Authors’ Note
With the exception of this note, the entire document is machine-written. We have carefully supervised its preparation to ensure that the proofs are readable and that previous literature is appropriately cited. We are releasing this manuscript on Hexagon to disseminate the result as quickly as possible because it may be of interest to the mathematical community. We welcome suggestions concerning mathematical corrections, omitted citations, and attribution of ideas.
P. L. was partially supported by NSF grant DMS-2450004.
Introduction
Spherical spin glasses are mean-field models of disordered systems, in which the energy is a random Gaussian function on a high-dimensional sphere. Following the work of Kirkpatrick, Thirumalai, and Wolynes [100–102], p p p p p p p ≥ 3 p \ge3 p ≥ 3 T d T_d T d T s T_s T s T s T_s T s T d T_d T d T s T_s T s T d T_d T d
In this article, we prove versions of these predictions in finite dimension N N N N N N p p p T s T_s T s T s < T < T d T_s < T < T_d T s < T < T d N N N T s T_s T s Theorem 1.2 and Corollary 1.3 ), and it extends to every initial law chosen independently of the disorder (Theorem 1.4 ). For the second question, we show for general finite mixtures that the generator has exponentially many exponentially small eigenvalues, that a fixed part of the spin lies in the corresponding slow modes, that the stationary spin correlation stays above a plateau up to an exponential time, that the dynamics started from most points of the Gibbs measure stays far from equilibrium, and that the two endpoints of the stationary dynamics share mutual information at least linear in N N N Theorems 1.5 to 1.7 ).
The model and the two thresholds
Let S N = { x ∈ R N : ∣ x ∣ 2 = N } S_N = \{x \in\mathbb{R}^N : \lvert x\rvert^2 = N\} S N = { x ∈ R N : ∣ x ∣ 2 = N } ν N \nu_N ν N S N S_N S N x , y ∈ R N x,y \in\mathbb{R}^N x , y ∈ R N R ( x , y ) = x ⋅ y / N R(x,y) = x \cdot y/N R ( x , y ) = x ⋅ y / N U ( x ) = x / N U(x) = x/\sqrt{N} U ( x ) = x / N finite mixture is a nonzero polynomial
ξ ( r ) = ∑ p = 2 P c p r p , c p ≥ 0. \xi(r) = \sum_{p=2}^{P} c_p r^p,\qquad c_p \ge0. ξ ( r ) = p = 2 ∑ P c p r p , c p ≥ 0. Its minimum degree d d d p p p c p > 0 c_p > 0 c p > 0 ξ \xi ξ H N H_N H N S N S_N S N
E H N ( x ) H N ( y ) = N ξ ( R ( x , y ) ) . \mathbb{E}H_N(x)H_N(y) = N\xi(R(x,y)). E H N ( x ) H N ( y ) = N ξ ( R ( x , y )) . For ξ ( r ) = r p \xi(r) = r^p ξ ( r ) = r p H N ( x ) = N − ( p − 1 ) / 2 ∑ i 1 , … , i p g i 1 ⋯ i p x i 1 ⋯ x i p H_N(x) = N^{-(p-1)/2} \sum_{i_1,\ldots,i_p} g_{i_1\cdots i_p}x_{i_1}\cdots x_{i_p} H N ( x ) = N − ( p − 1 ) /2 ∑ i 1 , … , i p g i 1 ⋯ i p x i 1 ⋯ x i p c p \sqrt{c_p} c p H N H_N H N β ≥ 0 \beta\ge0 β ≥ 0
π ( d x ) = e β H N ( x ) Z N , β ν N ( d x ) , Z N , β = ∫ S N e β H N d ν N , F ( β ) = lim N → ∞ 1 N E log Z N , β . \pi(dx) = \frac{e^{\beta H_N(x)}}{Z_{N,\beta}}\nu_N(dx),\qquad Z_{N,\beta} = \int_{S_N}e^{\beta H_N}\,d\nu_N,\qquad F(\beta) = \lim_{N\to\infty}\frac{1}{N}\mathbb{E}\log Z_{N,\beta}. π ( d x ) = Z N , β e β H N ( x ) ν N ( d x ) , Z N , β = ∫ S N e β H N d ν N , F ( β ) = N → ∞ lim N 1 E log Z N , β . The Gibbs measure favors large values of H N H_N H N − H N -H_N − H N F ( β ) F(\beta) F ( β ) S N S_N S N
L = Δ S N + β ∇ S N H N ⋅ ∇ S N , L = \Delta_{S_N} + \beta\nabla_{S_N}H_N \cdot\nabla_{S_N}, L = Δ S N + β ∇ S N H N ⋅ ∇ S N , that is, the solution of d X t = 2 P X t d W t + β ∇ S N H N ( X t ) d t − ( 1 − 1 / N ) X t d t dX_t = \sqrt{2}P_{X_t}\,dW_t + \beta\nabla_{S_N}H_N(X_t)\,dt - (1 - 1/N)X_t\,dt d X t = 2 P X t d W t + β ∇ S N H N ( X t ) d t − ( 1 − 1/ N ) X t d t W W W R N \mathbb{R}^N R N P x P_x P x S N S_N S N x x x π \pi π E ( f , f ) = ∫ ∣ ∇ S N f ∣ 2 d π \mathcal{E}(f,f) = \int\lvert\nabla_{S_N}f\rvert^2\,d\pi E ( f , f ) = ∫ ∣ ∇ S N f ∣ 2 d π − L -L − L 0 = λ 0 < λ 1 ≤ λ 2 ≤ ⋯ 0 = \lambda_0 < \lambda_1 \le\lambda_2 \le\cdots 0 = λ 0 < λ 1 ≤ λ 2 ≤ ⋯ P t = e t L P_t = e^{tL} P t = e t L P t ( x , ⋅ ) P_t(x,\cdot) P t ( x , ⋅ ) μ P t \mu P_t μ P t t t t μ \mu μ H N H_N H N ∥ μ − ν ∥ T V = sup A ∣ μ ( A ) − ν ( A ) ∣ \lVert\mu- \nu\rVert_{\mathrm{TV}} = \sup_A\lvert\mu(A) - \nu(A)\rvert ∥ μ − ν ∥ TV = sup A ∣ μ ( A ) − ν ( A )∣ T Δ − ∇ E ⋅ ∇ T\Delta- \nabla E \cdot\nabla T Δ − ∇ E ⋅ ∇ E = − H N E = -H_N E = − H N
The two thresholds are defined through the functions
A ( r ) = ( 1 − r ) ξ ′ ( r ) r , Φ β ( r ) = β 2 ξ ( r ) + r + log ( 1 − r ) , 0 < r < 1. A(r)=\frac{(1-r)\xi'(r)}{r},\qquad\Phi_\beta(r)=\beta^2\xi(r)+r+\log(1-r),\qquad0<r<1. A ( r ) = r ( 1 − r ) ξ ′ ( r ) , Φ β ( r ) = β 2 ξ ( r ) + r + log ( 1 − r ) , 0 < r < 1. The dynamical threshold β d = β d ( ξ ) \beta_{\mathrm d}=\beta_{\mathrm d}(\xi) β d = β d ( ξ ) static threshold β s = β s ( ξ ) \beta_{\mathrm s}=\beta_{\mathrm s}(\xi) β s = β s ( ξ )
β d − 2 = sup 0 < r < 1 A ( r ) , β s 2 = inf 0 < r < 1 − r − log ( 1 − r ) ξ ( r ) . (1) \beta_{\mathrm d}^{-2}=\sup_{0<r<1} A(r),\qquad\beta_{\mathrm s}^{2}=\inf_{0<r<1}\frac{-r-\log(1-r)}{\xi(r)}. \tag*{(1)} β d − 2 = 0 < r < 1 sup A ( r ) , β s 2 = 0 < r < 1 inf ξ ( r ) − r − log ( 1 − r ) . (1) For a pure model, we write β d ( p ) \beta_{\mathrm d}(p) β d ( p ) β s ( p ) \beta_{\mathrm s}(p) β s ( p ) ξ ( r ) = r p \xi(r)=r^p ξ ( r ) = r p
The static threshold is the point where the free energy leaves its annealed value. By Jensen’s inequality, we have F ( β ) ≤ lim N N − 1 log E Z N , β = β 2 ξ ( 1 ) / 2 F(\beta)\leq\lim_{N}N^{-1}\log\mathbb{E}Z_{N,\beta}=\beta^2\xi(1)/2 F ( β ) ≤ lim N N − 1 log E Z N , β = β 2 ξ ( 1 ) /2 Φ β ≤ 0 \Phi_\beta\leq0 Φ β ≤ 0 ( 0 , 1 ) (0,1) ( 0 , 1 ) β ≤ β s \beta\leq\beta_{\mathrm s} β ≤ β s Lemma 3.18 ). Following Huang and Sellke [90, Definition 3], we say that ( ξ , β ) (\xi,\beta) ( ξ , β ) strictly replica symmetric if Φ β < 0 \Phi_\beta<0 Φ β < 0 ( 0 , 1 ) (0,1) ( 0 , 1 ) β < β s \beta<\beta_{\mathrm s} β < β s β 2 ξ ′ ′ ( 0 ) < 1 \beta^2\xi''(0)<1 β 2 ξ ′′ ( 0 ) < 1 β > β s \beta>\beta_{\mathrm s} β > β s condensed phase .
The dynamical threshold enters through the derivative
Φ β ′ ( r ) = r 1 − r ( β 2 A ( r ) − 1 ) . \Phi_\beta'(r)=\frac{r}{1-r}\left(\beta^2A(r)-1\right). Φ β ′ ( r ) = 1 − r r ( β 2 A ( r ) − 1 ) . If β ≤ β d \beta\leq\beta_{\mathrm d} β ≤ β d Φ β \Phi_\beta Φ β Φ β ≤ 0 \Phi_\beta\leq0 Φ β ≤ 0 β d ≤ β s \beta_{\mathrm d}\leq\beta_{\mathrm s} β d ≤ β s β > β d \beta>\beta_{\mathrm d} β > β d { r ∈ ( 0 , 1 ) : β 2 A ( r ) > 1 } \{r\in(0,1):\beta^2A(r)>1\} { r ∈ ( 0 , 1 ) : β 2 A ( r ) > 1 } 1 1 1 A ( 1 − ) = 0 A(1^-)=0 A ( 1 − ) = 0
q ∗ = sup { r ∈ ( 0 , 1 ) : β 2 ( 1 − r ) ξ ′ ( r ) > r } . (2) q_*=\sup\{r\in(0,1):\beta^2(1-r)\xi'(r)>r\}. \tag*{(2)} q ∗ = sup { r ∈ ( 0 , 1 ) : β 2 ( 1 − r ) ξ ′ ( r ) > r } . (2) By continuity, q ∗ q_* q ∗ plateau equation
β 2 ( 1 − q ) ξ ′ ( q ) = q . (3) \beta^2(1-q)\xi'(q)=q. \tag*{(3)} β 2 ( 1 − q ) ξ ′ ( q ) = q . (3) It is the largest solution, unless β 2 A \beta^2A β 2 A 1 1 1 q ∗ q_* q ∗ β d < β < β s \beta_{\mathrm d}<\beta<\beta_{\mathrm s} β d < β < β s (3) is the predicted long-time limit of the equilibrium spin correlation (see § 2.6 ); below the static transition, the equilibrium plateau involves the parameters of the condensed phase instead. The threshold β d \beta_{\mathrm d} β d (3) has a solution in ( 0 , 1 ) (0,1) ( 0 , 1 ) ξ ( r ) = r p \xi(r)=r^p ξ ( r ) = r p p ≥ 3 p\geq3 p ≥ 3 A A A q d = ( p − 2 ) / ( p − 1 ) q_{\mathrm d}=(p-2)/(p-1) q d = ( p − 2 ) / ( p − 1 ) β d ( p ) − 2 = p ( p − 2 ) p − 2 / ( p − 1 ) p − 1 \beta_{\mathrm d}(p)^{-2}=p(p-2)^{p-2}/(p-1)^{p-1} β d ( p ) − 2 = p ( p − 2 ) p − 2 / ( p − 1 ) p − 1 p = 2 p=2 p = 2 A A A r ↓ 0 r\downarrow0 r ↓ 0 β d ( 2 ) = β s ( 2 ) = 1 / 2 \beta_{\mathrm d}(2)=\beta_{\mathrm s}(2)=1/\sqrt{2} β d ( 2 ) = β s ( 2 ) = 1/ 2 ξ ( r ) = r p / 2 \xi(r)=r^{p/2} ξ ( r ) = r p /2 β d < β < β s \beta_{\mathrm d}<\beta<\beta_{\mathrm s} β d < β < β s p ≥ 3 p\ge3 p ≥ 3 c 2 = 0 c_2=0 c 2 = 0 ξ ( r ) = r 2 / 2 + r 4 \xi(r)=r^2/2+r^4 ξ ( r ) = r 2 /2 + r 4 β d = β s = 1 \beta_{\mathrm d}=\beta_{\mathrm s}=1 β d = β s = 1
To describe the condensed phase, we recall the Crisanti–Sommers formula. For a probability measure ζ \zeta ζ [ 0 , 1 ] [0,1] [ 0 , 1 ] x ^ ζ ( t ) = ∫ ( 1 − max { s , t } ) ζ ( d s ) \widehat{x}_{\zeta}(t)=\int(1-\max\{s,t\})\zeta(\mathrm ds) x ζ ( t ) = ∫ ( 1 − max { s , t }) ζ ( d s ) F ( β ) F(\beta) F ( β ) ζ \zeta ζ x ^ ζ \widehat{x}_{\zeta} x ζ Definition 3.13 and Theorem 3.14 ). This formula was derived by Crisanti and Sommers [52] and proved by Talagrand for even mixtures [146, Theorem 1.1] and by Chen in general [46, Theorem 1]; Huang and Sellke gave a constructive proof [90]. The Parisi measure is δ 0 \delta_0 δ 0 β ≤ β s \beta\le\beta_{\mathrm s} β ≤ β s ζ \zeta ζ
V ζ ( r ) = ∫ 0 r r − v x ^ ζ ( v ) 2 d v − β 2 ξ ( r ) , 0 ≤ r < 1 , V_{\zeta}(r)=\int_0^r\frac{r-v}{\widehat{x}_{\zeta}(v)^2}\,\mathrm dv-\beta^2\xi(r),\qquad0\le r<1, V ζ ( r ) = ∫ 0 r x ζ ( v ) 2 r − v d v − β 2 ξ ( r ) , 0 ≤ r < 1 , is nonnegative and vanishes on the support of ζ \zeta ζ ζ \zeta ζ V ζ V_{\zeta} V ζ first-variation function and its zero set in [ 0 , 1 ) [0,1) [ 0 , 1 ) contact set . The contact set contains the support of ζ \zeta ζ ζ = x δ 0 + ( 1 − x ) δ Q \zeta=x\delta_0+(1-x)\delta_Q ζ = x δ 0 + ( 1 − x ) δ Q ζ \zeta ζ x ^ ζ \widehat{x}_{\zeta} x ζ x ^ \widehat{x} x V ζ V_{\zeta} V ζ V V V
Definition 1.1 (Strict one-step replica symmetry breaking). Let ξ \xi ξ β > 0 \beta>0 β > 0 x , Q ∈ ( 0 , 1 ) x,Q\in(0,1) x , Q ∈ ( 0 , 1 ) s = 1 − Q s=1-Q s = 1 − Q C = s + x Q C=s+xQ C = s + x Q
x ^ ( v ) = { C − x v , 0 ≤ v ≤ Q , 1 − v , Q ≤ v < 1 , V ( r ) = ∫ 0 r r − v x ^ ( v ) 2 d v − β 2 ξ ( r ) , 0 ≤ r < 1. \widehat{x}(v)=
\begin{cases}
C-xv, & 0\le v\le Q,\\
1-v, & Q\le v<1,
\end{cases}
\qquad
V(r)=\int_0^r\frac{r-v}{\widehat{x}(v)^2}\,\mathrm dv-\beta^2\xi(r),\qquad0\le r<1. x ( v ) = { C − xv , 1 − v , 0 ≤ v ≤ Q , Q ≤ v < 1 , V ( r ) = ∫ 0 r x ( v ) 2 r − v d v − β 2 ξ ( r ) , 0 ≤ r < 1. The pair ( ξ , β ) (\xi,\beta) ( ξ , β ) strictly 1RSB if there exist x , Q ∈ ( 0 , 1 ) x,Q\in(0,1) x , Q ∈ ( 0 , 1 ) x δ 0 + ( 1 − x ) δ Q x\delta_0+(1-x)\delta_Q x δ 0 + ( 1 − x ) δ Q { r ∈ [ 0 , 1 ) : V ( r ) = 0 } \{r\in[0,1):V(r)=0\} { r ∈ [ 0 , 1 ) : V ( r ) = 0 } { 0 , Q } \{0,Q\} { 0 , Q } Q Q Q nondegenerate if V ′ ′ ( Q ) > 0 V''(Q)>0 V ′′ ( Q ) > 0 § 5.2.3 ), ( ξ , β ) (\xi,\beta) ( ξ , β ) Q Q Q x , Q ∈ ( 0 , 1 ) x,Q\in(0,1) x , Q ∈ ( 0 , 1 )
V ( Q ) = V ′ ( Q ) = 0 , V ( r ) > 0 for r ∈ ( 0 , 1 ) ∖ { Q } , V ′ ′ ( Q ) > 0. (4) V(Q)=V'(Q)=0,\qquad V(r)>0\quad\text{for }r\in(0,1)\setminus\{Q\},\qquad V''(Q)>0. \tag*{(4)} V ( Q ) = V ′ ( Q ) = 0 , V ( r ) > 0 for r ∈ ( 0 , 1 ) ∖ { Q } , V ′′ ( Q ) > 0. (4) Note that V ( 0 ) = V ′ ( 0 ) = 0 V(0)=V'(0)=0 V ( 0 ) = V ′ ( 0 ) = 0 1RSB assumptions . They imply that x δ 0 + ( 1 − x ) δ Q x\delta_0+(1-x)\delta_Q x δ 0 + ( 1 − x ) δ Q x x x Q Q Q ( ξ , β ) (\xi,\beta) ( ξ , β ) Proposition 5.11 ). Since x ^ ( Q ) = 1 − Q \widehat{x}(Q)=1-Q x ( Q ) = 1 − Q
condition V ′ ′ ( Q ) > 0 V''(Q)>0 V ′′ ( Q ) > 0 β 2 ( 1 − Q ) 2 ξ ′ ′ ( Q ) < 1 \beta^2(1-Q)^2\xi''(Q)<1 β 2 ( 1 − Q ) 2 ξ ′′ ( Q ) < 1 ( 1 − Q ) − 2 − β 2 ξ ′ ′ ( Q ) (1-Q)^{-2}-\beta^2\xi''(Q) ( 1 − Q ) − 2 − β 2 ξ ′′ ( Q ) Q Q Q Q Q Q x x x { 0 } \{0\} { 0 } { 0 , 1 } \{0,1\} { 0 , 1 } β > β s \beta>\beta_{\mathrm{s}} β > β s
The uniform initial law and Gibbs-typical starting points
The two questions of this paper concern two different initial laws, and the distinction is essential. Three ways of measuring the distance to equilibrium at time t t t
sup x ∈ S N ∥ P t ( x , ⋅ ) − π ∥ T V , ∥ ν N P t − π ∥ T V , ∫ ∥ P t ( x , ⋅ ) − π ∥ T V π ( d x ) . \sup_{x\in S_N}\|P_t(x,\cdot)-\pi\|_{\mathrm{TV}},\qquad\|\nu_NP_t-\pi\|_{\mathrm{TV}},\qquad\int\|P_t(x,\cdot)-\pi\|_{\mathrm{TV}}\,\pi(\mathrm{d}x). x ∈ S N sup ∥ P t ( x , ⋅ ) − π ∥ TV , ∥ ν N P t − π ∥ TV , ∫ ∥ P t ( x , ⋅ ) − π ∥ TV π ( d x ) . An exponentially small spectral gap λ 1 \lambda_1 λ 1 ν N \nu_N ν N d ν N / d π \mathrm{d}\nu_N/\mathrm{d}\pi d ν N / d π N N N log N \log N log N § 2.7 ) therefore do not answer the first question.
The uniform law ν N \nu_N ν N ν N \nu_N ν N 1 / β 1/\beta 1/ β ν N \nu_N ν N uniform start .
If instead the initial point is drawn from π \pi π π P t = π \pi P_t=\pi π P t = π t t t X t X_t X t X 0 X_0 X 0 X t X_t X t P t ( X 0 , ⋅ ) P_t(X_0,\cdot) P t ( X 0 , ⋅ ) X 0 X_0 X 0 π \pi π E π R ( X 0 , X t ) \mathbb{E}_{\pi}R(X_0,X_t) E π R ( X 0 , X t ) X 0 X_0 X 0 X t X_t X t Gibbs-typical starting points . They depend on the disorder, and the example μ = π \mu=\pi μ = π Theorem 1.4 below says how much dependence is allowed.
Main results
Throughout, the covariance ξ \xi ξ β \beta β N → ∞ N \to\infty N → ∞ § 4.1.1 .
The condensed phase
Our first result concerns the dynamics started from the uniform law in the condensed phase.
Theorem 1.2 (Slow equilibration from the uniform law). Let ξ \xi ξ d ≥ 4 d \ge4 d ≥ 4 β > 0 \beta> 0 β > 0 ( ξ , β ) (\xi,\beta) ( ξ , β ) Q Q Q γ = γ ( ξ , β ) > 0 \gamma= \gamma(\xi,\beta) > 0 γ = γ ( ξ , β ) > 0
inf 0 ≤ t ≤ e γ N ∥ ν N P t − π ∥ T V ⟶ 1 in probability as N → ∞ . (5) \inf_{0 \le t \le e^{\gamma N}} \left\|\nu_N P_t-\pi\right\|_{\mathrm{TV}} \longrightarrow1 \qquad\text{in probability as } N \to\infty. \tag*{(5)} 0 ≤ t ≤ e γ N inf ∥ ν N P t − π ∥ TV ⟶ 1 in probability as N → ∞. (5) We make no assumption on the number of degrees with c p > 0 c_p > 0 c p > 0 Lemma 3.8 ), the infimum in (1.5) is the distance at the final time e γ N e^{\gamma N} e γ N ε > 0 \varepsilon> 0 ε > 0 N N N 1 − ε 1-\varepsilon 1 − ε B B B π ( B ) ≥ 1 − ε \pi(B) \ge1-\varepsilon π ( B ) ≥ 1 − ε ( ν N P t ) ( B ) ≤ ε (\nu_N P_t)(B) \le\varepsilon ( ν N P t ) ( B ) ≤ ε t ≤ e γ N t \le e^{\gamma N} t ≤ e γ N B B B B N , M B_{N,M} B N , M u u u H Q ( u ) = H N ( Q u ) H_Q(u)=H_N(\sqrt{Q}u) H Q ( u ) = H N ( Q u ) H Q ( u ) ∈ [ m N − M , m N + M ] H_Q(u) \in[m_N-M,m_N+M] H Q ( u ) ∈ [ m N − M , m N + M ] K Q ( u ) / N K_Q(u)/N K Q ( u ) / N K Q ( u ) K_Q(u) K Q ( u ) t ↦ H N ( t Q u ) t \mapsto H_N(t\sqrt{Q}u) t ↦ H N ( t Q u ) t = 1 t=1 t = 1 m N m_N m N M M M ε \varepsilon ε H Q H_Q H Q § 2.10 ).
The 1RSB assumptions can be checked in many cases. The following corollary lists the classes for which we verify them.
Corollary 1.3 (Strictly 1RSB models). The conclusion of Theorem 1.2 holds in each of the following cases.
(a) The pure model ξ ( r ) = r p \xi(r)=r^p ξ ( r ) = r p for every p ≥ 4 p \ge4 p ≥ 4 and every β > β s ( p ) \beta> \beta_s(p) β > β s ( p )
(b) Every finite mixture ξ \xi ξ with minimum degree at least four such that r ↦ ξ ′ ′ ( r ) − 1 / 2 r \mapsto\xi''(r)^{-1/2} r ↦ ξ ′′ ( r ) − 1/2 is strictly convex on ( 0 , 1 ] (0,1] ( 0 , 1 ] and every β > β s ( ξ ) \beta> \beta_s(\xi) β > β s ( ξ ) This holds in particular if the smallest and the largest degrees d d d and P P P with $c_p > 0 satisfy 6 ( d − 2 ) ( P − 2 ) > ( d + P − 5 ) 2 6(d-2)(P-2)>(d+P-5)^2 6 ( d − 2 ) ( P − 2 ) > ( d + P − 5 ) 2 for example, if all degrees with $c_p > 0 lie in { 4 , 5 , … , 11 } \{4,5,\ldots,11\} { 4 , 5 , … , 11 }
(c) The covariances ξ ( r ) = r d + ε ∑ d < p ≤ P , p e v e n b p r p \xi(r)=r^d+\varepsilon\sum_{d<p\le P,\ p\ \mathrm{even}}b_pr^p ξ ( r ) = r d + ε ∑ d < p ≤ P , p even b p r p where d ≥ 4 ∗ i s e v e n , ∗ P > d , β > β s ( d ) , ∗ t h e n u m b e r s ∗ d \ge4 *is even,* P>d, \beta>\beta_s(d), *the numbers* d ≥ 4 ∗ i se v e n , ∗ P > d , β > β s ( d ) , ∗ t h e n u mb er s ∗ satisfy ∑ p b p = 1 , ∗ a n d ∗ \sum_p b_p=1, *and* ∑ p b p = 1 , ∗ an d ∗ , ∗ f o r s o m e ∗ , *for some* , ∗ f or so m e ∗ that does not depend on ( b p ) (b_p) ( b p )
(d) The covariance ξ ( r ) = r 4 + b r 12 \xi(r)=r^4+br^{12} ξ ( r ) = r 4 + b r 12 β \beta β
t = 45 − 24 log 6 72 log 6 − 85 , b = t ( 10 9 ) 8 , β 2 = 15000 729 ( 4 + 12 t ) . t=\frac{45-24\log6}{72\log6-85},\qquad b=t\left(\frac{10}{9}\right)^8,\qquad\beta^2=\frac{15000}{729(4+12t)}. t = 72 log 6 − 85 45 − 24 log 6 , b = t ( 9 10 ) 8 , β 2 = 729 ( 4 + 12 t ) 15000 . Numerically, b ≈ 0.10546 b\approx0.10546 b ≈ 0.10546 β ≈ 2.12778 \beta\approx2.12778 β ≈ 2.12778 Q = 9 / 10 Q=9/10 Q = 9/10 x = 5 / 9 x=5/9 x = 5/9 ξ ′ ′ − 1 / 2 \xi^{\prime\prime-1/2} ξ ′′ − 1/2 ( 0 , 1 ] (0,1] ( 0 , 1 ] Proposition 5.24 ). In case (b), the degrees may have either parity, as in (a). The pure cubic model and models with a quadratic or cubic component are not covered; we return to this point in § 2.3 .
Next, we transfer Theorem 1.2 to other initial laws, with the same exponent γ \gamma γ
Theorem 1.4 (Other initial laws). Let ξ \xi ξ β \beta β Theorem 1.2 , let γ \gamma γ T N = e γ N T_N=e^{\gamma N} T N = e γ N
(i) For every N N N μ \mu μ S N S_N S N T ≥ 0 T\ge0 T ≥ 0
E [ 1 − ∥ μ P T − π ∥ T V ] ≤ E [ 1 − ∥ ν N P T − π ∥ T V ] . (6) \mathbb{E}\left[1-\left\|\mu P_T-\pi\right\|_{\mathrm{TV}}\right]\le\mathbb{E}\left[1-\left\|\nu_NP_T-\pi\right\|_{\mathrm{TV}}\right]. \tag*{(6)} E [ 1 − ∥ μ P T − π ∥ TV ] ≤ E [ 1 − ∥ ν N P T − π ∥ TV ] . (6) In particular, for every ε > 0 \varepsilon>0 ε > 0
sup μ P ( inf 0 ≤ t ≤ T N ∥ μ P t − π ∥ T V < 1 − ε ) ⟶ 0 , \sup_{\mu}\mathbb{P}\left(\inf_{0\le t\le T_N}\left\|\mu P_t-\pi\right\|_{\mathrm{TV}}<1-\varepsilon\right)\longrightarrow0, μ sup P ( 0 ≤ t ≤ T N inf ∥ μ P t − π ∥ TV < 1 − ε ) ⟶ 0 , where the supremum is over all Borel probability measures μ \mu μ S N S_N S N H N H_N H N
(ii) Let 0 ≤ κ < ( d − 2 ) / 2 0\le\kappa<(d-2)/2 0 ≤ κ < ( d − 2 ) /2 C 0 > 0 C_0>0 C 0 > 0 N N N μ N , 1 , … , μ N , K N \mu_{N,1},\ldots,\mu_{N,K_N} μ N , 1 , … , μ N , K N S N S_N S N H N H_N H N K N ≤ C 0 N κ K_N\le C_0N^\kappa K N ≤ C 0 N κ
inf { ∥ ( ∑ j = 1 K N w j μ N , j ) P t − π ∥ T V : w j ≥ 0 , ∑ j = 1 K N w j = 1 , 0 ≤ t ≤ T N } ⟶ 1 \inf\left\{\left\|\left(\sum_{j=1}^{K_N}w_j\mu_{N,j}\right)P_t-\pi\right\|_{\mathrm{TV}}:w_j\ge0,\ \sum_{j=1}^{K_N}w_j=1,\ 0\le t\le T_N\right\}\longrightarrow1 inf ⎩ ⎨ ⎧ ( j = 1 ∑ K N w j μ N , j ) P t − π TV : w j ≥ 0 , j = 1 ∑ K N w j = 1 , 0 ≤ t ≤ T N ⎭ ⎬ ⎫ ⟶ 1 in probability. If the collection is random and independent of H N H_N H N
(iii) Let H ↦ μ N H H\mapsto\mu_N^H H ↦ μ N H S N S_N S N D ( μ N H ∥ ν N ) / log N ⟶ 0 D(\mu_N^H\|\nu_N)/\log N\longrightarrow0 D ( μ N H ∥ ν N ) / log N ⟶ 0
inf 0 ≤ t ≤ T N ∥ μ N H P t − π ∥ T V ⟶ 1 in probability . \inf_{0\le t\le T_N}\left\|\mu_N^H P_t-\pi\right\|_{\mathrm{TV}}\longrightarrow1\qquad\text{in probability}. 0 ≤ t ≤ T N inf μ N H P t − π TV ⟶ 1 in probability . In part (i), we may take any deterministic sequence of initial laws, for instance point masses μ N = δ x N \mu_N=\delta_{x_N} μ N = δ x N
μ = π \mu=\pi μ = π π \pi π N κ N^\kappa N κ κ log N \kappa\log N κ log N ν N \nu_N ν N o ( log N ) o(\log N) o ( log N ) π \pi π ν N \nu_N ν N N N N
Below the static threshold
We now turn to the interval β d < β < β s \beta_d<\beta<\beta_s β d < β < β s q ∗ q_* q ∗ (2) . The centered spin Z N ( x ) = U ( x ) − π ( U ) Z_N(x)=U(x)-\pi(U) Z N ( x ) = U ( x ) − π ( U ) R N \mathbb{R}^N R N S N S_N S N
Theorem 1.5 (Slow relaxation of the equilibrium dynamics). Let ξ \xi ξ β d ( ξ ) < β s ( ξ ) \beta_d(\xi)<\beta_s(\xi) β d ( ξ ) < β s ( ξ ) β ∈ ( β d , β s ) \beta\in(\beta_d,\beta_s) β ∈ ( β d , β s ) q ∗ = sup { r ∈ ( 0 , 1 ) : β 2 ( 1 − r ) ξ ′ ( r ) > r } q_*=\sup\{r\in(0,1):\beta^2(1-r)\xi'(r)>r\} q ∗ = sup { r ∈ ( 0 , 1 ) : β 2 ( 1 − r ) ξ ′ ( r ) > r } 0 < b < q ∗ 2 0<b<q_*^2 0 < b < q ∗ 2 κ , γ , a , c , C > 0 \kappa,\gamma,a,c,C>0 κ , γ , a , c , C > 0 ξ \xi ξ β \beta β b b b N N N 1 − C e − a N 1-Ce^{-aN} 1 − C e − a N
(i) λ ⌊ e κ N ⌋ ≤ e − 2 γ N \lambda_{\lfloor e^{\kappa N}\rfloor}\le e^{-2\gamma N} λ ⌊ e κ N ⌋ ≤ e − 2 γ N
(ii) With Z N = U − π ( U ) Z_N=U-\pi(U) Z N = U − π ( U ) Π N = 1 ( 0 , e − 2 γ N ] ( − L ) \Pi_N=\mathbf{1}_{(0,e^{-2\gamma N}]}(-L) Π N = 1 ( 0 , e − 2 γ N ] ( − L )
∥ Π N Z N ∥ L 2 ( π ; R N ) 2 ≥ b . \|\Pi_N Z_N\|_{L^2(\pi;\mathbb{R}^N)}^2\ge b. ∥ Π N Z N ∥ L 2 ( π ; R N ) 2 ≥ b . (iii) For the stationary diffusion, with X 0 ∼ π X_0\sim\pi X 0 ∼ π E π R ( X 0 , X t ) − ∣ π ( U ) ∣ 2 ≥ b \mathbb{E}_\pi R(X_0,X_t)-|\pi(U)|^2\ge b E π R ( X 0 , X t ) − ∣ π ( U ) ∣ 2 ≥ b 0 ≤ t ≤ e γ N 0\le t\le e^{\gamma N} 0 ≤ t ≤ e γ N
(iv) There exists a Borel set X N ⊂ S N \mathcal{X}_N\subset S_N X N ⊂ S N π ( X N ) ≥ 1 − C e − a N \pi(\mathcal{X}_N)\ge1-Ce^{-aN} π ( X N ) ≥ 1 − C e − a N ∥ P t ( x , ⋅ ) − π ∥ T V ≥ 1 − C e − a N \|P_t(x,\cdot)-\pi\|_{\mathrm{TV}}\ge1-Ce^{-aN} ∥ P t ( x , ⋅ ) − π ∥ TV ≥ 1 − C e − a N x ∈ X N x\in\mathcal{X}_N x ∈ X N 0 ≤ t ≤ e γ N 0\le t\le e^{\gamma N} 0 ≤ t ≤ e γ N
(v) For the stationary diffusion in the given field, the mutual information satisfies I ( X 0 ; X t ) ≥ c N I(X_0;X_t)\ge cN I ( X 0 ; X t ) ≥ c N 0 < t ≤ e γ N 0<t\le e^{\gamma N} 0 < t ≤ e γ N
We allow a quadratic term and odd degrees, and we make no assumption on the shape of A A A q ∗ q_* q ∗ (3) , and A A A π \pi π
Conclusion (i) asserts that exponentially many eigenvalues are exponentially small, and (ii) shows that a fixed part of the centered spin lies in the span of the corresponding eigenfunctions, which we call slow modes. Since π \pi π E π R ( X 0 , X t ) − ∣ π ( U ) ∣ 2 = ⟨ Z N , P t Z N ⟩ L 2 ( π ; R N ) \mathbb{E}_\pi R(X_0,X_t)-|\pi(U)|^2=\langle Z_N,P_tZ_N\rangle_{L^2(\pi;\mathbb{R}^N)} E π R ( X 0 , X t ) − ∣ π ( U ) ∣ 2 = ⟨ Z N , P t Z N ⟩ L 2 ( π ; R N ) t t t b b b e γ N e^{\gamma N} e γ N e − e γ N e^{-e^{\gamma N}} e − e γ N x x x π \pi π e γ N e^{\gamma N} e γ N π \pi π X t X_t X t c N cN c N Lemma 3.12 , I ( X 0 ; X t ) = ∫ D ( P t ( x , ⋅ ) ∥ π ) π ( d x ) I(X_0;X_t)=\int D(P_t(x,\cdot)\mathbin\Vert\pi)\,\pi(dx) I ( X 0 ; X t ) = ∫ D ( P t ( x , ⋅ ) ∥ π ) π ( d x )
The plateau level in (ii) and (iii) is q ∗ 2 q_*^2 q ∗ 2 (3) , which is q ∗ q_* q ∗ (3) . In the next theorem, we reach every level below q ∗ q_* q ∗
Theorem 1.6 (Correlation bounds at every level below q ∗ q_* q ∗ ξ \xi ξ β \beta β q ∗ q_* q ∗ Theorem 1.5 . For every 0 < b < q ∗ 0<b<q_* 0 < b < q ∗ κ , γ , a , c , C > 0 \kappa,\gamma,a,c,C>0 κ , γ , a , c , C > 0 ξ \xi ξ β \beta β b b b Theorem 1.5 hold with probability tending to one, and conclusions (i), (iv), and (v) hold for all sufficiently large N N N 1 − C e − a N 1-Ce^{-aN} 1 − C e − a N
We do not prove a matching upper bound: the theorem does not show that the correlation stays close to q ∗ q_* q ∗ q ∗ q_* q ∗
Our last main result describes the slow modes as β \beta β β d \beta_d β d ξ ( r ) = ∑ p ≥ 2 c p r p \xi(r)=\sum_{p\geq2}c_pr^p ξ ( r ) = ∑ p ≥ 2 c p r p c p ≥ 0 c_p\geq0 c p ≥ 0
∑ p ≥ 2 p 5 / 2 c p < ∞ . (7) \sum_{p\geq2}p^{5/2}\sqrt{c_p}<\infty. \tag*{(7)} p ≥ 2 ∑ p 5/2 c p < ∞. (7) For such ξ \xi ξ H N H_N H N A A A A A A [ 0 , 1 ] [0,1] [ 0 , 1 ] A ( 0 ) = ξ ′ ′ ( 0 ) A(0)=\xi''(0) A ( 0 ) = ξ ′′ ( 0 ) A ( 1 ) = 0 A(1)=0 A ( 1 ) = 0
Theorem 1.7 (Small eigenvalues near the dynamical threshold). Let ξ \xi ξ β d ( ξ ) < β s ( ξ ) \beta_d(\xi)<\beta_s(\xi) β d ( ξ ) < β s ( ξ ) A max = sup ( 0 , 1 ) A = β d − 2 A_{\max}=\sup_{(0,1)}A=\beta_d^{-2} A m a x = sup ( 0 , 1 ) A = β d − 2 A A A [ 0 , 1 ] [0,1] [ 0 , 1 ] ( 0 , 1 ) (0,1) ( 0 , 1 ) q d q_d q d m ≥ 1 m\geq1 m ≥ 1 a > 0 a>0 a > 0
A ( q d + y ) = A max [ 1 − a y 2 m + O ( y 2 m + 1 ) ] ( y → 0 ) . A(q_d+y)=A_{\max}\left[1-ay^{2m}+O\left(y^{2m+1}\right)\right]\qquad(y\to0). A ( q d + y ) = A m a x [ 1 − a y 2 m + O ( y 2 m + 1 ) ] ( y → 0 ) . For β > β d \beta>\beta_d β > β d ε = β 2 A max − 1 \varepsilon=\beta^2A_{\max}-1 ε = β 2 A m a x − 1
C d = ( 2 m q d ( 2 m + 1 ) ( 1 − q d ) ) − 1 / ( 2 m ) a − 1 / ( 2 m ) . C_d=\left(\frac{2m q_d}{(2m+1)(1-q_d)}\right)^{-1/(2m)}a^{-1/(2m)}. C d = ( ( 2 m + 1 ) ( 1 − q d ) 2 m q d ) − 1/ ( 2 m ) a − 1/ ( 2 m ) . There exists κ ∗ > 0 \kappa_*>0 κ ∗ > 0 ϑ ∈ ( 0 , 1 ) \vartheta\in(0,1) ϑ ∈ ( 0 , 1 ) ε 0 ( ϑ ) > 0 \varepsilon_0(\vartheta)>0 ε 0 ( ϑ ) > 0 ε ∈ ( 0 , ε 0 ( ϑ ) ) \varepsilon\in(0,\varepsilon_0(\vartheta)) ε ∈ ( 0 , ε 0 ( ϑ )) a ′ , C > 0 a',C>0 a ′ , C > 0 N N N 1 − C e − a ′ N 1-Ce^{-a'N} 1 − C e − a ′ N
λ ⌈ e κ ∗ N ⌉ ≤ exp [ − ( 1 − ϑ ) C d ε 1 + 1 / ( 2 m ) N ] . \lambda_{\lceil e^{\kappa_*N}\rceil}\leq\exp\left[-(1-\vartheta)C_d\varepsilon^{1+1/(2m)}N\right]. λ ⌈ e κ ∗ N ⌉ ≤ exp [ − ( 1 − ϑ ) C d ε 1 + 1/ ( 2 m ) N ] . The number of slow modes stays exponential in N N N κ ∗ \kappa_* κ ∗ ε \varepsilon ε ε \varepsilon ε q d q_d q d 2 m 2m 2 m ξ \xi ξ m = 1 m=1 m = 1 3 / 2 3/2 3/2 C d ε 1 + 1 / ( 2 m ) C_d\varepsilon^{1+1/(2m)} C d ε 1 + 1/ ( 2 m ) § 2.6 ). For the pure cubic model, q d = 1 / 2 q_d=1/2 q d = 1/2 A max = 3 / 4 A_{\max}=3/4 A m a x = 3/4 m = 1 m=1 m = 1 C d = 1 / 3 C_d=1/3 C d = 1/3 a ′ a' a ′ C C C N N N ε \varepsilon ε ϑ \vartheta ϑ
Scope
We record here what the theorems do not assert. The convergence in Theorem 1.2 is in probability; we do not show that the probability of failure is exponentially small. The covariance and the temperature are fixed before N → ∞ N\to\infty N → ∞ γ \gamma γ Theorem 1.2 is uniform near β s \beta_s β s Theorem 1.7 . The proof of Theorem 1.2 requires minimum degree at least four: for a component of degree two or three, an error term in our hitting estimate for the caps around critical points does not tend to zero (Remark 13.2). In particular, the pure cubic model is not covered in the condensed phase, although it is covered by Theorems 1.5 to 1.7 . In Theorem 1.4 (ii), the endpoint κ = ( d − 2 ) / 2 \kappa=(d-2)/2 κ = ( d − 2 ) /2 β d < β ≤ β s \beta_d<\beta\le\beta_s β d < β ≤ β s § 2.7 . In Theorem 1.6 , the probability in (ii) and (iii) tends to one without a rate. Theorem 1.7 bounds eigenvalues only. Finally, the exponent κ \kappa κ Theorem 1.5 (i) is a lower bound for the growth rate of the number of slow modes; we do not identify it with the complexity of metastable states.
Background from physics
The distinction between a temperature of dynamical arrest and a lower static, or Kauzmann, temperature goes back to Kirkpatrick and Thirumalai, who connected the dynamical equations of p p p ? ]. Kirkpatrick, Thirumalai, and Wolynes developed the corresponding activated picture for liquids in finite dimension [? ]. For the spherical p p p ? ], and Crisanti, Horner, and Sommers analyzed the dynamics and found a dynamical transition above the static one [54]. In our notation, these transitions occur at β d \beta_{\mathrm d} β d β s \beta_{\mathrm s} β s
Between the two transitions, the correlation of the equilibrium dynamics, computed in the limit N → ∞ N \to\infty N → ∞ T s T_{\mathrm s} T s Q Q Q β 2 ξ ′ ( Q ) = Q / ( ( 1 − Q ) ( 1 − Q + x Q ) ) \beta^2 \xi'(Q) = Q/((1-Q)(1-Q+xQ)) β 2 ξ ′ ( Q ) = Q / (( 1 − Q ) ( 1 − Q + x Q )) q ∗ q_* q ∗ β 2 A \beta^2 A β 2 A q ∗ q_* q ∗ q ∗ q_* q ∗ β 2 ( 1 − q ) 2 ξ ′ ′ ( q ) = 1 \beta^2(1-q)^2\xi''(q)=1 β 2 ( 1 − q ) 2 ξ ′′ ( q ) = 1 q d q_{\mathrm d} q d A A A β = β d \beta=\beta_{\mathrm d} β = β d q = q d q=q_{\mathrm d} q = q d β > β d \beta>\beta_{\mathrm d} β > β d
Cugliandolo and Kurchan solved the dynamics started from a configuration chosen independently of the disorder, in the limit where N → ∞ N \to\infty N → ∞ T d T_{\mathrm d} T d 2 ( p − 1 ) / p 2\sqrt{(p-1)/p} 2 ( p − 1 ) / p ξ ( r ) = r p \xi(r)=r^p ξ ( r ) = r p 3 + 4 3+4 3 + 4
These works concern the dynamical mean-field equations and numerical evidence, rather than finite-N N N ( p + s ) (p+s) ( p + s )
Franz and Parisi introduced the free energy of configurations constrained to have a prescribed overlap with a reference configuration drawn from equilibrium [73]; see also [74]. In the replica-symmetric phase, their potential is ( F ( β ) − F F P ( r ) ) / β (F(\beta)-F_{\mathrm{FP}}(r))/\beta ( F ( β ) − F FP ( r )) / β F F P F_{\mathrm{FP}} F FP F F P F_{\mathrm{FP}} F FP s + p s+p s + p
The clustering and condensation transitions were formulated for random constraint satisfaction problems by Krzakala, Montanari, Ricci-Tersenghi, Semerjian, and Zdeborová [106]. Mézard and Montanari related the dynamical transition to reconstruction on trees [117], and Montanari and Semerjian proved inequalities between point-to-set correlation lengths and relaxation times, with exponential lower bounds on relaxation times for sparse models [120]. Biroli and Kurchan defined metastable states and their configurational entropy through the low-lying spectrum of the Fokker–Planck operator [33], and Tănase-Nicola and Kurchan made this construction explicit [148]; related ideas appear in the work of Gaveau and Schulman [79]. Biroli derived dynamical TAP equations [32]. Lopatin and Ioffe used dynamical instantons to study barriers between metastable states in the spherical p p p T d T_d T d ∣ T − T d ∣ 3 / 2 |T-T_d|^{3/2} ∣ T − T d ∣ 3/2
We now compare our results with these predictions. Theorem 1.2 is consistent with the aging picture of [56], in which the dynamics from a disorder-independent start does not reach the states that carry the Gibbs measure. It is a statement of a different kind: it holds in finite dimension up to times e γ N e^{\gamma N} e γ N Theorems 1.5 and 1.6 are lower bounds for the stationary correlation at levels below q ∗ q_* q ∗ q ∗ q_* q ∗ κ \kappa κ N N N § 2.7 ). For m = 1 m=1 m = 1 3 / 2 3/2 3/2 Theorem 1.7 matches the barrier exponent of the cubic normal form [72], and 1 + 1 / ( 2 m ) 1+1/(2m) 1 + 1/ ( 2 m ) 2 m 2m 2 m β ↓ β d \beta\downarrow\beta_d β ↓ β d T d T_d T d
Mathematical background
Statics and the geometry of the Gibbs measure
The free energy of spherical models was computed by Crisanti and Sommers [52]. Talagrand proved their formula for even mixtures [146], Chen extended it to general mixtures [46], and Huang and Sellke gave a constructive proof of the lower bound [90]. The ground state energy was computed by Chen and Sen [48] and by Jagannath and Tobasco [96], and properties of Parisi measures and contact sets of spherical models were studied in [9, 10, 97]. The TAP approach was introduced by Thouless, Anderson, and Palmer [149], and it was applied to spherical pure models by Kurchan, Parisi, and Virasoro [108] and by Crisanti and Sommers [53]. Subag developed a generalized TAP approach for spherical models [143]. He introduced multisamplable overlaps, values q q q q q q
The expected number of critical points of spherical spin glasses was computed with the Kac–Rice formula by Auffinger, Ben Arous, and Černý for pure models [12] and by Auffinger and Ben Arous for mixtures [8]. For pure models, Subag proved that the number of critical points at low energies concentrates, by a second moment computation [140], and Subag and Zeitouni identified the extremal process of critical values [144]. In [141], Subag showed that for pure models at sufficiently low temperature, the Gibbs measure concentrates on thin bands around the deepest critical points. Ben Arous, Subag, and Zeitouni obtained a similar description for mixtures close to pure models at low temperature [30]. Dembo and Subag proved a decomposition of this kind for generic mixtures that are strictly k k k β > β s \beta> \beta_{\mathrm{s}} β > β s k = 1 k = 1 k = 1 H N H_N H N q N \sqrt{qN} q N q q q o ( N ) o(N) o ( N )
Dynamics
The rigorous study of Langevin dynamics for spin glasses was initiated by Ben Arous and Guionnet, who proved large deviations for the empirical measure of paths and derived self-consistent limiting equations for soft spins [23, 24]. Ben Arous, Dembo, and Guionnet proved aging for the spherical 2-spin model [27] and derived the Cugliandolo–Kurchan equations for mixed p p p N → ∞ N \to\infty N → ∞ β d < β < β s \beta_{\mathrm{d}} < \beta< \beta_{\mathrm{s}} β d < β < β s N → ∞ N \to\infty N → ∞ N N N
In finite dimension, Ben Arous, Gheissari, and Jagannath proved differential inequalities for the energy and the gradient along the dynamics, uniformly over initial points [29]. Sellke studied pure models at sufficiently low temperature. From every initial point, the energy of the dynamics rises above the threshold energy minus any fixed η > 0 \eta> 0 η > 0 δ ( p , β 0 ) \delta(p, \beta_0) δ ( p , β 0 ) β ≤ β 0 \beta\le\beta_0 β ≤ β 0
Exponentially slow relaxation has been studied through the spectral gap. Gheissari and Jagannath proved for pure models with p ≥ 3 p \ge3 p ≥ 3 p ≥ 4 p \ge4 p ≥ 4 p ≥ 4 p \ge4 p ≥ 4 T B B M T_{\mathrm{BBM}} T BBM p ≥ 4 p \ge4 p ≥ 4 β ≤ β d ( 3 ) \beta\le\beta_{\mathrm{d}}(3) β ≤ β d ( 3 ) p ≥ 4 p \ge4 p ≥ 4 § 2.7 . Auffinger, El Alaoui, and Sellke studied mixtures with a discontinuous replica symmetry breaking transition and with β s < ζ ′ ′ ( 0 ) − 1 / 2 \beta_{\mathrm{s}} < \zeta^{\prime\prime}(0)^{-1/2} β s < ζ ′′ ( 0 ) − 1/2 β s \beta_s β s β \beta β 1 − ε 1-\varepsilon 1 − ε e c N e^{cN} e c N 1 − e − c N 1-e^{-cN} 1 − e − c N β → ∞ \beta\to\infty β → ∞ N N N N → ∞ N \to\infty N → ∞ β \beta β β \beta β β \beta β x ↦ − x x \mapsto-x x ↦ − x § 2.2 . For Ising models, exponentially slow worst-case mixing at low temperature was proved by Sellke for the Sherrington–Kirkpatrick model [137] and by Kouraich and Warzel for p p p
At high temperature, sampling and fast mixing are now well understood. Huang, Montanari, and Pham gave a polynomial-time algorithm, based on stochastic localization, that samples from the Gibbs measure with vanishing error in total variation when β 2 ζ ′ ′ ( q ) < ( 1 − q ) − 2 \beta^2 \zeta''(q) < (1-q)^{-2} β 2 ζ ′′ ( q ) < ( 1 − q ) − 2 q ∈ [ 0 , 1 ) q \in[0,1) q ∈ [ 0 , 1 ) [93 ] . Huang, Mohanty, Rajaraman, and Wu proved that simulated annealing with Langevin dynamics samples from the Gibbs measure up to the same stochastic localization threshold [94]. Their algorithm starts from the uniform law and runs through a sequence of temperatures, and weak Poincaré inequalities control its warm starts. In a different regime, Huang and Sellke proved mixing in logarithmic time at sufficiently low temperature for strongly topologically trivial landscapes with a unique local maximum, which include suitable external fields [92]. For Ising spins, spectral gaps and fast mixing at high temperature were proved in [2, 5, 21, 67].
The relation between metastable sets and small eigenvalues of reversible dynamics has a long history. It goes back to the exit and cycle theory of Freidlin and Wentzell [75] and to Witten’s spectral approach to Morse theory [153] and the semiclassical analysis of multiwell operators [87]. It was developed in the potential-theoretic approach of Bovier, Eckhoff, Gayrard, and Klein [35–39] and in the work of Helffer, Klein, and Nier [88]; see the monograph [34]. The works on reversible diffusions and the semiclassical works give sharp asymptotics as a noise parameter tends to zero, while the works on Markov chains and disordered mean-field models also treat growing systems. Our results concern spherical spin glasses in high dimension, with exponentially many slow modes.
Comparison with the closest results
We begin with the dynamics started from the uniform law. For pure spherical p p p t t t max ∥ f ∥ ∞ ≤ 1 ∫ ( P t f − π ( f ) ) 2 d ν N ≤ 1 / e \max_{\lVert f\rVert_\infty\le1}\int(P_t f-\pi(f))^2\,d\nu_N\le1/e max ∥ f ∥ ∞ ≤ 1 ∫ ( P t f − π ( f ) ) 2 d ν N ≤ 1/ e f = 21 A − 1 f=2\mathbf{1}_A-1 f = 2 1 A − 1 4 ∥ ν N P t − π ∥ T V 2 4\lVert\nu_NP_t-\pi\rVert_{\mathrm{TV}}^2 4 ∥ ν N P t − π ∥ TV 2 e γ N e^{\gamma N} e γ N p ≥ 4 p\ge4 p ≥ 4 β d < β ≤ β s \beta_d<\beta\le\beta_s β d < β ≤ β s p = 3 p=3 p = 3 N N N p ≥ 4 p\ge4 p ≥ 4 β > β s \beta>\beta_s β > β s
The closest antecedent of Theorem 1.5 is the work of El Alaoui, Montanari, and Sellke [65]. For pure models, they proved that with probability at least 1 − e − c N 1-e^{-cN} 1 − e − c N p p p β \beta β ( C , β s ( p ) ) (C,\beta_{\mathrm{s}}(p)) ( C , β s ( p )) C C C p ≥ 3 p \ge3 p ≥ 3 0.9999 0.9999 0.9999 e c N e^{cN} e c N − L -L − L e − c N e^{-cN} e − c N 1 − e − c N 1-e^{-cN} 1 − e − c N inf 0 ≤ t ≤ e c N R ( X 0 , X t ) ≥ 1 − Δ 2 \inf_{0\le t\le e^{cN}} R(X_0,X_t) \ge1-\Delta^2 inf 0 ≤ t ≤ e c N R ( X 0 , X t ) ≥ 1 − Δ 2 Δ \Delta Δ 1 − 2 Δ 2 1-2\Delta^2 1 − 2 Δ 2 β \beta β Theorem 1.5 extends them in three ways. First, it applies to every finite mixture, including a quadratic term and odd degrees, and to every β ∈ ( β d , β s ) \beta\in(\beta_{\mathrm{d}},\beta_{\mathrm{s}}) β ∈ ( β d , β s ) q ∗ 2 q_*^2 q ∗ 2 q ∗ 2 q_*^2 q ∗ 2 1 − C e − a N 1-Ce^{-aN} 1 − C e − a N q ∗ q_* q ∗ Theorem 1.6 ). Third, it adds the conclusions on Gibbs-typical starting points and on stationary information. For finite and infinite mixtures, Theorem 1.7 gives eigenvalue bounds close to β d \beta_{\mathrm{d}} β d N N N § 2.10 ).
In the limit N → ∞ N \to\infty N → ∞ β d < β < β s \beta_{\mathrm{d}} < \beta< \beta_{\mathrm{s}} β d < β < β s Theorems 1.5 and 1.6 hold in finite dimension up to exponential times, but they are only lower bounds. In their regimes, [13, 25] show that one eigenvalue is exponentially small; in Theorem 1.5 , we obtain exponentially many for every β ∈ ( β d , β s ) \beta\in(\beta_{\mathrm{d}},\beta_{\mathrm{s}}) β ∈ ( β d , β s )
Ideas of the proofs
A detailed overview of the proofs is given in Section 2. Here we describe the main steps and indicate where the tools come from.
The condensed phase. Let ( ξ , β ) (\xi,\beta) ( ξ , β ) x x x Q Q Q Q Q Q Theorem 1.2 rests on a description of the Gibbs measure that goes back to the work of Subag on pure models at low temperature [141].
Consider the center field H Q ( u ) = H N ( Q u ) H_Q(u)=H_N(\sqrt{Q}u) H Q ( u ) = H N ( Q u ) S N S_N S N m N = N E 0 + O ( log N ) m_N=NE_0+O(\log N) m N = N E 0 + O ( log N ) K Q ( u ) / N K_Q(u)/N K Q ( u ) / N K 0 K_0 K 0 H Q H_Q H Q { x ∈ S N : ∣ R ( x , u ) − Q ∣ ≤ b } \{x\in S_N:|R(x,u)-\sqrt{Q}|\le b\} { x ∈ S N : ∣ R ( x , u ) − Q ∣ ≤ b } u u u ν N \nu_N ν N N − 1 / 2 N^{-1/2} N − 1/2 e γ N e^{\gamma N} e γ N { x : ∣ R ( x , u ) ∣ ≥ a } \{x:|R(x,u)|\ge a\} { x : ∣ R ( x , u ) ∣ ≥ a } a < Q − b a<\sqrt{Q}-b a < Q − b Theorem 1.2 , using the contraction of total variation and a first moment computation with the Kac–Rice formula. We choose the exponent γ \gamma γ
Statics (Section 5). We identify the Parisi measure under the 1RSB assumptions through the Crisanti–Sommers optimality criterion [146, Proposition 2.1], [90, Lemma 1.2], and the stationarity equations for two atoms are the classical one-step equations of Crisanti and Sommers [52]. When ξ ′ ′ − 1 / 2 \xi^{\prime\prime-1/2} ξ ′′ − 1/2 β > β s \beta>\beta_s β > β s β \beta β Theorem 5.7 ). Subag proved that every zero is multisamplable, and the converse for generic mixtures [143, Theorem 10]. Under the 1RSB assumptions, the multisamplable overlaps are 0 0 0 Q Q Q H Q H_Q H Q E 0 E_0 E 0 H Q H_Q H Q m N m_N m N K 0 K_0 K 0
The directing measure (Section 6). Every limit of the overlap array of independent Gibbs samples is generated by a random measure on the unit ball of a Hilbert space, the directing measure [63, 123]. We show that it is carried by vectors of squared norm Q Q Q Theorem 6.3 ). We exclude other positive squared norms with the characterization of multisamplable overlaps, and an atom at the origin with a single identity of Ghirlanda–Guerra type, which we derive from the self-averaging argument of Panchenko [124]. This argument uses no ultrametricity and no Poisson–Dirichlet structure.
Counts (§ 6.2 and 9 to 11). We use the Kac–Rice formula [3, 14, 116, 139] with weights that are bounded Borel functions of the whole field, such as hitting probabilities of the diffusion. The other inputs are the law of the field at a point of a smaller sphere [30] and bounds on all moments of the Hessian determinant. For pure models, the bound on the counts follows from a first moment computation and the tightness of the centered maximum [144]. For mixtures, we follow the conditional trial argument of Huang and Sellke [90, Section 3.3] to show that a critical point near the ground state has no other high-energy points at intermediate overlap. We then bound the second moment of the number of critical points in a fixed window, in the spirit of [140]. The proof uses a local factorization of the two-point Kac–Rice integrand and an elementary bound for spherical codes in a cap [18, 129].
