the constant gives α 2 ω ( T ) E T , u F = α 2 ( k − s ) E T , u F \frac{\alpha}{2}\omega(T)\mathbb{E}_{T,u}F=\frac{\alpha}{2}(k-s)\mathbb{E}_{T,u}F 2 α ω ( T ) E T , u F = 2 α ( k − s ) E T , u F − α 2 S q ( F ) -\frac{\alpha}{2}\mathcal{S}_q(F) − 2 α S q ( F ) q v ^ v ^ = q v q_{\widehat{v}\widehat{v}}=q_v q v v = q v q r ^ r ^ = r q_{\widehat{r}\widehat{r}}=r q r r = r α 2 [ − k + ( k − s ) ] E T , u F − α 2 S q ( F ) \frac{\alpha}{2}[-k+(k-s)]\mathbb{E}_{T,u}F-\frac{\alpha}{2}\mathcal{S}_q(F) 2 α [ − k + ( k − s )] E T , u F − 2 α S q ( F ) (147) , we see that the constants α s / 2 \alpha s/2 α s /2 − α k / 2 -\alpha k/2 − α k /2 α ( k − s ) / 2 \alpha(k-s)/2 α ( k − s ) /2 S R − − S q = S Δ \mathcal{S}_{R^-}-\mathcal{S}_q=\mathcal{S}_{\Delta} S R − − S q = S Δ
∂ u E r e f [ F D T ] Z s ( u ) = α 2 S Δ ( F ) . (148) \frac{\partial_u\mathbb{E}_{\mathrm{ref}}[F\mathcal{D}_T]}{Z_s(u)}=\frac{\alpha}{2}\mathcal{S}_{\Delta}(F). \tag*{(148)} Z s ( u ) ∂ u E ref [ F D T ] = 2 α S Δ ( F ) . (148) In words, the row inserts a rest overlap, the last field inserts the level of the corresponding pair, and their difference is an overlap defect.
Step 4: the normalizer. We compute the derivative of Z s ( u ) = E r e f e s U ( 0 , 0 , 0 ) Z_s(u)=\mathbb{E}_{\mathrm{ref}}e^{sU(0,0,0)} Z s ( u ) = E ref e s U ( 0 , 0 , 0 ) (39) ; now the only active coordinates are those of the row. The row contributes α s 2 n ∑ i E u [ ∂ θ i 2 U ( 0 ) + s ( ∂ θ i U ( 0 ) ) 2 ] \frac{\alpha s}{2n}\sum_i\mathbb{E}_u[\partial_{\theta_i}^2U(0)+s(\partial_{\theta_i}U(0))^2] 2 n α s ∑ i E u [ ∂ θ i 2 U ( 0 ) + s ( ∂ θ i U ( 0 ) ) 2 ] a ( 0 ) − s = ν ( { 0 } ) a(0)-s=\nu(\{0\}) a ( 0 ) − s = ν ({ 0 }) (139) that
∂ θ i 2 U ( 0 ) + s ( ∂ θ i U ( 0 ) ) 2 = 1 − ∫ [ 0 , 1 ] E [ π i ( r ) 2 ∣ G 0 ] ν ( d r ) . \partial_{\theta_i}^2U(0)+s\bigl(\partial_{\theta_i}U(0)\bigr)^2=1-\int_{[0,1]}\mathbb{E}\left[\pi_i(r)^2\mid\mathcal{G}_0\right]\nu(\mathrm{d}r). ∂ θ i 2 U ( 0 ) + s ( ∂ θ i U ( 0 ) ) 2 = 1 − ∫ [ 0 , 1 ] E [ π i ( r ) 2 ∣ G 0 ] ν ( d r ) . Here, as in [4 (2.11)] , the term s ( ∂ θ i U ) 2 s(\partial_{\theta_i}U)^2 s ( ∂ θ i U ) 2 s s s ν ( { 0 } ) \nu(\{0\}) ν ({ 0 }) E u E [ π i ( r ) 2 ∣ G 0 ] = E 2 , r , u [ τ 1 2 ϵ 1 τ 2 2 ϵ 2 ] \mathbb{E}_u\mathbb{E}[\pi_i(r)^2\mid\mathcal{G}_0]=\mathbb{E}_{2,r,u}[\tau_1^2\epsilon_1\tau_2^2\epsilon_2] E u E [ π i ( r ) 2 ∣ G 0 ] = E 2 , r , u [ τ 1 2 ϵ 1 τ 2 2 ϵ 2 ] α s 2 [ 1 − ∫ E 2 , r , u [ ϵ 1 ϵ 2 R 12 ] ν ( d r ) ] \frac{\alpha s}{2}\left[1-\int\mathbb{E}_{2,r,u}[\epsilon_1\epsilon_2R_{12}]\nu(\mathrm{d}r)\right] 2 α s [ 1 − ∫ E 2 , r , u [ ϵ 1 ϵ 2 R 12 ] ν ( d r ) ] b ( u ) b(u) b ( u ) s b ′ ( u ) E u ∂ b U ( 0 ) = − α s E u ∂ b U ( 0 ) sb'(u)\mathbb{E}_u\partial_bU(0)=-\alpha s\mathbb{E}_u\partial_bU(0) s b ′ ( u ) E u ∂ b U ( 0 ) = − α s E u ∂ b U ( 0 ) − α s 2 [ 1 − ∫ r E 2 , r , u [ ϵ 1 ϵ 2 ] ν ( d r ) ] -\frac{\alpha s}{2}\left[1-\int r\mathbb{E}_{2,r,u}[\epsilon_1\epsilon_2]\nu(\mathrm{d}r)\right] − 2 α s [ 1 − ∫ r E 2 , r , u [ ϵ 1 ϵ 2 ] ν ( d r ) ]
d d u log Z s ( u ) = − α s 2 ∫ E 2 , r , u D 12 ν ( d r ) . (149) \frac{\mathrm{d}}{\mathrm{d}u}\log Z_s(u)=-\frac{\alpha s}{2}\int\mathbb{E}_{2,r,u}D_{12}\,\nu(\mathrm{d}r). \tag*{(149)} d u d log Z s ( u ) = − 2 α s ∫ E 2 , r , u D 12 ν ( d r ) . (149) Inserting (10.20) and (10.21) into (10.18), we obtain (10.17). In its last term, E 2 , r , u \mathbb{E}_{2,r,u} E 2 , r , u
Step 5: coefficients and regularity. Recall that ω ( T ) = k − s ≤ k \omega(\mathcal{T})=k-s\leq k ω ( T ) = k − s ≤ k a ≤ 1 a\leq1 a ≤ 1 ν ( [ 0 , 1 ] ) ≤ 1 \nu([0,1])\leq1 ν ([ 0 , 1 ]) ≤ 1 k ( k − 1 ) k(k-1) k ( k − 1 ) 2 k 2 2k^2 2 k 2 k 2 k^2 k 2 k k k k k k 1 1 1 4 k 2 + k + 1 ≤ 4 k 2 + 2 k + 1 4k^2+k+1\leq4k^2+2k+1 4 k 2 + k + 1 ≤ 4 k 2 + 2 k + 1
We now show that each extended-tree moment on the right side of (10.17) is continuous in u u u ( α u / n ) 1 / 2 g (\alpha u/n)^{1/2}g ( α u / n ) 1/2 g b ( u ) \sqrt{b(u)} b ( u ) u u u u u u ω \omega ω ν \nu ν u u u 2 ∥ F ∥ ∞ 2\lVert F\rVert_\infty 2 ∥ F ∥ ∞ u ∈ [ 0 , 1 ] u\in[0,1] u ∈ [ 0 , 1 ] u ↦ E T , u F u\mapsto\mathbb{E}_{\mathcal{T},u}F u ↦ E T , u F C 1 C^1 C 1 F D F_D F D u u u k k k C k α ∥ F ∥ ∞ C_k\alpha\lVert F\rVert_\infty C k α ∥ F ∥ ∞ v , w v,w v , w r r r ω \omega ω ν \nu ν C 2 C^2 C 2
Corollary 10.6. (Comparison of nonnegative observables.) For every deterministic profile, every marked tree T \mathcal{T} T k k k F F F
E T , u F ≤ K β , k E T , v F ( 0 ≤ u , v ≤ 1 ) , K β , k = exp ( β 2 ( 4 k 2 + 2 k + 1 ) ) . (150) \mathbb{E}_{\mathcal{T},u}F\leq K_{\beta,k}\mathbb{E}_{\mathcal{T},v}F\qquad(0\leq u,v\leq1),\qquad K_{\beta,k}=\exp\bigl(\beta^2(4k^2+2k+1)\bigr). \tag*{(150)} E T , u F ≤ K β , k E T , v F ( 0 ≤ u , v ≤ 1 ) , K β , k = exp ( β 2 ( 4 k 2 + 2 k + 1 ) ) . (150) Proof. Since ∣ D a b ∣ ≤ 2 \lvert D_{ab}\rvert\leq2 ∣ D ab ∣ ≤ 2 F ≥ 0 F\geq0 F ≥ 0 F F F E T , u F \mathbb{E}_{\mathcal{T},u}F E T , u F α ≤ β 2 \alpha\leq\beta^2 α ≤ β 2
∣ ∂ u E T , u F ∣ ≤ β 2 ( 4 k 2 + 2 k + 1 ) E T , u F , 0 ≤ u ≤ 1. \left\lvert\partial_u\mathbb{E}_{\mathcal{T},u}F\right\rvert\leq\beta^2(4k^2+2k+1)\mathbb{E}_{\mathcal{T},u}F,\qquad0\leq u\leq1. ∣ ∂ u E T , u F ∣ ≤ β 2 ( 4 k 2 + 2 k + 1 ) E T , u F , 0 ≤ u ≤ 1. By Gronwall’s inequality on the interval between u u u v v v E T , u F ≤ e β 2 ( 4 k 2 + 2 k + 1 ) ∣ u − v ∣ E T , v F \mathbb{E}_{T,u}F \le e^{\beta^{2}(4k^{2}+2k+1)|u-v|}\mathbb{E}_{T,v}F E T , u F ≤ e β 2 ( 4 k 2 + 2 k + 1 ) ∣ u − v ∣ E T , v F (150) holds, also in the case E T , v F = 0 \mathbb{E}_{T,v}F=0 E T , v F = 0 u u u
Products of three overlap defects
In the Taylor remainder of the cavity covariance, and in the variance comparison of Lemma 12.1, we differentiate tree moments of observables that already contain overlap defects. Each differentiation adds one more defect. We show that integrated moments of products of three defects are controlled by the integrated third absolute moment
M 3 ( u ) = ∫ E 2 , q , u ∣ R 12 − − q ∣ 3 ν ( d q ) (151) \mathcal{M}_{3}(u)=\int\mathbb{E}_{2,q,u}\lvert R_{12}^{-}-q\rvert^{3}\,\nu(\mathrm{d}q) \tag*{(151)} M 3 ( u ) = ∫ E 2 , q , u ∣ R 12 − − q ∣ 3 ν ( d q ) (151) of the overlap defect of two descendants. In Lemma 12.3, we bound M 3 \mathcal{M}_{3} M 3 A A A J J J H 3 H_{3} H 3
Terms and marks. Fix integers k 0 ≥ 1 k_{0}\ge1 k 0 ≥ 1 j ≥ 0 j\ge0 j ≥ 0 p ∈ [ 0 , 1 ] p\in[0,1] p ∈ [ 0 , 1 ] T p T_{p} T p k 0 k_{0} k 0 p p p F p F_{p} F p p p p ∫ ν ( d p ) E T p , u F p \int\nu(\mathrm{d}p)\mathbb{E}_{T_{p},u}F_{p} ∫ ν ( d p ) E T p , u F p j j j (145) , with the product rule for products of moments over independent trees. Each application multiplies by α / 2 \alpha/2 α /2 (145) . Each term produced after j j j ( α / 2 ) j (\alpha/2)^{j} ( α /2 ) j p p p v v v w w w r r r r r r (145) . The integrand is a product of expectations over independent trees, whose shapes are determined by the marks. The signed measure collects the coefficients 1 1 1 − 2 ω v -2\omega_{v} − 2 ω v ω v ω w \omega_{v}\omega_{w} ω v ω w a ( q v ) ω v a(q_{v})\omega_{v} a ( q v ) ω v ω v ν ( d r ) \omega_{v}\nu(\mathrm{d}r) ω v ν ( d r ) s ν ( d r ) s\nu(\mathrm{d}r) s ν ( d r ) m m m ( α / 2 ) j (\alpha/2)^{j} ( α /2 ) j m m m p p p q v q_{v} q v q w q_{w} q w r r r
Lemma 10.7 (Three-defect bound). For every deterministic profile, every such term containing three defects Δ 1 , Δ 2 , Δ 3 \Delta_{1},\Delta_{2},\Delta_{3} Δ 1 , Δ 2 , Δ 3
∫ E ∣ Δ 1 Δ 2 Δ 3 ∣ d m ≤ C M 3 ( u ) . (152) \int\mathbb{E}\lvert\Delta_{1}\Delta_{2}\Delta_{3}\rvert\,\mathrm{d}m \le C\mathcal{M}_{3}(u). \tag*{(152)} ∫ E ∣ Δ 1 Δ 2 Δ 3 ∣ d m ≤ C M 3 ( u ) . (152) Here E \mathbb{E} E C C C
The bounds on the marks in Step 1 of the proof are a counterpart, for an arbitrary profile, of the bounds on split levels in [54, Lemma 2.8].
Proof. The proof has three steps. We show that the level of every mark has a marginal under m \mathbf{m} m ν \nu ν
Step 1: marginals of the marks. Each differentiation adds at most two leaves to one tree, or creates an independent tree with two leaves. After i i i K i = k 0 + 2 i K_i = k_0 + 2i K i = k 0 + 2 i k 0 + i k_0 + i k 0 + i m i \mathbf{m}_i m i m \mathbf{m} m i ≤ j i \le j i ≤ j i i i m i \mathbf{m}_i m i C i C_i C i C i ν C_i\nu C i ν C i C_i C i k 0 k_0 k 0 i i i i = 0 i = 0 i = 0 p p p m 0 = ν ( d p ) \mathbf{m}_0 = \nu(dp) m 0 = ν ( d p )
For the induction step, let K = K i K = K_i K = K i ω \omega ω K K K v ↦ q v v \mapsto q_v v ↦ q v K K K ν \nu ν ν ( { q v } ) \nu(\{q_v\}) ν ({ q v }) K ν K\nu K ν ( l v − 1 ) a ( q v ) ≤ l v − 1 (l_v - 1)a(q_v) \le l_v - 1 ( l v − 1 ) a ( q v ) ≤ l v − 1 ∑ v ( l v − 1 ) δ q v \sum_v (l_v - 1)\delta_{q_v} ∑ v ( l v − 1 ) δ q v v v v l v ≥ 2 l_v \ge2 l v ≥ 2 ∑ v ( l v − 1 ) ≤ K \sum_v (l_v - 1) \le K ∑ v ( l v − 1 ) ≤ K q v q_v q v m i \mathbf{m}_i m i ( k 0 + i ) ( 4 K 2 + 2 K + 1 ) (k_0 + i)(4K^2 + 2K + 1) ( k 0 + i ) ( 4 K 2 + 2 K + 1 ) k 0 + i k_0 + i k 0 + i ( k 0 + i ) ( 4 K 2 + 2 K + 1 ) (k_0 + i)(4K^2 + 2K + 1) ( k 0 + i ) ( 4 K 2 + 2 K + 1 ) v v v w w w ω \omega ω K [ ν + ∑ r ′ δ r ′ ] K[\nu+ \sum_{r'}\delta_{r'}] K [ ν + ∑ r ′ δ r ′ ] r ′ r' r ′ r r r ν ( d r ) \nu(dr) ν ( d r ) ( q v , 1 ] (q_v,1] ( q v , 1 ] s ν ( d r ) s\nu(dr) s ν ( d r ) ν \nu ν a ( q v ) ≤ 1 a(q_v) \le1 a ( q v ) ≤ 1 r > q v r > q_v r > q v B B B r n e w r_{\mathrm{new}} r new
m i + 1 ( r n e w ∈ B ) ≤ C [ m i ( 1 ) ν ( B ) + ∑ r ′ m i ( r ′ ∈ B ) ] ≤ C i + 1 ν ( B ) , \mathbf{m}_{i+1}(r_{\mathrm{new}} \in B) \le C\left[\mathbf{m}_i(1)\nu(B) + \sum_{r'} \mathbf{m}_i(r' \in B)\right] \le C_{i+1}\nu(B), m i + 1 ( r new ∈ B ) ≤ C [ m i ( 1 ) ν ( B ) + r ′ ∑ m i ( r ′ ∈ B ) ] ≤ C i + 1 ν ( B ) , where C C C k 0 k_0 k 0 i i i
Step 2: levels of pairs. In a tree of such a term, every vertex with at least two outgoing edges or leaves lies at the level of a mark. This level is the original split level p p p q v q_v q v r r r a , b a,b a , b q a b q_{ab} q ab χ ( q a b ) ≤ ∑ r ′ χ ( r ′ ) \chi(q_{ab}) \le\sum_{r'}\chi(r') χ ( q ab ) ≤ ∑ r ′ χ ( r ′ ) χ ≥ 0 \chi\ge0 χ ≥ 0 r ′ r' r ′
∫ χ ( q a b ) d m ≤ C ∫ χ d ν . \int\chi(q_{ab})\,d\mathbf{m} \le C\int\chi\,d\nu. ∫ χ ( q ab ) d m ≤ C ∫ χ d ν . Step 3: Hölder’s inequality. As in [54, Remark 2.4], we apply Hölder’s inequality unconditionally under the positive tree laws, and we bound each factor after deleting all leaves except one pair. On the product of the probability space of the term and the mark space with m \mathfrak{m} m
∫ E ∣ Δ 1 Δ 2 Δ 3 ∣ d m ≤ ∏ i = 1 3 ( ∫ E ∣ Δ i ∣ 3 d m ) 1 / 3 . \int\mathbb{E}|\Delta_1\Delta_2\Delta_3|\,\mathrm{d}\mathfrak{m} \le\prod_{i=1}^{3}\left(\int\mathbb{E}|\Delta_i|^3\,\mathrm{d}\mathfrak{m}\right)^{1/3}. ∫ E ∣ Δ 1 Δ 2 Δ 3 ∣ d m ≤ i = 1 ∏ 3 ( ∫ E ∣ Δ i ∣ 3 d m ) 1/3 . If we delete all leaves except the two leaves of Δ i \Delta_i Δ i q i q_i q i E ∣ Δ i ∣ 3 = E 2 , q i , u ∣ R 12 − − q i ∣ 3 \mathbb{E}|\Delta_i|^3=\mathbb{E}_{2,q_i,u}|R_{12}^{-}-q_i|^3 E ∣ Δ i ∣ 3 = E 2 , q i , u ∣ R 12 − − q i ∣ 3 χ ( q ) = E 2 , q , u ∣ R 12 − − q ∣ 3 \chi(q)=\mathbb{E}_{2,q,u}|R_{12}^{-}-q|^3 χ ( q ) = E 2 , q , u ∣ R 12 − − q ∣ 3 ( C M 3 ( u ) ) 1 / 3 (C\mathcal{M}_3(u))^{1/3} ( C M 3 ( u ) ) 1/3
The same proof applies to products of finitely many independent initial tree moments whose leaves split at p p p ν ( d p ) \nu(\mathrm{d}p) ν ( d p ) ν ( d p ) \nu(\mathrm{d}p) ν ( d p )
The cavity covariance. In particular, Lemma 10.7 applies to the covariance between the squared conditional magnetization of the last spin and the rest self-overlap, and to its Taylor remainder,
C p p ( u ) = Cov u ( m N ( p ) 2 , A p − ) , R c a v = ∫ ν ( d p ) ∫ 0 1 ( 1 − u ) C p p ′ ′ ( u ) d u . C_{pp}(u)=\operatorname{Cov}_{u}\left(m_N(p)^2,A_p^{-}\right),\qquad
\mathcal{R}_{\mathrm{cav}}=\int\nu(\mathrm{d}p)\int_0^1(1-u)C_{pp}^{\prime\prime}(u)\,\mathrm{d}u. C pp ( u ) = Cov u ( m N ( p ) 2 , A p − ) , R cav = ∫ ν ( d p ) ∫ 0 1 ( 1 − u ) C pp ′′ ( u ) d u . Representing conditional expectations by leaves, we can write C p p = Ψ p − Ψ p ε Ψ p τ C_{pp}=\Psi_p-\Psi_p^{\varepsilon}\Psi_p^{\tau} C pp = Ψ p − Ψ p ε Ψ p τ
Ψ p ( u ) = E 4 , p , u [ ε 1 ε 2 ( R 34 − − p ) ] , Ψ p ε ( u ) = E 2 , p , u ε 1 ε 2 , Ψ p τ ( u ) = E 2 , p , u ( R 12 − − p ) . \Psi_p(u)=\mathbb{E}_{4,p,u}\left[\varepsilon_1\varepsilon_2\left(R_{34}^{-}-p\right)\right],\qquad
\Psi_p^{\varepsilon}(u)=\mathbb{E}_{2,p,u}\varepsilon_1\varepsilon_2,\qquad
\Psi_p^{\tau}(u)=\mathbb{E}_{2,p,u}\left(R_{12}^{-}-p\right). Ψ p ( u ) = E 4 , p , u [ ε 1 ε 2 ( R 34 − − p ) ] , Ψ p ε ( u ) = E 2 , p , u ε 1 ε 2 , Ψ p τ ( u ) = E 2 , p , u ( R 12 − − p ) . Indeed, by conditional independence given G p \mathcal{G}_p G p Ψ p = E u [ m N ( p ) 2 ( A p − − p ) ] \Psi_p=\mathbb{E}_u[m_N(p)^2(A_p^{-}-p)] Ψ p = E u [ m N ( p ) 2 ( A p − − p )] Ψ p ε = E u m N ( p ) 2 \Psi_p^{\varepsilon}=\mathbb{E}_u m_N(p)^2 Ψ p ε = E u m N ( p ) 2 Ψ p τ = E u ( A p − − p ) \Psi_p^{\tau}=\mathbb{E}_u(A_p^{-}-p) Ψ p τ = E u ( A p − − p ) p p p p p p u u u C 2 ( [ 0 , 1 ] ) C^2([0,1]) C 2 ([ 0 , 1 ]) C α C_\alpha C α C α 2 C_\alpha^2 C α 2 C p p ∈ C 2 ( [ 0 , 1 ] ) C_{pp}\in C^2([0,1]) C pp ∈ C 2 ([ 0 , 1 ]) ∣ C p p ′ ′ ∣ |C_{pp}^{\prime\prime}| ∣ C pp ′′ ∣ β \beta β p p p R c a v \mathcal{R}_{\mathrm{cav}} R cav
R c a v = ∫ [ C p p ( 1 ) − C p p ( 0 ) − C p p ′ ( 0 ) ] ν ( d p ) . \mathcal{R}_{\mathrm{cav}}=\int\left[C_{pp}(1)-C_{pp}(0)-C_{pp}^{\prime}(0)\right]\nu(\mathrm{d}p). R cav = ∫ [ C pp ( 1 ) − C pp ( 0 ) − C pp ′ ( 0 ) ] ν ( d p ) . Lemma 10.8 (Taylor remainder). For every deterministic profile and every u ∈ [ 0 , 1 ] u\in[0,1] u ∈ [ 0 , 1 ]
∫ ∣ C p p ′ ′ ( u ) ∣ ν ( d p ) ≤ C β M 3 ( u ) , ∣ R c a v ∣ ≤ C β sup 0 ≤ w ≤ 1 M 3 ( w ) . (153) \int|C_{pp}^{\prime\prime}(u)|\nu(\mathrm{d}p)\le C_\beta\mathcal{M}_3(u),\qquad
|\mathcal{R}_{\mathrm{cav}}|\le C_\beta\sup_{0\le w\le1}\mathcal{M}_3(w). \tag*{(153)} ∫ ∣ C pp ′′ ( u ) ∣ ν ( d p ) ≤ C β M 3 ( u ) , ∣ R cav ∣ ≤ C β 0 ≤ w ≤ 1 sup M 3 ( w ) . (153) Proof. We have
C p p ′ ′ = Ψ p ′ ′ − ( Ψ p ϵ ) ′ ′ Ψ p τ − 2 ( Ψ p ϵ ) ′ ( Ψ p τ ) ′ − Ψ p ϵ ( Ψ p τ ) ′ ′ . C^{\prime\prime}_{pp}=\Psi^{\prime\prime}_{p}-(\Psi^{\epsilon}_{p})^{\prime\prime}\Psi^{\tau}_{p}-2(\Psi^{\epsilon}_{p})^{\prime}(\Psi^{\tau}_{p})^{\prime}-\Psi^{\epsilon}_{p}(\Psi^{\tau}_{p})^{\prime\prime}. C pp ′′ = Ψ p ′′ − ( Ψ p ϵ ) ′′ Ψ p τ − 2 ( Ψ p ϵ ) ′ ( Ψ p τ ) ′ − Ψ p ϵ ( Ψ p τ ) ′′ . Each application of (145) adds a factor α / 2 ≤ β 2 / 2 \alpha/2\leq\beta^{2}/2 α /2 ≤ β 2 /2 D D D Ψ p \Psi_{p} Ψ p Ψ p τ \Psi^{\tau}_{p} Ψ p τ Ψ p ϵ \Psi^{\epsilon}_{p} Ψ p ϵ
Table 2.
Products of moments, and the last term of (145) , are products over independent trees. Every term of C p p ′ ′ C^{\prime\prime}_{pp} C pp ′′ ( α / 2 ) 2 (\alpha/2)^2 ( α /2 ) 2 Lemma 10.7 , or in its extension to products of independent initial moments stated after its proof, with at most four initial leaves and j = 2 j=2 j = 2 p p p ν \nu ν Lemma 10.7 and this extension, we obtain the first bound in (153) with C β = C β 4 C_{\beta}=C\beta^4 C β = C β 4 ( 1 − u ) d u (1-u)\,du ( 1 − u ) d u ∣ R c a v ∣ ≤ C β 4 ∫ 0 1 ( 1 − u ) M 3 ( u ) d u ≤ C β 4 sup w M 3 ( w ) |R_{\mathrm{cav}}|\leq C\beta^4\int_0^1(1-u)\mathcal{M}_3(u)\,du\leq C\beta^4\sup_w\mathcal{M}_3(w) ∣ R cav ∣ ≤ C β 4 ∫ 0 1 ( 1 − u ) M 3 ( u ) d u ≤ C β 4 sup w M 3 ( w )
The derivative at the independent endpoint
In Section 10, we constructed the cavity interpolation P u \mathbb{P}_u P u 0 ≤ u ≤ 1 0 \le u \le1 0 ≤ u ≤ 1 u = 1 u = 1 u = 1 N N N u = 0 u = 0 u = 0 ϵ \epsilon ϵ n = N − 1 n = N - 1 n = N − 1
C p p ( u ) = Cov u ( m N ( p ) 2 , A p − ) C_{pp}(u) = \operatorname{Cov}_u\left(m_N(p)^2,A_p^-\right) C pp ( u ) = Cov u ( m N ( p ) 2 , A p − ) between the squared conditional magnetization of the last spin and the rest self-overlap. By Lemma 10.5, it is twice continuously differentiable in u u u § 9.5.1 ), and Lemma 10.8 bounds its Taylor remainder. The covariance vanishes at u = 0 u = 0 u = 0
The main result is Lemma 11.2. It states that after integration against ν \nu ν C p p ′ ( 0 ) C_{pp}'(0) C pp ′ ( 0 ) α / d \alpha/d α / d J a , d s c J_{a,d}^{\mathrm{sc}} J a , d sc Σ − \Sigma^- Σ − α P / 2 \alpha P/2 α P /2 A A A J J J N − 2 N^{-2} N − 2 ν ( { 1 } ) = 0 \nu(\{1\}) = 0 ν ({ 1 }) = 0
The computation uses the exact density (10.9) of the cavity law and the parameter responses (10.11) from Section 10, the Gaussian heat identity of Lemma 4.3, and the explicit form (9.19) of the scalar response operator. The error bounds also use the kernel bounds of Lemma 9.4, the posterior variance bound (10.4), and the comparison inequality of Corollary 10.6.