Concentration on bands (Sections 9 and 12). Using the estimates of Subag for replicated bands with a fixed number of replicas [143, Lemmas 17–19 and Proposition 24], together with the directing measure, we place a Gibbs sample in a band around a critical point of nearly maximal value. We then pass to critical points in a fixed window, with Laplace estimates for the partition functions of bands and the lower tail of normalized partition functions [93 Lemma 3.2 and Remark 3.3] . For pure models at sufficiently low temperature, the concentration of the Gibbs measure on bands around the deepest critical points was proved by Subag [141, Theorem 1], and for generic mixtures that are strictly one-step replica symmetry breaking, the decomposition of Dembo and Subag places almost all of the Gibbs mass in narrow bands around critical points (see § 2.7 ). We prove that the bands around the critical points in a fixed window carry almost all of the Gibbs mass, for pure models with p ≥ 4 p \ge4 p ≥ 4 β > β s \beta> \beta_s β > β s o ( N ) o(N) o ( N )
The hitting estimate (Section 8). Fix a point u u u H Q H_Q H Q u u u H N H_N H N N N N u u u ∣ R ( x , u ) ∣ d \lvert R(x,u)\rvert^d ∣ R ( x , u ) ∣ d u u u u u u S N S_N S N u u u e γ N e^{\gamma N} e γ N ν N \nu_N ν N N N N Lemma 8.11 ). The resulting error N 1 − d / 2 + N 3 − d N^{1-d/2}+N^{3-d} N 1 − d /2 + N 3 − d d ≥ 4 d \ge4 d ≥ 4
Determinant weights and other initial laws (Sections 4, § 6.2 , and 13). The hitting estimate for the caps under critical-point conditioning concerns ordinary Gaussian conditioning, while the Kac–Rice formula weights each critical point by the absolute determinant of its Hessian. We transfer the estimate with Hölder’s inequality and the moment bounds for the determinant, as in the conditional Cauchy–Schwarz argument of [90, Corollary 3.16]. The transfer turns the bound N − ( d − 2 ) / 2 N^{-(d-2)/2} N − ( d − 2 ) /2 N − ( d − 2 ) ( 1 − 1 / r ) / 2 N^{-(d-2)(1-1/r)/2} N − ( d − 2 ) ( 1 − 1/ r ) /2 r > 1 r>1 r > 1 κ = ( d − 2 ) / 2 \kappa=(d-2)/2 κ = ( d − 2 ) /2 Theorem 1.4 (ii). Part (i) of Theorem 1.4 is an instance of the classical symmetrization over a compact group, the same averaging that underlies the Hunt–Stein theorem in statistics [113, Section 8.5]: the expected distance to equilibrium is convex and rotation invariant as a function of the initial law, and ν N \nu_N ν N ν N \nu_N ν N e γ N e^{\gamma N} e γ N
Below the static threshold. The proof of Theorem 1.5 has a probabilistic part and a deterministic part. In the probabilistic part, we study the constrained pressure F F P ( r ) F_{\mathrm{FP}}(r) F FP ( r ) π \pi π r r r β ≤ β s \beta\le\beta_s β ≤ β s β > β s \beta>\beta_s β > β s F F P F_{\mathrm{FP}} F FP q ∗ q_* q ∗ r ζ ′ ′ ( r ) / ζ ′ ( r ) r\zeta''(r)/\zeta'(r) r ζ ′′ ( r ) / ζ ′ ( r ) x x x { u : q 1 ≤ R ( x , u ) ≤ q 2 } \{u:q_1\le R(x,u)\le q_2\} { u : q 1 ≤ R ( x , u ) ≤ q 2 } q ∗ q_* q ∗ { u : R ( x , u ) > q 2 } \{u:R(x,u)>q_2\} { u : R ( x , u ) > q 2 } [93 Lemma 7.7] .
In the deterministic part, we use one observable, the overlap-weighted Gibbs mean B N B_N B N x ∈ S N x\in S_N x ∈ S N B N ( x ) B_N(x) B N ( x ) U U U u u u R ( x , u ) < q 1 R(x,u)<q_1 R ( x , u ) < q 1 U ⋅ B N ≥ q 1 − o ( 1 ) U \cdot B_N \ge q_1 - o(1) U ⋅ B N ≥ q 1 − o ( 1 ) B N B_N B N q 1 2 q_1^2 q 1 2 q 1 q_1 q 1 q ∗ q_* q ∗ B N B_N B N
For Theorem 1.6, we note that ∣ B N ( x ) ∣ 2 |B_N(x)|^2 ∣ B N ( x ) ∣ 2 q ∗ q_* q ∗ [93 Lemma 3.2] , and this step has no rate. For Theorem 1.7, we expand A A A q d q_d q d
The tools above are classical or published. Our proofs establish or combine the following ingredients: the hitting estimate for the caps under critical-point conditioning and its comparison of resolvents; the concentration of the Gibbs measure on bands around critical points in a fixed window, for pure models with p ≥ 4 p \ge4 p ≥ 4 β > β s \beta> \beta_s β > β s Q Q Q q ∗ q_* q ∗
Organization of the paper
Section 2 gives an overview of the proofs. Part I consists of Section 3, which fixes conventions for constants, for statements that hold with high probability, and for the order in which parameters are chosen (Section 3.1), and collects facts on the model, the dynamics, Markov semigroups, and statics that both halves of the paper use. The two halves are independent of each other; a reader interested in one of them can skip the other.
Part II proves Theorems 1.2 and 1.4 and Corollary 1.3 . Section 4 reduces separation to three properties of critical points and bands, and treats other initial laws. Section 5 derives the static consequences of the 1RSB assumptions and verifies these assumptions for the classes of Corollary 1.3 , with finite computations for example (d) in Section A. Section 6 proves that the directing measure is carried by vectors of squared norm Q Q Q § 6.2 contains the Kac–Rice formula and moment bounds for determinants, and Section 8 proves the conditioned hitting estimate. Section 9 proves bounds on counts and the concentration of the Gibbs measure on bands for pure models, and Sections 10 to 12 do so for mixtures. Section 13 assembles the proofs.
Part III proves Theorems 1.5 to 1.7 . Section 14 studies the constrained pressure, Section 15 contains deterministic estimates for counting slow modes and for the dynamics started from Gibbs-typical points, together with the overlap-weighted Gibbs mean, and Section 16 proves Theorem 1.5 . Section 17 proves Theorem 1.6 , Section 18 develops the infinite-mixture inputs and proves Theorem 1.7 . Notation specific to a section is introduced at its start.
Proof overview
In this section, we explain the main steps of the proofs and the ideas behind them. We do not restate the theorems. For each step, we point to the result where it is proved. In § 3.1 , we compare the two kinds of slowness that the two halves of the paper address. We explain the proofs of Theorems 1.2 and 1.4 in § 3.2 and the proofs of Theorems 1.5 to 1.7 in § 3.10 ; these two parts can be read in either order. § 3.14 lists the dependencies among the main propositions.
Both proofs have the same structure. We first show, by a short deterministic argument, that a few properties of the Gibbs measure, or of the critical points of the field, imply the dynamical conclusion. We then prove these properties in separate sections, with tools from the statics of spherical spin glasses. In the condensed phase, the deterministic argument is the reduction of Section 4, and most of the work goes into its inputs. Below the static threshold, the probabilistic input is a set of estimates on the constrained pressure (Section 14), and most of the new arguments are deterministic (Section 15).
We write ≈ \approx ≈
Two kinds of slowness
We can measure the distance to equilibrium from a worst-case starting point, from the uniform law ν N \nu_N ν N § 2.2 ). The two halves of the paper concern the second and the third of these. We use different mechanisms for them.
We first explain why an exponentially small spectral gap does not by itself slow down the dynamics started from ν N \nu_N ν N f = d ν N / d π = Z N , β e − β H N f = \mathrm{d}\nu_N / \mathrm{d}\pi= Z_{N,\beta}e^{-\beta H_N} f = d ν N / d π = Z N , β e − β H N ν N P t \nu_NP_t ν N P t π \pi π P t f P_tf P t f ( ϕ j ) j ≥ 0 (\phi_j)_{j\geq0} ( ϕ j ) j ≥ 0 − L -L − L L 2 ( π ) L^2(\pi) L 2 ( π ) ϕ 0 = 1 \phi_0 = 1 ϕ 0 = 1 P t f − 1 P_tf - 1 P t f − 1
∥ ν N P t − π ∥ T V = 1 2 ∫ ∣ P t f − 1 ∣ d π ≤ 1 2 ( ∑ j ≥ 1 e − 2 λ j t ⟨ f , ϕ j ⟩ 2 ) 1 / 2 . (8) \lVert\nu_NP_t-\pi\rVert_{\mathrm{TV}} = \frac{1}{2}\int\lvert P_tf-1\rvert\,\mathrm{d}\pi\leq\frac{1}{2}\left(\sum_{j\geq1}e^{-2\lambda_jt}\langle f,\phi_j\rangle^2\right)^{1/2}. \tag*{(8)} ∥ ν N P t − π ∥ TV = 2 1 ∫ ∣ P t f − 1 ∣ d π ≤ 2 1 ( j ≥ 1 ∑ e − 2 λ j t ⟨ f , ϕ j ⟩ 2 ) 1/2 . (8) We see that a small eigenvalue λ j \lambda_j λ j ν N \nu_N ν N ⟨ f , ϕ j ⟩ \langle f,\phi_j\rangle ⟨ f , ϕ j ⟩ (8) . The example of [82] recalled in § 2.2 shows that a small spectral gap can coexist with fast mixing from the uniform law. To bound the distance from below, we need a different argument.
We use the variational form of the total variation distance. Since ∥ μ − π ∥ T V = sup A ( π ( A ) − μ ( A ) ) \|\mu-\pi\|_{\mathrm{TV}}=\sup_A(\pi(A)-\mu(A)) ∥ μ − π ∥ TV = sup A ( π ( A ) − μ ( A )) ν N P t \nu_NP_t ν N P t π \pi π A A A π ( A ) \pi(A) π ( A ) ( ν N P t ) ( A ) (\nu_NP_t)(A) ( ν N P t ) ( A ) ν N \nu_N ν N A A A π ( A ) \pi(A) π ( A ) ( ν N P t ) ( A ) (\nu_NP_t)(A) ( ν N P t ) ( A ) ν N ( A ) \nu_N(A) ν N ( A ) Lemma 3.8 ), it suffices to exhibit such a set at the final time. In the condensed phase, we take for A A A H N H_N H N ν N \nu_N ν N
For Gibbs-typical starting points, the law of X t X_t X t π \pi π X 0 X_0 X 0 X t X_t X t B : S N → R N B:S_N\to\mathbb{R}^N B : S N → R N
E π ∣ B ( X t ) − B ( X 0 ) ∣ 2 = 2 ⟨ B , ( 1 − P t ) B ⟩ ≤ 2 t E v e c ( B ) (9) \mathbb{E}_{\pi}\left|B(X_t)-B(X_0)\right|^2=2\langle B,(1-P_t)B\rangle\leq2t\,\mathcal{E}_{\mathrm{vec}}(B) \tag*{(9)} E π ∣ B ( X t ) − B ( X 0 ) ∣ 2 = 2 ⟨ B , ( 1 − P t ) B ⟩ ≤ 2 t E vec ( B ) (9) (Lemma 15.8). For every fixed ρ > 0 \rho>0 ρ > 0 t t t P π { ∣ B ( X t ) − B ( X 0 ) ∣ > ρ } ≤ 2 t E v e c ( B ) / ρ 2 \mathbb{P}_{\pi}\{|B(X_t)-B(X_0)|>\rho\}\leq2t\mathcal{E}_{\mathrm{vec}}(B)/\rho^2 P π { ∣ B ( X t ) − B ( X 0 ) ∣ > ρ } ≤ 2 t E vec ( B ) / ρ 2 t E v e c ( B ) → 0 t\mathcal{E}_{\mathrm{vec}}(B)\to0 t E vec ( B ) → 0 B ( x ) B(x) B ( x ) x x x U ( x ) ⋅ B ( x ) ≥ q > 0 U(x)\cdot B(x)\geq q>0 U ( x ) ⋅ B ( x ) ≥ q > 0 x x x B ( X t ) B(X_t) B ( X t ) Theorem 1.5 from it.
The condensed phase
Let ( ξ , β ) (\xi,\beta) ( ξ , β ) x , Q x,Q x , Q ({xref}Definition 1.1`). Recall the center field H Q ( u ) = H N ( Q u ) H_Q(u)=H_N(\sqrt{Qu}) H Q ( u ) = H N ( Q u ) K Q ( u ) K_Q(u) K Q ( u ) t ↦ H N ( t Q u ) t \mapsto H_N(t\sqrt{Q}u) t ↦ H N ( t Q u ) t = 1 t=1 t = 1 u ∈ S N u \in S_N u ∈ S N Theorem 1.2 rests on a picture of the Gibbs measure that Subag established for pure models at sufficiently low temperature [141]. In this picture, the Gibbs measure is carried by the bands B u = { x ∈ S N : ∣ R ( x , u ) − Q ∣ ≤ b } B_u = \{x \in S_N : \lvert R(x,u) - \sqrt{Q}\rvert\le b\} B u = { x ∈ S N : ∣ R ( x , u ) − Q ∣ ≤ b } u u u H Q H_Q H Q H Q H_Q H Q
We can see the reason in the static results of Section 5. By Proposition 5.11 (5), we have
F ( β ) = β E 0 + 1 2 log ( 1 − Q ) + β 2 2 ψ Q ( 1 ) , (10) F(\beta) = \beta E_0 + \frac{1}{2}\log(1-Q) + \frac{\beta^2}{2}\psi_Q(1), \tag*{(10)} F ( β ) = β E 0 + 2 1 log ( 1 − Q ) + 2 β 2 ψ Q ( 1 ) , (10) where E 0 E_0 E 0 N − 1 max S N H Q N^{-1}\max_{S_N} H_Q N − 1 max S N H Q ψ Q ( t ) = ξ ( Q + ( 1 − Q ) t ) − ξ ( Q ) − ( 1 − Q ) ξ ′ ( Q ) t \psi_Q(t) = \xi(Q + (1-Q)t) - \xi(Q) - (1-Q)\xi'(Q)t ψ Q ( t ) = ξ ( Q + ( 1 − Q ) t ) − ξ ( Q ) − ( 1 − Q ) ξ ′ ( Q ) t u u u H Q ( u ) = N E H_Q(u) = NE H Q ( u ) = N E Q \sqrt{Q} Q u u u N E NE N E N ψ Q N\psi_Q N ψ Q Q \sqrt{Q} Q log ( 1 − Q ) / 2 \log(1-Q)/2 log ( 1 − Q ) /2 N − 1 / 2 N^{-1/2} N − 1/2 Q \sqrt{Q} Q ( 1 − Q ) N / 2 + o ( N ) (1-Q)^{N/2+o(N)} ( 1 − Q ) N /2 + o ( N ) Proposition 5.11 (2)), which suggests that its partition function is close to its mean e N β 2 ψ Q ( 1 ) / 2 e^{N\beta^2\psi_Q(1)/2} e N β 2 ψ Q ( 1 ) /2
∫ B u e β H N d ν N ≈ e β H Q ( u ) ( 1 − Q ) N / 2 e N β 2 ψ Q ( 1 ) / 2 ≈ e N F ( β ) e β ( H Q ( u ) − N E 0 ) . (11) \int_{B_u} e^{\beta H_N}\,d\nu_N \approx e^{\beta H_Q(u)}(1-Q)^{N/2}e^{N\beta^2\psi_Q(1)/2} \approx e^{NF(\beta)}e^{\beta(H_Q(u)-NE_0)}. \tag*{(11)} ∫ B u e β H N d ν N ≈ e β H Q ( u ) ( 1 − Q ) N /2 e N β 2 ψ Q ( 1 ) /2 ≈ e N F ( β ) e β ( H Q ( u ) − N E 0 ) . (11) Comparing with (10) , we see that a single band around a center of nearly maximal value carries a Gibbs weight comparable to the whole partition function, to leading exponential order. We control the number of centers with the Kac–Rice formula. For non-pure mixtures, Theorem 10.6 shows that the expected number of local maxima of H Q H_Q H Q U U U m N + t m_N+t m N + t e − c 0 t e^{-c_0t} e − c 0 t t t t c 0 = β x c_0=\beta x c 0 = β x m N m_N m N m N m_N m N m N = N E 0 − ( log N ) / ( 2 c 0 ) + O ( 1 ) m_N=NE_0-(\log N)/(2c_0)+O(1) m N = N E 0 − ( log N ) / ( 2 c 0 ) + O ( 1 ) u u u m N − H Q ( u ) m_N-H_Q(u) m N − H Q ( u ) v v v e c 0 v e − β v e^{c_0v}e^{-\beta v} e c 0 v e − β v v v v x < 1 x<1 x < 1 c 0 / β = x c_0/\beta=x c 0 / β = x
On the other hand, a point drawn from ν N \nu_N ν N N − 1 / 2 N^{-1/2} N − 1/2 a < Q − b a<\sqrt{Q}-b a < Q − b C a ( u ) = { x : ∣ R ( x , u ) ∣ ≥ a } C_a(u)=\{x:\lvert R(x,u)\rvert\ge a\} C a ( u ) = { x : ∣ R ( x , u )∣ ≥ a } a a a ± u \pm u ± u B u B_u B u C a ( u ) C_a(u) C a ( u ) ν N \nu_N ν N C a ( u ) C_a(u) C a ( u ) B u B_u B u e γ N e^{\gamma N} e γ N C a ( u ) C_a(u) C a ( u )
The separation criterion
For M > 0 M > 0 M > 0 C N , M \mathcal{C}_{N,M} C N , M H Q H_Q H Q [ m N − M , m N + M ] [m_N - M, m_N + M] [ m N − M , m N + M ] U U U K 0 K_0 K 0 B N , M B_{N,M} B N , M T N T_N T N θ N ( H , u ) = P ν N H { X t ∈ C a ( u ) for some t ≤ T N } \theta_N(H,u) = \mathbb{P}^{H}_{\nu_N}\{X_t \in C_a(u)\text{ for some }t \leq T_N\} θ N ( H , u ) = P ν N H { X t ∈ C a ( u ) for some t ≤ T N } ν N \nu_N ν N C a ( u ) C_a(u) C a ( u ) T N T_N T N B N , M B_{N,M} B N , M
inf 0 ≤ t ≤ T N ∥ ν N P t − π ∥ T V ≥ 1 − π ( B N , M c ) − ∑ u ∈ C N , M θ N ( H N , u ) (12) \inf_{0 \leq t \leq T_N}\left\|\nu_N P_t-\pi\right\|_{\mathrm{TV}} \geq1-\pi\left(B^c_{N,M}\right)-\sum_{u\in\mathcal{C}_{N,M}}\theta_N(H_N,u) \tag*{(12)} 0 ≤ t ≤ T N inf ∥ ν N P t − π ∥ TV ≥ 1 − π ( B N , M c ) − u ∈ C N , M ∑ θ N ( H N , u ) (12) (Proposition 4.5). It suffices to prove three properties. The first is that the bands carry almost all of the Gibbs mass, in the sense that E π ( B N , M c ) \mathbb{E}\pi(B^c_{N,M}) E π ( B N , M c ) N → ∞ N\to\infty N → ∞ M → ∞ M\to\infty M → ∞ (12) tends to zero for each fixed M M M
The third property concerns an average over critical points of a random field. We compute it with the Kac–Rice formula, which expresses it through the law of the field conditioned to have a critical point at a fixed u u u D u = ∣ det ∇ S N 2 H Q ( u ) ∣ D_u=\left|\det\nabla^2_{S_N}H_Q(u)\right| D u = det ∇ S N 2 H Q ( u ) E h , k , u \mathbb{E}_{h,k,u} E h , k , u H Q ( u ) = N h H_Q(u)=Nh H Q ( u ) = N h K Q ( u ) = N k K_Q(u)=Nk K Q ( u ) = N k ∇ S N H Q ( u ) = 0 \nabla_{S_N}H_Q(u)=0 ∇ S N H Q ( u ) = 0 Theorem 7.17 and rotation invariance, we can write this average as
E ∑ u ∈ C N , M θ N ( H N , u ) = c N ∫ W N , M p N ( h , k ) E h , k , u [ D u θ N ( H N , u ) ] d h d k , (13) \mathbb{E}\sum_{u\in\mathcal{C}_{N,M}}\theta_N(H_N,u)=c_N\int_{\mathcal{W}_{N,M}}p_N(h,k)\mathbb{E}_{h,k,u}\left[D_u\theta_N(H_N,u)\right]\,dh\,dk, \tag*{(13)} E u ∈ C N , M ∑ θ N ( H N , u ) = c N ∫ W N , M p N ( h , k ) E h , k , u [ D u θ N ( H N , u ) ] d h d k , (13) where W N , M \mathcal{W}_{N,M} W N , M p N p_N p N c N c_N c N S N S_N S N K Q = p H Q K_Q=pH_Q K Q = p H Q θ N \theta_N θ N u u u § 6.2 impose continuity or tameness hypotheses. We prove the required formula for Borel weights in the coefficient space of the polynomial field, with an explicit version of the conditional law (Proposition 7.16 and Theorem 7.17 ).
The second difficulty is the determinant. The conditioned hitting estimate, which we describe in § 3.2 , bounds q N = sup h , k , u E h , k , u θ N q_N=\sup_{h,k,u}\mathbb{E}_{h,k,u}\theta_N q N = sup h , k , u E h , k , u θ N D u D_u D u D u D_u D u k k k 2 b Q 2\sqrt{b_Q} 2 b Q b Q = Q 2 ζ ′ ′ ( Q ) b_Q = Q^2 \zeta''(Q) b Q = Q 2 ζ ′′ ( Q ) D u / E h , k , u D u D_u/\mathbb{E}_{h,k,u}D_u D u / E h , k , u D u N N N r > 1 r > 1 r > 1
E h , k , u [ D u θ N ] ≤ K r 1 / r q N 1 − 1 / r E h , k , u D u (14) \mathbb{E}_{h,k,u}\left[D_u\theta_N\right] \le K_r^{1/r}q_N^{1-1/r}\mathbb{E}_{h,k,u}D_u \tag*{(14)} E h , k , u [ D u θ N ] ≤ K r 1/ r q N 1 − 1/ r E h , k , u D u (14) (Lemma 7.22). We insert (2.7) into (2.6) and apply (2.6) once more with θ N = 1 \theta_N = 1 θ N = 1 K r 1 / r q N 1 − 1 / r E ∣ C N , M ∣ K_r^{1/r}q_N^{1-1/r}\mathbb{E}\lvert C_{N,M}\rvert K r 1/ r q N 1 − 1/ r E ∣ C N , M ∣ q N ≤ C N − ( d − 2 ) / 2 q_N \le CN^{-(d-2)/2} q N ≤ C N − ( d − 2 ) /2 C M , r N − ( d − 2 ) ( 1 − 1 / r ) / 2 C_{M,r}N^{-(d-2)(1-1/r)/2} C M , r N − ( d − 2 ) ( 1 − 1/ r ) /2
The order in which we choose the parameters is part of the argument. The concentration property lets M M M N N N T N = e γ N T_N = e^{\gamma N} T N = e γ N M M M M M M γ \gamma γ M M M K = [ E 0 − 1 , E 0 + 1 ] × U \mathcal{K} = [E_0 - 1, E_0 + 1] \times U K = [ E 0 − 1 , E 0 + 1 ] × U N N N U U U b b b a a a γ \gamma γ M M M N N N M M M
Statics
We now explain where Q Q Q E 0 E_0 E 0 K 0 K_0 K 0 c 0 c_0 c 0 ζ \zeta ζ ζ \zeta ζ V ζ V_\zeta V ζ ζ \zeta ζ x δ 0 + ( 1 − x ) δ Q x\delta_0 + (1-x)\delta_Q x δ 0 + ( 1 − x ) δ Q { V = 0 } \{V = 0\} { V = 0 } { 0 , Q } \{0,Q\} { 0 , Q } Q Q Q V ′ ′ ( Q ) > 0 V''(Q) > 0 V ′′ ( Q ) > 0
The first variation also describes the geometry of the Gibbs measure. Following Subag [143], we say that an overlap q q q k k k k k k π \pi π q q q V β V_\beta V β β \beta β q q q V β ( q ) = 0 V_\beta(q) = 0 V β ( q ) = 0 q q q T β ( q ) T_\beta(q) T β ( q ) β \beta β q \sqrt{q} q log ( 1 − q ) / 2 \log(1-q)/2 log ( 1 − q ) /2 ψ q \psi_q ψ q T β ( q ) ≤ F ( β ) T_\beta(q) \le F(\beta) T β ( q ) ≤ F ( β ) q q q F ( β ) F(\beta) F ( β ) T β ( q ) T_\beta(q) T β ( q ) V β ( q ) > 0 V_\beta(q) > 0 V β ( q ) > 0 T β ( q ) ≤ F ( β ) − V β ( q ) 2 / ( 8 β 3 M q ) T_\beta(q) \le F(\beta) - V_\beta(q)^2/(8\beta^3 M_q) T β ( q ) ≤ F ( β ) − V β ( q ) 2 / ( 8 β 3 M q ) M q > 0 M_q > 0 M q > 0 Theorem 5.7 (3)). If V β ( q ) = 0 V_\beta(q) = 0 V β ( q ) = 0 V β V_\beta V β
We collect the static consequences of the 1RSB assumptions in Proposition 5.11 . The multisamplable overlaps are 0 and Q Q Q Q Q Q H Q H_Q H Q E 0 E_0 E 0 K 0 K_0 K 0 c 0 = β x < β c_0 = \beta x < \beta c 0 = β x < β K 0 K_0 K 0 y = Q / ( β ( 1 − Q ) ) y = Q/(\beta(1 - Q)) y = Q / ( β ( 1 − Q ))
K 0 − 2 b ˉ Q = ( y − b ˉ Q ) 2 y > 0. (15) K_0 - 2\sqrt{\bar{b}_Q} = \frac{\left(y - \sqrt{\bar{b}_Q}\right)^2}{y} > 0. \tag*{(15)} K 0 − 2 b ˉ Q = y ( y − b ˉ Q ) 2 > 0. (15) This gap matters for the following reason. Conditionally on H Q ( u ) = N h H_Q(u) = Nh H Q ( u ) = N h K Q ( u ) = N k K_Q(u) = Nk K Q ( u ) = N k ∇ S N H Q ( u ) = 0 \nabla_{\mathrm{S}_N} H_Q(u) = 0 ∇ S N H Q ( u ) = 0 ∇ S N 2 H Q ( u ) \nabla^2_{\mathrm{S}_N} H_Q(u) ∇ S N 2 H Q ( u ) [ − 2 b ˉ Q , 2 b ˉ Q ] [-2\sqrt{\bar{b}_Q}, 2\sqrt{\bar{b}_Q}] [ − 2 b ˉ Q , 2 b ˉ Q ] − k -k − k Lemma 5.14 ). For a deterministic u u u k k k K 0 K_0 K 0 1 − C e − c N 1 - Ce^{-cN} 1 − C e − c N Corollary 7.6 ). The statements about all high critical points are established separately in Proposition 10.10 and Lemma 9.10 . The normalized determinant moments used in (2.7) follow from Proposition 7.7 .
The directing measure
The concentration argument of § 3.2 places a Gibbs sample in the band of a center by finding additional independent replicas whose overlaps with each other and with the given sample are all close to Q Q Q Theorem 6.2 ): every limit in distribution of the overlap array of independent Gibbs samples is generated by drawing a random measure G G G v 1 , v 2 , … v_1, v_2, \ldots v 1 , v 2 , … G G G R i j = v i ⋅ v j R_{ij} = v_i \cdot v_j R ij = v i ⋅ v j i ≠ j i \ne j i = j Theorem 6.3 that, under the 1RSB assumptions, G G G Q Q Q v 1 v_1 v 1 Q Q Q
We have to exclude two possibilities. Directing vectors of a squared norm q ∈ ( 0 , 1 ) ∖ { Q } q \in(0,1) \setminus\{Q\} q ∈ ( 0 , 1 ) ∖ { Q } q q q Lemmas 5.8 and 6.5 ); we exclude vectors of norm one with an entropy estimate (Lemma 6.4). An atom of G G G ζ ( R ) \zeta(R) ζ ( R ) n ≥ 1 n \ge1 n ≥ 1 f f f ( R i j ) 1 ≤ i < j ≤ n (R_{ij})_{1 \le i<j \le n} ( R ij ) 1 ≤ i < j ≤ n
E [ f ζ ( R 1 , n + 1 ) ] = ( 1 − x ) ζ ( Q ) n E f + 1 n ∑ j = 2 n E [ f ζ ( R 1 j ) ] . (16) \mathbb{E}\left[f\zeta(R_{1,n+1})\right] = \frac{(1-x)\zeta(Q)}{n}\mathbb{E}f + \frac{1}{n}\sum_{j=2}^{n}\mathbb{E}\left[f\zeta(R_{1j})\right]. \tag*{(16)} E [ f ζ ( R 1 , n + 1 ) ] = n ( 1 − x ) ζ ( Q ) E f + n 1 j = 2 ∑ n E [ f ζ ( R 1 j ) ] . (16) Let a n a_n a n ζ ( R 1 j ) = 0 \zeta(R_{1j})=0 ζ ( R 1 j ) = 0 2 ≤ j ≤ n 2 \le j \le n 2 ≤ j ≤ n f f f ∣ R i j ∣ ≤ Q |R_{ij}| \le Q ∣ R ij ∣ ≤ Q i ≠ j i \ne j i = j ζ ( R 1 , n + 1 ) ≤ ζ ( Q ) \zeta(R_{1,n+1}) \le\zeta(Q) ζ ( R 1 , n + 1 ) ≤ ζ ( Q ) ζ ( Q ) ( a n − a n + 1 ) \zeta(Q)(a_n-a_{n+1}) ζ ( Q ) ( a n − a n + 1 ) a n + 1 ≤ ( 1 − ( 1 − x ) / n ) a n a_{n+1} \le(1-(1-x)/n)a_n a n + 1 ≤ ( 1 − ( 1 − x ) / n ) a n a n ≤ n − ( 1 − x ) → 0 a_n \le n^{-(1-x)} \to0 a n ≤ n − ( 1 − x ) → 0 ζ ( R 1 j ) = 0 \zeta(R_{1j})=0 ζ ( R 1 j ) = 0 j j j G G G G G G
Critical points near the maximum
We prove the second property of Section 2.2, bounded counts in fixed windows, in Section 9 for pure models and in Sections 10 and 11 for non-pure mixtures, together with the existence of at least one center, which the concentration argument needs. The following limits for the complexity in two variables and for the normalized intensity concern non-pure mixtures. For pure models, K Q = p H Q K_Q=pH_Q K Q = p H Q H Q H_Q H Q e e e k k k e e e k k k e = E 0 e=E_0 e = E 0 K 0 K_0 K 0 e e e ( E 0 , K 0 ) (E_0,K_0) ( E 0 , K 0 ) − c 0 -c_0 − c 0 a < b a<b a < b
E # { u : ∇ S N H Q ( u ) = 0 , K Q ( u ) / N ∈ U , H Q ( u ) − m N ∈ [ a , b ] } ⟶ e − c 0 a − e − c 0 b (17) \mathbb{E}\#\left\{u:\nabla_{S_N}H_Q(u)=0,\ K_Q(u)/N\in U,\ H_Q(u)-m_N\in[a,b]\right\}\longrightarrow e^{-c_0a}-e^{-c_0b} \tag*{(17)} E # { u : ∇ S N H Q ( u ) = 0 , K Q ( u ) / N ∈ U , H Q ( u ) − m N ∈ [ a , b ] } ⟶ e − c 0 a − e − c 0 b (17) (Theorem 10.6). In particular, the expected count in every fixed window is bounded, and the probability that the maximum of H Q H_Q H Q m N + L m_N+L m N + L e − c 0 L e^{-c_0L} e − c 0 L
For existence, we need a second moment. For pure models, existence follows from the tightness of the centered maximum, which is a consequence of the work of Subag and Zeitouni [144] (Lemma 9.8 and Proposition 9.1). For non-pure mixtures, we bound the second moment of the number C N ( L ) C_N(L) C N ( L ) [ m N − L , m N + L ] [m_N-L,m_N+L] [ m N − L , m N + L ] r r r ± 1 \pm1 ± 1
Overlaps r ≥ ε r \ge\varepsilon r ≥ ε ( E 0 , K 0 ) (E_0,K_0) ( E 0 , K 0 )
the conditional trial argument of Huang and Sellke [90, Section 3.3]. We show that, conditionally on a critical point at u u u ( E 0 , K 0 ) (E_0,K_0) ( E 0 , K 0 ) 1 − e − c N 1-e^{-cN} 1 − e − c N ε \varepsilon ε 1 − η 1-\eta 1 − η u u u N E 0 NE_0 N E 0 N N N 1 − η 1-\eta 1 − η u u u ε \varepsilon ε
Overlaps r ≤ − δ r \le-\delta r ≤ − δ For even mixtures, these pairs are reflections of pairs with r ≥ δ r \ge\delta r ≥ δ ε < δ 2 \varepsilon< \delta^2 ε < δ 2 v 1 , … , v k v_1,\ldots,v_k v 1 , … , v k − δ -\delta − δ v 0 v_0 v 0 k δ ≤ ∣ v 1 + ⋯ + v k ∣ ≤ ( k + k ( k − 1 ) ε ) 1 / 2 k\delta\le\lvert v_1+\cdots+v_k\rvert\le(k+k(k-1)\varepsilon)^{1/2} k δ ≤ ∣ v 1 + ⋯ + v k ∣ ≤ ( k + k ( k − 1 ) ε ) 1/2 ( 1 − ε ) / ( δ 2 − ε ) (1-\varepsilon)/(\delta^2-\varepsilon) ( 1 − ε ) / ( δ 2 − ε )
Overlaps ∣ r ∣ < δ \lvert r\rvert< \delta ∣ r ∣ < δ We show that, for ∣ r ∣ ≤ N − α \lvert r\rvert\le N^{-\alpha} ∣ r ∣ ≤ N − α 1 / 4 < α < 1 / 2 1/4<\alpha<1/2 1/4 < α < 1/2 1 + O ( N 1 − 4 α + N − 1 / 2 ) 1+O(N^{1-4\alpha}+N^{-1/2}) 1 + O ( N 1 − 4 α + N − 1/2 ) ∣ r ∣ ≤ N − α \lvert r\rvert\le N^{-\alpha} ∣ r ∣ ≤ N − α f N ( r ) = ( 1 + o ( 1 ) ) N / ( 2 π ) e − N r 2 / 2 f_N(r)=(1+o(1))\sqrt{N/(2\pi)}e^{-Nr^2/2} f N ( r ) = ( 1 + o ( 1 )) N / ( 2 π ) e − N r 2 /2 r r r N − α ≤ ∣ r ∣ < δ N^{-\alpha}\le\lvert r\rvert<\delta N − α ≤ ∣ r ∣ < δ C N e − N r 2 / 4 C\sqrt{N}e^{-Nr^2/4} C N e − N r 2 /4
Combining the three ranges, we find that E C N ( L ) 2 ≤ ( 1 + o ( 1 ) ) ( E C N ( L ) ) 2 + ( 1 + K ) E C N ( L ) \mathbb{E}C_N(L)^2\le(1+o(1))(\mathbb{E}C_N(L))^2+(1+K)\mathbb{E}C_N(L) E C N ( L ) 2 ≤ ( 1 + o ( 1 )) ( E C N ( L ) ) 2 + ( 1 + K ) E C N ( L ) K K K L L L E C N ( L ) → e c 0 L − e − c 0 L \mathbb{E}C_N(L)\to e^{c_0L}-e^{-c_0L} E C N ( L ) → e c 0 L − e − c 0 L 1 − ( 1 + K ) / ( e c 0 L − e − c 0 L ) − o ( 1 ) 1-(1+K)/(e^{c_0L}-e^{-c_0L})-o(1) 1 − ( 1 + K ) / ( e c 0 L − e − c 0 L ) − o ( 1 )
Gibbs concentration on bands
We prove the concentration property in Section 9 for pure models (Theorem 9.2) and in Section 12 for non-pure mixtures (Theorem 12.22). For pure models at sufficiently low temperature, this concentration was proved by Subag [141, Theorem 1]; we need it at every temperature of the condensed phase and in a fixed window. For non-pure mixtures, the three inputs below verify Proposition 4.9. The pure proof instead starts from a coarse cap cover, lets the energy deficit per spin tend to zero, removes the caps around critical points with a large energy deficit, and passes from the caps of a fixed window to bands of width of order N − 1 / 2 N^{-1/2} N − 1/2
The first step is a coarse cover: for every fixed energy tolerance τ > 0 \tau> 0 τ > 0 H Q H_Q H Q N ( E 0 − τ ) N(E_0-\tau) N ( E 0 − τ ) Theorem 6.3 , among a fixed number of additional independent replicas, a Gibbs sample typically has n n n Q Q Q N E 0 NE_0 N E 0 H Q H_Q H Q Proposition 9.15 and Lemma 12.7). Writing σ ˉ = n − 1 ∑ i σ i \bar{\sigma}=n^{-1}\sum_i \sigma^i σ ˉ = n − 1 ∑ i σ i q ˉ = ∣ σ ˉ ∣ 2 / N \bar{q}=|\bar{\sigma}|^2/N q ˉ = ∣ σ ˉ ∣ 2 / N q ˉ \bar{q} q ˉ Q + ( 1 − Q ) / n Q+(1-Q)/n Q + ( 1 − Q ) / n u = σ ˉ / q ˉ u=\bar{\sigma}/\sqrt{\bar{q}} u = σ ˉ / q ˉ S N S_N S N H Q ( u ) = H N ( Q / q ˉ σ ˉ ) H_Q(u)=H_N(\sqrt{Q/\bar{q}}\,\bar{\sigma}) H Q ( u ) = H N ( Q / q ˉ σ ˉ ) n n n q ˉ \bar{q} q ˉ Q Q Q N → ∞ N\to\infty N → ∞ u u u H Q H_Q H Q
For non-pure mixtures, the coarse band family can contain exponentially many critical points, and the next two inputs pass to a fixed window. Let Z H ( D ) = ∫ D e β H d ν N Z_H(D)=\int_D e^{\beta H}\,\mathrm{d}\nu_N Z H ( D ) = ∫ D e β H d ν N u u u [ m N − L , m N + L ] [m_N-L,m_N+L] [ m N − L , m N + L ] Z H ( B u ) ≥ δ D N e β ( H Q ( u ) − m N ) Z_H(B_u)\geq\delta D_N e^{\beta(H_Q(u)-m_N)} Z H ( B u ) ≥ δ D N e β ( H Q ( u ) − m N ) D N D_N D N Z N , β ≥ δ e − β L D N Z_{N,\beta}\geq\delta e^{-\beta L}D_N Z N , β ≥ δ e − β L D N max H Q ≤ m N + M \max H_Q\leq m_N+M max H Q ≤ m N + M
π ( B N , M c ) ≤ π ( [ ⋃ u c o a r s e B u ] c ) + e β L δ D N ∑ u c o a r s e H Q ( u ) < m N − M Z H ( B u ) (18) \pi\left(B^c_{N,M}\right)\leq\pi\left(\left[\bigcup_{u\ \mathrm{coarse}}B_u\right]^c\right)+\frac{e^{\beta L}}{\delta D_N}\sum_{\substack{u\ \mathrm{coarse}\\H_Q(u)<m_N-M}}Z_H(B_u) \tag*{(18)} π ( B N , M c ) ≤ π ( [ u coarse ⋃ B u ] c ) + δ D N e β L u coarse H Q ( u ) < m N − M ∑ Z H ( B u ) (18) (Remark 12.23). In the second step, we bound the expected sum. In the third step, we show that light bands, those with Z H ( B u ) < δ D N e β ( H Q ( u ) − m N ) Z_H(B_u)<\delta D_N e^{\beta(H_Q(u)-m_N)} Z H ( B u ) < δ D N e β ( H Q ( u ) − m N )
For the bands of centers below the window, we use the Kac–Rice formula with the band partition function as a weight. We show that the Palm expectation of Z H ( B u ) Z_H(B_u) Z H ( B u ) E E E C 1 D N e β ( E − m N ) + C 2 N ( E / N − E 0 ) 2 C_1D_Ne^{\beta(E-m_N)+C_2N(E/N-E_0)^2} C 1 D N e β ( E − m N ) + C 2 N ( E / N − E 0 ) 2 m N − v m_N-v m N − v C e c + v Ce^{c_+v} C e c + v c 0 < c + < β c_0<c_+<\beta c 0 < c + < β 0 < τ ≤ τ 0 = min { ε 0 , ( β − c + ) / ( 4 C 2 ) } 0<\tau\leq\tau_0=\min\{\varepsilon_0,(\beta-c_+)/(4C_2)\} 0 < τ ≤ τ 0 = min { ε 0 , ( β − c + ) / ( 4 C 2 )} C D N e − a 0 M CD_Ne^{-a_0M} C D N e − a 0 M a 0 = ( β − c + ) / 2 > 0 a_0=(\beta-c_+)/2>0 a 0 = ( β − c + ) /2 > 0 e c 0 v e − β v e^{c_0v}e^{-\beta v} e c 0 v e − β v c 0 < β c_0 < \beta c 0 < β N N N M M M N → ∞ N \to\infty N → ∞ N − 1 / 2 N^{-1/2} N − 1/2 ( N / 2 π ) 1 / 2 (N/2\pi)^{1/2} ( N /2 π ) 1/2
The bound on light bands rests on the structure of the field on the band. At latitude exactly Q \sqrt{Q} Q N ψ Q N\psi_Q N ψ Q β 2 ψ Q ′ ′ ( 0 ) < 1 \beta^2\psi_Q''(0) < 1 β 2 ψ Q ′′ ( 0 ) < 1 [93 Lemma 3.2 and Remark 3.3] implies that its normalized partition function is tight away from zero (Lemma 3.19 ). We pass from this latitude to the band with Jensen’s inequality over a symmetric latitude interval of width N − 1 / 2 N^{-1/2} N − 1/2 Proposition 4.9 .
For pure models, the radial derivative is a multiple of the value, K Q = p H Q K_Q = pH_Q K Q = p H Q 1 / T 1/T 1/ T e − T e^{-T} e − T β 2 ψ Q ( 1 ) < 1 / 2 \beta^2\psi_Q(1) < 1/2 β 2 ψ Q ( 1 ) < 1/2 p ≥ 4 p \ge4 p ≥ 4 β > β s ( p ) \beta> \beta_s(p) β > β s ( p ) Lemma 5.22 ). The contribution of the bands around critical points with a large energy deficit decays exponentially in the deficit, at a rate that is positive because x < 1 x < 1 x < 1 Q \sqrt{Q} Q Lemma 9.3 and the proof of Theorem 9.2 ).
The conditioned hitting estimate
The hitting estimate for the caps under critical-point conditioning, Theorem 8.2 , is the main dynamical input of Part II. For a finite mixture of minimum degree d ≥ 4 d \ge4 d ≥ 4 Q ∈ ( 0 , 1 ] Q \in(0,1] Q ∈ ( 0 , 1 ] β ≥ 0 \beta\ge0 β ≥ 0 h ˉ , k ˉ \bar h,\bar k h ˉ , k ˉ a ∗ > 0 a_* > 0 a ∗ > 0 a ∈ ( 0 , a ∗ ] a \in(0,a_*] a ∈ ( 0 , a ∗ ] γ , c , C > 0 \gamma,c,C > 0 γ , c , C > 0 N N N
E h , k , u [ P ν N H N ( sup 0 ≤ t ≤ e γ N ∣ R ( X t , u ) ∣ ≥ a ) ] ≤ C ( N 1 − d / 2 + N 3 − d ) + e − c N , (19) \mathbb{E}_{h,k,u}\left[\mathbb{P}_{\nu_N}^{H_N}\left(\sup_{0\le t\le e^{\gamma N}} \left|R(X_t,u)\right| \ge a\right)\right] \le C\left(N^{1-d/2}+N^{3-d}\right)+e^{-cN}, \tag*{(19)} E h , k , u [ P ν N H N ( 0 ≤ t ≤ e γ N sup ∣ R ( X t , u ) ∣ ≥ a ) ] ≤ C ( N 1 − d /2 + N 3 − d ) + e − c N , (19) uniformly in u u u ( h , k ) (h,k) ( h , k ) ∣ h ∣ ≤ h ˉ |h| \le\bar h ∣ h ∣ ≤ h ˉ ∣ k ∣ ≤ k ˉ |k| \le\bar k ∣ k ∣ ≤ k ˉ k = p h k = ph k = p h p p p § 6.3.2 explains the proof in detail; here we describe its main points.
We begin with the unconditioned field. For an axis u u u S N S_N S N C a ( u ) C_a(u) C a ( u ) ∫ ν N { u : ∣ R ( x , u ) ∣ ≥ a } π ( d x ) \int\nu_N\{u : |R(x,u)| \ge a\}\,\pi(\mathrm{d}x) ∫ ν N { u : ∣ R ( x , u ) ∣ ≥ a } π ( d x ) ν N \nu_N ν N ν N ( C a ( v ) ) \nu_N(C_a(v)) ν N ( C a ( v )) v ∈ S N v \in S_N v ∈ S N H N + δ h , k , u H_N+\delta_{h,k,u} H N + δ h , k , u N N N u u u Proposition 8.7 ). We cannot treat it as a small perturbation. However, it vanishes to high order at the equator of u u u Ω N \Omega_N Ω N Lemma 8.6 , whose complement has probability at most e − N e^{-N} e − N r = R ( x , u ) r=R(x,u) r = R ( x , u ) V x ( u ) = N − 1 / 2 ∇ S N H Q ( u ) ⋅ x V_x(u)=N^{-1/2}\nabla_{S_N}H_Q(u)\cdot x V x ( u ) = N − 1/2 ∇ S N H Q ( u ) ⋅ x
∣ δ h , k , u ( x ) ∣ ≤ C N ∣ r ∣ d + C N ∣ r ∣ d − 1 ∣ V x ( u ) ∣ . (20) |\delta_{h,k,u}(x)| \le CN|r|^d+C\sqrt{N}|r|^{d-1}|V_x(u)|. \tag*{(20)} ∣ δ h , k , u ( x ) ∣ ≤ C N ∣ r ∣ d + C N ∣ r ∣ d − 1 ∣ V x ( u ) ∣. (20) A fixed point x x x N − 1 / 2 N^{-1/2} N − 1/2 V x ( u ) V_x(u) V x ( u ) N 1 − d / 2 N^{1-d/2} N 1 − d /2 (19) . Fix the field in Ω N \Omega_N Ω N α > 0 \alpha>0 α > 0 χ \chi χ [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] [ − 2 , 2 ] [-2,2] [ − 2 , 2 ] s = ∇ S N H Q ( u ) ⋅ x / N s=\nabla_{S_N}H_Q(u)\cdot x/N s = ∇ S N H Q ( u ) ⋅ x / N δ α ( u , x ) = χ ( r / α ) χ ( s / α ) δ h , k , u ( x ) \delta_\alpha(u,x)=\chi(r/\alpha)\chi(s/\alpha)\delta_{h,k,u}(x) δ α ( u , x ) = χ ( r / α ) χ ( s / α ) δ h , k , u ( x ) w ( u , x ) = β δ α ( u , x ) w(u,x)=\beta\delta_\alpha(u,x) w ( u , x ) = β δ α ( u , x ) x x x Ω N \Omega_N Ω N
η = E u ∣ e w − 1 ∣ = O ( N 1 − d / 2 ) , ϵ = E u [ e w ( 1 − e − w ) 2 ] = O ( N 2 − d ) . (21) \eta=\mathbb{E}_u|e^w-1|=O(N^{1-d/2}),\qquad\epsilon=\mathbb{E}_u\left[e^w(1-e^{-w})^2\right]=O(N^{2-d}). \tag*{(21)} η = E u ∣ e w − 1∣ = O ( N 1 − d /2 ) , ϵ = E u [ e w ( 1 − e − w ) 2 ] = O ( N 2 − d ) . (21) There is an apparent difficulty here, since the point x x x ∇ S N H Q ( u ) \nabla_{S_N}H_Q(u) ∇ S N H Q ( u ) V x ( u ) V_x(u) V x ( u ) x x x u u u { u : R ( x , u ) = r } \{u:R(x,u)=r\} { u : R ( x , u ) = r } V x V_x V x
The main obstacle is the length e γ N e^{\gamma N} e γ N τ \tau τ κ \kappa κ Lemma 8.11 bounds the Laplace transform E e − λ τ \mathbb{E}e^{-\lambda\tau} E e − λ τ
η + ϵ + ϵ ( 1 + H 0 ) = O ( N 1 − d / 2 + N 3 − d ) , (22) \eta+\sqrt{\epsilon}+\epsilon(1+\mathcal{H}_0)=O(N^{1-d/2}+N^{3-d}), \tag*{(22)} η + ϵ + ϵ ( 1 + H 0 ) = O ( N 1 − d /2 + N 3 − d ) , (22) where H 0 \mathcal{H}_0 H 0 ν N \nu_N ν N H 0 \mathcal{H}_0 H 0 N N N ∣ H N ∣ ≤ K N |H_N|\le KN ∣ H N ∣ ≤ K N λ = e − γ N \lambda= e^{-\gamma N} λ = e − γ N e γ N e^{\gamma N} e γ N γ \gamma γ f 0 f_0 f 0 H 0 H_0 H 0 λ ( λ + A 0 ) − 1 f 0 \lambda(\lambda+ A_0)^{-1}f_0 λ ( λ + A 0 ) − 1 f 0 A 0 A_0 A 0 d = 4 d = 4 d = 4 (22) are of order N − 1 N^{-1} N − 1 d = 3 d = 3 d = 3
Two further steps return to the true process. Recall that we have truncated the correction at a fixed scale α \alpha α r r r s s s C N α d CN^{\alpha d} C N α d α / 4 \alpha/4 α /4 e − c N α 2 e^{-cN\alpha^2} e − c N α 2 d > 2 d > 2 d > 2 α \alpha α γ \gamma γ Proposition 8.12 ). The true and truncated processes coincide until ∣ r ∣ |r| ∣ r ∣ ∣ s ∣ |s| ∣ s ∣ α \alpha α ∣ r ∣ = α |r| = \alpha ∣ r ∣ = α a = α / 2 a = \alpha/2 a = α /2 ∣ s ∣ = α |s| = \alpha ∣ s ∣ = α ∇ S N H Q ( u ) \nabla_{S_N} H_Q(u) ∇ S N H Q ( u ) u u u u u u s t s_t s t a Q / N a_Q/N a Q / N Ω N \Omega_N Ω N P ( Ω N ) \mathbb{P}(\Omega_N) P ( Ω N ) Lemma 8.13 ). Finally, we pass from a uniform axis to every deterministic axis by rotation invariance.
Other initial laws
We deduce Theorem 1.4 from Theorem 1.2 and the polynomial bound of Proposition 13.1, by three arguments that use only the rotation invariance of the disorder (§ 5.1.5 ). For part (i), let μ \mu μ μ ↦ E ∥ μ P T − π ∥ T V \mu\mapsto\mathbb{E}\left\|\mu P_T-\pi\right\|_{\mathrm{TV}} μ ↦ E ∥ μ P T − π ∥ TV μ \mu μ μ \mu μ ν N \nu_N ν N
E [ 1 − ∥ μ P T − π ∥ T V ] ≤ E [ 1 − ∥ ν N P T − π ∥ T V ] (23) \mathbb{E}\left[1-\left\|\mu P_T-\pi\right\|_{\mathrm{TV}}\right]\leq\mathbb{E}\left[1-\left\|\nu_NP_T-\pi\right\|_{\mathrm{TV}}\right] \tag*{(23)} E [ 1 − ∥ μ P T − π ∥ TV ] ≤ E [ 1 − ∥ ν N P T − π ∥ TV ] (23) (Theorem 4.11 ). This is an instance of the classical symmetrization over a compact group, the same averaging that underlies the Hunt–Stein theorem [113, Section 8.5]. In particular, ν N \nu_N ν N
For parts (ii) and (iii), the same invariance shows that the expected mass of the band union at time T N T_N T N Lemma 4.13 ). Proposition 13.1 implies that this mass is at most q N , M ≤ C M , r N − ( d − 2 ) ( 1 − 1 / r ) / 2 q_{N,M} \le C_{M,r}N^{-(d-2)(1-1/r)/2} q N , M ≤ C M , r N − ( d − 2 ) ( 1 − 1/ r ) /2 K N ≤ C 0 N κ K_N \le C_0N^\kappa K N ≤ C 0 N κ K N q N , M → 0 K_Nq_{N,M} \to0 K N q N , M → 0 κ < ( d − 2 ) / 2 \kappa< (d-2)/2 κ < ( d − 2 ) /2 r r r Corollary 4.14 ). The loss of the factor 1 − 1 / r 1-1/r 1 − 1/ r κ = ( d − 2 ) / 2 \kappa=(d-2)/2 κ = ( d − 2 ) /2 μ \mu μ ν N \nu_N ν N q q q ν N P T N \nu_NP_{T_N} ν N P T N μ P T N \mu P_{T_N} μ P T N ( D ( μ ∥ ν N ) + log 2 ) / log ( 1 / q ) (D(\mu\Vert\nu_N)+\log2)/\log(1/q) ( D ( μ ∥ ν N ) + log 2 ) / log ( 1/ q ) Lemma 3.9 and Corollary 4.15 ). For any fixed r > 1 r>1 r > 1 0 < τ < ( d − 2 ) ( 1 − 1 / r ) / 2 0<\tau<(d-2)(1-1/r)/2 0 < τ < ( d − 2 ) ( 1 − 1/ r ) /2 q ≤ N − τ q\le N^{-\tau} q ≤ N − τ D ( μ ∥ ν N ) / log N → 0 D(\mu\Vert\nu_N)/\log N\to0 D ( μ ∥ ν N ) / log N → 0
Below the static threshold
Let β d < β < β s \beta_d<\beta<\beta_s β d < β < β s q ∗ q_* q ∗ (2) . The proof of Theorem 1.5 has a probabilistic part, in which we describe the Gibbs measure as seen from a Gibbs-typical point, and a deterministic part, in which we build one observable from this description and derive the five conclusions from it.