Standing assumptions. We keep the assumptions of Section 10: N ≥ 2 N \ge2 N ≥ 2 n = N − 1 n = N - 1 n = N − 1 0 < ρ ≤ 1 0 < \rho\le1 0 < ρ ≤ 1 t = 1 − ρ t = 1 - \rho t = 1 − ρ c = β 2 ρ c = \beta^2\rho c = β 2 ρ 0 < s = λ / β ≤ 1 / 2 0 < s = \lambda/\beta\le1/2 0 < s = λ / β ≤ 1/2 α \alpha α d d d b ( u ) b(u) b ( u ) 0 ≤ α ≤ β 2 0 \le\alpha\le\beta^2 0 ≤ α ≤ β 2 d ∈ [ β 2 / 2 , β 2 ] d \in[\beta^2/2,\beta^2] d ∈ [ β 2 /2 , β 2 ] μ = s δ 0 + ν \mu= s\delta_0 + \nu μ = s δ 0 + ν a ( q ) = μ ( [ 0 , q ] ) a(q) = \mu([0,q]) a ( q ) = μ ([ 0 , q ]) ν ( { 1 } ) = 0 \nu(\{1\}) = 0 ν ({ 1 }) = 0 H ν \mathcal{H}_\nu H ν J a , d s c J_{a,d}^{\mathrm{sc}} J a , d sc
Linear response at the independent endpoint
The product law at the independent endpoint
In this section, we take all expectations at u = 0 u=0 u = 0 § 9.1.1 that P 0 \mathbb{P}_0 P 0 n n n E − \mathcal{E}^{-} E − c c c μ \mu μ d d d E − \mathbb{E}^{-} E − Var − \operatorname{Var}^{-} Var − Cov − \operatorname{Cov}^{-} Cov − E 0 \mathbb{E}^{0} E 0 Var 0 \operatorname{Var}^{0} Var 0 Cov 0 \operatorname{Cov}^{0} Cov 0 G q − \mathcal{G}^{-}_{q} G q − G q 0 \mathcal{G}^{0}_{q} G q 0 u = 0 u=0 u = 0 (130) splits as U ( q , x , y ) = U − ( q , x ) + U 0 ( q , y ) U(q,x,y)=U^{-}(q,x)+U^{0}(q,y) U ( q , x , y ) = U − ( q , x ) + U 0 ( q , y )
M ( r , x ) = ∇ x U − ( r , x ) , M i ( r , x ) = ∂ x i U − ( r , x ) , M(r,x)=\nabla_xU^{-}(r,x), \qquad M_i(r,x)=\partial_{x_i}U^{-}(r,x), M ( r , x ) = ∇ x U − ( r , x ) , M i ( r , x ) = ∂ x i U − ( r , x ) , A − ( r , x ) = ∣ M ( r , x ) ∣ 2 n , m ( r , y ) = ∂ y U 0 ( r , y ) , A^{-}(r,x)=\frac{\lvert M(r,x)\rvert^2}{n}, \qquad m(r,y)=\partial_yU^{0}(r,y), A − ( r , x ) = n ∣ M ( r , x ) ∣ 2 , m ( r , y ) = ∂ y U 0 ( r , y ) , so that M r = M ( r , X r ) M_r=M(r,X_r) M r = M ( r , X r ) A r − = A − ( r , X r ) A^{-}_r=A^{-}(r,X_r) A r − = A − ( r , X r ) H r − = ∇ x 2 U − ( r , X r ) H^{-}_r=\nabla_x^2U^{-}(r,X_r) H r − = ∇ x 2 U − ( r , X r ) m r = m ( r , Y r ) m_r=m(r,Y_r) m r = m ( r , Y r ) h r = ∂ y m ( r , Y r ) h_r=\partial_ym(r,Y_r) h r = ∂ y m ( r , Y r ) f p = m p 2 f_p=m_p^2 f p = m p 2 f ˉ p = f p − E 0 f p \bar{f}_p=f_p-\mathbb{E}^{0}f_p f ˉ p = f p − E 0 f p g r ℓ ( y ) = P r , ℓ 0 ( m ( ℓ , ⋅ ) 2 ) ( y ) g_r^{\ell}(y)=P^{0}_{r,\ell}(m(\ell,\cdot)^2)(y) g r ℓ ( y ) = P r , ℓ 0 ( m ( ℓ , ⋅ ) 2 ) ( y ) Γ = Γ a , d \Gamma=\Gamma_{a,d} Γ = Γ a , d Γ ( p ) = E 0 m p 2 \Gamma(p)=\mathbb{E}^{0}m_p^2 Γ ( p ) = E 0 m p 2 g r ℓ ( Y r ) = E 0 [ m ℓ 2 ∣ G r 0 ] g_r^{\ell}(Y_r)=\mathbb{E}^{0}[m_{\ell}^2\mid\mathcal{G}^{0}_r] g r ℓ ( Y r ) = E 0 [ m ℓ 2 ∣ G r 0 ] m r = E 0 [ ϵ ∣ G r 0 ] m_r=\mathbb{E}^{0}[\epsilon\mid\mathcal{G}^{0}_r] m r = E 0 [ ϵ ∣ G r 0 ] r ≤ ℓ r\leq\ell r ≤ ℓ P r , ℓ − P^{-}_{r,\ell} P r , ℓ − ( P r , ℓ − A ℓ − ) ( X r ) = E − [ A ℓ − ∣ G r − ] (P^{-}_{r,\ell}A^{-}_{\ell})(X_r)=\mathbb{E}^{-}[A^{-}_{\ell}\mid\mathcal{G}^{-}_r] ( P r , ℓ − A ℓ − ) ( X r ) = E − [ A ℓ − ∣ G r − ] a r = a r ( 0 ) a_r=a_r(0) a r = a r ( 0 ) j r = j r ( 0 ) j_r=j_r(0) j r = j r ( 0 ) u = 0 u=0 u = 0
Note that the rest Hamiltonian E − \mathcal{E}^{-} E − τ ↦ − τ \tau\mapsto-\tau τ ↦ − τ U − ( 0 , ⋅ ) U^{-}(0,\cdot) U − ( 0 , ⋅ ) M 0 = 0 M_0=0 M 0 = 0
A 0 − = 0. (154) A^{-}_0=0. \tag*{(154)} A 0 − = 0. (154) Since the spatial derivatives of U − U^{-} U − r ↦ A r − r\mapsto A^{-}_r r ↦ A r − [ 0 , 1 ] [0,1] [ 0 , 1 ]
The rest self-overlap. We define
Σ − ( p , q ) = Cov − ( A p − , A q − ) , A ˉ p − = A p − − E − A p − , \Sigma^{-}(p,q)=\operatorname{Cov}^{-}(A^{-}_p,A^{-}_q), \qquad\bar{A}^{-}_p=A^{-}_p-\mathbb{E}^{-}A^{-}_p, Σ − ( p , q ) = Cov − ( A p − , A q − ) , A ˉ p − = A p − − E − A p − , Θ ( r ; ℓ , p ) = Cov − ( P r , ℓ − A ℓ − , A ˉ p − ) ( r ≤ ℓ ) , \Theta(r;\ell,p)=\operatorname{Cov}^{-}(P^{-}_{r,\ell}A^{-}_{\ell},\bar{A}^{-}_p) \qquad(r\leq\ell), Θ ( r ; ℓ , p ) = Cov − ( P r , ℓ − A ℓ − , A ˉ p − ) ( r ≤ ℓ ) , where we recall that P r , ℓ − A ℓ − P^{-}_{r,\ell}A^{-}_{\ell} P r , ℓ − A ℓ − E − [ A ℓ − ∣ G r − ] \mathbb{E}^{-}[A^{-}_{\ell}\mid\mathcal{G}^{-}_r] E − [ A ℓ − ∣ G r − ] Σ − \Sigma^{-} Σ − Σ − ( 0 , p ) = 0 \Sigma^{-}(0,p)=0 Σ − ( 0 , p ) = 0 (154) . If r ≥ p r\geq p r ≥ p A ˉ p − \bar{A}^{-}_p A ˉ p − G r − \mathcal{G}^{-}_r G r −
Θ ( r ; ℓ , p ) = E − [ E − [ A ℓ − ∣ G r − ] A ˉ p − ] = E − [ A ℓ − A ˉ p − ] = Σ − ( ℓ , p ) ( r ≥ p ) . (155) \Theta(r;\ell,p)=\mathbb{E}^{-}\left[\mathbb{E}^{-}[A^{-}_{\ell}\mid\mathcal{G}^{-}_r]\bar{A}^{-}_p\right]=\mathbb{E}^{-}[A^{-}_{\ell}\bar{A}^{-}_p]=\Sigma^{-}(\ell,p) \qquad(r\geq p). \tag*{(155)} Θ ( r ; ℓ , p ) = E − [ E − [ A ℓ − ∣ G r − ] A ˉ p − ] = E − [ A ℓ − A ˉ p − ] = Σ − ( ℓ , p ) ( r ≥ p ) . (155) Fluctuations along the path. Let τ \tau τ r , ℓ ∈ [ 0 , 1 ] r,\ell\in[0,1] r , ℓ ∈ [ 0 , 1 ]
x r = τ ⋅ M r n − A r − , y r ℓ = M r ⋅ M ℓ n − A r ∧ ℓ − , x_r=\frac{\tau\cdot M_r}{n}-A_r^{-},\qquad y_{r\ell}=\frac{M_r\cdot M_\ell}{n}-A_{r\wedge\ell}^{-}, x r = n τ ⋅ M r − A r − , y r ℓ = n M r ⋅ M ℓ − A r ∧ ℓ − , random variables of the rest system; the subscripts distinguish them from the field variables x x x y y y M q = E − [ τ ∣ G q − ] M_q=\mathbb E^{-}[\tau\mid\mathcal G_q^{-}] M q = E − [ τ ∣ G q − ] r ≤ ℓ r\le\ell r ≤ ℓ
C r − = E − [ C ℓ − ∣ G r − ] + E − [ ( M ℓ − M r ) ( M ℓ − M r ) T ∣ G r − ] . (156) C_r^{-}=\mathbb E^{-}[C_\ell^{-}\mid\mathcal G_r^{-}]+\mathbb E^{-}\left[(M_\ell-M_r)(M_\ell-M_r)^{\mathsf T}\mid\mathcal G_r^{-}\right]. \tag*{(156)} C r − = E − [ C ℓ − ∣ G r − ] + E − [ ( M ℓ − M r ) ( M ℓ − M r ) T ∣ G r − ] . (156) In particular, E − [ C ℓ − ∣ G r − ] ⪯ C r − \mathbb E^{-}[C_\ell^{-}\mid\mathcal G_r^{-}]\preceq C_r^{-} E − [ C ℓ − ∣ G r − ] ⪯ C r − Cov − ( M ℓ ∣ G r − ) ⪯ C r − \operatorname{Cov}^{-}(M_\ell\mid\mathcal G_r^{-})\preceq C_r^{-} Cov − ( M ℓ ∣ G r − ) ⪯ C r − x r = M r ⋅ ( τ − M r ) / n x_r=M_r\cdot(\tau-M_r)/n x r = M r ⋅ ( τ − M r ) / n y r ℓ = M r ⋅ ( M ℓ − M r ) / n y_{r\ell}=M_r\cdot(M_\ell-M_r)/n y r ℓ = M r ⋅ ( M ℓ − M r ) / n r ≤ ℓ r\le\ell r ≤ ℓ G r − \mathcal G_r^{-} G r −
E − [ x r ∣ G r − ] = 0 , E − x r 2 = j r − , E − y r ℓ = 0 , E − y r ℓ 2 ≤ j r ∧ ℓ . (157) \mathbb E^{-}[x_r\mid\mathcal G_r^{-}]=0,\qquad\mathbb E^{-}x_r^2=j_r^{-},\qquad\mathbb E^{-}y_{r\ell}=0,\qquad\mathbb E^{-}y_{r\ell}^2\le j_{r\wedge\ell}. \tag*{(157)} E − [ x r ∣ G r − ] = 0 , E − x r 2 = j r − , E − y r ℓ = 0 , E − y r ℓ 2 ≤ j r ∧ ℓ . (157) Indeed, E − [ x r 2 ∣ G r − ] = M r T C r − M r / n 2 \mathbb E^{-}[x_r^2\mid\mathcal G_r^{-}]=M_r^{\mathsf T}C_r^{-}M_r/n^2 E − [ x r 2 ∣ G r − ] = M r T C r − M r / n 2 r ≤ ℓ r\le\ell r ≤ ℓ M ℓ − M r M_\ell-M_r M ℓ − M r C r − C_r^{-} C r − G r − \mathcal G_r^{-} G r − E − [ y r ℓ 2 ∣ G r − ] ≤ M r T C r − M r / n 2 \mathbb E^{-}[y_{r\ell}^2\mid\mathcal G_r^{-}]\le M_r^{\mathsf T}C_r^{-}M_r/n^2 E − [ y r ℓ 2 ∣ G r − ] ≤ M r T C r − M r / n 2 r > ℓ r>\ell r > ℓ
The error terms. We set v p 0 = Var 0 ( f p ) v_p^0=\operatorname{Var}^{0}(f_p) v p 0 = Var 0 ( f p )
a p H = E − Tr ( ( H p − ) 2 ) n 2 , b p H = E − [ M p T H p − M p ] n 2 , b r , p H = E − [ M r T H p − M p ] n 2 , a_p^H=\frac{\mathbb E^{-}\operatorname{Tr}\left((H_p^{-})^2\right)}{n^2},\qquad
b_p^H=\frac{\mathbb E^{-}[M_p^{\mathsf T}H_p^{-}M_p]}{n^2},\qquad
b_{r,p}^H=\frac{\mathbb E^{-}[M_r^{\mathsf T}H_p^{-}M_p]}{n^2}, a p H = n 2 E − Tr ( ( H p − ) 2 ) , b p H = n 2 E − [ M p T H p − M p ] , b r , p H = n 2 E − [ M r T H p − M p ] , z p , ℓ = 1 n E − [ M p ⋅ ∇ ( P p , ℓ − A ℓ − ) ( X p ) ] ( ℓ ≥ p ) , z_{p,\ell}=\frac{1}{n}\mathbb E^{-}\left[M_p\cdot\nabla\left(P_{p,\ell}^{-}A_\ell^{-}\right)(X_p)\right]\qquad(\ell\ge p), z p , ℓ = n 1 E − [ M p ⋅ ∇ ( P p , ℓ − A ℓ − ) ( X p ) ] ( ℓ ≥ p ) , where ∇ ( P p , ℓ − A ℓ − ) \nabla(P_{p,\ell}^{-}A_\ell^{-}) ∇ ( P p , ℓ − A ℓ − ) x ↦ ( P p , ℓ − A ℓ − ) ( x ) x\mapsto(P_{p,\ell}^{-}A_\ell^{-})(x) x ↦ ( P p , ℓ − A ℓ − ) ( x )
T p ( 1 ) = 2 E 0 [ m p h p ϵ ] E − [ x p A ‾ p − ] − 2 ∫ E 0 [ m p h p m r ] E − [ y p r A ‾ p − ] ν ( d r ) , (158) T_p^{(1)}=2\mathbb E^0[m_ph_p\epsilon]\mathbb E^{-}[x_p\overline A_p^{-}]-2\int\mathbb E^0[m_ph_pm_r]\mathbb E^{-}[y_{pr}\overline A_p^{-}]\,\nu(\mathrm{d}r), \tag*{(158)} T p ( 1 ) = 2 E 0 [ m p h p ϵ ] E − [ x p A p − ] − 2 ∫ E 0 [ m p h p m r ] E − [ y p r A p − ] ν ( d r ) , (158) T p ( 2 ) = − ∫ E 0 [ f ‾ p m r ] E − [ x r A ‾ p − ] ν ( d r ) + 1 2 ∬ E 0 [ f ‾ p m r m ℓ ] E − [ y r ℓ A ‾ p − ] ν ( d r ) ν ( d ℓ ) , (159) T_p^{(2)}=-\int\mathbb E^0[\overline f_pm_r]\mathbb E^{-}[x_r\overline A_p^{-}]\,\nu(\mathrm{d}r)
+\frac{1}{2}\iint\mathbb E^0[\overline f_pm_rm_\ell]\mathbb E^{-}[y_{r\ell}\overline A_p^{-}]\,\nu(\mathrm{d}r)\nu(\mathrm{d}\ell), \tag*{(159)} T p ( 2 ) = − ∫ E 0 [ f p m r ] E − [ x r A p − ] ν ( d r ) + 2 1 ∬ E 0 [ f p m r m ℓ ] E − [ y r ℓ A p − ] ν ( d r ) ν ( d ℓ ) , (159) T p ( 3 ) = v p 0 a p H + [ 4 E 0 ( m p 2 h p ) + 2 ( 1 − a ( p ) ) v p 0 ] b p H − ∫ ( p , 1 ] Cov 0 ( f p , g p ℓ ( Y p ) ) z p , ℓ ν ( d ℓ ) − 2 ∫ E 0 [ f ‾ p m p m r ] b r , p H ν ( d r ) . (160) T_p^{(3)}=v_p^0a_p^H+\left[4\mathbb E^0(m_p^2h_p)+2(1-a(p))v_p^0\right]b_p^H
-\int_{(p,1]}\operatorname{Cov}^{0}(f_p,g_p^\ell(Y_p))z_{p,\ell}\,\nu(\mathrm{d}\ell)
-2\int\mathbb E^0[\overline f_pm_pm_r]b_{r,p}^H\,\nu(\mathrm{d}r). \tag*{(160)} T p ( 3 ) = v p 0 a p H + [ 4 E 0 ( m p 2 h p ) + 2 ( 1 − a ( p )) v p 0 ] b p H − ∫ ( p , 1 ] Cov 0 ( f p , g p ℓ ( Y p )) z p , ℓ ν ( d ℓ ) − 2 ∫ E 0 [ f p m p m r ] b r , p H ν ( d r ) . (160) and T p = T p ( 1 ) + T p ( 2 ) + T p ( 3 ) T_p=T_p^{(1)}+T_p^{(2)}+T_p^{(3)} T p = T p ( 1 ) + T p ( 2 ) + T p ( 3 ) ν \nu ν [ 0 , 1 ] [0,1] [ 0 , 1 ] ν \nu ν 0 0 0 T p ( 1 ) T_p^{(1)} T p ( 1 ) T p ( 2 ) T_p^{(2)} T p ( 2 ) x r x_r x r y r ℓ y_{r\ell} y r ℓ T p ( 3 ) T_p^{(3)} T p ( 3 ) A p − A_p^{-} A p −
The conditional future correction. For p , ℓ > r p,\ell>r p , ℓ > r
Λ r ( p , ℓ ) = Cov r 0 ( g p r ( Y r ) , g r ℓ ( Y r ) ) , Σ ‾ r ( p , ℓ ) = E − Cov − ( A p − , A ℓ − ∣ G r − ) , \Lambda_r(p,\ell)=\operatorname{Cov}^{0}_r(g^r_p(Y_r),g^\ell_r(Y_r)),\qquad\overline{\Sigma}_r(p,\ell)=\mathbb{E}^{-}\operatorname{Cov}^{-}(A_p^-,A_\ell^-\mid\mathcal{G}_r^-), Λ r ( p , ℓ ) = Cov r 0 ( g p r ( Y r ) , g r ℓ ( Y r )) , Σ r ( p , ℓ ) = E − Cov − ( A p − , A ℓ − ∣ G r − ) , and
P = ∫ ν ( d r ) ∬ p , ℓ > r Λ r ( p , ℓ ) Σ ‾ r ( p , ℓ ) ν ( d p ) ν ( d ℓ ) . (161) \mathcal{P}=\int\nu(\mathrm{d}r)\iint_{p,\ell>r}\Lambda_r(p,\ell)\overline{\Sigma}_r(p,\ell)\nu(\mathrm{d}p)\nu(\mathrm{d}\ell). \tag*{(161)} P = ∫ ν ( d r ) ∬ p , ℓ > r Λ r ( p , ℓ ) Σ r ( p , ℓ ) ν ( d p ) ν ( d ℓ ) . (161) Note that Λ r \Lambda_r Λ r Σ ‾ r \overline{\Sigma}_r Σ r Lemma 11.2 that P ≥ 0 \mathcal{P}\ge0 P ≥ 0 − α P / 2 -\alpha\mathcal{P}/2 − α P /2 Θ \Theta Θ Σ ‾ − \overline{\Sigma}_- Σ −
The scalar response operator applied to Σ ‾ − . \overline{\Sigma}_-. Σ − . For a bounded Borel function φ \varphi φ [ 0 , 1 ] [0,1] [ 0 , 1 ] p ∈ [ 0 , 1 ] p\in[0,1] p ∈ [ 0 , 1 ] (106) , in the order written there, are
( J 1 ) E 0 [ h p 2 + 2 ( 1 − a ( p ) ) m p 2 h p ] φ ( p ) , ( J 2 ) − ∫ ( p , 1 ] E 0 [ m p ∂ y g p ℓ ( Y p ) ] φ ( ℓ ) ν ( d ℓ ) , ( J 3 ) − 2 ∫ E 0 [ m p h p m r ] φ ( p ∧ r ) ν ( d r ) , ( J 4 ) − ∫ E 0 [ f ‾ p ε m r ] φ ( r ) ν ( d r ) , ( J 5 ) 1 2 ∫ a ( r ) E 0 [ f ‾ p m r 2 ] φ ( r ) ν ( d r ) , ( J 6 ) 1 2 ∬ E 0 [ f ‾ p m r m ℓ ] φ ( r ∧ ℓ ) ν ( d r ) ν ( d ℓ ) , ( J 7 ) 1 2 ∫ ν ( d r ) ∫ ( r , 1 ] E 0 [ f ‾ p g r ℓ ( Y r ) ] φ ( ℓ ) ν ( d ℓ ) . \begin{aligned}
(\mathrm{J}1)\quad&\mathbb{E}^{0}[h_p^2+2(1-a(p))m_p^2h_p]\varphi(p),\\
(\mathrm{J}2)\quad&-\int_{(p,1]}\mathbb{E}^{0}[m_p\partial_y g_p^\ell(Y_p)]\varphi(\ell)\nu(\mathrm{d}\ell),\\
(\mathrm{J}3)\quad&-2\int\mathbb{E}^{0}[m_ph_pm_r]\varphi(p\wedge r)\nu(\mathrm{d}r),\\
(\mathrm{J}4)\quad&-\int\mathbb{E}^{0}[\overline{f}_p\varepsilon m_r]\varphi(r)\nu(\mathrm{d}r),\\
(\mathrm{J}5)\quad&\frac{1}{2}\int a(r)\mathbb{E}^{0}[\overline{f}_pm_r^2]\varphi(r)\nu(\mathrm{d}r),\\
(\mathrm{J}6)\quad&\frac{1}{2}\iint\mathbb{E}^{0}[\overline{f}_pm_rm_\ell]\varphi(r\wedge\ell)\nu(\mathrm{d}r)\nu(\mathrm{d}\ell),\\
(\mathrm{J}7)\quad&\frac{1}{2}\int\nu(\mathrm{d}r)\int_{(r,1]}\mathbb{E}^{0}[\overline{f}_pg_r^\ell(Y_r)]\varphi(\ell)\nu(\mathrm{d}\ell).
\end{aligned} ( J 1 ) ( J 2 ) ( J 3 ) ( J 4 ) ( J 5 ) ( J 6 ) ( J 7 ) E 0 [ h p 2 + 2 ( 1 − a ( p )) m p 2 h p ] φ ( p ) , − ∫ ( p , 1 ] E 0 [ m p ∂ y g p ℓ ( Y p )] φ ( ℓ ) ν ( d ℓ ) , − 2 ∫ E 0 [ m p h p m r ] φ ( p ∧ r ) ν ( d r ) , − ∫ E 0 [ f p ε m r ] φ ( r ) ν ( d r ) , 2 1 ∫ a ( r ) E 0 [ f p m r 2 ] φ ( r ) ν ( d r ) , 2 1 ∬ E 0 [ f p m r m ℓ ] φ ( r ∧ ℓ ) ν ( d r ) ν ( d ℓ ) , 2 1 ∫ ν ( d r ) ∫ ( r , 1 ] E 0 [ f p g r ℓ ( Y r )] φ ( ℓ ) ν ( d ℓ ) . and their sum is d − 1 ( J a , d φ ) ( p ) d^{-1}(J_{a,d}\varphi)(p) d − 1 ( J a , d φ ) ( p ) β \beta β p ∈ [ 0 , 1 ] p\in[0,1] p ∈ [ 0 , 1 ] ( J a , d s c Σ ‾ − ) ( p , p ) (J^{\mathrm{sc}}_{a,d}\overline{\Sigma}_-)(p,p) ( J a , d sc Σ − ) ( p , p ) p p p J a , d Σ ‾ − ( ⋅ , p ) J_{a,d}\overline{\Sigma}_-(\cdot,p) J a , d Σ − ( ⋅ , p ) J a , d J_{a,d} J a , d Σ ‾ − \overline{\Sigma}_- Σ − J a , d s c J^{\mathrm{sc}}_{a,d} J a , d sc φ \varphi φ φ ( 0 ) = 0 \varphi(0)=0 φ ( 0 ) = 0 m p m_p m p f ‾ p \overline{f}_p f p m 0 = f ‾ 0 = 0 m_0=\overline{f}_0=0 m 0 = f 0 = 0 p = 0 p=0 p = 0 h 0 2 φ ( 0 ) = 0 h_0^2\varphi(0)=0 h 0 2 φ ( 0 ) = 0 ν ( { 1 } ) = 0 \nu(\{1\})=0 ν ({ 1 }) = 0 J a , d φ J_{a,d}\varphi J a , d φ H ν \mathcal{H}_\nu H ν J a , d s c φ J^{\mathrm{sc}}_{a,d}\varphi J a , d sc φ φ ∈ H ν \varphi\in\mathcal{H}_\nu φ ∈ H ν φ = Σ ‾ − ( ⋅ , p ) \varphi=\overline{\Sigma}_-(\cdot,p) φ = Σ − ( ⋅ , p ) p p p
The endpoint derivative
Lemma 11.1 (Endpoint derivative at a fixed level). Let μ = s δ 0 + ν \mu=s\delta_0+\nu μ = s δ 0 + ν ν ( { 1 } ) = 0 \nu(\{1\})=0 ν ({ 1 }) = 0 p ∈ [ 0 , 1 ] p\in[0,1] p ∈ [ 0 , 1 ] α > 0 \alpha>0 α > 0
C p p ′ ′ ( 0 ) α = 1 d ( J a , d s c Σ ‾ − ) ( p , p ) + T p + 1 2 ∫ ν ( d r ) ∫ ( r , 1 ] Cov 0 ( f p , g r ℓ ( Y r ) ) [ Θ ( r ; ℓ , p ) − Σ ‾ − ( ℓ , p ) ] ν ( d ℓ ) . (162) \frac{C''_{pp}(0)}{\alpha}=\frac{1}{d}(J^{\mathrm{sc}}_{a,d}\overline{\Sigma}_-)(p,p)+T_p
+\frac{1}{2}\int\nu(\mathrm{d}r)\int_{(r,1]}\operatorname{Cov}^{0}(f_p,g_r^\ell(Y_r))\left[\Theta(r;\ell,p)-\overline{\Sigma}_-(\ell,p)\right]\nu(\mathrm{d}\ell). \tag*{(162)} α C pp ′′ ( 0 ) = d 1 ( J a , d sc Σ − ) ( p , p ) + T p + 2 1 ∫ ν ( d r ) ∫ ( r , 1 ] Cov 0 ( f p , g r ℓ ( Y r )) [ Θ ( r ; ℓ , p ) − Σ − ( ℓ , p ) ] ν ( d ℓ ) . (162) :::`
If α = 0 \alpha= 0 α = 0 C p p ′ ( 0 ) = 0 C'_{pp}(0) = 0 C pp ′ ( 0 ) = 0
The first term on the right side of (11.9) is the scalar response applied to Σ − ( ⋅ , p ) \Sigma^{-}(\cdot,p) Σ − ( ⋅ , p ) φ \varphi φ T p T_p T p E − [ A ℓ − ∣ G r − ] \mathbb{E}^{-}[A_{\ell}^{-} \mid\mathcal{G}_{r}^{-}] E − [ A ℓ − ∣ G r − ] Σ − ( ℓ , p ) \Sigma^{-}(\ell,p) Σ − ( ℓ , p ) Θ ( r ; ℓ , p ) \Theta(r;\ell,p) Θ ( r ; ℓ , p ) p p p − P / 2 -\mathcal{P}/2 − P /2
Lemma 11.2 (Integrated linear response). Let μ = s δ 0 + ν \mu= s\delta_0 + \nu μ = s δ 0 + ν ν ( { 1 } ) = 0 \nu(\{1\}) = 0 ν ({ 1 }) = 0 V − = ∫ Var − ( A p − ) ν ( d p ) V^{-} = \int\operatorname{Var}^{-}(A_p^{-})\nu(\mathrm{d}p) V − = ∫ Var − ( A p − ) ν ( d p )
∫ C p p ′ ( 0 ) ν ( d p ) = α [ 1 d Tr ν ( J a , d s c Σ − ) − 1 2 P + ∫ T p ν ( d p ) ] , P ≥ 0 , (163) \int C'_{pp}(0)\nu(\mathrm{d}p) = \alpha\left[\frac{1}{d}\operatorname{Tr}_{\nu}(J^{\mathrm{sc}}_{a,d}\Sigma^{-}) - \frac{1}{2}\mathcal{P} + \int T_p\nu(\mathrm{d}p)\right], \qquad\mathcal{P} \ge0, \tag*{(163)} ∫ C pp ′ ( 0 ) ν ( d p ) = α [ d 1 Tr ν ( J a , d sc Σ − ) − 2 1 P + ∫ T p ν ( d p ) ] , P ≥ 0 , (163) where P \mathcal{P} P Tr ν ( J a , d s c Σ − ) = ∫ ( J a , d s c Σ − ) ( p , p ) ν ( d p ) \operatorname{Tr}_{\nu}(J^{\mathrm{sc}}_{a,d}\Sigma^{-}) = \int(J^{\mathrm{sc}}_{a,d}\Sigma^{-})(p,p)\nu(\mathrm{d}p) Tr ν ( J a , d sc Σ − ) = ∫ ( J a , d sc Σ − ) ( p , p ) ν ( d p ) T p T_p T p Σ − \Sigma^{-} Σ − L 2 ( ν ) L^2(\nu) L 2 ( ν ) V − V^{-} V − H ν \mathcal{H}_{\nu} H ν A ˉ − \bar{A}^{-} A ˉ − p ↦ A ˉ p − p \mapsto\bar{A}^{-}_p p ↦ A ˉ p − Tr ν ( J a , d s c Σ − ) \operatorname{Tr}_{\nu}(J^{\mathrm{sc}}_{a,d}\Sigma^{-}) Tr ν ( J a , d sc Σ − ) E − ⟨ A ˉ − , J a , d s c A ˉ − ⟩ L 2 ( ν ) \mathbb{E}^{-}\langle\bar{A}^{-},J^{\mathrm{sc}}_{a,d}\bar{A}^{-}\rangle_{L^2(\nu)} E − ⟨ A ˉ − , J a , d sc A ˉ − ⟩ L 2 ( ν ) J a , d s c Σ − J^{\mathrm{sc}}_{a,d}\Sigma^{-} J a , d sc Σ − H ν \mathcal{H}_{\nu} H ν
Proof of the endpoint derivative at a fixed level
Proof of Lemma 11.1. If α = 0 \alpha= 0 α = 0 α u / n \alpha u/n α u / n b ( u ) = d b(u) = d b ( u ) = d u u u P u = P 0 P_u = P_0 P u = P 0 u u u C p p C_{pp} C pp α > 0 \alpha> 0 α > 0 p ∈ [ 0 , 1 ] p \in[0,1] p ∈ [ 0 , 1 ] C p p ′ ( 0 ) C'_{pp}(0) C pp ′ ( 0 ) N N N U U U
Step 1: reduction to fixed couplings. For w ≥ 0 w \ge0 w ≥ 0 b ∈ [ c , d ] b \in[c,d] b ∈ [ c , d ] P w , b P_{w,b} P w , b w w w α u / n \alpha u/n α u / n b b b b ( u ) b(u) b ( u ) P u = P α u / n , b ( u ) P_u = P_{\alpha u/n,b(u)} P u = P α u / n , b ( u ) b ( u ) ∈ [ c , d ] b(u) \in[c,d] b ( u ) ∈ [ c , d ] E w , b \mathbb{E}_{w,b} E w , b Cov w , b \operatorname{Cov}_{w,b} Cov w , b
C ( w , b ) = Cov w , b ( m N ( p ) 2 , A p − ) , so that C p p ( u ) = C ( α u / n , b ( u ) ) . C(w,b) = \operatorname{Cov}_{w,b}\left(m_N(p)^2,A_p^{-}\right), \qquad\text{so that} \qquad C_{pp}(u) = C(\alpha u/n,b(u)). C ( w , b ) = Cov w , b ( m N ( p ) 2 , A p − ) , so that C pp ( u ) = C ( α u / n , b ( u )) . At w = 0 w = 0 w = 0 U U U ( q , x ) (q,x) ( q , x ) ( q , y ) (q,y) ( q , y ) u = 0 u = 0 u = 0 d d d b b b P 0 , b P_{0,b} P 0 , b m N ( p ) m_N(p) m N ( p ) A p − A_p^{-} A p −
C ( 0 , b ) = 0 for every b ∈ [ c , d ] . (164) C(0,b) = 0 \qquad\text{for every } b \in[c,d]. \tag*{(164)} C ( 0 , b ) = 0 for every b ∈ [ c , d ] . (164) Applying Lemma 4.3 with the row variance w w w w ↦ C ( w , b ) w \mapsto C(w,b) w ↦ C ( w , b ) [ 0 , α / n ] [0,\alpha/n] [ 0 , α / n ] ( 0 , α / n ] (0,\alpha/n] ( 0 , α / n ] w = 0 w=0 w = 0 ∂ w C \partial_w C ∂ w C ( w , b ) (w,b) ( w , b ) w g \sqrt{w}g w g b B \sqrt{b}B b B 0 < u ≤ 1 0<u\leq1 0 < u ≤ 1 w w w w u ∈ ( 0 , α u / n ) w_u\in(0,\alpha u/n) w u ∈ ( 0 , α u / n )