The constrained pressure and the equality at q ∗ q_* q ∗
For a reference point x x x r r r u u u R ( x , u ) R(x,u) R ( x , u ) r r r x x x π \pi π F F P F_{\mathrm{FP}} F FP F = β 2 ξ ( 1 ) / 2 F=\beta^2\xi(1)/2 F = β 2 ξ ( 1 ) /2 h > 0 h>0 h > 0 [ r − h , r + h ] ⊂ ( − 1 , 1 ) [r-h,r+h]\subset(-1,1) [ r − h , r + h ] ⊂ ( − 1 , 1 )
π { u : R ( x , u ) ∈ [ r − h , r + h ] } ≍ exp { N [ max v ∈ [ r − h , r + h ] F F P ( v ) − F ] } . (24) \pi\{u:R(x,u)\in[r-h,r+h]\}\asymp\exp\left\{N\left[\max_{v\in[r-h,r+h]}F_{\mathrm{FP}}(v)-F\right]\right\}. \tag*{(24)} π { u : R ( x , u ) ∈ [ r − h , r + h ]} ≍ exp { N [ v ∈ [ r − h , r + h ] max F FP ( v ) − F ] } . (24) The midpoint exponent F F P ( r ) − F F_{\mathrm{FP}}(r)-F F FP ( r ) − F h h h N N N x x x x x x S N S_N S N β N ξ ( R ( x , ⋅ ) ) \beta N\xi(R(x,\cdot)) β N ξ ( R ( x , ⋅ )) π \pi π η r ( t ) = ξ ( r 2 + ( 1 − r 2 ) t ) − ξ ( r 2 ) \eta_r(t)=\xi(r^2+(1-r^2)t)-\xi(r^2) η r ( t ) = ξ ( r 2 + ( 1 − r 2 ) t ) − ξ ( r 2 ) n = N − 1 n=N-1 n = N − 1 { u : R ( x , u ) = r } \{u:R(x,u)=r\} { u : R ( x , u ) = r } β N ξ ( r ) + W + Y \beta N\xi(r)+W+Y β N ξ ( r ) + W + Y W W W N ξ ( r 2 ) N\xi(r^2) N ξ ( r 2 ) Y Y Y N η r ( R n ) = n ( N / n ) η r ( R n ) N\eta_r(R_n)=n(N/n)\eta_r(R_n) N η r ( R n ) = n ( N / n ) η r ( R n )
F F P ( r ) = β 2 ξ ( r ) + 1 2 log ( 1 − r 2 ) + P β ( η r ) (25) F_{\mathrm{FP}}(r)=\beta^2\xi(r)+\frac{1}{2}\log(1-r^2)+\mathcal{P}_{\beta}(\eta_r) \tag*{(25)} F FP ( r ) = β 2 ξ ( r ) + 2 1 log ( 1 − r 2 ) + P β ( η r ) (25) (Proposition 14.7). For r ≠ 0 r \ne0 r = 0 η r \eta_r η r ξ ~ \tilde{\xi} ξ ~ r = 0 r=0 r = 0 η 0 = ξ ~ \eta_0=\tilde{\xi} η 0 = ξ ~
The infimum P β ( η r ) \mathcal{P}_{\beta}(\eta_r) P β ( η r ) r ∈ [ 0 , 1 ) r \in[0,1) r ∈ [ 0 , 1 ) η r \eta_r η r δ r / ( 1 + r ) \delta_{r/(1+r)} δ r / ( 1 + r ) F F P ( r ) ≤ Ψ ( r ) F_{\mathrm{FP}}(r) \le\Psi(r) F FP ( r ) ≤ Ψ ( r )
Ψ ( r ) = F + 1 2 Φ β ( r ) , Ψ ′ ( r ) = r 2 ( 1 − r ) ( β 2 A ( r ) − 1 ) . (26) \Psi(r)=F+\frac{1}{2}\Phi_{\beta}(r), \qquad\Psi'(r)=\frac{r}{2(1-r)}\left(\beta^2 A(r)-1\right). \tag*{(26)} Ψ ( r ) = F + 2 1 Φ β ( r ) , Ψ ′ ( r ) = 2 ( 1 − r ) r ( β 2 A ( r ) − 1 ) . (26) For pure models, this is the trial bound of [65, Proposition 3.9]. By the definition of q ∗ q_* q ∗ ( q ∗ − ε 0 , q ∗ ) (q_*-\varepsilon_0,q_*) ( q ∗ − ε 0 , q ∗ ) [ q ∗ , 1 ) [q_*,1) [ q ∗ , 1 ) F F P ( q ∗ ) = Ψ ( q ∗ ) F_{\mathrm{FP}}(q_*)=\Psi(q_*) F FP ( q ∗ ) = Ψ ( q ∗ ) δ r 0 \delta_{r_0} δ r 0 r 0 = q ∗ / ( 1 + q ∗ ) r_0=q_* /(1+q_*) r 0 = q ∗ / ( 1 + q ∗ ) η q ∗ \eta_{q_*} η q ∗ r 0 r_0 r 0 β 2 A − 1 \beta^2 A-1 β 2 A − 1 q ∗ q_* q ∗ r 0 r_0 r 0 ξ ~ ′ ( v ) / ( v − q ∗ 2 ) \tilde{\xi}'(v)/(v-q_*^2) ξ ~ ′ ( v ) / ( v − q ∗ 2 ) ( q ∗ 2 , q ∗ ] (q_*^2,q_*] ( q ∗ 2 , q ∗ ] χ ( v ) = v ξ ~ ′ ′ ( v ) / ξ ~ ′ ( v ) \chi(v)=v\tilde{\xi}''(v)/\tilde{\xi}'(v) χ ( v ) = v ξ ~ ′′ ( v ) / ξ ~ ′ ( v ) p − 1 p-1 p − 1 p c p v p − 1 / ξ ~ ′ ( v ) pc_pv^{p-1}/\tilde{\xi}'(v) p c p v p − 1 / ξ ~ ′ ( v ) v v v A ′ ( q ∗ ) ≤ 0 A'(q_*)\le0 A ′ ( q ∗ ) ≤ 0 q ∗ q_* q ∗
Fix an admissible collar 0 < q 1 < q 2 < q ∗ 0<q_1<q_2<q_* 0 < q 1 < q 2 < q ∗ β 2 A ( r ) > 1 \beta^2A(r)>1 β 2 A ( r ) > 1 ( q 1 , q ∗ ) (q_1,q_*) ( q 1 , q ∗ ) Ψ \Psi Ψ [ q 1 , q ∗ ] [q_1,q_*] [ q 1 , q ∗ ] [ q ∗ , 1 ) [q_*,1) [ q ∗ , 1 ) q ∗ q_* q ∗ [ q 1 , 1 ) [q_1,1) [ q 1 , 1 ) F − J F-J F − J q ∗ q_* q ∗ J = − Φ β ( q ∗ ) / 2 > 0 J=-\Phi_{\beta}(q_*)/2>0 J = − Φ β ( q ∗ ) /2 > 0 [ q 1 , q 2 ] [q_1,q_2] [ q 1 , q 2 ] F − J − G F-J-G F − J − G G = Ψ ( q ∗ ) − Ψ ( q 2 ) > 0 G=\Psi(q_*)-\Psi(q_2)>0 G = Ψ ( q ∗ ) − Ψ ( q 2 ) > 0 0 < δ < min { J , G } / 8 0<\delta<\min\{J,G\}/8 0 < δ < min { J , G } /8 1 − C e − a N 1-Ce^{-aN} 1 − C e − a N G \mathcal{G} G π ( G c ) ≤ e − a N \pi(\mathcal{G}^{c})\le e^{-aN} π ( G c ) ≤ e − a N x ∈ G x\in\mathcal{G} x ∈ G
π { u : R ( x , u ) ≥ q 2 } ≥ e − N ( J + 2 δ ) , π { u : q 1 ≤ R ( x , u ) ≤ q 2 } ≤ e − N ( J + G − 2 δ ) . (27) \pi\{u:R(x,u)\ge q_2\}\ge e^{-N(J+2\delta)}, \qquad\pi\{u:q_1\le R(x,u)\le q_2\}\le e^{-N(J+G-2\delta)}. \tag*{(27)} π { u : R ( x , u ) ≥ q 2 } ≥ e − N ( J + 2 δ ) , π { u : q 1 ≤ R ( x , u ) ≤ q 2 } ≤ e − N ( J + G − 2 δ ) . (27) In words, around a Gibbs-typical point there is a region of Gibbs mass about e − N J e^{-NJ} e − N J q ∗ q_* q ∗
The second input is a uniform bound on the Gibbs masses of caps. By the replica concentration of Huang, Montanari, and Pham [93 Lemma 7.7] , for each fixed s > 0 s>0 s > 0 π \pi π s s s 2 e − c N 2e^{-cN} 2 e − c N 1 − e − c N 1-e^{-cN} 1 − e − c N
sup v ∈ S N π { x : R ( x , v ) ≥ s } ≤ [ 2 − s 2 s 2 ( π ⊗ π ) { R > s 2 / 2 } ] 1 / 2 . (28) \sup_{v\in S_N}\pi\{x:R(x,v)\ge s\}\le\left[\frac{2-s^2}{s^2}(\pi\otimes\pi)\{R>s^2/2\}\right]^{1/2}. \tag*{(28)} v ∈ S N sup π { x : R ( x , v ) ≥ s } ≤ [ s 2 2 − s 2 ( π ⊗ π ) { R > s 2 /2 } ] 1/2 . (28) The axis may depend on the field, and we need no union bound over axes. In the same way, we find that ∣ π ( U ) ∣ 2 ≤ s + e − c N |\pi(U)|^2 \le s + e^{-cN} ∣ π ( U ) ∣ 2 ≤ s + e − c N
The overlap-weighted Gibbs mean
The estimates (27) suggest an observable that records the region around x x x h : [ − 1 , 1 ] → [ 0 , K ] h : [-1,1] \to[0,K] h : [ − 1 , 1 ] → [ 0 , K ] h = K h = K h = K [ − 1 , q 1 ] [-1,q_1] [ − 1 , q 1 ] h = 0 h = 0 h = 0 [ q 2 , 1 ] [q_2,1] [ q 2 , 1 ] ψ u ( x ) = e − N h ( R ( x , u ) ) \psi_u(x) = e^{-Nh(R(x,u))} ψ u ( x ) = e − N h ( R ( x , u ))
B N ( x ) = ∫ U ( u ) w x ( d u ) , w x ( d u ) = ψ u ( x ) 2 π ( d u ) ∫ ψ u ′ ( x ) 2 π ( d u ′ ) . (29) B_N(x) = \int U(u)\,w_x(\mathrm{d}u), \qquad w_x(\mathrm{d}u) = \frac{\psi_u(x)^2\pi(\mathrm{d}u)}{\int\psi_{u'}(x)^2\pi(\mathrm{d}u')}. \tag*{(29)} B N ( x ) = ∫ U ( u ) w x ( d u ) , w x ( d u ) = ∫ ψ u ′ ( x ) 2 π ( d u ′ ) ψ u ( x ) 2 π ( d u ) . (29) In words, B N ( x ) B_N(x) B N ( x ) U U U u u u R ( x , u ) < q 1 R(x,u) < q_1 R ( x , u ) < q 1 w x w_x w x x x x ∣ B N ∣ ≤ 1 |B_N| \le1 ∣ B N ∣ ≤ 1
We show that B N B_N B N x ∈ G x \in\mathcal{G} x ∈ G (27) (Lemma 15.7). The first is that its gradient is small. Differentiating (29) , we find that ∇ B N ( x ) \nabla B_N(x) ∇ B N ( x ) w x w_x w x ( U ( u ) − B N ( x ) ) ⊗ ∇ log ψ u ( x ) (U(u) - B_N(x)) \otimes\nabla\log\psi_u(x) ( U ( u ) − B N ( x )) ⊗ ∇ log ψ u ( x ) ∇ log ψ u ( x ) \nabla\log\psi_u(x) ∇ log ψ u ( x ) u u u x x x w x w_x w x d N ( x ) d_N(x) d N ( x ) (29) , we obtain
∥ ∇ B N ( x ) ∥ H S 2 ≤ 4 N ∥ h ′ ∥ ∞ 2 π { u : q 1 ≤ R ( x , u ) ≤ q 2 } d N ( x ) ≤ 4 N ∥ h ′ ∥ ∞ 2 e − N ( G − 4 δ ) , (30) \lVert\nabla B_N(x)\rVert_{\mathrm{HS}}^2 \le4N\lVert h'\rVert_\infty^2 \frac{\pi\{u:q_1 \le R(x,u) \le q_2\}}{d_N(x)} \le4N\lVert h'\rVert_\infty^2 e^{-N(G-4\delta)}, \tag*{(30)} ∥ ∇ B N ( x ) ∥ HS 2 ≤ 4 N ∥ h ′ ∥ ∞ 2 d N ( x ) π { u : q 1 ≤ R ( x , u ) ≤ q 2 } ≤ 4 N ∥ h ′ ∥ ∞ 2 e − N ( G − 4 δ ) , (30) where we used d N ( x ) ≥ π { u : R ( x , u ) ≥ q 2 } d_N(x) \ge\pi\{u:R(x,u) \ge q_2\} d N ( x ) ≥ π { u : R ( x , u ) ≥ q 2 } (27) . The second property is alignment. The references with R ( x , u ) ≤ q 1 R(x,u) \le q_1 R ( x , u ) ≤ q 1 w x w_x w x e − 2 K N / d N ( x ) e^{-2KN}/d_N(x) e − 2 K N / d N ( x ) 2 K = J + 2 δ + 1 2K = J + 2\delta+ 1 2 K = J + 2 δ + 1 U ( x ) ⋅ B N ( x ) ≥ q 1 − ( 1 + q 1 ) e − N U(x) \cdot B_N(x) \ge q_1 - (1 + q_1)e^{-N} U ( x ) ⋅ B N ( x ) ≥ q 1 − ( 1 + q 1 ) e − N
The definition averages over all references u u u G \mathcal{G} G (27) may fail. The function B N B_N B N S N S_N S N ∥ ∇ B N ∥ H S 2 ≤ 4 N ∥ h ′ ∥ ∞ 2 \lVert\nabla B_N\rVert_{\mathrm{HS}}^2 \le4N\lVert h'\rVert_\infty^2 ∥ ∇ B N ∥ HS 2 ≤ 4 N ∥ h ′ ∥ ∞ 2 ∣ ∇ ψ u ∣ 2 ≤ N ∥ h ′ ∥ ∞ 2 ψ u 2 |\nabla\psi_u|^2 \le N\lVert h'\rVert_\infty^2\psi_u^2 ∣∇ ψ u ∣ 2 ≤ N ∥ h ′ ∥ ∞ 2 ψ u 2 w x w_x w x x x x G \mathcal{G} G
E v e c ( B N ) ≤ 4 N ∥ h ′ ∥ ∞ 2 ( e − N ( G − 4 δ ) + e − a N ) . (31) \mathcal{E}_{\mathrm{vec}}(B_N) \le4N\lVert h'\rVert_\infty^2\left(e^{-N(G-4\delta)} + e^{-aN}\right). \tag*{(31)} E vec ( B N ) ≤ 4 N ∥ h ′ ∥ ∞ 2 ( e − N ( G − 4 δ ) + e − a N ) . (31) From one observable to five conclusions
On the event of Proposition 14.13 and Corollary 14.16, the rest of the proof is deterministic (Sections 15 and 16). We list in Table 2.1 the tool that we use for each conclusion of Theorem 1.5 .
Table 2.1. The five conclusions of Theorem 1.5 and the deterministic results that derive them from the energy and the alignment of the overlap-weighted Gibbs mean.
For (i), we use the averaged variational principle of Harrell and Stubbe [86, Theorem 3.1] in the following form (Lemma 15.1). Suppose that a family of nonnegative trial functions f z f_z f z L 2 ( π ) L^2(\pi) L 2 ( π )
0 < T = ∫ ∥ f z ∥ 2 2 μ ( d z ) < ∞ 0<T=\int\|f_z\|_2^2\,\mu(\mathrm{d}z)<\infty 0 < T = ∫ ∥ f z ∥ 2 2 μ ( d z ) < ∞ r T rT r T π ( f z ) ≤ R \pi(f_z)\le R π ( f z ) ≤ R C C C Λ > r \Lambda>r Λ > r ( 1 − r / Λ ) T / ( R C ) (1-r/\Lambda)T/(RC) ( 1 − r /Λ ) T / ( R C ) f z ( x ) = α ( [ N [ 1 − ∣ B N ( x ) − z ∣ 2 / ρ 2 ] ] ) f_z(x)=\alpha([N[1-|B_N(x)-z|^2/\rho^2]]) f z ( x ) = α ([ N [ 1 − ∣ B N ( x ) − z ∣ 2 / ρ 2 ]]) z z z R N \mathbb{R}^N R N B N B_N B N ρ \rho ρ z z z B N B_N B N U U U G G G G G G (28) bounds its Gibbs mass by e − c N e^{-cN} e − c N ∫ f z ( x ) μ ( d z ) ≤ 1 \int f_z(x)\,\mu(\mathrm{d}z)\le1 ∫ f z ( x ) μ ( d z ) ≤ 1 x x x 1 / 2 ≤ ∫ ∥ f z ∥ 2 2 μ ( d z ) ≤ 1 1/2\le\int\|f_z\|_2^2\,\mu(\mathrm{d}z)\le1 1/2 ≤ ∫ ∥ f z ∥ 2 2 μ ( d z ) ≤ 1 C N 2 ρ − 2 E v e c ( B N ) CN^2\rho^{-2}\mathcal{E}_{\mathrm{vec}}(B_N) C N 2 ρ − 2 E vec ( B N ) (31) . This shows that at least e c N e^{cN} e c N e − 2 γ N e^{-2\gamma N} e − 2 γ N R R R C C C
For (ii) and (iii), let Z N = U − π ( U ) Z_N=U-\pi(U) Z N = U − π ( U ) B ‾ N = B N − π ( B N ) \overline{B}_N=B_N-\pi(B_N) B N = B N − π ( B N ) Π Λ = 1 ( 0 , Λ ] ( − L ) \Pi_\Lambda=\mathbf{1}_{(0,\Lambda]}(-L) Π Λ = 1 ( 0 , Λ ] ( − L ) B ‾ N \overline{B}_N B N Λ \Lambda Λ E v e c ( B N ) / Λ \mathcal{E}_{\mathrm{vec}}(B_N)/\Lambda E vec ( B N ) /Λ
∥ Π Λ Z N ∥ 2 ∥ B ‾ N ∥ 2 ≥ ⟨ Z N , B ‾ N ⟩ − ( E v e c ( B N ) / Λ ) 1 / 2 (32) \|\Pi_\Lambda Z_N\|_2\|\overline{B}_N\|_2\ge\langle Z_N,\overline{B}_N\rangle-\left(\mathcal{E}_{\mathrm{vec}}(B_N)/\Lambda\right)^{1/2} \tag*{(32)} ∥ Π Λ Z N ∥ 2 ∥ B N ∥ 2 ≥ ⟨ Z N , B N ⟩ − ( E vec ( B N ) /Λ ) 1/2 (32) (Lemma 15.5). The alignment implies that ⟨ U , B N ⟩ ≥ q 1 − o ( 1 ) \langle U,B_N\rangle\ge q_1-o(1) ⟨ U , B N ⟩ ≥ q 1 − o ( 1 ) π ( U ) \pi(U) π ( U ) ⟨ Z N , B ‾ N ⟩ ≥ q 0 \langle Z_N,\overline{B}_N\rangle\ge q_0 ⟨ Z N , B N ⟩ ≥ q 0 q 0 < q 1 q_0<q_1 q 0 < q 1 q 1 q_1 q 1 ∥ B ‾ N ∥ 2 ≤ 1 \|\overline{B}_N\|_2\le1 ∥ B N ∥ 2 ≤ 1 Λ = e − 2 γ N \Lambda=e^{-2\gamma N} Λ = e − 2 γ N ∥ Π Λ Z N ∥ 2 2 ≥ q 0 2 − o ( 1 ) \|\Pi_\Lambda Z_N\|_2^2\ge q_0^2-o(1) ∥ Π Λ Z N ∥ 2 2 ≥ q 0 2 − o ( 1 ) b < q ∗ 2 b<q_*^2 b < q ∗ 2 q 0 q_0 q 0 q 0 2 > b q_0^2>b q 0 2 > b ⟨ Z N , P t Z N ⟩ ≥ e − Λ t ∥ Π Λ Z N ∥ 2 2 \langle Z_N,P_tZ_N\rangle\ge e^{-\Lambda t}\|\Pi_\Lambda Z_N\|_2^2 ⟨ Z N , P t Z N ⟩ ≥ e − Λ t ∥ Π Λ Z N ∥ 2 2 e γ N e^{\gamma N} e γ N
Conclusion (iv) follows from (2.2) and Markov’s inequality. Let T = e γ N T=e^{\gamma N} T = e γ N x x x e − c N e^{-cN} e − c N P T ( x , ⋅ ) P_T(x,\cdot) P T ( x , ⋅ ) 1 − e − γ N 1-e^{-\gamma N} 1 − e − γ N D x = { y : ∣ B N ( y ) − B N ( x ) ∣ ≤ ρ } D_x=\{y:|B_N(y)-B_N(x)|\le\rho\} D x = { y : ∣ B N ( y ) − B N ( x ) ∣ ≤ ρ } x ∈ G x\in G x ∈ G D x D_x D x G c \mathcal{G}^{c} G c B N ( x ) B_N(x) B N ( x ) w = e − c N w=e^{-cN} w = e − c N D x D_x D x T T T ∥ P T ( x , ⋅ ) − π ∥ T V ≥ 1 − e − γ N − w \lVert P_T(x,\cdot)-\pi\rVert_{\mathrm{TV}}\ge1-e^{-\gamma N}-w ∥ P T ( x , ⋅ ) − π ∥ TV ≥ 1 − e − γ N − w 0 ≤ t ≤ T 0\le t\le T 0 ≤ t ≤ T T T T D ( P T ( x , ⋅ ) ∥ π ) ≥ ( 1 − e − γ N ) log ( 1 / w ) − log 2 D(P_T(x,\cdot)\mathbin\Vert\pi)\ge(1-e^{-\gamma N})\log(1/w)-\log2 D ( P T ( x , ⋅ ) ∥ π ) ≥ ( 1 − e − γ N ) log ( 1/ w ) − log 2 0 < t ≤ T 0<t\le T 0 < t ≤ T c N cN c N N N N I ( X 0 ; X t ) = ∫ D ( P t ( x , ⋅ ) ∥ π ) π ( d x ) I(X_0;X_t)=\int D(P_t(x,\cdot)\mathbin\Vert\pi)\,\pi(dx) I ( X 0 ; X t ) = ∫ D ( P t ( x , ⋅ ) ∥ π ) π ( d x )
Correlation bounds at every level below q ∗ q_* q ∗
The proof above gives levels below q ∗ 2 q_*^2 q ∗ 2 q ∗ q_* q ∗ β 2 A \beta^2 A β 2 A § 2.1 ). Theorem 1.6 gives lower bounds approaching q ∗ q_* q ∗ (32) , where we used ∥ B N ‾ ∥ 2 2 ≤ 1 \lVert\overline{B_N}\rVert_2^2\le1 ∥ B N ∥ 2 2 ≤ 1 (29) that ∣ B N ( x ) ∣ 2 \lvert B_N(x)\rvert^2 ∣ B N ( x ) ∣ 2 w x w_x w x q ∗ q_* q ∗ x x x q ∗ q_* q ∗ ∥ B N ‾ ∥ 2 2 ≤ q ∗ + o ( 1 ) \lVert\overline{B_N}\rVert_2^2\le q_*+o(1) ∥ B N ∥ 2 2 ≤ q ∗ + o ( 1 ) (32) , for a fixed sufficiently small 0 < ϑ < min { 1 , q 1 / 3 } 0<\vartheta<\min\{1,q_1/3\} 0 < ϑ < min { 1 , q 1 /3 } γ \gamma γ
∥ Π Λ N Z N ∥ 2 2 ≥ ( q 1 − 3 ϑ ) 2 q ∗ + ϑ , Λ N = e − 2 γ N . (33) \lVert\Pi_{\Lambda_N}Z_N\rVert_2^2\ge\frac{(q_1-3\vartheta)^2}{q_*+\vartheta},\qquad\Lambda_N=e^{-2\gamma N}. \tag*{(33)} ∥ Π Λ N Z N ∥ 2 2 ≥ q ∗ + ϑ ( q 1 − 3 ϑ ) 2 , Λ N = e − 2 γ N . (33) Choosing q 1 q_1 q 1 q ∗ q_* q ∗ ϑ \vartheta ϑ b < q ∗ b<q_* b < q ∗ ∫ ∣ B N ∣ 2 d π → q ∗ \int\lvert B_N\rvert^2\,d\pi\to q_* ∫ ∣ B N ∣ 2 d π → q ∗
We prove it under the planted law. We write each u u u r = R ( x , u ) r=R(x,u) r = R ( x , u ) y y y x ⊥ x^\perp x ⊥ R ( u , u ′ ) = r r ′ + ( ( 1 − r 2 ) ( 1 − r ′ 2 ) ) 1 / 2 R n ( y , y ′ ) R(u,u')=rr'+((1-r^2)(1-r'^2))^{1/2}R_n(y,y') R ( u , u ′ ) = r r ′ + (( 1 − r 2 ) ( 1 − r ′2 ) ) 1/2 R n ( y , y ′ ) q ∗ q_* q ∗ w x w_x w x R n ( y , y ′ ) R_n(y,y') R n ( y , y ′ ) t 0 = q ∗ / ( 1 + q ∗ ) t_0=q_*/(1+q_*) t 0 = q ∗ / ( 1 + q ∗ ) q ∗ 2 + ( 1 − q ∗ 2 ) t 0 = q ∗ q_*^2+(1-q_*^2)t_0=q_* q ∗ 2 + ( 1 − q ∗ 2 ) t 0 = q ∗ q ∗ q_* q ∗ q ∗ q_* q ∗ t 0 t_0 t 0 η q ∗ \eta_{q_*} η q ∗ t t t t 2 t^2 t 2 s 1 s_1 s 1 s 2 s_2 s 2 s k s_k s k s = ( 1 , 1 ) s=(1,1) s = ( 1 , 1 ) k k k β 2 a k 2 ( 1 − t 0 k ) \beta^2a_k^2(1-t_0^k) β 2 a k 2 ( 1 − t 0 k ) a k 2 > 0 a_k^2>0 a k 2 > 0 t k t^k t k k k k t 0 t_0 t 0 t 0 2 t_0^2 t 0 2 ( ξ , β ) (\xi,\beta) ( ξ , β ) β 2 E ′ ′ ( 0 ) < 1 \beta^2\mathcal{E}''(0)<1 β 2 E ′′ ( 0 ) < 1 [93 Lemma 3.2, Remark 3.3, and Corollary 3.5] , which transfers the estimate to the original disorder. Neither the convex derivative convergence nor the likelihood transfer supplies a rate. The derivative argument does not give exponentially small planted failure probabilities, so the exponential transfer of Lemma 14.3 cannot be applied; this is why (ii) and (iii) of Theorem 1.6 are proved only with probability tending to one.
Eigenvalues near the dynamical threshold
The proof of Theorem 1.7 reuses the estimates above.
For Theorem 1.7, we count slow modes with smooth cap cutoffs f u ( x ) = φ ( R ( x , u ) ) f_u(x)=\varphi(R(x,u)) f u ( x ) = φ ( R ( x , u )) φ \varphi φ ( − ∞ , q 1 ] (-\infty,q_1] ( − ∞ , q 1 ] [ q 2 , ∞ ) [q_2,\infty) [ q 2 , ∞ ) 1 G ( u ) π ( d u ) \mathbf{1}_G(u)\pi(\mathrm{d}u) 1 G ( u ) π ( d u ) G G G β ↓ β d \beta\downarrow\beta_d β ↓ β d q d q_d q d A A A ε = β 2 A max − 1 \varepsilon=\beta^2A_{\max}-1 ε = β 2 A m a x − 1 r = q d + h z r=q_d+hz r = q d + h z h = ( ε / a ) 1 / ( 2 m ) h=(\varepsilon/a)^{1/(2m)} h = ( ε / a ) 1/ ( 2 m ) { β 2 A > 1 } \{\beta^2A>1\} { β 2 A > 1 } q ∗ q_* q ∗ Ψ ′ ( r ) ≈ q d ε ( 1 − z 2 m ) / ( 2 ( 1 − q d ) ) \Psi'(r)\approx q_d\varepsilon(1-z^{2m})/(2(1-q_d)) Ψ ′ ( r ) ≈ q d ε ( 1 − z 2 m ) / ( 2 ( 1 − q d )) z ∈ [ − 1 , 1 ] z\in[-1,1] z ∈ [ − 1 , 1 ]
Ψ ( q ∗ ) − Ψ ( q − ) ≈ q d 2 ( 1 − q d ) h ε ∫ − 1 1 ( 1 − z 2 m ) d z = C d ε 1 + 1 / ( 2 m ) , (34) \Psi(q_*)-\Psi(q_-)\approx\frac{q_d}{2(1-q_d)}h\varepsilon\int_{-1}^{1}(1-z^{2m})\,\mathrm{d}z=C_d\varepsilon^{1+1/(2m)}, \tag*{(34)} Ψ ( q ∗ ) − Ψ ( q − ) ≈ 2 ( 1 − q d ) q d h ε ∫ − 1 1 ( 1 − z 2 m ) d z = C d ε 1 + 1/ ( 2 m ) , (34) where q − q_- q − J J J T T T R R R C C C e κ ∗ N e^{\kappa_*N} e κ ∗ N κ ∗ \kappa_* κ ∗ ε \varepsilon ε ε > 0 \varepsilon>0 ε > 0 N → ∞ N\to\infty N → ∞ N N N ε \varepsilon ε ϑ \vartheta ϑ
Dependencies among the main steps
We list in Tables 2.2 and 2.3 the principal proof inputs for each main result, including selected intermediate results, in the order discussed above. We suppress most general preliminary facts, but retain the Crisanti–Sommers optimality criterion where it is a central variational input.
Table 2. Dependencies in Part II.
Table 2.3. Dependencies in Part III.
Preliminaries
Preliminaries
This section collects the definitions and general facts that are used in both halves of the paper. § 4.1.1 fixes our conventions for constants, probabilities, and the order in which parameters are chosen. § 4.1.2 defines the model and records two standard Gaussian facts: the regularity of the field and of its derivatives (Lemma 3.2) and the concentration of free energies over deterministic domains (Lemma 3.3 ). § 4.1.3 introduces the Langevin dynamics on the sphere (Definition 3.4 and Lemma 3.5) and the spectral calculus of its generator (Lemma 3.6 ). § 4.1.4 contains facts that hold for every Markov semigroup with an invariant law. These are the contraction of total variation and relative entropy (Lemma 3.8 ), binary data processing (Lemma 3.9 ), an endpoint criterion for separation in total variation (Proposition 3.10 ), and the chain identity for the mutual information between the two endpoints (Lemma 3.12 ). § 4.1.5 recalls the Crisanti–Sommers functional, the Parisi formula, and the first-variation characterization of the minimizer (Definition 3.13 , Theorem 3.14 , and Lemma 3.15 ), and it defines the thresholds β d \beta_{\mathrm d} β d β s \beta_{\mathrm s} β s Definition 3.16 and Lemma 3.18 ). Finally, § 4.1.6 states two results of Huang, Montanari, and Pham in our notation (Lemmas 3.19 and 3.20 ).
The facts in § 4.1.3 and § 4.1.4 concern a fixed field, and no randomness enters. They are used in Sections 4 and 13 for the condensed phase and in Sections 15 and 16 below the static threshold. The field estimates of § 4.1.2 are used in Part II, in the Kac–Rice formula and in the proof that the Gibbs measure concentrates on bands, and in Section 14. The static facts of § 4.1.5 enter Sections 5, 14, and 18. Lemma 3.19 is used in Sections 12 and 17, and Lemma 3.20 in Section 14. We prove the elementary statements and give precise citations for the others. Facts that are used in a single section are stated there.
Conventions
Throughout, N ≥ 2 N \ge2 N ≥ 2 ξ \xi ξ β \beta β N → ∞ N \to\infty N → ∞ N → ∞ N \to\infty N → ∞ ξ \xi ξ β \beta β N N N O ( ⋅ ) O(\cdot) O ( ⋅ ) o ( ⋅ ) o(\cdot) o ( ⋅ ) N N N H H H N N N H H H C r , N , H C_{r,N,H} C r , N , H
Probabilities. We refer to the randomness of the field as the disorder and write P \mathbb{P} P E \mathbb{E} E for all sufficiently large N N N N 0 N_0 N 0 N ≥ N 0 N \ge N_0 N ≥ N 0 N N N with probability at least 1 − C e − a N 1-Ce^{-aN} 1 − C e − a N C , a > 0 C,a>0 C , a > 0 1 − C e − a N 1-Ce^{-aN} 1 − C e − a N N N N with probability tending to one if its probability tends to one as N → ∞ N \to\infty N → ∞ 1 − C e − a N 1-Ce^{-aN} 1 − C e − a N
Order of parameters. When a statement or proof involves several parameters, they are chosen in a fixed order. First ξ \xi ξ β \beta β C C C c c c a a a γ \gamma γ κ \kappa κ N 0 N_0 N 0 β \beta β β d \beta_d β d β s \beta_s β s
Measures and information. For a fixed field, we write P μ \mathbb{P}_{\mu} P μ E μ \mathbb{E}_{\mu} E μ μ \mu μ P x \mathbb{P}_x P x E x \mathbb{E}_x E x μ = δ x \mu=\delta_x μ = δ x μ \mu μ ν \nu ν ∥ μ − ν ∥ T V = sup A ∣ μ ( A ) − ν ( A ) ∣ \lVert\mu-\nu\rVert_{\mathrm{TV}}=\sup_A\lvert\mu(A)-\nu(A)\rvert ∥ μ − ν ∥ TV = sup A ∣ μ ( A ) − ν ( A )∣ D ( μ ∥ ν ) = ∫ log ( d μ d ν ) d μ D(\mu\Vert\nu)=\int\log\left(\frac{\mathrm{d}\mu}{\mathrm{d}\nu}\right)\mathrm{d}\mu D ( μ ∥ ν ) = ∫ log ( d ν d μ ) d μ μ ≪ ν \mu\ll\nu μ ≪ ν D ( μ ∥ ν ) = ∞ D(\mu\Vert\nu)=\infty D ( μ ∥ ν ) = ∞ X X X Y Y Y p , a ∈ [ 0 , 1 ] p,a\in[0,1] p , a ∈ [ 0 , 1 ]
d ( p ∥ a ) = p log p a + ( 1 − p ) log 1 − p 1 − a , h ( p ) = − p log p − ( 1 − p ) log ( 1 − p ) , d(p\Vert a)=p\log\frac{p}{a}+(1-p)\log\frac{1-p}{1-a},\qquad h(p)=-p\log p-(1-p)\log(1-p), d ( p ∥ a ) = p log a p + ( 1 − p ) log 1 − a 1 − p , h ( p ) = − p log p − ( 1 − p ) log ( 1 − p ) , with 0 log 0 = 0 0\log0=0 0 log 0 = 0 d ( p ∥ a ) = ∞ d(p\Vert a)=\infty d ( p ∥ a ) = ∞ a ∈ { 0 , 1 } a\in\{0,1\} a ∈ { 0 , 1 } p ≠ a p\ne a p = a 0 log ( 0 / a ) = 0 0\log(0/a)=0 0 log ( 0/ a ) = 0 a ∈ [ 0 , 1 ] a\in[0,1] a ∈ [ 0 , 1 ] 0 ⋅ ∞ = 0 0\cdot\infty=0 0 ⋅ ∞ = 0 p log ( p / 0 ) = ∞ p\log(p/0)=\infty p log ( p /0 ) = ∞ p > 0 p>0 p > 0
The model
Definition 3.1 (Model). Let S N = { x ∈ R N : ∣ x ∣ 2 = N } S_N=\{x\in\mathbb{R}^N:\lvert x\rvert^2=N\} S N = { x ∈ R N : ∣ x ∣ 2 = N } ν N \nu_N ν N S N S_N S N x , y ∈ R N x,y\in\mathbb{R}^N x , y ∈ R N R ( x , y ) = x ⋅ y / N R(x,y)=x\cdot y/N R ( x , y ) = x ⋅ y / N U ( x ) = x / N U(x)=x/\sqrt{N} U ( x ) = x / N
ξ ( r ) = ∑ p = 2 P c p r p , c p ≥ 0 ( 2 ≤ p ≤ P ) . \xi(r) = \sum_{p=2}^{P} c_p r^p,\qquad c_p \ge0\quad(2 \le p \le P). ξ ( r ) = p = 2 ∑ P c p r p , c p ≥ 0 ( 2 ≤ p ≤ P ) . Its minimum degree is d = min { p : c p > 0 } d = \min\{p : c_p > 0\} d = min { p : c p > 0 } ξ \xi ξ pure if exactly one coefficient is nonzero and non-pure if at least two coefficients are nonzero. The field H N H_N H N S N S_N S N E H N ( x ) H N ( y ) = N ξ ( R ( x , y ) ) \mathbb{E}H_N(x)H_N(y) = N\xi(R(x,y)) E H N ( x ) H N ( y ) = N ξ ( R ( x , y )) β ≥ 0 \beta\ge0 β ≥ 0 Gibbs measure π = π N , β \pi= \pi_{N,\beta} π = π N , β partition function Z N , β Z_{N,\beta} Z N , β
π ( d x ) = e β H N ( x ) Z N , β ν N ( d x ) , Z N , β = ∫ S N e β H N d ν N . \pi(\mathrm{d}x) = \frac{e^{\beta H_N(x)}}{Z_{N,\beta}}\,\nu_N(\mathrm{d}x),\qquad Z_{N,\beta} = \int_{S_N} e^{\beta H_N}\,\mathrm{d}\nu_N. π ( d x ) = Z N , β e β H N ( x ) ν N ( d x ) , Z N , β = ∫ S N e β H N d ν N . The free energy is F N ( β ) = N − 1 E log Z N , β F_N(\beta) = N^{-1}\mathbb{E}\log Z_{N,\beta} F N ( β ) = N − 1 E log Z N , β F ( β ) = lim N → ∞ F N ( β ) F(\beta) = \lim_{N\to\infty}F_N(\beta) F ( β ) = lim N → ∞ F N ( β ) Theorem 3.14 .
We allow odd degrees and a quadratic term. Note that the Gibbs measure favors large values of H N H_N H N f f f S N S_N S N π ( f ) = ∫ f d π \pi(f) = \int f\,\mathrm{d}\pi π ( f ) = ∫ f d π π ( U ) ∈ R N \pi(U) \in\mathbb{R}^N π ( U ) ∈ R N x , y ∈ S N x,y \in S_N x , y ∈ S N ∣ U ( x ) ∣ = 1 \lvert U(x)\rvert= 1 ∣ U ( x )∣ = 1 R ( x , x ) = 1 R(x,x) = 1 R ( x , x ) = 1 ∣ R ( x , y ) ∣ ≤ 1 \lvert R(x,y)\rvert\le1 ∣ R ( x , y )∣ ≤ 1 x ∈ S N x \in S_N x ∈ S N H N ( x ) H_N(x) H N ( x ) N ξ ( 1 ) N\xi(1) N ξ ( 1 )
E Z N , β = ∫ S N E e β H N ( x ) ν N ( d x ) = e N β 2 ξ ( 1 ) / 2 . (35) \mathbb{E}Z_{N,\beta} = \int_{S_N}\mathbb{E}e^{\beta H_N(x)}\,\nu_N(\mathrm{d}x) = e^{N\beta^2\xi(1)/2}. \tag*{(35)} E Z N , β = ∫ S N E e β H N ( x ) ν N ( d x ) = e N β 2 ξ ( 1 ) /2 . (35) Lemma 3.2 (Representation, rotation invariance, and regularity of the field). Let ξ \xi ξ g i 1 … i p ( p ) g^{(p)}_{i_1\ldots i_p} g i 1 … i p ( p ) 2 ≤ p ≤ P 2 \le p \le P 2 ≤ p ≤ P 1 ≤ i 1 , … , i p ≤ N 1 \le i_1,\ldots,i_p \le N 1 ≤ i 1 , … , i p ≤ N
H N ( x ) = ∑ p = 2 P c p N − ( p − 1 ) / 2 ∑ i 1 , … , i p = 1 N g i 1 … i p ( p ) x i 1 ⋯ x i p , x ∈ R N . (36) H_N(x) = \sum_{p=2}^{P}\sqrt{c_p}\,N^{-(p-1)/2}\sum_{i_1,\ldots,i_p=1}^{N}g^{(p)}_{i_1\ldots i_p}x_{i_1}\cdots x_{i_p},\qquad x\in\mathbb{R}^N. \tag*{(36)} H N ( x ) = p = 2 ∑ P c p N − ( p − 1 ) /2 i 1 , … , i p = 1 ∑ N g i 1 … i p ( p ) x i 1 ⋯ x i p , x ∈ R N . (36) For all x , y ∈ R N x,y \in\mathbb{R}^N x , y ∈ R N E H N ( x ) H N ( y ) = N ξ ( R ( x , y ) ) \mathbb{E}H_N(x)H_N(y) = N\xi(R(x,y)) E H N ( x ) H N ( y ) = N ξ ( R ( x , y )) H N H_N H N S N S_N S N
For every g ∈ O ( N ) g \in O(N) g ∈ O ( N ) H N ∘ g − 1 H_N\circ g^{-1} H N ∘ g − 1 H N H_N H N
There exist K , c > 0 K,c > 0 K , c > 0 ξ \xi ξ N ≥ 2 N \ge2 N ≥ 2 1 − e − c N 1-e^{-cN} 1 − e − c N
sup ∣ x ∣ ≤ N ∣ H N ( x ) ∣ ≤ K N , sup ∣ x ∣ ≤ N ∣ ∇ H N ( x ) ∣ ≤ K N , \sup_{\lvert x\rvert\le\sqrt{N}}\lvert H_N(x)\rvert\le KN,\qquad
\sup_{\lvert x\rvert\le\sqrt{N}}\lvert\nabla H_N(x)\rvert\le K\sqrt{N}, ∣ x ∣ ≤ N sup ∣ H N ( x )∣ ≤ K N , ∣ x ∣ ≤ N sup ∣ ∇ H N ( x )∣ ≤ K N , sup ∣ x ∣ ≤ N ∥ ∇ 2 H N ( x ) ∥ op ≤ K , sup ∣ x ∣ ≤ N ∥ ∇ 3 H N ( x ) ∥ op ≤ K / N . (37) \sup_{\lvert x\rvert\le\sqrt{N}}\lVert\nabla^2H_N(x)\rVert_{\operatorname{op}} \le K,\qquad
\sup_{\lvert x\rvert\le\sqrt{N}}\lVert\nabla^3H_N(x)\rVert_{\operatorname{op}} \le K/\sqrt{N}. \tag*{(37)} ∣ x ∣ ≤ N sup ∥ ∇ 2 H N ( x ) ∥ op ≤ K , ∣ x ∣ ≤ N sup ∥ ∇ 3 H N ( x ) ∥ op ≤ K / N . (37) and, on the same event,
sup x ∈ S N ∣ ∇ S N H N ( x ) ∣ ≤ K N , sup x ∈ S N ∥ ∇ S N 2 H N ( x ) ∥ o p ≤ 2 K . (38) \sup_{x \in S_N} \left|\nabla_{S_N} H_N(x)\right| \le K\sqrt{N}, \qquad\sup_{x \in S_N} \left\|\nabla_{S_N}^{2} H_N(x)\right\|_{\mathrm{op}} \le2K. \tag*{(38)} x ∈ S N sup ∣ ∇ S N H N ( x ) ∣ ≤ K N , x ∈ S N sup ∇ S N 2 H N ( x ) op ≤ 2 K . (38) Here ∇ k \nabla^k ∇ k k k k ∥ T ∥ o p = sup { ∣ T [ v 1 , … , v k ] ∣ : ∣ v 1 ∣ , … , ∣ v k ∣ ≤ 1 } \|T\|_{\mathrm{op}} = \sup\{|T[v_1,\ldots,v_k]| : |v_1|,\ldots,|v_k| \le1\} ∥ T ∥ op = sup { ∣ T [ v 1 , … , v k ] ∣ : ∣ v 1 ∣ , … , ∣ v k ∣ ≤ 1 } k k k T T T ∇ S N \nabla_{S_N} ∇ S N ∇ S N 2 \nabla_{S_N}^{2} ∇ S N 2 S N S_N S N
Proof. For (1), since the variables g i 1 … i p ( p ) g_{i_1\ldots i_p}^{(p)} g i 1 … i p ( p )
E H N ( x ) H N ( y ) = ∑ p = 2 P c p N − ( p − 1 ) ∑ i 1 , … , i p = 1 N x i 1 y i 1 ⋯ x i p y i p = ∑ p = 2 P c p N 1 − p ( x ⋅ y ) p = N ξ ( R ( x , y ) ) . \begin{aligned}
\mathbb{E}H_N(x)H_N(y) = \sum_{p=2}^{P} c_p N^{-(p-1)} \sum_{i_1,\ldots,i_p=1}^{N} x_{i_1}y_{i_1}\cdots x_{i_p}y_{i_p} \\
&= \sum_{p=2}^{P} c_p N^{1-p}(x\cdot y)^p = N\xi(R(x,y)).
\end{aligned} E H N ( x ) H N ( y ) = p = 2 ∑ P c p N − ( p − 1 ) i 1 , … , i p = 1 ∑ N x i 1 y i 1 ⋯ x i p y i p = p = 2 ∑ P c p N 1 − p ( x ⋅ y ) p = N ξ ( R ( x , y )) . For (2), note that g − 1 g^{-1} g − 1 H N ∘ g − 1 H_N \circ g^{-1} H N ∘ g − 1 R N \mathbb{R}^N R N H N H_N H N P P P R N \mathbb{R}^N R N { 0 , 1 , … , P } N \{0,1,\ldots,P\}^N { 0 , 1 , … , P } N P P P P P P H N ∘ g − 1 H_N \circ g^{-1} H N ∘ g − 1 H N H_N H N
For (3), we use [93 Proposition 3.6] , which is quoted there from [91]. Their Hamiltonian [93 (1.1)] is (3.2) with γ p 2 = c p \gamma_p^2 = c_p γ p 2 = c p ∑ p 2 p γ p 2 < ∞ \sum_p 2^p\gamma_p^2 < \infty ∑ p 2 p γ p 2 < ∞ c p c_p c p k k k A A A ∥ A ∥ o p , N = N ( k − 2 ) / 2 ∥ A ∥ o p \|A\|_{\mathrm{op},N} = N^{(k-2)/2}\|A\|_{\mathrm{op}} ∥ A ∥ op , N = N ( k − 2 ) /2 ∥ A ∥ op ∣ x ∣ ≤ N |x| \le\sqrt{N} ∣ x ∣ ≤ N C k C_k C k c > 0 c > 0 c > 0 1 − e − c N 1-e^{-cN} 1 − e − c N sup ∣ x ∣ ≤ N N ( k − 2 ) / 2 ∥ ∇ k H N ( x ) ∥ o p ≤ C k \sup_{|x|\le\sqrt{N}} N^{(k-2)/2}\|\nabla^k H_N(x)\|_{\mathrm{op}} \le C_k sup ∣ x ∣ ≤ N N ( k − 2 ) /2 ∥ ∇ k H N ( x ) ∥ op ≤ C k k ≥ 0 k \ge0 k ≥ 0 k = 0 , 1 , 2 , 3 k=0,1,2,3 k = 0 , 1 , 2 , 3 K = max ( C 0 , C 1 , C 2 , C 3 ) K=\max(C_0,C_1,C_2,C_3) K = max ( C 0 , C 1 , C 2 , C 3 )
For (3.4), let x ∈ S N x \in S_N x ∈ S N P x = I − x x T / N P_x = I-xx^{\mathsf{T}}/N P x = I − x x T / N x x x ∇ S N H N ( x ) = P x ∇ H N ( x ) \nabla_{S_N}H_N(x)=P_x\nabla H_N(x) ∇ S N H N ( x ) = P x ∇ H N ( x ) x x x
∇ S N 2 H N ( x ) [ v , w ] = ∇ 2 H N ( x ) [ v , w ] − N − 1 ( x ⋅ ∇ H N ( x ) ) v ⋅ w \nabla_{S_N}^{2}H_N(x)[v,w] = \nabla^{2}H_N(x)[v,w] - N^{-1}(x\cdot\nabla H_N(x))\,v\cdot w ∇ S N 2 H N ( x ) [ v , w ] = ∇ 2 H N ( x ) [ v , w ] − N − 1 ( x ⋅ ∇ H N ( x )) v ⋅ w for tangent vectors v v v w w w ∣ ∇ S N H N ( x ) ∣ ≤ K N \left|\nabla_{S_N}H_N(x)\right| \le K\sqrt{N} ∣ ∇ S N H N ( x ) ∣ ≤ K N ∥ ∇ S N 2 H N ( x ) ∥ o p ≤ K + N − 1 ∣ x ∣ ∣ ∇ H N ( x ) ∣ ≤ 2 K \left\|\nabla_{S_N}^{2}H_N(x)\right\|_{\mathrm{op}} \le K+N^{-1}|x||\nabla H_N(x)| \le2K ∇ S N 2 H N ( x ) op ≤ K + N − 1 ∣ x ∣∣∇ H N ( x ) ∣ ≤ 2 K
We always use the extension (3.2) when Euclidean derivatives of H N H_N H N
Lemma 3.3 (Concentration of free energies). Let ξ \xi ξ β > 0 \beta> 0 β > 0 B ⊂ S N B \subset S_N B ⊂ S N ν N ( B ) > 0 \nu_N(B) > 0 ν N ( B ) > 0 h : S N → R h:S_N \to\mathbb{R} h : S N → R
Y = log ∫ B e β H N + h d ν N Y = \log\int_B e^{\beta H_N+h}\,\mathrm{d}\nu_N Y = log ∫ B e β H N + h d ν N is integrable and, for every t > 0 t > 0 t > 0
P ( ∣ Y − E Y ∣ ≥ t N ) ≤ 2 exp ( − t 2 N 2 β 2 ξ ( 1 ) ) . (39) \mathbb{P}\left(\lvert Y-\mathbb{E}Y\rvert\ge tN\right) \le2\exp\left(-\frac{t^2N}{2\beta^2\xi(1)}\right). \tag*{(39)} P ( ∣ Y − E Y ∣ ≥ tN ) ≤ 2 exp ( − 2 β 2 ξ ( 1 ) t 2 N ) . (39) Proof. Let g g g g i 1 … i p ( p ) g_{i_1\ldots i_p}^{(p)} g i 1 … i p ( p ) H N ( x ) = g ⋅ Ψ ( x ) H_N(x)=g\cdot\Psi(x) H N ( x ) = g ⋅ Ψ ( x ) Ψ ( x ) \Psi(x) Ψ ( x ) c p N − ( p − 1 ) / 2 x i 1 ⋯ x i p \sqrt{c_p}N^{-(p-1)/2}x_{i_1}\cdots x_{i_p} c p N − ( p − 1 ) /2 x i 1 ⋯ x i p ∣ Ψ ( x ) ∣ 2 = N ξ ( 1 ) \lvert\Psi(x)\rvert^2=N\xi(1) ∣ Ψ ( x ) ∣ 2 = N ξ ( 1 ) x ∈ S N x\in S_N x ∈ S N g ↦ Y g\mapsto Y g ↦ Y β \beta β Ψ \Psi Ψ B B B e β H N + h e^{\beta H_N+h} e β H N + h N ξ ( 1 ) \sqrt{N\xi(1)} N ξ ( 1 ) Y Y Y β N ξ ( 1 ) \beta\sqrt{N\xi(1)} β N ξ ( 1 ) g g g ∣ Y − log ν N ( B ) ∣ ≤ ∥ h ∥ ∞ + β ∣ g ∣ N ξ ( 1 ) \lvert Y-\log\nu_N(B)\rvert\le\lVert h\rVert_\infty+\beta\lvert g\rvert\sqrt{N\xi(1)} ∣ Y − log ν N ( B )∣ ≤ ∥ h ∥ ∞ + β ∣ g ∣ N ξ ( 1 ) Y Y Y L L L Y Y Y P ( ∣ Y − E Y ∣ ≥ s ) ≤ 2 e − s 2 / ( 2 L 2 ) \mathbb{P}(\lvert Y-\mathbb{E}Y\rvert\ge s)\le2e^{-s^2/(2L^2)} P (∣ Y − E Y ∣ ≥ s ) ≤ 2 e − s 2 / ( 2 L 2 ) s = t N s=tN s = tN L 2 = β 2 N ξ ( 1 ) L^2=\beta^2N\xi(1) L 2 = β 2 N ξ ( 1 )
The bounded function h h h N η ( R ( x , y ) ) N\eta(R(x,y)) N η ( R ( x , y )) η \eta η ξ ( 1 ) \xi(1) ξ ( 1 ) η ( 1 ) \eta(1) η ( 1 ) η ≡ 0 \eta\equiv0 η ≡ 0 B B B
Langevin dynamics on the sphere
We equip S N S_N S N R N \mathbb{R}^N R N ∇ S N \nabla_{S_N} ∇ S N Δ S N \Delta_{S_N} Δ S N y ∈ R N y\in\mathbb{R}^N y ∈ R N P y = I − y y T / N P_y=I-yy^{\mathsf{T}}/N P y = I − y y T / N y ∈ S N y\in S_N y ∈ S N y y y
Definition 3.4 (Langevin dynamics). Let β ≥ 0 \beta\ge0 β ≥ 0 H H H S N S_N S N H H H H N H_N H N π \pi π e β H e^{\beta H} e β H ν N \nu_N ν N generator L = L β , H L=L_{\beta,H} L = L β , H Dirichlet form E \mathcal{E} E f , g f,g f , g S N S_N S N
L f = Δ S N f + β ∇ S N H ⋅ ∇ S N f , E ( f , g ) = ∫ S N ∇ S N f ⋅ ∇ S N g d π , Lf=\Delta_{S_N}f+\beta\nabla_{S_N}H\cdot\nabla_{S_N}f,\qquad\mathcal{E}(f,g)=\int_{S_N}\nabla_{S_N}f\cdot\nabla_{S_N}g\,\mathrm{d}\pi, L f = Δ S N f + β ∇ S N H ⋅ ∇ S N f , E ( f , g ) = ∫ S N ∇ S N f ⋅ ∇ S N g d π , and we write E ( f ) = E ( f , f ) \mathcal{E}(f)=\mathcal{E}(f,f) E ( f ) = E ( f , f ) D ( E ) \mathcal{D}(\mathcal{E}) D ( E ) semigroup is P t = e t L P_t=e^{tL} P t = e t L t ≥ 0 t\ge0 t ≥ 0 − L -L − L λ 0 ≤ λ 1 ≤ λ 2 ≤ ⋯ \lambda_0\le\lambda_1\le\lambda_2\le\cdots λ 0 ≤ λ 1 ≤ λ 2 ≤ ⋯ B = ( B 1 , … , B m ) B=(B_1,\ldots,B_m) B = ( B 1 , … , B m ) D ( E ) \mathcal{D}(\mathcal{E}) D ( E ) vector energy , is E v e c ( B ) = ∑ i = 1 m E ( B i ) \mathcal{E}_{\mathrm{vec}}(B)=\sum_{i=1}^m\mathcal{E}(B_i) E vec ( B ) = ∑ i = 1 m E ( B i ) Langevin diffusion ( X t ) t ≥ 0 (X_t)_{t\ge0} ( X t ) t ≥ 0 S N S_N S N
We work in the real Hilbert space L 2 ( π ) L^2(\pi) L 2 ( π ) ⟨ f , g ⟩ = ∫ f g d π \langle f,g\rangle=\int fg\,\mathrm{d}\pi ⟨ f , g ⟩ = ∫ f g d π ∥ f ∥ 2 \|f\|_2 ∥ f ∥ 2 ∥ B ∥ 2 2 = ∫ ∣ B ∣ 2 d π \|B\|_2^2=\int|B|^2\,\mathrm{d}\pi ∥ B ∥ 2 2 = ∫ ∣ B ∣ 2 d π H 1 ( S N ) H^1(S_N) H 1 ( S N ) S N S_N S N L 2 ( ν N ) L^2(\nu_N) L 2 ( ν N )
Lemma 3.5 (Basic properties of the dynamics). Let β ≥ 0 \beta\ge0 β ≥ 0 H H H S N S_N S N π \pi π L L L E \mathcal{E} E
Integration by parts. For smooth f f f g g g ∫ f L g d π = − E ( f , g ) = ∫ g L f d π \int fLg\,\mathrm{d}\pi=-\mathcal{E}(f,g)=\int gLf\,\mathrm{d}\pi ∫ f Lg d π = − E ( f , g ) = ∫ g L f d π
Self-adjoint generator. The form E \mathcal{E} E L 2 ( π ) L^2(\pi) L 2 ( π ) D ( E ) = H 1 ( S N ) \mathcal{D}(\mathcal{E})=H^1(S_N) D ( E ) = H 1 ( S N ) A A A L 2 ( π ) L^2(\pi) L 2 ( π ) D ( A 1 / 2 ) = D ( E ) \mathcal{D}(A^{1/2})=\mathcal{D}(\mathcal{E}) D ( A 1/2 ) = D ( E ) E ( f ) = ∥ A 1 / 2 f ∥ 2 2 \mathcal{E}(f)=\|A^{1/2}f\|_2^2 E ( f ) = ∥ A 1/2 f ∥ 2 2 − L -L − L A A A
Discrete spectrum. The operator A A A ( ϕ j ) j ≥ 0 (\phi_j)_{j\ge0} ( ϕ j ) j ≥ 0 L 2 ( π ) L^2(\pi) L 2 ( π ) 0 = λ 0 < λ 1 ≤ λ 2 ≤ ⋯ 0=\lambda_0<\lambda_1\le\lambda_2\le\cdots 0 = λ 0 < λ 1 ≤ λ 2 ≤ ⋯ λ j → ∞ \lambda_j\to\infty λ j → ∞ A ϕ j = λ j ϕ j A\phi_j=\lambda_j\phi_j A ϕ j = λ j ϕ j j j j A A A ϕ 0 = 1 \phi_0=1 ϕ 0 = 1 f ∈ L 2 ( π ) f\in L^2(\pi) f ∈ L 2 ( π ) f ^ j = ⟨ f , ϕ j ⟩ \widehat{f}_j=\langle f,\phi_j\rangle f j = ⟨ f , ϕ j ⟩ ∥ f ∥ 2 2 = ∑ j f ^ j 2 \|f\|_2^2=\sum_j\widehat{f}_j^2 ∥ f ∥ 2 2 = ∑ j f j 2 f ∈ D ( E ) f\in\mathcal{D}(\mathcal{E}) f ∈ D ( E ) ∑ j λ j f ^ j 2 < ∞ \sum_j\lambda_j\widehat{f}_j^2<\infty ∑ j λ j f j 2 < ∞ E ( f ) = ∑ j λ j f ^ j 2 \mathcal{E}(f)=\sum_j\lambda_j\widehat{f}_j^2 E ( f ) = ∑ j λ j f j 2
Semigroup. The semigroup P t = e − t A P_t=e^{-tA} P t = e − t A ( t , x , y ) ↦ p t ( x , y ) (t,x,y)\mapsto p_t(x,y) ( t , x , y ) ↦ p t ( x , y ) ( 0 , ∞ ) × S N × S N (0,\infty)\times S_N\times S_N ( 0 , ∞ ) × S N × S N p t ≥ 0 p_t\ge0 p t ≥ 0 p t ( x , y ) = p t ( y , x ) p_t(x,y)=p_t(y,x) p t ( x , y ) = p t ( y , x ) ∫ p t ( x , y ) π ( d y ) = 1 \int p_t(x,y)\,\pi(\mathrm{d}y)=1 ∫ p t ( x , y ) π ( d y ) = 1 P t ( x , d y ) = p t ( x , y ) π ( d y ) P_t(x,\mathrm{d}y)=p_t(x,y)\pi(\mathrm{d}y) P t ( x , d y ) = p t ( x , y ) π ( d y ) t > 0 t>0 t > 0 P 0 ( x , ⋅ ) = δ x P_0(x,\cdot)=\delta_x P 0 ( x , ⋅ ) = δ x P t f ( x ) = ∫ f ( y ) P t ( x , d y ) P_tf(x)=\int f(y)P_t(x,\mathrm{d}y) P t f ( x ) = ∫ f ( y ) P t ( x , d y ) f ∈ L 2 ( π ) f\in L^2(\pi) f ∈ L 2 ( π ) x ∈ S N x\in S_N x ∈ S N t > 0 t>0 t > 0 π \pi π π ( d x ) P t ( x , d y ) = π ( d y ) P t ( y , d x ) \pi(\mathrm{d}x)P_t(x,\mathrm{d}y)=\pi(\mathrm{d}y)P_t(y,\mathrm{d}x) π ( d x ) P t ( x , d y ) = π ( d y ) P t ( y , d x )
The diffusion. Let ( W t ) t ≥ 0 (W_t)_{t\ge0} ( W t ) t ≥ 0 R N \mathbb{R}^N R N x ∈ S N x\in S_N x ∈ S N
d X t = 2 P X t d W t + β ∇ S N H ( X t ) d t − N − 1 N X t d t , X 0 = x , (40) \mathrm{d}X_t=\sqrt{2}P_{X_t}\,\mathrm{d}W_t+\beta\nabla_{S_N}H(X_t)\,\mathrm{d}t-\frac{N-1}{N}X_t\,\mathrm{d}t,\qquad X_0=x, \tag*{(40)} d X t = 2 P X t d W t + β ∇ S N H ( X t ) d t − N N − 1 X t d t , X 0 = x , (40) with coefficients extended smoothly and with compact support off a neighborhood of S N S_N S N S N S_N S N
P x ( X t ∈ ⋅ ) = P t ( x , ⋅ ) \mathbb{P}_x(X_t \in\cdot) = P_t(x, \cdot) P x ( X t ∈ ⋅ ) = P t ( x , ⋅ ) t ≥ 0 t \ge0 t ≥ 0 f f f S N S_N S N f ( X t ) − f ( x ) − ∫ 0 t L f ( X s ) d s f(X_t) - f(x) - \int_0^t Lf(X_s)\,\mathrm{d}s f ( X t ) − f ( x ) − ∫ 0 t L f ( X s ) d s
All of these facts are standard for a smooth elliptic operator on a compact connected manifold that is symmetric with respect to a smooth positive density. We refer to [17, Section 1.7] for the correspondence between closed forms, self-adjoint operators, and spectral decompositions. We give the short arguments below. The diffusion is the spherical Langevin dynamics studied, up to normalization, in [25, 65, 81]. The factor 2 \sqrt{2} 2 L L L 1 / 2 1/2 1/2
Proof. In this proof, ∇ = ∇ S N \nabla= \nabla_{S_N} ∇ = ∇ S N Δ = Δ S N \Delta= \Delta_{S_N} Δ = Δ S N (40) . We use three classical facts. The first is the Rellich–Kondrachov theorem: the inclusion H 1 ( S N ) ⊂ L 2 ( ν N ) H^1(S_N) \subset L^2(\nu_N) H 1 ( S N ) ⊂ L 2 ( ν N ) 1 − L 1-L 1 − L r ≥ 0 r \ge0 r ≥ 0 ϕ \phi ϕ A A A λ \lambda λ ∥ ϕ ∥ 2 = 1 \|\phi\|_2 = 1 ∥ ϕ ∥ 2 = 1