C p p ′ ( u ) = C ( α u / n , b ( u ) ) − C ( 0 , b ( u ) ) = α u n ∂ w C ( w u , b ( u ) ) . C'_{pp}(u)=C(\alpha u/n,b(u))-C(0,b(u))=\frac{\alpha u}{n}\partial_w C(w_u,b(u)). C pp ′ ( u ) = C ( α u / n , b ( u )) − C ( 0 , b ( u )) = n α u ∂ w C ( w u , b ( u )) . Dividing by u u u u ↓ 0 u\downarrow0 u ↓ 0
C p p ′ ( 0 ) = α n ∂ w C ( 0 , d ) . (165) C'_{pp}(0)=\frac{\alpha}{n}\partial_w C(0,d). \tag*{(165)} C pp ′ ( 0 ) = n α ∂ w C ( 0 , d ) . (165) In particular, the variation of the last-field coefficient b ( u ) b(u) b ( u ) C p p ′ ( 0 ) C'_{pp}(0) C pp ′ ( 0 )
We now compute ∂ w C ( 0 , d ) \partial_w C(0,d) ∂ w C ( 0 , d ) θ ∈ R n \theta\in\mathbb{R}^n θ ∈ R n P θ \mathbb{P}^{\theta} P θ d d d α u / n ϵ ∑ i g i N τ i \sqrt{\alpha u/n}\,\epsilon\sum_i g_{iN}\tau_i α u / n ϵ ∑ i g i N τ i E ( σ ; x , y ) \mathcal{E}(\sigma;x,y) E ( σ ; x , y ) ϵ θ ⋅ τ \epsilon\theta\cdot\tau ϵ θ ⋅ τ U ( q , x , y ; θ ) U(q,x,y;\theta) U ( q , x , y ; θ ) E θ \mathbb{E}^{\theta} E θ Cov θ \operatorname{Cov}^{\theta} Cov θ P θ \mathbb{P}^{\theta} P θ
Z ( θ ) = E r e f e s U ( 0 , 0 , 0 ; θ ) . Z(\theta)=\mathbb{E}_{\mathrm{ref}}e^{sU(0,0,0;\theta)}. Z ( θ ) = E ref e s U ( 0 , 0 , 0 ; θ ) . for its outer normalizer, which is an expectation over the couplings g i j g_{ij} g ij 1 ≤ i < j ≤ n 1\leq i<j\leq n 1 ≤ i < j ≤ n P θ \mathbb{P}^{\theta} P θ m N ( p ) = ∂ y U ( p , X p , Y p ; θ ) m_N(p)=\partial_yU(p,X_p,Y_p;\theta) m N ( p ) = ∂ y U ( p , X p , Y p ; θ ) M p = ∇ x U ( p , X p , Y p ; θ ) M_p=\nabla_xU(p,X_p,Y_p;\theta) M p = ∇ x U ( p , X p , Y p ; θ ) U ( ⋅ ; θ ) U(\cdot;\theta) U ( ⋅ ; θ ) P w , d \mathbb{P}_{w,d} P w , d g i j g_{ij} g ij θ = w ( g 1 N , … , g n N ) \theta=\sqrt{w}(g_{1N},\ldots,g_{nN}) θ = w ( g 1 N , … , g n N ) θ \theta θ P θ \mathbb{P}^{\theta} P θ e s U ( 0 , 0 , 0 ) e^{sU(0,0,0)} e s U ( 0 , 0 , 0 ) θ \theta θ Z ( θ ) / E Z ( θ ) Z(\theta)/\mathbb{E}Z(\theta) Z ( θ ) / E Z ( θ ) N ( 0 , w I n ) N(0,wI_n) N ( 0 , w I n ) G G G
E w , d G = E [ Z ( θ ) E θ G ] E Z ( θ ) , θ ∼ N ( 0 , w I n ) . (166) \mathbb{E}_{w,d}G=\frac{\mathbb{E}[Z(\theta)\mathbb{E}^{\theta}G]}{\mathbb{E}Z(\theta)},\qquad\theta\sim N(0,wI_n). \tag*{(166)} E w , d G = E Z ( θ ) E [ Z ( θ ) E θ G ] , θ ∼ N ( 0 , w I n ) . (166) This mixture is not the Gaussian average of the laws P θ \mathbb{P}^{\theta} P θ θ \theta θ Z ( θ ) Z(\theta) Z ( θ ) w = 0 w=0 w = 0
Let f = m N ( p ) 2 f=m_N(p)^2 f = m N ( p ) 2 G G G Φ G ( θ ) = E θ G \Phi_G(\theta)=\mathbb{E}^{\theta}G Φ G ( θ ) = E θ G ϵ \epsilon ϵ − ϵ -\epsilon − ϵ U ( 1 , x , y ; θ ) = U ( 1 , x , − y ; − θ ) U(1,x,y;\theta)=U(1,x,-y;-\theta) U ( 1 , x , y ; θ ) = U ( 1 , x , − y ; − θ ) y ↦ − y y\mapsto-y y ↦ − y U ( q , x , y ; θ ) = U ( q , x , − y ; − θ ) U(q,x,y;\theta)=U(q,x,-y;-\theta) U ( q , x , y ; θ ) = U ( q , x , − y ; − θ ) q q q ( ϵ , Y ) ↦ ( − ϵ , − Y ) (\epsilon,Y)\mapsto(-\epsilon,-Y) ( ϵ , Y ) ↦ ( − ϵ , − Y ) P θ \mathbb{P}^{\theta} P θ P − θ \mathbb{P}^{-\theta} P − θ U ( 0 , 0 , 0 ; θ ) U(0,0,0;\theta) U ( 0 , 0 , 0 ; θ ) M p M_p M p m N ( p ) m_N(p) m N ( p )
Z , Φ f , Φ A p , Φ f A p Z,\qquad\Phi_f,\qquad\Phi_{A_p},\qquad\Phi_{fA_p} Z , Φ f , Φ A p , Φ f A p are even in θ \theta θ θ = 0 \theta= 0 θ = 0 N N N C e C ∣ θ ∣ C e^{C|\theta|} C e C ∣ θ ∣ C C C N N N (139) , we have ∣ ∂ θ i U ∣ ≤ 1 |\partial_{\theta_i}U| \le1 ∣ ∂ θ i U ∣ ≤ 1 0 ≤ ∂ θ i 2 U ≤ 1 0 \le\partial_{\theta_i}^2 U \le1 0 ≤ ∂ θ i 2 U ≤ 1 U U U ( θ , x , y ) (\theta,x,y) ( θ , x , y ) N N N P θ \mathbb{P}^{\theta} P θ θ \theta θ
C exp { C ( 1 + ∑ 1 ≤ i < j ≤ n ∣ g i j ∣ + ∣ θ ∣ + sup q ≤ 1 ∣ X q ∣ + sup q ≤ 1 ∣ Y q ∣ ) } C \exp\left\{C\left(1+\sum_{1\le i<j\le n}|g_{ij}|+|\theta|+\sup_{q\le1}|X_q|+\sup_{q\le1}|Y_q|\right)\right\} C exp { C ( 1 + 1 ≤ i < j ≤ n ∑ ∣ g ij ∣ + ∣ θ ∣ + q ≤ 1 sup ∣ X q ∣ + q ≤ 1 sup ∣ Y q ∣ ) } under the Gaussian reference law of the couplings and the paths. Constants that we use only to justify differentiation may depend on N N N
Let F ∈ C 2 ( R n ) F\in C^2(\mathbb{R}^n) F ∈ C 2 ( R n ) ∣ ∇ 2 F ( θ ) ∣ ≤ C e C ∣ θ ∣ |\nabla^2F(\theta)|\le Ce^{C|\theta|} ∣ ∇ 2 F ( θ ) ∣ ≤ C e C ∣ θ ∣ g g g R n \mathbb{R}^n R n w > 0 w>0 w > 0 w ↦ E F ( w g ) w\mapsto\mathbb{E}F(\sqrt{w}\,g) w ↦ E F ( w g ) E Δ F ( w g ) / 2 \mathbb{E}\,\Delta F(\sqrt{w}\,g)/2 E Δ F ( w g ) /2 Δ F ( 0 ) / 2 \Delta F(0)/2 Δ F ( 0 ) /2 w ↓ 0 w\downarrow0 w ↓ 0 w = 0 w=0 w = 0 Δ F ( 0 ) / 2 \Delta F(0)/2 Δ F ( 0 ) /2 F = Z Φ G F=Z\Phi_G F = Z Φ G F = Z F=Z F = Z (166) . By the quotient rule, we have
∂ w E w , d G ∣ w = 0 = Δ ( Z Φ G ) ( 0 ) − Φ G ( 0 ) Δ Z ( 0 ) 2 Z ( 0 ) = 1 2 Δ Φ G ( 0 ) + ∇ Z ( 0 ) ⋅ ∇ Φ G ( 0 ) Z ( 0 ) = 1 2 Δ Φ G ( 0 ) . \left.\partial_w\mathbb{E}_{w,d}G\right|_{w=0}
=\frac{\Delta(Z\Phi_G)(0)-\Phi_G(0)\Delta Z(0)}{2Z(0)}
=\frac{1}{2}\Delta\Phi_G(0)+\frac{\nabla Z(0)\cdot\nabla\Phi_G(0)}{Z(0)}
=\frac{1}{2}\Delta\Phi_G(0). ∂ w E w , d G ∣ w = 0 = 2 Z ( 0 ) Δ ( Z Φ G ) ( 0 ) − Φ G ( 0 ) Δ Z ( 0 ) = 2 1 Δ Φ G ( 0 ) + Z ( 0 ) ∇ Z ( 0 ) ⋅ ∇ Φ G ( 0 ) = 2 1 Δ Φ G ( 0 ) . where in the second equality we used Δ ( Z Φ G ) = Z Δ Φ G + 2 ∇ Z ⋅ ∇ Φ G + Φ G Δ Z \Delta(Z\Phi_G)=Z\Delta\Phi_G+2\nabla Z\cdot\nabla\Phi_G+\Phi_G\Delta Z Δ ( Z Φ G ) = Z Δ Φ G + 2∇ Z ⋅ ∇ Φ G + Φ G Δ Z ∇ Z ( 0 ) = 0 \nabla Z(0)=0 ∇ Z ( 0 ) = 0 G ∈ { f , A p ˉ , f A p ˉ } G\in\{f,A_{\bar p},fA_{\bar p}\} G ∈ { f , A p ˉ , f A p ˉ } ∇ Φ f ⋅ ∇ Φ A p ˉ \nabla\Phi_f\cdot\nabla\Phi_{A_{\bar p}} ∇ Φ f ⋅ ∇ Φ A p ˉ Δ ( Φ f Φ A p ˉ ) / 2 \Delta(\Phi_f\Phi_{A_{\bar p}})/2 Δ ( Φ f Φ A p ˉ ) /2 θ = 0 \theta=0 θ = 0 ∇ Φ f ( 0 ) = 0 \nabla\Phi_f(0)=0 ∇ Φ f ( 0 ) = 0
∂ w C ( 0 , d ) = 1 2 Δ Φ f A p ˉ ( 0 ) − 1 2 Δ Φ f ( 0 ) Φ A p ˉ ( 0 ) − 1 2 Φ f ( 0 ) Δ Φ A p ˉ ( 0 ) = 1 2 Δ θ Cov θ ( f , A p ˉ ) ∣ θ = 0 . \partial_w C(0,d)=\frac{1}{2}\Delta\Phi_{fA_{\bar p}}(0)-\frac{1}{2}\Delta\Phi_f(0)\Phi_{A_{\bar p}}(0)-\frac{1}{2}\Phi_f(0)\Delta\Phi_{A_{\bar p}}(0)=\left.\frac{1}{2}\Delta_\theta\operatorname{Cov}^{\theta}(f,A_{\bar p})\right|_{\theta=0}. ∂ w C ( 0 , d ) = 2 1 Δ Φ f A p ˉ ( 0 ) − 2 1 Δ Φ f ( 0 ) Φ A p ˉ ( 0 ) − 2 1 Φ f ( 0 ) Δ Φ A p ˉ ( 0 ) = 2 1 Δ θ Cov θ ( f , A p ˉ ) θ = 0 . Inserting this into (165) , we obtain
C p p ′ ( 0 ) = α 2 n ∑ i = 1 n ∂ θ i 2 Cov θ ( f , A p ˉ ) ∣ θ = 0 . (167) C_{pp}'(0)=\frac{\alpha}{2n}\sum_{i=1}^{n}\left.\partial_{\theta_i}^{2}\operatorname{Cov}^{\theta}(f,A_{\bar p})\right|_{\theta=0}. \tag*{(167)} C pp ′ ( 0 ) = 2 n α i = 1 ∑ n ∂ θ i 2 Cov θ ( f , A p ˉ ) θ = 0 . (167) Step 2: derivatives in one coordinate of the row. For θ = 0 \theta=0 θ = 0 P θ \mathbb{P}^{\theta} P θ P 0 \mathbb{P}_0 P 0 U ( q , x , y ; 0 ) = U − ( q , x ) + U 0 ( q , y ) U(q,x,y;0)=U^{-}(q,x)+U^{0}(q,y) U ( q , x , y ; 0 ) = U − ( q , x ) + U 0 ( q , y ) E \mathbb{E} E Cov \operatorname{Cov} Cov P 0 \mathbb{P}_0 P 0
U ˙ i ( r , x , y ) = ∂ θ i U ( r , x , y ; θ ) ∣ θ = 0 , U ¨ i ( r , x , y ) = ∂ θ i 2 U ( r , x , y ; θ ) ∣ θ = 0 . \dot{U}_i(r,x,y)=\left.\partial_{\theta_i}U(r,x,y;\theta)\right|_{\theta=0},\qquad\ddot{U}_i(r,x,y)=\left.\partial_{\theta_i}^{2}U(r,x,y;\theta)\right|_{\theta=0}. U ˙ i ( r , x , y ) = ∂ θ i U ( r , x , y ; θ ) ∣ θ = 0 , U ¨ i ( r , x , y ) = ∂ θ i 2 U ( r , x , y ; θ ) θ = 0 . Under P 0 \mathbb{P}_0 P 0 ( τ , ϵ ) (\tau,\epsilon) ( τ , ϵ ) G r \mathcal{G}_r G r τ \tau τ G r − \mathcal{G}_r^{-} G r − ϵ \epsilon ϵ G r 0 \mathcal{G}_r^{0} G r 0 π i ( r ) = E [ τ i ϵ ∣ G r ] \pi_i(r)=\mathbb{E}[\tau_i\epsilon\mid\mathcal{G}_r] π i ( r ) = E [ τ i ϵ ∣ G r ] (139) therefore equals M i ( r , X r ) m r M_i(r,X_r)m_r M i ( r , X r ) m r ℓ ≥ r \ell\ge r ℓ ≥ r E [ π i ( ℓ ) 2 ∣ G r ] = ( P r , ℓ M i ( ℓ , ⋅ ) 2 ) ( X r ) g r ℓ ( Y r ) \mathbb{E}[\pi_i(\ell)^2 \mid\mathcal{G}_r] = \left(P_{r,\ell}M_i(\ell,\cdot)^2\right)(X_r)g_r^{\ell}(Y_r) E [ π i ( ℓ ) 2 ∣ G r ] = ( P r , ℓ M i ( ℓ , ⋅ ) 2 ) ( X r ) g r ℓ ( Y r )
U ¨ i ( r , x , y ) = M i ( r , x ) m ( r , y ) , (168) \ddot{U}_i(r,x,y)=M_i(r,x)m(r,y), \tag*{(168)} U ¨ i ( r , x , y ) = M i ( r , x ) m ( r , y ) , (168) U ¨ i ( r , x , y ) = 1 − a ( r ) M i ( r , x ) 2 m ( r , y ) 2 − ∫ ( r , 1 ] ( P r , ℓ M i ( ℓ , ⋅ ) 2 ) ( x ) g r ℓ ( y ) ν ( d ℓ ) . (169) \ddot{U}_i(r,x,y)=1-a(r)M_i(r,x)^2m(r,y)^2-\int_{(r,1]}\left(P_{r,\ell}M_i(\ell,\cdot)^2\right)(x)g_r^{\ell}(y)\,\nu(\mathrm{d}\ell). \tag*{(169)} U ¨ i ( r , x , y ) = 1 − a ( r ) M i ( r , x ) 2 m ( r , y ) 2 − ∫ ( r , 1 ] ( P r , ℓ M i ( ℓ , ⋅ ) 2 ) ( x ) g r ℓ ( y ) ν ( d ℓ ) . (169) Since ∑ i M i ( ℓ , ⋅ ) 2 = n A − ( ℓ , ⋅ ) \sum_i M_i(\ell,\cdot)^2=nA^{-}(\ell,\cdot) ∑ i M i ( ℓ , ⋅ ) 2 = n A − ( ℓ , ⋅ ) i i i
U ¨ a v ( r , x , y ) = 1 n ∑ i = 1 n U ¨ i ( r , x , y ) = 1 − a ( r ) A − ( r , x ) m ( r , y ) 2 − ∫ ( r , 1 ] ( P r , ℓ A − ) ( x ) g r ℓ ( y ) ν ( d ℓ ) . (170) \ddot{U}_{\mathrm{av}}(r,x,y)=\frac{1}{n}\sum_{i=1}^{n}\ddot{U}_i(r,x,y)=1-a(r)A^{-}(r,x)m(r,y)^2-\int_{(r,1]}\left(P_{r,\ell}A^{-}\right)(x)g_r^{\ell}(y)\,\nu(\mathrm{d}\ell). \tag*{(170)} U ¨ av ( r , x , y ) = n 1 i = 1 ∑ n U ¨ i ( r , x , y ) = 1 − a ( r ) A − ( r , x ) m ( r , y ) 2 − ∫ ( r , 1 ] ( P r , ℓ A − ) ( x ) g r ℓ ( y ) ν ( d ℓ ) . (170) By Lemma 10.3 with one leaf, applied with the row fixed at θ \theta θ P θ \mathbb{P}^{\theta} P θ
ϖ θ = 1 Z ( θ ) exp { E − ( τ ) + ϵ θ ⋅ τ + X 1 ⋅ τ + Y 1 ϵ − ∫ [ 0 , 1 ] U ( q , X q , Y q ; θ ) ν ( d q ) } \varpi_{\theta}=\frac{1}{Z(\theta)}\exp\left\{\mathcal{E}^{-}(\tau)+\epsilon\theta\cdot\tau+X_1\cdot\tau+Y_1\epsilon-\int_{[0,1]}U(q,X_q,Y_q;\theta)\,\nu(\mathrm{d}q)\right\} ϖ θ = Z ( θ ) 1 exp { E − ( τ ) + ϵ θ ⋅ τ + X 1 ⋅ τ + Y 1 ϵ − ∫ [ 0 , 1 ] U ( q , X q , Y q ; θ ) ν ( d q ) } with respect to the Gaussian reference law of the couplings g i j g_{ij} g ij X X X Y Y Y c ( q ∧ q ′ ) I n c(q\wedge q')I_n c ( q ∧ q ′ ) I n d ( q ∧ q ′ ) d(q\wedge q') d ( q ∧ q ′ ) σ = ( τ , ϵ ) \sigma=(\tau,\epsilon) σ = ( τ , ϵ ) ∇ Z ( 0 ) = 0 \nabla Z(0)=0 ∇ Z ( 0 ) = 0 θ i \theta_i θ i ϖ θ \varpi_\theta ϖ θ θ = 0 \theta=0 θ = 0
s i = ∂ θ i log ϖ θ ∣ θ = 0 = τ i ϵ − ∫ M i ( r ) m r ν ( d r ) , M i ( r ) = M i ( r , X r ) , s_i=\left.\partial_{\theta_i}\log\varpi_\theta\right|_{\theta=0}=\tau_i\epsilon-\int M_i(r)m_r\,\nu(\mathrm{d}r),\qquad M_i(r)=M_i(r,X_r), s i = ∂ θ i log ϖ θ ∣ θ = 0 = τ i ϵ − ∫ M i ( r ) m r ν ( d r ) , M i ( r ) = M i ( r , X r ) , where we used (11.15), and the second derivative satisfies
∂ θ i 2 ϖ θ ϖ θ ∣ θ = 0 = s i 2 + ∂ θ i 2 log ϖ θ ∣ θ = 0 = s i 2 − ∫ U ¨ i ( r , X r , Y r ) ν ( d r ) − ∂ θ i 2 log Z ( 0 ) . (171) \left.\frac{\partial_{\theta_i}^2\varpi_\theta}{\varpi_\theta}\right|_{\theta=0}=s_i^2+\left.\partial_{\theta_i}^2\log\varpi_\theta\right|_{\theta=0}=s_i^2-\int\ddot{U}_i(r,X_r,Y_r)\,\nu(\mathrm{d}r)-\left.\partial_{\theta_i}^2\log Z(0)\right.. \tag*{(171)} ϖ θ ∂ θ i 2 ϖ θ θ = 0 = s i 2 + ∂ θ i 2 log ϖ θ θ = 0 = s i 2 − ∫ U ¨ i ( r , X r , Y r ) ν ( d r ) − ∂ θ i 2 log Z ( 0 ) . (171) This quantity has mean zero under P 0 \mathbb{P}_0 P 0 ϖ θ \varpi_\theta ϖ θ θ \theta θ
An observable G θ G_\theta G θ P θ \mathbb{P}^{\theta} P θ θ \theta θ U ( ⋅ ; θ ) U(\cdot;\theta) U ( ⋅ ; θ ) G i G_i G i G i i G_{ii} G ii θ i \theta_i θ i θ = 0 \theta=0 θ = 0
∂ θ i 2 E θ G θ ∣ θ = 0 = E [ G i i + 2 G i s i + G ∂ θ i 2 ϖ θ ϖ θ ∣ θ = 0 ] . \left.\partial_{\theta_i}^2\mathbb{E}^{\theta}G_\theta\right|_{\theta=0}=\mathbb{E}\left[G_{ii}+2G_is_i+G\left.\frac{\partial_{\theta_i}^2\varpi_\theta}{\varpi_\theta}\right|_{\theta=0}\right]. ∂ θ i 2 E θ G θ θ = 0 = E [ G ii + 2 G i s i + G ϖ θ ∂ θ i 2 ϖ θ θ = 0 ] . This formula holds for G = f A p − G=fA^{-}_p G = f A p − f f f A p − A^{-}_p A p − θ = 0 \theta=0 θ = 0 f = f p f=f_p f = f p f f f f p , i f_{p,i} f p , i f p , i i f_{p,ii} f p , ii A p − A^{-}_p A p − A ‾ p , i − \overline{A}^{-}_{p,i} A p , i − A ‾ p , i i − \overline{A}^{-}_{p,ii} A p , ii − E θ f \mathbb{E}^{\theta}f E θ f E θ A p − \mathbb{E}^{\theta}A^{-}_p E θ A p − θ = 0 \theta=0 θ = 0 Cov θ ( f , A p − ) = E θ [ f A p − ] − E θ f E θ A p − \operatorname{Cov}^{\theta}(f,A^{-}_p)=\mathbb{E}^{\theta}[fA^{-}_p]-\mathbb{E}^{\theta}f\mathbb{E}^{\theta}A^{-}_p Cov θ ( f , A p − ) = E θ [ f A p − ] − E θ f E θ A p − ∂ θ i 2 Cov θ ( f , A p − ) \partial_{\theta_i}^2\operatorname{Cov}^{\theta}(f,A^{-}_p) ∂ θ i 2 Cov θ ( f , A p − ) θ = 0 \theta=0 θ = 0
E [ f p , i i A ‾ p − + f ‾ p A ‾ p , i i − + 2 f p , i A ‾ p , i − ] + 2 E [ s i ( f p , i A ‾ p − + f ‾ p A ‾ p , i − ) ] + E [ f ‾ p A ‾ p − ∂ θ i 2 ϖ θ ϖ θ ∣ θ = 0 ] . \mathbb{E}\left[f_{p,ii}\overline{A}^{-}_p+\overline{f}_p\overline{A}^{-}_{p,ii}+2f_{p,i}\overline{A}^{-}_{p,i}\right]+2\mathbb{E}\left[s_i\left(f_{p,i}\overline{A}^{-}_p+\overline{f}_p\overline{A}^{-}_{p,i}\right)\right]+\mathbb{E}\left[\overline{f}_p\overline{A}^{-}_p\left.\frac{\partial_{\theta_i}^2\varpi_\theta}{\varpi_\theta}\right|_{\theta=0}\right]. E [ f p , ii A p − + f p A p , ii − + 2 f p , i A p , i − ] + 2 E [ s i ( f p , i A p − + f p A p , i − ) ] + E [ f p A p − ϖ θ ∂ θ i 2 ϖ θ θ = 0 ] . Since P 0 \mathbb{P}_0 P 0 E [ f ˉ p A ˉ p − ] = 0 \mathbb{E}[\bar{f}_p\bar{A}_p^-]=0 E [ f ˉ p A ˉ p − ] = 0 ∂ θ i 2 log Z ( 0 ) \partial_{\theta_i}^2\log Z(0) ∂ θ i 2 log Z ( 0 ) i i i
C p p ′ ( 0 ) α = 1 2 n ∑ i { E [ f p , i i A ˉ p − + f ˉ p A ˉ p , i i − + 2 f p , i A ˉ p , i − + 2 E [ s i ( f p , i A ˉ p − + f ˉ p A ˉ p , i − ) ] ] + E [ f ˉ p A ˉ p − ( s i 2 − ∫ U ¨ i d ν ) ] } . (172) \frac{C_{pp}'(0)}{\alpha}=\frac{1}{2n}\sum_i\left\{\mathbb{E}\left[f_{p,ii}\bar{A}_p^-+\bar{f}_p\bar{A}_{p,ii}^-+2f_{p,i}\bar{A}_{p,i}^-+2\mathbb{E}\left[s_i(f_{p,i}\bar{A}_p^-+\bar{f}_p\bar{A}_{p,i}^-)\right]\right]+\mathbb{E}\left[\bar{f}_p\bar{A}_p^-\left(s_i^2-\int\ddot{U}_i\,\mathrm{d}\nu\right)\right]\right\}. \tag*{(172)} α C pp ′ ( 0 ) = 2 n 1 i ∑ { E [ f p , ii A ˉ p − + f ˉ p A ˉ p , ii − + 2 f p , i A ˉ p , i − + 2 E [ s i ( f p , i A ˉ p − + f ˉ p A ˉ p , i − ) ] ] + E [ f ˉ p A ˉ p − ( s i 2 − ∫ U ¨ i d ν ) ] } . (172) We sort the right side into four groups, each including the factor 1 / ( 2 n ) 1/(2n) 1/ ( 2 n ) i i i
(a) the terms f p , i i A ˉ p − f_{p,ii}\bar{A}_p^- f p , ii A ˉ p −
(b) the terms 2 s i f p , i A ˉ p − 2s_if_{p,i}\bar{A}_p^- 2 s i f p , i A ˉ p −
(c) the terms f ˉ p A ˉ p − ( s i 2 − ∫ U ¨ i d ν ) \bar{f}_p\bar{A}_p^-(s_i^2-\int\ddot{U}_i\,\mathrm{d}\nu) f ˉ p A ˉ p − ( s i 2 − ∫ U ¨ i d ν )
(d) the three terms containing a derivative of A ˉ p − \bar{A}_p^- A ˉ p − f ˉ p A ˉ p , i i − \bar{f}_p\bar{A}_{p,ii}^- f ˉ p A ˉ p , ii − 2 f p , i A ˉ p , i − 2f_{p,i}\bar{A}_{p,i}^- 2 f p , i A ˉ p , i − 2 s i f ˉ p A ˉ p , i − 2s_i\bar{f}_p\bar{A}_{p,i}^- 2 s i f ˉ p A ˉ p , i −
We now compute the explicit derivatives. We have f = ( ∂ y U ( p , X p , Y p ; θ ) ) 2 f=(\partial_yU(p,X_p,Y_p;\theta))^2 f = ( ∂ y U ( p , X p , Y p ; θ ) ) 2 A ˉ p − = ∣ ∇ x U ( p , X p , Y p ; θ ) ∣ 2 / n \bar{A}_p^-=\lvert\nabla_xU(p,X_p,Y_p;\theta)\rvert^2/n A ˉ p − = ∣ ∇ x U ( p , X p , Y p ; θ ) ∣ 2 / n ∂ y U ˙ i ( p ) = M i ( p ) h p \partial_y\dot{U}_i(p)=M_i(p)h_p ∂ y U ˙ i ( p ) = M i ( p ) h p ∇ x U ˙ i ( p ) = m p H p − e i \nabla_x\dot{U}_i(p)=m_pH_p^-e_i ∇ x U ˙ i ( p ) = m p H p − e i θ i \theta_i θ i
f p , i = 2 m p h p M i ( p ) , f p , i i = 2 M i ( p ) 2 h p 2 + 2 m p ∂ y U ¨ i ( p ) , f_{p,i}=2m_ph_pM_i(p),\qquad f_{p,ii}=2M_i(p)^2h_p^2+2m_p\partial_y\ddot{U}_i(p), f p , i = 2 m p h p M i ( p ) , f p , ii = 2 M i ( p ) 2 h p 2 + 2 m p ∂ y U ¨ i ( p ) , A ˉ p , i − = 2 m p n ( H p − M p ) i , A ˉ p , i i − = 2 n { m p 2 ∣ H p − e i ∣ 2 + M p ⋅ ∇ x U ¨ i ( p ) } . (173) \bar{A}_{p,i}^-=\frac{2m_p}{n}(H_p^-M_p)_i,\qquad\bar{A}_{p,ii}^-=\frac{2}{n}\left\{m_p^2\lvert H_p^-e_i\rvert^2+M_p\cdot\nabla_x\ddot{U}_i(p)\right\}. \tag*{(173)} A ˉ p , i − = n 2 m p ( H p − M p ) i , A ˉ p , ii − = n 2 { m p 2 ∣ H p − e i ∣ 2 + M p ⋅ ∇ x U ¨ i ( p ) } . (173) where U ¨ i ( p ) \ddot{U}_i(p) U ¨ i ( p ) ( p , X p , Y p ) (p,X_p,Y_p) ( p , X p , Y p ) e i e_i e i i i i R n \mathbb{R}^n R n
Step 3: the groups (a) and (b). The identities (11.20) and (11.17), together with ∂ y ( m 2 ) = 2 m h \partial_y(m^2)=2mh ∂ y ( m 2 ) = 2 mh
1 2 n ∑ i f p , i i = A ˉ p − h p 2 + m p ∂ y U ¨ a v ( p ) = ( h p 2 − 2 a ( p ) m p 2 h p ) A ˉ p − − ∫ ( p , 1 ] m p ∂ y g p ℓ ( Y p ) ( P p , ℓ − A ℓ − ) ( X p ) ν ( d ℓ ) . \begin{aligned}
\frac{1}{2n}\sum_i f_{p,ii}=\bar{A}_p^-h_p^2+m_p\partial_y\ddot{U}_{av}(p) \\
&=\left(h_p^2-2a(p)m_p^2h_p\right)\bar{A}_p^--\int_{(p,1]}m_p\partial_yg_p^\ell(Y_p)\left(P_{p,\ell}^-\!A_\ell^-\right)(X_p)\,\nu(\mathrm{d}\ell).
\end{aligned} 2 n 1 i ∑ f p , ii = A ˉ p − h p 2 + m p ∂ y U ¨ a v ( p ) = ( h p 2 − 2 a ( p ) m p 2 h p ) A ˉ p − − ∫ ( p , 1 ] m p ∂ y g p ℓ ( Y p ) ( P p , ℓ − A ℓ − ) ( X p ) ν ( d ℓ ) . We multiply by A ˉ p − \bar{A}_p^- A ˉ p − E − [ A p − A ˉ p − ] = Σ − ( p , p ) \mathbb{E}^-[A_p^-\bar{A}_p^-]=\Sigma^-(p,p) E − [ A p − A ˉ p − ] = Σ − ( p , p ) E − [ ( P p , ℓ − A ℓ − ) A ˉ p − ] = Θ ( p ; ℓ , p ) = Σ − ( ℓ , p ) \mathbb{E}^-[(P_{p,\ell}^-A_\ell^-)\bar{A}_p^-]=\Theta(p;\ell,p)=\Sigma^-(\ell,p) E − [( P p , ℓ − A ℓ − ) A ˉ p − ] = Θ ( p ; ℓ , p ) = Σ − ( ℓ , p )
( E 0 [ h p 2 ] − 2 a ( p ) E 0 [ m p 2 h p ] ) Σ − ( p , p ) − ∫ ( p , 1 ] E 0 [ m p ∂ y g p ℓ ( Y p ) ] Σ − ( ℓ , p ) ν ( d ℓ ) . (174) \left(\mathbb{E}^0[h_p^2]-2a(p)\mathbb{E}^0[m_p^2h_p]\right)\Sigma^-(p,p)-\int_{(p,1]}\mathbb{E}^0[m_p\partial_yg_p^\ell(Y_p)]\Sigma^-(\ell,p)\,\nu(\mathrm{d}\ell). \tag*{(174)} ( E 0 [ h p 2 ] − 2 a ( p ) E 0 [ m p 2 h p ] ) Σ − ( p , p ) − ∫ ( p , 1 ] E 0 [ m p ∂ y g p ℓ ( Y p )] Σ − ( ℓ , p ) ν ( d ℓ ) . (174) For group (b), we insert the definitions of s i s_i s i x p x_p x p y p r = y r p y_{pr}=y_{rp} y p r = y r p
1 n ∑ i s i f p , i = 2 m p h p [ ε τ ⋅ M p n − ∫ m r M r ⋅ M p n ν ( d r ) ] = 2 m p h p [ ε ( A ˉ p − + x p ) − ∫ m r ( A ˉ p ∧ r − + y p r ) ν ( d r ) ] . \begin{aligned}
\frac{1}{n}\sum_i s_if_{p,i}=2m_ph_p\left[\varepsilon\frac{\tau\cdot M_p}{n}-\int m_r\frac{M_r\cdot M_p}{n}\,\nu(\mathrm{d}r)\right] \\
&=2m_ph_p\left[\varepsilon(\bar{A}_p^-+x_p)-\int m_r(\bar{A}_{p\wedge r}^-+y_{pr})\,\nu(\mathrm{d}r)\right].