∥ ϕ ∥ H 2 r ( S N ) ≤ C r , N , H ( 1 + λ ) r . \|\phi\|_{H^{2r}(S_N)} \le C_{r,N,H}(1+\lambda)^r. ∥ ϕ ∥ H 2 r ( S N ) ≤ C r , N , H ( 1 + λ ) r . Together with Sobolev embedding in local charts [64, Theorem 12.16], applied with 2 r > k + ( N − 1 ) / 2 2r > k + (N-1)/2 2 r > k + ( N − 1 ) /2 ∥ ϕ ∥ C k ≤ C k , N , H ( 1 + λ ) m k \|\phi\|_{C^k} \le C_{k,N,H}(1+\lambda)^{m_k} ∥ ϕ ∥ C k ≤ C k , N , H ( 1 + λ ) m k m k m_k m k ρ = d π / d ν N \rho= \mathrm{d}\pi/\mathrm{d}\nu_N ρ = d π / d ν N ∇ ρ = β ρ ∇ H \nabla\rho= \beta\rho\nabla H ∇ ρ = β ρ ∇ H e − β osc H ≤ ρ ≤ e β osc H e^{-\beta\operatorname{osc} H} \le\rho\le e^{\beta\operatorname{osc} H} e − β osc H ≤ ρ ≤ e β osc H osc H = max H − min H \operatorname{osc} H = \max H - \min H osc H = max H − min H
Integration by parts. By the divergence theorem on the closed manifold S N S_N S N
∫ f Δ g ρ d ν N = − ∫ ∇ ( f ρ ) ⋅ ∇ g d ν N = − ∫ ∇ f ⋅ ∇ g ρ d ν N − β ∫ f ∇ H ⋅ ∇ g ρ d ν N . \int f\Delta g\,\rho\,\mathrm{d}\nu_N = -\int\nabla(f\rho)\cdot\nabla g\,\mathrm{d}\nu_N = -\int\nabla f\cdot\nabla g\,\rho\,\mathrm{d}\nu_N - \beta\int f\nabla H\cdot\nabla g\,\rho\,\mathrm{d}\nu_N. ∫ f Δ g ρ d ν N = − ∫ ∇ ( f ρ ) ⋅ ∇ g d ν N = − ∫ ∇ f ⋅ ∇ g ρ d ν N − β ∫ f ∇ H ⋅ ∇ g ρ d ν N . where in the last step we used ∇ ρ = β ρ ∇ H \nabla\rho= \beta\rho\nabla H ∇ ρ = β ρ ∇ H ∫ f L g d π = − E ( f , g ) \int fLg\,\mathrm{d}\pi= -\mathcal{E}(f,g) ∫ f Lg d π = − E ( f , g ) E \mathcal{E} E
The form and the operator. By the bounds on ρ \rho ρ ( E ( f ) + ∥ f ∥ 2 2 ) 1 / 2 (\mathcal{E}(f) + \|f\|_2^2)^{1/2} ( E ( f ) + ∥ f ∥ 2 2 ) 1/2 H 1 ( S N ) H^1(S_N) H 1 ( S N ) H 1 ( S N ) H^1(S_N) H 1 ( S N ) E \mathcal{E} E D ( E ) = H 1 ( S N ) D(\mathcal{E}) = H^1(S_N) D ( E ) = H 1 ( S N ) A A A g ∈ D ( A ) g \in D(A) g ∈ D ( A ) A g = k Ag = k A g = k g ∈ D ( E ) g \in D(\mathcal{E}) g ∈ D ( E ) E ( f , g ) = ⟨ f , k ⟩ \mathcal{E}(f,g) = \langle f,k\rangle E ( f , g ) = ⟨ f , k ⟩ f ∈ D ( E ) f \in D(\mathcal{E}) f ∈ D ( E ) D ( A 1 / 2 ) = D ( E ) D(A^{1/2}) = D(\mathcal{E}) D ( A 1/2 ) = D ( E ) E ( f ) = ∥ A 1 / 2 f ∥ 2 2 \mathcal{E}(f) = \|A^{1/2}f\|_2^2 E ( f ) = ∥ A 1/2 f ∥ 2 2 D ( E ) D(\mathcal{E}) D ( E ) g g g D ( A ) D(A) D ( A ) A g = − L g Ag = -Lg A g = − Lg
The spectrum. If A g + g = k Ag+g=k A g + g = k E ( g ) + ∥ g ∥ 2 2 = ⟨ g , k ⟩ ≤ ∥ g ∥ 2 ∥ k ∥ 2 \mathcal{E}(g)+\lVert g\rVert_{2}^{2}=\langle g,k\rangle\leq\lVert g\rVert_{2}\lVert k\rVert_{2} E ( g ) + ∥ g ∥ 2 2 = ⟨ g , k ⟩ ≤ ∥ g ∥ 2 ∥ k ∥ 2 ( A + 1 ) − 1 (A+1)^{-1} ( A + 1 ) − 1 L 2 ( π ) L^{2}(\pi) L 2 ( π ) H 1 ( S N ) H^{1}(S_N) H 1 ( S N ) L 2 ( π ) L^{2}(\pi) L 2 ( π ) L 2 ( ν N ) L^{2}(\nu_N) L 2 ( ν N ) ( A + 1 ) − 1 (A+1)^{-1} ( A + 1 ) − 1 A A A λ j ≥ 0 \lambda_j\geq0 λ j ≥ 0 A f = 0 Af=0 A f = 0 E ( f ) = ⟨ f , A f ⟩ = 0 \mathcal{E}(f)=\langle f,Af\rangle=0 E ( f ) = ⟨ f , A f ⟩ = 0 f f f S N S_N S N f f f A A A λ 0 = 0 \lambda_0=0 λ 0 = 0 ϕ 0 = 1 \phi_0=1 ϕ 0 = 1 λ 1 > 0 \lambda_1>0 λ 1 > 0 ∥ f ∥ 2 \lVert f\rVert^2 ∥ f ∥ 2 E ( f ) \mathcal{E}(f) E ( f ) A 1 / 2 f = ∑ j λ j 1 / 2 f ^ j ϕ j A^{1/2}f=\sum_j\lambda_j^{1/2}\hat{f}_j\phi_j A 1/2 f = ∑ j λ j 1/2 f ^ j ϕ j f ∈ D ( A 1 / 2 ) f\in D(A^{1/2}) f ∈ D ( A 1/2 ) A A A f ∈ D ( A ) f\in D(A) f ∈ D ( A ) f f f − L -L − L A A A A A A
The semigroup. We define p t ( x , y ) = ∑ j e − λ j t ϕ j ( x ) ϕ j ( y ) p_t(x,y)=\sum_j e^{-\lambda_jt}\phi_j(x)\phi_j(y) p t ( x , y ) = ∑ j e − λ j t ϕ j ( x ) ϕ j ( y ) C ∞ ( ( 0 , ∞ ) × S N × S N ) C^\infty((0,\infty)\times S_N\times S_N) C ∞ (( 0 , ∞ ) × S N × S N ) ρ \rho ρ − L -L − L − Δ -\Delta − Δ λ j ≥ e − 2 β osc H λ j ( − Δ ) \lambda_j\geq e^{-2\beta\operatorname{osc}H}\lambda_j(-\Delta) λ j ≥ e − 2 β osc H λ j ( − Δ ) Lemma 3.6 (5), whose proof uses only (3). The eigenvalues of − Δ -\Delta − Δ k ( k + N − 2 ) / N k(k+N-2)/N k ( k + N − 2 ) / N k ≥ 0 k\geq0 k ≥ 0 k k k ∑ j ( 1 + λ j ) − m < ∞ \sum_j(1+\lambda_j)^{-m}<\infty ∑ j ( 1 + λ j ) − m < ∞ m m m ∥ ϕ j ∥ C k \lVert\phi_j\rVert_{C^k} ∥ ϕ j ∥ C k f ∈ L 2 ( π ) f\in L^2(\pi) f ∈ L 2 ( π ) t > 0 t>0 t > 0 P t f = ∑ j e − λ j t f ^ j ϕ j P_tf=\sum_j e^{-\lambda_jt}\hat{f}_j\phi_j P t f = ∑ j e − λ j t f ^ j ϕ j ∑ j e − λ j t f ^ j ϕ j ( x ) = ∫ f ( y ) p t ( x , y ) π ( d y ) \sum_j e^{-\lambda_jt}\hat{f}_j\phi_j(x)=\int f(y)p_t(x,y)\,\pi(\mathrm{d}y) ∑ j e − λ j t f ^ j ϕ j ( x ) = ∫ f ( y ) p t ( x , y ) π ( d y ) x x x P t f P_tf P t f p t p_t p t ∫ p t ( x , y ) π ( d y ) = P t 1 ( x ) = 1 \int p_t(x,y)\,\pi(\mathrm{d}y)=P_t1(x)=1 ∫ p t ( x , y ) π ( d y ) = P t 1 ( x ) = 1 e − ( s + t ) A = e − s A e − t A e^{-(s+t)A}=e^{-sA}e^{-tA} e − ( s + t ) A = e − s A e − t A p t p_t p t p t ≥ 0 p_t\geq0 p t ≥ 0
The diffusion. We extend σ ( y ) = 2 P y \sigma(y)=\sqrt{2}P_y σ ( y ) = 2 P y b ( y ) = β ∇ H ( y ) − ( N − 1 ) y / N b(y)=\beta\nabla H(y)-(N-1)y/N b ( y ) = β ∇ H ( y ) − ( N − 1 ) y / N S N S_N S N R N \mathbb{R}^N R N u ( y ) = ∣ y ∣ 2 − N u(y)=|y|^2-N u ( y ) = ∣ y ∣ 2 − N u ( X t ) u(X_t) u ( X t ) 2 y ⋅ b ( y ) + Tr ( σ σ T ) ( y ) 2y\cdot b(y)+\operatorname{Tr}(\sigma\sigma^{\mathsf{T}})(y) 2 y ⋅ b ( y ) + Tr ( σ σ T ) ( y ) 2 y T σ ( y ) 2y^{\mathsf{T}}\sigma(y) 2 y T σ ( y ) y = X t y=X_t y = X t S N S_N S N y T P y = 0 y^{\mathsf{T}}P_y=0 y T P y = 0 y ⋅ ∇ H ( y ) = 0 y\cdot\nabla H(y)=0 y ⋅ ∇ H ( y ) = 0 Tr ( 2 P y 2 ) = 2 ( N − 1 ) \operatorname{Tr}(2P_y^2)=2(N-1) Tr ( 2 P y 2 ) = 2 ( N − 1 ) y ∈ S N y\in S_N y ∈ S N S N S_N S N u u u S N S_N S N u ( y ) α ( y ) u(y)\alpha(y) u ( y ) α ( y ) u ( y ) γ ( y ) u(y)\gamma(y) u ( y ) γ ( y ) α \alpha α γ \gamma γ u ( X t ) u(X_t) u ( X t ) S N S_N S N u ( X t ) = 0 u(X_t)=0 u ( X t ) = 0 t t t X X X
Next, let F F F R N \mathbb{R}^N R N y ∈ S N y\in S_N y ∈ S N N \sqrt{N} N Δ ( F ∣ S N ) ( y ) = Tr ( P y ∇ 2 F ( y ) P y ) − ( N − 1 ) y ⋅ ∇ F ( y ) / N \Delta(F|_{S_N})(y)=\operatorname{Tr}(P_y\nabla^2F(y)P_y)-(N-1)y\cdot\nabla F(y)/N Δ ( F ∣ S N ) ( y ) = Tr ( P y ∇ 2 F ( y ) P y ) − ( N − 1 ) y ⋅ ∇ F ( y ) / N ∇ \nabla ∇ ∇ 2 \nabla^2 ∇ 2
d F ( X t ) = L F ( X t ) d t + 2 ∇ F ( X t ) ⋅ P X t d W t , (41) \mathrm{d}F(X_t)=LF(X_t)\,\mathrm{d}t+\sqrt{2}\,\nabla F(X_t)\cdot P_{X_t}\,\mathrm{d}W_t, \tag*{(41)} d F ( X t ) = L F ( X t ) d t + 2 ∇ F ( X t ) ⋅ P X t d W t , (41) which proves the martingale property. We now apply (41) to e λ j t ϕ j ( X t ) e^{\lambda_jt}\phi_j(X_t) e λ j t ϕ j ( X t ) ϕ j \phi_j ϕ j R N \mathbb{R}^N R N L ϕ j = − λ j ϕ j L\phi_j=-\lambda_j\phi_j L ϕ j = − λ j ϕ j e λ j t ϕ j ( X t ) e^{\lambda_jt}\phi_j(X_t) e λ j t ϕ j ( X t ) E x ϕ j ( X t ) = e − λ j t ϕ j ( x ) = P t ϕ j ( x ) \mathbb{E}_x\phi_j(X_t)=e^{-\lambda_jt}\phi_j(x)=P_t\phi_j(x) E x ϕ j ( X t ) = e − λ j t ϕ j ( x ) = P t ϕ j ( x ) x x x X t X_t X t P x \mathbb{P}_x P x p t ( x , y ) π ( d y ) p_t(x,y)\,\pi(\mathrm{d}y) p t ( x , y ) π ( d y ) S N S_N S N f f f f ^ j = λ j − k ⟨ ( − L ) k f , ϕ j ⟩ \hat{f}_j=\lambda_j^{-k}\langle(-L)^k f,\phi_j\rangle f ^ j = λ j − k ⟨( − L ) k f , ϕ j ⟩ j ≥ 1 j\geq1 j ≥ 1 k k k f f f P x ( X t ∈ ⋅ ) = P t ( x , ⋅ ) \mathbb{P}_x(X_t\in\cdot)=P_t(x,\cdot) P x ( X t ∈ ⋅ ) = P t ( x , ⋅ ) p t ≥ 0 p_t\geq0 p t ≥ 0
Lemma 3.6 (Spectral calculus). In the setting of Lemma 3.5, let f ∈ L 2 ( π ) f\in L^2(\pi) f ∈ L 2 ( π ) and t ≥ 0 t\geq0 t ≥ 0
We have P t f = ∑ j e − t λ j f ^ j ϕ j P_tf=\sum_j e^{-t\lambda_j}\hat{f}_j\phi_j P t f = ∑ j e − t λ j f ^ j ϕ j ⟨ f , P t f ⟩ = ∑ j e − t λ j f ^ j 2 \langle f,P_tf\rangle=\sum_j e^{-t\lambda_j}\hat{f}_j^2 ⟨ f , P t f ⟩ = ∑ j e − t λ j f ^ j 2
If f ∈ D ( E ) f\in D(\mathcal{E}) f ∈ D ( E ) ∥ f − P t f ∥ 2 2 ≤ t E ( f ) \|f-P_tf\|_2^2\leq t\mathcal{E}(f) ∥ f − P t f ∥ 2 2 ≤ t E ( f ) ⟨ f , ( 1 − P t ) f ⟩ ≤ t E ( f ) \langle f,(1-P_t)f\rangle\leq t\mathcal{E}(f) ⟨ f , ( 1 − P t ) f ⟩ ≤ t E ( f )
If X 0 X_0 X 0 π \pi π E π [ f ( X 0 ) g ( X t ) ] = ⟨ f , P t g ⟩ \mathbb{E}_\pi[f(X_0)g(X_t)]=\langle f,P_tg\rangle E π [ f ( X 0 ) g ( X t )] = ⟨ f , P t g ⟩ g ∈ L 2 ( π ) g\in L^2(\pi) g ∈ L 2 ( π )
E π ∣ f ( X t ) − f ( X 0 ) ∣ 2 = 2 ⟨ f , ( 1 − P t ) f ⟩ ≤ 2 t E ( f ) if f ∈ D ( E ) . \mathbb{E}_\pi\lvert f(X_t)-f(X_0)\rvert^2=2\langle f,(1-P_t)f\rangle\leq2t\mathcal{E}(f)\qquad\text{if }f\in D(\mathcal{E}). E π ∣ f ( X t ) − f ( X 0 ) ∣ 2 = 2 ⟨ f , ( 1 − P t ) f ⟩ ≤ 2 t E ( f ) if f ∈ D ( E ) . For Λ > 0 \Lambda>0 Λ > 0 Π Λ = 1 ( 0 , Λ ] ( A ) \Pi_\Lambda=1_{(0,\Lambda]}(A) Π Λ = 1 ( 0 , Λ ] ( A ) Π Λ f = ∑ j : 0 < λ j ≤ Λ f ^ j ϕ j \Pi_\Lambda f=\sum_{j:0<\lambda_j\leq\Lambda}\hat{f}_j\phi_j Π Λ f = ∑ j : 0 < λ j ≤ Λ f ^ j ϕ j rank Π Λ = # { j ≥ 1 : λ j ≤ Λ } \operatorname{rank}\Pi_\Lambda=\#\{j\geq1:\lambda_j\leq\Lambda\} rank Π Λ = # { j ≥ 1 : λ j ≤ Λ } k ≥ 0 k\geq0 k ≥ 0 λ k ≤ Λ \lambda_k\leq\Lambda λ k ≤ Λ rank Π Λ ≥ k \operatorname{rank}\Pi_\Lambda\geq k rank Π Λ ≥ k
If π ( f ) = 0 \pi(f)=0 π ( f ) = 0 ⟨ f , P t f ⟩ ≥ e − t Λ ∥ Π Λ f ∥ 2 2 \langle f,P_tf\rangle\geq e^{-t\Lambda}\|\Pi_\Lambda f\|_2^2 ⟨ f , P t f ⟩ ≥ e − t Λ ∥ Π Λ f ∥ 2 2 f ∈ D ( E ) f\in D(\mathcal{E}) f ∈ D ( E ) ∥ f − π ( f ) − Π Λ f ∥ 2 2 ≤ E ( f ) / Λ \|f-\pi(f)-\Pi_\Lambda f\|_2^2\leq\mathcal{E}(f)/\Lambda ∥ f − π ( f ) − Π Λ f ∥ 2 2 ≤ E ( f ) /Λ
Min–max principle. For every $k\geq 0,
λ k = min V max f ∈ V ∖ { 0 } E ( f ) ∥ f ∥ 2 2 , \lambda_k=\min_{V}\max_{f\in V\setminus\{0\}}\frac{\mathcal{E}(f)}{\|f\|_2^2}, λ k = V min f ∈ V ∖ { 0 } max ∥ f ∥ 2 2 E ( f ) , where V V V ( k + 1 ) (k+1) ( k + 1 ) D ( E ) D(\mathcal{E}) D ( E ) ϕ 0 , … , ϕ k \phi_0,\ldots,\phi_k ϕ 0 , … , ϕ k V V V
Items (1)–(3) and the two estimates for functions in item (4) extend to vector-valued functions, by applying the operators componentwise and summing over components, with E v e c \mathcal{E}_{\mathrm{vec}} E vec E \mathcal{E} E A A A
E π ∣ B ( X t ) − B ( X 0 ) ∣ 2 ≤ 2 t E v e c ( B ) . \mathbb{E}_\pi\lvert B(X_t)-B(X_0)\rvert^2\leq2t\mathcal{E}_{\mathrm{vec}}(B). E π ∣ B ( X t ) − B ( X 0 ) ∣ 2 ≤ 2 t E vec ( B ) . Proof. Item (1) is the spectral theorem for P t = e − t A P_t=e^{-tA} P t = e − t A
∥ f − P t f ∥ 2 2 = ∑ j ( 1 − e − t λ j ) 2 f ^ j 2 ≤ ∑ j t λ j f ^ j 2 = t E ( f ) , \|f-P_tf\|_2^2=\sum_j(1-e^{-t\lambda_j})^2\hat{f}_j^2\leq\sum_j t\lambda_j\hat{f}_j^2=t\mathcal{E}(f), ∥ f − P t f ∥ 2 2 = j ∑ ( 1 − e − t λ j ) 2 f ^ j 2 ≤ j ∑ t λ j f ^ j 2 = t E ( f ) , where we used ( 1 − e − u ) 2 ≤ u (1-e^{-u})^2\le u ( 1 − e − u ) 2 ≤ u u ≥ 0 u\ge0 u ≥ 0 u ≤ 1 u\le1 u ≤ 1 1 − e − u ≤ u 1-e^{-u}\le u 1 − e − u ≤ u u ≥ 1 u\ge1 u ≥ 1 1 − e − u ≤ u 1-e^{-u}\le u 1 − e − u ≤ u
For (3), recall from Lemma 3.5(4)–(5) that the joint law of ( X 0 , X t ) (X_0,X_t) ( X 0 , X t ) P π \mathbb{P}_{\pi} P π π ( d x ) P t ( x , d y ) \pi(\mathrm{d}x)P_t(x,\mathrm{d}y) π ( d x ) P t ( x , d y ) E π [ f ( X 0 ) g ( X t ) ] = ∫ f ( x ) P t g ( x ) π ( d x ) = ⟨ f , P t g ⟩ \mathbb{E}_{\pi}[f(X_0)g(X_t)]=\int f(x)P_tg(x)\,\pi(\mathrm{d}x)=\langle f,P_tg\rangle E π [ f ( X 0 ) g ( X t )] = ∫ f ( x ) P t g ( x ) π ( d x ) = ⟨ f , P t g ⟩ π \pi π
E π ∣ f ( X t ) − f ( X 0 ) ∣ 2 = 2 ∥ f ∥ 2 2 − 2 ⟨ f , P t f ⟩ = 2 ⟨ f , ( 1 − P t ) f ⟩ , \mathbb{E}_{\pi}|f(X_t)-f(X_0)|^2=2\|f\|_2^2-2\langle f,P_tf\rangle=2\langle f,(1-P_t)f\rangle, E π ∣ f ( X t ) − f ( X 0 ) ∣ 2 = 2∥ f ∥ 2 2 − 2 ⟨ f , P t f ⟩ = 2 ⟨ f , ( 1 − P t ) f ⟩ , and the bound follows from (2).
For (4), note that the range of Π Λ \Pi_{\Lambda} Π Λ ϕ j \phi_j ϕ j j ≥ 1 j\ge1 j ≥ 1 λ j ≤ Λ \lambda_j\le\Lambda λ j ≤ Λ λ 0 = 0 \lambda_0=0 λ 0 = 0 λ k ≤ Λ \lambda_k\le\Lambda λ k ≤ Λ k + 1 k+1 k + 1 j ≥ 0 j\ge0 j ≥ 0 λ j ≤ Λ \lambda_j\le\Lambda λ j ≤ Λ rank Π Λ ≥ k \operatorname{rank}\Pi_{\Lambda}\ge k rank Π Λ ≥ k π ( f ) = 0 \pi(f)=0 π ( f ) = 0 f ^ 0 = 0 \widehat f_0=0 f 0 = 0 ⟨ f , P t f ⟩ \langle f,P_tf\rangle ⟨ f , P t f ⟩ 0 < λ j ≤ Λ 0<\lambda_j\le\Lambda 0 < λ j ≤ Λ ⟨ f , P t f ⟩ ≥ e − t Λ ∥ Π Λ f ∥ 2 2 \langle f,P_tf\rangle\ge e^{-t\Lambda}\|\Pi_{\Lambda}f\|_2^2 ⟨ f , P t f ⟩ ≥ e − t Λ ∥ Π Λ f ∥ 2 2 f ^ 0 = π ( f ) \widehat f_0=\pi(f) f 0 = π ( f ) f − π ( f ) − Π Λ f = ∑ j : λ j > Λ f ^ j ϕ j f-\pi(f)-\Pi_{\Lambda}f=\sum_{j:\lambda_j>\Lambda}\widehat f_j\phi_j f − π ( f ) − Π Λ f = ∑ j : λ j > Λ f j ϕ j Λ − 1 ∑ j λ j f ^ j 2 = E ( f ) / Λ \Lambda^{-1}\sum_j\lambda_j\widehat f_j^2=\mathcal{E}(f)/\Lambda Λ − 1 ∑ j λ j f j 2 = E ( f ) /Λ
For (5), let V V V ( k + 1 ) (k+1) ( k + 1 ) D ( E ) \mathcal{D}(\mathcal{E}) D ( E ) f f f ϕ 0 , … , ϕ k − 1 \phi_0,\ldots,\phi_{k-1} ϕ 0 , … , ϕ k − 1 k = 0 k=0 k = 0 f f f E ( f ) = ∑ j ≥ k λ j f ^ j 2 ≥ λ k ∥ f ∥ 2 2 \mathcal{E}(f)=\sum_{j\ge k}\lambda_j\widehat f_j^2\ge\lambda_k\|f\|_2^2 E ( f ) = ∑ j ≥ k λ j f j 2 ≥ λ k ∥ f ∥ 2 2 f f f ϕ 0 , … , ϕ k \phi_0,\ldots,\phi_k ϕ 0 , … , ϕ k E ( f ) = ∑ j ≤ k λ j f ^ j 2 ≤ λ k ∥ f ∥ 2 2 \mathcal{E}(f)=\sum_{j\le k}\lambda_j\widehat f_j^2\le\lambda_k\|f\|_2^2 E ( f ) = ∑ j ≤ k λ j f j 2 ≤ λ k ∥ f ∥ 2 2
Markov semigroups and relative entropy
In this section, E E E F F F K K K E E E F F F E E E μ K ( ⋅ ) = ∫ K ( x , ⋅ ) μ ( d x ) \mu K(\cdot)=\int K(x,\cdot)\,\mu(\mathrm{d}x) μ K ( ⋅ ) = ∫ K ( x , ⋅ ) μ ( d x ) F F F K f ( x ) = ∫ f ( y ) K ( x , d y ) Kf(x)=\int f(y)K(x,\mathrm{d}y) K f ( x ) = ∫ f ( y ) K ( x , d y ) E E E E E E E E E ( P t ) t ≥ 0 (P_t)_{t\ge0} ( P t ) t ≥ 0 E E E P 0 ( x , ⋅ ) = δ x P_0(x,\cdot)=\delta_x P 0 ( x , ⋅ ) = δ x P s + t = P s P t P_{s+t}=P_sP_t P s + t = P s P t s , t ≥ 0 s,t\ge0 s , t ≥ 0 π \pi π invariant if π P t = π \pi P_t=\pi π P t = π t t t S N S_N S N π \pi π
The following facts are classical. For data processing, we refer to [55] and [127, Theorem 2.17 and Corollary 2.18]. We include the short proofs.
Lemma 3.8 (Contraction). Let K K K E E E F F F μ , ν \mu,\nu μ , ν E E E
∥ μ K − ν K ∥ T V ≤ ∥ μ − ν ∥ T V , D ( μ K ∥ ν K ) ≤ D ( μ ∥ ν ) . \lVert\mu K-\nu K\rVert_{\mathrm{TV}}\le\lVert\mu-\nu\rVert_{\mathrm{TV}},\qquad D(\mu K\mathbin\Vert\nu K)\le D(\mu\mathbin\Vert\nu). ∥ μ K − ν K ∥ TV ≤ ∥ μ − ν ∥ TV , D ( μ K ∥ ν K ) ≤ D ( μ ∥ ν ) . Let ( P t ) (P_t) ( P t ) π \pi π μ \mu μ t ↦ ∥ μ P t − π ∥ T V t\mapsto\lVert\mu P_t-\pi\rVert_{\mathrm{TV}} t ↦ ∥ μ P t − π ∥ TV t ↦ D ( μ P t ∥ π ) t\mapsto D(\mu P_t\mathbin\Vert\pi) t ↦ D ( μ P t ∥ π ) [ 0 , ∞ ) [0,\infty) [ 0 , ∞ )
Proof. Let E + E_+ E + μ − ν \mu-\nu μ − ν ∥ μ − ν ∥ T V = ( μ − ν ) ( E + ) = − ( μ − ν ) ( E + c ) \lVert\mu-\nu\rVert_{\mathrm{TV}}=(\mu-\nu)(E_+) =-(\mu-\nu)(E_+^{\mathrm{c}}) ∥ μ − ν ∥ TV = ( μ − ν ) ( E + ) = − ( μ − ν ) ( E + c ) u u u 0 ≤ u ≤ 1 0\le u\le1 0 ≤ u ≤ 1 ( μ − ν ) ( E + c ) ≤ ∫ u d ( μ − ν ) ≤ ( μ − ν ) ( E + ) (\mu-\nu)(E_+^{\mathrm{c}})\le\int u\,\mathrm{d}(\mu-\nu)\le(\mu-\nu)(E_+) ( μ − ν ) ( E + c ) ≤ ∫ u d ( μ − ν ) ≤ ( μ − ν ) ( E + ) ∣ ∫ u d ( μ − ν ) ∣ ≤ ∥ μ − ν ∥ T V \left|\int u\,\mathrm{d}(\mu-\nu)\right|\le\lVert\mu-\nu\rVert_{\mathrm{TV}} ∫ u d ( μ − ν ) ≤ ∥ μ − ν ∥ TV u = K ( ⋅ , A ) u=K(\cdot,A) u = K ( ⋅ , A ) ∣ μ K ( A ) − ν K ( A ) ∣ ≤ ∥ μ − ν ∥ T V |\mu K(A)-\nu K(A)|\le\lVert\mu-\nu\rVert_{\mathrm{TV}} ∣ μ K ( A ) − ν K ( A ) ∣ ≤ ∥ μ − ν ∥ TV A ⊂ F A\subset F A ⊂ F
For relative entropy, we use the Donsker–Varadhan formula D ( μ ∥ ν ) = sup u { ∫ u d μ − log ∫ e u d ν } D(\mu\mathbin\Vert\nu)=\sup_u\left\{\int u\,\mathrm{d}\mu-\log\int e^u\,\mathrm{d}\nu\right\} D ( μ ∥ ν ) = sup u { ∫ u d μ − log ∫ e u d ν } u u u F F F K ( e u ) ≥ e K u K(e^u)\ge e^{Ku} K ( e u ) ≥ e K u
∫ u d ( μ K ) − log ∫ e u d ( ν K ) ≤ ∫ K u d μ − log ∫ e K u d ν ≤ D ( μ ∥ ν ) . \int u\,\mathrm{d}(\mu K)-\log\int e^u\,\mathrm{d}(\nu K)\le\int Ku\,\mathrm{d}\mu-\log\int e^{Ku}\,\mathrm{d}\nu\le D(\mu\mathbin\Vert\nu). ∫ u d ( μ K ) − log ∫ e u d ( ν K ) ≤ ∫ K u d μ − log ∫ e K u d ν ≤ D ( μ ∥ ν ) . In the last inequality, we used that K u Ku K u E E E u u u D ( μ K ∥ ν K ) ≤ D ( μ ∥ ν ) D(\mu K\mathbin\Vert\nu K)\le D(\mu\mathbin\Vert\nu) D ( μ K ∥ ν K ) ≤ D ( μ ∥ ν )
For the last assertion, let s , t ≥ 0 s,t\ge0 s , t ≥ 0 μ P t + s = ( μ P t ) P s \mu P_{t+s}=(\mu P_t)P_s μ P t + s = ( μ P t ) P s π = π P s \pi=\pi P_s π = π P s K = P s K=P_s K = P s ∥ μ P t + s − π ∥ T V ≤ ∥ μ P t − π ∥ T V \lVert\mu P_{t+s}-\pi\rVert_{\mathrm{TV}}\le\lVert\mu P_t-\pi\rVert_{\mathrm{TV}} ∥ μ P t + s − π ∥ TV ≤ ∥ μ P t − π ∥ TV D ( μ P t + s ∥ π ) ≤ D ( μ P t ∥ π ) D(\mu P_{t+s}\mathbin\Vert\pi)\le D(\mu P_t\mathbin\Vert\pi) D ( μ P t + s ∥ π ) ≤ D ( μ P t ∥ π )
Lemma 3.9 (Binary data processing). Let μ \mu μ ν \nu ν B B B p = μ ( B ) p=\mu(B) p = μ ( B ) a = ν ( B ) a=\nu(B) a = ν ( B )
D ( μ ∥ ν ) ≥ d ( p ∥ a ) ≥ p log 1 a − log 2. (42) D(\mu\mathbin\Vert\nu)\ge d(p\mathbin\Vert a)\ge p\log\frac{1}{a}-\log2. \tag*{(42)} D ( μ ∥ ν ) ≥ d ( p ∥ a ) ≥ p log a 1 − log 2. (42) In particular, if 0 < a < 1 0<a<1 0 < a < 1 p ≤ ( D ( μ ∥ ν ) + log 2 ) / log ( 1 / a ) p\le\bigl(D(\mu\mathbin\Vert\nu)+\log2\bigr)/\log(1/a) p ≤ ( D ( μ ∥ ν ) + log 2 ) / log ( 1/ a ) a = 0 a=0 a = 0 D ( μ ∥ ν ) < ∞ D(\mu\mathbin\Vert\nu)<\infty D ( μ ∥ ν ) < ∞ p = 0 p=0 p = 0
Proof. Let K K K x x x 1 B ( x ) ∈ { 0 , 1 } 1_B(x)\in\{0,1\} 1 B ( x ) ∈ { 0 , 1 } μ K \mu K μ K ν K \nu K ν K p p p a a a Lemma 3.8 , applied to this kernel
into { 0 , 1 } \{0,1\} { 0 , 1 } D ( μ ∥ ν ) ≥ D ( μ K ∥ ν K ) = d ( p ∥ a ) D(\mu\Vert\nu) \ge D(\mu K\Vert\nu K)=d(p\Vert a) D ( μ ∥ ν ) ≥ D ( μ K ∥ ν K ) = d ( p ∥ a ) 0 < a < 1 0<a<1 0 < a < 1
d ( p ∥ a ) = p log 1 a + ( 1 − p ) log 1 1 − a − h ( p ) ≥ p log 1 a − log 2 , d(p\Vert a)=p\log\frac{1}{a}+(1-p)\log\frac{1}{1-a}-h(p)\ge p\log\frac{1}{a}-\log2, d ( p ∥ a ) = p log a 1 + ( 1 − p ) log 1 − a 1 − h ( p ) ≥ p log a 1 − log 2 , since the middle term is nonnegative and h ( p ) ≤ log 2 h(p)\le\log2 h ( p ) ≤ log 2 a = 0 < p a=0<p a = 0 < p d ( p ∥ a ) = ∞ = p log ( 1 / a ) d(p\Vert a)=\infty=p\log(1/a) d ( p ∥ a ) = ∞ = p log ( 1/ a ) a = p = 0 a=p=0 a = p = 0 − log 2 -\log2 − log 2 0 ⋅ ∞ = 0 0\cdot\infty=0 0 ⋅ ∞ = 0 a = 1 a=1 a = 1 − log 2 -\log2 − log 2 d ( p ∥ a ) ≥ 0 d(p\Vert a)\ge0 d ( p ∥ a ) ≥ 0
The next proposition starts from a bound on the mass that the law at a single time T T T B B B [ 0 , T ] [0,T] [ 0 , T ]
Proposition 3.10 (Separation in total variation and transfer by relative entropy). Let ( P t ) (P_t) ( P t ) E E E π \pi π ν \nu ν E E E T ≥ 0 T\ge0 T ≥ 0 B ⊂ E B\subset E B ⊂ E a = ( ν P T ) ( B ) a=(\nu P_T)(B) a = ( ν P T ) ( B ) η = π ( B c ) \eta=\pi(B^c) η = π ( B c )
inf 0 ≤ t ≤ T ∥ ν P t − π ∥ T V ≥ 1 − η − a . (43) \inf_{0\le t\le T}\lVert\nu P_t-\pi\rVert_{\mathrm{TV}}\ge1-\eta-a. \tag*{(43)} 0 ≤ t ≤ T inf ∥ ν P t − π ∥ TV ≥ 1 − η − a . (43) Let μ \mu μ D ( μ ∥ ν ) < ∞ D(\mu\Vert\nu)<\infty D ( μ ∥ ν ) < ∞ 0 < a < 1 0<a<1 0 < a < 1
inf 0 ≤ t ≤ T ∥ μ P t − π ∥ T V ≥ 1 − η − min { 1 , D ( μ ∥ ν ) + log 2 log ( 1 / a ) } . (44) \inf_{0\le t\le T}\lVert\mu P_t-\pi\rVert_{\mathrm{TV}}\ge1-\eta-\min\left\{1,\frac{D(\mu\Vert\nu)+\log2}{\log(1/a)}\right\}. \tag*{(44)} 0 ≤ t ≤ T inf ∥ μ P t − π ∥ TV ≥ 1 − η − min { 1 , log ( 1/ a ) D ( μ ∥ ν ) + log 2 } . (44) If a = 0 a=0 a = 0 ( μ P T ) ( B ) = 0 (\mu P_T)(B)=0 ( μ P T ) ( B ) = 0 inf 0 ≤ t ≤ T ∥ μ P t − π ∥ T V ≥ 1 − η \inf_{0\le t\le T}\lVert\mu P_t-\pi\rVert_{\mathrm{TV}}\ge1-\eta inf 0 ≤ t ≤ T ∥ μ P t − π ∥ TV ≥ 1 − η
Proof. Let ρ \rho ρ E E E t ↦ ∥ ρ P t − π ∥ T V t\mapsto\lVert\rho P_t-\pi\rVert_{\mathrm{TV}} t ↦ ∥ ρ P t − π ∥ TV B B B T T T
inf 0 ≤ t ≤ T ∥ ρ P t − π ∥ T V = ∥ ρ P T − π ∥ T V ≥ π ( B ) − ( ρ P T ) ( B ) = 1 − η − ( ρ P T ) ( B ) . (45) \inf_{0\le t\le T}\lVert\rho P_t-\pi\rVert_{\mathrm{TV}}=\lVert\rho P_T-\pi\rVert_{\mathrm{TV}}\ge\pi(B)-(\rho P_T)(B)=1-\eta-(\rho P_T)(B). \tag*{(45)} 0 ≤ t ≤ T inf ∥ ρ P t − π ∥ TV = ∥ ρ P T − π ∥ TV ≥ π ( B ) − ( ρ P T ) ( B ) = 1 − η − ( ρ P T ) ( B ) . (45) With ρ = ν \rho=\nu ρ = ν
Now let ρ = μ \rho=\mu ρ = μ p = ( μ P T ) ( B ) p=(\mu P_T)(B) p = ( μ P T ) ( B ) K = P T K=P_T K = P T D ( μ P T ∥ ν P T ) ≤ D ( μ ∥ ν ) D(\mu P_T\Vert\nu P_T)\le D(\mu\Vert\nu) D ( μ P T ∥ ν P T ) ≤ D ( μ ∥ ν ) 0 < a < 1 0<a<1 0 < a < 1 μ P T \mu P_T μ P T ν P T \nu P_T ν P T B B B p ≤ ( D ( μ ∥ ν ) + log 2 ) / log ( 1 / a ) p\le(D(\mu\Vert\nu)+\log2)/\log(1/a) p ≤ ( D ( μ ∥ ν ) + log 2 ) / log ( 1/ a ) p ≤ 1 p\le1 p ≤ 1 a = 0 a=0 a = 0 D ( μ P T ∥ ν P T ) < ∞ D(\mu P_T\Vert\nu P_T)<\infty D ( μ P T ∥ ν P T ) < ∞ p = 0 p=0 p = 0
and Proposition 3.10 is applied for each fixed realization. For example, suppose that ( ν P T ) ( B ) ≤ N − ζ (\nu P_T)(B) \le N^{-\zeta} ( ν P T ) ( B ) ≤ N − ζ ζ > 0 \zeta> 0 ζ > 0
( μ P T ) ( B ) ≤ D ( μ ∥ ν ) + log 2 ζ log N . (\mu P_T)(B) \le\frac{D(\mu\Vert\nu)+\log2}{\zeta\log N}. ( μ P T ) ( B ) ≤ ζ log N D ( μ ∥ ν ) + log 2 . In particular, a field-dependent initial law μ \mu μ B B B T T T D ( μ ∥ ν ) D(\mu\Vert\nu) D ( μ ∥ ν ) ν \nu ν o ( log N ) o(\log N) o ( log N ) T T T T T T
The last lemma of this section is the chain identity for mutual information, known as the golden formula [127, Theorem 4.1], with the reference law π \pi π
Lemma 3.12 (Endpoint mutual information). Let K K K E E E π \pi π μ \mu μ E E E ( X 0 , X 1 ) (X_0,X_1) ( X 0 , X 1 ) μ ( d x ) K ( x , d y ) \mu(\mathrm{d}x)K(x,\mathrm{d}y) μ ( d x ) K ( x , d y ) [ 0 , ∞ ] [0,\infty] [ 0 , ∞ ]
∫ D ( K ( x , ⋅ ) ∥ π ) μ ( d x ) = I ( X 0 ; X 1 ) + D ( μ K ∥ π ) . (46) \int D(K(x,\cdot)\Vert\pi)\,\mu(\mathrm{d}x)=I(X_0;X_1)+D(\mu K\Vert\pi). \tag*{(46)} ∫ D ( K ( x , ⋅ ) ∥ π ) μ ( d x ) = I ( X 0 ; X 1 ) + D ( μ K ∥ π ) . (46) In particular, if D ( μ K ∥ π ) < ∞ D(\mu K\Vert\pi)<\infty D ( μ K ∥ π ) < ∞ I ( X 0 ; X 1 ) = ∫ D ( K ( x , ⋅ ) ∥ π ) μ ( d x ) − D ( μ K ∥ π ) I(X_0;X_1)=\int D(K(x,\cdot)\Vert\pi)\,\mu(\mathrm{d}x)-D(\mu K\Vert\pi) I ( X 0 ; X 1 ) = ∫ D ( K ( x , ⋅ ) ∥ π ) μ ( d x ) − D ( μ K ∥ π ) π K = π \pi K=\pi π K = π μ = π \mu=\pi μ = π
I ( X 0 ; X 1 ) = ∫ D ( K ( x , ⋅ ) ∥ π ) π ( d x ) . (47) I(X_0;X_1)=\int D(K(x,\cdot)\Vert\pi)\,\pi(\mathrm{d}x). \tag*{(47)} I ( X 0 ; X 1 ) = ∫ D ( K ( x , ⋅ ) ∥ π ) π ( d x ) . (47) For a Markov semigroup, we write I μ ( X 0 ; X t ) I_\mu(X_0;X_t) I μ ( X 0 ; X t ) ( X 0 , X t ) (X_0,X_t) ( X 0 , X t ) X 0 X_0 X 0 μ \mu μ K = P t K=P_t K = P t I ( X 0 ; X t ) = I π ( X 0 ; X t ) I(X_0;X_t)=I_\pi(X_0;X_t) I ( X 0 ; X t ) = I π ( X 0 ; X t )
Proof. Let Γ ( d x , d y ) = μ ( d x ) K ( x , d y ) \Gamma(\mathrm{d}x,\mathrm{d}y)=\mu(\mathrm{d}x)K(x,\mathrm{d}y) Γ ( d x , d y ) = μ ( d x ) K ( x , d y ) μ 1 = μ K \mu_1=\mu K μ 1 = μ K I ( X 0 ; X 1 ) = D ( Γ ∥ μ ⊗ μ 1 ) I(X_0;X_1)=D(\Gamma\Vert\mu\otimes\mu_1) I ( X 0 ; X 1 ) = D ( Γ∥ μ ⊗ μ 1 )
D ( Γ ∥ μ ⊗ π ) = ∫ D ( K ( x , ⋅ ) ∥ π ) μ ( d x ) , (48) D(\Gamma\Vert\mu\otimes\pi)=\int D(K(x,\cdot)\Vert\pi)\,\mu(\mathrm{d}x), \tag*{(48)} D ( Γ∥ μ ⊗ π ) = ∫ D ( K ( x , ⋅ ) ∥ π ) μ ( d x ) , (48) D ( Γ ∥ μ ⊗ π ) = D ( Γ ∥ μ ⊗ μ 1 ) + D ( μ 1 ∥ π ) . (49) D(\Gamma\Vert\mu\otimes\pi)=D(\Gamma\Vert\mu\otimes\mu_1)+D(\mu_1\Vert\pi). \tag*{(49)} D ( Γ∥ μ ⊗ π ) = D ( Γ∥ μ ⊗ μ 1 ) + D ( μ 1 ∥ π ) . (49) We begin with (3.14). Suppose that Γ ≪ μ ⊗ π \Gamma\ll\mu\otimes\pi Γ ≪ μ ⊗ π k k k μ \mu μ x x x K ( x , A ) = ∫ A k ( x , y ) π ( d y ) K(x,A)=\int_A k(x,y)\,\pi(\mathrm{d}y) K ( x , A ) = ∫ A k ( x , y ) π ( d y ) A A A A A A K ( x , ⋅ ) ≪ π K(x,\cdot)\ll\pi K ( x , ⋅ ) ≪ π k ( x , ⋅ ) k(x,\cdot) k ( x , ⋅ ) μ \mu μ x x x k log k + e − 1 ≥ 0 k\log k+e^{-1}\ge0 k log k + e − 1 ≥ 0 Γ \Gamma Γ μ ⊗ π \mu\otimes\pi μ ⊗ π K ( x , ⋅ ) ≪ π K(x,\cdot)\ll\pi K ( x , ⋅ ) ≪ π μ \mu μ x x x Γ \Gamma Γ μ ⊗ π \mu\otimes\pi μ ⊗ π
For (3.15), suppose first that μ 1 ≪̸ π \mu_1 \not\ll\pi μ 1 ≪ π μ 1 \mu_1 μ 1 Γ \Gamma Γ Lemma 3.8 , applied to the projection kernel from E × E E \times E E × E E E E D ( Γ ∥ μ ⊗ π ) ≥ D ( μ 1 ∥ π ) = ∞ D(\Gamma\Vert\mu\otimes\pi) \ge D(\mu_1\Vert\pi) = \infty D ( Γ∥ μ ⊗ π ) ≥ D ( μ 1 ∥ π ) = ∞ μ 1 ≪ π \mu_1 \ll\pi μ 1 ≪ π r r r μ ⊗ μ 1 \mu\otimes\mu_1 μ ⊗ μ 1 r ( y ) r(y) r ( y ) μ ⊗ π \mu\otimes\pi μ ⊗ π Γ ≪ μ ⊗ μ 1 \Gamma\ll\mu\otimes\mu_1 Γ ≪ μ ⊗ μ 1 Γ ≪ μ ⊗ π \Gamma\ll\mu\otimes\pi Γ ≪ μ ⊗ π Γ ≪ μ ⊗ π \Gamma\ll\mu\otimes\pi Γ ≪ μ ⊗ π ( μ ⊗ μ 1 ) ( E ′ ) = 0 (\mu\otimes\mu_1)(E') = 0 ( μ ⊗ μ 1 ) ( E ′ ) = 0 E ′ ⊂ E × E E' \subset E \times E E ′ ⊂ E × E r ( y ) = 0 r(y) = 0 r ( y ) = 0 ( μ ⊗ π ) (\mu\otimes\pi) ( μ ⊗ π ) ( x , y ) ∈ E ′ (x,y) \in E' ( x , y ) ∈ E ′ E ′ ∩ { ( x , y ) : r ( y ) > 0 } E' \cap\{(x,y) : r(y) > 0\} E ′ ∩ {( x , y ) : r ( y ) > 0 } ( μ ⊗ π ) (\mu\otimes\pi) ( μ ⊗ π ) Γ \Gamma Γ Γ ( { ( x , y ) : r ( y ) = 0 } ) = μ 1 ( { r = 0 } ) = 0 \Gamma(\{(x,y) : r(y) = 0\}) = \mu_1(\{r = 0\}) = 0 Γ ({( x , y ) : r ( y ) = 0 }) = μ 1 ({ r = 0 }) = 0 Γ ( E ′ ) = 0 \Gamma(E') = 0 Γ ( E ′ ) = 0 Γ ≪ μ ⊗ π \Gamma\ll\mu\otimes\pi Γ ≪ μ ⊗ π Γ ≪ μ ⊗ μ 1 \Gamma\ll\mu\otimes\mu_1 Γ ≪ μ ⊗ μ 1 k k k Γ \Gamma Γ μ ⊗ μ 1 \mu\otimes\mu_1 μ ⊗ μ 1 Γ \Gamma Γ k ( x , y ) r ( y ) k(x,y)r(y) k ( x , y ) r ( y ) μ ⊗ π \mu\otimes\pi μ ⊗ π
D ( Γ ∥ μ ⊗ π ) = ∫ log k d Γ + ∫ log r ( y ) Γ ( d x , d y ) = D ( Γ ∥ μ ⊗ μ 1 ) + D ( μ 1 ∥ π ) . D(\Gamma\Vert\mu\otimes\pi) = \int\log k\,\mathrm{d}\Gamma+ \int\log r(y)\,\Gamma(\mathrm{d}x,\mathrm{d}y) = D(\Gamma\Vert\mu\otimes\mu_1) + D(\mu_1\Vert\pi). D ( Γ∥ μ ⊗ π ) = ∫ log k d Γ + ∫ log r ( y ) Γ ( d x , d y ) = D ( Γ∥ μ ⊗ μ 1 ) + D ( μ 1 ∥ π ) . The splitting of the integral is legitimate because the negative parts of log k \log k log k log r \log r log r Γ \Gamma Γ ∫ ( log k ) − d Γ = ∫ k ( log k ) − d ( μ ⊗ μ 1 ) ≤ e − 1 \int(\log k)^{-}\,\mathrm{d}\Gamma= \int k(\log k)^{-}\,\mathrm{d}(\mu\otimes\mu_1) \le e^{-1} ∫ ( log k ) − d Γ = ∫ k ( log k ) − d ( μ ⊗ μ 1 ) ≤ e − 1
The remaining assertions follow from (3.12). If D ( μ K ∥ π ) < ∞ D(\mu K\Vert\pi) < \infty D ( μ K ∥ π ) < ∞ π K = π \pi K = \pi π K = π μ = π \mu= \pi μ = π D ( μ K ∥ π ) = 0 D(\mu K\Vert\pi) = 0 D ( μ K ∥ π ) = 0
For the Langevin semigroup, P t ( x , ⋅ ) P_t(x,\cdot) P t ( x , ⋅ ) p t ( x , ⋅ ) p_t(x,\cdot) p t ( x , ⋅ ) t > 0 t > 0 t > 0 t = 0 t = 0 t = 0 π ⊗ π \pi\otimes\pi π ⊗ π I ( X 0 ; X 0 ) = ∞ I(X_0;X_0) = \infty I ( X 0 ; X 0 ) = ∞
The Crisanti–Sommers functional and the static thresholds
The free energy of a spherical spin glass is given by a variational formula over probability measures on [ 0 , 1 ) [0,1) [ 0 , 1 )
Definition 3.13 (Crisanti–Sommers functional). Let M \mathcal{M} M ζ \zeta ζ [ 0 , 1 ) [0,1) [ 0 , 1 ) ζ ( [ 0 , q ^ ] ) = 1 \zeta([0,\hat{q}]) = 1 ζ ([ 0 , q ^ ]) = 1 q ^ < 1 \hat{q} < 1 q ^ < 1 ζ ∈ M \zeta\in\mathcal{M} ζ ∈ M
x ^ ζ ( t ) = ∫ t 1 ζ ( [ 0 , s ] ) d s = ∫ ( 1 − max ( t , s ) ) ζ ( d s ) , 0 ≤ t ≤ 1. \hat{x}_{\zeta}(t) = \int_t^1 \zeta([0,s])\,\mathrm{d}s = \int(1-\max(t,s))\,\zeta(\mathrm{d}s), \qquad0 \le t \le1. x ^ ζ ( t ) = ∫ t 1 ζ ([ 0 , s ]) d s = ∫ ( 1 − max ( t , s )) ζ ( d s ) , 0 ≤ t ≤ 1. A covariance polynomial is a polynomial η ( t ) = ∑ p = 1 P a p t p \eta(t)=\sum_{p=1}^{P}a_pt^p η ( t ) = ∑ p = 1 P a p t p a p ≥ 0 a_p\ge0 a p ≥ 0 η \eta η β ≥ 0 \beta\ge0 β ≥ 0 Crisanti–Sommers functional is
P β ( ζ ; η ) = 1 2 [ β 2 ( η ( 1 ) − ∫ η d ζ ) + ∫ 0 q ^ d t x ^ ζ ( t ) + log ( 1 − q ^ ) ] , (50) \mathcal{P}_{\beta}(\zeta;\eta)=\frac{1}{2}\left[\beta^2\left(\eta(1)-\int\eta\,\mathrm{d}\zeta\right)+\int_{0}^{\hat{q}}\frac{\mathrm{d}t}{\widehat{x}_{\zeta}(t)}+\log(1-\hat{q})\right], \tag*{(50)} P β ( ζ ; η ) = 2 1 [ β 2 ( η ( 1 ) − ∫ η d ζ ) + ∫ 0 q ^ x ζ ( t ) d t + log ( 1 − q ^ ) ] , (50) where q ^ < 1 \hat{q}<1 q ^ < 1 ζ ( [ 0 , q ^ ] ) = 1 \zeta([0,\hat{q}])=1 ζ ([ 0 , q ^ ]) = 1 P β ( η ) = inf ζ ∈ M P β ( ζ ; η ) \mathcal{P}_{\beta}(\eta)=\inf_{\zeta\in\mathcal{M}}\mathcal{P}_{\beta}(\zeta;\eta) P β ( η ) = inf ζ ∈ M P β ( ζ ; η )
The second expression for x ^ ζ \widehat{x}_{\zeta} x ζ x ^ ζ \widehat{x}_{\zeta} x ζ − ζ ( [ 0 , t ] ) -\zeta([0,t]) − ζ ([ 0 , t ]) x ^ ζ ( t ) = 1 − t \widehat{x}_{\zeta}(t)=1-t x ζ ( t ) = 1 − t t ≥ q ^ t\ge\hat{q} t ≥ q ^ ζ \zeta ζ [ 0 , q ^ ] [0,\hat{q}] [ 0 , q ^ ] x ^ ζ ( t ) ≥ 1 − max ( t , q ^ ) > 0 \widehat{x}_{\zeta}(t)\ge1-\max(t,\hat{q})>0 x ζ ( t ) ≥ 1 − max ( t , q ^ ) > 0 t < 1 t<1 t < 1 q ^ \hat{q} q ^ ∫ q ^ q ^ ′ d t / ( 1 − t ) = log ( 1 − q ^ ) − log ( 1 − q ^ ′ ) \int_{\hat{q}}^{\hat{q}'}\mathrm{d}t/(1-t)=\log(1-\hat{q})-\log(1-\hat{q}') ∫ q ^ q ^ ′ d t / ( 1 − t ) = log ( 1 − q ^ ) − log ( 1 − q ^ ′ ) η ( 0 ) = 0 \eta(0)=0 η ( 0 ) = 0
η ( 1 ) − ∫ η d ζ = ∫ 0 1 η ′ ( t ) ζ ( [ 0 , t ] ) d t = η ′ ( 0 ) x ^ ζ ( 0 ) + ∫ 0 1 η ′ ′ ( t ) x ^ ζ ( t ) d t . (51) \eta(1)-\int\eta\,\mathrm{d}\zeta=\int_{0}^{1}\eta'(t)\zeta([0,t])\,\mathrm{d}t=\eta'(0)\widehat{x}_{\zeta}(0)+\int_{0}^{1}\eta''(t)\widehat{x}_{\zeta}(t)\,\mathrm{d}t. \tag*{(51)} η ( 1 ) − ∫ η d ζ = ∫ 0 1 η ′ ( t ) ζ ([ 0 , t ]) d t = η ′ ( 0 ) x ζ ( 0 ) + ∫ 0 1 η ′′ ( t ) x ζ ( t ) d t . (51) For the first equality, we wrote η ( 1 ) − η ( s ) = ∫ s 1 η ′ ( t ) d t \eta(1)-\eta(s)=\int_s^1\eta'(t)\,\mathrm{d}t η ( 1 ) − η ( s ) = ∫ s 1 η ′ ( t ) d t x ^ ζ ′ ( t ) = − ζ ( [ 0 , t ] ) \widehat{x}_{\zeta}'(t)=-\zeta([0,t]) x ζ ′ ( t ) = − ζ ([ 0 , t ]) x ^ ζ ( 1 ) = 0 \widehat{x}_{\zeta}(1)=0 x ζ ( 1 ) = 0 P β ( ζ ; η ) \mathcal{P}_{\beta}(\zeta;\eta) P β ( ζ ; η ) P ( x ; ξ ) \mathcal{P}(x;\xi) P ( x ; ξ ) ξ = β 2 η \xi=\beta^2\eta ξ = β 2 η x ( t ) = ζ ( [ 0 , t ] ) x(t)=\zeta([0,t]) x ( t ) = ζ ([ 0 , t ])
Two choices of ζ \zeta ζ ζ = δ 0 \zeta=\delta_0 ζ = δ 0 x ^ δ 0 ( t ) = 1 − t \widehat{x}_{\delta_0}(t)=1-t x δ 0 ( t ) = 1 − t P β ( δ 0 ; η ) = β 2 η ( 1 ) / 2 \mathcal{P}_{\beta}(\delta_0;\eta)=\beta^2\eta(1)/2 P β ( δ 0 ; η ) = β 2 η ( 1 ) /2 ζ = x δ 0 + ( 1 − x ) δ Q \zeta=x\delta_0+(1-x)\delta_Q ζ = x δ 0 + ( 1 − x ) δ Q 0 < x < 1 0<x<1 0 < x < 1 0 < Q < 1 0<Q<1 0 < Q < 1 x ^ ζ ( t ) = 1 − Q + x ( Q − t ) \widehat{x}_{\zeta}(t)=1-Q+x(Q-t) x ζ ( t ) = 1 − Q + x ( Q − t ) t ≤ Q t\le Q t ≤ Q x ^ ζ ( t ) = 1 − t \widehat{x}_{\zeta}(t)=1-t x ζ ( t ) = 1 − t t ≥ Q t\ge Q t ≥ Q x ^ \widehat{x} x
Theorem 3.14 (Parisi formula; [46, 146]). Let η \eta η β ≥ 0 \beta\ge0 β ≥ 0 G N G_N G N S N S_N S N E G N ( x ) G N ( y ) = N η ( R ( x , y ) ) \mathbb{E}G_N(x)G_N(y)=N\eta(R(x,y)) E G N ( x ) G N ( y ) = N η ( R ( x , y ))
lim N → ∞ 1 N E log ∫ S N e β G N d ν N = P β ( η ) . \lim_{N\to\infty}\frac{1}{N}\mathbb{E}\log\int_{S_N}e^{\beta G_N}\,\mathrm{d}\nu_N=\mathcal{P}_{\beta}(\eta). N → ∞ lim N 1 E log ∫ S N e β G N d ν N = P β ( η ) . In particular, F ( β ) = P β ( ξ ) F(\beta)=\mathcal{P}_{\beta}(\xi) F ( β ) = P β ( ξ ) ξ \xi ξ β ≥ 0 \beta\ge0 β ≥ 0
For even η \eta η β p = β a p \beta_p=\beta\sqrt{a_p} β p = β a p N − 1 log ∫ e β G N d ν N N^{-1}\log\int e^{\beta G_N}\,\mathrm{d}\nu_N N − 1 log ∫ e β G N d ν N ∫ e β G N d ν N \int e^{\beta G_N}\,\mathrm{d}\nu_N ∫ e β G N d ν N G N G_N G N p p p η \eta η
The minimizer of the Crisanti–Sommers functional is characterized by its first variation. For ζ ∈ M \zeta\in\mathcal{M} ζ ∈ M
f ζ ( s ) = β 2 η ( s ) − ∫ 0 s s − t x ^ ζ ( t ) 2 d t , 0 ≤ s < 1 , (52) f_{\zeta}(s) = \beta^{2}\eta(s) - \int_{0}^{s} \frac{s-t}{\widehat{x}_{\zeta}(t)^{2}}\,\mathrm{d}t,\qquad0 \le s < 1, \tag*{(52)} f ζ ( s ) = β 2 η ( s ) − ∫ 0 s x ζ ( t ) 2 s − t d t , 0 ≤ s < 1 , (52) and let S ζ = { s ∈ [ 0 , 1 ) : f ζ ( s ) = sup [ 0 , 1 ) f ζ } S_{\zeta} = \{s \in[0,1): f_{\zeta}(s) = \sup_{[0,1)} f_{\zeta}\} S ζ = { s ∈ [ 0 , 1 ) : f ζ ( s ) = sup [ 0 , 1 ) f ζ } η = ξ \eta= \xi η = ξ ζ = x δ 0 + ( 1 − x ) δ Q \zeta= x\delta_{0} + (1-x)\delta_{Q} ζ = x δ 0 + ( 1 − x ) δ Q − f ζ -f_{\zeta} − f ζ V V V
Lemma 3.15 (First variation and the optimality criterion). Let η \eta η β ≥ 0 \beta\ge0 β ≥ 0
For every ζ ∈ M \zeta\in\mathcal{M} ζ ∈ M f ζ f_{\zeta} f ζ [ 0 , 1 ) [0,1) [ 0 , 1 ) − ∞ -\infty − ∞ 1 1 1 S ζ S_{\zeta} S ζ [ 0 , 1 ) [0,1) [ 0 , 1 )
For ζ , μ ∈ M \zeta,\mu\in\mathcal{M} ζ , μ ∈ M ε ↦ P β ( ( 1 − ε ) ζ + ε μ ; η ) \varepsilon\mapsto\mathcal{P}_{\beta}((1-\varepsilon)\zeta+\varepsilon\mu;\eta) ε ↦ P β (( 1 − ε ) ζ + ε μ ; η ) [ 0 , 1 ] [0,1] [ 0 , 1 ] ε = 0 \varepsilon=0 ε = 0 − ∫ f ζ d ( μ − ζ ) / 2 -\int f_{\zeta}\,\mathrm{d}(\mu-\zeta)/2 − ∫ f ζ d ( μ − ζ ) /2
P β ( μ ; η ) − P β ( ζ ; η ) ≥ 1 2 ( ∫ f ζ d ζ − ∫ f ζ d μ ) . (53) \mathcal{P}_{\beta}(\mu;\eta)-\mathcal{P}_{\beta}(\zeta;\eta) \ge\frac{1}{2}\left(\int f_{\zeta}\,\mathrm{d}\zeta-\int f_{\zeta}\,\mathrm{d}\mu\right). \tag*{(53)} P β ( μ ; η ) − P β ( ζ ; η ) ≥ 2 1 ( ∫ f ζ d ζ − ∫ f ζ d μ ) . (53) The functional P β ( ⋅ ; η ) \mathcal{P}_{\beta}(\cdot\,;\eta) P β ( ⋅ ; η ) M \mathcal{M} M ζ ∈ M \zeta\in\mathcal{M} ζ ∈ M ζ ( S ζ ) = 1 \zeta(S_{\zeta})=1 ζ ( S ζ ) = 1
We call the characterization in (3) the Crisanti–Sommers optimality criterion . Together with existence and uniqueness, it is due to Talagrand [146, Proposition 2.1]. In the present normalization it is [90, Lemma 1.2]; see also [143, Proposition 9]. For η = ξ \eta=\xi η = ξ Parisi measure of ( ξ , β ) (\xi,\beta) ( ξ , β )
Proof. For (1), continuity is clear. Fix q ^ < 1 \widehat{q}<1 q < 1 ζ ( [ 0 , q ^ ] ) = 1 \zeta([0,\widehat{q}])=1 ζ ([ 0 , q ]) = 1 q ^ < s < 1 \widehat{q}<s<1 q < s < 1 x ^ ζ ( t ) = 1 − t \widehat{x}_{\zeta}(t)=1-t x ζ ( t ) = 1 − t [ q ^ , s ] [\widehat{q},s] [ q , s ]
f ζ ( s ) ≤ β 2 η ( 1 ) − ∫ q ^ s s − t ( 1 − t ) 2 d t = β 2 η ( 1 ) − log 1 − q ^ 1 − s + 1 − 1 − s 1 − q ^ . f_{\zeta}(s) \le\beta^{2}\eta(1)-\int_{\widehat{q}}^{s}\frac{s-t}{(1-t)^{2}}\,\mathrm{d}t
=\beta^{2}\eta(1)-\log\frac{1-\widehat{q}}{1-s}+1-\frac{1-s}{1-\widehat{q}}. f ζ ( s ) ≤ β 2 η ( 1 ) − ∫ q s ( 1 − t ) 2 s − t d t = β 2 η ( 1 ) − log 1 − s 1 − q + 1 − 1 − q 1 − s . The right side tends to − ∞ -\infty − ∞ s → 1 s\to1 s → 1 f ζ f_{\zeta} f ζ S ζ S_{\zeta} S ζ [ 0 , s 0 ] [0,s_{0}] [ 0 , s 0 ] s 0 < 1 s_{0}<1 s 0 < 1
For (2), let ζ ε = ( 1 − ε ) ζ + ε μ \zeta_{\varepsilon}=(1-\varepsilon)\zeta+\varepsilon\mu ζ ε = ( 1 − ε ) ζ + ε μ q ^ < 1 \widehat{q}<1 q < 1 ζ ( [ 0 , q ^ ] ) = μ ( [ 0 , q ^ ] ) = 1 \zeta([0,\widehat{q}])=\mu([0,\widehat{q}])=1 ζ ([ 0 , q ]) = μ ([ 0 , q ]) = 1 ε ↦ x ^ ζ ε = x ^ ζ + ε ( x ^ μ − x ^ ζ ) \varepsilon\mapsto\widehat{x}_{\zeta_{\varepsilon}}=\widehat{x}_{\zeta}+\varepsilon(\widehat{x}_{\mu}-\widehat{x}_{\zeta}) ε ↦ x ζ ε = x ζ + ε ( x μ − x ζ ) x ^ ζ ε ≥ 1 − q ^ \widehat{x}_{\zeta_{\varepsilon}}\ge1-\widehat{q} x ζ ε ≥ 1 − q [ 0 , q ^ ] [0,\widehat{q}] [ 0 , q ] ε ∈ [ 0 , 1 ] \varepsilon\in[0,1] ε ∈ [ 0 , 1 ] ε \varepsilon ε ε \varepsilon ε u ↦ 1 / u u\mapsto1/u u ↦ 1/ u ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) 1 − q ^ 1-\widehat{q} 1 − q ε = 0 \varepsilon=0 ε = 0
− 1 2 [ β 2 ∫ η d ( μ − ζ ) + ∫ 0 q ^ x ^ μ ( t ) − x ^ ζ ( t ) x ^ ζ ( t ) 2 d t ] = − 1 2 ∫ G ( s ) ( μ − ζ ) ( d s ) , -\frac{1}{2}\left[\beta^{2}\int\eta\,\mathrm{d}(\mu-\zeta)+\int_{0}^{\widehat{q}}\frac{\widehat{x}_{\mu}(t)-\widehat{x}_{\zeta}(t)}{\widehat{x}_{\zeta}(t)^{2}}\,\mathrm{d}t\right]
=-\frac{1}{2}\int G(s)\,(\mu-\zeta)(\mathrm{d}s), − 2 1 [ β 2 ∫ η d ( μ − ζ ) + ∫ 0 q x ζ ( t ) 2 x μ ( t ) − x ζ ( t ) d t ] = − 2 1 ∫ G ( s ) ( μ − ζ ) ( d s ) , @page 68/200 starts_mid=0 ends_mid=1 where G ( s ) = β 2 η ( s ) + ∫ 0 q ^ ( 1 − max ( t , s ) ) x ^ ζ ( t ) − 2 d t G(s)=\beta^{2}\eta(s)+\int_{0}^{\widehat{q}}(1-\max(t,s))\widehat{x}_{\zeta}(t)^{-2}\,\mathrm{d}t G ( s ) = β 2 η ( s ) + ∫ 0 q ( 1 − max ( t , s )) x ζ ( t ) − 2 d t x ^ ζ \widehat{x}_{\zeta} x ζ s ≤ q ^ s\leq\widehat{q} s ≤ q 1 − max ( t , s ) = ( 1 − t ) − ( s − t ) + 1-\max(t,s)=(1-t)-(s-t)_{+} 1 − max ( t , s ) = ( 1 − t ) − ( s − t ) + G ( s ) = f ζ ( s ) + ∫ 0 q ^ ( 1 − t ) x ^ ζ ( t ) − 2 d t G(s)=f_{\zeta}(s)+\int_{0}^{\widehat{q}}(1-t)\widehat{x}_{\zeta}(t)^{-2}\,\mathrm{d}t G ( s ) = f ζ ( s ) + ∫ 0 q ( 1 − t ) x ζ ( t ) − 2 d t s s s μ − ζ \mu-\zeta μ − ζ [ 0 , q ^ ] [0,\widehat{q}] [ 0 , q ] − ∫ f ζ d ( μ − ζ ) / 2 -\int f_{\zeta}\,\mathrm{d}(\mu-\zeta)/2 − ∫ f ζ d ( μ − ζ ) /2 [ 0 , 1 ] [0,1] [ 0 , 1 ] 0 0 0 ε = 1 \varepsilon=1 ε = 1 (53) .