\end{aligned} n 1 i ∑ s i f p , i = 2 m p h p [ ε n τ ⋅ M p − ∫ m r n M r ⋅ M p ν ( d r ) ] = 2 m p h p [ ε ( A ˉ p − + x p ) − ∫ m r ( A ˉ p ∧ r − + y p r ) ν ( d r ) ] . We multiply by A ˉ p − \bar{A}^{-}_{p} A ˉ p − E 0 [ ϵ ∣ G p 0 ] = m p \mathbb{E}^{0}[\epsilon\mid\mathcal{G}^{0}_{p}] = m_{p} E 0 [ ϵ ∣ G p 0 ] = m p E 0 [ m p h p ϵ ] = E 0 [ m p 2 h p ] \mathbb{E}^{0}[m_{p}h_{p}\epsilon] = \mathbb{E}^{0}[m_{p}^{2}h_{p}] E 0 [ m p h p ϵ ] = E 0 [ m p 2 h p ] A − A^{-} A − x p x_{p} x p y p r y_{pr} y p r T p ( 1 ) T_{p}^{(1)} T p ( 1 ) (158) . Therefore, group (b) equals
2 E 0 [ m p 2 h p ] Σ − ( p , p ) − 2 ∫ E 0 [ m p h p m r ] Σ − ( p ∧ r , p ) ν ( d r ) + T p ( 1 ) . (175) 2\mathbb{E}^{0}[m_{p}^{2}h_{p}]\Sigma^{-}(p,p)-2\int\mathbb{E}^{0}[m_{p}h_{p}m_{r}]\Sigma^{-}(p\wedge r,p)\,\nu(\mathrm{d}r)+T_{p}^{(1)}. \tag*{(175)} 2 E 0 [ m p 2 h p ] Σ − ( p , p ) − 2 ∫ E 0 [ m p h p m r ] Σ − ( p ∧ r , p ) ν ( d r ) + T p ( 1 ) . (175) The coefficients of Σ − ( p , p ) \Sigma^{-}(p,p) Σ − ( p , p ) (174) and (175) add up to E 0 [ h p 2 + 2 ( 1 − a ( p ) ) m p 2 h p ] \mathbb{E}^{0}[h_{p}^{2}+2(1-a(p))m_{p}^{2}h_{p}] E 0 [ h p 2 + 2 ( 1 − a ( p )) m p 2 h p ]
Step 4: the group (c). We average the last factor of (172) over i i i τ i 2 = ϵ 2 = 1 \tau_{i}^{2}=\epsilon^{2}=1 τ i 2 = ϵ 2 = 1 x r x_{r} x r y r ℓ y_{r\ell} y r ℓ (170) , and ν ( [ 0 , 1 ] ) = 1 − s \nu([0,1])=1-s ν ([ 0 , 1 ]) = 1 − s
1 n ∑ i ( ξ i 2 − ∫ U ¨ i d ν ) = s − 2 ∫ ϵ m r ( A ˉ r − + x r ) ν ( d r ) + ∬ m r m ℓ ( A ˉ r ∧ ℓ − + y r ℓ ) ν ( d r ) ν ( d ℓ ) \frac{1}{n}\sum_{i}\left(\xi_{i}^{2}-\int\ddot{U}_{i}\,\mathrm{d}\nu\right)=s-2\int\epsilon m_{r}(\bar{A}_{r}^{-}+x_{r})\,\nu(\mathrm{d}r)+\iint m_{r}m_{\ell}(\bar{A}_{r\wedge\ell}^{-}+y_{r\ell})\,\nu(\mathrm{d}r)\nu(\mathrm{d}\ell) n 1 i ∑ ( ξ i 2 − ∫ U ¨ i d ν ) = s − 2 ∫ ϵ m r ( A ˉ r − + x r ) ν ( d r ) + ∬ m r m ℓ ( A ˉ r ∧ ℓ − + y r ℓ ) ν ( d r ) ν ( d ℓ ) + ∫ a ( r ) m r 2 A ˉ r − ν ( d r ) + ∫ ν ( d r ) ∫ ( r , 1 ] g r ℓ ( Y r ) ( P r , ℓ − A ℓ − ) ( X r ) ν ( d ℓ ) . (176) \qquad+\int a(r)m_{r}^{2}\bar{A}_{r}^{-}\,\nu(\mathrm{d}r)+\int\nu(\mathrm{d}r)\int_{(r,1]}g_{r}^{\ell}(Y_{r})(P_{r,\ell}^{-}A_{\ell}^{-})(X_{r})\,\nu(\mathrm{d}\ell). \tag*{(176)} + ∫ a ( r ) m r 2 A ˉ r − ν ( d r ) + ∫ ν ( d r ) ∫ ( r , 1 ] g r ℓ ( Y r ) ( P r , ℓ − A ℓ − ) ( X r ) ν ( d ℓ ) . (176) Here the constant s s s 1 − ν ( [ 0 , 1 ] ) 1-\nu([0,1]) 1 − ν ([ 0 , 1 ]) τ i 2 ϵ 2 \tau_{i}^{2}\epsilon^{2} τ i 2 ϵ 2 ξ i 2 \xi_{i}^{2} ξ i 2 (170) integrates to ν ( [ 0 , 1 ] ) \nu([0,1]) ν ([ 0 , 1 ]) (176) by f ˉ p A ˉ p − / 2 \bar{f}_{p}\bar{A}_{p}^{-}/2 f ˉ p A ˉ p − /2 s E [ f ˉ p A ˉ p − ] / 2 = 0 s\mathbb{E}[\bar{f}_{p}\bar{A}_{p}^{-}]/2=0 s E [ f ˉ p A ˉ p − ] /2 = 0 A − A^{-} A − E − [ A q − A ˉ p − ] = Σ − ( q , p ) \mathbb{E}^{-}[A_{q}^{-}\bar{A}_{p}^{-}]=\Sigma^{-}(q,p) E − [ A q − A ˉ p − ] = Σ − ( q , p ) q = r q=r q = r q = r ∧ ℓ q=r\wedge\ell q = r ∧ ℓ E − [ ( P r , ℓ − A ℓ − ) A ˉ p − ] = Θ ( r ; ℓ , p ) \mathbb{E}^{-}[(P_{r,\ell}^{-}A_{\ell}^{-})\bar{A}_{p}^{-}]=\Theta(r;\ell,p) E − [( P r , ℓ − A ℓ − ) A ˉ p − ] = Θ ( r ; ℓ , p ) x r x_{r} x r y r ℓ y_{r\ell} y r ℓ T p ( 2 ) T_{p}^{(2)} T p ( 2 ) (159) . Collecting these terms, we find that group (c) equals
− ∫ E 0 [ f ˉ p ϵ m r ] Σ − ( r , p ) ν ( d r ) + 1 2 ∬ E 0 [ f ˉ p m r m ℓ ] Σ − ( r ∧ ℓ , p ) ν ( d r ) ν ( d ℓ ) -\int\mathbb{E}^{0}[\bar{f}_{p}\epsilon m_{r}]\Sigma^{-}(r,p)\,\nu(\mathrm{d}r)+\frac{1}{2}\iint\mathbb{E}^{0}[\bar{f}_{p}m_{r}m_{\ell}]\Sigma^{-}(r\wedge\ell,p)\,\nu(\mathrm{d}r)\nu(\mathrm{d}\ell) − ∫ E 0 [ f ˉ p ϵ m r ] Σ − ( r , p ) ν ( d r ) + 2 1 ∬ E 0 [ f ˉ p m r m ℓ ] Σ − ( r ∧ ℓ , p ) ν ( d r ) ν ( d ℓ ) + 1 2 ∫ a ( r ) E 0 [ f ˉ p m r 2 ] Σ − ( r , p ) ν ( d r ) + 1 2 ∫ ν ( d r ) ∫ ( r , 1 ] E 0 [ f ˉ p g r ℓ ( Y r ) ] Θ ( r ; ℓ , p ) ν ( d ℓ ) + T p ( 2 ) . (177) \qquad+\frac{1}{2}\int a(r)\mathbb{E}^{0}[\bar{f}_{p}m_{r}^{2}]\Sigma^{-}(r,p)\,\nu(\mathrm{d}r)+\frac{1}{2}\int\nu(\mathrm{d}r)\int_{(r,1]}\mathbb{E}^{0}[\bar{f}_{p}g_{r}^{\ell}(Y_{r})]\Theta(r;\ell,p)\,\nu(\mathrm{d}\ell)+T_{p}^{(2)}. \tag*{(177)} + 2 1 ∫ a ( r ) E 0 [ f ˉ p m r 2 ] Σ − ( r , p ) ν ( d r ) + 2 1 ∫ ν ( d r ) ∫ ( r , 1 ] E 0 [ f ˉ p g r ℓ ( Y r )] Θ ( r ; ℓ , p ) ν ( d ℓ ) + T p ( 2 ) . (177) Step 5: the group (d). We compute the three terms in turn. First, by (173) , ∑ i ∣ H p − e i ∣ 2 = Tr ( ( H p − ) 2 ) \sum_{i}|H_{p}^{-}e_{i}|^{2}=\operatorname{Tr}((H_{p}^{-})^{2}) ∑ i ∣ H p − e i ∣ 2 = Tr (( H p − ) 2 ) (170) , we have
1 2 n ∑ i A p , i i − = m p 2 n 2 Tr ( ( H p − ) 2 ) + 1 n M p ⋅ ∇ x U ¨ a v ( p ) . \frac{1}{2n}\sum_{i}A_{p,ii}^{-}=\frac{m_{p}^{2}}{n^{2}}\operatorname{Tr}((H_{p}^{-})^{2})+\frac{1}{n}M_{p}\cdot\nabla_{x}\ddot{U}_{\mathrm{av}}(p). 2 n 1 i ∑ A p , ii − = n 2 m p 2 Tr (( H p − ) 2 ) + n 1 M p ⋅ ∇ x U ¨ av ( p ) . We differentiate (170) in x x x ∇ x A − ( p , x ) = 2 H − ( p , x ) M ( p , x ) / n \nabla_{x}A^{-}(p,x)=2H^{-}(p,x)M(p,x)/n ∇ x A − ( p , x ) = 2 H − ( p , x ) M ( p , x ) / n H − ( p , x ) = ∇ x 2 U − ( p , x ) H^{-}(p,x)=\nabla_{x}^{2}U^{-}(p,x) H − ( p , x ) = ∇ x 2 U − ( p , x )
M p ⋅ ∇ x U ¨ a v ( p ) = − 2 a ( p ) m p 2 n M p T H p − M p − ∫ ( p , 1 ] g p ℓ ( Y p ) M p ⋅ ∇ ( P p , ℓ − A ℓ − ) ( X p ) ν ( d ℓ ) . M_{p}\cdot\nabla_{x}\ddot{U}_{\mathrm{av}}(p)=-\frac{2a(p)m_{p}^{2}}{n}M_{p}^{\mathsf{T}}H_{p}^{-}M_{p}-\int_{(p,1]}g_{p}^{\ell}(Y_{p})M_{p}\cdot\nabla(P_{p,\ell}^{-}A_{\ell}^{-})(X_{p})\,\nu(\mathrm{d}\ell). M p ⋅ ∇ x U ¨ av ( p ) = − n 2 a ( p ) m p 2 M p T H p − M p − ∫ ( p , 1 ] g p ℓ ( Y p ) M p ⋅ ∇ ( P p , ℓ − A ℓ − ) ( X p ) ν ( d ℓ ) . We multiply by f ˉ p \bar{f}_p f ˉ p E 0 [ f ˉ p m p 2 ] = Var 0 ( f p ) = v p 0 \mathbb{E}^{0}[\bar{f}_p m_p^2] = \operatorname{Var}^{0}(f_p) = v_p^{0} E 0 [ f ˉ p m p 2 ] = Var 0 ( f p ) = v p 0 E 0 [ f ˉ p g p ℓ ( Y p ) ] = Cov 0 ( f p , g p ℓ ( Y p ) ) \mathbb{E}^{0}[\bar{f}_p g_p^{\ell}(Y_p)] = \operatorname{Cov}^{0}(f_p,g_p^{\ell}(Y_p)) E 0 [ f ˉ p g p ℓ ( Y p )] = Cov 0 ( f p , g p ℓ ( Y p ))
1 2 n ∑ i E [ f ˉ p A p , i i − ] = v p 0 a p H − 2 a ( p ) v p 0 b p H − ∫ ( p , 1 ] Cov 0 ( f p , g p ℓ ( Y p ) ) z p , ℓ ν ( d ℓ ) . \frac{1}{2n}\sum_i \mathbb{E}[\bar{f}_p A_{p,ii}^{-}] = v_p^{0}a_p^{H} - 2a(p)v_p^{0}b_p^{H} - \int_{(p,1]} \operatorname{Cov}^{0}(f_p,g_p^{\ell}(Y_p))z_{p,\ell}\,\nu(\mathrm{d}\ell). 2 n 1 i ∑ E [ f ˉ p A p , ii − ] = v p 0 a p H − 2 a ( p ) v p 0 b p H − ∫ ( p , 1 ] Cov 0 ( f p , g p ℓ ( Y p )) z p , ℓ ν ( d ℓ ) . Second, (173) gives n − 1 ∑ i f p , i A p , i − = 4 n − 2 m p 2 h p M p T H p − M p n^{-1}\sum_i f_{p,i}A_{p,i}^{-} = 4n^{-2}m_p^2h_p M_p^{T}H_p^{-}M_p n − 1 ∑ i f p , i A p , i − = 4 n − 2 m p 2 h p M p T H p − M p
1 2 n ∑ i E [ 2 f p , i A p , i − ] = 4 E 0 [ m p 2 h p ] b p H . \frac{1}{2n}\sum_i \mathbb{E}[2f_{p,i}A_{p,i}^{-}] = 4\mathbb{E}^{0}[m_p^2h_p]b_p^{H}. 2 n 1 i ∑ E [ 2 f p , i A p , i − ] = 4 E 0 [ m p 2 h p ] b p H . Third, by the definition of s i s_i s i
1 n ∑ i s i A p , i − = 2 m p n 2 [ ϵ τ T H p − M p − ∫ m r M r T H p − M p ν ( d r ) ] . \frac{1}{n}\sum_i s_i A_{p,i}^{-} = \frac{2m_p}{n^2}\left[\epsilon\,\tau^{T}H_p^{-}M_p - \int m_r M_r^{T}H_p^{-}M_p\,\nu(\mathrm{d}r)\right]. n 1 i ∑ s i A p , i − = n 2 2 m p [ ϵ τ T H p − M p − ∫ m r M r T H p − M p ν ( d r ) ] . We multiply by f ˉ p \bar{f}_p f ˉ p ϵ \epsilon ϵ G p 0 G_p^{0} G p 0 τ \tau τ G p − G_p^{-} G p − E 0 [ f ˉ p m p ϵ ] = E 0 [ f ˉ p m p 2 ] = v p 0 \mathbb{E}^{0}[\bar{f}_p m_p\epsilon] = \mathbb{E}^{0}[\bar{f}_p m_p^2] = v_p^{0} E 0 [ f ˉ p m p ϵ ] = E 0 [ f ˉ p m p 2 ] = v p 0 E − [ τ T H p − M p ] = E − [ M p T H p − M p ] \mathbb{E}^{-}[\tau^{T}H_p^{-}M_p] = \mathbb{E}^{-}[M_p^{T}H_p^{-}M_p] E − [ τ T H p − M p ] = E − [ M p T H p − M p ]
1 2 n ∑ i E [ 2 s i f ˉ p A p , i − ] = 2 v p 0 b p H − 2 ∫ E 0 [ f ˉ p m p m r ] b r , p H ν ( d r ) . \frac{1}{2n}\sum_i \mathbb{E}[2s_i\bar{f}_p A_{p,i}^{-}] = 2v_p^{0}b_p^{H} - 2\int\mathbb{E}^{0}[\bar{f}_p m_pm_r]b_{r,p}^{H}\,\nu(\mathrm{d}r). 2 n 1 i ∑ E [ 2 s i f ˉ p A p , i − ] = 2 v p 0 b p H − 2 ∫ E 0 [ f ˉ p m p m r ] b r , p H ν ( d r ) . Adding the three terms, we find that group (d) equals
v p 0 a p H + [ 4 E 0 ( m p 2 h p ) + 2 ( 1 − a ( p ) ) v p 0 ] b p H − ∫ ( p , 1 ] Cov 0 ( f p , g p ℓ ( Y p ) ) z p , ℓ ν ( d ℓ ) − 2 ∫ E 0 [ f ˉ p m p m r ] b r , p H ν ( d r ) , (178) v_p^{0}a_p^{H} + [4\mathbb{E}^{0}(m_p^2h_p) + 2(1-a(p))v_p^{0}]b_p^{H}
- \int_{(p,1]} \operatorname{Cov}^{0}(f_p,g_p^{\ell}(Y_p))z_{p,\ell}\,\nu(\mathrm{d}\ell)
- 2\int\mathbb{E}^{0}[\bar{f}_pm_pm_r]b_{r,p}^{H}\,\nu(\mathrm{d}r), \tag*{(178)} v p 0 a p H + [ 4 E 0 ( m p 2 h p ) + 2 ( 1 − a ( p )) v p 0 ] b p H − ∫ ( p , 1 ] Cov 0 ( f p , g p ℓ ( Y p )) z p , ℓ ν ( d ℓ ) − 2 ∫ E 0 [ f ˉ p m p m r ] b r , p H ν ( d r ) , (178) which is T p ( 3 ) T_p^{(3)} T p ( 3 ) (160) .
Step 6: comparison with the scalar response operator. By (172) , C p p ′ ( 0 ) / α C_{pp}'(0)/\alpha C pp ′ ( 0 ) / α (174) , (175) , (177) , and (178) . We take φ = Σ − ( ⋅ , p ) \varphi= \Sigma^{-}(\mathord{\cdot},p) φ = Σ − ( ⋅ , p )
In detail, the integral in (174) is (J2), and the integral in (175) is (J3), with φ ( p ∧ r ) = Σ − ( p ∧ r , p ) \varphi(p\wedge r)=\Sigma^{-}(p\wedge r,p) φ ( p ∧ r ) = Σ − ( p ∧ r , p ) (177) are (J4), (J6), and (J5). In the fourth term of (177) , we write Θ ( r ; ℓ , p ) = Σ − ( ℓ , p ) + [ Θ ( r ; ℓ , p ) − Σ − ( ℓ , p ) ] \Theta(r;\ell,p)=\Sigma^{-}(\ell,p)+[\Theta(r;\ell,p)-\Sigma^{-}(\ell,p)] Θ ( r ; ℓ , p ) = Σ − ( ℓ , p ) + [ Θ ( r ; ℓ , p ) − Σ − ( ℓ , p )] E 0 [ f ˉ p g r ℓ ( Y r ) ] = Cov 0 ( f p , g r ℓ ( Y r ) ) \mathbb{E}^{0}[\bar{f}_pg_r^{\ell}(Y_r)]=\operatorname{Cov}^{0}(f_p,g_r^{\ell}(Y_r)) E 0 [ f ˉ p g r ℓ ( Y r )] = Cov 0 ( f p , g r ℓ ( Y r )) (162) . The remaining terms are T p ( 1 ) T_p^{(1)} T p ( 1 ) T p ( 2 ) T_p^{(2)} T p ( 2 ) T p ( 3 ) T_p^{(3)} T p ( 3 ) d − 1 ( J a , d Σ − ( ⋅ , p ) ) ( p ) = d − 1 ( J a , d s c Σ − ) ( p , p ) d^{-1}(J_{a,d}\Sigma^{-}(\mathord{\cdot},p))(p)=d^{-1}(J_{a,d}^{\mathrm{sc}}\Sigma^{-})(p,p) d − 1 ( J a , d Σ − ( ⋅ , p )) ( p ) = d − 1 ( J a , d sc Σ − ) ( p , p ) (162) follows.
Atoms of ν \nu ν (137) and the exact parameter responses (139) . At an atom, the formulas use the right-continuous value a ( p ) a(p) a ( p ) ( p , 1 ] (p,1] ( p , 1 ] ( r , 1 ] (r,1] ( r , 1 ] (106) does, and the double integrals over [ 0 , 1 ] 2 [0,1]^2 [ 0 , 1 ] 2
Table 11.1. The terms of (11.9), with φ = Σ − ( ⋅ , p ) \varphi=\Sigma^{-}(\cdot,p) φ = Σ − ( ⋅ , p )
Proof of the integrated linear response
Proof of Lemma 11.2. The proof has three steps. First, we show that the last line of (11.9), integrated over p p p − P / 2 -\mathcal{P}/2 − P /2 P ≥ 0 \mathcal{P}\ge0 P ≥ 0 Σ − \Sigma^{-} Σ −
Step 1: the conditional future correction. If r ≥ p r\ge p r ≥ p ℓ > r \ell>r ℓ > r p , ℓ > r p,\ell>r p , ℓ > r g r p ( Y r ) = E 0 [ f p ∣ G r 0 ] g_r^p(Y_r)=\mathbb{E}^0[f_p\mid\mathcal{G}_r^0] g r p ( Y r ) = E 0 [ f p ∣ G r 0 ]
Cov 0 ( f p , g r ℓ ( Y r ) ) = Cov 0 ( g r p ( Y r ) , g r ℓ ( Y r ) ) = Λ r ( p , ℓ ) . \operatorname{Cov}^{0}(f_p,g_r^\ell(Y_r))=\operatorname{Cov}^{0}(g_r^p(Y_r),g_r^\ell(Y_r))=\Lambda_r(p,\ell). Cov 0 ( f p , g r ℓ ( Y r )) = Cov 0 ( g r p ( Y r ) , g r ℓ ( Y r )) = Λ r ( p , ℓ ) . By the law of total covariance given G r − \mathcal{G}_r^{-} G r −
Σ − ( ℓ , p ) = E − Cov − ( A ℓ − , A p − ∣ G r − ) + Cov − ( E − [ A ℓ − ∣ G r − ] , E − [ A p − ∣ G r − ] ) = Σ r − ( p , ℓ ) + Θ ( r ; ℓ , p ) , \begin{aligned}
\Sigma^{-}(\ell,p)=\mathbb{E}^{-}\operatorname{Cov}^{-}(A_\ell^{-},A_p^{-}\mid\mathcal{G}_r^{-})+\operatorname{Cov}^{-}\left(\mathbb{E}^{-}[A_\ell^{-}\mid\mathcal{G}_r^{-}],\mathbb{E}^{-}[A_p^{-}\mid\mathcal{G}_r^{-}]\right) \\
&=\Sigma_r^{-}(p,\ell)+\Theta(r;\ell,p),
\end{aligned} Σ − ( ℓ , p ) = E − Cov − ( A ℓ − , A p − ∣ G r − ) + Cov − ( E − [ A ℓ − ∣ G r − ] , E − [ A p − ∣ G r − ] ) = Σ r − ( p , ℓ ) + Θ ( r ; ℓ , p ) , where in the second equality we used that the covariance of the G r − \mathcal{G}_r^{-} G r − E − [ A ℓ − ∣ G r − ] \mathbb{E}^{-}[A_\ell^{-}\mid\mathcal{G}_r^{-}] E − [ A ℓ − ∣ G r − ] A p − A_p^{-} A p − E − [ A p − ∣ G r − ] \mathbb{E}^{-}[A_p^{-}\mid\mathcal{G}_r^{-}] E − [ A p − ∣ G r − ] Θ ( r ; ℓ , p ) − Σ − ( ℓ , p ) = − Σ r − ( p , ℓ ) \Theta(r;\ell,p)-\Sigma^{-}(\ell,p)=-\Sigma_r^{-}(p,\ell) Θ ( r ; ℓ , p ) − Σ − ( ℓ , p ) = − Σ r − ( p , ℓ ) ν ( d p ) \nu(\mathrm{d}p) ν ( d p )
1 2 ∫ ν ( d p ) ∫ ν ( d r ) ∫ ( r , 1 ] Cov 0 ( f p , g r ℓ ( Y r ) ) [ Θ ( r ; ℓ , p ) − Σ − ( ℓ , p ) ] ν ( d ℓ ) = − 1 2 P . (179) \frac{1}{2}\int\nu(\mathrm{d}p)\int\nu(\mathrm{d}r)\int_{(r,1]} \operatorname{Cov}^{0}(f_p,g_r^\ell(Y_r))\left[\Theta(r;\ell,p)-\Sigma^{-}(\ell,p)\right]\nu(\mathrm{d}\ell)=-\frac{1}{2}\mathcal{P}. \tag*{(179)} 2 1 ∫ ν ( d p ) ∫ ν ( d r ) ∫ ( r , 1 ] Cov 0 ( f p , g r ℓ ( Y r )) [ Θ ( r ; ℓ , p ) − Σ − ( ℓ , p ) ] ν ( d ℓ ) = − 2 1 P . (179) To show that P ≥ 0 \mathcal{P}\ge0 P ≥ 0 r ∈ [ 0 , 1 ] r\in[0,1] r ∈ [ 0 , 1 ]
Ξ r = ∫ ( r , 1 ] ( g r p ( Y r ) − E 0 g r p ( Y r ) ) ( A p − − E − [ A p − ∣ G r − ] ) ν ( d p ) . \Xi_r=\int_{(r,1]}(g_r^p(Y_r)-\mathbb{E}^{0}g_r^p(Y_r))(A_p^{-}-\mathbb{E}^{-}[A_p^{-}\mid\mathcal{G}_r^{-}])\nu(\mathrm{d}p). Ξ r = ∫ ( r , 1 ] ( g r p ( Y r ) − E 0 g r p ( Y r )) ( A p − − E − [ A p − ∣ G r − ]) ν ( d p ) . We expand the square of Ξ r \Xi_r Ξ r E − [ ( A p − − E − [ A p − ∣ G r ] ) ( A ℓ − − E − [ A ℓ − ∣ G r ] ) ] = Σ r − ( p , ℓ ) \mathbb{E}^{-}[(A_p^{-}-\mathbb{E}^{-}[A_p^{-}\mid\mathcal{G}_r])(A_\ell^{-}-\mathbb{E}^{-}[A_\ell^{-}\mid\mathcal{G}_r])] = \Sigma_r^{-}(p,\ell) E − [( A p − − E − [ A p − ∣ G r ]) ( A ℓ − − E − [ A ℓ − ∣ G r ])] = Σ r − ( p , ℓ )
E Ξ r 2 = ∬ p , ℓ > r Λ r ( p , ℓ ) Σ r − ( p , ℓ ) ν ( d p ) ν ( d ℓ ) , and hence P = ∫ E Ξ r 2 ν ( d r ) ≥ 0. \mathbb{E}\,\Xi_r^2=\iint_{p,\ell>r}\Lambda_r(p,\ell)\Sigma_r^{-}(p,\ell)\,\nu(dp)\nu(d\ell),
\qquad\text{and hence}\qquad
\mathcal{P}=\int\mathbb{E}\,\Xi_r^2\nu(dr)\ge0. E Ξ r 2 = ∬ p , ℓ > r Λ r ( p , ℓ ) Σ r − ( p , ℓ ) ν ( d p ) ν ( d ℓ ) , and hence P = ∫ E Ξ r 2 ν ( d r ) ≥ 0. Individual entries of Λ r \Lambda_r Λ r Σ r − \Sigma_r^{-} Σ r −
Step 2: the covariance operator and the trace. Let B = A ‾ − B=\overline{A}^{-} B = A − p ↦ A ‾ p − p\mapsto\overline{A}^{-}_p p ↦ A p − p ↦ A p − p\mapsto A^{-}_p p ↦ A p − [ 0 , 1 ] [0,1] [ 0 , 1 ] ν ( [ 0 , 1 ] ) ≤ 1 \nu([0,1])\le1 ν ([ 0 , 1 ]) ≤ 1 B B B L 2 ( ν ) L^2(\nu) L 2 ( ν ) ∥ B ∥ L 2 ( ν ) ≤ 1 \|B\|_{L^2(\nu)}\le1 ∥ B ∥ L 2 ( ν ) ≤ 1 Σ − = E − [ B ⊗ B ] \Sigma^{-}=\mathbb{E}^{-}[B\otimes B] Σ − = E − [ B ⊗ B ] ( B ⊗ B ) w = ⟨ B , w ⟩ B (B\otimes B)w=\langle B,w\rangle B ( B ⊗ B ) w = ⟨ B , w ⟩ B E − ∥ B ⊗ B ∥ 1 = E − ∥ B ∥ 2 < ∞ \mathbb{E}^{-}\|B\otimes B\|_1=\mathbb{E}^{-}\|B\|^2<\infty E − ∥ B ⊗ B ∥ 1 = E − ∥ B ∥ 2 < ∞ Σ − \Sigma^{-} Σ − Σ − ( p , q ) \Sigma^{-}(p,q) Σ − ( p , q ) E − ∥ B ∥ L 2 ( ν ) 2 = ∫ Var − ( A p − ) ν ( d p ) = V − \mathbb{E}^{-}\|B\|_{L^2(\nu)}^2=\int\operatorname{Var}^{-}(A^{-}_p)\nu(dp)=V^{-} E − ∥ B ∥ L 2 ( ν ) 2 = ∫ Var − ( A p − ) ν ( d p ) = V − (154) , B ( 0 ) = 0 B(0)=0 B ( 0 ) = 0 ν ( { 1 } ) = 0 \nu(\{1\})=0 ν ({ 1 }) = 0 B B B H ν \mathcal{H}_\nu H ν H ν \mathcal{H}_\nu H ν Σ − w = E − [ ⟨ B , w ⟩ B ] \Sigma^{-}w=\mathbb{E}^{-}[\langle B,w\rangle B] Σ − w = E − [⟨ B , w ⟩ B ] H ν \mathcal{H}_\nu H ν w ∈ L 2 ( ν ) w\in L^2(\nu) w ∈ L 2 ( ν ) Σ − \Sigma^{-} Σ − H ν \mathcal{H}_\nu H ν
By (9.22), J a , d s c J_{a,d}^{\mathrm{sc}} J a , d sc H ν \mathcal{H}_\nu H ν J a , d s c ( B ⊗ B ) J_{a,d}^{\mathrm{sc}}(B\otimes B) J a , d sc ( B ⊗ B ) ⟨ B , J a , d s c B ⟩ \langle B,J_{a,d}^{\mathrm{sc}}B\rangle ⟨ B , J a , d sc B ⟩ Σ − = E − [ B ⊗ B ] \Sigma^{-}=\mathbb{E}^{-}[B\otimes B] Σ − = E − [ B ⊗ B ] J a , d s c Σ − J_{a,d}^{\mathrm{sc}}\Sigma^{-} J a , d sc Σ − H ν \mathcal{H}_\nu H ν E − ⟨ B , J a , d s c B ⟩ \mathbb{E}^{-}\langle B,J_{a,d}^{\mathrm{sc}}B\rangle E − ⟨ B , J a , d sc B ⟩ B ∈ H ν B\in\mathcal{H}_\nu B ∈ H ν J a , d s c J_{a,d}^{\mathrm{sc}} J a , d sc J a , d J_{a,d} J a , d ⟨ B , J a , d s c B ⟩ = ⟨ B , J a , d B ⟩ = ∫ B ( p ) ( J a , d B ) ( p ) ν ( d p ) \langle B,J_{a,d}^{\mathrm{sc}}B\rangle=\langle B,J_{a,d}B\rangle=\int B(p)(J_{a,d}B)(p)\nu(dp) ⟨ B , J a , d sc B ⟩ = ⟨ B , J a , d B ⟩ = ∫ B ( p ) ( J a , d B ) ( p ) ν ( d p ) ( J a , d B ) ( p ) (J_{a,d}B)(p) ( J a , d B ) ( p ) ϕ = B \phi=B ϕ = B β \beta β ν \nu ν B B B p p p r r r ℓ \ell ℓ p ∧ r p\wedge r p ∧ r r ∧ ℓ r\wedge\ell r ∧ ℓ ∣ B ∣ ≤ 1 |B|\le1 ∣ B ∣ ≤ 1 ν \nu ν E − \mathbb{E}^{-} E − E − [ B ( p ) B ( q ) ] = Σ − ( q , p ) \mathbb{E}^{-}[B(p)B(q)]=\Sigma^{-}(q,p) E − [ B ( p ) B ( q )] = Σ − ( q , p )
E − ⟨ B , J a , d s c B ⟩ = ∫ ( J a , d s c Σ − ( ⋅ , p ) ) ( p ) ν ( d p ) = Tr ν ( J a , d s c Σ − ) . \mathbb{E}^{-}\langle B,J_{a,d}^{\mathrm{sc}}B\rangle
=\int(J_{a,d}^{\mathrm{sc}}\Sigma^{-}(\cdot,p))(p)\nu(dp)
=\operatorname{Tr}_{\nu}(J_{a,d}^{\mathrm{sc}}\Sigma^{-}). E − ⟨ B , J a , d sc B ⟩ = ∫ ( J a , d sc Σ − ( ⋅ , p )) ( p ) ν ( d p ) = Tr ν ( J a , d sc Σ − ) . Step 3: integration. Let α > 0 \alpha>0 α > 0 (162) are bounded measurable functions of p p p (162) against ν ( d p ) \nu(dp) ν ( d p ) (179) and the definition of Tr ν \operatorname{Tr}_{\nu} Tr ν α \alpha α (163) . If α = 0 \alpha=0 α = 0 C p p ′ ( 0 ) = 0 C_{pp}'(0)=0 C pp ′ ( 0 ) = 0 Lemma 11.1 , and both sides of (163) vanish.
The integrated response errors
If we bounded the error terms T p T_p T p Lemma 11.2 one level at a time, we would meet covariances such as E − [ x r A ‾ p − ] \mathbb{E}^{-}[x_r\overline{A}^{-}_p] E − [ x r A p − ] Var − ( A p − ) \operatorname{Var}^{-}(A^{-}_p) Var − ( A p − ) V − V^{-} V − p p p ν \nu ν A − A^{-} A − (132) applies. The resulting error is linear in A A A J J J
As in Section 11.1, E − \mathrm{E}^{-} E − E 0 \mathrm{E}^{0} E 0 u = 0 u=0 u = 0 a ˉ p = a ˉ p ( 0 ) \bar{a}_p=\bar{a}_p(0) a ˉ p = a ˉ p ( 0 ) j ˉ p = j ˉ p ( 0 ) \bar{j}_p=\bar{j}_p(0) j ˉ p = j ˉ p ( 0 ) Γ = Γ a , d \Gamma=\Gamma_{a,\mathrm{d}} Γ = Γ a , d
a = ∫ Γ ( p ) a ˉ p ν ( d p ) , j = ∫ Γ ( p ) j ˉ p ν ( d p ) . a=\int\Gamma(p)\bar{a}_p\,\nu(\mathrm{d}p),\qquad j=\int\Gamma(p)\bar{j}_p\,\nu(\mathrm{d}p). a = ∫ Γ ( p ) a ˉ p ν ( d p ) , j = ∫ Γ ( p ) j ˉ p ν ( d p ) . The quantities A A A J J J N N N μ \mu μ
Lemma 11.3 (Signed response estimate). Let μ = s δ 0 + ν \mu=s\delta_0+\nu μ = s δ 0 + ν ν ( { 1 } ) = 0 \nu(\{1\})=0 ν ({ 1 }) = 0
∣ ∫ T p ν ( d p ) ∣ ≤ a + 24 j (180) \left|\int T_p\nu(\mathrm{d}p)\right|\leq a+24j \tag*{(180)} ∫ T p ν ( d p ) ≤ a + 24 j (180) and, with K β , 3 K_{\beta,3} K β , 3
a ∣ ∫ T p ν ( d p ) ∣ ≤ 192 β 2 K β , 3 [ A + J + N − 2 ] . (181) a\left|\int T_p\nu(\mathrm{d}p)\right|\leq192\beta^2K_{\beta,3}\left[A+J+N^{-2}\right]. \tag*{(181)} a ∫ T p ν ( d p ) ≤ 192 β 2 K β , 3 [ A + J + N − 2 ] . (181) Proof. The proof has eight steps. In Step 1, we record two elementary inequalities. In Steps 2–6, we split ∫ T p ν ( d p ) \int T_p\nu(\mathrm{d}p) ∫ T p ν ( d p ) T ( 1 ) T^{(1)} T ( 1 ) T ( 2 ) T^{(2)} T ( 2 ) (160) , the explicit and score parts of its future-gradient term, and its last integral over r ≥ p r\geq p r ≥ p r < p r<p r < p Table 11.2 ) and obtain ((180) ). In Step 8, we compare a + j a+j a + j (181) ). Bounded observables and integrable Gaussian envelopes justify the exchanges of expectations and finite level integrals below; in Step 1, integrability follows from E − ξ 2 < ∞ \mathrm{E}^{-}\xi^2<\infty E − ξ 2 < ∞
Table 11.2. Bounds for the pieces of the integrated error terms in the proof of Lemma 11.3.
Step 1: two elementary inequalities. Let ξ \xi ξ E − ξ = 0 \mathrm{E}^{-}\xi=0 E − ξ = 0 E − ξ 2 < ∞ \mathrm{E}^{-}\xi^2<\infty E − ξ 2 < ∞ r ∈ [ 0 , 1 ] r\in[0,1] r ∈ [ 0 , 1 ] ψ \psi ψ [ 0 , 1 ] [0,1] [ 0 , 1 ] ψ ( q ) 2 ≤ Γ ( r ) Γ ( q ) \psi(q)^2\leq\Gamma(r)\Gamma(q) ψ ( q ) 2 ≤ Γ ( r ) Γ ( q ) q q q E − [ ξ Y ] = E − [ ξ ( Y − E − Y ) ] \mathrm{E}^{-}[\xi Y]=\mathrm{E}^{-}[\xi(Y-\mathrm{E}^{-}Y)] E − [ ξ Y ] = E − [ ξ ( Y − E − Y )] Y Y Y ξ \xi ξ u = 0 u=0 u = 0
∣ E − [ ξ ∫ ψ A − d ν ] ∣ ≤ ( E − ξ 2 ) 1 / 2 Var − ( ∫ ψ A − d ν ) 1 / 2 ≤ 2 Γ ( r ) ( E − ξ 2 ) 1 / 2 , (182) \left|\mathrm{E}^{-}\left[\xi\int\psi A^{-}\,\mathrm{d}\nu\right]\right|
\leq\left(\mathrm{E}^{-}\xi^2\right)^{1/2}\operatorname{Var}^{-}\left(\int\psi A^{-}\,\mathrm{d}\nu\right)^{1/2}
\leq2\Gamma(r)\left(\mathrm{E}^{-}\xi^2\right)^{1/2}, \tag*{(182)} E − [ ξ ∫ ψ A − d ν ] ≤ ( E − ξ 2 ) 1/2 Var − ( ∫ ψ A − d ν ) 1/2 ≤ 2Γ ( r ) ( E − ξ 2 ) 1/2 , (182) where in the last inequality we used 4 ∫ ψ 2 j − d ν ≤ 4 Γ ( r ) ∫ Γ j − d ν = 4 Γ ( r ) j 4\int\psi^2j^{-}\,\mathrm{d}\nu\leq4\Gamma(r)\int\Gamma j^{-}\,\mathrm{d}\nu=4\Gamma(r)j 4 ∫ ψ 2 j − d ν ≤ 4Γ ( r ) ∫ Γ j − d ν = 4Γ ( r ) j χ ≥ 0 \chi\geq0 χ ≥ 0
∫ Γ j − d ν ≤ j , ∬ χ ( r ∧ ℓ ) ν ( d r ) ν ( d ℓ ) ≤ 2 ∫ χ d ν , (183) \int\sqrt{\Gamma j^{-}}\,\mathrm{d}\nu\leq\sqrt{j},\qquad
\iint\chi(r\wedge\ell)\,\nu(\mathrm{d}r)\nu(\mathrm{d}\ell)\leq2\int\chi\,\mathrm{d}\nu, \tag*{(183)} ∫ Γ j − d ν ≤ j , ∬ χ ( r ∧ ℓ ) ν ( d r ) ν ( d ℓ ) ≤ 2 ∫ χ d ν , (183) where the first bound follows from the Cauchy–Schwarz inequality and ν ( [ 0 , 1 ] ) ≤ 1 \nu([0,1])\leq1 ν ([ 0 , 1 ]) ≤ 1 χ ( r ∧ ℓ ) ≤ χ ( r ) + χ ( ℓ ) \chi(r\wedge\ell)\leq\chi(r)+\chi(\ell) χ ( r ∧ ℓ ) ≤ χ ( r ) + χ ( ℓ )
Step 2: the term T ( 1 ) T^{(1)} T ( 1 ) The first term of (11.5) vanishes, since E − [ x p ∣ G p − ] = 0 \mathbb{E}^{-}[x_p \mid\mathcal{G}^{-}_p] = 0 E − [ x p ∣ G p − ] = 0 A ˉ p − \bar{A}^{-}_p A ˉ p − G p − \mathcal{G}^{-}_p G p − r ≥ p r \ge p r ≥ p y p r = M p ⋅ ( M r − M p ) / n y_{pr} = M_p \cdot(M_r - M_p)/n y p r = M p ⋅ ( M r − M p ) / n G p − \mathcal{G}^{-}_p G p − r < p r < p r < p E − [ x r ∣ G p − ] = M p ⋅ M r / n − A r − = y p r \mathbb{E}^{-}[x_r \mid\mathcal{G}^{-}_p] = M_p \cdot M_r/n - A^{-}_r = y_{pr} E − [ x r ∣ G p − ] = M p ⋅ M r / n − A r − = y p r E − [ y p r A ˉ p − ] = E − [ x r A ˉ p − ] = E − [ x r A p − ] \mathbb{E}^{-}[y_{pr}\bar{A}^{-}_p] = \mathbb{E}^{-}[x_r\bar{A}^{-}_p] = \mathbb{E}^{-}[x_r A^{-}_p] E − [ y p r A ˉ p − ] = E − [ x r A ˉ p − ] = E − [ x r A p − ] x r x_r x r p p p r r r
∫ T p ( 1 ) ν ( d p ) = − 2 ∫ E − [ x r ( ∫ ( r , 1 ] E 0 [ m p h p m r ] A ˉ p − ν ( d p ) ) ] ν ( d r ) . \int T^{(1)}_p \,\nu(dp) = -2 \int\mathbb{E}^{-}\left[x_r\left(\int_{(r,1]} \mathbb{E}^{0}[m_p h_p m_r]\,\bar{A}^{-}_p\,\nu(dp)\right)\right]\nu(dr). ∫ T p ( 1 ) ν ( d p ) = − 2 ∫ E − [ x r ( ∫ ( r , 1 ] E 0 [ m p h p m r ] A ˉ p − ν ( d p ) ) ] ν ( d r ) . For fixed r r r ψ ( p ) = 1 { p > r } E 0 [ m p h p m r ] \psi(p) = \mathbf{1}_{\{p>r\}}\mathbb{E}^{0}[m_p h_p m_r] ψ ( p ) = 1 { p > r } E 0 [ m p h p m r ] ψ 2 ≤ Γ ( r ) Γ \psi^2 \le\Gamma(r)\Gamma ψ 2 ≤ Γ ( r ) Γ E − x r 2 = j ˉ r \mathbb{E}^{-}x_r^2 = \bar{j}_r E − x r 2 = j ˉ r 2 ( Γ ( r ) j ˉ r ) 1 / 2 2(\Gamma(r)\bar{j}_r)^{1/2} 2 ( Γ ( r ) j ˉ r ) 1/2 r r r
∣ ∫ T p ( 1 ) ν ( d p ) ∣ ≤ 4 j ∫ Γ j ˉ d ν ≤ 4 j . \left|\int T^{(1)}_p \,\nu(dp)\right| \le4\sqrt{j}\int\sqrt{\Gamma\bar{j}}\,d\nu\le4j. ∫ T p ( 1 ) ν ( d p ) ≤ 4 j ∫ Γ j ˉ d ν ≤ 4 j . Step 3: the term T ( 2 ) T^{(2)} T ( 2 ) Since x r x_r x r p p p
− ∫ E − [ x r ( ∫ E 0 [ f ˉ p ϵ m r ] A ˉ p − ν ( d p ) ) ] ν ( d r ) . -\int\mathbb{E}^{-}\left[x_r\left(\int\mathbb{E}^{0}[\bar{f}_p\epsilon m_r]\,\bar{A}^{-}_p\,\nu(dp)\right)\right]\nu(dr). − ∫ E − [ x r ( ∫ E 0 [ f ˉ p ϵ m r ] A ˉ p − ν ( d p ) ) ] ν ( d r ) . The kernel ψ ( p ) = E 0 [ f ˉ p ϵ m r ] \psi(p) = \mathbb{E}^{0}[\bar{f}_p\epsilon m_r] ψ ( p ) = E 0 [ f ˉ p ϵ m r ] ψ 2 ≤ Γ ( r ) Γ \psi^2 \le\Gamma(r)\Gamma ψ 2 ≤ Γ ( r ) Γ 2 j ∫ Γ j ˉ d ν ≤ 2 j 2\sqrt{j}\int\sqrt{\Gamma\bar{j}}\,d\nu\le2j 2 j ∫ Γ j ˉ d ν ≤ 2 j y r ℓ y_{r\ell} y r ℓ E − y r ℓ 2 ≤ j ˉ r ∧ ℓ \mathbb{E}^{-}y_{r\ell}^2 \le\bar{j}_{r\wedge\ell} E − y r ℓ 2 ≤ j ˉ r ∧ ℓ ψ ( p ) = E 0 [ f ˉ p m r m ℓ ] \psi(p) = \mathbb{E}^{0}[\bar{f}_p m_r m_\ell] ψ ( p ) = E 0 [ f ˉ p m r m ℓ ] ψ 2 ≤ Γ ( r ∧ ℓ ) Γ \psi^2 \le\Gamma(r\wedge\ell)\Gamma ψ 2 ≤ Γ ( r ∧ ℓ ) Γ r ∧ ℓ r\wedge\ell r ∧ ℓ χ = Γ j ˉ \chi= \sqrt{\Gamma\bar{j}} χ = Γ j ˉ
1 2 ∬ 2 j χ ( r ∧ ℓ ) ν ( d r ) ν ( d ℓ ) ≤ 2 j ∫ χ d ν ≤ 2 j . \frac{1}{2}\iint2\sqrt{j}\chi(r\wedge\ell)\,\nu(dr)\nu(d\ell) \le2\sqrt{j}\int\chi\,d\nu\le2j. 2 1 ∬ 2 j χ ( r ∧ ℓ ) ν ( d r ) ν ( d ℓ ) ≤ 2 j ∫ χ d ν ≤ 2 j . Step 4: the first line of T ( 3 ) T^{(3)} T ( 3 ) By Lemma 4.2, as recalled in Section 10.1, we have 0 ⪯ H p − ⪯ C p − 0 \preceq H^{-}_p \preceq C^{-}_p 0 ⪯ H p − ⪯ C p − 0 ⪯ H ⪯ C 0 \preceq H \preceq C 0 ⪯ H ⪯ C Tr ( H 2 ) ≤ Tr ( H C ) ≤ Tr ( C 2 ) \operatorname{Tr}(H^2) \le\operatorname{Tr}(HC) \le\operatorname{Tr}(C^2) Tr ( H 2 ) ≤ Tr ( H C ) ≤ Tr ( C 2 ) Tr ( H ( C − H ) ) = Tr ( H 1 / 2 ( C − H ) H 1 / 2 ) ≥ 0 \operatorname{Tr}(H(C-H)) = \operatorname{Tr}(H^{1/2}(C-H)H^{1/2}) \ge0 Tr ( H ( C − H )) = Tr ( H 1/2 ( C − H ) H 1/2 ) ≥ 0 Tr ( ( C − H ) C ) = Tr ( C 1 / 2 ( C − H ) C 1 / 2 ) ≥ 0 \operatorname{Tr}((C-H)C) = \operatorname{Tr}(C^{1/2}(C-H)C^{1/2}) \ge0 Tr (( C − H ) C ) = Tr ( C 1/2 ( C − H ) C 1/2 ) ≥ 0 M p T H p − M p ≤ M p T C p − M p M_p^{\mathsf T}H^{-}_pM_p \le M_p^{\mathsf T}C^{-}_pM_p M p T H p − M p ≤ M p T C p − M p
0 ≤ a p H ≤ a p − , 0 ≤ b p H ≤ j ˉ p . 0 \le a^H_p \le a^{-}_p,\qquad0 \le b^H_p \le\bar{j}_p. 0 ≤ a p H ≤ a p − , 0 ≤ b p H ≤ j ˉ p . In addition, (9.16) implies that 0 ≤ v p 0 ≤ Γ ( p ) 0 \le v^0_p \le\Gamma(p) 0 ≤ v p 0 ≤ Γ ( p ) 0 ≤ E 0 [ m p 2 h p ] ≤ Γ ( p ) 0 \le\mathbb{E}^{0}[m_p^2h_p] \le\Gamma(p) 0 ≤ E 0 [ m p 2 h p ] ≤ Γ ( p ) 0 ≤ 1 − a ( p ) ≤ 1 0 \le1-a(p) \le1 0 ≤ 1 − a ( p ) ≤ 1 Γ ( p ) a p − + ( 4 + 2 ) Γ ( p ) j ˉ p \Gamma(p)a^{-}_p + (4+2)\Gamma(p)\bar{j}_p Γ ( p ) a p − + ( 4 + 2 ) Γ ( p ) j ˉ p ν \nu ν a + 6 j a+6j a + 6 j
Step 5: the future gradient in T ( 3 ) T^{(3)} T ( 3 ) We first compute z p , ℓ z_{p,\ell} z p , ℓ ℓ ≥ p \ell\ge p ℓ ≥ p G p − \mathcal{G}^{-}_p G p − ( X r − X p ) r ≥ p (X_r-X_p)_{r\ge p} ( X r − X p ) r ≥ p τ \tau τ (138) on [ p , 1 ] [p,1] [ p , 1 ] (137) , their conditional law has the density
exp { E − ( τ ) + X 1 ⋅ τ − a ( p ) U − ( p , X p ) − ∫ ( p , 1 ] U − ( r , X r ) ν ( d r ) } \exp\left\{\mathcal{E}^{-}(\tau)+X_{1}\cdot\tau-a(p)U^{-}(p,X_{p})-\int_{(p,1]}U^{-}(r,X_{r})\,\nu(\mathrm{d}r)\right\} exp { E − ( τ ) + X 1 ⋅ τ − a ( p ) U − ( p , X p ) − ∫ ( p , 1 ] U − ( r , X r ) ν ( d r ) } with respect to driftless Gaussian increments and counting measure on τ \tau τ X p = x X_{p}=x X p = x X r X_{r} X r r ≥ p r\ge p r ≥ p x x x τ − a ( p ) M p − ∫ ( p , 1 ] M r ν ( d r ) \tau-a(p)M_{p}-\int_{(p,1]}M_{r}\,\nu(\mathrm{d}r) τ − a ( p ) M p − ∫ ( p , 1 ] M r ν ( d r ) x x x A ℓ − = A − ( ℓ , X ℓ ) A^{-}_{\ell}=A^{-}(\ell,X_{\ell}) A ℓ − = A − ( ℓ , X ℓ ) 2 H ℓ − M ℓ / n 2H^{-}_{\ell}M_{\ell}/n 2 H ℓ − M ℓ / n ( P p , ℓ − A ℓ − ) ( x ) (P^{-}_{p,\ell}A^{-}_{\ell})(x) ( P p , ℓ − A ℓ − ) ( x ) N N N x x x
∇ ( P p , ℓ − A ℓ − ) ( X p ) = E − [ 2 n H ℓ − M ℓ + A ℓ − ( τ − a ( p ) M p − ∫ ( p , 1 ] M r ν ( d r ) ) | G p − ] . \nabla(P^{-}_{p,\ell}A^{-}_{\ell})(X_{p})=\mathbb{E}^{-}\left[\frac{2}{n}H^{-}_{\ell}M_{\ell}+A^{-}_{\ell}\left(\tau-a(p)M_{p}-\int_{(p,1]}M_{r}\,\nu(\mathrm{d}r)\right)\,\middle|\,\mathcal{G}^{-}_{p}\right]. ∇ ( P p , ℓ − A ℓ − ) ( X p ) = E − [ n 2 H ℓ − M ℓ + A ℓ − ( τ − a ( p ) M p − ∫ ( p , 1 ] M r ν ( d r ) ) G p − ] . Taking the inner product with M p / n M_{p}/n M p / n a ( p ) + ν ( ( p , 1 ] ) = 1 a(p)+\nu((p,1])=1 a ( p ) + ν (( p , 1 ]) = 1
z p , ℓ = 2 n 2 E − [ M p T H ℓ − M ℓ ] + E − [ ζ p A ℓ − ] , ζ p = M p ⋅ ( τ − M p ) n − ∫ ( p , 1 ] M p ⋅ ( M r − M p ) n ν ( d r ) . (184) \begin{aligned}
z_{p,\ell}&=\frac{2}{n^{2}}\mathbb{E}^{-}\left[M_{p}^{T}H^{-}_{\ell}M_{\ell}\right]+\mathbb{E}^{-}\left[\zeta_{p}A^{-}_{\ell}\right],\\
\zeta_{p}&=\frac{M_{p}\cdot(\tau-M_{p})}{n}-\int_{(p,1]}\frac{M_{p}\cdot(M_{r}-M_{p})}{n}\,\nu(\mathrm{d}r). \tag*{(184)}
\end{aligned} z p , ℓ ζ p = n 2 2 E − [ M p T H ℓ − M ℓ ] + E − [ ζ p A ℓ − ] , = n M p ⋅ ( τ − M p ) − ∫ ( p , 1 ] n M p ⋅ ( M r − M p ) ν ( d r ) . (184) We bound the explicit term. For ℓ ≥ p \ell\ge p ℓ ≥ p (156) implies that E − [ C ℓ − ∣ G p − ] ⪯ C p − \mathbb{E}^{-}[C^{-}_{\ell}\mid\mathcal{G}^{-}_{p}]\preceq C^{-}_{p} E − [ C ℓ − ∣ G p − ] ⪯ C p − H ℓ − ⪯ C ℓ − H^{-}_{\ell}\preceq C^{-}_{\ell} H ℓ − ⪯ C ℓ − H ℓ − H^{-}_{\ell} H ℓ − E − \mathbb{E}^{-} E −
2 n 2 ∣ E − [ M p T H ℓ − M ℓ ] ∣ ≤ 2 n 2 ( E − [ M p T C p − M p ] E − [ M ℓ T C ℓ − M ℓ ] ) 1 / 2 ≤ 2 ȷ ˉ p ȷ ˉ ℓ , \frac{2}{n^{2}}\left|\mathbb{E}^{-}[M_{p}^{T}H^{-}_{\ell}M_{\ell}]\right|
\le\frac{2}{n^{2}}\left(\mathbb{E}^{-}[M_{p}^{T}C^{-}_{p}M_{p}]\,\mathbb{E}^{-}[M_{\ell}^{T}C^{-}_{\ell}M_{\ell}]\right)^{1/2}
\le2\sqrt{\bar{\jmath}_{p}\bar{\jmath}_{\ell}}, n 2 2 E − [ M p T H ℓ − M ℓ ] ≤ n 2 2 ( E − [ M p T C p − M p ] E − [ M ℓ T C ℓ − M ℓ ] ) 1/2 ≤ 2 ˉ p ˉ ℓ , where in the last inequality we conditioned on G p − \mathcal{G}^{-}_{p} G p − ζ p \zeta_{p} ζ p G p − \mathcal{G}^{-}_{p} G p − E − [ τ ∣ G p − ] = M p \mathbb{E}^{-}[\tau\mid\mathcal{G}^{-}_{p}]=M_{p} E − [ τ ∣ G p − ] = M p E − [ M r ∣ G p − ] = M p \mathbb{E}^{-}[M_{r}\mid\mathcal{G}^{-}_{p}]=M_{p} E − [ M r ∣ G p − ] = M p r ≥ p r\ge p r ≥ p M p T C p − M p / n 2 M_{p}^{T}C^{-}_{p}M_{p}/n^{2} M p T C p − M p / n 2 (156) , each M p ⋅ ( M r − M p ) / n M_{p}\cdot(M_{r}-M_{p})/n M p ⋅ ( M r − M p ) / n M p T C p − M p / n 2 M_{p}^{T}C^{-}_{p}M_{p}/n^{2} M p T C p − M p / n 2 ν ( ( p , 1 ] ) ≤ 1 \nu((p,1])\le1 ν (( p , 1 ]) ≤ 1 L 2 L^{2} L 2 ( M p T C p − M p ) 1 / 2 / n (M_{p}^{T}C^{-}_{p}M_{p})^{1/2}/n ( M p T C p − M p ) 1/2 / n
E − ζ p = 0 , E − ζ p 2 ≤ 4 ȷ ˉ p . \mathbb{E}^{-}\zeta_{p}=0,\qquad\mathbb{E}^{-}\zeta_{p}^{2}\le4\bar{\jmath}_{p}. E − ζ p = 0 , E − ζ p 2 ≤ 4 ˉ p . Let K ( p , ℓ ) = 1 { ℓ > p } Cov 0 ( f p , g ℓ 0 ( Y p ) ) K(p,\ell)=\mathbf{1}_{\{\ell>p\}}\operatorname{Cov}^{0}(f_{p},g^{0}_{\ell}(Y_{p})) K ( p , ℓ ) = 1 { ℓ > p } Cov 0 ( f p , g ℓ 0 ( Y p )) (104) , we have K ( p , ℓ ) 2 ≤ Γ ( p ) Γ ( ℓ ) K(p,\ell)^{2}\le\Gamma(p)\Gamma(\ell) K ( p , ℓ ) 2 ≤ Γ ( p ) Γ ( ℓ ) (160) , integrated in p p p − ∬ K ( p , ℓ ) z p , ℓ ν ( d p ) ν ( d ℓ ) -\iint K(p,\ell)z_{p,\ell}\,\nu(\mathrm{d}p)\nu(\mathrm{d}\ell) − ∬ K ( p , ℓ ) z p , ℓ ν ( d p ) ν ( d ℓ ) (184) , we split it into an explicit part and a score part. Using (183) , we find that the explicit part is at most
2 ∬ Γ ( p ) ȷ ˉ p Γ ( ℓ ) ȷ ˉ ℓ ν ( d p ) ν ( d ℓ ) = 2 ( ∫ Γ ȷ ˉ d ν ) 2 ≤ 2 ȷ ˉ . 2\iint\sqrt{\Gamma(p)\bar{\jmath}_{p}}\sqrt{\Gamma(\ell)\bar{\jmath}_{\ell}}\,\nu(\mathrm{d}p)\nu(\mathrm{d}\ell)
=2\left(\int\sqrt{\Gamma\bar{\jmath}}\,\mathrm{d}\nu\right)^{2}\le2\bar{\jmath}. 2 ∬ Γ ( p ) ˉ p Γ ( ℓ ) ˉ ℓ ν ( d p ) ν ( d ℓ ) = 2 ( ∫ Γ ˉ d ν ) 2 ≤ 2 ˉ . The score part is − ∫ E − [ ζ p ∫ K ( p , ℓ ) A ν − ( d ℓ ) ] ν ( d p ) -\int\mathbb{E}^{-}[\zeta_p\int\mathcal{K}(p,\ell)A^{-}_{\nu}(d\ell)]\nu(dp) − ∫ E − [ ζ p ∫ K ( p , ℓ ) A ν − ( d ℓ )] ν ( d p ) (182) with ξ = ζ p \xi=\zeta_p ξ = ζ p ψ = K ( p , ⋅ ) \psi=\mathcal{K}(p,\cdot) ψ = K ( p , ⋅ ) p p p (183) , we bound it by ∫ 2 ( Γ ( p ) j ⋅ 4 j 1 / 2 ) ν ( d p ) ≤ 4 j ∫ Γ j d ν ≤ 4 j \int2(\Gamma(p)j\cdot4j^{1/2})\nu(dp)\le4\sqrt{j}\int\sqrt{\Gamma j}\,d\nu\le4j ∫ 2 ( Γ ( p ) j ⋅ 4 j 1/2 ) ν ( d p ) ≤ 4 j ∫ Γ j d ν ≤ 4 j
Step 6: the last integral in T ( 3 ) T^{(3)} T ( 3 ) If r ≥ p r\ge p r ≥ p G p − \mathcal{G}^{-}_{p} G p − E − [ M r T H p − M p ] = E − [ M p T H p − M p ] \mathbb{E}^{-}[M_r^{T}H^{-}_{p}M_p]=\mathbb{E}^{-}[M_p^{T}H^{-}_{p}M_p] E − [ M r T H p − M p ] = E − [ M p T H p − M p ] b r , p H = b p H ∈ [ 0 , j p ] b^{H}_{r,p}=b^{H}_{p}\in[0,j_p] b r , p H = b p H ∈ [ 0 , j p ] (102) , the kernel satisfies ∣ E − [ f p m p m r ] ∣ ≤ Γ ( p ) Γ ( p ∧ r ) = Γ ( p ) \lvert\mathbb{E}^{-}[f^{p}m_pm_r]\rvert\le\sqrt{\Gamma(p)\Gamma(p\wedge r)}=\Gamma(p) ∣ E − [ f p m p m r ]∣ ≤ Γ ( p ) Γ ( p ∧ r ) = Γ ( p ) 2 ∫ Γ ( p ) j p ν ( [ p , 1 ] ) ν ( d p ) ≤ 2 j 2\int\Gamma(p)j_p\nu([p,1])\nu(dp)\le2j 2 ∫ Γ ( p ) j p ν ([ p , 1 ]) ν ( d p ) ≤ 2 j r < p r<p r < p
∣ b r , p H ∣ ≤ n − 2 ( E − [ M r T C p − M r ] E − [ M p T C p − M p ] ) 1 / 2 ≤ j r j p . \lvert b^{H}_{r,p}\rvert\le n^{-2}\left(\mathbb{E}^{-}[M_r^{T}C^{-}_{p}M_r]\mathbb{E}^{-}[M_p^{T}C^{-}_{p}M_p]\right)^{1/2}\le\sqrt{j_rj_p}. ∣ b r , p H ∣ ≤ n − 2 ( E − [ M r T C p − M r ] E − [ M p T C p − M p ] ) 1/2 ≤ j r j p . where we used E − [ C p − ∣ G r − ] ⪯ C r − \mathbb{E}^{-}[C^{-}_{p}\mid\mathcal{G}^{-}_{r}]\preceq C^{-}_{r} E − [ C p − ∣ G r − ] ⪯ C r − Γ ( r ) Γ ( p ) \sqrt{\Gamma(r)\Gamma(p)} Γ ( r ) Γ ( p ) (102) , and by (183) , this part also contributes at most 2 ( ∫ Γ j d ν ) 2 ≤ 2 j 2(\int\sqrt{\Gamma j}\,d\nu)^2\le2j 2 ( ∫ Γ j d ν ) 2 ≤ 2 j
Step 7: the sum. Table 11.2 collects the bounds of Steps 2–6, with their running total.
Summing the bounds, we obtain ∣ ∫ T p ν ( d p ) ∣ ≤ a + ( 4 + 2 + 2 + 6 + 2 + 4 + 2 + 2 ) j = a + 24 j \lvert\int T_p\,\nu(dp)\rvert\le a+(4+2+2+6+2+4+2+2)j=a+24j ∣ ∫ T p ν ( d p )∣ ≤ a + ( 4 + 2 + 2 + 6 + 2 + 4 + 2 + 2 ) j = a + 24 j (180) .
Step 8: transfer to the full system. Fix q ∈ [ 0 , 1 ] q\in[0,1] q ∈ [ 0 , 1 ] q q q τ 1 , τ 2 , τ 3 \tau^{1},\tau^{2},\tau^{3} τ 1 , τ 2 , τ 3 G q \mathcal{G}_q G q M q M_q M q C q − C^{-}_q C q − D = τ 1 − τ 2 D=\tau^{1}-\tau^{2} D = τ 1 − τ 2 2 C q − 2C^{-}_q 2 C q − τ 3 \tau^{3} τ 3 C q − + M q M q T C^{-}_q+M_qM_q^{T} C q − + M q M q T E [ ( D ⋅ τ 3 ) 2 ∣ G q ] = Tr ( 2 C q − ( C q − + M q M q T ) ) = 2 Tr ( ( C q − ) 2 ) + 2 M q T C q − M q \mathbb{E}[(D\cdot\tau^{3})^2\mid\mathcal{G}_q]=\operatorname{Tr}(2C^{-}_q(C^{-}_q+M_qM_q^{T}))=2\operatorname{Tr}((C^{-}_q)^2)+2M_q^{T}C^{-}_qM_q E [( D ⋅ τ 3 ) 2 ∣ G q ] = Tr ( 2 C q − ( C q − + M q M q T )) = 2 Tr (( C q − ) 2 ) + 2 M q T C q − M q
a q − ( u ) + j q − ( u ) = 1 2 E 3 , q , u [ ( ( τ 1 − τ 2 ) ⋅ τ 3 n ) 2 ] ( 0 ≤ u ≤ 1 ) . a^{-}_{q}(u)+j^{-}_{q}(u)=\frac{1}{2}\mathbb{E}_{3,q,u}\left[\left(\frac{(\tau^{1}-\tau^{2})\cdot\tau^{3}}{n}\right)^2\right]\qquad(0\le u\le1). a q − ( u ) + j q − ( u ) = 2 1 E 3 , q , u [ ( n ( τ 1 − τ 2 ) ⋅ τ 3 ) 2 ] ( 0 ≤ u ≤ 1 ) . The observable on the right is nonnegative and bounded, and it does not depend on u u u k = 3 k=3 k = 3 u = 0 u=0 u = 0 K β , 3 K_{\beta,3} K β , 3 u = 1 u=1 u = 1
At u = 1 u = 1 u = 1 m q ∈ R N m_q \in\mathbb{R}^N m q ∈ R N C q C_q C q a q t h a_q^{\mathrm{th}} a q th j q j_q j q C ˉ q \bar{C}_q C ˉ q n × n n \times n n × n C q C_q C q ( M q , 0 ) = m q − m N ( q ) e N (M_q,0) = m_q - m_N(q)e_N ( M q , 0 ) = m q − m N ( q ) e N e N e_N e N M q T C ˉ q M q = ( M q , 0 ) T C q ( M q , 0 ) M_q^{T}\bar{C}_qM_q = (M_q,0)^{T}C_q(M_q,0) M q T C ˉ q M q = ( M q , 0 ) T C q ( M q , 0 ) Tr ( ( C ˉ q ) 2 ) ≤ Tr ( C q 2 ) \operatorname{Tr}((\bar{C}_q)^2) \leq\operatorname{Tr}(C_q^2) Tr (( C ˉ q ) 2 ) ≤ Tr ( C q 2 ) C q ⪰ 0 C_q \succeq0 C q ⪰ 0
( M q , 0 ) T C q ( M q , 0 ) ≤ 2 m q T C q m q + 2 m N ( q ) 2 ( C q ) N N . (M_q,0)^{T}C_q(M_q,0) \leq2m_q^{T}C_qm_q + 2m_N(q)^2(C_q)_{NN}. ( M q , 0 ) T C q ( M q , 0 ) ≤ 2 m q T C q m q + 2 m N ( q ) 2 ( C q ) N N . and m N ( q ) 2 ( C q ) N N ≤ 1 m_N(q)^2(C_q)_{NN} \leq1 m N ( q ) 2 ( C q ) N N ≤ 1 n 2 n^2 n 2
a q ( 1 ) ≤ ( N n ) 2 a q t h , j q ( 1 ) ≤ 2 ( N n ) 2 j q + 2 n 2 . a_q(1) \leq\left(\frac{N}{n}\right)^2 a_q^{\mathrm{th}}, \qquad j_q(1) \leq2\left(\frac{N}{n}\right)^2 j_q + \frac{2}{n^2}. a q ( 1 ) ≤ ( n N ) 2 a q th , j q ( 1 ) ≤ 2 ( n N ) 2 j q + n 2 2 . We multiply the comparison by Γ ( q ) ∈ [ 0 , 1 ] \Gamma(q) \in[0,1] Γ ( q ) ∈ [ 0 , 1 ] ν \nu ν ν ( [ 0 , 1 ] ) ≤ 1 \nu([0,1]) \leq1 ν ([ 0 , 1 ]) ≤ 1 n ≥ N / 2 n \geq N/2 n ≥ N /2
a + j ≤ K β , 3 [ ( N n ) 2 A + 2 ( N n ) 2 J + 2 n 2 ] ≤ 8 K β , 3 [ A + J + N − 2 ] . a+j \leq K_{\beta,3}\left[\left(\frac{N}{n}\right)^2 A + 2\left(\frac{N}{n}\right)^2 J + \frac{2}{n^2}\right] \leq8K_{\beta,3}\left[A+J+N^{-2}\right]. a + j ≤ K β , 3 [ ( n N ) 2 A + 2 ( n N ) 2 J + n 2 2 ] ≤ 8 K β , 3 [ A + J + N − 2 ] . Since α ≤ β 2 \alpha\leq\beta^2 α ≤ β 2 α ∣ ∫ T p ν ( d p ) ∣ ≤ 24 β 2 ( a + j ) \alpha\left|\int T_p\,\nu(dp)\right| \leq24\beta^2(a+j) α ∫ T p ν ( d p ) ≤ 24 β 2 ( a + j ) 192 = 24 ⋅ 8 192 = 24 \cdot8 192 = 24 ⋅ 8 □ \square □
Closing the cavity inequality
In this section, we complete the cavity argument of Sections 10 and 11. The main result is Proposition 12.4 : if the scalar response operator at a minimizing profile satisfies J a , d s c ⪯ ( 1 + ε N ) I \mathcal{J}^{\mathrm{sc}}_{a,d} \preceq(1+\varepsilon_N)I J a , d sc ⪯ ( 1 + ε N ) I 0 ≤ ε N ≤ ρ / 4 0 \le\varepsilon_N \le\rho/4 0 ≤ ε N ≤ ρ /4 ( 3 ρ / 4 ) V ≤ C c a v [ N − 1 + A + J + H 3 ] (3\rho/4)\mathsf{V} \le C_{\mathrm{cav}}[N^{-1}+A+J+H_3] ( 3 ρ /4 ) V ≤ C cav [ N − 1 + A + J + H 3 ] Corollary 12.5 , we absorb the cubic moment H 3 H_3 H 3 D N ( ρ ) D_N(\rho) D N ( ρ ) C β log 2 ( 2 + N ) / ( N λ ρ 2 ) C_\beta\log^2(2+N)/(N\lambda\rho^2) C β log 2 ( 2 + N ) / ( N λ ρ 2 ) 1 / N ≤ ρ ≤ 1 1/N \le\rho\le1 1/ N ≤ ρ ≤ 1 Theorem 1.1 , and we verify the stability assumption there with Proposition 9.9 .