For (3), suppose first that ζ ( S ζ ) = 1 \zeta(S_{\zeta})=1 ζ ( S ζ ) = 1 ∫ f ζ d ζ = max f ζ ≥ ∫ f ζ d μ \int f_{\zeta}\,\mathrm{d}\zeta=\max f_{\zeta}\geq\int f_{\zeta}\,\mathrm{d}\mu ∫ f ζ d ζ = max f ζ ≥ ∫ f ζ d μ μ ∈ M \mu\in\mathcal{M} μ ∈ M (53) implies that ζ \zeta ζ ζ \zeta ζ r ∈ [ 0 , 1 ) r\in[0,1) r ∈ [ 0 , 1 ) μ = δ r \mu=\delta_{r} μ = δ r ε = 0 \varepsilon=0 ε = 0 f ζ ( r ) ≤ ∫ f ζ d ζ f_{\zeta}(r)\leq\int f_{\zeta}\,\mathrm{d}\zeta f ζ ( r ) ≤ ∫ f ζ d ζ r r r max f ζ ≤ ∫ f ζ d ζ ≤ max f ζ \max f_{\zeta}\leq\int f_{\zeta}\,\mathrm{d}\zeta\leq\max f_{\zeta} max f ζ ≤ ∫ f ζ d ζ ≤ max f ζ f ζ = max f ζ f_{\zeta}=\max f_{\zeta} f ζ = max f ζ ζ \zeta ζ ζ ( S ζ ) = 1 \zeta(S_{\zeta})=1 ζ ( S ζ ) = 1 ζ ≠ μ \zeta\neq\mu ζ = μ x ^ ζ ′ ( t ) = − ζ ( [ 0 , t ] ) \widehat{x}_{\zeta}'(t)=-\zeta([0,t]) x ζ ′ ( t ) = − ζ ([ 0 , t ]) x ^ ζ \widehat{x}_{\zeta} x ζ x ^ μ \widehat{x}_{\mu} x μ [ 0 , q ^ ] [0,\widehat{q}] [ 0 , q ] u ↦ 1 / u u\mapsto1/u u ↦ 1/ u ζ \zeta ζ μ \mu μ
Definition 3.16 (Static and dynamical thresholds). Let ζ \zeta ζ β ≥ 0 \beta\geq0 β ≥ 0 0 < r < 1 0<r<1 0 < r < 1
A ( r ) = ( 1 − r ) ζ ′ ( r ) r , Φ β ( r ) = β 2 ζ ( r ) + r + log ( 1 − r ) . A(r)=\frac{(1-r)\zeta'(r)}{r},\qquad\Phi_{\beta}(r)=\beta^{2}\zeta(r)+r+\log(1-r). A ( r ) = r ( 1 − r ) ζ ′ ( r ) , Φ β ( r ) = β 2 ζ ( r ) + r + log ( 1 − r ) . The dynamical threshold β d = β d ( ζ ) \beta_{\mathrm{d}}=\beta_{\mathrm{d}}(\zeta) β d = β d ( ζ ) static threshold β s = β s ( ζ ) \beta_{\mathrm{s}}=\beta_{\mathrm{s}}(\zeta) β s = β s ( ζ )
β d 2 = sup 0 < r < 1 A ( r ) , β s 2 = inf 0 < r < 1 − r − log ( 1 − r ) ζ ( r ) . \beta_{\mathrm{d}}^{2}=\sup_{0<r<1}A(r),\qquad\beta_{\mathrm{s}}^{2}=\inf_{0<r<1}\frac{-r-\log(1-r)}{\zeta(r)}. β d 2 = 0 < r < 1 sup A ( r ) , β s 2 = 0 < r < 1 inf ζ ( r ) − r − log ( 1 − r ) . The pair ( ζ , β ) (\zeta,\beta) ( ζ , β ) strictly replica symmetric if Φ β ( r ) < 0 \Phi_{\beta}(r)<0 Φ β ( r ) < 0 r ∈ ( 0 , 1 ) r\in(0,1) r ∈ ( 0 , 1 )
In the proof of Lemma 3.18 below, we show that Φ β \Phi_{\beta} Φ β f δ 0 f_{\delta_{0}} f δ 0 η = ζ \eta=\zeta η = ζ ( ζ , β ) (\zeta,\beta) ( ζ , β ) δ 0 \delta_{0} δ 0 S δ 0 = { 0 } S_{\delta_{0} }=\{0\} S δ 0 = { 0 } β 2 ζ ′ ′ ( 0 ) < 1 \beta^{2}\zeta''(0)<1 β 2 ζ ′′ ( 0 ) < 1 ( ζ , β ) (\zeta,\beta) ( ζ , β ) β 2 ζ ′ ′ ( 0 ) < 1 \beta^{2}\zeta''(0)<1 β 2 ζ ′′ ( 0 ) < 1
positive and continuous on ( 0 , 1 ) (0,1) ( 0 , 1 ) ∞ \infty ∞ 1 1 1 1 / ξ ′ ′ ( 0 ) 1/\xi^{\prime\prime}(0) 1/ ξ ′′ ( 0 ) 0 0 0 ∞ \infty ∞ ξ ′ ′ ( 0 ) = 0 \xi^{\prime\prime}(0)=0 ξ ′′ ( 0 ) = 0 Φ β ( r ) → 0 \Phi_\beta(r)\to0 Φ β ( r ) → 0 r → 0 r\to0 r → 0
Φ β ′ ( r ) = β 2 ξ ′ ( r ) − r 1 − r = r 1 − r ( β 2 A ( r ) − 1 ) , 0 < r < 1. (54) \Phi_\beta'(r)=\beta^2\xi'(r)-\frac{r}{1-r}=\frac{r}{1-r}\left(\beta^2A(r)-1\right),\qquad0<r<1. \tag*{(54)} Φ β ′ ( r ) = β 2 ξ ′ ( r ) − 1 − r r = 1 − r r ( β 2 A ( r ) − 1 ) , 0 < r < 1. (54) If β ≤ β d \beta\le\beta_d β ≤ β d β 2 A ≤ 1 \beta^2A\le1 β 2 A ≤ 1 (54) implies that Φ β \Phi_\beta Φ β ( 0 , 1 ) (0,1) ( 0 , 1 ) β 2 ≤ ( − r − log ( 1 − r ) ) / ξ ( r ) \beta^2\le(-r-\log(1-r))/\xi(r) β 2 ≤ ( − r − log ( 1 − r )) / ξ ( r ) r r r β ≤ β s \beta\le\beta_s β ≤ β s β d ≤ β s \beta_d\le\beta_s β d ≤ β s β > β s \beta>\beta_s β > β s β d < β < β s \beta_d<\beta<\beta_s β d < β < β s
Lemma 3.18 (The replica-symmetric regime). Let ξ \xi ξ β ≥ 0 \beta\ge0 β ≥ 0
We have F ( β ) ≤ β 2 ξ ( 1 ) / 2 F(\beta)\le\beta^2\xi(1)/2 F ( β ) ≤ β 2 ξ ( 1 ) /2 F ( β ) = β 2 ξ ( 1 ) / 2 F(\beta)=\beta^2\xi(1)/2 F ( β ) = β 2 ξ ( 1 ) /2 Φ β ( r ) ≤ 0 \Phi_\beta(r)\le0 Φ β ( r ) ≤ 0 r ∈ ( 0 , 1 ) r\in(0,1) r ∈ ( 0 , 1 ) β ≤ β s \beta\le\beta_s β ≤ β s δ 0 \delta_0 δ 0 ( ξ , β ) (\xi,\beta) ( ξ , β )
If β < β s \beta<\beta_s β < β s ( ξ , β ) (\xi,\beta) ( ξ , β ) β 2 ξ ′ ′ ( 0 ) < 1 \beta^2\xi^{\prime\prime}(0)<1 β 2 ξ ′′ ( 0 ) < 1
If 0 < β ≤ β s 0<\beta\le\beta_s 0 < β ≤ β s ε > 0 \varepsilon>0 ε > 0 N N N
P ( ∣ 1 N log Z N , β − β 2 ξ ( 1 ) 2 ∣ > ε ) ≤ 2 exp ( − ε 2 N 8 β 2 ξ ( 1 ) ) . (55) \mathbb{P}\left(\left|\frac{1}{N}\log Z_{N,\beta}-\frac{\beta^2\xi(1)}{2}\right|>\varepsilon\right)\le2\exp\left(-\frac{\varepsilon^2N}{8\beta^2\xi(1)}\right). \tag*{(55)} P ( N 1 log Z N , β − 2 β 2 ξ ( 1 ) > ε ) ≤ 2 exp ( − 8 β 2 ξ ( 1 ) ε 2 N ) . (55) For even mixtures, Talagrand proved that the strict form of the criterion in (1) is sufficient and that the non-strict form is necessary [146, Proposition 2.3]; see also [13, (5.2)] for the formula for β s \beta_s β s Theorem 3.14 and Lemma 3.15 , and so it applies to all finite mixtures, including those with odd degrees.
Proof. For (1), by Jensen’s inequality and (35) , we have F N ( β ) ≤ N − 1 log E Z N , β = β 2 ξ ( 1 ) / 2 F_N(\beta)\le N^{-1}\log\mathbb{E}Z_{N,\beta}=\beta^2\xi(1)/2 F N ( β ) ≤ N − 1 log E Z N , β = β 2 ξ ( 1 ) /2 N N N F ( β ) ≤ β 2 ξ ( 1 ) / 2 F(\beta)\le\beta^2\xi(1)/2 F ( β ) ≤ β 2 ξ ( 1 ) /2 ζ = δ 0 \zeta=\delta_0 ζ = δ 0 x ^ δ 0 ( t ) = 1 − t \hat{x}_{\delta_0}(t)=1-t x ^ δ 0 ( t ) = 1 − t
∫ 0 s s − t ( 1 − t ) 2 d t = ∫ 0 s ( 1 1 − t − 1 − s ( 1 − t ) 2 ) d t = − log ( 1 − s ) − s . \int_0^s\frac{s-t}{(1-t)^2}\,\mathrm{d}t=\int_0^s\left(\frac{1}{1-t}-\frac{1-s}{(1-t)^2}\right)\,\mathrm{d}t=-\log(1-s)-s. ∫ 0 s ( 1 − t ) 2 s − t d t = ∫ 0 s ( 1 − t 1 − ( 1 − t ) 2 1 − s ) d t = − log ( 1 − s ) − s . Comparing with (52) for η = ξ \eta=\xi η = ξ f δ 0 = Φ β f_{\delta_0}=\Phi_\beta f δ 0 = Φ β f δ 0 ( 0 ) = 0 f_{\delta_0}(0)=0 f δ 0 ( 0 ) = 0 Lemma 3.15 (3), δ 0 \delta_0 δ 0 0 ∈ S δ 0 0\in S_{\delta_0} 0 ∈ S δ 0 Φ β ≤ 0 \Phi_\beta\le0 Φ β ≤ 0 ( 0 , 1 ) (0,1) ( 0 , 1 ) P β ( δ 0 ; ξ ) = β 2 ξ ( 1 ) / 2 P_\beta(\delta_0;\xi)=\beta^2\xi(1)/2 P β ( δ 0 ; ξ ) = β 2 ξ ( 1 ) /2 Theorem 3.14 and the uniqueness of the minimizer, we find that F ( β ) = β 2 ξ ( 1 ) / 2 F(\beta)=\beta^2\xi(1)/2 F ( β ) = β 2 ξ ( 1 ) /2 δ 0 \delta_0 δ 0 ξ ( r ) > 0 \xi(r)>0 ξ ( r ) > 0 0 < r < 1 0<r<1 0 < r < 1 Φ β ( r ) ≤ 0 \Phi_\beta(r)\le0 Φ β ( r ) ≤ 0 β 2 ≤ ( − r − log ( 1 − r ) ) / ξ ( r ) \beta^2\le(-r-\log(1-r))/\xi(r) β 2 ≤ ( − r − log ( 1 − r )) / ξ ( r ) r ∈ ( 0 , 1 ) r\in(0,1) r ∈ ( 0 , 1 ) β ≤ β s \beta\le\beta_s β ≤ β s
For (2), let β < β s \beta<\beta_s β < β s r ∈ ( 0 , 1 ) r\in(0,1) r ∈ ( 0 , 1 ) β 2 < β s 2 ≤ ( − r − log ( 1 − r ) ) / ξ ( r ) \beta^2<\beta_s^2\le(-r-\log(1-r))/\xi(r) β 2 < β s 2 ≤ ( − r − log ( 1 − r )) / ξ ( r ) Φ β ( r ) < 0 \Phi_\beta(r)<0 Φ β ( r ) < 0 ξ ′ ′ ( 0 ) = 2 c 2 > 0 \xi^{\prime\prime}(0)=2c_2>0 ξ ′′ ( 0 ) = 2 c 2 > 0 ( − r − log ( 1 − r ) ) / ξ ( r ) → 1 / ξ ′ ′ ( 0 ) (-r-\log(1-r))/\xi(r)\to1/\xi^{\prime\prime}(0) ( − r − log ( 1 − r )) / ξ ( r ) → 1/ ξ ′′ ( 0 ) r → 0 r\to0 r → 0 Remark 3.17 . This implies β s 2 ≤ 1 / ξ ′ ′ ( 0 ) \beta_s^2\le1/\xi^{\prime\prime}(0) β s 2 ≤ 1/ ξ ′′ ( 0 ) β 2 ξ ′ ′ ( 0 ) < 1 \beta^2\xi^{\prime\prime}(0)<1 β 2 ξ ′′ ( 0 ) < 1 ξ ′ ′ ( 0 ) = 0 \xi^{\prime\prime}(0)=0 ξ ′′ ( 0 ) = 0 β 2 ξ ′ ′ ( 0 ) < 1 \beta^2\xi^{\prime\prime}(0)<1 β 2 ξ ′′ ( 0 ) < 1
For (3), note that F N ( β ) → β 2 ξ ( 1 ) / 2 F_N(\beta)\to\beta^2\xi(1)/2 F N ( β ) → β 2 ξ ( 1 ) /2 Theorem 3.14 . Then ∣ F N ( β ) − β 2 ξ ( 1 ) / 2 ∣ ≤ ε / 2 |F_N(\beta)-\beta^2\xi(1)/2|\le\varepsilon/2 ∣ F N ( β ) − β 2 ξ ( 1 ) /2∣ ≤ ε /2 N N N N N N (55) is contained in the event ∣ log Z N , β − E log Z N , β ∣ > ε N / 2 |\log Z_{N,\beta}-\mathbb{E}\log Z_{N,\beta}|>\varepsilon N/2 ∣ log Z N , β − E log Z N , β ∣ > εN /2 Lemma 3.3 with B = S N B=S_N B = S N h = 0 h=0 h = 0 t = ε / 2 t=\varepsilon/2 t = ε /2 (55) . □ \square □
Two inputs from Huang, Montanari, and Pham
We state two results of [93 ] in our notation. Both hold when ( ξ , β ) (\xi,\beta) ( ξ , β ) β 2 ξ ′ ′ ( 0 ) < 1 \beta^2\xi''(0)<1 β 2 ξ ′′ ( 0 ) < 1 Lemma 3.18 (2) is the case for every β < β s \beta<\beta_s β < β s [93 ] , the temperature is absorbed into the covariance. The field has covariance N η ( R ( x , y ) ) N\eta(R(x,y)) N η ( R ( x , y )) η ( t ) = ∑ p ≥ 2 γ p 2 t p \eta(t)=\sum_{p\ge2}\gamma_p^2t^p η ( t ) = ∑ p ≥ 2 γ p 2 t p ∑ p 2 p γ p 2 < ∞ \sum_p2^p\gamma_p^2<\infty ∑ p 2 p γ p 2 < ∞ e G e^G e G N \sqrt{N} N [93 (1.1)–(1.2)] . For our model at inverse temperature β \beta β η = β 2 ξ \eta=\beta^2\xi η = β 2 ξ G = β H N G=\beta H_N G = β H N [93 (2.21)] , reproduced as [93 (7.9)] , reads
η ′ ′ ( 0 ) < 1 , η ( t ) + t + log ( 1 − t ) < 0 for all t ∈ ( 0 , 1 ) . (56) \eta''(0)<1,\qquad\eta(t)+t+\log(1-t)<0\quad\text{for all }t\in(0,1). \tag*{(56)} η ′′ ( 0 ) < 1 , η ( t ) + t + log ( 1 − t ) < 0 for all t ∈ ( 0 , 1 ) . (56) For η = β 2 ξ \eta=\beta^2\xi η = β 2 ξ ( ξ , β ) (\xi,\beta) ( ξ , β ) β 2 ξ ′ ′ ( 0 ) < 1 \beta^2\xi''(0)<1 β 2 ξ ′′ ( 0 ) < 1
Lemma 3.19 (Normalized partition function; [93 Lemma 3.2 and Remark 3.3] ). Let η ( t ) = ∑ p = 2 P a p t p \eta(t)=\sum_{p=2}^P a_pt^p η ( t ) = ∑ p = 2 P a p t p (56) . For n ≥ 2 n\ge2 n ≥ 2 G n G_n G n S n S_n S n n η ( R ( x , y ) ) n\eta(R(x,y)) n η ( R ( x , y ))
W n = e − n η ( 1 ) / 2 ∫ S n e G n d ν n , W_n=e^{-n\eta(1)/2}\int_{S_n}e^{G_n}\,d\nu_n, W n = e − n η ( 1 ) /2 ∫ S n e G n d ν n , so that E W n = 1 \mathbb{E}W_n=1 E W n = 1 W n W_n W n e σ g − σ 2 / 2 e^{\sigma g-\sigma^2/2} e σ g − σ 2 /2 g g g σ 2 = − log ( 1 − η ′ ′ ( 0 ) ) / 2 \sigma^2=-\log(1-\eta''(0))/2 σ 2 = − log ( 1 − η ′′ ( 0 )) /2
The lower tail of W n W_n W n lim t ↓ 0 lim sup n → ∞ P ( W n < t ) = 0 \lim_{t\downarrow0}\limsup_{n\to\infty}\mathbb{P}(W_n<t)=0 lim t ↓ 0 lim sup n → ∞ P ( W n < t ) = 0
For each n n n Q n Q_n Q n G n G_n G n P n P_n P n d P n / d Q n = W n dP_n/dQ_n=W_n d P n / d Q n = W n A n A_n A n t > 0 t>0 t > 0
Q n ( A n ) ≤ t − 1 P n ( A n ) + Q n ( W n < t ) . Q_n(A_n)\le t^{-1}P_n(A_n)+Q_n(W_n<t). Q n ( A n ) ≤ t − 1 P n ( A n ) + Q n ( W n < t ) . In particular, P n ( A n ) → 0 P_n(A_n)\to0 P n ( A n ) → 0 Q n ( A n ) → 0 Q_n(A_n)\to0 Q n ( A n ) → 0
If ( ξ , β ) (\xi,\beta) ( ξ , β ) β 2 ξ ′ ′ ( 0 ) < 1 \beta^2\xi''(0)<1 β 2 ξ ′′ ( 0 ) < 1 Z N , β / E Z N , β Z_{N,\beta}/\mathbb{E}Z_{N,\beta} Z N , β / E Z N , β e σ g − σ 2 / 2 e^{\sigma g-\sigma^2/2} e σ g − σ 2 /2 σ 2 = − log ( 1 − β 2 ξ ′ ′ ( 0 ) ) / 2 \sigma^2=-\log(1-\beta^2\xi''(0))/2 σ 2 = − log ( 1 − β 2 ξ ′′ ( 0 )) /2
Proof. The convergence in distribution is [93 Lemma 3.2] , applied with their mixture equal to η \eta η γ p 2 = a p \gamma_p^2=a_p γ p 2 = a p [93 Remark 3.3] , it is proved there under (56) alone. The law of W n W_n W n G n G_n G n [93 (1.1)] may be replaced by any other. As in (35) , Fubini’s theorem gives E W n = 1 \mathbb{E}W_n=1 E W n = 1 { w ≤ t } \{w\le t\} { w ≤ t } lim sup n P ( W n < t ) ≤ P ( e σ g − σ 2 / 2 ≤ t ) \limsup_n\mathbb{P}(W_n<t)\le\mathbb{P}(e^{\sigma g-\sigma^2/2}\le t) lim sup n P ( W n < t ) ≤ P ( e σ g − σ 2 /2 ≤ t ) t ↓ 0 t\downarrow0 t ↓ 0 e σ g − σ 2 / 2 > 0 e^{\sigma g-\sigma^2/2}>0 e σ g − σ 2 /2 > 0
Q n ( A n ) ≤ Q n ( A n ∩ { W n ≥ t } ) + Q n ( W n < t ) ≤ t − 1 ∫ A n W n d Q n + Q n ( W n < t ) , Q_n(A_n)\le Q_n(A_n\cap\{W_n\ge t\})+Q_n(W_n<t)\le t^{-1}\int_{A_n}W_n\,dQ_n+Q_n(W_n<t), Q n ( A n ) ≤ Q n ( A n ∩ { W n ≥ t }) + Q n ( W n < t ) ≤ t − 1 ∫ A n W n d Q n + Q n ( W n < t ) , and the integral equals P n ( A n ) P_n(A_n) P n ( A n ) n → ∞ n \to\infty n → ∞ t ↓ 0 t \downarrow0 t ↓ 0 n = N n=N n = N η = β 2 ξ \eta=\beta^2\xi η = β 2 ξ G N = β H N G_N=\beta H_N G N = β H N E Z N , β \mathbb{E}Z_{N,\beta} E Z N , β
Item (2) is the one-sided contiguity of [93 Corollary 3.5] , which deduces it from [93 Lemma 3.2] by Le Cam’s first lemma. In Section 12, the lemma is applied in dimension n = N − 1 n=N-1 n = N − 1
Lemma 3.20 (Replica concentration; [93 Section 7.1 and Lemma 7.7] ). Let ξ \xi ξ β > 0 \beta>0 β > 0 ( ξ , β ) (\xi,\beta) ( ξ , β ) β 2 ξ ′ ′ ( 0 ) < 1 \beta^2\xi^{\prime\prime}(0)<1 β 2 ξ ′′ ( 0 ) < 1 s ∈ ( 0 , 1 ) s\in(0,1) s ∈ ( 0 , 1 ) c s > 0 c_s>0 c s > 0 N N N 1 − e − c s N 1-e^{-c_sN} 1 − e − c s N T s ⊂ S N T_s\subset S_N T s ⊂ S N
π ( T s c ) ≤ e − c s N and π ( { y : ∣ R ( x , y ) ∣ > s } ) ≤ e − c s N for every x ∈ T s . (57) \pi(T_s^c)\le e^{-c_sN}\quad\text{and}\quad\pi(\{y:|R(x,y)|>s\})\le e^{-c_sN}\quad\text{for every }x\in T_s. \tag*{(57)} π ( T s c ) ≤ e − c s N and π ({ y : ∣ R ( x , y ) ∣ > s }) ≤ e − c s N for every x ∈ T s . (57) On the same event,
( π ⊗ π ) ( { ( x , y ) : ∣ R ( x , y ) ∣ > s } ) ≤ 2 e − c s N . (58) (\pi\otimes\pi)(\{(x,y):|R(x,y)|>s\})\le2e^{-c_sN}. \tag*{(58)} ( π ⊗ π ) ({( x , y ) : ∣ R ( x , y ) ∣ > s }) ≤ 2 e − c s N . (58) Proof. We apply the results of [93 Section 7.1] to the model [93 (7.7)–(7.8)] with mixture η = β 2 ξ \eta=\beta^2\xi η = β 2 ξ u = 0 u=0 u = 0 [93 (7.9)] is (3.22), which holds by hypothesis. For u = 0 u=0 u = 0 ∥ u ∥ ≤ u ( δ ) N \lVert u\rVert\le u(\delta)\sqrt{N} ∥ u ∥ ≤ u ( δ ) N [93 Lemma 7.7] holds for every δ \delta δ s ∈ ( 0 , 1 ) s\in(0,1) s ∈ ( 0 , 1 ) δ = s \delta=s δ = s T ( s ) T(s) T ( s ) [93 (7.31)] consists of the points x ∈ S N x\in S_N x ∈ S N
∫ { y : ∣ R ( x , y ) ∣ > s } e β H N ( y ) ν N ( d y ) < e − c 1 ( s ) N min { Z N , β , e N β 2 ξ ( 1 ) / 2 } , \int_{\{y:|R(x,y)|>s\}} e^{\beta H_N(y)}\,\nu_N(\mathrm{d}y)<e^{-c_1(s)N}\min\{Z_{N,\beta},e^{N\beta^2\xi(1)/2}\}, ∫ { y : ∣ R ( x , y ) ∣ > s } e β H N ( y ) ν N ( d y ) < e − c 1 ( s ) N min { Z N , β , e N β 2 ξ ( 1 ) /2 } , for a constant c 1 ( s ) > 0 c_1(s)>0 c 1 ( s ) > 0 Z N , β Z_{N,\beta} Z N , β x ∈ T ( s ) x\in T(s) x ∈ T ( s ) π ( { y : ∣ R ( x , y ) ∣ > s } ) < e − c 1 ( s ) N \pi(\{y:|R(x,y)|>s\})<e^{-c_1(s)N} π ({ y : ∣ R ( x , y ) ∣ > s }) < e − c 1 ( s ) N [93 (7.34)] , with probability at least 1 − e − c 2 ( s ) N 1-e^{-c_2(s)N} 1 − e − c 2 ( s ) N π ( T ( s ) ) ≥ 1 − e − c 2 ( s ) N \pi(T(s))\ge1-e^{-c_2(s)N} π ( T ( s )) ≥ 1 − e − c 2 ( s ) N T ( s ) T(s) T ( s ) x ↦ ∫ { ∣ R ( x , y ) ∣ > s } e β H N ( y ) ν N ( d y ) x\mapsto\int_{\{|R(x,y)|>s\}}e^{\beta H_N(y)}\nu_N(\mathrm{d}y) x ↦ ∫ { ∣ R ( x , y ) ∣ > s } e β H N ( y ) ν N ( d y ) T s = T ( s ) T_s=T(s) T s = T ( s ) c s = min ( c 1 ( s ) , c 2 ( s ) ) c_s=\min(c_1(s),c_2(s)) c s = min ( c 1 ( s ) , c 2 ( s )) x ∈ T s x\in T_s x ∈ T s T s c T_s^c T s c
The proof of [93 Lemma 7.7] uses that N − 1 log Z N , β → β 2 ξ ( 1 ) / 2 N^{-1}\log Z_{N,\beta}\to\beta^2\xi(1)/2 N − 1 log Z N , β → β 2 ξ ( 1 ) /2 log Z N , β \log Z_{N,\beta} log Z N , β Φ β < 0 \Phi_\beta<0 Φ β < 0 ( 0 , 1 ) (0,1) ( 0 , 1 ) β ≤ β s \beta\le\beta_s β ≤ β s s s s s s s s ↓ 0 s\downarrow0 s ↓ 0
II The condensed phase
Total variation lower bounds from Gibbs concentration and hitting estimates
In this section, we reduce the separation from equilibrium of the dynamics started from the uniform law to estimates on the critical points of the center field H Q H_Q H Q Proposition 4.5 . It shows that separation up to a deterministic time T N T_N T N ν N \nu_N ν N T N T_N T N Proposition 4.9 reduces the first property to three estimates on single bands. In § 5.1.5 , we use the rotation invariance of the disorder to transfer the conclusion from the uniform initial law to every initial law independent of the field (Theorem 4.11 ), to field-dependent choices among polynomially many prescribed laws (Corollary 4.14 ), and to field-dependent initial laws whose relative entropy with respect to ν N \nu_N ν N o ( log N ) o(\log N) o ( log N ) Corollary 4.15 ).
No estimate on the critical points or on the Gibbs measure is proved here. The only property of the disorder that we use is the invariance of its law under rotations. From § 6.4 , we use the representation of H N H_N H N Lemma 3.8 ), binary data processing (Lemma 3.9 ), and the endpoint estimate of Proposition 3.10 . The results of this section are applied in Section 13, where the three properties above are verified using Sections 5 to 12.
Setting and notation
Throughout this part, ξ ( r ) = ∑ p c p r p \xi(r)=\sum_p c_p r^p ξ ( r ) = ∑ p c p r p c p ≥ 0 c_p\geq0 c p ≥ 0 P = { p : c p > 0 } \mathcal{P}=\{p:c_p>0\} P = { p : c p > 0 } N ≥ 2 N\geq2 N ≥ 2 H N \mathcal{H}_N H N R N \mathbb{R}^N R N P \mathcal{P} P ∣ ⋅ ∣ |\cdot| ∣ ⋅ ∣ H N H_N H N H N \mathcal{H}_N H N P H \mathbb{P}_{H} P H E H \mathbb{E}_{H} E H H H H H N \mathcal{H}_N H N g ∈ O ( N ) g\in O(N) g ∈ O ( N ) H g = H ∘ g − 1 H^g=H\circ g^{-1} H g = H ∘ g − 1 μ \mu μ S N S_N S N g ∗ μ = μ ∘ g − 1 g_*\mu=\mu\circ g^{-1} g ∗ μ = μ ∘ g − 1 g g g H N H_N H N R ( x , y ) R(x,y) R ( x , y )
H N g = d H N for every g ∈ O ( N ) . (59) H_N^g \overset{d}{=} H_N \qquad\text{for every } g \in O(N). \tag*{(59)} H N g = d H N for every g ∈ O ( N ) . (59) Fix β ≥ 0 \beta\ge0 β ≥ 0 H ∈ H N H \in\mathcal{H}_N H ∈ H N π H = π N , β H \pi^H = \pi^H_{N,\beta} π H = π N , β H P t H = e t L β , H P_t^H = e^{tL_{\beta,H}} P t H = e t L β , H P t H ( x , A ) P_t^H(x,A) P t H ( x , A ) μ \mu μ S N S_N S N ( μ P t H ) ( A ) = ∫ μ ( d x ) P t H ( x , A ) (\mu P_t^H)(A) = \int\mu(dx)P_t^H(x,A) ( μ P t H ) ( A ) = ∫ μ ( d x ) P t H ( x , A ) P μ H P_\mu^H P μ H ( X t ) t ≥ 0 (X_t)_{t\ge0} ( X t ) t ≥ 0 L β , H L_{\beta,H} L β , H μ \mu μ P x H P_x^H P x H μ = δ x \mu= \delta_x μ = δ x D ⊂ S N D \subset S_N D ⊂ S N
Z H ( D ) = ∫ D e β H d ν N , Z_H(D) = \int_D e^{\beta H} \,d\nu_N, Z H ( D ) = ∫ D e β H d ν N , so that Z N , β = Z H ( S N ) Z_{N,\beta} = Z_H(S_N) Z N , β = Z H ( S N ) π H ( D ) = Z H ( D ) / Z H ( S N ) \pi^H(D) = Z_H(D)/Z_H(S_N) π H ( D ) = Z H ( D ) / Z H ( S N )
Fix Q ∈ ( 0 , 1 ) Q \in(0,1) Q ∈ ( 0 , 1 ) H Q ( u ) = H ( Q u ) H_Q(u) = H(\sqrt{Q}u) H Q ( u ) = H ( Q u ) K Q ( u ) = d d t H ( t Q u ) ∣ t = 1 K_Q(u) = \left.\frac{d}{dt}H(t\sqrt{Q}u)\right|_{t=1} K Q ( u ) = d t d H ( t Q u ) t = 1 u ∈ S N u \in S_N u ∈ S N H H H K Q ( u ) = Q u ⋅ ∇ H ( Q u ) K_Q(u) = \sqrt{Q}u \cdot\nabla H(\sqrt{Q}u) K Q ( u ) = Q u ⋅ ∇ H ( Q u ) ∇ \nabla ∇ ∇ S N \nabla_{S_N} ∇ S N S N S_N S N
Definition 4.1 (Critical points in a window and their bands). Let Q ∈ ( 0 , 1 ) Q \in(0,1) Q ∈ ( 0 , 1 ) U ⊂ R U \subset\mathbb{R} U ⊂ R ( m N ) N ≥ 2 (m_N)_{N\ge2} ( m N ) N ≥ 2 b ∈ ( 0 , Q ) b \in(0,\sqrt{Q}) b ∈ ( 0 , Q ) H ∈ H N H \in\mathcal{H}_N H ∈ H N M > 0 M>0 M > 0
C N , M ( H ) = { u ∈ S N : ∇ S N H Q ( u ) = 0 , ∣ H Q ( u ) − m N ∣ ≤ M , K Q ( u ) / N ∈ U } . C_{N,M}(H) = \{u \in S_N : \nabla_{S_N} H_Q(u)=0,\ |H_Q(u)-m_N|\le M,\ K_Q(u)/N \in U\}. C N , M ( H ) = { u ∈ S N : ∇ S N H Q ( u ) = 0 , ∣ H Q ( u ) − m N ∣ ≤ M , K Q ( u ) / N ∈ U } . For u ∈ S N u \in S_N u ∈ S N u u u B u = { x ∈ S N : ∣ R ( x , u ) − Q ∣ ≤ b } B_u = \{x \in S_N : |R(x,u)-\sqrt{Q}|\le b\} B u = { x ∈ S N : ∣ R ( x , u ) − Q ∣ ≤ b }
B N , M ( H ) = ⋃ u ∈ C N , M ( H ) B u . B_{N,M}(H) = \bigcup_{u\in C_{N,M}(H)} B_u. B N , M ( H ) = u ∈ C N , M ( H ) ⋃ B u . For a ∈ ( 0 , 1 ) a \in(0,1) a ∈ ( 0 , 1 ) u ∈ S N u \in S_N u ∈ S N C a ( u ) = { x ∈ S N : ∣ R ( x , u ) ∣ ≥ a } C_a(u) = \{x \in S_N : |R(x,u)|\ge a\} C a ( u ) = { x ∈ S N : ∣ R ( x , u ) ∣ ≥ a }
The elements of C N , M ( H ) C_{N,M}(H) C N , M ( H ) H Q H_Q H Q [ m N − M , m N + M ] [m_N-M,m_N+M] [ m N − M , m N + M ] U U U U U U K 0 K_0 K 0 m N = N E 0 + O ( log N ) m_N = NE_0 + O(\log N) m N = N E 0 + O ( log N ) E 0 E_0 E 0 K 0 K_0 K 0 U U U ( m N ) (m_N) ( m N ) b b b S N S_N S N ξ \xi ξ H Q ( − u ) = H Q ( u ) H_Q(-u)=H_Q(u) H Q ( − u ) = H Q ( u ) K Q ( − u ) = K Q ( u ) K_Q(-u)=K_Q(u) K Q ( − u ) = K Q ( u ) C N , M ( H ) = − C N , M ( H ) C_{N,M}(H)=-C_{N,M}(H) C N , M ( H ) = − C N , M ( H ) B u B_u B u B − u = − B u B_{-u}=-B_u B − u = − B u a < Q − b a<\sqrt{Q}-b a < Q − b
B u ⊂ C a ( u ) , B_u \subset C_a(u), B u ⊂ C a ( u ) , since every x ∈ B u x \in B_u x ∈ B u R ( x , u ) ≥ Q − b > a R(x,u)\ge\sqrt{Q}-b>a R ( x , u ) ≥ Q − b > a
We say that the bands carry almost all of the Gibbs mass if
lim M → ∞ lim sup N → ∞ E H [ π H ( B N , M ( H ) c ) ] = 0. (60) \lim_{M\to\infty}\limsup_{N\to\infty}\mathbb{E}_{H}\left[\pi^{H}\left(B_{N,M}(H)^{c}\right)\right]=0. \tag*{(60)} M → ∞ lim N → ∞ lim sup E H [ π H ( B N , M ( H ) c ) ] = 0. (60) Because 0 ≤ π H ( B N , M ( H ) c ) ≤ 1 0\leq\pi^{H}\left(B_{N,M}(H)^{c}\right)\leq1 0 ≤ π H ( B N , M ( H ) c ) ≤ 1
lim M → ∞ lim sup N → ∞ P H { π H ( B N , M ( H ) c ) > η } = 0 \lim_{M\to\infty}\limsup_{N\to\infty}\mathbb{P}_{H}\left\{\pi^{H}\left(B_{N,M}(H)^{c}\right)>\eta\right\}=0 M → ∞ lim N → ∞ lim sup P H { π H ( B N , M ( H ) c ) > η } = 0 for every η > 0 \eta>0 η > 0 E Y ≤ η + P { Y > η } \mathbb{E}Y\leq\eta+\mathbb{P}\{Y>\eta\} E Y ≤ η + P { Y > η } Y ∈ [ 0 , 1 ] Y\in[0,1] Y ∈ [ 0 , 1 ] (60) : M M M N N N M M M N N N
Measurability and rotations
The arguments below integrate quantities such as P T H ( x , B N , M ( H ) ) P^{H}_{T}(x,B_{N,M}(H)) P T H ( x , B N , M ( H ))
Lemma 4.2 (Measurable path laws and rotation covariance). Let N ≥ 2 N\geq2 N ≥ 2 β ≥ 0 \beta\geq0 β ≥ 0 T ∈ ( 0 , ∞ ) T\in(0,\infty) T ∈ ( 0 , ∞ ) H ∈ H N H\in\mathcal{H}_{N} H ∈ H N x ∈ S N x\in S_{N} x ∈ S N P x H P^{H}_{x} P x H ( X t ) t ∈ [ 0 , T ] (X_{t})_{t\in[0,T]} ( X t ) t ∈ [ 0 , T ] C ( [ 0 , T ] , S N ) C([0,T],S_{N}) C ([ 0 , T ] , S N )
(a) The map ( H , x ) ↦ P x H (H,x)\mapsto P^{H}_{x} ( H , x ) ↦ P x H H N × S N \mathcal{H}_{N}\times S_{N} H N × S N C ( [ 0 , T ] , S N ) C([0,T],S_{N}) C ([ 0 , T ] , S N ) ( H , x ) ↦ P t H ( x , A ) (H,x)\mapsto P^{H}_{t}(x,A) ( H , x ) ↦ P t H ( x , A ) t ∈ [ 0 , T ] t\in[0,T] t ∈ [ 0 , T ] A ⊂ S N A\subset S_{N} A ⊂ S N
(b) For every g ∈ O ( N ) g\in O(N) g ∈ O ( N ) x ∈ S N x\in S_{N} x ∈ S N t ≥ 0 t\geq0 t ≥ 0 A ⊂ S N A\subset S_{N} A ⊂ S N
P t H g ( g x , g A ) = P t H ( x , A ) , π H g = g ∗ π H . (61) P^{Hg}_{t}(gx,gA)=P^{H}_{t}(x,A),\qquad\pi^{Hg}=g_{*}\pi^{H}. \tag*{(61)} P t H g ( g x , g A ) = P t H ( x , A ) , π H g = g ∗ π H . (61) (c) For every Borel probability kernel H ↦ μ H H\mapsto\mu^{H} H ↦ μ H H N \mathcal{H}_{N} H N S N S_{N} S N t ≥ 0 t\geq0 t ≥ 0 H ↦ ∥ μ H P t H − π H ∥ T V H\mapsto\left\|\mu^{H}P^{H}_{t}-\pi^{H}\right\|_{\mathrm{TV}} H ↦ μ H P t H − π H TV
inf 0 ≤ t ≤ T ∥ μ H P t H − π H ∥ T V = ∥ μ H P T H − π H ∥ T V . (62) \inf_{0\leq t\leq T}\left\|\mu^{H}P^{H}_{t}-\pi^{H}\right\|_{\mathrm{TV}}
=
\left\|\mu^{H}P^{H}_{T}-\pi^{H}\right\|_{\mathrm{TV}}. \tag*{(62)} 0 ≤ t ≤ T inf μ H P t H − π H TV = μ H P T H − π H TV . (62) Proof. We realize the diffusion as the solution of an Itô equation in R N \mathbb{R}^{N} R N H H H ( H , x ) (H,x) ( H , x ) g g g
The stochastic differential equation. For x ∈ R N x\in\mathbb{R}^{N} x ∈ R N Π x = Id − x x T / N \Pi_{x}=\operatorname{Id}-xx^{T}/N Π x = Id − x x T / N x ∈ S N x\in S_{N} x ∈ S N S N S_{N} S N x x x χ : R N → [ 0 , 1 ] \chi:\mathbb{R}^{N}\to[0,1] χ : R N → [ 0 , 1 ] ∣ x ∣ \lvert x\rvert ∣ x ∣ { N / 2 ≤ ∣ x ∣ 2 ≤ 2 N } \{N/2\leq\lvert x\rvert^{2}\leq2N\} { N /2 ≤ ∣ x ∣ 2 ≤ 2 N } { N / 4 ≤ ∣ x ∣ 2 ≤ 4 N } \{N/4\leq\lvert x\rvert^{2}\leq4N\} { N /4 ≤ ∣ x ∣ 2 ≤ 4 N } W W W R N \mathbb{R}^{N} R N
d X t = 2 χ ( X t ) Π X t d W t + χ ( X t ) ( β Π X t ∇ H ( X t ) − N − 1 N X t ) d t , X 0 = x ∈ S N . (63) \mathrm{d}X_{t}=\sqrt{2}\,\chi(X_{t})\Pi_{X_{t}}\,\mathrm{d}W_{t}+\chi(X_{t})\left(\beta\Pi_{X_{t}}\nabla H(X_{t})-\frac{N-1}{N}X_{t}\right)\mathrm{d}t,\qquad X_{0}=x\in S_{N}. \tag*{(63)} d X t = 2 χ ( X t ) Π X t d W t + χ ( X t ) ( β Π X t ∇ H ( X t ) − N N − 1 X t ) d t , X 0 = x ∈ S N . (63) For x ∈ S N x \in S_N x ∈ S N Π x ∇ H ( x ) = ∇ S N H ( x ) \Pi_x \nabla H(x) = \nabla_{S_N} H(x) Π x ∇ H ( x ) = ∇ S N H ( x ) (63) is the equation (40) of Lemma 3.5, with a particular extension of its coefficients off a neighborhood of S N S_N S N (63) has a unique strong solution X H , x X^{H,x} X H , x S N S_N S N L β , H L_{\beta,H} L β , H x x x P x H \mathbb{P}_x^H P x H X H , x X^{H,x} X H , x
We now show that X H , x X^{H,x} X H , x ( H , x ) (H,x) ( H , x ) (63) are smooth with compact support in x x x H H H R > 0 R > 0 R > 0 C = C ( N , β , R ) C = C(N,\beta,R) C = C ( N , β , R ) ∣ H ∣ , ∣ H ′ ∣ ≤ R |H|, |H'| \le R ∣ H ∣ , ∣ H ′ ∣ ≤ R C C C x x x C ∣ H − H ′ ∣ C|H-H'| C ∣ H − H ′ ∣ H H H H ′ H' H ′ X t ( 0 ) = x X_t^{(0)} = x X t ( 0 ) = x X ( k + 1 ) X^{(k+1)} X ( k + 1 ) X ( k ) X^{(k)} X ( k ) (63) . Using Doob’s maximal inequality and the Itô isometry, we find
E sup s ≤ t ∣ X s ( k + 1 ) − X s ( k ) ∣ 2 ≤ C ′ ∫ 0 t E sup v ≤ s ∣ X v ( k ) − X v ( k − 1 ) ∣ 2 d s . \mathbb{E}\sup_{s\le t}\left|X_s^{(k+1)}-X_s^{(k)}\right|^2 \le C'\int_0^t\mathbb{E}\sup_{v\le s}\left|X_v^{(k)}-X_v^{(k-1)}\right|^2\,\mathrm{d}s. E s ≤ t sup X s ( k + 1 ) − X s ( k ) 2 ≤ C ′ ∫ 0 t E v ≤ s sup X v ( k ) − X v ( k − 1 ) 2 d s . so the left side is at most C ′ ′ ( C ′ t ) k / k ! C''(C't)^k/k! C ′′ ( C ′ t ) k / k ! [ 0 , T ] [0,T] [ 0 , T ] X H , x X^{H,x} X H , x k k k ( X ( k ) , W ) (X^{(k)},W) ( X ( k ) , W ) W W W (63) is a limit in probability of Riemann sums, which are continuous functions of ( X ( k ) , W ) (X^{(k)},W) ( X ( k ) , W ) X H , x X^{H,x} X H , x
E sup t ≤ T ∣ X t H , x − X t H ′ , x ′ ∣ 2 ≤ C T ( ∣ x − x ′ ∣ 2 + ∣ H − H ′ ∣ 2 ) , ∣ H ∣ , ∣ H ′ ∣ ≤ R . (64) \mathbb{E}\sup_{t\le T}\left|X_t^{H,x}-X_t^{H',x'}\right|^2 \le C_T\left(|x-x'|^2+|H-H'|^2\right), \qquad|H|,|H'|\le R. \tag*{(64)} E t ≤ T sup X t H , x − X t H ′ , x ′ 2 ≤ C T ( ∣ x − x ′ ∣ 2 + ∣ H − H ′ ∣ 2 ) , ∣ H ∣ , ∣ H ′ ∣ ≤ R . (64) where both solutions are driven by the same W W W C T C_T C T N N N β \beta β R R R T T T
Measurability. If ( H j , x j ) → ( H , x ) (H_j,x_j)\to(H,x) ( H j , x j ) → ( H , x ) (64) implies that X H j , x j → X H , x X^{H_j,x_j}\to X^{H,x} X H j , x j → X H , x C ( [ 0 , T ] , R N ) C([0,T],\mathbb{R}^N) C ([ 0 , T ] , R N ) P x j H j → P x H \mathbb{P}_{x_j}^{H_j}\to\mathbb{P}_x^H P x j H j → P x H ( H , x ) ↦ ∫ Φ d P x H (H,x)\mapsto\int\Phi\,\mathrm{d}\mathbb{P}_x^H ( H , x ) ↦ ∫ Φ d P x H Φ \Phi Φ C ( [ 0 , T ] , S N ) C([0,T],S_N) C ([ 0 , T ] , S N ) Φ \Phi Φ
Rotations. Fix g ∈ O ( N ) g\in O(N) g ∈ O ( N ) χ \chi χ ∣ x ∣ |x| ∣ x ∣ Π g x = g Π x g − 1 \Pi_{gx}=g\Pi_xg^{-1} Π g x = g Π x g − 1 ∇ H g ( g x ) = g ∇ H ( x ) \nabla H^g(gx)=g\nabla H(x) ∇ H g ( g x ) = g ∇ H ( x ) g X H , x gX^{H,x} g X H , x (63) with H g H^g H g H H H g x gx g x g W gW g W W W W g X H , x gX^{H,x} g X H , x (63) for the field H g H^g H g g x gx g x g W gW g W g W gW g W g X H , x gX^{H,x} g X H , x X H g , g x X^{H^g,gx} X H g , g x t t t (61) . For the second identity, we use the invariance of ν N \nu_N ν N g g g
∫ g A e β H g d ν N = ∫ g A e β H ( g − 1 y ) ν N ( d y ) = ∫ A e β H d ν N . \int_{gA}e^{\beta H^g}\,\mathrm{d}\nu_N=\int_{gA}e^{\beta H(g^{-1}y)}\,\nu_N(\mathrm{d}y)=\int_Ae^{\beta H}\,\mathrm{d}\nu_N. ∫ g A e β H g d ν N = ∫ g A e β H ( g − 1 y ) ν N ( d y ) = ∫ A e β H d ν N . Taking A = S N A=S_N A = S N π H g ( g A ) = π H ( A ) \pi^{H^g}(gA)=\pi^H(A) π H g ( g A ) = π H ( A )
Total variation. Let A \mathcal{A} A S N S_N S N λ , ρ \lambda,\rho λ , ρ S N S_N S N ∣ λ ( A ) − ρ ( A ) ∣ |\lambda(A)-\rho(A)| ∣ λ ( A ) − ρ ( A ) ∣ A ∈ A A\in\mathcal{A} A ∈ A A A A ε > 0 \varepsilon>0 ε > 0 K K K O O O K ⊂ A ⊂ O K\subset A\subset O K ⊂ A ⊂ O ( λ + ρ ) ( O ∖ K ) < ε (\lambda+\rho)(O\setminus K)<\varepsilon ( λ + ρ ) ( O ∖ K ) < ε O O O K K K A ′ ∈ A A'\in\mathcal{A} A ′ ∈ A ∣ λ ( A ) − λ ( A ′ ) ∣ + ∣ ρ ( A ) − ρ ( A ′ ) ∣ ≤ ( λ + ρ ) ( O ∖ K ) < ε |\lambda(A)-\lambda(A')|+|\rho(A)-\rho(A')|\leq(\lambda+\rho)(O\setminus K)<\varepsilon ∣ λ ( A ) − λ ( A ′ ) ∣ + ∣ ρ ( A ) − ρ ( A ′ ) ∣ ≤ ( λ + ρ ) ( O ∖ K ) < ε A ∈ A A\in\mathcal{A} A ∈ A H ↦ ( μ H H P t H ) ( A ) H\mapsto(\mu_H^H P_t^H)(A) H ↦ ( μ H H P t H ) ( A ) H ↦ π H ( A ) H\mapsto\pi^H(A) H ↦ π H ( A ) H ↦ ∥ μ H H P t H − π H ∥ T V H\mapsto\lVert\mu_H^H P_t^H-\pi^H\rVert_{\mathrm{TV}} H ↦ ∥ μ H H P t H − π H ∥ TV π H \pi^H π H P t H P_t^H P t H t ↦ ∥ μ H H P t H − π H ∥ T V t\mapsto\lVert\mu_H^H P_t^H-\pi^H\rVert_{\mathrm{TV}} t ↦ ∥ μ H H P t H − π H ∥ TV
Lemma 4.3 (Measurable and equivariant critical sets). Let Q Q Q U U U ( m N ) (m_N) ( m N ) b b b M > 0 M>0 M > 0 a ∈ ( 0 , 1 ) a\in(0,1) a ∈ ( 0 , 1 ) T ∈ ( 0 , ∞ ) T\in(0,\infty) T ∈ ( 0 , ∞ )
(a) The sets { ( H , u ) : u ∈ C N , M ( H ) } \{(H,u):u\in C_{N,M}(H)\} {( H , u ) : u ∈ C N , M ( H )} { ( H , x ) : x ∈ B N , M ( H ) } \{(H,x):x\in B_{N,M}(H)\} {( H , x ) : x ∈ B N , M ( H )} H N × S N \mathcal{H}_N\times S_N H N × S N g ∈ O ( N ) g\in O(N) g ∈ O ( N ) H ∈ H N H\in\mathcal{H}_N H ∈ H N
C N , M ( H g ) = g C N , M ( H ) , B N , M ( H g ) = g B N , M ( H ) . (65) C_{N,M}(H^g)=gC_{N,M}(H),\qquad B_{N,M}(H^g)=gB_{N,M}(H). \tag*{(65)} C N , M ( H g ) = g C N , M ( H ) , B N , M ( H g ) = g B N , M ( H ) . (65) (b) The function θ ( H , u ) = P ν N H { X t ∈ C a ( u ) for some t ∈ [ 0 , T ] } \theta(H,u)=\mathbb{P}_{\nu_N^H}\{X_t\in C_a(u)\text{ for some }t\in[0,T]\} θ ( H , u ) = P ν N H { X t ∈ C a ( u ) for some t ∈ [ 0 , T ]} H N × S N \mathcal{H}_N\times S_N H N × S N P ν N H { X t ∈ C a ( u ) for some t ∈ [ 0 , T ] and some u ∈ C N , M ( H ) } \mathbb{P}_{\nu_N^H}\{X_t\in C_a(u)\text{ for some }t\in[0,T]\text{ and some }u\in C_{N,M}(H)\} P ν N H { X t ∈ C a ( u ) for some t ∈ [ 0 , T ] and some u ∈ C N , M ( H )} H H H
(c) The set of H H H C N , M ( H ) C_{N,M}(H) C N , M ( H ) S ( H ) = ∑ u ∈ C N , M ( H ) θ ( H , u ) ∈ [ 0 , ∞ ] S(H)=\sum_{u\in C_{N,M}(H)}\theta(H,u)\in[0,\infty] S ( H ) = ∑ u ∈ C N , M ( H ) θ ( H , u ) ∈ [ 0 , ∞ ] H H H + ∞ +\infty + ∞ C N , M ( H ) C_{N,M}(H) C N , M ( H ) H H H
sup 0 ≤ t ≤ T ( ν N P t H ) ( ⋃ u ∈ C N , M ( H ) C a ( u ) ) ≤ P ν N H { X t ∈ C a ( u ) for some t ≤ T and u ∈ C N , M ( H ) } ≤ S ( H ) . (66) \sup_{0\leq t\leq T}\left(\nu_NP_t^H\right)\left(\bigcup_{u\in C_{N,M}(H)}C_a(u)\right)\leq\mathbb{P}_{\nu_N^H}\{X_t\in C_a(u)\text{ for some }t\leq T\text{ and }u\in C_{N,M}(H)\}\leq S(H). \tag*{(66)} 0 ≤ t ≤ T sup ( ν N P t H ) u ∈ C N , M ( H ) ⋃ C a ( u ) ≤ P ν N H { X t ∈ C a ( u ) for some t ≤ T and u ∈ C N , M ( H )} ≤ S ( H ) . (66) Proof. For (a), we first note that the maps ( H , u ) ↦ H Q ( u ) (H,u)\mapsto H_Q(u) ( H , u ) ↦ H Q ( u ) K Q ( u ) K_Q(u) K Q ( u ) ∇ S N H Q ( u ) \nabla_{S_N}H_Q(u) ∇ S N H Q ( u ) u u u H H H U U U S N S_N S N u u u { ( H , u , x ) : u ∈ C N , M ( H ) , ∣ R ( x , u ) − Q ∣ ≤ b } \{(H,u,x):u\in C_{N,M}(H),\ |R(x,u)-\sqrt{Q}|\leq b\} {( H , u , x ) : u ∈ C N , M ( H ) , ∣ R ( x , u ) − Q ∣ ≤ b } ( H j , x j ) → ( H , x ) (H_j,x_j)\to(H,x) ( H j , x j ) → ( H , x ) u j u_j u j u j u_j u j u u u ( H , u , x ) (H,u,x) ( H , u , x ) ( H g ) Q ( v ) = H Q ( g − 1 v ) (H^g)_Q(v)=H_Q(g^{-1}v) ( H g ) Q ( v ) = H Q ( g − 1 v ) v ∈ S N v\in S_N v ∈ S N
( H g ) Q ( g u ) = H Q ( u ) , ( K g ) Q ( g u ) = K Q ( u ) , ∇ S N ( H g ) Q ( g u ) = g ∇ S N H Q ( u ) , (H^g)_Q(gu)=H_Q(u),\qquad(K^g)_Q(gu)=K_Q(u),\qquad\nabla_{S_N}(H^g)_Q(gu)=g\nabla_{S_N}H_Q(u), ( H g ) Q ( g u ) = H Q ( u ) , ( K g ) Q ( g u ) = K Q ( u ) , ∇ S N ( H g ) Q ( g u ) = g ∇ S N H Q ( u ) , where ( K g ) Q (K^g)_Q ( K g ) Q H g H^g H g u ∈ C N , M ( H ) u\in C_{N,M}(H) u ∈ C N , M ( H ) g u ∈ C N , M ( H g ) gu\in C_{N,M}(H^g) g u ∈ C N , M ( H g ) R ( g x , g u ) = R ( x , u ) R(gx,gu)=R(x,u) R ( g x , g u ) = R ( x , u ) B g u = g B u B_{gu}=gB_u B g u = g B u ( ω , u ) ↦ max t ≤ T ∣ R ( ω ( t ) , u ) ∣ (\omega,u) \mapsto\max_{t\le T} \lvert R(\omega(t),u)\rvert ( ω , u ) ↦ max t ≤ T ∣ R ( ω ( t ) , u )∣ C ( [ 0 , T ] , S N ) × S N C([0,T],S_N)\times S_N C ([ 0 , T ] , S N ) × S N { ( ω , u ) : max t ≤ T ∣ R ( ω ( t ) , u ) ∣ ≥ a } \{(\omega,u):\max_{t\le T}\lvert R(\omega(t),u)\rvert\ge a\} {( ω , u ) : max t ≤ T ∣ R ( ω ( t ) , u )∣ ≥ a } P x H P_x^H P x H ν N ( d x ) \nu_N(\mathrm{d}x) ν N ( d x ) θ \theta θ ( H , ω ) (H,\omega) ( H , ω ) u ∈ C N , M ( H ) u\in C_{N,M}(H) u ∈ C N , M ( H ) max t ≤ T ∣ R ( ω ( t ) , u ) ∣ ≥ a \max_{t\le T}\lvert R(\omega(t),u)\rvert\ge a max t ≤ T ∣ R ( ω ( t ) , u )∣ ≥ a A = { ( ω , u ) : max t ≤ T ∣ R ( ω ( t ) , u ) ∣ ≥ a } \mathcal{A}=\{(\omega,u):\max_{t\le T}\lvert R(\omega(t),u)\rvert\ge a\} A = {( ω , u ) : max t ≤ T ∣ R ( ω ( t ) , u )∣ ≥ a }
For (c), let O ⊂ S N O\subset S_N O ⊂ S N O k = { x ∈ S N : dist ( x , O c ) ≥ 1 / k } O_k=\{x\in S_N:\operatorname{dist}(x,O^c)\ge1/k\} O k = { x ∈ S N : dist ( x , O c ) ≥ 1/ k } l ∈ N l\in\mathbb{N} l ∈ N { # ( C N , M ( H ) ∩ O ) ≥ l } \{\#(C_{N,M}(H)\cap O)\ge l\} { # ( C N , M ( H ) ∩ O ) ≥ l } k k k u 1 , … , u l ∈ C N , M ( H ) ∩ O k u_1,\ldots,u_l\in C_{N,M}(H)\cap O_k u 1 , … , u l ∈ C N , M ( H ) ∩ O k ∣ u i − u j ∣ ≥ 1 / k \lvert u_i-u_j\rvert\ge1/k ∣ u i − u j ∣ ≥ 1/ k i ≠ j i\ne j i = j ( S N ) l (S_N)^l ( S N ) l H ↦ # ( C N , M ( H ) ∩ O ) H\mapsto\#(C_{N,M}(H)\cap O) H ↦ # ( C N , M ( H ) ∩ O ) O O O F F F C N , M ( H ) C_{N,M}(H) C N , M ( H ) F l = { H : # C N , M ( H ) ≤ l } F_l=\{H:\#C_{N,M}(H)\le l\} F l = { H : # C N , M ( H ) ≤ l } n H = ∑ u ∈ C N , M ( H ) δ u n_H=\sum_{u\in C_{N,M}(H)}\delta_u n H = ∑ u ∈ C N , M ( H ) δ u A A A H ↦ n H ( A ) 1 F l ( H ) H\mapsto n_H(A)1_{F_l}(H) H ↦ n H ( A ) 1 F l ( H ) θ \theta θ 1 A 1 ( H ) 1 A 2 ( u ) 1_{A_1}(H)1_{A_2}(u) 1 A 1 ( H ) 1 A 2 ( u ) H ↦ ∫ θ ( H , u ) n H ( d u ) = S ( H ) H\mapsto\int\theta(H,u)\,n_H(\mathrm{d}u)=S(H) H ↦ ∫ θ ( H , u ) n H ( d u ) = S ( H ) F l F_l F l F F F θ ( H , u ) ≥ P ν N H { X 0 ∈ C a ( u ) } = ν N ( C a ( u ) ) \theta(H,u)\ge P^H_{\nu_N}\{X_0\in C_a(u)\}=\nu_N(C_a(u)) θ ( H , u ) ≥ P ν N H { X 0 ∈ C a ( u )} = ν N ( C a ( u )) ν N ( C a ( u ) ) > 0 \nu_N(C_a(u))>0 ν N ( C a ( u )) > 0 u u u a < 1 a<1 a < 1 ν N \nu_N ν N S = + ∞ S=+\infty S = + ∞ F F F S S S X t ∈ ⋃ u C a ( u ) X_t\in\bigcup_u C_a(u) X t ∈ ⋃ u C a ( u ) t ≤ T t\le T t ≤ T C N , M ( H ) C_{N,M}(H) C N , M ( H )
Separation from Gibbs concentration, counts, and hitting bounds
Each of the separation statements in this part ends with the same order of limits: a window half-width M M M N N N
Lemma 4.4 (Testing the band union). Assume that the bands carry almost all of the Gibbs mass , (4.2). Let Δ N ( H ) ∈ [ 0 , 1 ] \Delta_N(H)\in[0,1] Δ N ( H ) ∈ [ 0 , 1 ] M > 0 M>0 M > 0 E N , M ( H ) ≥ 0 \mathcal{E}_{N,M}(H)\ge0 E N , M ( H ) ≥ 0 E N , M → 0 \mathcal{E}_{N,M}\to0 E N , M → 0 P H P_H P H N → ∞ N\to\infty N → ∞
Δ N ( H ) ≥ 1 − π H ( B N , M ( H ) c ) − E N , M ( H ) for all N and H . \Delta_N(H)\ge1-\pi^H(B_{N,M}(H)^c)-\mathcal{E}_{N,M}(H)\qquad\text{for all }N\text{ and }H. Δ N ( H ) ≥ 1 − π H ( B N , M ( H ) c ) − E N , M ( H ) for all N and H . Then Δ N → 1 \Delta_N\to1 Δ N → 1 P H P_H P H
Proof. Fix ε , δ ∈ ( 0 , 1 ) \varepsilon,\delta\in(0,1) ε , δ ∈ ( 0 , 1 ) M M M lim sup N E H π H ( B N , M c ) < ε δ / 4 \limsup_N\mathbb{E}_H\pi^H(B_{N,M}^c)<\varepsilon\delta/4 lim sup N E H π H ( B N , M c ) < ε δ /4 lim sup N P H { π H ( B N , M c ) > ε / 2 } ≤ δ / 2 \limsup_NP_H\{\pi^H(B_{N,M}^c)>\varepsilon/2\}\le\delta/2 lim sup N P H { π H ( B N , M c ) > ε /2 } ≤ δ /2 M M M P H { E N , M > ε / 2 } → 0 P_H\{\mathcal{E}_{N,M}>\varepsilon/2\}\to0 P H { E N , M > ε /2 } → 0 Δ N ≥ 1 − ε \Delta_N\ge1-\varepsilon Δ N ≥ 1 − ε lim sup N P H { Δ N < 1 − ε } ≤ δ / 2 \limsup_NP_H\{\Delta_N<1-\varepsilon\}\le\delta/2 lim sup N P H { Δ N < 1 − ε } ≤ δ /2 δ \delta δ
In the applications, Δ N = inf 0 ≤ t ≤ T N ∥ μ H P t H − π H ∥ T V \Delta_N=\inf_{0\leq t\leq T_N}\lVert\mu^H P_t^H-\pi^H\rVert_{\mathrm{TV}} Δ N = inf 0 ≤ t ≤ T N ∥ μ H P t H − π H ∥ TV μ H \mu^H μ H Proposition 3.10 implies that Δ N ≥ 1 − π H ( B N , M c ) − ( μ H P T N H ) ( B N , M ) \Delta_N\geq1-\pi^H(B_{N,M}^c)-(\mu^H P_{T_N}^H)(B_{N,M}) Δ N ≥ 1 − π H ( B N , M c ) − ( μ H P T N H ) ( B N , M )
Proposition 4.5. (Separation from Gibbs concentration, counts, and hitting bounds.) Let β > 0 \beta>0 β > 0 Q Q Q U U U ( m N ) (m_N) ( m N ) b b b Definition 4.1 , let a ∈ ( 0 , Q − b ) a\in(0,\sqrt{Q}-b) a ∈ ( 0 , Q − b ) ( T N ) (T_N) ( T N ) u ∈ S N u\in S_N u ∈ S N
θ N ( H , u ) = P ν N H { X t ∈ C a ( u ) for some t ∈ [ 0 , T N ] } . \theta_N(H,u)=\mathbb{P}_{\nu_N^H}\{X_t\in C_a(u)\text{ for some }t\in[0,T_N]\}. θ N ( H , u ) = P ν N H { X t ∈ C a ( u ) for some t ∈ [ 0 , T N ]} . Assume the following.