We rely on the following earlier results. Section 10 provides the laws of finitely many descendants with Lemma 10.5 , the comparison of nonnegative observables in Corollary 10.6, the three-defect bound of Lemma 10.7 , and the Taylor remainder of Lemma 10.8 . Section 11 provides the integrated linear response of Lemma 11.2 and the signed response estimate of Lemma 11.3 , and Lemma 9.7 states that J a , d s c \mathcal{J}^{\mathrm{sc}}_{a,d} J a , d sc (22) , the conditional decomposition in the proof of Lemma 3.5 , the trace bound (23) , the bound on J J J Proposition 6.4 , the cubic moment bound of Corollary 5.3, and the defect identity (29) .
Standing assumptions and notation. We keep the setting of Section 10: N ≥ 2 N \ge2 N ≥ 2 n = N − 1 n=N-1 n = N − 1 0 < ρ ≤ 1 0<\rho\le1 0 < ρ ≤ 1 t = 1 − ρ t=1-\rho t = 1 − ρ c = β 2 ρ c=\beta^2\rho c = β 2 ρ 0 < s = λ / β ≤ 1 / 2 0<s=\lambda/\beta\le1/2 0 < s = λ / β ≤ 1/2 α \alpha α d d d b ( u ) b(u) b ( u ) (129) . The profile μ = s δ 0 + ν \mu=s\delta_0+\nu μ = s δ 0 + ν a a a μ \mu μ F t , s \mathcal{F}_{t,s} F t , s ν ( { 1 } ) = 0 \nu(\{1\})=0 ν ({ 1 }) = 0 C β C_\beta C β β \beta β N N N ρ \rho ρ s s s
We write E 1 \mathbb{E}_1 E 1 Var 1 \operatorname{Var}_1 Var 1 Cov 1 \operatorname{Cov}_1 Cov 1 P u \mathbb{P}_u P u u = 1 u=1 u = 1 § 9.1.1 that P 1 \mathbb{P}_1 P 1 § 4.1.3 for the full system, so that E 1 = E \mathbb{E}_1=\mathbb{E} E 1 = E u = 0 u=0 u = 0 E − \mathbb{E}^- E − Var − \operatorname{Var}^- Var − A p ˉ = ∣ M p ˉ ∣ 2 / n A_{\bar p}=|M_{\bar p}|^2/n A p ˉ = ∣ M p ˉ ∣ 2 / n m N ( p ) m_N(p) m N ( p ) E k , q , u \mathbb{E}_{k,q,u} E k , q , u Δ a b \Delta_{ab} Δ ab D a b D_{ab} D ab (133) , and the integrated third moment M 3 ( u ) \mathcal{M}_3(u) M 3 ( u ) (151) . In § 9.5.1 , we defined the covariance C p p ( u ) = Cov u ( m N ( p ) 2 , A p ˉ ) C_{pp}(u)=\operatorname{Cov}_u(m_N(p)^2,A_{\bar p}) C pp ( u ) = Cov u ( m N ( p ) 2 , A p ˉ ) R c a v \mathcal{R}_{\mathrm{cav}} R cav § 9.6.1 , we defined the covariance Σ − \Sigma^- Σ − A ˉ p ˉ \bar A_{\bar p} A ˉ p ˉ T p T_p T p P \mathcal{P} P (161) . Lemma 11.2 uses the rest variance V − = ∫ Var − ( A p − ) ν ( d p ) V^{-}=\int\operatorname{Var}^{-}(A_{p}^{-})\,\nu(dp) V − = ∫ Var − ( A p − ) ν ( d p ) P ≥ 0 \mathcal{P}\ge0 P ≥ 0
Exchangeability. The tilted law of Section 3.3 is invariant under a simultaneous permutation of the N N N π \pi π { 1 , … , N } \{1,\ldots,N\} { 1 , … , N } W π W^{\pi} W π x π x^{\pi} x π W W W W ↦ W π W\mapsto W^{\pi} W ↦ W π U ( q , x π ; W π ) = U ( q , x ; W ) U(q,x^{\pi};W^{\pi})=U(q,x;W) U ( q , x π ; W π ) = U ( q , x ; W ) ( m 1 ( p ) , … , m N ( p ) ) (m_{1}(p),\ldots,m_{N}(p)) ( m 1 ( p ) , … , m N ( p )) m i ( p ) = E [ σ i ∣ G p ] m_{i}(p)=\mathbb{E}[\sigma_{i}\mid\mathcal{G}_{p}] m i ( p ) = E [ σ i ∣ G p ] S p = N − 1 ∑ i m i ( p ) 2 S_{p}=N^{-1}\sum_{i}m_{i}(p)^{2} S p = N − 1 ∑ i m i ( p ) 2 ( S p , m i ( p ) 2 ) (S_{p},m_{i}(p)^{2}) ( S p , m i ( p ) 2 ) 1 ≤ i ≤ N 1\le i\le N 1 ≤ i ≤ N p ∈ [ 0 , 1 ] p\in[0,1] p ∈ [ 0 , 1 ]
E 1 m N ( p ) 2 = E S p , Cov 1 ( S p , m N ( p ) 2 ) = 1 N ∑ i = 1 N Cov ( S p , m i ( p ) 2 ) = Var ( S p ) . (185) \mathbb{E}_{1}m_{N}(p)^{2}=\mathbb{E}S_{p},\qquad\operatorname{Cov}_{1}(S_{p},m_{N}(p)^{2})=\frac{1}{N}\sum_{i=1}^{N}\operatorname{Cov}(S_{p},m_{i}(p)^{2})=\operatorname{Var}(S_{p}). \tag*{(185)} E 1 m N ( p ) 2 = E S p , Cov 1 ( S p , m N ( p ) 2 ) = N 1 i = 1 ∑ N Cov ( S p , m i ( p ) 2 ) = Var ( S p ) . (185) At u = 1 u=1 u = 1 M p M_{p} M p n n n m p m_{p} m p
N S p = m N ( p ) 2 + n A p − . (186) NS_{p}=m_{N}(p)^{2}+nA_{p}^{-}. \tag*{(186)} N S p = m N ( p ) 2 + n A p − . (186) Exchangeability of the sites is used in the same way in the cavity method of [81, Section 1.6].
The mechanism of the contraction
Before giving the proofs, we explain how the inequality closes. In Lemma 12.2 below, we prove the identity
V = V s i t e N + n N [ α d Tr ν ( J a , d s c Σ − ) − α 2 P + α ∫ T p ν ( d p ) + R c a v ] . V=\frac{V^{\mathrm{site}}}{N}+\frac{n}{N}\left[\frac{\alpha}{d}\operatorname{Tr}_{\nu}\left(J^{\mathrm{sc}}_{a,d}\Sigma^{-}\right)-\frac{\alpha}{2}\mathcal{P}+\alpha\int T_{p}\,\nu(dp)+R_{\mathrm{cav}}\right]. V = N V site + N n [ d α Tr ν ( J a , d sc Σ − ) − 2 α P + α ∫ T p ν ( d p ) + R cav ] . where 0 ≤ V s i t e ≤ 1 / 4 0\le V^{\mathrm{site}}\le1/4 0 ≤ V site ≤ 1/4 V − = Tr Σ − V^{-}=\operatorname{Tr}\Sigma^{-} V − = Tr Σ − Tr ν ( J a , d s c Σ − ) \operatorname{Tr}_{\nu}(J^{\mathrm{sc}}_{a,d}\Sigma^{-}) Tr ν ( J a , d sc Σ − ) − α P / 2 -\alpha\mathcal{P}/2 − α P /2 A A A J J J H 3 H_{3} H 3 N − 1 N^{-1} N − 1 J a , d s c ⪯ ( 1 + ε N ) I J^{\mathrm{sc}}_{a,d}\preceq(1+\varepsilon_{N})I J a , d sc ⪯ ( 1 + ε N ) I ( α / d ) ( 1 + ε N ) V − (\alpha/d)(1+\varepsilon_{N})V^{-} ( α / d ) ( 1 + ε N ) V −
The contraction comes from the factor α / d \alpha/d α / d a , b a,b a , b q a b q_{ab} q ab τ a , τ b \tau^{a},\tau^{b} τ a , τ b ϵ a \epsilon_{a} ϵ a ϵ b \epsilon_{b} ϵ b ( α u / n ) τ a ⋅ τ b = α u R a b − (\alpha u/n)\tau^{a}\cdot\tau^{b}=\alpha uR^{-}_{ab} ( α u / n ) τ a ⋅ τ b = α u R ab − b ( u ) q a b b(u)q_{ab} b ( u ) q ab
b ( u ) q a b + α u R a b − = d q a b + α u ( R a b − − q a b ) . b(u)q_{ab}+\alpha uR^{-}_{ab}=dq_{ab}+\alpha u\left(R^{-}_{ab}-q_{ab}\right). b ( u ) q ab + α u R ab − = d q ab + α u ( R ab − − q ab ) . Only the part α u ( R a b − − q a b ) \alpha u(R^{-}_{ab}-q_{ab}) α u ( R ab − − q ab ) α u \alpha u α u α \alpha α q a b q_{ab} q ab d d d (129) , the ratio of the two satisfies
α d = β 2 t n / N β 2 ( N − t ) / N = t N − 1 N − t ≤ t = 1 − ρ . (187) \frac{\alpha}{d}=\frac{\beta^{2}tn/N}{\beta^{2}(N-t)/N}=t\frac{N-1}{N-t}\leq t=1-\rho. \tag*{(187)} d α = β 2 ( N − t ) / N β 2 t n / N = t N − t N − 1 ≤ t = 1 − ρ . (187) The remaining part of the field has deterministic covariance; at u = 1 u=1 u = 1 b ( 1 ) = c = β 2 ρ b(1)=c=\beta^{2}\rho b ( 1 ) = c = β 2 ρ V ≤ t ( 1 + ϵ N ) V − + errors V\leq t(1+\epsilon_N)V^{-}+\text{errors} V ≤ t ( 1 + ϵ N ) V − + errors
To close the inequality, we must compare V − V^{-} V − V V V V − ≤ K V + errors V^{-}\leq KV+\text{errors} V − ≤ K V + errors K > 1 K>1 K > 1 K = K β , k K=K_{\beta,k} K = K β , k V ≤ t K ( 1 + ϵ N ) V + errors V\leq tK(1+\epsilon_N)V+\text{errors} V ≤ t K ( 1 + ϵ N ) V + errors V V V t K ( 1 + ϵ N ) < 1 tK(1+\epsilon_N)<1 t K ( 1 + ϵ N ) < 1 ρ ≤ 1 − 1 / K \rho\leq1-1/K ρ ≤ 1 − 1/ K V − ≤ V + errors V^{-}\leq V+\text{errors} V − ≤ V + errors V ≤ ( 1 − ρ ) ( 1 + ϵ N ) V + errors V\leq(1-\rho)(1+\epsilon_N)V+\text{errors} V ≤ ( 1 − ρ ) ( 1 + ϵ N ) V + errors ( 1 − ρ ) ( 1 + ρ / 4 ) ≤ 1 − 3 ρ / 4 (1-\rho)(1+\rho/4)\leq1-3\rho/4 ( 1 − ρ ) ( 1 + ρ /4 ) ≤ 1 − 3 ρ /4 ρ \rho ρ ϵ N ≤ ρ / 4 \epsilon_N\leq\rho/4 ϵ N ≤ ρ /4
The comparison with coefficient one has two parts. At the independent endpoint u = 0 u=0 u = 0 E − ( A p − − p ) 2 \mathbb{E}^{-}(A_p^{-}-p)^2 E − ( A p − − p ) 2 p p p u u u Lemma 10.7 bounds them by the third moment M 3 \mathcal{M}_3 M 3 u = 1 u=1 u = 1 E S p = p \mathbb{E}S_p=p E S p = p Var ( S p ) \operatorname{Var}(S_p) Var ( S p ) 1 / ( 4 n 2 ) 1/(4n^2) 1/ ( 4 n 2 ) Lemma 12.3 bounds M 3 \mathcal{M}_3 M 3 A A A J J J H 3 H_3 H 3 N − 3 N^{-3} N − 3 H 3 H_3 H 3 V V V
The rest of this section carries out this plan. In Section 12.2, we prove Lemma 12.1 and the trace identity of Lemma 12.2 . In Section 12.3, we bound M 3 \mathcal{M}_3 M 3 Lemma 12.3 . In Section 12.4, we prove Proposition 12.4 and Corollary 12.5 .
Comparison with coefficient one and the trace identity
Lemma 12.1 (Four-descendant variance comparison). Let μ = s δ 0 + ν \mu=s\delta_0+\nu μ = s δ 0 + ν F t , s \mathcal{F}_{t,s} F t , s
V − ≤ V + C β sup 0 ≤ u ≤ 1 M 3 ( u ) + 1 4 n 2 . (188) V^{-}\leq V+C_{\beta}\sup_{0\leq u\leq1}\mathcal{M}_3(u)+\frac{1}{4n^2}. \tag*{(188)} V − ≤ V + C β 0 ≤ u ≤ 1 sup M 3 ( u ) + 4 n 2 1 . (188) Proof. For each level p p p E u ( A p − − p ) 2 \mathbb{E}_u(A_p^{-}-p)^2 E u ( A p − − p ) 2 u = 0 u=0 u = 0 u = 1 u=1 u = 1 Lemmas 10.5 and 10.7 . In the second step, we evaluate it at u = 1 u=1 u = 1 μ \mu μ
Step 1: The interpolation. Fix p ∈ [ 0 , 1 ] p \in[0,1] p ∈ [ 0 , 1 ] p p p ( 1 , 2 ) (1,2) ( 1 , 2 ) ( 3 , 4 ) (3,4) ( 3 , 4 ) p p p Δ 12 = R 12 − − p \Delta_{12}=R_{12}^{-}-p Δ 12 = R 12 − − p Δ 34 = R 34 − − p \Delta_{34}=R_{34}^{-}-p Δ 34 = R 34 − − p
F p = Δ 12 Δ 34 = ( R 12 − − p ) ( R 34 − − p ) . F_p=\Delta_{12}\Delta_{34}=(R_{12}^{-}-p)(R_{34}^{-}-p). F p = Δ 12 Δ 34 = ( R 12 − − p ) ( R 34 − − p ) . Given G p \mathcal{G}_p G p σ \sigma σ G p \mathcal{G}_p G p P u \mathbb{P}_u P u E 4 , p , u [ R 12 − ∣ G p ] = ∣ M p ∣ 2 / n = A p − \mathbb{E}_{4,p,u}[R_{12}^{-}\mid\mathcal{G}_p]=|M_p|^2/n=A_p^{-} E 4 , p , u [ R 12 − ∣ G p ] = ∣ M p ∣ 2 / n = A p − R 34 − R_{34}^{-} R 34 − G p \mathcal{G}_p G p E 4 , p , u [ F p ∣ G p ] = ( A p − − p ) 2 \mathbb{E}_{4,p,u}[F_p\mid\mathcal{G}_p]=(A_p^{-}-p)^2 E 4 , p , u [ F p ∣ G p ] = ( A p − − p ) 2
V p ( u ) = E 4 , p , u F p = E u ( A p − − p ) 2 , V ( u ) = ∫ V p ( u ) ν ( d p ) . \mathcal{V}_p(u)=\mathbb{E}_{4,p,u}F_p=\mathbb{E}_u(A_p^{-}-p)^2,\qquad\mathcal{V}(u)=\int\mathcal{V}_p(u)\,\nu(dp). V p ( u ) = E 4 , p , u F p = E u ( A p − − p ) 2 , V ( u ) = ∫ V p ( u ) ν ( d p ) . Since ∣ F p ∣ ≤ 4 |F_p|\leq4 ∣ F p ∣ ≤ 4 V p ∈ C 2 ( [ 0 , 1 ] ) \mathcal{V}_p\in C^2([0,1]) V p ∈ C 2 ([ 0 , 1 ]) V p ′ ( u ) \mathcal{V}_p'(u) V p ′ ( u ) α / 2 \alpha/2 α /2 F p F_p F p ϵ x ϵ y \epsilon_x\epsilon_y ϵ x ϵ y s V p ( u ) ∫ E 2 , r , u D 12 ν ( d r ) s\mathcal{V}_p(u)\int\mathbb{E}_{2,r,u}D_{12}\,\nu(dr) s V p ( u ) ∫ E 2 , r , u D 12 ν ( d r ) V p ( u ) \mathcal{V}_p(u) V p ( u ) F p F_p F p ν ( d p ) \nu(dp) ν ( d p ) α ≤ β 2 \alpha\leq\beta^2 α ≤ β 2
∫ ∣ V p ′ ( u ) ∣ ν ( d p ) ≤ C β M 3 ( u ) ( 0 ≤ u ≤ 1 ) . (189) \int|\mathcal{V}_p'(u)|\,\nu(dp)\leq C_\beta\mathcal{M}_3(u)\qquad(0\leq u\leq1). \tag*{(189)} ∫ ∣ V p ′ ( u ) ∣ ν ( d p ) ≤ C β M 3 ( u ) ( 0 ≤ u ≤ 1 ) . (189) For each p p p V p ( 0 ) = V p ( 1 ) − ∫ 0 1 V p ′ ( u ) d u \mathcal{V}_p(0)=\mathcal{V}_p(1)-\int_0^1\mathcal{V}_p'(u)\,du V p ( 0 ) = V p ( 1 ) − ∫ 0 1 V p ′ ( u ) d u N N N V p ′ ( u ) \mathcal{V}_p'(u) V p ′ ( u ) p p p u u u ν ( d p ) \nu(dp) ν ( d p ) ∣ ∫ 0 1 V p ′ ( u ) d u ∣ \left|\int_0^1\mathcal{V}_p'(u)\,du\right| ∫ 0 1 V p ′ ( u ) d u ∫ 0 1 ∣ V p ′ ( u ) ∣ d u \int_0^1|\mathcal{V}_p'(u)|\,du ∫ 0 1 ∣ V p ′ ( u ) ∣ d u p p p u u u
V ( 0 ) ≤ V ( 1 ) + C β sup 0 ≤ u ≤ 1 M 3 ( u ) . (190) \mathcal{V}(0)\leq\mathcal{V}(1)+C_\beta\sup_{0\leq u\leq1}\mathcal{M}_3(u). \tag*{(190)} V ( 0 ) ≤ V ( 1 ) + C β 0 ≤ u ≤ 1 sup M 3 ( u ) . (190) At u = 0 u=0 u = 0 A p − A_p^{-} A p − p p p V − ≤ V ( 0 ) V^{-}\leq\mathcal{V}(0) V − ≤ V ( 0 ) E − A p − = p \mathbb{E}^{-}A_p^{-}=p E − A p − = p μ \mu μ
Step 2: The full endpoint. We write f p s i t e = m N ( p ) 2 f_p^{\mathrm{site}}=m_N(p)^2 f p site = m N ( p ) 2 n A p − = N S p − f p s i t e nA_p^{-}=NS_p-f_p^{\mathrm{site}} n A p − = N S p − f p site p ∈ supp ν p\in\operatorname{supp}\nu p ∈ supp ν E S p = p \mathbb{E}S_p=p E S p = p E 1 f p s i t e = p \mathbb{E}_1 f_p^{\mathrm{site}}=p E 1 f p site = p E 1 A p − = ( N p − p ) / n = p \mathbb{E}_1 A_p^{-}=(Np-p)/n=p E 1 A p − = ( N p − p ) / n = p
Var 1 ( A p − ) = N 2 Var ( S p ) − 2 N Cov 1 ( S p , f p s i t e ) + Var 1 ( f p s i t e ) n 2 = N 2 − 2 N n 2 Var ( S p ) + Var 1 ( f p s i t e ) n 2 ≤ Var ( S p ) + 1 4 n 2 . \operatorname{Var}_1(A_p^{-})=\frac{N^2\operatorname{Var}(S_p)-2N\operatorname{Cov}_1(S_p,f_p^{\mathrm{site}})+\operatorname{Var}_1(f_p^{\mathrm{site}})}{n^2}
=\frac{N^2-2N}{n^2}\operatorname{Var}(S_p)+\frac{\operatorname{Var}_1(f_p^{\mathrm{site}})}{n^2}\leq\operatorname{Var}(S_p)+\frac{1}{4n^2}. Var 1 ( A p − ) = n 2 N 2 Var ( S p ) − 2 N Cov 1 ( S p , f p site ) + Var 1 ( f p site ) = n 2 N 2 − 2 N Var ( S p ) + n 2 Var 1 ( f p site ) ≤ Var ( S p ) + 4 n 2 1 . In the last step, we used N 2 − 2 N = n 2 − 1 N^2-2N=n^2-1 N 2 − 2 N = n 2 − 1 Var 1 ( f p s i t e ) ≤ 1 / 4 \operatorname{Var}_1(f_p^{\mathrm{site}})\leq1/4 Var 1 ( f p site ) ≤ 1/4 0 ≤ f p s i t e ≤ 1 0\leq f_p^{\mathrm{site}}\leq1 0 ≤ f p site ≤ 1 E 1 A p − = p \mathbb{E}_1 A_p^{-}=p E 1 A p − = p V p ( 1 ) \mathcal{V}_p(1) V p ( 1 ) ν \nu ν supp ν \operatorname{supp}\nu supp ν V ( 1 ) ≤ V + 1 / ( 4 n 2 ) \mathcal{V}(1)\leq V+1/(4n^2) V ( 1 ) ≤ V + 1/ ( 4 n 2 ) V − ≤ V ( 0 ) V^{-}\leq\mathcal{V}(0) V − ≤ V ( 0 )
The next lemma expresses V V V
Lemma 12.2 (Trace identity for the full variance). Let the profile be deterministic with ν ( { 1 } ) = 0 \nu(\{1\})=0 ν ({ 1 }) = 0 and let
V s i t e = ∫ Var 1 ( m N ( p ) 2 ) ν ( d p ) . V^{\mathrm{site}}=\int\operatorname{Var}_1\left(m_N(p)^2\right)\nu(\mathrm{d}p). V site = ∫ Var 1 ( m N ( p ) 2 ) ν ( d p ) . Then 0 ≤ V s i t e ≤ 1 / 4 0\leq V^{\mathrm{site}}\leq1/4 0 ≤ V site ≤ 1/4 and
V = V s i t e N + n N [ α d Tr ν ( J a , d s c Σ − ) − α 2 P + α ∫ T p ν ( d p ) + R c a v ] . (191) V=\frac{V^{\mathrm{site}}}{N}+\frac{n}{N}\left[\frac{\alpha}{d}\operatorname{Tr}_{\nu}(J_{a,d}^{\mathrm{sc}}\Sigma^{-})-\frac{\alpha}{2}\mathcal{P}+\alpha\int T_p\,\nu(\mathrm{d}p)+\mathcal{R}_{\mathrm{cav}}\right]. \tag*{(191)} V = N V site + N n [ d α Tr ν ( J a , d sc Σ − ) − 2 α P + α ∫ T p ν ( d p ) + R cav ] . (191) Proof. Each variance in the definition of V s i t e V^{\mathrm{site}} V site [ 0 , 1 / 4 ] [0,1/4] [ 0 , 1/4 ] 0 ≤ m N ( p ) 2 ≤ 1 0\leq m_N(p)^2\leq1 0 ≤ m N ( p ) 2 ≤ 1 ν ( [ 0 , 1 ] ) ≤ 1 \nu([0,1])\leq1 ν ([ 0 , 1 ]) ≤ 1 V s i t e V^{\mathrm{site}} V site p ∈ [ 0 , 1 ] p\in[0,1] p ∈ [ 0 , 1 ]
Var ( S p ) = Cov 1 ( S p , m N ( p ) 2 ) = 1 N Var 1 ( m N ( p ) 2 ) + n N C p p ( 1 ) . (192) \operatorname{Var}(S_p)=\operatorname{Cov}_1(S_p,m_N(p)^2)=\frac{1}{N}\operatorname{Var}_1(m_N(p)^2)+\frac{n}{N}C_{pp}(1). \tag*{(192)} Var ( S p ) = Cov 1 ( S p , m N ( p ) 2 ) = N 1 Var 1 ( m N ( p ) 2 ) + N n C pp ( 1 ) . (192) At u = 0 u=0 u = 0 P 0 \mathbb{P}_0 P 0 m N ( p ) m_N(p) m N ( p ) A p − A_p^{-} A p − C p p ( 0 ) = 0 C_{pp}(0)=0 C pp ( 0 ) = 0 C p p ∈ C 2 ( [ 0 , 1 ] ) C_{pp}\in C^2([0,1]) C pp ∈ C 2 ([ 0 , 1 ])
C p p ( 1 ) = C p p ( 0 ) + ∫ 0 1 ( 1 − u ) C p p ′ ′ ( u ) d u . C_{pp}(1)=C_{pp}(0)+\int_0^1(1-u)C_{pp}''(u)\,\mathrm{d}u. C pp ( 1 ) = C pp ( 0 ) + ∫ 0 1 ( 1 − u ) C pp ′′ ( u ) d u . At fixed N N N p p p ν ( d p ) \nu(\mathrm{d}p) ν ( d p ) R c a v \mathcal{R}_{\mathrm{cav}} R cav ν ( { 1 } ) = 0 \nu(\{1\})=0 ν ({ 1 }) = 0 C p p ′ ( 0 ) C_{pp}'(0) C pp ′ ( 0 ) ν ( d p ) \nu(\mathrm{d}p) ν ( d p ) V = ∫ Var ( S p ) ν ( d p ) V=\int\operatorname{Var}(S_p)\nu(\mathrm{d}p) V = ∫ Var ( S p ) ν ( d p )
The third moment of the rest overlap
The remaining input to Lemma 10.8 and Lemma 12.1 is the integrated third moment M 3 \mathcal{M}_3 M 3 A A A J J J H 3 H_3 H 3
Lemma 12.3 (Third moment of the rest overlap). For every deterministic profile,
sup 0 ≤ u ≤ 1 M 3 ( u ) ≤ K β , 2 [ 64 ( A + J ) + 16 H 3 + 32 N − 3 ] . \sup_{0 \le u \le1} \mathcal{M}_3(u) \le K_{\beta,2}\left[64(A+J)+16H_3+32N^{-3}\right]. 0 ≤ u ≤ 1 sup M 3 ( u ) ≤ K β , 2 [ 64 ( A + J ) + 16 H 3 + 32 N − 3 ] . where K β , 2 = e 21 β 2 K_{\beta,2}=e^{21\beta^2} K β , 2 = e 21 β 2 A A A J J J H 3 H_3 H 3
Proof. The proof has three steps: an exact formula for the conditional second moment of R q − S q R_q-S_q R q − S q u = 1 u=1 u = 1
Step 1: The full overlap. Fix q ∈ [ 0 , 1 ] q\in[0,1] q ∈ [ 0 , 1 ] σ 1 , σ 2 \sigma^1,\sigma^2 σ 1 , σ 2 G q \mathcal{G}_q G q σ \sigma σ R q = σ 1 ⋅ σ 2 / N R_q=\sigma^1\cdot\sigma^2/N R q = σ 1 ⋅ σ 2 / N δ a = σ a − m q \delta^a=\sigma^a-m_q δ a = σ a − m q
R q − S q = δ 1 ⋅ δ 2 + m q ⋅ δ 1 + m q ⋅ δ 2 N . R_q-S_q=\frac{\delta^1\cdot\delta^2+m_q\cdot\delta^1+m_q\cdot\delta^2}{N}. R q − S q = N δ 1 ⋅ δ 2 + m q ⋅ δ 1 + m q ⋅ δ 2 . Given G q \mathcal{G}_q G q δ 1 \delta^1 δ 1 δ 2 \delta^2 δ 2 C q C_q C q L 2 L^2 L 2 E [ ( δ 1 ⋅ δ 2 ) ( m q ⋅ δ 1 ) ∣ G q ] = 0 \mathbb{E}[(\delta^1\cdot\delta^2)(m_q\cdot\delta^1)\mid\mathcal{G}_q]=0 E [( δ 1 ⋅ δ 2 ) ( m q ⋅ δ 1 ) ∣ G q ] = 0 δ 2 \delta^2 δ 2 δ 1 \delta^1 δ 1
E [ ( R q − S q ) 2 ∣ G q ] = Tr C q 2 + 2 m q T C q m q N 2 . \mathbb{E}\left[(R_q-S_q)^2\mid\mathcal{G}_q\right]=\frac{\operatorname{Tr}C_q^2+2m_q^{\mathsf T}C_qm_q}{N^2}. E [ ( R q − S q ) 2 ∣ G q ] = N 2 Tr C q 2 + 2 m q T C q m q . This is the conditional orthogonality in the proof of Lemma 3.5 (see also [4 Section 2.5, before (2.17)] ); it does not use the contact identity and holds for every deterministic profile. Taking expectations, we have E ( R q − S q ) 2 = a q t h + 2 j q \mathbb{E}(R_q-S_q)^2=a_q^{\mathrm{th}}+2j_q E ( R q − S q ) 2 = a q th + 2 j q ∣ R q ∣ ≤ 1 |R_q|\le1 ∣ R q ∣ ≤ 1 0 ≤ S q ≤ 1 0\le S_q\le1 0 ≤ S q ≤ 1 ∣ R q − S q ∣ ≤ 2 |R_q-S_q|\le2 ∣ R q − S q ∣ ≤ 2 ∣ R q − S q ∣ 3 ≤ 2 ( R q − S q ) 2 |R_q-S_q|^3\le2(R_q-S_q)^2 ∣ R q − S q ∣ 3 ≤ 2 ( R q − S q ) 2 ν \nu ν
∫ E ∣ R q − S q ∣ 3 ν ( d q ) ≤ 2 ( A + 2 J ) . (193) \int\mathbb{E}|R_q-S_q|^3\,\nu(\mathrm{d}q)\le2(A+2J). \tag*{(193)} ∫ E ∣ R q − S q ∣ 3 ν ( d q ) ≤ 2 ( A + 2 J ) . (193) Step 2: The rest overlap at u = 1 u=1 u = 1 Since P 1 \mathbb{P}_1 P 1 q q q E 2 , q , 1 \mathbb{E}_{2,q,1} E 2 , q , 1 ( σ 1 , σ 2 ) (\sigma^1,\sigma^2) ( σ 1 , σ 2 ) N R q = n R 12 − + ϵ 1 ϵ 2 NR_q=nR_{12}^{-}+\epsilon_1\epsilon_2 N R q = n R 12 − + ϵ 1 ϵ 2
∣ R 12 − − R q ∣ = ∣ R 12 − − ϵ 1 ϵ 2 ∣ N ≤ 2 N . |R_{12}^{-}-R_q|=\frac{|R_{12}^{-}-\epsilon_1\epsilon_2|}{N}\le\frac{2}{N}. ∣ R 12 − − R q ∣ = N ∣ R 12 − − ϵ 1 ϵ 2 ∣ ≤ N 2 . By the convexity of x ↦ ∣ x ∣ 3 x \mapsto|x|^3 x ↦ ∣ x ∣ 3 ∣ x + y ∣ 3 ≤ 4 ( ∣ x ∣ 3 + ∣ y ∣ 3 ) |x+y|^3 \le4(|x|^3+|y|^3) ∣ x + y ∣ 3 ≤ 4 ( ∣ x ∣ 3 + ∣ y ∣ 3 ) ( R 12 − − R q ) + ( R q − q ) (R_{12}^{-}-R_q)+(R_q-q) ( R 12 − − R q ) + ( R q − q ) ( R q − S q ) + ( S q − q ) (R_q-S_q)+(S_q-q) ( R q − S q ) + ( S q − q )
∣ R 12 − − q ∣ 3 ≤ 32 N − 3 + 16 ∣ R q − S q ∣ 3 + 16 ∣ S q − q ∣ 3 . |R_{12}^{-}-q|^3 \le32N^{-3}+16|R_q-S_q|^3+16|S_q-q|^3. ∣ R 12 − − q ∣ 3 ≤ 32 N − 3 + 16∣ R q − S q ∣ 3 + 16∣ S q − q ∣ 3 . We take expectations and integrate against ν \nu ν (193) and H 3 = ∫ E ∣ S q − q ∣ 3 ν ( d q ) H_3=\int\mathbb{E}|S_q-q|^3\,\nu(dq) H 3 = ∫ E ∣ S q − q ∣ 3 ν ( d q )
M 3 ( 1 ) ≤ 32 N − 3 + 32 A + 64 J + 16 H 3 ≤ 64 ( A + J ) + 16 H 3 + 32 N − 3 . \mathcal{M}_3(1) \le32N^{-3}+32A+64J+16H_3 \le64(A+J)+16H_3+32N^{-3}. M 3 ( 1 ) ≤ 32 N − 3 + 32 A + 64 J + 16 H 3 ≤ 64 ( A + J ) + 16 H 3 + 32 N − 3 . Step 3: Comparison. For each q q q ∣ R 12 − − q ∣ 3 |R_{12}^{-}-q|^3 ∣ R 12 − − q ∣ 3 8 8 8 q q q k = 2 k=2 k = 2 E 2 , q , u ∣ R 12 − − q ∣ 3 ≤ K β , 2 E 2 , q , 1 ∣ R 12 − − q ∣ 3 \mathbb{E}_{2,q,u}|R_{12}^{-}-q|^3 \le K_{\beta,2}\mathbb{E}_{2,q,1}|R_{12}^{-}-q|^3 E 2 , q , u ∣ R 12 − − q ∣ 3 ≤ K β , 2 E 2 , q , 1 ∣ R 12 − − q ∣ 3 u ∈ [ 0 , 1 ] u\in[0,1] u ∈ [ 0 , 1 ] q q q ν \nu ν M 3 ( u ) ≤ K β , 2 M 3 ( 1 ) \mathcal{M}_3(u)\le K_{\beta,2}\mathcal{M}_3(1) M 3 ( u ) ≤ K β , 2 M 3 ( 1 )
Closing the variance inequality
We now combine the trace identity, the error bounds, and the comparison with coefficient one. The stability assumption below concerns the scalar response operator of Section 9 with the cavity coefficient d d d (129) , acting on the scalar transport space H ν \mathcal{H}_\nu H ν
Proposition 12.4 (Self-overlap contraction). Let N ≥ 2 N\ge2 N ≥ 2 0 < ρ ≤ 1 0<\rho\le1 0 < ρ ≤ 1 0 < s ≤ 1 / 2 0<s\le1/2 0 < s ≤ 1/2 μ = s δ 0 + ν \mu=s\delta_0+\nu μ = s δ 0 + ν F t , s \mathcal{F}_{t,s} F t , s a a a d = β 2 ( 1 − t / N ) d=\beta^2(1-t/N) d = β 2 ( 1 − t / N )
T a , d s c ⪯ ( 1 + ε N ) I on H ν , 0 ≤ ε N ≤ ρ / 4. (194) \mathcal{T}^{\mathrm{sc}}_{a,d}\preceq(1+\varepsilon_N)I \text{ on }\mathcal{H}_\nu,\qquad0\le\varepsilon_N\le\rho/4. \tag*{(194)} T a , d sc ⪯ ( 1 + ε N ) I on H ν , 0 ≤ ε N ≤ ρ /4. (194) Then, with a constant C c a v C_{\mathrm{cav}} C cav β \beta β
3 ρ 4 V ≤ C c a v [ N − 1 + A + J + H 3 ] . (195) \frac{3\rho}{4}V\le C_{\mathrm{cav}}\left[N^{-1}+A+J+H_3\right]. \tag*{(195)} 4 3 ρ V ≤ C cav [ N − 1 + A + J + H 3 ] . (195) Proof. The proof has three steps. We first collect the bounds on the error terms in (191) and on V − V^- V − (194) . Finally, we insert the comparison (188) into (191) and absorb the resulting multiple of V V V
Step 1: The errors. As μ \mu μ (22) implies that supp ν ⊂ [ 0 , 1 ) \operatorname{supp}\nu\subset[0,1) supp ν ⊂ [ 0 , 1 ) ν ( { 1 } ) = 0 \nu(\{1\})=0 ν ({ 1 }) = 0 Lemmas 11.2 , 11.3 , and 12.2 apply. Let
ϵ = H 3 + A + J + N − 3 . \epsilon=H_3+A+J+N^{-3}. ϵ = H 3 + A + J + N − 3 . We claim that
α ∣ ∫ T p ν ( d p ) ∣ ≤ C β [ A + J + N − 2 ] , ∣ R c a v ∣ ≤ C β ϵ , V − ≤ V + C β ϵ + 1 4 n 2 . (196) \alpha\left|\int T_p\,\nu(dp)\right|\le C_\beta\left[A+J+N^{-2}\right],\qquad|\mathcal{R}_{\mathrm{cav}}|\le C_\beta\epsilon,\qquad V^-\le V+C_\beta\epsilon+\frac{1}{4n^2}. \tag*{(196)} α ∫ T p ν ( d p ) ≤ C β [ A + J + N − 2 ] , ∣ R cav ∣ ≤ C β ϵ , V − ≤ V + C β ϵ + 4 n 2 1 . (196) The first bound is (181) , with C β = 192 β 2 K β , 3 C_\beta=192\beta^2K_{\beta,3} C β = 192 β 2 K β , 3 Lemma 10.8 , we have ∣ R c a v ∣ ≤ C β sup w M 3 ( w ) |\mathcal{R}_{\mathrm{cav}}|\le C_\beta\sup_w\mathcal{M}_3(w) ∣ R cav ∣ ≤ C β sup w M 3 ( w ) Lemma 12.3 bounds this supremum by 64 K β , 2 ϵ 64K_{\beta,2}\epsilon 64 K β , 2 ϵ M 3 \mathcal{M}_3 M 3 (188) .
Step 2: The trace pairing. Let A ˉ − \bar{A}^{-} A ˉ − p ↦ A ˉ p − p \mapsto\bar{A}^{-}_{p} p ↦ A ˉ p − Lemma 11.2 , the trace pairing is Tr ν ( J a , d s c Σ − ) = E − ⟨ A ˉ − , J a , d s c A ˉ − ⟩ L 2 ( ν ) \operatorname{Tr}_{\nu}(J^{\mathrm{sc}}_{a,d}\Sigma^{-}) = \mathbb{E}^{-}\langle\bar{A}^{-}, J^{\mathrm{sc}}_{a,d}\bar{A}^{-}\rangle_{L^{2}(\nu)} Tr ν ( J a , d sc Σ − ) = E − ⟨ A ˉ − , J a , d sc A ˉ − ⟩ L 2 ( ν ) A ˉ − \bar{A}^{-} A ˉ − H ν \mathcal{H}_{\nu} H ν A ˉ 0 − = 0 \bar{A}^{-}_{0}=0 A ˉ 0 − = 0 (154) and ν ( { 1 } ) = 0 \nu(\{1\})=0 ν ({ 1 }) = 0 E − ∥ A ˉ − ∥ L 2 ( ν ) 2 = ∫ Var − ( A p − ) ν ( d p ) = V − \mathbb{E}^{-}\|\bar{A}^{-}\|_{L^{2}(\nu)}^{2}=\int\operatorname{Var}^{-}(A^{-}_{p})\,\nu(\mathrm{d}p)=V^{-} E − ∥ A ˉ − ∥ L 2 ( ν ) 2 = ∫ Var − ( A p − ) ν ( d p ) = V − J a , d s c J^{\mathrm{sc}}_{a,d} J a , d sc H ν \mathcal{H}_{\nu} H ν (194) states that ⟨ v , J a , d s c v ⟩ L 2 ( ν ) ≤ ( 1 + ε N ) ∥ v ∥ L 2 ( ν ) 2 \langle v,J^{\mathrm{sc}}_{a,d}v\rangle_{L^{2}(\nu)}\leq(1+\varepsilon_{N})\|v\|_{L^{2}(\nu)}^{2} ⟨ v , J a , d sc v ⟩ L 2 ( ν ) ≤ ( 1 + ε N ) ∥ v ∥ L 2 ( ν ) 2 v ∈ H ν v\in\mathcal{H}_{\nu} v ∈ H ν v = A ˉ − v=\bar{A}^{-} v = A ˉ −
Tr ν ( J a , d s c Σ − ) ≤ ( 1 + ε N ) V − . \operatorname{Tr}_{\nu}(J^{\mathrm{sc}}_{a,d}\Sigma^{-})\leq(1+\varepsilon_{N})V^{-}. Tr ν ( J a , d sc Σ − ) ≤ ( 1 + ε N ) V − . Equivalently, we may test (194) on the eigenvectors of Σ − \Sigma^{-} Σ − H ν \mathcal{H}_{\nu} H ν α / d \alpha/d α / d [ 0 , t ] [0,t] [ 0 , t ] ( 1 + ε N ) V − ≥ 0 (1+\varepsilon_{N})V^{-}\geq0 ( 1 + ε N ) V − ≥ 0
α d Tr ν ( J a , d s c Σ − ) ≤ t ( 1 + ε N ) V − , (197) \frac{\alpha}{d}\operatorname{Tr}_{\nu}(J^{\mathrm{sc}}_{a,d}\Sigma^{-})\leq t(1+\varepsilon_{N})V^{-}, \tag*{(197)} d α Tr ν ( J a , d sc Σ − ) ≤ t ( 1 + ε N ) V − , (197) whatever the sign of the trace.
Step 3: Absorption. The term − α P / 2 -\alpha\mathcal{P}/2 − α P /2 (191) is nonpositive, because P ≥ 0 \mathcal{P}\geq0 P ≥ 0 Lemma 11.2 . By (196) and (197) , the bracket in (191) is at most
t ( 1 + ε N ) V − + C β [ A + J + N − 2 + ϵ ] . t(1+\varepsilon_{N})V^{-}+C_{\beta}\left[A+J+N^{-2}+\epsilon\right]. t ( 1 + ε N ) V − + C β [ A + J + N − 2 + ϵ ] . and this upper bound is nonnegative. As 0 < n / N ≤ 1 0<n/N\leq1 0 < n / N ≤ 1 n / N n/N n / N V s i t e ≤ 1 / 4 V^{\mathrm{site}}\leq1/4 V site ≤ 1/4 (191) , we have
V ≤ 1 4 N + t ( 1 + ε N ) V − + C β [ A + J + N − 2 + ϵ ] . V\leq\frac{1}{4N}+t(1+\varepsilon_{N})V^{-}+C_{\beta}\left[A+J+N^{-2}+\epsilon\right]. V ≤ 4 N 1 + t ( 1 + ε N ) V − + C β [ A + J + N − 2 + ϵ ] . The stability assumption (194) implies that
t ( 1 + ε N ) ≤ ( 1 − ρ ) ( 1 + ρ 4 ) = 1 − 3 ρ 4 − ρ 2 4 ≤ 1 − 3 ρ 4 ≤ 1. t(1+\varepsilon_{N})\leq(1-\rho)\left(1+\frac{\rho}{4}\right)=1-\frac{3\rho}{4}-\frac{\rho^{2}}{4}\leq1-\frac{3\rho}{4}\leq1. t ( 1 + ε N ) ≤ ( 1 − ρ ) ( 1 + 4 ρ ) = 1 − 4 3 ρ − 4 ρ 2 ≤ 1 − 4 3 ρ ≤ 1. Inserting the bound on V − V^{-} V − (196) , using t ( 1 + ε N ) V ≤ ( 1 − 3 ρ / 4 ) V t(1+\varepsilon_{N})V\leq(1-3\rho/4)V t ( 1 + ε N ) V ≤ ( 1 − 3 ρ /4 ) V t ( 1 + ε N ) t(1+\varepsilon_{N}) t ( 1 + ε N )
V ≤ ( 1 − 3 ρ 4 ) V + 1 4 N + 1 4 n 2 + C β [ A + J + N − 2 + ϵ ] . V\leq\left(1-\frac{3\rho}{4}\right)V+\frac{1}{4N}+\frac{1}{4n^{2}}+C_{\beta}\left[A+J+N^{-2}+\epsilon\right]. V ≤ ( 1 − 4 3 ρ ) V + 4 N 1 + 4 n 2 1 + C β [ A + J + N − 2 + ϵ ] . We move ( 1 − 3 ρ / 4 ) V (1-3\rho/4)V ( 1 − 3 ρ /4 ) V ( 4 n 2 ) − 1 (4n^{2})^{-1} ( 4 n 2 ) − 1 N − 2 N^{-2} N − 2 N − 3 N^{-3} N − 3 N − 1 N^{-1} N − 1 n ≥ N / 2 n\geq N/2 n ≥ N /2 (195) , with a constant C c a v C_{\mathrm{cav}} C cav β \beta β
By Corollary 5.3, H 3 H_{3} H 3 V V V (195) , we obtain the bound on the defect that we use in Section 13.
Corollary 12.5 (Defect on the stable interval). Let N ≥ 2 N \ge2 N ≥ 2 1 / N ≤ ρ ≤ 1 1/N \le\rho\le1 1/ N ≤ ρ ≤ 1 0 < s = λ / β ≤ 1 / 2 0 < s = \lambda/\beta\le1/2 0 < s = λ / β ≤ 1/2
(a) Every minimizer μ = s δ 0 + ν \mu= s\delta_0 + \nu μ = s δ 0 + ν F t , s \mathcal{F}_{t,s} F t , s (194) for some ϵ N \epsilon_N ϵ N a a a
β ( A + 2 J + V ) ≤ C β log 2 ( 2 + N ) N λ ρ 2 , (198) \beta(A + 2J + V) \le C_\beta\frac{\log^2(2+N)}{N\lambda\rho^2}, \tag*{(198)} β ( A + 2 J + V ) ≤ C β N λ ρ 2 log 2 ( 2 + N ) , (198) where A A A J J J V V V μ \mu μ C β C_\beta C β β \beta β
(b) If the defect D N ( ρ ) D_N(\rho) D N ( ρ ) ρ \rho ρ (29) and some minimizer at ρ \rho ρ (194) , then D N ( ρ ) D_N(\rho) D N ( ρ ) (198) .
(c) If 1 / N ≤ ρ 0 ≤ 1 1/N \le\rho_0 \le1 1/ N ≤ ρ 0 ≤ 1 ρ ∈ [ ρ 0 , 1 ] \rho\in[\rho_0,1] ρ ∈ [ ρ 0 , 1 ] ρ \rho ρ (194) , then
∫ ρ 0 1 D N ( ρ ) d ρ ≤ C β log 2 ( 2 + N ) N λ ρ 0 . (199) \int_{\rho_0}^{1} D_N(\rho)\,\mathrm{d}\rho\le C_\beta\frac{\log^2(2+N)}{N\lambda\rho_0}. \tag*{(199)} ∫ ρ 0 1 D N ( ρ ) d ρ ≤ C β N λ ρ 0 log 2 ( 2 + N ) . (199) Proof. We first prove (a). Fix C c a v C_{\mathrm{cav}} C cav (195) , and let κ β = min { 1 / 4 , ( 4 C c a v ) − 1 } \kappa_\beta= \min\{1/4,(4C_{\mathrm{cav}})^{-1}\} κ β = min { 1/4 , ( 4 C cav ) − 1 } β > 1 \beta> 1 β > 1 ρ ≥ 1 / N \rho\ge1/N ρ ≥ 1/ N M = N c = β 2 N ρ M = Nc = \beta^2N\rho M = N c = β 2 N ρ 1 < β 2 ≤ M ≤ β 2 N 1 < \beta^2 \le M \le\beta^2N 1 < β 2 ≤ M ≤ β 2 N κ β ρ ∈ ( 0 , 1 / 4 ] \kappa_\beta\rho\in(0,1/4] κ β ρ ∈ ( 0 , 1/4 ] z = κ β ρ z = \kappa_\beta\rho z = κ β ρ (47) gives
H 3 ≤ κ β ρ V + 425 log ( 2 M ) M s log 2 κ β ρ . H_3 \le\kappa_\beta\rho V + \frac{425\log(2M)}{Ms}\log\frac{2}{\kappa_\beta\rho}. H 3 ≤ κ β ρ V + M s 425 log ( 2 M ) log κ β ρ 2 . We insert this bound into (195) . The resulting term C c a v κ β ρ V C_{\mathrm{cav}}\kappa_\beta\rho V C cav κ β ρ V ρ V / 4 \rho V/4 ρ V /4 ( ρ / 2 ) V (\rho/2)V ( ρ /2 ) V
ρ V ≤ 2 C c a v [ N − 1 + A + J + 425 log ( 2 M ) M s log 2 κ β ρ ] . \rho V \le2C_{\mathrm{cav}}\left[N^{-1}+A+J+\frac{425\log(2M)}{Ms}\log\frac{2}{\kappa_\beta\rho}\right]. ρ V ≤ 2 C cav [ N − 1 + A + J + M s 425 log ( 2 M ) log κ β ρ 2 ] . By (23) and (55) , we have A ≤ ( M s ) − 1 A \le(Ms)^{-1} A ≤ ( M s ) − 1 J ≤ 341 ( 1 + log + M ) 2 / ( M s ) J \le341(1+\log_+ M)^2/(Ms) J ≤ 341 ( 1 + log + M ) 2 / ( M s ) 1 ≤ M ≤ β 2 N 1 \le M \le\beta^2N 1 ≤ M ≤ β 2 N ρ ≥ 1 / N \rho\ge1/N ρ ≥ 1/ N
log ( 2 M ) ≤ log ( 2 β 2 ) + log N , 1 + log + M ≤ 1 + log β 2 + log N , \log(2M) \le\log(2\beta^2)+\log N,\qquad1+\log_+ M \le1+\log\beta^2+\log N, log ( 2 M ) ≤ log ( 2 β 2 ) + log N , 1 + log + M ≤ 1 + log β 2 + log N , and log ( 2 / ( κ β ρ ) ) ≤ log ( 2 / κ β ) + log N \log(2/(\kappa_\beta\rho)) \le\log(2/\kappa_\beta)+\log N log ( 2/ ( κ β ρ )) ≤ log ( 2/ κ β ) + log N C β log ( 2 + N ) C_\beta\log(2+N) C β log ( 2 + N ) log ( 2 + N ) ≥ log 4 > 1 \log(2+N) \ge\log4 > 1 log ( 2 + N ) ≥ log 4 > 1 M s = β 2 N ρ ⋅ λ / β = β N ρ λ Ms = \beta^2N\rho\cdot\lambda/\beta= \beta N\rho\lambda M s = β 2 N ρ ⋅ λ / β = β N ρ λ
β V ≤ C β N ρ + C β log 2 ( 2 + N ) N λ ρ 2 , β ( A + 2 J ) ≤ C β log 2 ( 2 + N ) N λ ρ . \beta V \le\frac{C_\beta}{N\rho}+\frac{C_\beta\log^2(2+N)}{N\lambda\rho^2},\qquad\beta(A+2J) \le\frac{C_\beta\log^2(2+N)}{N\lambda\rho}. β V ≤ N ρ C β + N λ ρ 2 C β log 2 ( 2 + N ) , β ( A + 2 J ) ≤ N λ ρ C β log 2 ( 2 + N ) . Finally, ρ ≤ 1 \rho\le1 ρ ≤ 1 λ = β s ≤ β / 2 \lambda=\beta s\le\beta/2 λ = β s ≤ β /2 log ( 2 + N ) ≥ 1 \log(2+N)\ge1 log ( 2 + N ) ≥ 1
1 N ρ = λ ρ N λ ρ 2 ≤ β 2 ⋅ log 2 ( 2 + N ) N λ ρ 2 , log 2 ( 2 + N ) N λ ρ ≤ log 2 ( 2 + N ) N λ ρ 2 . \frac{1}{N\rho}=\frac{\lambda\rho}{N\lambda\rho^2}\le\frac{\beta}{2}\cdot\frac{\log^2(2+N)}{N\lambda\rho^2},\qquad\frac{\log^2(2+N)}{N\lambda\rho}\le\frac{\log^2(2+N)}{N\lambda\rho^2}. N ρ 1 = N λ ρ 2 λ ρ ≤ 2 β ⋅ N λ ρ 2 log 2 ( 2 + N ) , N λ ρ log 2 ( 2 + N ) ≤ N λ ρ 2 log 2 ( 2 + N ) . which proves (198) .
For (b), recall from (29) that, at every ρ \rho ρ D N ( ρ ) \mathcal{D}_N(\rho) D N ( ρ ) D N ( ρ ) = β ( A + 2 J + V ) \mathcal{D}_N(\rho) = \beta(A + 2J + V) D N ( ρ ) = β ( A + 2 J + V ) ρ \rho ρ (194) , and (a) gives the bound for D N ( ρ ) \mathcal{D}_N(\rho) D N ( ρ )
For (c), recall from § 4.1 that D N \mathcal{D}_N D N ρ \rho ρ ρ ∈ [ ρ 0 , 1 ] \rho\in[\rho_0,1] ρ ∈ [ ρ 0 , 1 ] D N ( ρ ) ≤ C β log 2 ( 2 + N ) / ( N λ ρ 2 ) \mathcal{D}_N(\rho) \leq C_\beta\log^2(2+N)/(N\lambda\rho^2) D N ( ρ ) ≤ C β log 2 ( 2 + N ) / ( N λ ρ 2 ) ∫ ρ 0 1 ρ − 2 d ρ = ρ 0 − 1 − 1 ≤ ρ 0 − 1 \int_{\rho_0}^{1}\rho^{-2}\,\mathrm{d}\rho=\rho_0^{-1}-1\leq\rho_0^{-1} ∫ ρ 0 1 ρ − 2 d ρ = ρ 0 − 1 − 1 ≤ ρ 0 − 1 (199) .
Part V
V Proof of the main theorem
Proof of the main theorem
In this section, we prove Theorem 1.1 . By (29) , four times the fractional-moment error e N , β ( λ ) e_{N,\beta}(\lambda) e N , β ( λ ) D N ( ρ ) D_N(\rho) D N ( ρ ) ρ ∈ ( 0 , 1 ] \rho\in(0,1] ρ ∈ ( 0 , 1 ] ρ 0 \rho_0 ρ 0 ρ 0 = min { 1 , K η 1 / 7 } \rho_0 = \min\{1,K\eta^{1/7}\} ρ 0 = min { 1 , K η 1/7 } K K K β \beta β η = e N , β ( λ ) + L s λ 5 \eta= e_{N,\beta}(\lambda) + L_s\lambda^5 η = e N , β ( λ ) + L s λ 5 e N , β ( λ ) e_{N,\beta}(\lambda) e N , β ( λ ) λ = N − 7 / 50 \lambda= N^{-7/50} λ = N − 7/50 N − 7 / 10 N^{-7/10} N − 7/10 e N , β ( N − 7 / 50 ) e_{N,\beta}(N^{-7/50}) e N , β ( N − 7/50 ) 7 / 10 7/10 7/10
We use results from Section 3 (the defect identity (29) , the bound (30) , the trial budget of Lemma 3.8, the trace bound (23) , the decomposition of Lemma 3.5, and the reduction of the mean bias in Proposition 3.9), from Sections 5 and 6 (Propositions 5.2 and 6.4), from Section 8 (Corollary 8.4), from Section 9 (Proposition 9.9), and from Section 12 (Corollary 12.5). Before the proof of Theorem 1.1 , we list these inputs with the parameters at which they are used.
Standing assumptions and notation. Throughout this section, β > 1 \beta> 1 β > 1 C β C_\beta C β β \beta β λ 0 ≤ min { 1 , β / 4 } \lambda_0 \le\min\{1,\beta/4\} λ 0 ≤ min { 1 , β /4 } L s ≥ 1 L_s \ge1 L s ≥ 1 (31) holds for 0 < λ ≤ 2 λ 0 0 < \lambda\le2\lambda_0 0 < λ ≤ 2 λ 0 c = β 2 ρ c = \beta^2\rho c = β 2 ρ M = N c M = Nc M = N c s = λ / β s = \lambda/\beta s = λ / β μ = s δ 0 + ν \mu= s\delta_0 + \nu μ = s δ 0 + ν log + x = log ( max { 1 , x } ) \log_+ x = \log(\max\{1,x\}) log + x = log ( max { 1 , x }) x ≥ 0 x \ge0 x ≥ 0 trial budget at ρ \rho ρ η ≥ 0 \eta\ge0 η ≥ 0 (32) . Since every minimizer γ ∗ \gamma_* γ ∗ F ρ F_\rho F ρ min γ F ρ ( γ ) \min_\gamma F_\rho(\gamma) min γ F ρ ( γ ) η \eta η ρ \rho ρ (63) .
The interval near ρ = 0 \rho= 0 ρ = 0
We estimate the three terms of D N ( ρ ) = β ( A + 2 J + V ) D_N(\rho) = \beta(A + 2J + V) D N ( ρ ) = β ( A + 2 J + V ) (23) and (55) , we have at every minimizer
A ≤ 1 M s , J ≤ 341 ( 1 + log + M ) 2 M s . (200) A \le\frac{1}{Ms}, \qquad J \le\frac{341(1+\log_+ M)^2}{Ms}. \tag*{(200)} A ≤ M s 1 , J ≤ M s 341 ( 1 + log + M ) 2 . (200) To estimate the self-overlap variance V V V [4 Proposition 7.7] , with Proposition 5.2 in place of [4 Lemma 3.3] and Corollary 8.4 in place of [4 Lemma 7.6] .
Proposition 13.1 (The interval near ρ = 0 \rho= 0 ρ = 0 N ≥ 2 N \ge2 N ≥ 2 0 < λ ≤ λ 0 0 < \lambda\le\lambda_0 0 < λ ≤ λ 0 0 < ρ ≤ 1 0 < \rho\le1 0 < ρ ≤ 1 η > 0 \eta> 0 η > 0 ρ \rho ρ (32) . Let ω ( ρ , η ) = min { 1 , ( η / ρ ) 1 / 6 } \omega(\rho,\eta) = \min\{1,(\eta/\rho)^{1/6}\} ω ( ρ , η ) = min { 1 , ( η / ρ ) 1/6 } (86) . Then every minimizing profile μ \mu μ
β ( A + 2 J + V ) ≤ C β [ log 2 ( 2 + N ) N ρ λ + ( N ρ ) − 2 / 3 + ( N ρ ) − 2 / 3 λ − 1 / 3 ω ( ρ , η ) 2 / 3 ] (201) \beta(A + 2J + V) \le C_\beta\left[\frac{\log^2(2 + N)}{N\rho\lambda} + (N\rho)^{-2/3} + (N\rho)^{-2/3}\lambda^{-1/3}\omega(\rho,\eta)^{2/3}\right] \tag*{(201)} β ( A + 2 J + V ) ≤ C β [ N ρ λ log 2 ( 2 + N ) + ( N ρ ) − 2/3 + ( N ρ ) − 2/3 λ − 1/3 ω ( ρ , η ) 2/3 ] (201) β ( A + 2 J + V ) ≤ C β ( N ρ ) − 1 / 2 , (202) \beta(A + 2J + V) \le C_\beta(N\rho)^{-1/2}, \tag*{(202)} β ( A + 2 J + V ) ≤ C β ( N ρ ) − 1/2 , (202) where A A A J J J V V V μ \mu μ D N ( ρ ) D_N(\rho) D N ( ρ ) D N ( ρ ) D_N(\rho) D N ( ρ ) 0 < ρ 0 ≤ 1 0 < \rho_0 \le1 0 < ρ 0 ≤ 1 η \eta η ρ ∈ ( 0 , ρ 0 ] \rho\in(0,\rho_0] ρ ∈ ( 0 , ρ 0 ]
∫ 0 ρ 0 D N ( ρ ) d ρ ≤ C β [ N − 1 + log 3 ( 2 + N ) N λ + N − 2 / 3 ρ 0 1 / 3 + N − 2 / 3 λ − 1 / 3 η 1 / 9 ρ 0 2 / 9 ] . (203) \int_0^{\rho_0} D_N(\rho)\,\mathrm{d}\rho\le C_\beta\left[N^{-1} + \frac{\log^3(2 + N)}{N\lambda} + N^{-2/3}\rho_0^{1/3} + N^{-2/3}\lambda^{-1/3}\eta^{1/9}\rho_0^{2/9}\right]. \tag*{(203)} ∫ 0 ρ 0 D N ( ρ ) d ρ ≤ C β [ N − 1 + N λ log 3 ( 2 + N ) + N − 2/3 ρ 0 1/3 + N − 2/3 λ − 1/3 η 1/9 ρ 0 2/9 ] . (203) The constants depend only on β \beta β N N N ρ \rho ρ λ \lambda λ η \eta η ν \nu ν
Proof. The proof has five steps. We partition the quantile variable into dyadic shells and bound the diameter of the image of each shell by the root localization. We then apply Proposition 5.2 on each shell and sum over the shells to bound V V V ρ \rho ρ
Step 1: Shells and localization. Fix a minimizing profile μ = s δ 0 + ν \mu= s\delta_0 + \nu μ = s δ 0 + ν Q Q Q ν \nu ν (19) . Recall that Q Q Q ( 0 , 1 − s ) (0,1-s) ( 0 , 1 − s ) ν \nu ν a ( Q ( v ) ) ≥ s + v a(Q(v)) \ge s + v a ( Q ( v )) ≥ s + v 0 < v < 1 − s 0 < v < 1-s 0 < v < 1 − s
V = ∫ Var ( S q ) ν ( d q ) = ∫ 0 1 − s Var ( S Q ( v ) ) d v . (204) V = \int\operatorname{Var}(S_q)\,\nu(\mathrm{d}q) = \int_0^{1-s} \operatorname{Var}(S_{Q(v)})\,\mathrm{d}v. \tag*{(204)} V = ∫ Var ( S q ) ν ( d q ) = ∫ 0 1 − s Var ( S Q ( v ) ) d v . (204) For the dyadic levels u = 2 j s < 1 u = 2^j s < 1 u = 2 j s < 1 j ≥ 0 j \ge0 j ≥ 0
I u = { v ∈ ( 0 , 1 − s ) : u ≤ s + v < 2 u } , h u = ∣ I u ∣ . I_u = \{v \in(0,1-s): u \le s+v < 2u\}, \qquad h_u = |I_u|. I u = { v ∈ ( 0 , 1 − s ) : u ≤ s + v < 2 u } , h u = ∣ I u ∣. (Here u u u v ∈ ( 0 , 1 − s ) v \in(0,1-s) v ∈ ( 0 , 1 − s ) s < s + v < 1 s < s+v < 1 s < s + v < 1 j ≥ 0 j \ge0 j ≥ 0 2 j s ≤ s + v < 2 j + 1 s 2^j s \le s+v < 2^{j+1}s 2 j s ≤ s + v < 2 j + 1 s j j j u = 2 j s < 1 u = 2^j s < 1 u = 2 j s < 1 ( 0 , 1 − s ) (0,1-s) ( 0 , 1 − s ) I u I_u I u h u ≤ u h_u \le u h u ≤ u a ( Q ( v ) ) ≥ s + v ≥ u a(Q(v)) \ge s+v \ge u a ( Q ( v )) ≥ s + v ≥ u v ∈ I u v \in I_u v ∈ I u I u I_u I u u u u ν \nu ν
We write ω = ω ( ρ , η ) \omega= \omega(\rho,\eta) ω = ω ( ρ , η ) ℓ ( I u ) = ess sup v , w ∈ I u ∣ Q ( v ) − Q ( w ) ∣ \ell(I_u) = \operatorname{ess\,sup}_{v,w\in I_u} \lvert Q(v) - Q(w)\rvert ℓ ( I u ) = ess sup v , w ∈ I u ∣ Q ( v ) − Q ( w )∣ I u I_u I u § 5.1.3 . Since μ \mu μ 0 < λ ≤ λ 0 0 < \lambda\le\lambda_0 0 < λ ≤ λ 0 η \eta η γ ∗ = β μ ( [ 0 , ⋅ ] ) \gamma_* = \beta\mu([0,\cdot]) γ ∗ = β μ ([ 0 , ⋅ ]) Corollary 8.4 implies that 0 ≤ Q ( v ) ≤ C Q ( s + v + ω ) 0 \le Q(v) \le C_Q(s+v+\omega) 0 ≤ Q ( v ) ≤ C Q ( s + v + ω ) 0 < v < 1 − s 0 < v < 1-s 0 < v < 1 − s C Q C_Q C Q β \beta β v ∈ I u v \in I_u v ∈ I u s + v < 2 u s+v < 2u s + v < 2 u Q ≥ 0 Q \ge0 Q ≥ 0
ℓ ( I u ) ≤ C Q ( 2 u + ω ) ≤ 2 C Q ( u + ω ) . \ell(I_u) \le C_Q(2u+\omega) \le2C_Q(u+\omega). ℓ ( I u ) ≤ C Q ( 2 u + ω ) ≤ 2 C Q ( u + ω ) . Step 2: The variance on one shell. We use Proposition 5.2 on I u I_u I u u u u h u h_u h u ℓ ( I u ) \ell(I_u) ℓ ( I u ) ϵ u = ( M u ) − 1 \epsilon_u = (Mu)^{-1} ϵ u = ( M u ) − 1 ϵ u ( 1 + log + ( h u / ϵ u ) ) ≤ ( M u ) − 1 ( 1 + log + ( M u 2 ) ) \epsilon_u(1+\log_+(h_u/\epsilon_u)) \le(Mu)^{-1}(1+\log_+(Mu^2)) ϵ u ( 1 + log + ( h u / ϵ u )) ≤ ( M u ) − 1 ( 1 + log + ( M u 2 )) h u / ϵ u = M u h u ≤ M u 2 h_u/\epsilon_u = Muh_u \le Mu^2 h u / ϵ u = M u h u ≤ M u 2 h u ≤ u h_u \le u h u ≤ u ( u + ω ) 2 / 3 ≤ u 2 / 3 + ω 2 / 3 (u+\omega)^{2/3} \le u^{2/3}+\omega^{2/3} ( u + ω ) 2/3 ≤ u 2/3 + ω 2/3
( ϵ u ℓ ( I u ) ) 2 / 3 h u 1 / 3 ≤ ( 2 C Q ) 2 / 3 ( M u ) − 2 / 3 ( u + ω ) 2 / 3 u 1 / 3 ≤ ( 2 C Q ) 2 / 3 M − 2 / 3 ( u 1 / 3 + u − 1 / 3 ω 2 / 3 ) . \begin{aligned}
(\epsilon_u\ell(I_u))^{2/3}h_u^{1/3} \le(2C_Q)^{2/3}(Mu)^{-2/3}(u+\omega)^{2/3}u^{1/3} \\
&\le(2C_Q)^{2/3}M^{-2/3}(u^{1/3}+u^{-1/3}\omega^{2/3}).