(i) The bands carry almost all of the Gibbs mass, (60) .
(ii) For every fixed M > 0 M>0 M > 0 A M = lim sup N → ∞ E H ∣ C N , M ( H ) ∣ < ∞ A_M=\limsup_{N\to\infty}\mathbb{E}_H|C_{N,M}(H)|<\infty A M = lim sup N → ∞ E H ∣ C N , M ( H ) ∣ < ∞
(iii) For every fixed M > 0 M>0 M > 0 q N , M ≥ 0 q_{N,M}\geq0 q N , M ≥ 0 lim N → ∞ q N , M = 0 \lim_{N\to\infty}q_{N,M}=0 lim N → ∞ q N , M = 0 N N N
E H ∑ u ∈ C N , M ( H ) θ N ( H , u ) ≤ q N , M E H ∣ C N , M ( H ) ∣ . (67) \mathbb{E}_H\sum_{u\in C_{N,M}(H)}\theta_N(H,u)\leq q_{N,M}\mathbb{E}_H|C_{N,M}(H)|. \tag*{(67)} E H u ∈ C N , M ( H ) ∑ θ N ( H , u ) ≤ q N , M E H ∣ C N , M ( H ) ∣. (67) For M > 0 M>0 M > 0 S N , M ( H ) = ∑ u ∈ C N , M ( H ) θ N ( H , u ) S_{N,M}(H)=\sum_{u\in C_{N,M}(H)}\theta_N(H,u) S N , M ( H ) = ∑ u ∈ C N , M ( H ) θ N ( H , u ) lim N → ∞ E H S N , M = 0 \lim_{N\to\infty}\mathbb{E}_H S_{N,M}=0 lim N → ∞ E H S N , M = 0 M M M H H H
sup 0 ≤ t ≤ T N ( ν N P t H ) ( B N , M ( H ) ) ≤ S N , M ( H ) , \sup_{0\leq t\leq T_N}(\nu_NP_t^H)(B_{N,M}(H))\leq S_{N,M}(H), 0 ≤ t ≤ T N sup ( ν N P t H ) ( B N , M ( H )) ≤ S N , M ( H ) , inf 0 ≤ t ≤ T N ∥ ν N P t H − π H ∥ T V ≥ 1 − π H ( B N , M ( H ) c ) − S N , M ( H ) . (68) \inf_{0\leq t\leq T_N}\lVert\nu_NP_t^H-\pi^H\rVert_{\mathrm{TV}}\geq1-\pi^H(B_{N,M}(H)^c)-S_{N,M}(H). \tag*{(68)} 0 ≤ t ≤ T N inf ∥ ν N P t H − π H ∥ TV ≥ 1 − π H ( B N , M ( H ) c ) − S N , M ( H ) . (68) In particular, inf 0 ≤ t ≤ T N ∥ ν N P t H − π H ∥ T V → 1 \inf_{0\leq t\leq T_N}\lVert\nu_NP_t^H-\pi^H\rVert_{\mathrm{TV}}\to1 inf 0 ≤ t ≤ T N ∥ ν N P t H − π H ∥ TV → 1 P H \mathbb{P}_H P H
Proof. Fix M M M E H S N , M ≤ q N , M ( A M + 1 ) \mathbb{E}_H S_{N,M}\leq q_{N,M}(A_M+1) E H S N , M ≤ q N , M ( A M + 1 ) N N N a < Q − b a<\sqrt{Q}-b a < Q − b B u ⊂ C a ( u ) B_u\subset C_a(u) B u ⊂ C a ( u ) B N , M ( H ) ⊂ ⋃ u ∈ C N , M ( H ) C a ( u ) B_{N,M}(H)\subset\bigcup_{u\in C_{N,M}(H)}C_a(u) B N , M ( H ) ⊂ ⋃ u ∈ C N , M ( H ) C a ( u ) (68) now follows from (66) . For every t ≤ T N t\leq T_N t ≤ T N B N , M ( H ) B_{N,M}(H) B N , M ( H )
∥ ν N P t H − π H ∥ T V ≥ π H ( B N , M ( H ) ) − ( ν N P t H ) ( B N , M ( H ) ) \lVert\nu_NP_t^H-\pi^H\rVert_{\mathrm{TV}}\geq\pi^H(B_{N,M}(H))-(\nu_NP_t^H)(B_{N,M}(H)) ∥ ν N P t H − π H ∥ TV ≥ π H ( B N , M ( H )) − ( ν N P t H ) ( B N , M ( H )) ≥ 1 − π H ( B N , M ( H ) c ) − S N , M ( H ) . \geq1-\pi^H(B_{N,M}(H)^c)-S_{N,M}(H). ≥ 1 − π H ( B N , M ( H ) c ) − S N , M ( H ) . which is the second bound. Using Markov’s inequality, we find that S N , M → 0 S_{N,M}\to0 S N , M → 0 P H \mathbb{P}_H P H M M M Lemma 4.4 , applied with Δ N = inf t ≤ T N ∥ ν N P t H − π H ∥ T V \Delta_N=\inf_{t\leq T_N}\lVert\nu_NP_t^H-\pi^H\rVert_{\mathrm{TV}} Δ N = inf t ≤ T N ∥ ν N P t H − π H ∥ TV E N , M = S N , M \mathcal{E}_{N,M}=S_{N,M} E N , M = S N , M Δ N \Delta_N Δ N Lemma 4.2 (c), and S N , M S_{N,M} S N , M
Remark 4.6 (The weights in hypothesis (iii)). The left side of (67) is the expected sum of the hitting probabilities over the critical points in the window. By the marked Kac–Rice formula of Theorem 7.17 , we can write it as the integral, against the intensity measure of C N , M C_{N,M} C N , M θ N ( H , u ) \theta_N(H,u) θ N ( H , u ) u u u ∣ det ∇ S N 2 H Q ( u ) ∣ \lvert\det\nabla^2_{S_N}H_Q(u)\rvert ∣ det ∇ S N 2 H Q ( u )∣ E H ∣ C N , M ∣ \mathbb{E}_H\lvert C_{N,M}\rvert E H ∣ C N , M ∣ (67) holds whenever this determinant-weighted conditional hitting probability is at most q N , M q_{N,M} q N , M H Q ( u ) H_Q(u) H Q ( u ) K Q ( u ) K_Q(u) K Q ( u ) ∇ S N H Q ( u ) \nabla_{S_N}H_Q(u) ∇ S N H Q ( u ) (67) , at the cost of a power, using the moments of the normalized determinant from Proposition 7.7. This is carried out in Proposition 13.1.
Remark 4.7 (The order of the parameters). In Proposition 4.5, the times T N T_N T N M M M M M M T N T_N T N T N = e γ N T_N=e^{\gamma N} T N = e γ N γ \gamma γ ( h , k ) = ( H Q ( u ) / N , K Q ( u ) / N ) (h,k)=(H_Q(u)/N,K_Q(u)/N) ( h , k ) = ( H Q ( u ) / N , K Q ( u ) / N ) m N / N m_N/N m N / N [ ( m N − M ) / N , ( m N + M ) / N ] × U [(m_N-M)/N,(m_N+M)/N]\times U [( m N − M ) / N , ( m N + M ) / N ] × U N N N M M M a a a γ \gamma γ M M M N N N M M M (60) lets M M M N N N
Remark 4.8 (An endpoint formulation). For separation alone, the bound on the whole path in (68) is more than needed. If lim N → ∞ E H ( ν N P N H ) ( B N , M ( H ) ) = 0 \lim_{N\to\infty}\mathbb{E}_H(\nu_NP_N^{H})(B_{N,M}(H))=0 lim N → ∞ E H ( ν N P N H ) ( B N , M ( H )) = 0 M M M
Concentration on the bands of a fixed window from a coarse cover
The hypothesis (60) concerns critical points in a fixed window [ m N − M , m N + M ] [m_N-M,m_N+M] [ m N − M , m N + M ] Z H ( B u ) Z_H(B_u) Z H ( B u ) H Q ( u ) < m N − M H_Q(u)<m_N-M H Q ( u ) < m N − M M M M
Proposition 4.9 (Concentration on the bands of a fixed window from a coarse cover). Let β > 0 \beta> 0 β > 0 Q Q Q U U U ( m N ) (m_N) ( m N ) b b b
lim L → ∞ lim inf N → ∞ P H { C N , L ( H ) ≠ ∅ and max S N H Q ≤ m N + L } = 1. (69) \lim_{L\to\infty}\liminf_{N\to\infty}\mathbb{P}_{\mathcal{H}}\left\{C_{N,L}(H)\ne\varnothing\ \text{and}\ \max_{S_N} H_Q \le m_N+L\right\}=1. \tag*{(69)} L → ∞ lim N → ∞ lim inf P H { C N , L ( H ) = ∅ and S N max H Q ≤ m N + L } = 1. (69) Let D N > 0 D_N>0 D N > 0 C N c o ( H ) C_N^{\mathrm{co}}(H) C N co ( H ) u u u H Q H_Q H Q K Q ( u ) / N ∈ U K_Q(u)/N\in U K Q ( u ) / N ∈ U { ( H , u ) : u ∈ C N c o ( H ) } \{(H,u):u\in C_N^{\mathrm{co}}(H)\} {( H , u ) : u ∈ C N co ( H )}
(i) (Coarse cover.)
lim N → ∞ E H π H ( [ ⋃ u ∈ C N c o ( H ) B u ] c ) = 0. \lim_{N\to\infty}\mathbb{E}_{H}\,\pi^H\left(\left[\bigcup_{u\in C_N^{\mathrm{co}}(H)}B_u\right]^c\right)=0. N → ∞ lim E H π H u ∈ C N co ( H ) ⋃ B u c = 0. (ii) (Light bands are rare.) For every fixed L > 0 L>0 L > 0
lim δ ↓ 0 lim sup N → ∞ E H ∑ u ∈ C N , L ( H ) 1 { Z H ( B u ) < δ D N e β ( H Q ( u ) − m N ) } = 0. \lim_{\delta\downarrow0}\limsup_{N\to\infty}\mathbb{E}_{H}\sum_{u\in C_{N,L}(H)}\mathbf{1}\left\{Z_H(B_u)<\delta D_Ne^{\beta(H_Q(u)-m_N)}\right\}=0. δ ↓ 0 lim N → ∞ lim sup E H u ∈ C N , L ( H ) ∑ 1 { Z H ( B u ) < δ D N e β ( H Q ( u ) − m N ) } = 0. (iii) (Critical points below the window.) There exist C ∗ , c ∗ > 0 C_*,c_*>0 C ∗ , c ∗ > 0 M > 0 M>0 M > 0
lim sup N → ∞ D N − 1 E H ∑ u ∈ C N c o ( H ) H Q ( u ) < m N − M Z H ( B u ) ≤ C ∗ e − c ∗ M . \limsup_{N\to\infty}D_N^{-1}\mathbb{E}_{H}\sum_{\substack{u\in C_N^{\mathrm{co}}(H)\\H_Q(u)<m_N-M}}Z_H(B_u)\le C_*e^{-c_*M}. N → ∞ lim sup D N − 1 E H u ∈ C N co ( H ) H Q ( u ) < m N − M ∑ Z H ( B u ) ≤ C ∗ e − c ∗ M . Then the bands carry almost all of the Gibbs mass, (60) .
We call the band B u B_u B u u ∈ C N , L ( H ) u\in C_{N,L}(H) u ∈ C N , L ( H ) Z H ( B u ) < δ D N e β ( H Q ( u ) − m N ) Z_H(B_u)<\delta D_Ne^{\beta(H_Q(u)-m_N)} Z H ( B u ) < δ D N e β ( H Q ( u ) − m N ) H H H ( H , u ) ↦ Z H ( B u ) (H,u)\mapsto Z_H(B_u) ( H , u ) ↦ Z H ( B u ) B u B_u B u ν N \nu_N ν N N N N L L L δ \delta δ f ( H , u ) = δ D N e β ( H Q ( u ) − m N ) − Z H ( B u ) f(H,u)=\delta D_Ne^{\beta(H_Q(u)-m_N)}-Z_H(B_u) f ( H , u ) = δ D N e β ( H Q ( u ) − m N ) − Z H ( B u ) l l l u ∈ C N , L ( H ) u\in C_{N,L}(H) u ∈ C N , L ( H ) f ( H , u ) > 0 f(H,u)>0 f ( H , u ) > 0 k ≥ 1 k\ge1 k ≥ 1 ( S N ) l (S_N)^l ( S N ) l u i ∈ C N , L ( H ) u_i\in C_{N,L}(H) u i ∈ C N , L ( H ) f ( H , u i ) ≥ 1 / k f(H,u_i)\ge1/k f ( H , u i ) ≥ 1/ k ∣ u i − u j ∣ ≥ 1 / k \lvert u_i-u_j\rvert\ge1/k ∣ u i − u j ∣ ≥ 1/ k i ≠ j i\ne j i = j C N c o C_N^{\mathrm{co}} C N co u j u_j u j H N \mathcal{H}_N H N ( H , x ) (H,x) ( H , x )
Proof. We first use (ii) and the existence part of (69) to bound Z H ( S N ) Z_H(S_N) Z H ( S N )
A lower bound for the partition function. Fix ε ∈ ( 0 , 1 ) \varepsilon\in(0,1) ε ∈ ( 0 , 1 ) L L L lim sup N → ∞ P H { C N , L ( H ) = ∅ } < ε \limsup_{N\to\infty}\mathbb{P}_H\{C_{N,L}(H)=\varnothing\}<\varepsilon lim sup N → ∞ P H { C N , L ( H ) = ∅ } < ε L L L δ > 0 \delta>0 δ > 0 u ∈ C N , L ( H ) u\in C_{N,L}(H) u ∈ C N , L ( H ) Z H ( B u ) < δ D N e β ( H Q ( u ) − m N ) Z_H(B_u)<\delta D_N e^{\beta(H_Q(u)-m_N)} Z H ( B u ) < δ D N e β ( H Q ( u ) − m N ) lim sup \limsup lim sup N N N ε \varepsilon ε u u u lim sup \limsup lim sup ε \varepsilon ε C N , L ( H ) C_{N,L}(H) C N , L ( H ) u u u H Q ( u ) ≥ m N − L H_Q(u)\ge m_N-L H Q ( u ) ≥ m N − L
Z H ( S N ) ≥ Z H ( B u ) ≥ δ D N e β ( H Q ( u ) − m N ) ≥ η D N , η = δ e − β L . Z_H(S_N)\ge Z_H(B_u)\ge\delta D_N e^{\beta(H_Q(u)-m_N)}\ge\eta D_N,\qquad\eta=\delta e^{-\beta L}. Z H ( S N ) ≥ Z H ( B u ) ≥ δ D N e β ( H Q ( u ) − m N ) ≥ η D N , η = δ e − β L . We conclude that
lim sup N → ∞ P H { Z H ( S N ) < η D N } < 2 ε . (70) \limsup_{N\to\infty}\mathbb{P}_H\{Z_H(S_N)<\eta D_N\}<2\varepsilon. \tag*{(70)} N → ∞ lim sup P H { Z H ( S N ) < η D N } < 2 ε . (70) This uses only the expected number of light bands; no band is selected in a way that depends on its determinant or on the field.
Critical points below the window. Let G N , M = { max S N H Q ≤ m N + M } G_{N,M}=\{\max_{S_N}H_Q\le m_N+M\} G N , M = { max S N H Q ≤ m N + M } lim M → ∞ lim sup N → ∞ P H ( G N , M c ) = 0 \lim_{M\to\infty}\limsup_{N\to\infty}\mathbb{P}_H(G_{N,M}^{\mathrm c})=0 lim M → ∞ lim sup N → ∞ P H ( G N , M c ) = 0 G N , M G_{N,M} G N , M u ∈ C N c o ( H ) ∖ C N , M ( H ) u\in C_N^{\mathrm{co}}(H)\setminus C_{N,M}(H) u ∈ C N co ( H ) ∖ C N , M ( H ) H Q ( u ) < m N − M H_Q(u)<m_N-M H Q ( u ) < m N − M u u u K Q ( u ) / N ∈ U K_Q(u)/N\in U K Q ( u ) / N ∈ U u ∉ C N , M ( H ) u\notin C_{N,M}(H) u ∈ / C N , M ( H ) ∣ H Q ( u ) − m N ∣ > M \lvert H_Q(u)-m_N\rvert>M ∣ H Q ( u ) − m N ∣ > M H Q ( u ) ≤ m N + M H_Q(u)\le m_N+M H Q ( u ) ≤ m N + M G N , M G_{N,M} G N , M G N , M G_{N,M} G N , M
B N , M ( H ) c ⊂ [ ⋃ u ∈ C N c o ( H ) B u ] c ∪ ⋃ u ∈ C N c o ( H ) H Q ( u ) < m N − M B u . B_{N,M}(H)^{\mathrm c}\subset\left[\bigcup_{u\in C_N^{\mathrm{co}}(H)}B_u\right]^{\mathrm c}\cup\bigcup_{\substack{u\in C_N^{\mathrm{co}}(H)\\H_Q(u)<m_N-M}}B_u. B N , M ( H ) c ⊂ u ∈ C N co ( H ) ⋃ B u c ∪ u ∈ C N co ( H ) H Q ( u ) < m N − M ⋃ B u . and on G N , M ∩ { Z H ( S N ) ≥ η D N } G_{N,M}\cap\{Z_H(S_N)\ge\eta D_N\} G N , M ∩ { Z H ( S N ) ≥ η D N }
π H ( B N , M ( H ) c ) ≤ π H ( [ ⋃ u ∈ C N c o ( H ) B u ] c ) + 1 η D N ∑ u ∈ C N c o ( H ) H Q ( u ) < m N − M Z H ( B u ) . (71) \pi^H(B_{N,M}(H)^{\mathrm c})\le\pi^H\left(\left[\bigcup_{u\in C_N^{\mathrm{co}}(H)}B_u\right]^{\mathrm c}\right)+\frac{1}{\eta D_N}\sum_{\substack{u\in C_N^{\mathrm{co}}(H)\\H_Q(u)<m_N-M}}Z_H(B_u). \tag*{(71)} π H ( B N , M ( H ) c ) ≤ π H u ∈ C N co ( H ) ⋃ B u c + η D N 1 u ∈ C N co ( H ) H Q ( u ) < m N − M ∑ Z H ( B u ) . (71) A point covered by several bands is counted several times in the last sum, which only increases it; we therefore do not need the bands to be disjoint.
Conclusion. We bound π H ( B N , M c ) \pi^H(B_{N,M}^{\mathrm c}) π H ( B N , M c ) G N , M ∩ { Z H ( S N ) ≥ η D N } G_{N,M}\cap\{Z_H(S_N)\ge\eta D_N\} G N , M ∩ { Z H ( S N ) ≥ η D N } N → ∞ N\to\infty N → ∞ M M M
lim sup N → ∞ E H π H ( B N , M ( H ) c ) ≤ 2 ε + lim sup N → ∞ P H ( G N , M c ) + η − 1 C ∗ e − c ∗ M . \limsup_{N\to\infty}\mathbb{E}_H\pi^H(B_{N,M}(H)^{\mathrm c})\le2\varepsilon+\limsup_{N\to\infty}\mathbb{P}_H(G_{N,M}^{\mathrm c})+\eta^{-1}C_*e^{-c_*M}. N → ∞ lim sup E H π H ( B N , M ( H ) c ) ≤ 2 ε + N → ∞ lim sup P H ( G N , M c ) + η − 1 C ∗ e − c ∗ M . Letting M → ∞ M\to\infty M → ∞ L L L δ \delta δ η \eta η 2 ε 2\varepsilon 2 ε ε \varepsilon ε L L L δ \delta δ M M M N → ∞ N\to\infty N → ∞
Remark 4.10 (On hypothesis (ii)). Hypothesis (ii) bounds the expected number of critical points in a fixed window whose band is light. Suppose that the counts in the window C N , L C_{N,L} C N , L { Z H ( B u ) < δ D N e β ( H Q ( u ) − m N ) } \{Z_H(B_u)<\delta D_N e^{\beta(H_Q(u)-m_N)}\} { Z H ( B u ) < δ D N e β ( H Q ( u ) − m N ) } ρ L ( δ ) + o N ( 1 ) \rho_L(\delta)+o_N(1) ρ L ( δ ) + o N ( 1 ) ρ L ( δ ) → 0 \rho_L(\delta)\to0 ρ L ( δ ) → 0 δ ↓ 0 \delta\downarrow0 δ ↓ 0 ( A L + o N ( 1 ) ) ( ρ L ( δ ) + o N ( 1 ) ) (A_L+o_N(1))(\rho_L(\delta)+o_N(1)) ( A L + o N ( 1 )) ( ρ L ( δ ) + o N ( 1 )) C N , L ( H ) C_{N,L}(H) C N , L ( H ) C N , L ( H ) C_{N,L}(H) C N , L ( H )
Initial laws other than the uniform law
We now transfer separation from the uniform initial law to other initial laws. The first result uses only the rotation invariance (4.1). It is an instance of the classical symmetrization over a compact group: the expected distance to equilibrium is a convex and rotation-invariant function of the initial law, so it is smallest at the rotation average of any initial law, which is ν N \nu_N ν N
Theorem 4.11 (Initial laws independent of the field). Let β > 0 \beta>0 β > 0 N ≥ 2 N\ge2 N ≥ 2 μ \mu μ S N S_N S N T ≥ 0 T\ge0 T ≥ 0
E H [ 1 − ∥ μ P T H − π H ∥ T V ] ≤ E H [ 1 − ∥ ν N P T H − π H ∥ T V ] . (72) \mathbb{E}_H\left[1-\left\|\mu P_T^H-\pi^H\right\|_{\mathrm{TV}}\right]\le\mathbb{E}_H\left[1-\left\|\nu_NP_T^H-\pi^H\right\|_{\mathrm{TV}}\right]. \tag*{(72)} E H [ 1 − μ P T H − π H TV ] ≤ E H [ 1 − ν N P T H − π H TV ] . (72) Further, for deterministic times T N T_N T N ε > 0 \varepsilon>0 ε > 0
sup μ P H { inf 0 ≤ t ≤ T N ∥ μ P t H − π H ∥ T V < 1 − ε } ≤ ε − 1 E H [ 1 − ∥ ν N P T N H − π H ∥ T V ] , (73) \sup_{\mu}\mathbb{P}^H\left\{\inf_{0\le t\le T_N}\left\|\mu P_t^H-\pi^H\right\|_{\mathrm{TV}}<1-\varepsilon\right\}\le\varepsilon^{-1}\mathbb{E}_H\left[1-\left\|\nu_NP_{T_N}^H-\pi^H\right\|_{\mathrm{TV}}\right], \tag*{(73)} μ sup P H { 0 ≤ t ≤ T N inf μ P t H − π H TV < 1 − ε } ≤ ε − 1 E H [ 1 − ν N P T N H − π H TV ] , (73) where the supremum is over all Borel probability measures μ \mu μ S N S_N S N inf 0 ≤ t ≤ T N ∥ ν N P t H − π H ∥ T V → 1 \inf_{0\le t\le T_N}\left\|\nu_NP_t^H-\pi^H\right\|_{\mathrm{TV}}\to1 inf 0 ≤ t ≤ T N ν N P t H − π H TV → 1 P H \mathbb{P}_H P H ε > 0 \varepsilon>0 ε > 0 μ N \mu_N μ N S N S_N S N H N H_N H N H N H_N H N μ N \mu_N μ N
Proof. We show that the expected distance is invariant under rotations of the initial law and convex in it, and then average over the rotation group.
Averaging over rotations. Let d g dg d g O ( N ) O(N) O ( N ) f f f S N S_N S N φ ( x ) = ∫ f ( g x ) d g \varphi(x)=\int f(gx)\,dg φ ( x ) = ∫ f ( g x ) d g h ∈ O ( N ) h\in O(N) h ∈ O ( N ) φ ( h x ) = ∫ f ( g h x ) d g = φ ( x ) \varphi(hx)=\int f(ghx)\,dg=\varphi(x) φ ( h x ) = ∫ f ( g h x ) d g = φ ( x ) O ( N ) O(N) O ( N ) S N S_N S N φ \varphi φ ν N \nu_N ν N ν N \nu_N ν N ∫ f d ν N \int f\,\mathrm{d}\nu_N ∫ f d ν N f f f ∫ ∫ f ( g x ) μ ( d x ) d g = ∫ f d ν N \int\!\int f(gx)\,\mu(\mathrm{d}x)\,\mathrm{d}g=\int f\,\mathrm{d}\nu_N ∫ ∫ f ( g x ) μ ( d x ) d g = ∫ f d ν N
∫ O ( N ) g ∗ μ d g = ν N . (74) \int_{O(N)}g_*\mu\,\mathrm{d}g=\nu_N. \tag*{(74)} ∫ O ( N ) g ∗ μ d g = ν N . (74) Invariance and convexity. For a probability measure λ \lambda λ S N S_N S N F ( λ ) = E H ∥ λ P T H − π H ∥ T V F(\lambda)=\mathbb{E}_H\left\|\lambda P_T^H-\pi^H\right\|_{\mathrm{TV}} F ( λ ) = E H λ P T H − π H TV g ∈ O ( N ) g\in O(N) g ∈ O ( N ) ( g ∗ μ ) P T H g = g ∗ ( μ P T H ) (g_*\mu)P_T^{H^g}=g_*(\mu P_T^H) ( g ∗ μ ) P T H g = g ∗ ( μ P T H ) π H g = g ∗ π H \pi^{H^g}=g_*\pi^H π H g = g ∗ π H g ∗ g_* g ∗
∥ ( g ∗ μ ) P T H g − π H g ∥ T V = ∥ μ P T H − π H ∥ T V . \left\|(g_*\mu)P_T^{H^g}-\pi^{H^g}\right\|_{\mathrm{TV}}=\left\|\mu P_T^H-\pi^H\right\|_{\mathrm{TV}}. ( g ∗ μ ) P T H g − π H g TV = μ P T H − π H TV . Taking expectations and using (4.1), we obtain F ( g ∗ μ ) = F ( μ ) F(g_*\mu)=F(\mu) F ( g ∗ μ ) = F ( μ ) ν N P T H = ∫ ( g ∗ μ ) P T H d g \nu_NP_T^H=\int(g_*\mu)P_T^H\,\mathrm{d}g ν N P T H = ∫ ( g ∗ μ ) P T H d g H H H A A A
∣ ( ν N P T H ) ( A ) − π H ( A ) ∣ ≤ ∫ ∣ ( ( g ∗ μ ) P T H ) ( A ) − π H ( A ) ∣ d g ≤ ∫ ∥ ( g ∗ μ ) P T H − π H ∥ T V d g . \left|(\nu_NP_T^H)(A)-\pi^H(A)\right|\leq\int\left|((g_*\mu)P_T^H)(A)-\pi^H(A)\right|\,\mathrm{d}g\leq\int\left\|(g_*\mu)P_T^H-\pi^H\right\|_{\mathrm{TV}}\,\mathrm{d}g. ( ν N P T H ) ( A ) − π H ( A ) ≤ ∫ (( g ∗ μ ) P T H ) ( A ) − π H ( A ) d g ≤ ∫ ( g ∗ μ ) P T H − π H TV d g . Taking the supremum over A A A ∥ ν N P T H − π H ∥ T V ≤ ∫ ∥ ( g ∗ μ ) P T H − π H ∥ T V d g \left\|\nu_NP_T^H-\pi^H\right\|_{\mathrm{TV}}\leq\int\left\|(g_*\mu)P_T^H-\pi^H\right\|_{\mathrm{TV}}\,\mathrm{d}g ν N P T H − π H TV ≤ ∫ ( g ∗ μ ) P T H − π H TV d g ( H , g ) (H,g) ( H , g ) ( H , g , x ) ↦ P T H ( g x , A ) (H,g,x)\mapsto P_T^H(gx,A) ( H , g , x ) ↦ P T H ( g x , A ) A \mathcal{A} A F ( ν N ) ≤ ∫ F ( g ∗ μ ) d g = F ( μ ) F(\nu_N)\leq\int F(g_*\mu)\,\mathrm{d}g=F(\mu) F ( ν N ) ≤ ∫ F ( g ∗ μ ) d g = F ( μ )
Uniformity. Recall from (4.4) that the infimum over t ≤ T N t\leq T_N t ≤ T N T N T_N T N 1 − ∥ μ P T N H − π H ∥ T V 1-\left\|\mu P_{T_N}^H-\pi^H\right\|_{\mathrm{TV}} 1 − μ P T N H − π H TV μ \mu μ μ \mu μ Y N = 1 − ∥ ν N P T N H − π H ∥ T V ∈ [ 0 , 1 ] Y_N=1-\left\|\nu_NP_{T_N}^H-\pi^H\right\|_{\mathrm{TV}}\in[0,1] Y N = 1 − ν N P T N H − π H TV ∈ [ 0 , 1 ] E H Y N → 0 \mathbb{E}_H Y_N\to0 E H Y N → 0 E H Y N ≤ η + P H { Y N > η } \mathbb{E}_H Y_N\leq\eta+\mathbb{P}_H\{Y_N>\eta\} E H Y N ≤ η + P H { Y N > η } η > 0 \eta>0 η > 0 μ N \mu_N μ N H N H_N H N μ N \mu_N μ N ( H , μ ) ↦ ∥ μ P T N H − π H ∥ T V (H,\mu)\mapsto\left\|\mu P_{T_N}^H-\pi^H\right\|_{\mathrm{TV}} ( H , μ ) ↦ μ P T N H − π H TV μ ↦ ∫ f d μ \mu\mapsto\int f\,\mathrm{d}\mu μ ↦ ∫ f d μ f f f
Lemma 4.13 (Equal expected mass from every starting point). Let T ≥ 0 T \ge0 T ≥ 0 B ( H ) ⊂ S N B(H) \subset S_N B ( H ) ⊂ S N { ( H , x ) : x ∈ B ( H ) } \{(H,x): x \in B(H)\} {( H , x ) : x ∈ B ( H )} B ( H g ) = g B ( H ) B(H^g) = gB(H) B ( H g ) = g B ( H ) g ∈ O ( N ) g \in O(N) g ∈ O ( N ) H ∈ H N H \in\mathcal{H}_N H ∈ H N q = E H ( ν N P T H ) ( B ( H N ) ) q = \mathbb{E}_H(\nu_N P_T^H)(B(H_N)) q = E H ( ν N P T H ) ( B ( H N )) x ∈ S N x \in S_N x ∈ S N μ \mu μ S N S_N S N
E H P T H ( x , B ( H ) ) = q , E H ( μ P T H ) ( B ( H ) ) = q . (75) \mathbb{E}_H P_T^H(x,B(H)) = q,\qquad\mathbb{E}_H(\mu P_T^H)(B(H)) = q. \tag*{(75)} E H P T H ( x , B ( H )) = q , E H ( μ P T H ) ( B ( H )) = q . (75) Proof. The function ϱ ( H , x ) = P T H ( x , B ( H ) ) \varrho(H,x) = P_T^H(x,B(H)) ϱ ( H , x ) = P T H ( x , B ( H )) H N × S N \mathcal{H}_N \times S_N H N × S N B B B B B B ϱ ( H g , g x ) = P T H g ( g x , g B ( H ) ) = ϱ ( H , x ) \varrho(H^g,gx) = P_T^{H^g}(gx,gB(H)) = \varrho(H,x) ϱ ( H g , g x ) = P T H g ( g x , g B ( H )) = ϱ ( H , x ) E H ϱ ( H , g x ) = E H ϱ ( H g , g x ) = E H ϱ ( H , x ) \mathbb{E}_H\varrho(H,gx) = \mathbb{E}_H\varrho(H^g,gx) = \mathbb{E}_H\varrho(H,x) E H ϱ ( H , g x ) = E H ϱ ( H g , g x ) = E H ϱ ( H , x ) O ( N ) O(N) O ( N ) S N S_N S N x ↦ E H ϱ ( H , x ) x \mapsto\mathbb{E}_H\varrho(H,x) x ↦ E H ϱ ( H , x ) ν N \nu_N ν N q q q μ \mu μ □ \square □
For the remaining results, we let Q Q Q U U U ( m N ) (m_N) ( m N ) b b b B N , M ( H ) B_{N,M}(H) B N , M ( H ) B N , M ( H g ) = g B N , M ( H ) B_{N,M}(H^g) = gB_{N,M}(H) B N , M ( H g ) = g B N , M ( H ) T N T_N T N
q N , M = E H ( ν N P T N H ) ( B N , M ( H ) ) . q_{N,M} = \mathbb{E}_H\left(\nu_N P_{T_N}^H\right)(B_{N,M}(H)). q N , M = E H ( ν N P T N H ) ( B N , M ( H )) . In Proposition 13.1, this endpoint mass is shown to be at most a negative power of N N N M M M
q N , M ≤ C M N − ζ for all sufficiently large N , (76) q_{N,M} \le C_M N^{-\zeta}\qquad\text{for all sufficiently large }N, \tag*{(76)} q N , M ≤ C M N − ζ for all sufficiently large N , (76) where ζ > 0 \zeta> 0 ζ > 0 M > 0 M > 0 M > 0 C M C_M C M
The next result allows a field-dependent choice among K N K_N K N K N K_N K N
Corollary 4.14 (Selection among prescribed initial laws). Let β > 0 \beta> 0 β > 0 ( T N ) (T_N) ( T N ) N N N μ N , 1 , … , μ N , K N \mu_{N,1},\ldots,\mu_{N,K_N} μ N , 1 , … , μ N , K N S N S_N S N H H H K N q N , M → 0 K_Nq_{N,M} \to0 K N q N , M → 0 N → ∞ N \to\infty N → ∞ M > 0 M > 0 M > 0
inf { ∥ ( ∑ j = 1 K N w j μ N , j ) P t H − π H ∥ T V : w j ≥ 0 , ∑ j = 1 K N w j = 1 , 0 ≤ t ≤ T N } ⟶ 1 (77) \inf\left\{\left\|\left(\sum_{j=1}^{K_N}w_j\mu_{N,j}\right)P_t^H-\pi^H\right\|_{\mathrm{TV}}: w_j\ge0,\ \sum_{j=1}^{K_N}w_j=1,\ 0\le t\le T_N\right\}\longrightarrow1 \tag*{(77)} inf ⎩ ⎨ ⎧ ( j = 1 ∑ K N w j μ N , j ) P t H − π H TV : w j ≥ 0 , j = 1 ∑ K N w j = 1 , 0 ≤ t ≤ T N ⎭ ⎬ ⎫ ⟶ 1 (77) in P H \mathbb{P}_H P H w j = w j ( H ) w_j = w_j(H) w j = w j ( H ) μ N , j \mu_{N,j} μ N , j H N H_N H N K N K_N K N
Proof. Fix M > 0 M > 0 M > 0 a j ( H ) = ( μ N , j P N H ) ( B N , M ( H ) ) a_j(H) = (\mu_{N,j}P_N^H)(B_{N,M}(H)) a j ( H ) = ( μ N , j P N H ) ( B N , M ( H )) Lemma 4.13 , E H a j = q N , M \mathbb{E}_H a_j = q_{N,M} E H a j = q N , M j j j
E H max j ≤ K N a j ( H ) ≤ ∑ j ≤ K N E H a j ( H ) = K N q N , M ⟶ 0. \mathbb{E}_H \max_{j \le K_N} a_j(H) \le\sum_{j \le K_N} \mathbb{E}_H a_j(H) = K_N q_{N,M} \longrightarrow0. E H j ≤ K N max a j ( H ) ≤ j ≤ K N ∑ E H a j ( H ) = K N q N , M ⟶ 0. For every choice of weights, the mass of B N , M ( H ) B_{N,M}(H) B N , M ( H ) ( ∑ j w j μ N , j ) P N H (\sum_j w_j\mu_{N,j})P_N^H ( ∑ j w j μ N , j ) P N H ∑ j w j a j ( H ) ≤ max j a j ( H ) \sum_j w_j a_j(H) \le\max_j a_j(H) ∑ j w j a j ( H ) ≤ max j a j ( H ) Proposition 3.10 to each mixture, we find that the infimum Δ N ( H ) \Delta_N(H) Δ N ( H ) (77) satisfies
Δ N ( H ) ≥ 1 − π H ( B N , M ( H ) c ) − max j ≤ K N a j ( H ) . \Delta_N(H) \ge1 - \pi^H\left(B_{N,M}(H)^c\right) - \max_{j \le K_N} a_j(H). Δ N ( H ) ≥ 1 − π H ( B N , M ( H ) c ) − j ≤ K N max a j ( H ) . Then Lemma 4.4 , with E N , M = max j a j \mathcal{E}_{N,M} = \max_j a_j E N , M = max j a j (77) . The infimum Δ N \Delta_N Δ N H H H (62) it is the infimum over the weights of the distance at T N T_N T N ∑ j ∣ w j − w j ′ ∣ \sum_j |w_j - w'_j| ∑ j ∣ w j − w j ′ ∣ w w w w ′ w' w ′ E H max j a j ≤ K N q N , M \mathbb{E}_H \max_j a_j \le K_N q_{N,M} E H max j a j ≤ K N q N , M
Finally, we measure the dependence of the initial law on the field by its relative entropy with respect to ν N \nu_N ν N
Corollary 4.15 (Field-dependent initial laws with small relative entropy). Let β > 0 \beta> 0 β > 0 ( T N ) (T_N) ( T N ) (60) , and that (76) holds for some ζ > 0 \zeta> 0 ζ > 0 H ↦ μ N H H \mapsto\mu_N^H H ↦ μ N H H N \mathcal{H}_N H N S N \mathcal{S}_N S N
D ( μ N H ∥ ν N ) log N ⟶ 0 in P H -probability . (78) \frac{D(\mu_N^H \Vert\nu_N)}{\log N} \longrightarrow0 \quad\text{in } \mathbb{P}_H\text{-probability}. \tag*{(78)} log N D ( μ N H ∥ ν N ) ⟶ 0 in P H -probability . (78) Then inf 0 ≤ t ≤ T N ∥ μ N H P t H − π H ∥ T V ⟶ 1 \inf_{0 \le t \le T_N} \lVert\mu_N^H P_t^H - \pi^H \rVert_{\mathrm{TV}} \longrightarrow1 inf 0 ≤ t ≤ T N ∥ μ N H P t H − π H ∥ TV ⟶ 1 P H \mathbb{P}_H P H
Proof. Fix M > 0 M > 0 M > 0 τ ∈ ( 0 , ζ ) \tau\in(0,\zeta) τ ∈ ( 0 , ζ ) a N , M ( H ) = ( ν N P N H ) ( B N , M ( H ) ) a_{N,M}(H) = (\nu_NP_N^H)(B_{N,M}(H)) a N , M ( H ) = ( ν N P N H ) ( B N , M ( H )) (76) , we have
P H { a N , M > N − τ } ≤ C M N τ − ζ ⟶ 0. \mathbb{P}_H\{a_{N,M} > N^{-\tau}\} \le C_M N^{\tau-\zeta} \longrightarrow0. P H { a N , M > N − τ } ≤ C M N τ − ζ ⟶ 0. Fix a field H H H a N , M ( H ) ≤ N − τ a_{N,M}(H) \le N^{-\tau} a N , M ( H ) ≤ N − τ D ( μ N H ∥ ν N ) < ∞ D(\mu_N^H\Vert\nu_N) < \infty D ( μ N H ∥ ν N ) < ∞ a N , M ( H ) = 0 a_{N,M}(H) = 0 a N , M ( H ) = 0 Proposition 3.10 we have
inf 0 ≤ t ≤ T N ∥ μ N H P t H − π H ∥ T V ≥ 1 − π H ( B N , M ( H ) c ) . \inf_{0 \le t \le T_N} \lVert\mu_N^H P_t^H - \pi^H \rVert_{\mathrm{TV}} \ge1 - \pi^H\left(B_{N,M}(H)^c\right). 0 ≤ t ≤ T N inf ∥ μ N H P t H − π H ∥ TV ≥ 1 − π H ( B N , M ( H ) c ) . If 0 < a N , M ( H ) ≤ N − τ < 1 0 < a_{N,M}(H) \le N^{-\tau} < 1 0 < a N , M ( H ) ≤ N − τ < 1 Proposition 3.10 implies that
inf 0 ≤ t ≤ T N ∥ μ N H P t H − π H ∥ T V ≥ 1 − π H ( B N , M ( H ) c ) − D ( μ N H ∥ ν N ) + log 2 τ log N . \inf_{0 \le t \le T_N} \lVert\mu_N^H P_t^H - \pi^H \rVert_{\mathrm{TV}} \ge1 - \pi^H\left(B_{N,M}(H)^c\right) - \frac{D(\mu_N^H\Vert\nu_N) + \log2}{\tau\log N}. 0 ≤ t ≤ T N inf ∥ μ N H P t H − π H ∥ TV ≥ 1 − π H ( B N , M ( H ) c ) − τ log N D ( μ N H ∥ ν N ) + log 2 . Let E N , M ( H ) \mathcal{E}_{N,M}(H) E N , M ( H ) a N , M ( H ) ≤ N − τ a_{N,M}(H) \le N^{-\tau} a N , M ( H ) ≤ N − τ D ( μ N H ∥ ν N ) < ∞ D(\mu_{N}^{H}\mathbin{\|}\nu_{N}) < \infty D ( μ N H ∥ ν N ) < ∞ E N , M ( H ) = 1 \mathcal{E}_{N,M}(H) = 1 E N , M ( H ) = 1 E N , M → 0 \mathcal{E}_{N,M} \to0 E N , M → 0 □ \square □
Statics of the condensed phase
In this section, we prove the static results on which Part II rests. The central object is the first-variation function V β V_\beta V β (79) below, whose zero set is the contact set. In Theorem 5.7 , we show for every finite mixture and every β > 0 \beta> 0 β > 0 q ∈ [ 0 , 1 ) q \in[0,1) q ∈ [ 0 , 1 ) Definition 5.4 ) if and only if V β ( q ) = 0 V_\beta(q) = 0 V β ( q ) = 0 q N qN q N Lemma 5.8 restates this gap as an exponential bound for replicas with mutual overlaps close to q q q
In § 5.2.3 , we specialize to pairs ( ξ , β ) (\xi,\beta) ( ξ , β ) Q Q Q Definition 1.1 , which we restate as Definition 5.9 . Such pairs are condensed (Lemma 5.10 ). Proposition 5.11 collects the static consequences of these 1RSB assumptions. The Parisi measure has two atoms 0 0 0 Q Q Q Q Q Q β 2 ψ Q ′ ′ ( 0 ) < 1 \beta^2\psi_Q^{\prime\prime}(0) < 1 β 2 ψ Q ′′ ( 0 ) < 1 H Q H_Q H Q E 0 E_0 E 0 K 0 K_0 K 0 K 0 K_0 K 0 2 b Q 2\sqrt{b_Q} 2 b Q 0 0 0 Q Q Q H Q H_Q H Q E 0 E_0 E 0 K 0 K_0 K 0 § 5.2.5 shows that the 1RSB assumptions persist under small changes of ξ \xi ξ β \beta β § 5.2.6 verifies the 1RSB assumptions for the classes of Corollary 1.3 . These are the covariances for which ξ ′ ′ − 1 / 2 \xi^{\prime\prime-1/2} ξ ′′ − 1/2 Proposition 5.17 ), in particular those in a degree window (Proposition 5.18 ), the pure models (Proposition 5.20 ), the small even perturbations of pure even models (Proposition 5.23), and the explicit covariance r 4 + b r 12 r^4 + br^{12} r 4 + b r 12 Proposition 5.24 , for which ξ ′ ′ − 1 / 2 \xi^{\prime\prime-1/2} ξ ′′ − 1/2
From § 6.4 , we use the representation of the field (Lemma 3.2), the Crisanti–Sommers functional, the Parisi formula, and the Crisanti–Sommers optimality criterion (Definition 3.13 , Theorem 3.14 , and Lemma 3.15 ), and the thresholds and the replica-symmetric regime (Definition 3.16 and Lemma 3.18 ). No other result of the paper is used here. The results of this section are used as follows. Section 6 uses Theorem 5.7 , Lemma 5.8, Proposition 5.11, and Corollary 5.16. The radius Q \sqrt{Q} Q E 0 E_0 E 0 K 0 K_0 K 0 § 6.2 and 9 to 12. The bound of Lemma 5.22 is used in Section 9, and the strict replica symmetry of the residual model in Section 12. The membership statements of § 5.2.6 enter only the proof of Corollary 1.3 in Section 13.
Theorem 5.7 belongs to Subag’s generalized TAP theory [143]. That every zero of V β V_\beta V β V β V_\beta V β (87) for both directions. The ground-state formula that we use is due to Chen and Sen [48] and, independently, to Jagannath and Tobasco [96]. The two-atom stationarity equations are the classical one-step equations of Crisanti and Sommers [52]. The bound on the number of contacts under strict convexity of ξ ′ ′ − 1 / 2 \xi^{\prime\prime-1/2} ξ ′′ − 1/2
Notation. Throughout the section, ξ ( r ) = ∑ p = 2 P c p r p \xi(r) = \sum_{p=2}^{P} c_p r^p ξ ( r ) = ∑ p = 2 P c p r p c P > 0 c_P > 0 c P > 0 β > 0 \beta> 0 β > 0 M \mathcal{M} M x ^ ζ \hat{x}_{\zeta} x ^ ζ P β ( ζ ; η ) \mathcal{P}_{\beta}(\zeta;\eta) P β ( ζ ; η ) ζ ∈ M \zeta\in\mathcal{M} ζ ∈ M
V ζ ( r ) = ∫ 0 r r − t x ^ ζ ( t ) 2 d t − β 2 ξ ( r ) , 0 ≤ r < 1. (79) V_{\zeta}(r) = \int_0^r \frac{r-t}{\hat{x}_{\zeta}(t)^2}\,\mathrm{d}t - \beta^2 \xi(r), \qquad0 \le r < 1. \tag*{(79)} V ζ ( r ) = ∫ 0 r x ^ ζ ( t ) 2 r − t d t − β 2 ξ ( r ) , 0 ≤ r < 1. (79) In the notation of (3.18) with η = ξ \eta= \xi η = ξ V ζ = − f ζ V_{\zeta} = -f_{\zeta} V ζ = − f ζ x ^ ζ \hat{x}_{\zeta} x ^ ζ [ 0 , 1 ) [0,1) [ 0 , 1 ) V ζ V_{\zeta} V ζ [ 0 , 1 ) [0,1) [ 0 , 1 )
V ζ ( 0 ) = V ζ ′ ( 0 ) = 0 , V ζ ′ ( r ) = ∫ 0 r d t x ^ ζ ( t ) 2 − β 2 ξ ′ ( r ) , V ζ ′ ′ ( r ) = 1 x ^ ζ ( r ) 2 − β 2 ξ ′ ′ ( r ) . (80) V_{\zeta}(0) = V_{\zeta}'(0) = 0, \qquad V_{\zeta}'(r) = \int_0^r \frac{\mathrm{d}t}{\hat{x}_{\zeta}(t)^2} - \beta^2 \xi'(r), \qquad V_{\zeta}''(r) = \frac{1}{\hat{x}_{\zeta}(r)^2} - \beta^2 \xi''(r). \tag*{(80)} V ζ ( 0 ) = V ζ ′ ( 0 ) = 0 , V ζ ′ ( r ) = ∫ 0 r x ^ ζ ( t ) 2 d t − β 2 ξ ′ ( r ) , V ζ ′′ ( r ) = x ^ ζ ( r ) 2 1 − β 2 ξ ′′ ( r ) . (80) We write ζ β \zeta_{\beta} ζ β ( ξ , β ) (\xi,\beta) ( ξ , β ) P β ( ⋅ ; ξ ) \mathcal{P}_{\beta}(\cdot\,;\xi) P β ( ⋅ ; ξ ) x ^ β = x ^ ζ β \hat{x}_{\beta} = \hat{x}_{\zeta_{\beta}} x ^ β = x ^ ζ β V β = V ζ β V_{\beta} = V_{\zeta_{\beta}} V β = V ζ β V ζ V_{\zeta} V ζ first-variation function of ζ \zeta ζ − f ζ -f_{\zeta} − f ζ V β ≥ 0 V_{\beta} \ge0 V β ≥ 0 V β V_{\beta} V β [ 0 , 1 ) [0,1) [ 0 , 1 ) contact set of ( ζ , β ) (\zeta,\beta) ( ζ , β ) contact points . This interpretation agrees with the coincidence set in the obstacle formulation of [97, Theorem 1.1], after absorbing β 2 \beta^2 β 2 F ( β ) = P β ( ζ β ; ξ ) F(\beta) = \mathcal{P}_{\beta}(\zeta_{\beta};\xi) F ( β ) = P β ( ζ β ; ξ ) ζ β \zeta_{\beta} ζ β V β V_{\beta} V β [ 0 , 1 ) [0,1) [ 0 , 1 ) ζ β \zeta_{\beta} ζ β
For 0 ≤ q < 1 0 \le q < 1 0 ≤ q < 1 residual mixture at q q q
ψ q ( t ) = ξ ( q + ( 1 − q ) t ) − ξ ( q ) − ( 1 − q ) ξ ′ ( q ) t = ∑ k = 2 P ( ∑ p = k P c p ( p k ) q p − k ) ( 1 − q ) k t k . (81) \psi_q(t)=\xi\bigl(q+(1-q)t\bigr)-\xi(q)-(1-q)\xi'(q)t=\sum_{k=2}^{P}\left(\sum_{p=k}^{P}c_p\binom{p}{k}q^{p-k}\right)(1-q)^kt^k. \tag*{(81)} ψ q ( t ) = ξ ( q + ( 1 − q ) t ) − ξ ( q ) − ( 1 − q ) ξ ′ ( q ) t = k = 2 ∑ P p = k ∑ P c p ( k p ) q p − k ( 1 − q ) k t k . (81) Its coefficients are nonnegative, it has no terms of degree zero or one, and its coefficient of t P t^P t P c P ( 1 − q ) P > 0 c_P(1-q)^P>0 c P ( 1 − q ) P > 0 ψ q \psi_q ψ q ψ 0 = ξ \psi_0=\xi ψ 0 = ξ 0 < Q < 1 0<Q<1 0 < Q < 1 u ∈ S N u\in S_N u ∈ S N
H Q ( u ) = H N ( Q u ) , K Q ( u ) = d d t H N ( t Q u ) ∣ t = 1 = u ⋅ ∇ H Q ( u ) , H_Q(u)=H_N(\sqrt{Q}u),\qquad K_Q(u)=\left.\frac{\mathrm{d}}{\mathrm{d}t}H_N(t\sqrt{Q}u)\right|_{t=1}=u\cdot\nabla H_Q(u), H Q ( u ) = H N ( Q u ) , K Q ( u ) = d t d H N ( t Q u ) t = 1 = u ⋅ ∇ H Q ( u ) , where H N H_N H N ∇ \nabla ∇ a Q = Q ξ ′ ( Q ) a_Q=Q\xi'(Q) a Q = Q ξ ′ ( Q ) b Q = Q 2 ξ ′ ′ ( Q ) b_Q=Q^2\xi''(Q) b Q = Q 2 ξ ′′ ( Q ) K Q ( u ) K_Q(u) K Q ( u ) radial derivative of H Q H_Q H Q u u u
Ground states and the generalized TAP functional
For 0 < q ≤ 1 0<q\le1 0 < q ≤ 1 u ↦ H N ( q u ) u\mapsto H_N(\sqrt{q}u) u ↦ H N ( q u ) S N S_N S N N ξ ( q R ( u , v ) ) N\xi(qR(u,v)) N ξ ( q R ( u , v )) H N H_N H N q N qN q N N N N
Definition 5.1 (Zero-temperature Crisanti–Sommers functional). Let 0 < q ≤ 1 0<q\le1 0 < q ≤ 1 admissible ground profile on [ 0 , q ] [0,q] [ 0 , q ]
ℓ ( v ) = L − ∫ 0 v α ( w ) d w , 0 ≤ v ≤ q , \ell(v)=L-\int_0^v\alpha(w)\,\mathrm{d}w,\qquad0\le v\le q, ℓ ( v ) = L − ∫ 0 v α ( w ) d w , 0 ≤ v ≤ q , where α : [ 0 , q ) → [ 0 , ∞ ) \alpha:[0,q)\to[0,\infty) α : [ 0 , q ) → [ 0 , ∞ ) L > ∫ 0 q α ( w ) d w L>\int_0^q\alpha(w)\,\mathrm{d}w L > ∫ 0 q α ( w ) d w
Q q ( ℓ ) = 1 2 ( ∫ 0 q ξ ′ ′ ( v ) ℓ ( v ) d v + ∫ 0 q d v ℓ ( v ) ) . (82) \mathcal{Q}_q(\ell)=\frac{1}{2}\left(\int_0^q\xi''(v)\ell(v)\,\mathrm{d}v+\int_0^q\frac{\mathrm{d}v}{\ell(v)}\right). \tag*{(82)} Q q ( ℓ ) = 2 1 ( ∫ 0 q ξ ′′ ( v ) ℓ ( v ) d v + ∫ 0 q ℓ ( v ) d v ) . (82) Note that an admissible profile is nonincreasing. It follows that ℓ ≥ ℓ ( q ) > 0 \ell\ge\ell(q)>0 ℓ ≥ ℓ ( q ) > 0 [ 0 , q ] [0,q] [ 0 , q ] Q q ( ℓ ) \mathcal{Q}_q(\ell) Q q ( ℓ ) Q q \mathcal{Q}_q Q q z ↦ 1 / z z\mapsto1/z z ↦ 1/ z ( 0 , ∞ ) (0,\infty) ( 0 , ∞ )
Theorem 5.2 (Ground-state energy; Chen–Sen). Let 0 < q ≤ 1 0<q\le1 0 < q ≤ 1
E ( q ) = lim N → ∞ 1 N E max u ∈ S N H N ( q u ) E(q)=\lim_{N\to\infty}\frac{1}{N}\mathbb{E}\max_{u\in S_N}H_N(\sqrt{q}u) E ( q ) = N → ∞ lim N 1 E u ∈ S N max H N ( q u ) exists, and E ( q ) = min ℓ Q q ( ℓ ) E(q)=\min_{\ell}\mathcal{Q}_q(\ell) E ( q ) = min ℓ Q q ( ℓ ) [ 0 , q ] [0,q] [ 0 , q ]
We also set E ( 0 ) = 0 E(0)=0 E ( 0 ) = 0 Theorem 5.2 is [48, Theorem 1] after a change of variables, as we now explain. Chen and Sen consider a field on S N S_N S N N η ( R ) N\eta(R) N η ( R ) η ( r ) = ∑ p ≥ 2 γ p 2 r p \eta(r)=\sum_{p\geq2}\gamma_p^2r^p η ( r ) = ∑ p ≥ 2 γ p 2 r p h h h h = 0 h=0 h = 0 η ( r ) = ξ ( q r ) \eta(r)=\xi(qr) η ( r ) = ξ ( q r ) u ↦ H N ( q u ) u\mapsto H_N(\sqrt{q}\,u) u ↦ H N ( q u ) ∑ p 2 p γ p 2 < ∞ \sum_p 2^p\gamma_p^2<\infty ∑ p 2 p γ p 2 < ∞ γ p \gamma_p γ p N − 1 max S N N^{-1}\max_{S_N} N − 1 max S N L 1 L^1 L 1
min 1 2 ( η ′ ( 1 ) L − ∫ 0 1 η ′ ′ ( r ) ∫ 0 r α ( w ) d w d r + ∫ 0 1 d r L − ∫ 0 r α ( w ) d w ) \min\frac{1}{2}\left(\eta'(1)L-\int_0^1\eta''(r)\int_0^r\alpha(w)\,\mathrm{d}w\,\mathrm{d}r+\int_0^1\frac{\mathrm{d}r}{L-\int_0^r\alpha(w)\,\mathrm{d}w}\right) min 2 1 ( η ′ ( 1 ) L − ∫ 0 1 η ′′ ( r ) ∫ 0 r α ( w ) d w d r + ∫ 0 1 L − ∫ 0 r α ( w ) d w d r ) where the minimum is over L > ∫ 0 1 α L>\int_0^1\alpha L > ∫ 0 1 α α \alpha α [ 0 , 1 ) [0,1) [ 0 , 1 ) b ( r ) = L − ∫ 0 r α ( w ) d w b(r)=L-\int_0^r\alpha(w)\,\mathrm{d}w b ( r ) = L − ∫ 0 r α ( w ) d w ℓ ( v ) = q b ( v / q ) \ell(v)=qb(v/q) ℓ ( v ) = q b ( v / q ) ℓ \ell ℓ [ 0 , q ] [0,q] [ 0 , q ] α ( ⋅ / q ) \alpha(\cdot/q) α ( ⋅ / q ) η ′ ( 0 ) = 0 \eta'(0)=0 η ′ ( 0 ) = 0 η ′ ′ ( r ) = q 2 ξ ′ ′ ( q r ) \eta''(r)=q^2\xi''(qr) η ′′ ( r ) = q 2 ξ ′′ ( q r ) v = q r v=qr v = q r
η ′ ( 1 ) L − ∫ 0 1 η ′ ′ ( r ) ∫ 0 r α ( w ) d w d r = ∫ 0 1 η ′ ′ ( r ) b ( r ) d r = ∫ 0 q ξ ′ ′ ( v ) ℓ ( v ) d v , \eta'(1)L-\int_0^1\eta''(r)\int_0^r\alpha(w)\,\mathrm{d}w\,\mathrm{d}r=\int_0^1\eta''(r)b(r)\,\mathrm{d}r=\int_0^q\xi''(v)\ell(v)\,\mathrm{d}v, η ′ ( 1 ) L − ∫ 0 1 η ′′ ( r ) ∫ 0 r α ( w ) d w d r = ∫ 0 1 η ′′ ( r ) b ( r ) d r = ∫ 0 q ξ ′′ ( v ) ℓ ( v ) d v , ∫ 0 1 d r b ( r ) = ∫ 0 q d v ℓ ( v ) . \int_0^1\frac{\mathrm{d}r}{b(r)}=\int_0^q\frac{\mathrm{d}v}{\ell(v)}. ∫ 0 1 b ( r ) d r = ∫ 0 q ℓ ( v ) d v . This shows that the objective of Chen and Sen at ( L , α ) (L,\alpha) ( L , α ) Q q ( ℓ ) Q_q(\ell) Q q ( ℓ ) Theorem 5.2 follows.
Definition 5.3 (Generalized TAP functional). For 0 ≤ q < 1 0\leq q<1 0 ≤ q < 1 F r e s ( β , q ) = P β ( ψ q ) F_{\mathrm{res}}(\beta,q)=P_\beta(\psi_q) F res ( β , q ) = P β ( ψ q ) Theorem 3.14 , this is the limiting free energy of a field with covariance N ψ q ( R ) N\psi_q(R) N ψ q ( R ) β \beta β generalized TAP functional is
T β ( q ) = β E ( q ) + 1 2 log ( 1 − q ) + F r e s ( β , q ) . (83) T_\beta(q)=\beta E(q)+\frac{1}{2}\log(1-q)+F_{\mathrm{res}}(\beta,q). \tag*{(83)} T β ( q ) = β E ( q ) + 2 1 log ( 1 − q ) + F res ( β , q ) . (83) Definition 5.4 (Multisamplable overlaps). For q ∈ [ 0 , 1 ) q\in[0,1) q ∈ [ 0 , 1 ) k ≥ 1 k\geq1 k ≥ 1 ρ > 0 \rho>0 ρ > 0
p N , k , ρ ( q ) = E π ⊗ k ( ∣ R ( σ i , σ j ) − q ∣ < ρ for all 1 ≤ i < j ≤ k ) . (84) p_{N,k,\rho}(q)=\mathbb{E}\,\pi^{\otimes k}\left(\left|R(\sigma^i,\sigma^j)-q\right|<\rho\text{ for all }1\leq i<j\leq k\right). \tag*{(84)} p N , k , ρ ( q ) = E π ⊗ k ( R ( σ i , σ j ) − q < ρ for all 1 ≤ i < j ≤ k ) . (84) where σ 1 , … , σ k \sigma^1,\ldots,\sigma^k σ 1 , … , σ k π \pi π q q q multisamplable if lim N → ∞ N − 1 log p N , k , ρ ( q ) = 0 \lim_{N\to\infty}N^{-1}\log p_{N,k,\rho}(q)=0 lim N → ∞ N − 1 log p N , k , ρ ( q ) = 0 k ≥ 1 k\geq1 k ≥ 1 ρ > 0 \rho>0 ρ > 0
Note that for k = 1 k=1 k = 1 k ≥ 2 k\geq2 k ≥ 2
Theorem 5.5 (Generalized TAP representation; Subag). Let 0 ≤ q < 1 0\leq q<1 0 ≤ q < 1
We have T β ( q ) ≤ F ( β ) T_\beta(q)\leq F(\beta) T β ( q ) ≤ F ( β ) q q q
If lim sup N → ∞ N − 1 log p N , k , ρ ( q ) = 0 \limsup_{N\to\infty} N^{-1}\log p_{N,k,\rho}(q)=0 lim sup N → ∞ N − 1 log p N , k , ρ ( q ) = 0 k ≥ 1 k\ge1 k ≥ 1 ρ > 0 \rho>0 ρ > 0 q q q
Part (1) is [143, Theorem 5, (1.24)], and part (2) is [143, footnote 4 to Definition 2, and Remark 25]. Subag’s mixtures ν ( t ) = ∑ p γ p 2 t p \nu(t)=\sum_p\gamma_p^2t^p ν ( t ) = ∑ p γ p 2 t p γ p 2 = c p \gamma_p^2=c_p γ p 2 = c p N \sqrt{N} N F ( β ) F(\beta) F ( β ) E ⋆ ( q ) E_\star(q) E ⋆ ( q ) N q Nq N q N N N E ( q ) E(q) E ( q ) F β ( q ) F_\beta(q) F β ( q ) ν q \nu_q ν q ψ q \psi_q ψ q F β ( q ) = F r e s ( β , q ) F_\beta(q)=F_{\mathrm{res}}(\beta,q) F β ( q ) = F res ( β , q )
Multisamplable overlaps and the contact set
We first show that 0 always belongs to the contact set. This is [143, Lemma 37] in our notation; we include the short proof.
Lemma 5.6 (Zero is a contact point). The minimum of V β V_\beta V β [ 0 , 1 ) [0,1) [ 0 , 1 ) V β ( 0 ) = 0 V_\beta(0)=0 V β ( 0 ) = 0 V β ≥ 0 V_\beta\ge0 V β ≥ 0 [ 0 , 1 ) [0,1) [ 0 , 1 ) ζ β \zeta_\beta ζ β ξ \xi ξ ζ β \zeta_\beta ζ β
Proof. Let a 0 a_0 a 0 ζ β \zeta_\beta ζ β 0 ≤ a 0 < 1 0\le a_0<1 0 ≤ a 0 < 1 a 0 a_0 a 0 V β V_\beta V β a 0 = 0 a_0=0 a 0 = 0 V β V_\beta V β V β ( 0 ) = 0 V_\beta(0)=0 V β ( 0 ) = 0 a 0 > 0 a_0>0 a 0 > 0 ζ β ( [ 0 , t ] ) = 0 \zeta_\beta([0,t])=0 ζ β ([ 0 , t ]) = 0 t < a 0 t<a_0 t < a 0 x ^ β \hat{x}_\beta x ^ β [ 0 , a 0 ] [0,a_0] [ 0 , a 0 ] x ^ β ( 0 ) \hat{x}_\beta(0) x ^ β ( 0 ) (80) , we see that the function G = − V β ′ G=-V_\beta' G = − V β ′
G ( r ) = β 2 ξ ′ ( r ) − r x ^ β ( 0 ) 2 , 0 ≤ r ≤ a 0 . G(r)=\beta^2\xi'(r)-\frac{r}{\hat{x}_\beta(0)^2}, \qquad0\le r\le a_0. G ( r ) = β 2 ξ ′ ( r ) − x ^ β ( 0 ) 2 r , 0 ≤ r ≤ a 0 . We have G ( 0 ) = 0 G(0)=0 G ( 0 ) = 0 ξ ′ ( 0 ) = 0 \xi'(0)=0 ξ ′ ( 0 ) = 0 G ( a 0 ) = 0 G(a_0)=0 G ( a 0 ) = 0 a 0 a_0 a 0 V β V_\beta V β G ′ ′ = β 2 ξ ′ ′ ′ ≥ 0 G''=\beta^2\xi''' \ge0 G ′′ = β 2 ξ ′′′ ≥ 0 [ 0 , a 0 ] [0,a_0] [ 0 , a 0 ] ξ \xi ξ G ≤ 0 G\le0 G ≤ 0 [ 0 , a 0 ] [0,a_0] [ 0 , a 0 ]
V β ( a 0 ) − V β ( 0 ) = − ∫ 0 a 0 G ( r ) d r ≥ 0. V_\beta(a_0)-V_\beta(0)=-\int_0^{a_0}G(r)\,\mathrm{d}r\ge0. V β ( a 0 ) − V β ( 0 ) = − ∫ 0 a 0 G ( r ) d r ≥ 0. Since a 0 a_0 a 0 V β ( a 0 ) ≤ V β ( 0 ) V_\beta(a_0)\le V_\beta(0) V β ( a 0 ) ≤ V β ( 0 ) V β V_\beta V β V β ( 0 ) = 0 V_\beta(0)=0 V β ( 0 ) = 0
If the minimum degree is at least three, then ξ ′ ′ ′ > 0 \xi'''>0 ξ ′′′ > 0 ( 0 , 1 ) (0,1) ( 0 , 1 ) G G G a 0 > 0 a_0>0 a 0 > 0 G < 0 G<0 G < 0 ( 0 , a 0 ) (0,a_0) ( 0 , a 0 ) V β ( a 0 ) > V β ( 0 ) V_\beta(a_0)>V_\beta(0) V β ( a 0 ) > V β ( 0 ) a 0 a_0 a 0 a 0 = 0 a_0=0 a 0 = 0
For q ∈ ( 0 , 1 ) q \in(0,1) q ∈ ( 0 , 1 ) ζ β , q \zeta_{\beta,q} ζ β , q [ 0 , 1 ) [0,1) [ 0 , 1 )
ζ β , q ( [ 0 , t ] ) = ζ β ( [ 0 , q + ( 1 − q ) t ] ) , 0 ≤ t ≤ 1. (85) \zeta_{\beta,q}([0,t])=\zeta_{\beta}([0,q+(1-q)t]), \qquad0\le t\le1. \tag*{(85)} ζ β , q ([ 0 , t ]) = ζ β ([ 0 , q + ( 1 − q ) t ]) , 0 ≤ t ≤ 1. (85) Equivalently, ζ β , q \zeta_{\beta,q} ζ β , q ζ β \zeta_{\beta} ζ β τ ( r ) = max { 0 , ( r − q ) / ( 1 − q ) } \tau(r)=\max\{0,(r-q)/(1-q)\} τ ( r ) = max { 0 , ( r − q ) / ( 1 − q )} ζ β , q ( { 0 } ) = ζ β ( [ 0 , q ] ) \zeta_{\beta,q}(\{0\})=\zeta_{\beta}([0,q]) ζ β , q ({ 0 }) = ζ β ([ 0 , q ]) ζ β , q ∈ M \zeta_{\beta,q}\in\mathcal{M} ζ β , q ∈ M
Theorem 5.7 (Multisamplable overlaps and the contact set). Let ξ \xi ξ β > 0 \beta>0 β > 0
An overlap q ∈ [ 0 , 1 ) q\in[0,1) q ∈ [ 0 , 1 ) V β ( q ) = 0 V_{\beta}(q)=0 V β ( q ) = 0
Let q ∈ ( 0 , 1 ) q\in(0,1) q ∈ ( 0 , 1 ) V β ( q ) = 0 V_{\beta}(q)=0 V β ( q ) = 0 Q q Q_q Q q ℓ q ( v ) = β x ^ β ( v ) \ell_q(v)=\beta\widehat{x}_{\beta}(v) ℓ q ( v ) = β x β ( v ) 0 ≤ v ≤ q 0\le v\le q 0 ≤ v ≤ q E ( q ) = Q q ( ℓ q ) E(q)=Q_q(\ell_q) E ( q ) = Q q ( ℓ q ) ζ β , q \zeta_{\beta,q} ζ β , q ( ψ q , β ) (\psi_q,\beta) ( ψ q , β )
Let q ∈ ( 0 , 1 ) q\in(0,1) q ∈ ( 0 , 1 ) V β ( q ) > 0 V_{\beta}(q)>0 V β ( q ) > 0
M q = 1 β 3 ∫ 0 q ( q − v ) 2 x ^ β ( v ) 3 d v . M_q=\frac{1}{\beta^3}\int_0^q\frac{(q-v)^2}{\widehat{x}_{\beta}(v)^3}\,\mathrm{d}v. M q = β 3 1 ∫ 0 q x β ( v ) 3 ( q − v ) 2 d v . Then 0 < M q ≤ q 3 / ( 3 β 3 x ^ β ( q ) 3 ) 0<M_q\le q^3/(3\beta^3\widehat{x}_{\beta}(q)^3) 0 < M q ≤ q 3 / ( 3 β 3 x β ( q ) 3 )
T β ( q ) ≤ F ( β ) − V β ( q ) 2 8 β 3 M q . (86) T_{\beta}(q)\le F(\beta)-\frac{V_{\beta}(q)^2}{8\beta^3M_q}. \tag*{(86)} T β ( q ) ≤ F ( β ) − 8 β 3 M q V β ( q ) 2 . (86) In particular, for every compact set K ⊂ ( 0 , 1 ) K\subset(0,1) K ⊂ ( 0 , 1 ) inf q ∈ K V β ( q ) 2 / M q > 0 \inf_{q\in K}V_{\beta}(q)^2/M_q>0 inf q ∈ K V β ( q ) 2 / M q > 0
Before the proof, we describe its structure. For every q ∈ ( 0 , 1 ) q\in(0,1) q ∈ ( 0 , 1 ) F ( β ) F(\beta) F ( β ) β Q q ( ℓ q ) \beta Q_q(\ell_q) β Q q ( ℓ q ) 1 2 log ( 1 − q ) \frac{1}{2}\log(1-q) 2 1 log ( 1 − q ) ψ q \psi_q ψ q ζ β , q \zeta_{\beta,q} ζ β , q ℓ q \ell_q ℓ q ζ β , q \zeta_{\beta,q} ζ β , q T β ( q ) T_{\beta}(q) T β ( q ) V β ( q ) > 0 V_{\beta}(q)>0 V β ( q ) > 0 ℓ q \ell_q ℓ q V β ( q ) 2 V_{\beta}(q)^2 V β ( q ) 2 V β ( q ) = 0 V_{\beta}(q)=0 V β ( q ) = 0 ℓ q \ell_q ℓ q ζ β , q \zeta_{\beta,q} ζ β , q
Proof. Throughout the proof, we fix q ∈ ( 0 , 1 ) q\in(0,1) q ∈ ( 0 , 1 ) s = 1 − q s=1-q s = 1 − q ζ = ζ β \zeta=\zeta_{\beta} ζ = ζ β x ^ = x ^ β \widehat{x}=\widehat{x}_{\beta} x = x β V = V β V=V_{\beta} V = V β P β ( ⋅ ; ψ q ) \mathcal{P}_{\beta}(\mathord{\cdot};\psi_q) P β ( ⋅ ; ψ q )
A decomposition at overlap $q. We use the variational split behind [143, proof of Lemma 34, (6.8)], evaluated here at trial profiles for every q q q ℓ q = β x ^ \ell_q=\beta\widehat{x} ℓ q = β x [ 0 , q ] [0,q] [ 0 , q ] β ζ ( [ 0 , ⋅ ] ) \beta\zeta([0,\mathord{\cdot}]) β ζ ([ 0 , ⋅ ]) ℓ q ( q ) = β x ^ ( q ) > 0 \ell_q(q)=\beta\hat{x}(q)>0 ℓ q ( q ) = β x ^ ( q ) > 0 (85) and the substitution w = q + s u w=q+su w = q + s u
x ^ ζ β , q ( t ) = ∫ t 1 ζ ( [ 0 , q + s u ] ) d u = 1 s ∫ q + s t 1 ζ ( [ 0 , w ] ) d w = x ^ ( q + s t ) s , 0 ≤ t ≤ 1. \hat{x}_{\zeta_{\beta,q}}(t)=\int_t^1\zeta([0,q+su])\,du=\frac{1}{s}\int_{q+st}^1\zeta([0,w])\,dw=\frac{\hat{x}(q+st)}{s},\qquad0\leq t\leq1. x ^ ζ β , q ( t ) = ∫ t 1 ζ ([ 0 , q + s u ]) d u = s 1 ∫ q + s t 1 ζ ([ 0 , w ]) d w = s x ^ ( q + s t ) , 0 ≤ t ≤ 1. Fix q ^ ∈ ( q , 1 ) \hat{q}\in(q,1) q ^ ∈ ( q , 1 ) ζ ( [ 0 , q ^ ] ) = 1 \zeta([0,\hat{q}])=1 ζ ([ 0 , q ^ ]) = 1 q ^ ′ = ( q ^ − q ) / s \hat{q}'=(\hat{q}-q)/s q ^ ′ = ( q ^ − q ) / s ζ β , q ( [ 0 , q ^ ′ ] ) = 1 \zeta_{\beta,q}([0,\hat{q}'])=1 ζ β , q ([ 0 , q ^ ′ ]) = 1 v = q + s t v=q+st v = q + s t
∫ 0 q ^ ′ d t x ^ ζ β , q ( t ) + log ( 1 − q ^ ′ ) = ∫ q q ^ d v x ^ ( v ) + log 1 − q ^ s . \int_0^{\hat{q}'}\frac{dt}{\hat{x}_{\zeta_{\beta,q}}(t)}+\log(1-\hat{q}')=\int_q^{\hat{q}}\frac{dv}{\hat{x}(v)}+\log\frac{1-\hat{q}}{s}. ∫ 0 q ^ ′ x ^ ζ β , q ( t ) d t + log ( 1 − q ^ ′ ) = ∫ q q ^ x ^ ( v ) d v + log s 1 − q ^ . For the energy terms, we use (51) for ψ q \psi_q ψ q ζ \zeta ζ ψ q ′ ( t ) = s ( ζ ′ ( q + s t ) − ζ ′ ( q ) ) \psi_q'(t)=s(\zeta'(q+st)-\zeta'(q)) ψ q ′ ( t ) = s ( ζ ′ ( q + s t ) − ζ ′ ( q ))
ψ q ( 1 ) − ∫ ψ q d ζ β , q = ∫ 0 1 ψ q ′ ( t ) ζ ( [ 0 , q + s t ] ) d t = ∫ q 1 ζ ′ ( v ) ζ ( [ 0 , v ] ) d v − ζ ′ ( q ) x ^ ( q ) . \psi_q(1)-\int\psi_q\,d\zeta_{\beta,q}=\int_0^1\psi_q'(t)\zeta([0,q+st])\,dt=\int_q^1\zeta'(v)\zeta([0,v])\,dv-\zeta'(q)\hat{x}(q). ψ q ( 1 ) − ∫ ψ q d ζ β , q = ∫ 0 1 ψ q ′ ( t ) ζ ([ 0 , q + s t ]) d t = ∫ q 1 ζ ′ ( v ) ζ ([ 0 , v ]) d v − ζ ′ ( q ) x ^ ( q ) . Inserting the last two displays into (50) , we find
P β ( ζ β , q ; ψ q ) = β 2 2 ( ∫ q 1 ζ ′ ( v ) ζ ( [ 0 , v ] ) d v − ζ ′ ( q ) x ^ ( q ) ) + 1 2 ∫ q q ^ d v x ^ ( v ) + 1 2 log ( 1 − q ^ ) − 1 2 log s . \mathcal{P}_{\beta}(\zeta_{\beta,q};\psi_q)=\frac{\beta^2}{2}\left(\int_q^1\zeta'(v)\zeta([0,v])\,dv-\zeta'(q)\hat{x}(q)\right)+\frac{1}{2}\int_q^{\hat{q}}\frac{dv}{\hat{x}(v)}+\frac{1}{2}\log(1-\hat{q})-\frac{1}{2}\log s. P β ( ζ β , q ; ψ q ) = 2 β 2 ( ∫ q 1 ζ ′ ( v ) ζ ([ 0 , v ]) d v − ζ ′ ( q ) x ^ ( q ) ) + 2 1 ∫ q q ^ x ^ ( v ) d v + 2 1 log ( 1 − q ^ ) − 2 1 log s . For the zero-temperature functional, we integrate by parts, using x ^ ′ ( v ) = − ζ ( [ 0 , v ] ) \hat{x}'(v)=-\zeta([0,v]) x ^ ′ ( v ) = − ζ ([ 0 , v ]) v v v ζ ′ ( 0 ) = 0 \zeta'(0)=0 ζ ′ ( 0 ) = 0
β Q q ( ℓ q ) = β 2 2 ( ζ ′ ( q ) x ^ ( q ) + ∫ 0 q ζ ′ ( v ) ζ ( [ 0 , v ] ) d v ) + 1 2 ∫ 0 q d v x ^ ( v ) . \beta Q_q(\ell_q)=\frac{\beta^2}{2}\left(\zeta'(q)\hat{x}(q)+\int_0^q\zeta'(v)\zeta([0,v])\,dv\right)+\frac{1}{2}\int_0^q\frac{dv}{\hat{x}(v)}. β Q q ( ℓ q ) = 2 β 2 ( ζ ′ ( q ) x ^ ( q ) + ∫ 0 q ζ ′ ( v ) ζ ([ 0 , v ]) d v ) + 2 1 ∫ 0 q x ^ ( v ) d v . We add the last two identities and 1 2 log s \frac{1}{2}\log s 2 1 log s ζ ′ ( q ) x ^ ( q ) \zeta'(q)\hat{x}(q) ζ ′ ( q ) x ^ ( q ) log s \log s log s [ 0 , 1 ] [0,1] [ 0 , 1 ] [ 0 , q ^ ] [0,\hat{q}] [ 0 , q ^ ] (51) for ζ \zeta ζ
F ( β ) = P β ( ζ ; ζ ) = β Q q ( ℓ q ) + 1 2 log s + P β ( ζ β , q ; ψ q ) . (87) F(\beta)=\mathcal{P}_{\beta}(\zeta;\zeta)=\beta Q_q(\ell_q)+\frac{1}{2}\log s+\mathcal{P}_{\beta}(\zeta_{\beta,q};\psi_q). \tag*{(87)} F ( β ) = P β ( ζ ; ζ ) = β Q q ( ℓ q ) + 2 1 log s + P β ( ζ β , q ; ψ q ) . (87) This holds for every q ∈ ( 0 , 1 ) q\in(0,1) q ∈ ( 0 , 1 ) q q q Theorem 5.2 , E ( q ) = min Q q ≤ Q q ( ℓ q ) E(q)=\min Q_q\leq Q_q(\ell_q) E ( q ) = min Q q ≤ Q q ( ℓ q ) F r e s ( β , q ) = min P β ( ⋅ ; ψ q ) ≤ P β ( ζ β , q ; ψ q ) F_{\mathrm{res}}(\beta,q)=\min\mathcal{P}_{\beta}(\cdot\,;\psi_q)\leq\mathcal{P}_{\beta}(\zeta_{\beta,q};\psi_q) F res ( β , q ) = min P β ( ⋅ ; ψ q ) ≤ P β ( ζ β , q ; ψ q ) (87) implies that
T β ( q ) ≤ F ( β ) − β ( Q q ( ℓ q ) − E ( q ) ) . (88) T_\beta(q)\leq F(\beta)-\beta\bigl(Q_q(\ell_q)-E(q)\bigr). \tag*{(88)} T β ( q ) ≤ F ( β ) − β ( Q q ( ℓ q ) − E ( q ) ) . (88) Away from the contacts. For ε ≥ 0 \varepsilon\geq0 ε ≥ 0 ℓ ε ( v ) = ℓ q ( v ) + ε ( q − v ) \ell_\varepsilon(v)=\ell_q(v)+\varepsilon(q-v) ℓ ε ( v ) = ℓ q ( v ) + ε ( q − v ) β ζ ( [ 0 , ⋅ ] ) + ε \beta\zeta([0,\cdot])+\varepsilon β ζ ([ 0 , ⋅ ]) + ε 0 0 0 β x ^ ( 0 ) + ε q \beta\hat{x}(0)+\varepsilon q β x ^ ( 0 ) + εq ℓ ε ( q ) = ℓ q ( q ) > 0 \ell_\varepsilon(q)=\ell_q(q)>0 ℓ ε ( q ) = ℓ q ( q ) > 0 ℓ ε \ell_\varepsilon ℓ ε ℓ ε ≥ ℓ q ≥ β x ^ ( q ) \ell_\varepsilon\geq\ell_q\geq\beta\hat{x}(q) ℓ ε ≥ ℓ q ≥ β x ^ ( q ) [ 0 , q ] [0,q] [ 0 , q ] ℓ ε \ell_\varepsilon ℓ ε β x ^ ( q ) > 0 \beta\hat{x}(q)>0 β x ^ ( q ) > 0 ε \varepsilon ε (82) , and we find
d d ε Q q ( ℓ ε ) ∣ ε = 0 = 1 2 ∫ 0 q ( ζ ′ ′ ( v ) − 1 β 2 x ^ ( v ) 2 ) ( q − v ) d v = − 1 2 β 2 ∫ 0 q V ′ ′ ( v ) ( q − v ) d v = − V ( q ) 2 β 2 . \left.\frac{d}{d\varepsilon}Q_q(\ell_\varepsilon)\right|_{\varepsilon=0}
=\frac{1}{2}\int_0^q\left(\zeta''(v)-\frac{1}{\beta^2\hat{x}(v)^2}\right)(q-v)\,dv
=-\frac{1}{2\beta^2}\int_0^q V''(v)(q-v)\,dv
=-\frac{V(q)}{2\beta^2}. d ε d Q q ( ℓ ε ) ε = 0 = 2 1 ∫ 0 q ( ζ ′′ ( v ) − β 2 x ^ ( v ) 2 1 ) ( q − v ) d v = − 2 β 2 1 ∫ 0 q V ′′ ( v ) ( q − v ) d v = − 2 β 2 V ( q ) . In the second step, we used (80) , and in the last step we integrated by parts and used V ( 0 ) = V ′ ( 0 ) = 0 V(0)=V'(0)=0 V ( 0 ) = V ′ ( 0 ) = 0 ε ≥ 0 \varepsilon\ge0 ε ≥ 0
d 2 d ε 2 Q q ( ℓ ε ) = ∫ 0 q ( q − v ) 2 ℓ ε ( v ) 3 d v ≤ ∫ 0 q ( q − v ) 2 ℓ q ( v ) 3 d v = M q . \frac{\mathrm{d}^{2}}{\mathrm{d}\varepsilon^{2}}Q_q(\ell_\varepsilon)=\int_{0}^{q}\frac{(q-v)^2}{\ell_\varepsilon(v)^3}\,\mathrm{d}v\le\int_{0}^{q}\frac{(q-v)^2}{\ell_q(v)^3}\,\mathrm{d}v=M_q. d ε 2 d 2 Q q ( ℓ ε ) = ∫ 0 q ℓ ε ( v ) 3 ( q − v ) 2 d v ≤ ∫ 0 q ℓ q ( v ) 3 ( q − v ) 2 d v = M q . Taylor’s theorem implies that Q q ( ℓ ε ) ≤ Q q ( ℓ q ) − ε V ( q ) / ( 2 β 2 ) + ε 2 M q / 2 Q_q(\ell_\varepsilon)\le Q_q(\ell_q)-\varepsilon V(q)/(2\beta^2)+\varepsilon^2M_q/2 Q q ( ℓ ε ) ≤ Q q ( ℓ q ) − ε V ( q ) / ( 2 β 2 ) + ε 2 M q /2 ε ≥ 0 \varepsilon\ge0 ε ≥ 0 V ( q ) > 0 V(q)>0 V ( q ) > 0 ε = V ( q ) / ( 2 β 2 M q ) \varepsilon=V(q)/(2\beta^2M_q) ε = V ( q ) / ( 2 β 2 M q )
E ( q ) ≤ Q q ( ℓ ε ) ≤ Q q ( ℓ q ) − V ( q ) 2 8 β 4 M q . E(q)\le Q_q(\ell_\varepsilon)\le Q_q(\ell_q)-\frac{V(q)^2}{8\beta^4M_q}. E ( q ) ≤ Q q ( ℓ ε ) ≤ Q q ( ℓ q ) − 8 β 4 M q V ( q ) 2 . Inserting this bound into (88) gives (86) . The bounds on M q M_q M q β x ^ ( q ) ≤ ℓ q ≤ β \beta\hat{x}(q)\le\ell_q\le\beta β x ^ ( q ) ≤ ℓ q ≤ β [ 0 , q ] [0,q] [ 0 , q ] ∫ 0 q ( q − v ) 2 d v = q 3 / 3 \int_0^q(q-v)^2\,\mathrm{d}v=q^3/3 ∫ 0 q ( q − v ) 2 d v = q 3 /3 K ⊂ ( 0 , 1 ) K\subset(0,1) K ⊂ ( 0 , 1 ) V V V M q ≤ q 3 / ( 3 β 3 x ^ ( max K ) 3 ) M_q\le q^3/(3\beta^3\hat{x}(\max K)^3) M q ≤ q 3 / ( 3 β 3 x ^ ( max K ) 3 )
At a contact: the ground profile. Suppose that V ( q ) = 0 V(q)=0 V ( q ) = 0 q q q V ≥ 0 V\ge0 V ≥ 0 Lemma 5.6 ), and we also have V ′ ( q ) = 0 V'(q)=0 V ′ ( q ) = 0 ℓ ( v ) = L − ∫ 0 v α ( w ) d w \ell(v)=L-\int_0^v\alpha(w)\,\mathrm{d}w ℓ ( v ) = L − ∫ 0 v α ( w ) d w [ 0 , q ] [0,q] [ 0 , q ] m m m [ 0 , q ) [0,q) [ 0 , q ) m ( [ 0 , v ] ) = α ( v ) m([0,v])=\alpha(v) m ([ 0 , v ]) = α ( v ) α ( 0 ) \alpha(0) α ( 0 ) 0 0 0 β ζ ( [ 0 , ⋅ ] ) \beta\zeta([0,\cdot]) β ζ ([ 0 , ⋅ ]) ℓ q \ell_q ℓ q m q = β ζ m_q=\beta\zeta m q = β ζ [ 0 , q ) [0,q) [ 0 , q ) ∫ 0 v α ( w ) d w = ∫ [ 0 , v ] ( v − t ) m ( d t ) \int_0^v\alpha(w)\,\mathrm{d}w=\int_{[0,v]}(v-t)m(\mathrm{d}t) ∫ 0 v α ( w ) d w = ∫ [ 0 , v ] ( v − t ) m ( d t )
ℓ ( v ) = L − ∫ [ 0 , v ] ( v − t ) m ( d t ) , ℓ q ( v ) = β x ^ ( 0 ) − ∫ [ 0 , v ] ( v − t ) m q ( d t ) . \ell(v)=L-\int_{[0,v]}(v-t)m(\mathrm{d}t),\qquad\ell_q(v)=\beta\hat{x}(0)-\int_{[0,v]}(v-t)m_q(\mathrm{d}t). ℓ ( v ) = L − ∫ [ 0 , v ] ( v − t ) m ( d t ) , ℓ q ( v ) = β x ^ ( 0 ) − ∫ [ 0 , v ] ( v − t ) m q ( d t ) . Using the convexity inequality 1 / z ≥ 1 / z 0 − ( z − z 0 ) / z 0 2 1/z\ge1/z_0-(z-z_0)/z_0^2 1/ z ≥ 1/ z 0 − ( z − z 0 ) / z 0 2 z , z 0 > 0 z,z_0>0 z , z 0 > 0 (80) , we find that
Q q ( ℓ ) − Q q ( ℓ q ) ≥ 1 2 ∫ 0 q ( ζ ′ ′ ( v ) − 1 ℓ q ( v ) 2 ) ( ℓ ( v ) − ℓ q ( v ) ) d v = − 1 2 β 2 ∫ 0 q V ′ ′ ( v ) ( ℓ ( v ) − ℓ q ( v ) ) d v . (89) \begin{aligned}
Q_q(\ell)-Q_q(\ell_q)&\ge\frac{1}{2}\int_0^q\left(\zeta''(v)-\frac{1}{\ell_q(v)^2}\right)(\ell(v)-\ell_q(v))\,\mathrm{d}v\\
&=-\frac{1}{2\beta^2}\int_0^q V''(v)(\ell(v)-\ell_q(v))\,\mathrm{d}v. \tag*{(89)}
\end{aligned} Q q ( ℓ ) − Q q ( ℓ q ) ≥ 2 1 ∫ 0 q ( ζ ′′ ( v ) − ℓ q ( v ) 2 1 ) ( ℓ ( v ) − ℓ q ( v )) d v = − 2 β 2 1 ∫ 0 q V ′′ ( v ) ( ℓ ( v ) − ℓ q ( v )) d v . (89) We compute the contribution of ℓ \ell ℓ L L L − L ( V ′ ( q ) − V ′ ( 0 ) ) = 0 -L(V'(q)-V'(0))=0 − L ( V ′ ( q ) − V ′ ( 0 )) = 0 0 ≤ t < q 0\le t<q 0 ≤ t < q
∫ t q V ′ ′ ( v ) ( v − t ) d v = ( q − t ) V ′ ( q ) − V ( q ) + V ( t ) = V ( t ) . \int_t^q V''(v)(v-t)\,\mathrm{d}v=(q-t)V'(q)-V(q)+V(t)=V(t). ∫ t q V ′′ ( v ) ( v − t ) d v = ( q − t ) V ′ ( q ) − V ( q ) + V ( t ) = V ( t ) . To exchange the order of integration below, note that ∫ t q ∣ V ′ ′ ( v ) ∣ ( v − t ) d v ≤ ∥ V ′ ′ ∥ ∞ , [ 0 , q ] ( q − t ) 2 / 2 \int_t^q|V''(v)|(v-t)\,\mathrm{d}v\le\lVert V''\rVert_{\infty,[0,q]}(q-t)^2/2 ∫ t q ∣ V ′′ ( v ) ∣ ( v − t ) d v ≤ ∥ V ′′ ∥ ∞ , [ 0 , q ] ( q − t ) 2 /2
∫ [ 0 , q ) ( q − t ) 2 m ( d t ) ≤ q ∫ [ 0 , q ) ( q − t ) m ( d t ) = q ∫ 0 q α ( w ) d w < ∞ . \int_{[0,q)}(q-t)^2m(\mathrm{d}t)\le q\int_{[0,q)}(q-t)m(\mathrm{d}t)=q\int_0^q\alpha(w)\,\mathrm{d}w<\infty. ∫ [ 0 , q ) ( q − t ) 2 m ( d t ) ≤ q ∫ [ 0 , q ) ( q − t ) m ( d t ) = q ∫ 0 q α ( w ) d w < ∞. Hence Fubini’s theorem applies to V ′ ′ ( v ) ( v − t ) V''(v)(v-t) V ′′ ( v ) ( v − t ) { 0 ≤ t ≤ v ≤ q } \{0\le t\le v\le q\} { 0 ≤ t ≤ v ≤ q } m ( d t ) d v m(\mathrm{d}t)\,\mathrm{d}v m ( d t ) d v
− ∫ 0 q V ′ ′ ( v ) ℓ ( v ) d v = ∫ [ 0 , q ) ( ∫ t q V ′ ′ ( v ) ( v − t ) d v ) m ( d t ) = ∫ [ 0 , q ) V ( t ) m ( d t ) . -\int_0^q V''(v)\ell(v)\,\mathrm{d}v=\int_{[0,q)}\left(\int_t^q V''(v)(v-t)\,\mathrm{d}v\right)m(\mathrm{d}t)=\int_{[0,q)}V(t)m(\mathrm{d}t). − ∫ 0 q V ′′ ( v ) ℓ ( v ) d v = ∫ [ 0 , q ) ( ∫ t q V ′′ ( v ) ( v − t ) d v ) m ( d t ) = ∫ [ 0 , q ) V ( t ) m ( d t ) . For ℓ q \ell_q ℓ q ∫ [ 0 , q ) V d m q \int_{[0,q)}V\,\mathrm{d}m_q ∫ [ 0 , q ) V d m q ζ \zeta ζ V V V Lemma 5.6 ). Inserting these two identities into (89) , we find
Q q ( ℓ ) − Q q ( ℓ q ) ≥ 1 2 β 2 ∫ [ 0 , q ) V ( t ) m ( d t ) ≥ 0. (90) Q_q(\ell)-Q_q(\ell_q)\ge\frac{1}{2\beta^2}\int_{[0,q)}V(t)m(\mathrm{d}t)\ge0. \tag*{(90)} Q q ( ℓ ) − Q q ( ℓ q ) ≥ 2 β 2 1 ∫ [ 0 , q ) V ( t ) m ( d t ) ≥ 0. (90) We see that ℓ q \ell_q ℓ q Q q Q_q Q q Q q ( ℓ ) = Q q ( ℓ q ) Q_q(\ell)=Q_q(\ell_q) Q q ( ℓ ) = Q q ( ℓ q ) (89) , that is, ∫ 0 q ( ℓ − ℓ q ) 2 / ( ℓ ℓ q 2 ) d v = 0 \int_0^q(\ell-\ell_q)^2/(\ell\ell_q^2)\,\mathrm{d}v=0 ∫ 0 q ( ℓ − ℓ q ) 2 / ( ℓ ℓ q 2 ) d v = 0 ℓ \ell ℓ ℓ q \ell_q ℓ q E ( q ) = Q q ( ℓ q ) E(q)=Q_q(\ell_q) E ( q ) = Q q ( ℓ q ) Theorem 5.2 .
At a contact: the residual measure. Still assuming V ( q ) = 0 V(q)=0 V ( q ) = 0 W = V ζ β , q W=V_{\zeta_{\beta,q}} W = V ζ β , q ψ q \psi_q ψ q ζ β , q \zeta_{\beta,q} ζ β , q
W ( t ) = ∫ 0 t t − u x ^ ζ β , q ( u ) 2 d u − β 2 ψ q ( t ) , 0 ≤ t < 1. W(t)=\int_0^t\frac{t-u}{\widehat{x}_{\zeta_{\beta,q}}(u)^2}\,\mathrm{d}u-\beta^2\psi_q(t),\qquad0\le t<1. W ( t ) = ∫ 0 t x ζ β , q ( u ) 2 t − u d u − β 2 ψ q ( t ) , 0 ≤ t < 1. Using x ^ ζ β , q ( u ) = x ^ ( q + s u ) / s \widehat{x}_{\zeta_{\beta,q}}(u)=\widehat{x}(q+su)/s x ζ β , q ( u ) = x ( q + s u ) / s v = q + s u v=q+su v = q + s u ∫ 0 t ( t − u ) x ^ ζ β , q ( u ) − 2 d u = ∫ q q + s t ( q + s t − v ) x ^ ( v ) − 2 d v \int_0^t(t-u)\widehat{x}_{\zeta_{\beta,q}}(u)^{-2}\,\mathrm{d}u=\int_q^{q+st}(q+st-v)\widehat{x}(v)^{-2}\,\mathrm{d}v ∫ 0 t ( t − u ) x ζ β , q ( u ) − 2 d u = ∫ q q + s t ( q + s t − v ) x ( v ) − 2 d v
V ( q + s t ) − V ( q ) − s t V ′ ( q ) = ∫ q q + s t q + s t − v x ^ ( v ) 2 d v − β 2 ( ζ ( q + s t ) − ζ ( q ) − s t ζ ′ ( q ) ) . V(q+st)-V(q)-stV'(q)=\int_q^{q+st}\frac{q+st-v}{\widehat{x}(v)^2}\,\mathrm{d}v-\beta^2\bigl(\zeta(q+st)-\zeta(q)-st\zeta'(q)\bigr). V ( q + s t ) − V ( q ) − s t V ′ ( q ) = ∫ q q + s t x ( v ) 2 q + s t − v d v − β 2 ( ζ ( q + s t ) − ζ ( q ) − s t ζ ′ ( q ) ) . We have V ( q ) = V ′ ( q ) = 0 V(q)=V'(q)=0 V ( q ) = V ′ ( q ) = 0
W ( t ) = V ( q + s t ) , 0 ≤ t < 1. (91) W(t)=V(q+st),\qquad0\le t<1. \tag*{(91)} W ( t ) = V ( q + s t ) , 0 ≤ t < 1. (91) In particular, W ≥ 0 = W ( 0 ) W\ge0=W(0) W ≥ 0 = W ( 0 ) ζ β , q \zeta_{\beta,q} ζ β , q ζ \zeta ζ τ ( r ) = max { 0 , ( r − q ) / s } \tau(r)=\max\{0,(r-q)/s\} τ ( r ) = max { 0 , ( r − q ) / s } r ≤ q r\le q r ≤ q W ( τ ( r ) ) = W ( 0 ) = 0 W(\tau(r))=W(0)=0 W ( τ ( r )) = W ( 0 ) = 0 ζ \zeta ζ r > q r>q r > q W ( τ ( r ) ) = V ( r ) = 0 W(\tau(r))=V(r)=0 W ( τ ( r )) = V ( r ) = 0 ζ \zeta ζ V V V ζ β , q \zeta_{\beta,q} ζ β , q W W W f ζ β , q = − W f_{\zeta_{\beta,q}}=-W f ζ β , q = − W ψ q \psi_q ψ q Lemma 3.15 (3), we find that ζ β , q \zeta_{\beta,q} ζ β , q ( ψ q , β ) (\psi_q,\beta) ( ψ q , β )
Conclusion. If q ∈ ( 0 , 1 ) q\in(0,1) q ∈ ( 0 , 1 ) V ( q ) > 0 V(q)>0 V ( q ) > 0 T β ( q ) < F ( β ) T_\beta(q)<F(\beta) T β ( q ) < F ( β ) q q q Theorem 5.5 (1). If q ∈ ( 0 , 1 ) q\in(0,1) q ∈ ( 0 , 1 ) V ( q ) = 0 V(q)=0 V ( q ) = 0 (87) are solved by the trial points, that is, E ( q ) = Q q ( ℓ q ) E(q)=Q_q(\ell_q) E ( q ) = Q q ( ℓ q ) F r e s ( β , q ) = P β ( ζ β , q ; ψ q ) F_{\mathrm{res}}(\beta,q)=\mathcal{P}_\beta(\zeta_{\beta,q};\psi_q) F res ( β , q ) = P β ( ζ β , q ; ψ q ) (87) gives T β ( q ) = F ( β ) T_\beta(q)=F(\beta) T β ( q ) = F ( β ) q q q Theorem 5.5 (1). Finally, V ( 0 ) = 0 V(0)=0 V ( 0 ) = 0 E ( 0 ) = 0 E(0)=0 E ( 0 ) = 0 ψ 0 = ξ \psi_0=\xi ψ 0 = ξ T β ( 0 ) = P β ( ξ ) = F ( β ) T_\beta(0)=\mathcal{P}_\beta(\xi)=F(\beta) T β ( 0 ) = P β ( ξ ) = F ( β ) Theorem 5.5 (1), 0 0 0
The next lemma restates the bound for overlaps outside the contact set in the form used in Section 6. The point is that the replica number, the tolerance, and the exponential rate are fixed before N → ∞ N \to\infty N → ∞
Lemma 5.8 (Replicas at an overlap outside the contact set). Let ξ \xi ξ β > 0 \beta> 0 β > 0 q ∈ ( 0 , 1 ) q \in(0,1) q ∈ ( 0 , 1 ) V β ( q ) > 0 V_\beta(q) > 0 V β ( q ) > 0 k ≥ 2 k \ge2 k ≥ 2 ρ ∈ ( 0 , 1 ) \rho\in(0,1) ρ ∈ ( 0 , 1 ) c > 0 c > 0 c > 0 ξ \xi ξ β \beta β q q q p N , k , ρ ( q ) ≤ e − c N p_{N,k,\rho}(q) \le e^{-cN} p N , k , ρ ( q ) ≤ e − c N N N N
Proof. By Theorem 5.7 (1), q q q lim sup N N − 1 log p N , k , ρ ( q ) \limsup_N N^{-1}\log p_{N,k,\rho}(q) lim sup N N − 1 log p N , k , ρ ( q ) p N , k , ρ ( q ) ≤ 1 p_{N,k,\rho}(q) \le1 p N , k , ρ ( q ) ≤ 1 Theorem 5.5 (2), there exist k ≥ 1 k \ge1 k ≥ 1 ρ > 0 \rho> 0 ρ > 0 − c -c − c c > 0 c > 0 c > 0 p N , k , ρ ( q ) ≤ e − c N p_{N,k,\rho}(q) \le e^{-cN} p N , k , ρ ( q ) ≤ e − c N N N N k ≥ 2 k \ge2 k ≥ 2 p N , 1 , ρ ( q ) = 1 p_{N,1,\rho}(q) = 1 p N , 1 , ρ ( q ) = 1 p N , k , ρ ( q ) p_{N,k,\rho}(q) p N , k , ρ ( q ) ρ \rho ρ ρ \rho ρ min { ρ , 1 / 2 } \min\{\rho,1/2\} min { ρ , 1/2 }
Strict one-step replica symmetry breaking
We restate Definition 1.1 .
Definition 5.9 (Strict one-step replica symmetry breaking). Let ξ \xi ξ β > 0 \beta> 0 β > 0 ( ξ , β ) (\xi,\beta) ( ξ , β ) 1 R S B 1RSB 1 R S B Q Q Q x , Q ∈ ( 0 , 1 ) x,Q \in(0,1) x , Q ∈ ( 0 , 1 ) s = 1 − Q s = 1-Q s = 1 − Q C = s + x Q C = s+xQ C = s + x Q
x ^ ( v ) = { C − x v , 0 ≤ v ≤ Q , 1 − v , Q ≤ v < 1 , (92) \hat{x}(v) =
\begin{cases}
C-xv, & 0 \le v \le Q,\\
1-v, & Q \le v < 1,
\end{cases} \tag*{(92)} x ^ ( v ) = { C − xv , 1 − v , 0 ≤ v ≤ Q , Q ≤ v < 1 , (92) and
V ( r ) = ∫ 0 r r − v x ^ ( v ) 2 d v − β 2 ξ ( r ) , 0 ≤ r < 1 , V(r) = \int_0^r \frac{r-v}{\hat{x}(v)^2}\,\mathrm{d}v-\beta^2\xi(r), \qquad0 \le r < 1, V ( r ) = ∫ 0 r x ^ ( v ) 2 r − v d v − β 2 ξ ( r ) , 0 ≤ r < 1 , we have
V ( Q ) = V ′ ( Q ) = 0 , V ( r ) > 0 for r ∈ ( 0 , 1 ) ∖ { Q } , V ′ ′ ( Q ) > 0. (93) V(Q)=V'(Q)=0,\qquad V(r)>0\quad\text{for }r\in(0,1)\setminus\{Q\},\qquad V''(Q)>0. \tag*{(93)} V ( Q ) = V ′ ( Q ) = 0 , V ( r ) > 0 for r ∈ ( 0 , 1 ) ∖ { Q } , V ′′ ( Q ) > 0. (93) We call x x x Q Q Q parameters , and we refer to the three conditions in (93) as the 1 R S B 1RSB 1 R S B assumptions .
The function x ^ \hat{x} x ^ x ^ ζ \hat{x}_{\zeta} x ^ ζ ζ = x δ 0 + ( 1 − x ) δ Q \zeta=x\delta_0+(1-x)\delta_Q ζ = x δ 0 + ( 1 − x ) δ Q Definition 3.13 , and V = V ζ V=V_{\zeta} V = V ζ V V V [ 0 , 1 ) [0,1) [ 0 , 1 ) V ( 0 ) = V ′ ( 0 ) = 0 V(0)=V'(0)=0 V ( 0 ) = V ′ ( 0 ) = 0 V ′ ′ ( Q ) = s − 2 − β 2 ξ ′ ′ ( Q ) V''(Q)=s^{-2}-\beta^2\xi''(Q) V ′′ ( Q ) = s − 2 − β 2 ξ ′′ ( Q ) Lemma 3.15 (3), as in the proof of Proposition 5.11 (1) below, the first two conditions in (93) imply that ζ \zeta ζ V = V β V=V_\beta V = V β ( ξ , β ) (\xi,\beta) ( ξ , β ) { 0 , Q } \{0,Q\} { 0 , Q } x δ 0 + ( 1 − x ) δ Q x\delta_0+(1-x)\delta_Q x δ 0 + ( 1 − x ) δ Q { 0 , Q } \{0,Q\} { 0 , Q } V = V β ≥ 0 V=V_\beta\ge0 V = V β ≥ 0 Lemma 5.6 , Q Q Q V V V 1 R S B 1RSB 1 R S B
measure has two atoms and the contact set is exactly its support. In particular, the parameters are determined by ( ξ , β ) (\xi,\beta) ( ξ , β ) Q Q Q V ′ ′ ( Q ) = s − 2 − β 2 ξ ′ ′ ( Q ) V''(Q)=s^{-2}-\beta^2\xi''(Q) V ′′ ( Q ) = s − 2 − β 2 ξ ′′ ( Q ) β 2 s 2 ξ ′ ′ ( Q ) < 1 \beta^2s^2\xi''(Q)<1 β 2 s 2 ξ ′′ ( Q ) < 1 Q Q Q x ^ ( Q ) − 2 − β 2 ξ ′ ′ ( Q ) \hat{x}(Q)^{-2}-\beta^2\xi''(Q) x ^ ( Q ) − 2 − β 2 ξ ′′ ( Q )
This terminology follows Huang and Sellke [90], who call a model strictly replica symmetric if its contact set is { 0 } \{0\} { 0 } Definition 3.16 ) and, at zero temperature, strictly 1RSB if the zero set of the corresponding function is { 0 , 1 } \{0,1\} { 0 , 1 } Definition 5.9 is the analogous notion at positive temperature, with the nondegeneracy condition added. Throughout Sections 5.3 and 5.4, when ( ξ , β ) (\xi,\beta) ( ξ , β )
z = x Q s , y = Q β s , c 0 = β x . (94) z=\frac{xQ}{s},\qquad y=\frac{Q}{\beta s},\qquad c_0=\beta x. \tag*{(94)} z = s x Q , y = β s Q , c 0 = β x . (94) In this section, the letter C C C C = s + x Q C=s+xQ C = s + x Q Definition 5.9 , never a generic constant.