\end{aligned} ( ϵ u ℓ ( I u ) ) 2/3 h u 1/3 ≤ ( 2 C Q ) 2/3 ( M u ) − 2/3 ( u + ω ) 2/3 u 1/3 ≤ ( 2 C Q ) 2/3 M − 2/3 ( u 1/3 + u − 1/3 ω 2/3 ) . We conclude that
∫ I u Var ( S Q ( v ) ) d v ≤ C β [ 1 + log + ( M u 2 ) M u + M − 2 / 3 ( u 1 / 3 + u − 1 / 3 ω 2 / 3 ) ] . (205) \int_{I_u} \operatorname{Var}(S_{Q(v)})\,\mathrm{d}v \le C_\beta\left[\frac{1+\log_+(Mu^2)}{Mu}+M^{-2/3}(u^{1/3}+u^{-1/3}\omega^{2/3})\right]. \tag*{(205)} ∫ I u Var ( S Q ( v ) ) d v ≤ C β [ M u 1 + log + ( M u 2 ) + M − 2/3 ( u 1/3 + u − 1/3 ω 2/3 ) ] . (205) The term u − 1 / 3 ω 2 / 3 u^{-1/3}\omega^{2/3} u − 1/3 ω 2/3
Step 3: Summation over the shells. For u = 2 j s u=2^js u = 2 j s log + ( M u 2 ) ≤ log + ( M s 2 ) + 2 j log 2 \log_+(Mu^2) \le\log_+(Ms^2)+2j\log2 log + ( M u 2 ) ≤ log + ( M s 2 ) + 2 j log 2 ∑ j ≥ 0 2 − j = ∑ j ≥ 0 j 2 − j = 2 \sum_{j\ge0}2^{-j}=\sum_{j\ge0}j2^{-j}=2 ∑ j ≥ 0 2 − j = ∑ j ≥ 0 j 2 − j = 2 (205) sum to at most
2 ( 1 + log + ( M s 2 ) ) + 4 log 2 M s ≤ 5 ( 1 + log + ( M s 2 ) ) M s , \frac{2(1+\log_+(Ms^2))+4\log2}{Ms}\le\frac{5(1+\log_+(Ms^2))}{Ms}, M s 2 ( 1 + log + ( M s 2 )) + 4 log 2 ≤ M s 5 ( 1 + log + ( M s 2 )) , where we used 4 log 2 < 3 4\log2<3 4 log 2 < 3 c 3 = ( 1 − 2 − 1 / 3 ) − 1 c_3=(1-2^{-1/3})^{-1} c 3 = ( 1 − 2 − 1/3 ) − 1 ∑ u < 1 u 1 / 3 ≤ c 3 \sum_{u<1}u^{1/3}\le c_3 ∑ u < 1 u 1/3 ≤ c 3 s s s ∑ u < 1 u − 1 / 3 ≤ c 3 s − 1 / 3 \sum_{u<1}u^{-1/3}\le c_3s^{-1/3} ∑ u < 1 u − 1/3 ≤ c 3 s − 1/3 ( 0 , 1 − s ) (0,1-s) ( 0 , 1 − s ) (205) and using (204) yields
V ≤ C β [ 1 + log + ( M s 2 ) M s + M − 2 / 3 ( 1 + s − 1 / 3 ω 2 / 3 ) ] . (206) V\le C_\beta\left[\frac{1+\log_+(Ms^2)}{Ms}+M^{-2/3}(1+s^{-1/3}\omega^{2/3})\right]. \tag*{(206)} V ≤ C β [ M s 1 + log + ( M s 2 ) + M − 2/3 ( 1 + s − 1/3 ω 2/3 ) ] . (206) Step 4: The pointwise bounds. We insert (200) and (206) into β ( A + 2 J + V ) \beta(A+2J+V) β ( A + 2 J + V )
M s = β N ρ λ , β M − 2 / 3 = β − 1 / 3 ( N ρ ) − 2 / 3 , β M − 2 / 3 s − 1 / 3 = ( N ρ ) − 2 / 3 λ − 1 / 3 . Ms=\beta N\rho\lambda,\qquad\beta M^{-2/3}=\beta^{-1/3}(N\rho)^{-2/3},\qquad\beta M^{-2/3}s^{-1/3}=(N\rho)^{-2/3}\lambda^{-1/3}. M s = β N ρ λ , β M − 2/3 = β − 1/3 ( N ρ ) − 2/3 , β M − 2/3 s − 1/3 = ( N ρ ) − 2/3 λ − 1/3 . As M ≤ β 2 N M\le\beta^2N M ≤ β 2 N M s 2 = N ρ λ 2 ≤ N Ms^2=N\rho\lambda^2\le N M s 2 = N ρ λ 2 ≤ N 1 + log + M 1+\log_+M 1 + log + M 1 + log + ( M s 2 ) 1+\log_+(Ms^2) 1 + log + ( M s 2 ) C β log ( 2 + N ) C_\beta\log(2+N) C β log ( 2 + N ) log ( 2 + N ) ≥ 1 \log(2+N)\ge1 log ( 2 + N ) ≥ 1 β A \beta A β A 2 β J 2\beta J 2 β J β \beta β C β log 2 ( 2 + N ) / ( N ρ λ ) C_{\beta}\log^{2}(2+N)/(N\rho\lambda) C β log 2 ( 2 + N ) / ( N ρ λ ) (206) give the last two terms of (201) . This proves (201) . No lower bound on M M M
For (202) , recall the first bound in (30) : at every minimizer, ∫ E ( R q − q ) 2 ν ( d q ) ≤ C min { 1 , M − 1 / 2 } \int\mathbb{E}(R_q-q)^2\,\nu(dq) \le C\min\{1,M^{-1/2}\} ∫ E ( R q − q ) 2 ν ( d q ) ≤ C min { 1 , M − 1/2 } C C C ν \nu ν (25) gives E ( R q − q ) 2 = a q t h + 2 j q + Var ( S q ) \mathbb{E}(R_q-q)^2=a_q^{\mathrm{th}}+2j_q+\operatorname{Var}(S_q) E ( R q − q ) 2 = a q th + 2 j q + Var ( S q ) ν \nu ν β ( A + 2 J + V ) = β ∫ E ( R q − q ) 2 ν ( d q ) ≤ C β M − 1 / 2 = C ( N ρ ) − 1 / 2 \beta(A+2J+V)=\beta\int\mathbb{E}(R_q-q)^2\,\nu(dq)\le C\beta M^{-1/2}=C(N\rho)^{-1/2} β ( A + 2 J + V ) = β ∫ E ( R q − q ) 2 ν ( d q ) ≤ C β M − 1/2 = C ( N ρ ) − 1/2 (202) .
At every ρ \rho ρ D N ( ρ ) D_N(\rho) D N ( ρ ) D N ( ρ ) = β ( A + 2 J + V ) D_N(\rho)=\beta(A+2J+V) D N ( ρ ) = β ( A + 2 J + V ) ρ \rho ρ (29) . This implies that the two bounds also hold for D N ( ρ ) D_N(\rho) D N ( ρ )
Step 5: Integration. Suppose that η \eta η ρ ∈ ( 0 , ρ 0 ] \rho\in(0,\rho_0] ρ ∈ ( 0 , ρ 0 ] (201) and (202) hold for D N ( ρ ) D_N(\rho) D N ( ρ ) ρ ∈ ( 0 , ρ 0 ] \rho\in(0,\rho_0] ρ ∈ ( 0 , ρ 0 ] (202) , whose singularity at ρ = 0 \rho=0 ρ = 0
∫ 0 min { ρ 0 , 1 / N } D N ( ρ ) d ρ ≤ C β N − 1 / 2 ∫ 0 1 / N ρ − 1 / 2 d ρ = 2 C β N − 1 . \int_{0}^{\min\{\rho_0,1/N\}}D_N(\rho)\,d\rho\le C_\beta N^{-1/2}\int_{0}^{1/N}\rho^{-1/2}\,d\rho=2C_\beta N^{-1}. ∫ 0 m i n { ρ 0 , 1/ N } D N ( ρ ) d ρ ≤ C β N − 1/2 ∫ 0 1/ N ρ − 1/2 d ρ = 2 C β N − 1 . If ρ 0 ≤ 1 / N \rho_0\le1/N ρ 0 ≤ 1/ N (203) . If ρ 0 > 1 / N \rho_0>1/N ρ 0 > 1/ N (201) over [ 1 / N , ρ 0 ] [1/N,\rho_0] [ 1/ N , ρ 0 ]
log 2 ( 2 + N ) N λ ∫ 1 / N ρ 0 d ρ ρ = log 2 ( 2 + N ) log ( N ρ 0 ) N λ ≤ log 3 ( 2 + N ) N λ , \frac{\log^2(2+N)}{N\lambda}\int_{1/N}^{\rho_0}\frac{d\rho}{\rho}=\frac{\log^2(2+N)\log(N\rho_0)}{N\lambda}\le\frac{\log^3(2+N)}{N\lambda}, N λ log 2 ( 2 + N ) ∫ 1/ N ρ 0 ρ d ρ = N λ log 2 ( 2 + N ) log ( N ρ 0 ) ≤ N λ log 3 ( 2 + N ) , and the second at most N − 2 / 3 ∫ 1 / N ρ 0 ρ − 2 / 3 d ρ = 3 N − 2 / 3 ρ 0 1 / 3 N^{-2/3}\int_{1/N}^{\rho_0}\rho^{-2/3}\,d\rho=3N^{-2/3}\rho_0^{1/3} N − 2/3 ∫ 1/ N ρ 0 ρ − 2/3 d ρ = 3 N − 2/3 ρ 0 1/3 ω ( ρ , η ) 2 / 3 = min { 1 , ( η / ρ ) 1 / 9 } ≤ ( η / ρ ) 1 / 9 \omega(\rho,\eta)^{2/3}=\min\{1,(\eta/\rho)^{1/9}\}\le(\eta/\rho)^{1/9} ω ( ρ , η ) 2/3 = min { 1 , ( η / ρ ) 1/9 } ≤ ( η / ρ ) 1/9 ρ > 0 \rho>0 ρ > 0
N − 2 / 3 λ − 1 / 3 η 1 / 9 ∫ 0 ρ 0 ρ − 2 / 3 − 1 / 9 d ρ = 9 2 N − 2 / 3 λ − 1 / 3 η 1 / 9 ρ 0 2 / 9 . N^{-2/3}\lambda^{-1/3}\eta^{1/9}\int_{0}^{\rho_0}\rho^{-2/3-1/9}\,d\rho=\frac{9}{2}N^{-2/3}\lambda^{-1/3}\eta^{1/9}\rho_0^{2/9}. N − 2/3 λ − 1/3 η 1/9 ∫ 0 ρ 0 ρ − 2/3 − 1/9 d ρ = 2 9 N − 2/3 λ − 1/3 η 1/9 ρ 0 2/9 . Adding the contributions, we obtain (203) . The constants in Steps 1–5 depend only on β \beta β C Q C_Q C Q (30) , and through powers of β \beta β
Closure at the critical tilt
We now fix the tilt. We take λ = N − 7 / 50 \lambda=N^{-7/50} λ = N − 7/50 λ 5 \lambda^5 λ 5 N − 7 / 10 N^{-7/10} N − 7/10 ρ 0 \rho_0 ρ 0 e N , β ( λ ) e_{N,\beta}(\lambda) e N , β ( λ ) N − 7 / 10 N^{-7/10} N − 7/10 x = η / N − 7 / 10 x=\eta/N^{-7/10} x = η / N − 7/10 x x x [4 Proposition 7.7] : there, the trial budget [4, (7.44)] also contains the unknown error, and the resulting inequality [4 (7.52)] is closed in Step 6 because the error enters only through sublinear powers. Here the cutoff produces the leading term
N − 2 / 3 η 1 / 21 . N^{-2/3}\eta^{1/21}. N − 2/3 η 1/21 . The inputs. The proof of Theorem 1.1 uses the following results, at the parameters listed. In each case, the hypotheses are verified in the step indicated.
Table 1 (PDF p. 195).
Proof of Theorem 1.1 . The proof has six steps. We fix the parameters and the common trial budget, define the cutoff, bound the integral of the defect above and below the cutoff, close the resulting inequality after normalization, and convert the bound on the fractional-moment error into the bound on the mean bias.
Step 1: Parameters and the trial budget. Let N 1 = max { 2 , ⌈ λ 0 − 50 / 7 ⌉ } N_1=\max\{2,\lceil\lambda_0^{-50/7}\rceil\} N 1 = max { 2 , ⌈ λ 0 − 50/7 ⌉} β \beta β N ≥ N 1 N\ge N_1 N ≥ N 1 N − 7 / 50 ≤ λ 0 N^{-7/50}\le\lambda_0 N − 7/50 ≤ λ 0
λ = N − 7 / 50 , r = N − 7 / 10 = λ 5 , e = e N , β ( λ ) , η = e + L s r . (207) \lambda=N^{-7/50},\qquad r=N^{-7/10}=\lambda^5,\qquad e=e_{N,\beta}(\lambda),\qquad\eta=e+L_{\mathrm{s}}r. \tag*{(207)} λ = N − 7/50 , r = N − 7/10 = λ 5 , e = e N , β ( λ ) , η = e + L s r . (207) Then 0 < λ ≤ λ 0 ≤ min { 1 , β / 4 } 0<\lambda\le\lambda_0\le\min\{1,\beta/4\} 0 < λ ≤ λ 0 ≤ min { 1 , β /4 } s = λ / β ≤ 1 / 4 s=\lambda/\beta\le1/4 s = λ / β ≤ 1/4 η \eta η ρ ∈ ( 0 , 1 ] \rho\in(0,1] ρ ∈ ( 0 , 1 ] e e e E β ( λ ) E_\beta(\lambda) E β ( λ ) F N , β c / β F^{{\mathrm{c}}}_{N,\beta}/\beta F N , β c / β § 4.1 ), which has finite exponential moments of every order; these moments enter (3.1). As e ≥ 0 e\ge0 e ≥ 0 L s ≥ 1 L_{\mathrm{s}}\ge1 L s ≥ 1
∞ > η ≥ L s λ 5 ≥ λ 5 > 0. (208) \infty>\eta\ge L_{\mathrm{s}}\lambda^5\ge\lambda^5>0. \tag*{(208)} ∞ > η ≥ L s λ 5 ≥ λ 5 > 0. (208) Step 2: The cutoff. Let C s t ≥ 1 C_{\mathrm{st}}\ge1 C st ≥ 1
K = 12 C s t , ρ 0 = min { 1 , K η 1 / 7 } . (209) K=12C_{\mathrm{st}},\qquad\rho_0=\min\{1,K\eta^{1/7}\}. \tag*{(209)} K = 12 C st , ρ 0 = min { 1 , K η 1/7 } . (209) In both cases ρ 0 < 1 \rho_0<1 ρ 0 < 1 ρ 0 = 1 \rho_0=1 ρ 0 = 1
ρ 0 ≤ K η 1 / 7 , ρ 0 ≥ min { 1 , K λ 5 / 7 } ≥ λ 5 / 7 = N − 1 / 10 ≥ N − 1 , \rho_0\le K\eta^{1/7},\qquad\rho_0\ge\min\{1,K\lambda^{5/7}\}\ge\lambda^{5/7}=N^{-1/10}\ge N^{-1}, ρ 0 ≤ K η 1/7 , ρ 0 ≥ min { 1 , K λ 5/7 } ≥ λ 5/7 = N − 1/10 ≥ N − 1 , where we used (13.9), K ≥ 1 K\ge1 K ≥ 1 λ ≤ 1 \lambda\le1 λ ≤ 1
Step 3: The stable interval. Suppose first that ρ 0 < 1 \rho_0 < 1 ρ 0 < 1 ρ ∈ [ ρ 0 , 1 ] \rho\in[\rho_0,1] ρ ∈ [ ρ 0 , 1 ] ρ ≥ K η 1 / 7 ≥ K λ 5 / 7 \rho\ge K\eta^{1/7} \ge K\lambda^{5/7} ρ ≥ K η 1/7 ≥ K λ 5/7 N − 1 ≤ λ ≤ 1 N^{-1} \le\lambda\le1 N − 1 ≤ λ ≤ 1
λ 2 ≤ λ 5 / 7 ≤ ρ K ′ , N − 1 ≤ λ 5 / 7 ≤ ρ K ′ , ρ − 3 / 4 η 1 / 4 ≤ ρ K 7 / 4 ≤ ρ K ′ . \lambda^2 \le\lambda^{5/7} \le\frac{\rho}{K'}, \qquad N^{-1} \le\lambda^{5/7} \le\frac{\rho}{K'}, \qquad\rho^{-3/4}\eta^{1/4} \le\frac{\rho}{K^{7/4}} \le\frac{\rho}{K'}. λ 2 ≤ λ 5/7 ≤ K ′ ρ , N − 1 ≤ λ 5/7 ≤ K ′ ρ , ρ − 3/4 η 1/4 ≤ K 7/4 ρ ≤ K ′ ρ . where the last bound follows from ρ 7 / 4 ≥ K 7 / 4 η 1 / 4 \rho^{7/4} \ge K^{7/4}\eta^{1/4} ρ 7/4 ≥ K 7/4 η 1/4 K ≥ 1 K \ge1 K ≥ 1 μ \mu μ ρ \rho ρ a a a Proposition 9.9 hold: N ≥ 2 N \ge2 N ≥ 2 0 < ρ ≤ 1 0 < \rho\le1 0 < ρ ≤ 1 0 < λ ≤ λ 0 0 < \lambda\le\lambda_0 0 < λ ≤ λ 0 η \eta η μ \mu μ d = β 2 ( 1 − t / N ) d = \beta^2(1-t/N) d = β 2 ( 1 − t / N )
J a , d s c ⪯ ( 1 + ε N ) I on H ν , 0 ≤ ε N ≤ C s t [ ρ − 3 / 4 η 1 / 4 + λ 2 + N − 1 ] ≤ 3 C s t ρ K = ρ 4 . J^{\mathrm{sc}}_{a,d} \preceq(1+\varepsilon_N)I \text{ on } \mathcal{H}_\nu,\qquad0 \le\varepsilon_N \le C_{\mathrm{st}}\left[\rho^{-3/4}\eta^{1/4}+\lambda^2+N^{-1}\right] \le\frac{3C_{\mathrm{st}}\rho}{K}=\frac{\rho}{4}. J a , d sc ⪯ ( 1 + ε N ) I on H ν , 0 ≤ ε N ≤ C st [ ρ − 3/4 η 1/4 + λ 2 + N − 1 ] ≤ K 3 C st ρ = 4 ρ . where ε N ≥ 0 \varepsilon_N \ge0 ε N ≥ 0 Proposition 9.9 . This is the stability condition (194) , at every minimizer and every ρ ∈ [ ρ 0 , 1 ] \rho\in[\rho_0,1] ρ ∈ [ ρ 0 , 1 ] N ≥ 2 N \ge2 N ≥ 2 1 / N ≤ ρ 0 ≤ 1 1/N \le\rho_0 \le1 1/ N ≤ ρ 0 ≤ 1 s ≤ 1 / 2 s \le1/2 s ≤ 1/2 ρ 0 = K η 1 / 7 ≥ λ 5 / 7 \rho_0=K\eta^{1/7}\ge\lambda^{5/7} ρ 0 = K η 1/7 ≥ λ 5/7 λ 12 / 7 = N − 6 / 25 \lambda^{12/7}=N^{-6/25} λ 12/7 = N − 6/25
∫ ρ 0 1 D N ( ρ ) d ρ ≤ C β log 2 ( 2 + N ) N λ ρ 0 ≤ C β log 2 ( 2 + N ) N λ 12 / 7 = C β N − 19 / 25 log 2 ( 2 + N ) . (210) \int_{\rho_0}^{1}D_N(\rho)\,\mathrm{d}\rho\le C_\beta\frac{\log^2(2+N)}{N\lambda\rho_0}\le C_\beta\frac{\log^2(2+N)}{N\lambda^{12/7}}=C_\beta N^{-19/25}\log^2(2+N). \tag*{(210)} ∫ ρ 0 1 D N ( ρ ) d ρ ≤ C β N λ ρ 0 log 2 ( 2 + N ) ≤ C β N λ 12/7 log 2 ( 2 + N ) = C β N − 19/25 log 2 ( 2 + N ) . (210) When ρ 0 = 1 \rho_0=1 ρ 0 = 1 (210) holds in both cases.
Step 4: The interval near zero. The hypotheses of Proposition 13.1 hold on ( 0 , ρ 0 ] (0,\rho_0] ( 0 , ρ 0 ] N ≥ 2 N\ge2 N ≥ 2 0 < λ ≤ λ 0 0<\lambda\le\lambda_0 0 < λ ≤ λ 0 0 < ρ 0 ≤ 1 0<\rho_0\le1 0 < ρ 0 ≤ 1 (208) , η > 0 \eta>0 η > 0 ρ ∈ ( 0 , ρ 0 ] \rho\in(0,\rho_0] ρ ∈ ( 0 , ρ 0 ] ρ 0 ≤ K η 1 / 7 \rho_0\le K\eta^{1/7} ρ 0 ≤ K η 1/7 ρ 0 < 1 \rho_0<1 ρ 0 < 1 ρ 0 = 1 \rho_0=1 ρ 0 = 1
ρ 0 1 / 3 ≤ K 1 / 3 η 1 / 21 , η 1 / 9 ρ 0 2 / 9 ≤ K 2 / 9 η 1 / 9 + 2 / 63 = K 2 / 9 η 1 / 7 . \rho_0^{1/3}\le K^{1/3}\eta^{1/21},\qquad\eta^{1/9}\rho_0^{2/9}\le K^{2/9}\eta^{1/9+2/63}=K^{2/9}\eta^{1/7}. ρ 0 1/3 ≤ K 1/3 η 1/21 , η 1/9 ρ 0 2/9 ≤ K 2/9 η 1/9 + 2/63 = K 2/9 η 1/7 . We insert these bounds into (203) , add (210) , and use 4 e = ∫ 0 1 D N ( ρ ) d ρ 4e=\int_0^1D_N(\rho)\,\mathrm{d}\rho 4 e = ∫ 0 1 D N ( ρ ) d ρ (29) . This gives
e ≤ C β [ N − 1 + N − 43 / 50 log 3 ( 2 + N ) + N − 19 / 25 log 2 ( 2 + N ) + N − 2 / 3 η 1 / 21 + N − 2 / 3 λ − 1 / 3 η 1 / 7 ] . e\le C_\beta\left[N^{-1}+N^{-43/50}\log^3(2+N)+N^{-19/25}\log^2(2+N)+N^{-2/3}\eta^{1/21}+N^{-2/3}\lambda^{-1/3}\eta^{1/7}\right]. e ≤ C β [ N − 1 + N − 43/50 log 3 ( 2 + N ) + N − 19/25 log 2 ( 2 + N ) + N − 2/3 η 1/21 + N − 2/3 λ − 1/3 η 1/7 ] . where we used N − 1 λ − 1 = N − 43 / 50 N^{-1}\lambda^{-1}=N^{-43/50} N − 1 λ − 1 = N − 43/50 1 > 43 / 50 > 19 / 25 1>43/50>19/25 1 > 43/50 > 19/25 log ( 2 + N ) ≥ 1 \log(2+N)\ge1 log ( 2 + N ) ≥ 1 3 N − 19 / 25 log 3 ( 2 + N ) 3N^{-19/25}\log^3(2+N) 3 N − 19/25 log 3 ( 2 + N )
e ≤ C β [ N − 19 / 25 log 3 ( 2 + N ) + N − 2 / 3 η 1 / 21 + N − 2 / 3 λ − 1 / 3 η 1 / 7 ] . (211) e\le C_\beta\left[N^{-19/25}\log^3(2+N)+N^{-2/3}\eta^{1/21}+N^{-2/3}\lambda^{-1/3}\eta^{1/7}\right]. \tag*{(211)} e ≤ C β [ N − 19/25 log 3 ( 2 + N ) + N − 2/3 η 1/21 + N − 2/3 λ − 1/3 η 1/7 ] . (211) Step 5: Normalization and closure. Let
x = η r = e r + L s , x=\frac{\eta}{r}=\frac{e}{r}+L_s, x = r η = r e + L s , which is finite by (208) and satisfies x ≥ L s ≥ 1 x\ge L_s\ge1 x ≥ L s ≥ 1 η = r x \eta=rx η = r x e e e r = N − 7 / 10 r=N^{-7/10} r = N − 7/10
Table 3.
N − 2 / 3 η 1 / 21 = N − 2 / 3 r 1 / 21 x 1 / 21 = N − 2 / 3 − 1 / 30 x 1 / 21 = r x 1 / 21 N^{-2/3}\eta^{1/21}=N^{-2/3}r^{1/21}x^{1/21}=N^{-2/3-1/30}x^{1/21}=rx^{1/21} N − 2/3 η 1/21 = N − 2/3 r 1/21 x 1/21 = N − 2/3 − 1/30 x 1/21 = r x 1/21 and
N − 2 / 3 λ − 1 / 3 η 1 / 7 = N − 2 / 3 + 7 / 150 − 1 / 10 x 1 / 7 = N − 18 / 25 x 1 / 7 = r N − 1 / 50 x 1 / 7 , N^{-2/3}\lambda^{-1/3}\eta^{1/7}=N^{-2/3+7/150-1/10}x^{1/7}=N^{-18/25}x^{1/7}=rN^{-1/50}x^{1/7}, N − 2/3 λ − 1/3 η 1/7 = N − 2/3 + 7/150 − 1/10 x 1/7 = N − 18/25 x 1/7 = r N − 1/50 x 1/7 , since 2 / 3 + 1 / 30 = 7 / 10 2/3+1/30=7/10 2/3 + 1/30 = 7/10 18 / 25 = 7 / 10 + 1 / 50 18/25=7/10+1/50 18/25 = 7/10 + 1/50 N − 19 / 25 / r = N − 3 / 50 N^{-19/25}/r=N^{-3/50} N − 19/25 / r = N − 3/50 r r r L s L_s L s
x ≤ L s + C β [ N − 3 / 50 log 3 ( 2 + N ) + x 1 / 21 + N − 1 / 50 x 1 / 7 ] ≤ C β x 1 / 7 . (212) x\le L_s+C_\beta\left[N^{-3/50}\log^{3}(2+N)+x^{1/21}+N^{-1/50}x^{1/7}\right]\le C_\beta x^{1/7}. \tag*{(212)} x ≤ L s + C β [ N − 3/50 log 3 ( 2 + N ) + x 1/21 + N − 1/50 x 1/7 ] ≤ C β x 1/7 . (212) In the last inequality, we used that sup N ≥ 2 N − 3 / 50 log 3 ( 2 + N ) \sup_{N\ge2}N^{-3/50}\log^{3}(2+N) sup N ≥ 2 N − 3/50 log 3 ( 2 + N ) log ( 2 + N ) \log(2+N) log ( 2 + N ) N N N N − 1 / 50 ≤ 1 N^{-1/50}\le1 N − 1/50 ≤ 1 1 ≤ x 1 / 7 1\le x^{1/7} 1 ≤ x 1/7 x 1 / 21 ≤ x 1 / 7 x^{1/21}\le x^{1/7} x 1/21 ≤ x 1/7 x ≥ 1 x\ge1 x ≥ 1 x 1 / 7 > 0 x^{1/7}>0 x 1/7 > 0 x 6 / 7 ≤ C β x^{6/7}\le C_\beta x 6/7 ≤ C β x ≤ C β 7 / 6 x\le C_\beta^{7/6} x ≤ C β 7/6
e ≤ η = r x ≤ C β N − 7 / 10 . e\le\eta=rx\le C_\beta N^{-7/10}. e ≤ η = r x ≤ C β N − 7/10 . This argument uses no preliminary bound on e e e η \eta η x x x
Step 6: The mean bias. By Proposition 3.9 , which applies since 0 < λ ≤ λ 0 0<\lambda\le\lambda_0 0 < λ ≤ λ 0
0 ≤ P β − p N , β β ≤ 32 L s λ 5 + 2 e ≤ ( 32 L s + 2 C β ) N − 7 / 10 . 0\le\frac{P_\beta-p_{N,\beta}}{\beta}\le32L_s\lambda^5+2e\le(32L_s+2C_\beta)N^{-7/10}. 0 ≤ β P β − p N , β ≤ 32 L s λ 5 + 2 e ≤ ( 32 L s + 2 C β ) N − 7/10 . Multiplying by β \beta β P β − p N , β ≤ C β N − 7 / 10 P_\beta-p_{N,\beta}\le C_\beta N^{-7/10} P β − p N , β ≤ C β N − 7/10 N ≥ N 1 N\ge N_1 N ≥ N 1 2 ≤ N < N 1 2\le N<N_1 2 ≤ N < N 1 P β − p N , β P_\beta-p_{N,\beta} P β − p N , β
C β s m a l l = max 2 ≤ N < N 1 N 7 / 10 ( P β − p N , β ) , C_\beta^{\mathrm{small}}=\max_{2\le N<N_1}N^{7/10}(P_\beta-p_{N,\beta}), C β small = 2 ≤ N < N 1 max N 7/10 ( P β − p N , β ) , with the maximum interpreted as zero when $N_1=2, is therefore finite and depends only on β \beta β C β C_\beta C β max { C β , C β s m a l l } \max\{C_\beta,C_\beta^{\mathrm{small}}\} max { C β , C β small } N ≥ 2 N\ge2 N ≥ 2
Remark 13.2. The leading term in (211) is N − 2 / 3 η 1 / 21 N^{-2/3}\eta^{1/21} N − 2/3 η 1/21 1 / 21 1/21 1/21 1 / 3 1/3 1/3 ρ 0 \rho_0 ρ 0 (203) and the exponent 1 / 7 1/7 1/7 η \eta η ρ − 3 / 4 η 1 / 4 ≲ ρ \rho^{-3/4}\eta^{1/4} \lesssim\rho ρ − 3/4 η 1/4 ≲ ρ η \eta η N − a N^{-a} N − a a = 2 / 3 + a / 21 a = 2/3 + a/21 a = 2/3 + a /21 20 a / 21 = 2 / 3 20a/21 = 2/3 20 a /21 = 2/3 a = 7 / 10 a = 7/10 a = 7/10 λ 5 \lambda^5 λ 5 λ = N − 7 / 50 \lambda= N^{-7/50} λ = N − 7/50 N − 1 / 50 N^{-1/50} N − 1/50 (203) contribute at most N − 3 / 50 log 3 ( 2 + N ) N^{-3/50}\log^3(2 + N) N − 3/50 log 3 ( 2 + N ) (210) on the stable interval has only a squared logarithm; the third logarithm comes from integrating the first term of (201) over [ 1 / N , ρ 0 ] [1/N,\rho_0] [ 1/ N , ρ 0 ] L s N − 7 / 10 ≤ η ≤ C β N − 7 / 10 L_sN^{-7/10} \le\eta\le C_\beta N^{-7/10} L s N − 7/10 ≤ η ≤ C β N − 7/10 N N N ρ 0 < 1 \rho_0 < 1 ρ 0 < 1 N − 1 / 10 N^{-1/10} N − 1/10
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