Lemma 5.10 (Strictly 1RSB pairs are condensed). If ( ξ , β ) (\xi,\beta) ( ξ , β ) Q Q Q β > β s ( ξ ) \beta>\beta_s(\xi) β > β s ( ξ )
Proof. For 0 ≤ v < Q 0\leq v<Q 0 ≤ v < Q x ^ ( v ) = 1 − Q + x ( Q − v ) < 1 − v \hat{x}(v)=1-Q+x(Q-v)<1-v x ^ ( v ) = 1 − Q + x ( Q − v ) < 1 − v x < 1 x<1 x < 1 V ( Q ) = 0 V(Q)=0 V ( Q ) = 0
β 2 ξ ( Q ) = ∫ 0 Q Q − v x ^ ( v ) 2 d v > ∫ 0 Q Q − v ( 1 − v ) 2 d v = − Q − log ( 1 − Q ) , \beta^2\xi(Q)=\int_0^Q\frac{Q-v}{\hat{x}(v)^2}\,\mathrm{d}v>\int_0^Q\frac{Q-v}{(1-v)^2}\,\mathrm{d}v=-Q-\log(1-Q), β 2 ξ ( Q ) = ∫ 0 Q x ^ ( v ) 2 Q − v d v > ∫ 0 Q ( 1 − v ) 2 Q − v d v = − Q − log ( 1 − Q ) , where the last integral was computed in the proof of Lemma 3.18 . Dividing by ξ ( Q ) > 0 \xi(Q)>0 ξ ( Q ) > 0 β s \beta_s β s Definition 3.16 , we obtain β 2 > β s 2 \beta^2>\beta_s^2 β 2 > β s 2
Proposition 5.11 (Static consequences of strict 1RSB). Let ( ξ , β ) (\xi,\beta) ( ξ , β ) x x x Q Q Q Q Q Q s s s C C C x ^ \hat{x} x ^ V V V z z z y y y c 0 c_0 c 0 Definition 5.9 and (5.16).
(Parisi measure.) The Parisi measure of ( ξ , β ) (\xi,\beta) ( ξ , β ) ζ β = x δ 0 + ( 1 − x ) δ Q \zeta_\beta=x\delta_0+(1-x)\delta_Q ζ β = x δ 0 + ( 1 − x ) δ Q x ^ β = x ^ \hat{x}_\beta=\hat{x} x ^ β = x ^ V β = V V_\beta=V V β = V
β 2 ξ ′ ( Q ) = Q s C , β 2 x 2 ξ ( Q ) = log C s − x Q C , (95) \beta^2\xi'(Q)=\frac{Q}{sC},\qquad\beta^2x^2\xi(Q)=\log\frac{C}{s}-\frac{xQ}{C}, \tag*{(95)} β 2 ξ ′ ( Q ) = s C Q , β 2 x 2 ξ ( Q ) = log s C − C x Q , (95) and
F ( β ) = 1 2 ( β 2 ( ξ ( 1 ) − ( 1 − x ) ξ ( Q ) ) + 1 x log C s + log s ) . (96) F(\beta)=\frac{1}{2}\left(\beta^2\left(\xi(1)-(1-x)\xi(Q)\right)+\frac{1}{x}\log\frac{C}{s}+\log s\right). \tag*{(96)} F ( β ) = 2 1 ( β 2 ( ξ ( 1 ) − ( 1 − x ) ξ ( Q ) ) + x 1 log s C + log s ) . (96) (Residual model.) We have β 2 s 2 ξ ′ ′ ( Q ) < 1 \beta^2s^2\xi''(Q)<1 β 2 s 2 ξ ′′ ( Q ) < 1
β 2 ψ Q ( t ) + t + log ( 1 − t ) = − V ( Q + s t ) < 0 , 0 < t < 1. (97) \beta^2\psi_Q(t)+t+\log(1-t)=-V(Q+st)<0,\qquad0<t<1. \tag*{(97)} β 2 ψ Q ( t ) + t + log ( 1 − t ) = − V ( Q + s t ) < 0 , 0 < t < 1. (97) There exists κ r e s = κ r e s ( ζ ~ , β ) > 0 \kappa_{\mathrm{res}}=\kappa_{\mathrm{res}}(\widetilde{\zeta},\beta)>0 κ res = κ res ( ζ , β ) > 0 V ( Q + s t ) ≥ κ r e s t 2 V(Q+st)\geq\kappa_{\mathrm{res}}t^2 V ( Q + s t ) ≥ κ res t 2 0 < t < 1 0<t<1 0 < t < 1 ( ψ Q , β ) (\psi_Q,\beta) ( ψ Q , β ) β 2 ψ Q ′ ′ ( 0 ) < 1 \beta^2\psi_Q''(0)<1 β 2 ψ Q ′′ ( 0 ) < 1 δ 0 \delta_0 δ 0 F r e s ( β , Q ) = β 2 ψ Q ( 1 ) / 2 F_{\mathrm{res}}(\beta,Q)=\beta^2\psi_Q(1)/2 F res ( β , Q ) = β 2 ψ Q ( 1 ) /2
(Ground state at radius Q \sqrt{Q} Q Q Q \mathcal{Q}_Q Q Q ℓ ( v ) = β x ^ ( v ) = β ( C − x v ) \ell(v)=\beta\hat{x}(v)=\beta(C-xv) ℓ ( v ) = β x ^ ( v ) = β ( C − xv ) 0 ≤ v ≤ Q 0\leq v\leq Q 0 ≤ v ≤ Q
E ( Q ) = E 0 , where E 0 = a Q + z ζ ~ ( Q ) y . (98) E(Q)=E_0,\qquad\text{where}\quad E_0=\frac{a_Q+z\widetilde{\zeta}(Q)}{y}. \tag*{(98)} E ( Q ) = E 0 , where E 0 = y a Q + z ζ ( Q ) . (98) The function g ( r ) = β − 2 V ( Q r ) g(r)=\beta^{-2}V(Qr) g ( r ) = β − 2 V ( Q r ) [ 0 , 1 ] [0,1] [ 0 , 1 ] g ( 0 ) = g ( 1 ) = 0 g(0)=g(1)=0 g ( 0 ) = g ( 1 ) = 0 g ( r ) > 0 g(r)>0 g ( r ) > 0 0 < r < 1 0<r<1 0 < r < 1
g ( r ) = ζ ~ ( Q ) − ζ ~ ( Q r ) − a Q ( 1 + z z 2 log ( 1 + ( 1 − r ) z ) − 1 − r z ) , 0 ≤ r ≤ 1. (99) g(r)=\widetilde{\zeta}(Q)-\widetilde{\zeta}(Qr)-a_Q\left(\frac{1+z}{z^2}\log(1+(1-r)z)-\frac{1-r}{z}\right),\qquad0\leq r\leq1. \tag*{(99)} g ( r ) = ζ ( Q ) − ζ ( Q r ) − a Q ( z 2 1 + z log ( 1 + ( 1 − r ) z ) − z 1 − r ) , 0 ≤ r ≤ 1. (99) (Energy and radial derivative.) Let K 0 = y + b Q / y K_0=y+b_Q/y K 0 = y + b Q / y
y 2 = ( 1 + z ) a Q , ζ ~ ( Q ) a Q = ( 1 + z ) log ( 1 + z ) − z z 2 , c 0 = z y < β , (100) y^2=(1+z)a_Q,\qquad\frac{\widetilde{\zeta}(Q)}{a_Q}=\frac{(1+z)\log(1+z)-z}{z^2},\qquad c_0=\frac{z}{y}<\beta, \tag*{(100)} y 2 = ( 1 + z ) a Q , a Q ζ ( Q ) = z 2 ( 1 + z ) log ( 1 + z ) − z , c 0 = y z < β , (100) and
b Q y 2 = β 2 s 2 ζ ~ ′ ′ ( Q ) < 1 , K 0 − 2 b Q = ( y − b Q ) 2 y > 0. (101) \frac{b_Q}{y^2}=\beta^2s^2\widetilde{\zeta}''(Q)<1,\qquad K_0-2\sqrt{b_Q}=\frac{(y-\sqrt{b_Q})^2}{y}>0. \tag*{(101)} y 2 b Q = β 2 s 2 ζ ′′ ( Q ) < 1 , K 0 − 2 b Q = y ( y − b Q ) 2 > 0. (101) (TAP identity.) We have
F ( β ) = β E 0 + 1 2 log s + β 2 2 ψ Q ( 1 ) , (102) F(\beta)=\beta E_0+\frac{1}{2}\log s+\frac{\beta^2}{2}\psi_Q(1), \tag*{(102)} F ( β ) = β E 0 + 2 1 log s + 2 β 2 ψ Q ( 1 ) , (102) the set of multisamplable overlaps in [ 0 , 1 ) [0,1) [ 0 , 1 ) { 0 , Q } \{0,Q\} { 0 , Q } (86) holds for every q ∈ ( 0 , 1 ) ∖ { Q } q\in(0,1)\setminus\{Q\} q ∈ ( 0 , 1 ) ∖ { Q } x ^ β = x ^ \hat{x}_\beta=\hat{x} x ^ β = x ^ M q M_q M q
In the language of [90, Definition 4], part (3) says that the field H Q H_Q H Q N ζ ~ ( Q R ) N\widetilde{\zeta}(QR) N ζ ( QR ) g g g r ↦ ζ ~ ( Q r ) r\mapsto\widetilde{\zeta}(Qr) r ↦ ζ ( Q r ) ℓ ( Q r ) / Q \ell(Qr)/Q ℓ ( Q r ) / Q { 0 , 1 } \{0,1\} { 0 , 1 }
Proof. We prove the five parts in order, except that part (3) uses the first two identities of (100) , which are proved in part (4) from part (1) alone.
Part (1). Let ζ = x δ 0 + ( 1 − x ) δ Q \zeta=x\delta_0+(1-x)\delta_Q ζ = x δ 0 + ( 1 − x ) δ Q x ^ ζ = x ^ \hat{x}_\zeta=\hat{x} x ^ ζ = x ^ V ζ = V V_\zeta=V V ζ = V f ζ = − V f_\zeta=-V f ζ = − V (52) . By (93) and V ( 0 ) = 0 V(0)=0 V ( 0 ) = 0 f ζ f_\zeta f ζ { 0 , Q } \{0,Q\} { 0 , Q } ζ \zeta ζ Lemma 3.15 (3) that ζ \zeta ζ
∫ 0 Q d v ( C − x v ) 2 = Q s C , ∫ 0 Q Q − v ( C − x v ) 2 d v = 1 x 2 ( log C s − x Q C ) , \int_0^Q\frac{dv}{(C-xv)^2}=\frac{Q}{sC},\qquad\int_0^Q\frac{Q-v}{(C-xv)^2}\,dv=\frac{1}{x^2}\left(\log\frac{C}{s}-\frac{xQ}{C}\right), ∫ 0 Q ( C − xv ) 2 d v = s C Q , ∫ 0 Q ( C − xv ) 2 Q − v d v = x 2 1 ( log s C − C x Q ) , where we used C − x Q = s C-xQ=s C − x Q = s (80) , we see that the equations V ′ ( Q ) = 0 V'(Q)=0 V ′ ( Q ) = 0 V ( Q ) = 0 V(Q)=0 V ( Q ) = 0 (95) . For (96) , we evaluate (50) at ζ \zeta ζ q ^ = Q \hat{q}=Q q ^ = Q ∫ ζ ~ d ζ = ( 1 − x ) ζ ~ ( Q ) \int\widetilde{\zeta}\,d\zeta=(1-x)\widetilde{\zeta}(Q) ∫ ζ d ζ = ( 1 − x ) ζ ( Q ) ∫ 0 Q d v / ( C − x v ) = x − 1 log ( C / s ) \int_0^Qdv/(C-xv)=x^{-1}\log(C/s) ∫ 0 Q d v / ( C − xv ) = x − 1 log ( C / s ) F ( β ) = P β ( ζ ; ζ ~ ) F(\beta)=P_\beta(\zeta;\widetilde{\zeta}) F ( β ) = P β ( ζ ; ζ )
Part (2). On [ Q , 1 ) [Q,1) [ Q , 1 ) x ^ ( v ) = 1 − v \hat{x}(v)=1-v x ^ ( v ) = 1 − v V ′ ′ ( v ) = ( 1 − v ) − 2 − β 2 ζ ′ ′ ( v ) V''(v)=(1-v)^{-2}-\beta^2\zeta''(v) V ′′ ( v ) = ( 1 − v ) − 2 − β 2 ζ ′′ ( v ) V ( Q ) = V ′ ( Q ) = 0 V(Q)=V'(Q)=0 V ( Q ) = V ′ ( Q ) = 0 0 ≤ t < 1 0\leq t<1 0 ≤ t < 1
V ( Q + s t ) = ∫ Q Q + s t Q + s t − v ( 1 − v ) 2 d v − β 2 ψ Q ( t ) = ∫ 0 t t − w ( 1 − w ) 2 d w − β 2 ψ Q ( t ) = − t − log ( 1 − t ) − β 2 ψ Q ( t ) . \begin{aligned}
V(Q+st)&=\int_Q^{Q+st}\frac{Q+st-v}{(1-v)^2}\,dv-\beta^2\psi_Q(t)\\
&=\int_0^t\frac{t-w}{(1-w)^2}\,dw-\beta^2\psi_Q(t)=-t-\log(1-t)-\beta^2\psi_Q(t).
\end{aligned} V ( Q + s t ) = ∫ Q Q + s t ( 1 − v ) 2 Q + s t − v d v − β 2 ψ Q ( t ) = ∫ 0 t ( 1 − w ) 2 t − w d w − β 2 ψ Q ( t ) = − t − log ( 1 − t ) − β 2 ψ Q ( t ) . where we substituted v = Q + s w v=Q+sw v = Q + s w V ′ ′ ( Q ) = s − 2 − β 2 ζ ′ ′ ( Q ) V''(Q)=s^{-2}-\beta^2\zeta''(Q) V ′′ ( Q ) = s − 2 − β 2 ζ ′′ ( Q ) V ′ ′ ( Q ) > 0 V''(Q)>0 V ′′ ( Q ) > 0 β 2 s 2 ζ ′ ′ ( Q ) < 1 \beta^2s^2\zeta''(Q)<1 β 2 s 2 ζ ′′ ( Q ) < 1 κ r e s = inf 0 < t < 1 V ( Q + s t ) / t 2 \kappa_{\mathrm{res}}=\inf_{0<t<1}V(Q+st)/t^2 κ res = inf 0 < t < 1 V ( Q + s t ) / t 2 t → 0 t\to0 t → 0 s 2 V ′ ′ ( Q ) / 2 > 0 s^2V''(Q)/2>0 s 2 V ′′ ( Q ) /2 > 0 t → 1 t\to1 t → 1 − log ( 1 − t ) → ∞ -\log(1-t)\to\infty − log ( 1 − t ) → ∞ t t t ψ Q ( t ) \psi_Q(t) ψ Q ( t ) ( 0 , 1 ) (0,1) ( 0 , 1 ) κ r e s > 0 \kappa_{\mathrm{res}}>0 κ res > 0 ψ Q ′ ′ ( 0 ) = s 2 ζ ′ ′ ( Q ) \psi_Q''(0)=s^2\zeta''(Q) ψ Q ′′ ( 0 ) = s 2 ζ ′′ ( Q ) ( ψ Q , β ) (\psi_Q,\beta) ( ψ Q , β ) β 2 ψ Q ′ ′ ( 0 ) < 1 \beta^2\psi_Q''(0)<1 β 2 ψ Q ′′ ( 0 ) < 1 δ 0 \delta_0 δ 0 F r e s ( β , Q ) = P β ( ψ Q ) = β 2 ψ Q ( 1 ) / 2 F_{\mathrm{res}}(\beta,Q)=\mathcal{P}_\beta(\psi_Q)=\beta^2\psi_Q(1)/2 F res ( β , Q ) = P β ( ψ Q ) = β 2 ψ Q ( 1 ) /2
Part (3). By (1), V β ( Q ) = V ( Q ) = 0 V_\beta(Q)=V(Q)=0 V β ( Q ) = V ( Q ) = 0 ℓ = β x ^ β = β x ^ \ell=\beta\hat{x}_\beta=\beta\hat{x} ℓ = β x ^ β = β x ^ Q Q \mathcal{Q}_Q Q Q E ( Q ) = Q Q ( ℓ ) E(Q)=\mathcal{Q}_Q(\ell) E ( Q ) = Q Q ( ℓ )
∫ 0 Q ζ ′ ′ ( v ) β ( C − x v ) d v = β ( ( C − x Q ) ζ ′ ( Q ) + x ζ ( Q ) ) = β ( s ζ ′ ( Q ) + x ζ ( Q ) ) . \int_0^Q\zeta''(v)\beta(C-xv)\,dv=\beta\bigl((C-xQ)\zeta'(Q)+x\zeta(Q)\bigr)=\beta\bigl(s\zeta'(Q)+x\zeta(Q)\bigr). ∫ 0 Q ζ ′′ ( v ) β ( C − xv ) d v = β ( ( C − x Q ) ζ ′ ( Q ) + x ζ ( Q ) ) = β ( s ζ ′ ( Q ) + x ζ ( Q ) ) . ∫ 0 Q d v β ( C − x v ) = 1 β x log C s . \int_0^Q\frac{dv}{\beta(C-xv)}=\frac{1}{\beta x}\log\frac{C}{s}. ∫ 0 Q β ( C − xv ) d v = β x 1 log s C . Using C / s = 1 + z C/s=1+z C / s = 1 + z β s ζ ′ ( Q ) = a Q / y \beta s\zeta'(Q)=a_Q/y β s ζ ′ ( Q ) = a Q / y β x = z / y \beta x=z/y β x = z / y 1 / ( β x ) = y / z 1/(\beta x)=y/z 1/ ( β x ) = y / z
2 Q Q ( ℓ ) = a Q + z ζ ( Q ) y + y z log ( 1 + z ) . 2\mathcal{Q}_Q(\ell)=\frac{a_Q+z\zeta(Q)}{y}+\frac{y}{z}\log(1+z). 2 Q Q ( ℓ ) = y a Q + z ζ ( Q ) + z y log ( 1 + z ) . By the first two identities in (5.22), we have ( y / z ) log ( 1 + z ) = ( 1 + z ) a Q log ( 1 + z ) / ( y z ) = ( a Q + z ζ ( Q ) ) / y (y/z)\log(1+z)=(1+z)a_Q\log(1+z)/(yz)=(a_Q+z\zeta(Q))/y ( y / z ) log ( 1 + z ) = ( 1 + z ) a Q log ( 1 + z ) / ( y z ) = ( a Q + z ζ ( Q )) / y
For g g g g ( 0 ) = β − 2 V ( 0 ) = 0 g(0)=\beta^{-2}V(0)=0 g ( 0 ) = β − 2 V ( 0 ) = 0 g ( 1 ) = β − 2 V ( Q ) = 0 g(1)=\beta^{-2}V(Q)=0 g ( 1 ) = β − 2 V ( Q ) = 0 g > 0 g>0 g > 0 ( 0 , 1 ) (0,1) ( 0 , 1 ) g ~ \tilde{g} g ~ g ~ ( 1 ) = 0 = g ( 1 ) \tilde{g}(1)=0=g(1) g ~ ( 1 ) = 0 = g ( 1 ) g ′ = g ~ ′ g'=\tilde{g}' g ′ = g ~ ′ [ 0 , 1 ] [0,1] [ 0 , 1 ]
g ′ ( r ) = Q β 2 V ′ ( Q r ) = Q β 2 ∫ 0 Q r d v ( C − x v ) 2 − Q ζ ′ ( Q r ) = Q 2 r β 2 C ( C − x Q r ) − Q ζ ′ ( Q r ) . g'(r)=\frac{Q}{\beta^2}V'(Qr)=\frac{Q}{\beta^2}\int_0^{Qr}\frac{dv}{(C-xv)^2}-Q\zeta'(Qr)=\frac{Q^2r}{\beta^2C(C-xQr)}-Q\zeta'(Qr). g ′ ( r ) = β 2 Q V ′ ( Q r ) = β 2 Q ∫ 0 Q r ( C − xv ) 2 d v − Q ζ ′ ( Q r ) = β 2 C ( C − x Q r ) Q 2 r − Q ζ ′ ( Q r ) . Since C = s ( 1 + z ) C=s(1+z) C = s ( 1 + z ) C − x Q r = s ( 1 + ( 1 − r ) z ) C-xQr=s(1+(1-r)z) C − x Q r = s ( 1 + ( 1 − r ) z ) Q 2 / ( β 2 s 2 ) = y 2 = ( 1 + z ) a Q Q^2/(\beta^2s^2)=y^2=(1+z)a_Q Q 2 / ( β 2 s 2 ) = y 2 = ( 1 + z ) a Q a Q r / ( 1 + ( 1 − r ) z ) a_Qr/(1+(1-r)z) a Q r / ( 1 + ( 1 − r ) z )
g ~ ′ ( r ) = − Q ζ ′ ( Q r ) − a Q ( − 1 + z z ( 1 + ( 1 − r ) z ) + 1 z ) = − Q ζ ′ ( Q r ) + a Q r 1 + ( 1 − r ) z , \tilde{g}'(r)=-Q\zeta'(Qr)-a_Q\left(-\frac{1+z}{z(1+(1-r)z)}+\frac{1}{z}\right)=-Q\zeta'(Qr)+\frac{a_Qr}{1+(1-r)z}, g ~ ′ ( r ) = − Q ζ ′ ( Q r ) − a Q ( − z ( 1 + ( 1 − r ) z ) 1 + z + z 1 ) = − Q ζ ′ ( Q r ) + 1 + ( 1 − r ) z a Q r , and we conclude that g ′ = g ~ ′ g'=\tilde{g}' g ′ = g ~ ′
Part (4). Recall that C = s ( 1 + z ) C=s(1+z) C = s ( 1 + z ) (95) reads β 2 ξ ′ ( Q ) = Q / ( s 2 ( 1 + z ) ) \beta^2\xi'(Q)=Q/(s^2(1+z)) β 2 ξ ′ ( Q ) = Q / ( s 2 ( 1 + z )) Q / β 2 Q/\beta^2 Q / β 2 a Q = y 2 / ( 1 + z ) a_Q=y^2/(1+z) a Q = y 2 / ( 1 + z ) log ( C / s ) = log ( 1 + z ) \log(C/s)=\log(1+z) log ( C / s ) = log ( 1 + z ) x Q / C = z / ( 1 + z ) x_Q/C=z/(1+z) x Q / C = z / ( 1 + z ) β 2 x 2 = z 2 / y 2 = z 2 / ( ( 1 + z ) a Q ) \beta^2x^2=z^2/y^2=z^2/((1+z)a_Q) β 2 x 2 = z 2 / y 2 = z 2 / (( 1 + z ) a Q ) (95) reads
z 2 ξ ′ ( Q ) ( 1 + z ) a Q = log ( 1 + z ) − z 1 + z , \frac{z^2\xi'(Q)}{(1+z)a_Q}=\log(1+z)-\frac{z}{1+z}, ( 1 + z ) a Q z 2 ξ ′ ( Q ) = log ( 1 + z ) − 1 + z z , which is the second identity in (100) . Next, z / y = ( x Q / s ) ( β s / Q ) = β x = c 0 z/y=(x_Q/s)(\beta s/Q)=\beta x=c_0 z / y = ( x Q / s ) ( β s / Q ) = β x = c 0 c 0 < β c_0<\beta c 0 < β x < 1 x<1 x < 1 b Q / y 2 = Q 2 ξ ′ ′ ( Q ) β 2 s 2 / Q 2 = β 2 s 2 ξ ′ ′ ( Q ) b_Q/y^2=Q^2\xi''(Q)\beta^2s^2/Q^2=\beta^2s^2\xi''(Q) b Q / y 2 = Q 2 ξ ′′ ( Q ) β 2 s 2 / Q 2 = β 2 s 2 ξ ′′ ( Q ) K 0 − 2 b Q = ( y 2 − 2 y b Q + b Q ) / y K_0-2\sqrt{b_Q}=(y^2-2y\sqrt{b_Q}+b_Q)/y K 0 − 2 b Q = ( y 2 − 2 y b Q + b Q ) / y y > b Q y>\sqrt{b_Q} y > b Q
Part (5). By (1), the contact set is the zero set of V V V { 0 , Q } \{0,Q\} { 0 , Q } (93) . Using Theorem 5.7 (1), we find that the multisamplable overlaps are 0 0 0 Q Q Q Theorem 5.7 (3) implies (86) for every other q ∈ ( 0 , 1 ) q\in(0,1) q ∈ ( 0 , 1 ) T β ( Q ) = F ( β ) T_\beta(Q)=F(\beta) T β ( Q ) = F ( β ) Theorem 5.5 (1). Inserting E ( Q ) = E 0 E(Q)=E_0 E ( Q ) = E 0 F r e s ( β , Q ) = β 2 ψ Q ( 1 ) / 2 F_{\mathrm{res}}(\beta,Q)=\beta^2\psi_Q(1)/2 F res ( β , Q ) = β 2 ψ Q ( 1 ) /2
Remark 5.12 (Comparison with Huang and Sellke). For the covariance η ( r ) = ξ ( Q r ) \eta(r)=\xi(Qr) η ( r ) = ξ ( Q r ) η ( 1 ) = ξ ( Q ) \eta(1)=\xi(Q) η ( 1 ) = ξ ( Q ) η ′ ( 1 ) = a Q \eta'(1)=a_Q η ′ ( 1 ) = a Q η ′ ′ ( 1 ) = b Q \eta''(1)=b_Q η ′′ ( 1 ) = b Q ℓ ( Q r ) / Q = ( 1 + z − z r ) / y \ell(Qr)/Q=(1+z-zr)/y ℓ ( Q r ) / Q = ( 1 + z − z r ) / y ( 1 + z ) / y (1+z)/y ( 1 + z ) / y c 0 = z / y c_0=z/y c 0 = z / y (100) that z z z ( ( 1 + z ) log ( 1 + z ) − z ) / z 2 = η ( 1 ) / η ′ ( 1 ) ((1+z)\log(1+z)-z)/z^2=\eta(1)/\eta'(1) (( 1 + z ) log ( 1 + z ) − z ) / z 2 = η ( 1 ) / η ′ ( 1 ) y = ( 1 + z ) η ′ ( 1 ) y=\sqrt{(1+z)\eta'(1)} y = ( 1 + z ) η ′ ( 1 ) z z z y y y L = ( 1 + z ) / y L=(1+z)/y L = ( 1 + z ) / y u = z / y u=z/y u = z / y ( η ′ ( 1 ) + z η ( 1 ) ) / y (\eta'(1)+z\eta(1))/y ( η ′ ( 1 ) + z η ( 1 )) / y E 0 E_0 E 0 0 0 0 1 1 1 ( 0 , 1 ) (0,1) ( 0 , 1 ) R 0 = y + η ′ ′ ( 1 ) / y R_0=y+\eta''(1)/y R 0 = y + η ′′ ( 1 ) / y K 0 K_0 K 0 H Q H_Q H Q u u u K Q ( u ) / N K_Q(u)/\sqrt{N} K Q ( u ) / N N \sqrt{N} N y 2 ≥ η ′ ′ ( 1 ) y^2\geq\eta''(1) y 2 ≥ η ′′ ( 1 ) b Q < y 2 b_Q<y^2 b Q < y 2 V ′ ′ ( Q ) > 0 V''(Q)>0 V ′′ ( Q ) > 0
Value, radial derivative, and Hessian at radius Q \sqrt{Q} Q
In this section, ( ξ , β ) (\xi,\beta) ( ξ , β ) x x x Q Q Q Q Q Q E 0 E_0 E 0 K 0 K_0 K 0 c 0 c_0 c 0 y y y z z z
Σ Q = ( ξ ( Q ) a Q a Q a Q + b Q ) . (103) \Sigma_Q=\begin{pmatrix}\xi(Q)&a_Q\\a_Q&a_Q+b_Q\end{pmatrix}. \tag*{(103)} Σ Q = ( ξ ( Q ) a Q a Q a Q + b Q ) . (103) For t > 2 t>2 t > 2
Ω ( t ) = ∫ − 2 2 log ( t − λ ) ρ s c ( d λ ) , ρ s c ( d λ ) = 4 − λ 2 2 π d λ , \Omega(t)=\int_{-2}^{2}\log(t-\lambda)\,\rho_{\mathrm{sc}}(\mathrm{d}\lambda),\qquad
\rho_{\mathrm{sc}}(\mathrm{d}\lambda)=\frac{\sqrt{4-\lambda^2}}{2\pi}\,\mathrm{d}\lambda, Ω ( t ) = ∫ − 2 2 log ( t − λ ) ρ sc ( d λ ) , ρ sc ( d λ ) = 2 π 4 − λ 2 d λ , and Ω ( 2 ) = lim t ↓ 2 Ω ( t ) \Omega(2)=\lim_{t\downarrow2}\Omega(t) Ω ( 2 ) = lim t ↓ 2 Ω ( t ) n ≥ 1 n\ge1 n ≥ 1 G n G_n G n n × n n\times n n × n 1 / n 1/n 1/ n 2 / n 2/n 2/ n ρ s c \rho_{\mathrm{sc}} ρ sc [ − 2 , 2 ] [-2,2] [ − 2 , 2 ]
Proposition 5.13 (Value and radial derivative). Let ( ζ , β ) (\zeta,\beta) ( ζ , β ) Q Q Q
For every u ∈ S N u\in S_N u ∈ S N ( H Q ( u ) , K Q ( u ) ) (H_Q(u),K_Q(u)) ( H Q ( u ) , K Q ( u )) N Σ Q N\Sigma_Q N Σ Q w p = c p Q p / ξ ( Q ) w_p=c_pQ^p/\xi(Q) w p = c p Q p / ξ ( Q )
det Σ Q = ξ ( Q ) 2 ( ∑ p p 2 w p − ( ∑ p p w p ) 2 ) . \det\Sigma_Q=\xi(Q)^2\left(\sum_p p^2w_p-\left(\sum_p pw_p\right)^2\right). det Σ Q = ξ ( Q ) 2 p ∑ p 2 w p − ( p ∑ p w p ) 2 . which is ξ ( Q ) 2 \xi(Q)^2 ξ ( Q ) 2 p p p ( w p ) (w_p) ( w p ) Σ Q \Sigma_Q Σ Q ξ \xi ξ
We have
Σ Q ( c 0 1 / y ) = ( E 0 K 0 ) . (104) \Sigma_Q\begin{pmatrix}c_0\\1/y\end{pmatrix}=\begin{pmatrix}E_0\\K_0\end{pmatrix}. \tag*{(104)} Σ Q ( c 0 1/ y ) = ( E 0 K 0 ) . (104) If ξ \xi ξ Σ Q − 1 ( E 0 , K 0 ) T = ( c 0 , 1 / y ) T \Sigma_Q^{-1}(E_0,K_0)^{\mathsf T}=(c_0,1/y)^{\mathsf T} Σ Q − 1 ( E 0 , K 0 ) T = ( c 0 , 1/ y ) T ξ ( r ) = c p r p \xi(r)=c_pr^p ξ ( r ) = c p r p K Q = p H Q K_Q=pH_Q K Q = p H Q K 0 = p E 0 K_0=pE_0 K 0 = p E 0 E 0 / ξ ( Q ) = c 0 + p / y E_0/\xi(Q)=c_0+p/y E 0 / ξ ( Q ) = c 0 + p / y
Suppose that ξ \xi ξ e ∈ R e\in\mathbb{R} e ∈ R k ≥ 2 b Q k\ge2\sqrt{b_Q} k ≥ 2 b Q
Θ ( e , k ) = 1 2 + 1 2 log b Q a Q − 1 2 ( e , k ) Σ Q − 1 ( e , k ) T + Ω ( k / b Q ) . (105) \Theta(e,k)=\frac{1}{2}+\frac{1}{2}\log\frac{b_Q}{a_Q}-\frac{1}{2}(e,k)\Sigma_Q^{-1}(e,k)^{\mathsf T}+\Omega(k/\sqrt{b_Q}). \tag*{(105)} Θ ( e , k ) = 2 1 + 2 1 log a Q b Q − 2 1 ( e , k ) Σ Q − 1 ( e , k ) T + Ω ( k / b Q ) . (105) Then
Θ ( E 0 , K 0 ) = 0 , ∂ e Θ ( E 0 , K 0 ) = − c 0 , ∂ k Θ ( E 0 , K 0 ) = 0 , \Theta(E_0,K_0)=0,\qquad\partial_e\Theta(E_0,K_0)=-c_0,\qquad\partial_k\Theta(E_0,K_0)=0, Θ ( E 0 , K 0 ) = 0 , ∂ e Θ ( E 0 , K 0 ) = − c 0 , ∂ k Θ ( E 0 , K 0 ) = 0 , and K 0 K_0 K 0 k ↦ Θ ( E 0 , k ) k\mapsto\Theta(E_0,k) k ↦ Θ ( E 0 , k ) [ 2 b Q , ∞ ) [2\sqrt{b_Q},\infty) [ 2 b Q , ∞ )
Suppose that ξ ( r ) = c p r p \xi(r)=c_pr^p ξ ( r ) = c p r p e ≥ 2 b Q / p e\ge2\sqrt{b_Q}/p e ≥ 2 b Q / p
Θ p u r e ( e ) = 1 2 + 1 2 log b Q a Q − e 2 2 ξ ( Q ) + Ω ( p e / b Q ) . \Theta_{\mathrm{pure}}(e)=\frac{1}{2}+\frac{1}{2}\log\frac{b_Q}{a_Q}-\frac{e^2}{2\xi(Q)}+\Omega(pe/\sqrt{b_Q}). Θ pure ( e ) = 2 1 + 2 1 log a Q b Q − 2 ξ ( Q ) e 2 + Ω ( p e / b Q ) . Then E 0 > 2 b Q / p E_0>2\sqrt{b_Q}/p E 0 > 2 b Q / p Θ p u r e ( E 0 ) = 0 \Theta_{\mathrm{pure}}(E_0)=0 Θ pure ( E 0 ) = 0 Θ p u r e ′ ( E 0 ) = − c 0 \Theta_{\mathrm{pure}}'(E_0)=-c_0 Θ pure ′ ( E 0 ) = − c 0
In Section 10, Θ \Theta Θ H Q H_Q H Q e e e k k k ( E 0 , K 0 ) (E_0,K_0) ( E 0 , K 0 ) c 0 c_0 c 0
Proof. For (1), let u ∈ S N u \in S_N u ∈ S N E H N ( t Q u ) H N ( t ′ Q u ) = N ζ ( t t ′ Q ) \mathbb{E}H_N(t\sqrt{Q}u)H_N(t'\sqrt{Q}u)=N\zeta(tt'Q) E H N ( t Q u ) H N ( t ′ Q u ) = N ζ ( t t ′ Q ) t , t ′ > 0 t,t'>0 t , t ′ > 0 ∣ u ∣ 2 = N \lvert u\rvert^2=N ∣ u ∣ 2 = N t ′ t' t ′ t t t t ′ t' t ′ t = t ′ = 1 t=t'=1 t = t ′ = 1
E H Q ( u ) K Q ( u ) = N Q ζ ′ ( Q ) = N a Q , E K Q ( u ) 2 = N ( Q ζ ′ ( Q ) + Q 2 ζ ′ ′ ( Q ) ) . \mathbb{E}H_Q(u)K_Q(u)=NQ\zeta'(Q)=Na_Q,\qquad\mathbb{E}K_Q(u)^2=N\bigl(Q\zeta'(Q)+Q^2\zeta''(Q)\bigr). E H Q ( u ) K Q ( u ) = N Q ζ ′ ( Q ) = N a Q , E K Q ( u ) 2 = N ( Q ζ ′ ( Q ) + Q 2 ζ ′′ ( Q ) ) . Differentiation under the expectation is justified because H N H_N H N w p = c p Q p / ζ ( Q ) w_p=c_pQ^p/\zeta(Q) w p = c p Q p / ζ ( Q ) ζ ′ ( Q ) = ζ ( Q ) ∑ p w p \zeta'(Q)=\zeta(Q)\sum_p w_p ζ ′ ( Q ) = ζ ( Q ) ∑ p w p a Q = ζ ( Q ) ∑ p p w p a_Q=\zeta(Q)\sum_p pw_p a Q = ζ ( Q ) ∑ p p w p a Q + b Q = ∑ p ( p + p ( p − 1 ) ) c p Q p = ζ ( Q ) ∑ p p 2 w p a_Q+b_Q=\sum_p\bigl(p+p(p-1)\bigr)c_pQ^p=\zeta(Q)\sum_p p^2w_p a Q + b Q = ∑ p ( p + p ( p − 1 ) ) c p Q p = ζ ( Q ) ∑ p p 2 w p det Σ Q \det\Sigma_Q det Σ Q p p p ( w p ) (w_p) ( w p ) w p w_p w p
For (2), we use c 0 = z / y c_0=z/y c 0 = z / y y 2 = ( 1 + z ) a Q y^2=(1+z)a_Q y 2 = ( 1 + z ) a Q (100) , and multiply:
Σ Q ( z / y 1 / y ) = 1 y ( z ζ ( Q ) + a Q z a Q + a Q + b Q ) = ( ( a Q + z ζ ( Q ) ) / y ( y 2 + b Q ) / y ) = ( E 0 K 0 ) . \Sigma_Q
\begin{pmatrix}
z/y\\
1/y
\end{pmatrix}
=
\frac{1}{y}
\begin{pmatrix}
z\zeta(Q)+a_Q\\
za_Q+a_Q+b_Q
\end{pmatrix}
=
\begin{pmatrix}
(a_Q+z\zeta(Q))/y\\
(y^2+b_Q)/y
\end{pmatrix}
=
\begin{pmatrix}
E_0\\
K_0
\end{pmatrix}. Σ Q ( z / y 1/ y ) = y 1 ( z ζ ( Q ) + a Q z a Q + a Q + b Q ) = ( ( a Q + z ζ ( Q )) / y ( y 2 + b Q ) / y ) = ( E 0 K 0 ) . If Σ Q \Sigma_Q Σ Q ζ = c p r p \zeta=c_pr^p ζ = c p r p K Q ( u ) = u ⋅ ∇ H Q ( u ) = p H Q ( u ) K_Q(u)=u\cdot\nabla H_Q(u)=pH_Q(u) K Q ( u ) = u ⋅ ∇ H Q ( u ) = p H Q ( u )
a Q = p ζ ( Q ) and b Q = p ( p − 1 ) ζ ( Q ) , a_Q=p\zeta(Q)\quad\text{and}\quad b_Q=p(p-1)\zeta(Q), a Q = pζ ( Q ) and b Q = p ( p − 1 ) ζ ( Q ) , so that
Σ Q = ζ ( Q ) ( 1 p p p 2 ) , \Sigma_Q=\zeta(Q)
\begin{pmatrix}
1 & p\\
p & p^2
\end{pmatrix}, Σ Q = ζ ( Q ) ( 1 p p p 2 ) , and (104) reads
E 0 = ζ ( Q ) ( c 0 + p / y ) and K 0 = p ζ ( Q ) ( c 0 + p / y ) = p E 0 . E_0=\zeta(Q)(c_0+p/y)\quad\text{and}\quad K_0=p\zeta(Q)(c_0+p/y)=pE_0. E 0 = ζ ( Q ) ( c 0 + p / y ) and K 0 = pζ ( Q ) ( c 0 + p / y ) = p E 0 . For (3) and (4), we first compute Ω \Omega Ω t > 2 t>2 t > 2 t = ω + 1 / ω t=\omega+1/\omega t = ω + 1/ ω ω > 1 \omega>1 ω > 1
Ω ′ ( t ) = ∫ ρ s c ( d λ ) t − λ = t − t 2 − 4 2 = 1 ω , Ω ( t ) = log ω + 1 2 ω 2 . (106) \Omega'(t)=\int\frac{\rho_{\mathrm{sc}}(d\lambda)}{t-\lambda}=\frac{t-\sqrt{t^2-4}}{2}=\frac{1}{\omega},\qquad\Omega(t)=\log\omega+\frac{1}{2\omega^2}. \tag*{(106)} Ω ′ ( t ) = ∫ t − λ ρ sc ( d λ ) = 2 t − t 2 − 4 = ω 1 , Ω ( t ) = log ω + 2 ω 2 1 . (106) The first identity is the classical Stieltjes transform of the semicircle law; the resulting logarithmic potential is also [30, (3.3)]; it follows from the substitution λ = 2 cos θ \lambda=2\cos\theta λ = 2 cos θ ∫ 0 π ( t − 2 cos θ ) − 1 d θ = π ( t 2 − 4 ) − 1 / 2 \int_0^\pi(t-2\cos\theta)^{-1}\,d\theta=\pi(t^2-4)^{-1/2} ∫ 0 π ( t − 2 cos θ ) − 1 d θ = π ( t 2 − 4 ) − 1/2 d d t ( log ω + 1 2 ω 2 ) = ( 1 ω − 1 ω 3 ) d ω d t = 1 ω \frac{d}{dt}\left(\log\omega+\frac{1}{2\omega^2}\right)=\left(\frac{1}{\omega}-\frac{1}{\omega^3}\right)\frac{d\omega}{dt}=\frac{1}{\omega} d t d ( log ω + 2 ω 2 1 ) = ( ω 1 − ω 3 1 ) d t d ω = ω 1 d t d ω = 1 − ω − 2 \frac{dt}{d\omega}=1-\omega^{-2} d ω d t = 1 − ω − 2 (106) equal log t + o ( 1 ) \log t+o(1) log t + o ( 1 ) t → ∞ t\to\infty t → ∞ Ω ′ ′ ( t ) = − ∫ ( t − λ ) − 2 ρ s c ( d λ ) < 0 \Omega''(t)=-\int(t-\lambda)^{-2}\rho_{\mathrm{sc}}(d\lambda)<0 Ω ′′ ( t ) = − ∫ ( t − λ ) − 2 ρ sc ( d λ ) < 0 t > 2 t>2 t > 2 Ω \Omega Ω [ 2 , ∞ ) [2,\infty) [ 2 , ∞ )
We have K 0 / b Q = ω + 1 / ω K_0/\sqrt{b_Q}=\omega+1/\omega K 0 / b Q = ω + 1/ ω ω = y / b Q > 1 \omega=y/\sqrt{b_Q}>1 ω = y / b Q > 1 (101) . Using (106) , we obtain
1 2 log b Q + Ω ( K 0 / b Q ) = log y + b Q 2 y 2 , 1 b Q Ω ′ ( K 0 / b Q ) = 1 y . (107) \frac{1}{2}\log b_Q+\Omega(K_0/\sqrt{b_Q})=\log y+\frac{b_Q}{2y^2},\qquad\frac{1}{\sqrt{b_Q}}\Omega'(K_0/\sqrt{b_Q})=\frac{1}{y}. \tag*{(107)} 2 1 log b Q + Ω ( K 0 / b Q ) = log y + 2 y 2 b Q , b Q 1 Ω ′ ( K 0 / b Q ) = y 1 . (107) By (104) , the quadratic term at ( E 0 , K 0 ) (E_0,K_0) ( E 0 , K 0 ) ( E 0 , K 0 ) Σ Q − 1 ( E 0 , K 0 ) T = c 0 E 0 + K 0 / y (E_0,K_0)\Sigma_Q^{-1}(E_0,K_0)^{\mathsf T}=c_0E_0+K_0/y ( E 0 , K 0 ) Σ Q − 1 ( E 0 , K 0 ) T = c 0 E 0 + K 0 / y
c 0 E 0 + K 0 y = z a Q + z 2 ζ ( Q ) y 2 + 1 + b Q y 2 . c_0E_0+\frac{K_0}{y}=\frac{za_Q+z^2\zeta(Q)}{y^2}+1+\frac{b_Q}{y^2}. c 0 E 0 + y K 0 = y 2 z a Q + z 2 ζ ( Q ) + 1 + y 2 b Q . When we insert this and (107) into (105) , the constant terms and the terms b Q / ( 2 y 2 ) b_Q/(2y^2) b Q / ( 2 y 2 )
Θ ( E 0 , K 0 ) = 1 2 log y 2 a Q − z a Q + z 2 ζ ( Q ) 2 y 2 = 1 2 log ( 1 + z ) − z 2 ( 1 + z ) ( 1 + z ζ ( Q ) a Q ) = 0. \Theta(E_0,K_0)=\frac{1}{2}\log\frac{y^2}{a_Q}-\frac{za_Q+z^2\zeta(Q)}{2y^2}=\frac{1}{2}\log(1+z)-\frac{z}{2(1+z)}\left(1+\frac{z\zeta(Q)}{a_Q}\right)=0. Θ ( E 0 , K 0 ) = 2 1 log a Q y 2 − 2 y 2 z a Q + z 2 ζ ( Q ) = 2 1 log ( 1 + z ) − 2 ( 1 + z ) z ( 1 + a Q z ζ ( Q ) ) = 0. In the second step, we used y 2 = ( 1 + z ) a Q y^{2}=(1+z)a_{Q} y 2 = ( 1 + z ) a Q (100) , which gives 1 + z ξ ( Q ) / a Q = ( 1 + z ) log ( 1 + z ) / z 1+z\xi(Q)/a_{Q}=(1+z)\log(1+z)/z 1 + z ξ ( Q ) / a Q = ( 1 + z ) log ( 1 + z ) / z 1 2 ( e , k ) Σ − 1 ( e , k ) T \frac{1}{2}(e,k)\Sigma^{-1}(e,k)^{\mathsf{T}} 2 1 ( e , k ) Σ − 1 ( e , k ) T ( E 0 , K 0 ) (E_{0},K_{0}) ( E 0 , K 0 ) Σ − 1 ( E 0 , K 0 ) T = ( c 0 , 1 / y ) T \Sigma^{-1}(E_{0},K_{0})^{\mathsf{T}}=(c_{0},1/y)^{\mathsf{T}} Σ − 1 ( E 0 , K 0 ) T = ( c 0 , 1/ y ) T (107) we find ∂ e Θ ( E 0 , K 0 ) = − c 0 \partial_{e}\Theta(E_{0},K_{0})=-c_{0} ∂ e Θ ( E 0 , K 0 ) = − c 0 ∂ k Θ ( E 0 , K 0 ) = − 1 / y + 1 / y = 0 \partial_{k}\Theta(E_{0},K_{0})=-1/y+1/y=0 ∂ k Θ ( E 0 , K 0 ) = − 1/ y + 1/ y = 0 Σ − 1 \Sigma^{-1} Σ − 1 Ω ′ ′ < 0 \Omega''<0 Ω ′′ < 0 2 2 2 k ↦ Θ ( E 0 , k ) k\mapsto\Theta(E_{0},k) k ↦ Θ ( E 0 , k ) ( 2 b Q , ∞ ) (2\sqrt{b_{Q}},\infty) ( 2 b Q , ∞ ) [ 2 b Q , ∞ ) [2\sqrt{b_{Q}},\infty) [ 2 b Q , ∞ ) K 0 K_{0} K 0
For (4), we have p E 0 = K 0 > 2 b Q pE_{0}=K_{0}>2\sqrt{b_{Q}} p E 0 = K 0 > 2 b Q (101) . By (2), E 0 2 / ξ ( Q ) = E 0 ( c 0 + p / y ) = c 0 E 0 + K 0 / y E_{0}^{2}/\xi(Q)=E_{0}(c_{0}+p/y)=c_{0}E_{0}+K_{0}/y E 0 2 / ξ ( Q ) = E 0 ( c 0 + p / y ) = c 0 E 0 + K 0 / y Θ ( E 0 , K 0 ) \Theta(E_{0},K_{0}) Θ ( E 0 , K 0 ) Θ p u r e ( E 0 ) = 0 \Theta_{\mathrm{pure}}(E_{0})=0 Θ pure ( E 0 ) = 0 (107) ,
Θ p u r e ′ ( E 0 ) = − E 0 ξ ( Q ) + p b Q Ω ′ ( K 0 b Q ) = − E 0 ξ ( Q ) + p y = − c 0 . \Theta'_{\mathrm{pure}}(E_{0})=-\frac{E_{0}}{\xi(Q)}+\frac{p}{\sqrt{b_{Q}}}\Omega'\left(\frac{K_{0}}{\sqrt{b_{Q}}}\right)=-\frac{E_{0}}{\xi(Q)}+\frac{p}{y}=-c_{0}. Θ pure ′ ( E 0 ) = − ξ ( Q ) E 0 + b Q p Ω ′ ( b Q K 0 ) = − ξ ( Q ) E 0 + y p = − c 0 . The next lemma gives the law of the Riemannian Hessian of H Q H_{Q} H Q r ↦ ξ ( Q r ) r\mapsto\xi(Qr) r ↦ ξ ( Q r ) ( N − 1 ) / N (N-1)/N ( N − 1 ) / N
Lemma 5.14 (Conditional law of the Hessian). Let ξ \xi ξ 0 < Q < 1 0<Q<1 0 < Q < 1 N ≥ 2 N\geq2 N ≥ 2 u ∈ S N u\in S_{N} u ∈ S N e 2 , … , e N e_{2},\ldots,e_{N} e 2 , … , e N { w : w ⋅ u = 0 } \{w:w\cdot u=0\} { w : w ⋅ u = 0 } ∇ S N 2 H Q ( u ) \nabla^{2}_{S_{N}}H_{Q}(u) ∇ S N 2 H Q ( u ) ( ∇ S N 2 H Q ( u ) [ e i , e j ] ) i , j = 2 N (\nabla^{2}_{S_{N}}H_{Q}(u)[e_{i},e_{j}])_{i,j=2}^{N} ( ∇ S N 2 H Q ( u ) [ e i , e j ] ) i , j = 2 N
The matrix ∇ S N 2 H Q ( u ) + N − 1 K Q ( u ) I N − 1 \nabla^{2}_{S_{N}}H_{Q}(u)+N^{-1}K_{Q}(u)I_{N-1} ∇ S N 2 H Q ( u ) + N − 1 K Q ( u ) I N − 1 ( H Q ( u ) , K Q ( u ) , ∇ S N H Q ( u ) ) (H_{Q}(u),K_{Q}(u),\nabla_{S_{N}}H_{Q}(u)) ( H Q ( u ) , K Q ( u ) , ∇ S N H Q ( u )) ( N − 1 ) b Q / N G N − 1 \sqrt{(N-1)b_{Q}/N}\,G_{N-1} ( N − 1 ) b Q / N G N − 1 H Q ( u ) = N e H_{Q}(u)=Ne H Q ( u ) = N e K Q ( u ) = N k K_{Q}(u)=Nk K Q ( u ) = N k ∇ S N H Q ( u ) = 0 \nabla_{S_{N}}H_{Q}(u)=0 ∇ S N H Q ( u ) = 0
∇ S N 2 H Q ( u ) = l a w ( N − 1 ) b Q N G N − 1 − k I N − 1 . (108) \nabla^{2}_{S_{N}}H_{Q}(u)\mathrel{\stackrel{\mathrm{law}}{=}}\sqrt{\frac{(N-1)b_{Q}}{N}}G_{N-1}-kI_{N-1}. \tag*{(108)} ∇ S N 2 H Q ( u ) = law N ( N − 1 ) b Q G N − 1 − k I N − 1 . (108) When ξ \xi ξ p p p K Q ( u ) K_{Q}(u) K Q ( u ) k = p e k=pe k = p e
If ( ξ , β ) (\xi,\beta) ( ξ , β ) Q Q Q g 0 = K 0 − 2 b Q = ( y − 2 / b Q ) 2 / y > 0 g_{0}=K_{0}-2\sqrt{b_{Q}}=(y-2/\sqrt{b_{Q}})^{2}/y>0 g 0 = K 0 − 2 b Q = ( y − 2/ b Q ) 2 / y > 0 U 0 = ( K 0 − g 0 / 4 , K 0 + g 0 / 4 ) U_{0}=(K_{0}-g_{0}/4,K_{0}+g_{0}/4) U 0 = ( K 0 − g 0 /4 , K 0 + g 0 /4 ) inf U 0 = 2 b Q + 3 g 0 / 4 \inf U_{0}=2\sqrt{b_{Q}}+3g_{0}/4 inf U 0 = 2 b Q + 3 g 0 /4
By (1), for each fixed k ∈ U 0 k\in U_{0} k ∈ U 0 b Q \sqrt{b_{Q}} b Q − k -k − k l e f t [ − k − 2 b Q , − k + 2 b Q ] left[-k-2\sqrt{b_{Q}},-k+2\sqrt{b_{Q}}] l e f t [ − k − 2 b Q , − k + 2 b Q ] − 3 g 0 / 4 -3g_{0}/4 − 3 g 0 /4 N N N § 6.2 . Part (2) itself is a deterministic statement about the location of U 0 U_0 U 0 Definition 7.3 , as Corollary 7.6 , which is the form used in the later sections.
Proof. For (1), by rotation invariance (Lemma 3.2(2)), we may assume that u = N e 1 u=\sqrt{N}e_1 u = N e 1 e 2 , … , e N e_2,\ldots,e_N e 2 , … , e N H Q H_Q H Q R N \mathbb{R}^N R N E H Q ( v ) H Q ( w ) = N η ( v ⋅ w / N ) \mathbb{E}H_Q(v)H_Q(w)=N\eta(v\cdot w/N) E H Q ( v ) H Q ( w ) = N η ( v ⋅ w / N ) η ( r ) = ξ ( Q r ) \eta(r)=\xi(Qr) η ( r ) = ξ ( Q r ) η ′ ( 1 ) = a Q \eta'(1)=a_Q η ′ ( 1 ) = a Q η ′ ′ ( 1 ) = b Q \eta''(1)=b_Q η ′′ ( 1 ) = b Q H Q H_Q H Q u u u N η ( v ⋅ w / N ) N\eta(v\cdot w/N) N η ( v ⋅ w / N ) v v v w w w v = w = u v=w=u v = w = u i , j , k , l ≥ 2 i,j,k,l\ge2 i , j , k , l ≥ 2
E [ ∂ i j H Q ( u ) ∂ k l H Q ( u ) ] = ∂ v i ∂ v j ∂ w k ∂ w l N η ( v ⋅ w N ) ∣ v = w = u = η ′ ′ ( 1 ) N ( δ i k δ j l + δ i l δ j k ) . \mathbb{E}\bigl[\partial_{ij}H_Q(u)\,\partial_{kl}H_Q(u)\bigr]
=\left.\partial_{v_i}\partial_{v_j}\partial_{w_k}\partial_{w_l}N\eta\left(\frac{v\cdot w}{N}\right)\right|_{v=w=u}
=\frac{\eta''(1)}{N}\left(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}\right). E [ ∂ ij H Q ( u ) ∂ k l H Q ( u ) ] = ∂ v i ∂ v j ∂ w k ∂ w l N η ( N v ⋅ w ) v = w = u = N η ′′ ( 1 ) ( δ ik δ j l + δ i l δ j k ) . Indeed, every other term of this fourth derivative contains a factor w i w_i w i w j w_j w j v k v_k v k v l v_l v l v = w = u v=w=u v = w = u ∂ i j H Q ( u ) \partial_{ij}H_Q(u) ∂ ij H Q ( u ) i , j ≥ 2 i,j\ge2 i , j ≥ 2 H Q ( u ) H_Q(u) H Q ( u ) ∂ 1 H Q ( u ) \partial_1H_Q(u) ∂ 1 H Q ( u ) ∂ k H Q ( u ) \partial_kH_Q(u) ∂ k H Q ( u ) k ≥ 2 k\ge2 k ≥ 2 w i w_i w i w j w_j w j ( ∂ i j H Q ( u ) ) i , j ≥ 2 (\partial_{ij}H_Q(u))_{i,j\ge2} ( ∂ ij H Q ( u ) ) i , j ≥ 2 H Q ( u ) H_Q(u) H Q ( u ) K Q ( u ) = N ∂ 1 H Q ( u ) K_Q(u)=\sqrt{N}\partial_1H_Q(u) K Q ( u ) = N ∂ 1 H Q ( u ) ∇ S N H Q ( u ) = ( ∂ k H Q ( u ) ) k ≥ 2 \nabla_{S_N}H_Q(u)=(\partial_kH_Q(u))_{k\ge2} ∇ S N H Q ( u ) = ( ∂ k H Q ( u ) ) k ≥ 2 b Q / N b_Q/N b Q / N 2 b Q / N 2b_Q/N 2 b Q / N ( N − 1 ) b Q / N G N − 1 \sqrt{(N-1)b_Q/N}G_{N-1} ( N − 1 ) b Q / N G N − 1
∇ S N 2 H Q ( u ) [ e i , e j ] = ∂ i j H Q ( u ) − N − 1 ( u ⋅ ∇ H Q ( u ) ) δ i j = ∂ i j H Q ( u ) − N − 1 K Q ( u ) δ i j . \nabla^2_{S_N}H_Q(u)[e_i,e_j]=\partial_{ij}H_Q(u)-N^{-1}(u\cdot\nabla H_Q(u))\delta_{ij}=\partial_{ij}H_Q(u)-N^{-1}K_Q(u)\delta_{ij}. ∇ S N 2 H Q ( u ) [ e i , e j ] = ∂ ij H Q ( u ) − N − 1 ( u ⋅ ∇ H Q ( u )) δ ij = ∂ ij H Q ( u ) − N − 1 K Q ( u ) δ ij . which proves the first assertion. The conditional law (108) follows from this independence. In the pure case, K Q = p H Q K_Q=pH_Q K Q = p H Q g 0 g_0 g 0 (101) , and K 0 − g 0 / 4 = 2 b Q + 3 g 0 / 4 K_0-g_0/4=2\sqrt{b_Q}+3g_0/4 K 0 − g 0 /4 = 2 b Q + 3 g 0 /4
Continuation of the 1RSB solution and the derivative of the free energy
In this section, we fix a finite set D \mathcal{D} D c = ( c p ) p ∈ D ∈ R D c=(c_p)_{p\in\mathcal{D}}\in\mathbb{R}^{\mathcal{D}} c = ( c p ) p ∈ D ∈ R D ξ c ( r ) = ∑ p ∈ D c p r p \xi_c(r)=\sum_{p\in\mathcal{D}}c_pr^p ξ c ( r ) = ∑ p ∈ D c p r p ( c , β ) (c,\beta) ( c , β ) c ∈ [ 0 , ∞ ) D ∖ { 0 } c\in[0,\infty)^{\mathcal{D}}\setminus\{0\} c ∈ [ 0 , ∞ ) D ∖ { 0 } β > 0 \beta>0 β > 0 V c , β , x , Q V_{c,\beta,x,Q} V c , β , x , Q V V V Definition 5.9 built from ξ c \xi_c ξ c β \beta β x , Q ∈ ( 0 , 1 ) x,Q\in(0,1) x , Q ∈ ( 0 , 1 )
Lemma 5.15 (Continuation of the 1RSB solution). Let c ˉ ∈ [ 0 , ∞ ) D ∖ { 0 } \bar{c}\in[0,\infty)^{\mathcal{D}}\setminus\{0\} c ˉ ∈ [ 0 , ∞ ) D ∖ { 0 } β ˉ > 0 \bar{\beta}>0 β ˉ > 0 ( ξ c ˉ , β ˉ ) (\xi_{\bar{c}},\bar{\beta}) ( ξ c ˉ , β ˉ ) x ˉ , Q ˉ \bar{x},\bar{Q} x ˉ , Q ˉ Q ˉ \bar{Q} Q ˉ U U U ( c ˉ , β ˉ ) (\bar{c},\bar{\beta}) ( c ˉ , β ˉ ) R D × ( 0 , ∞ ) \mathbb{R}^{\mathcal{D}}\times(0,\infty) R D × ( 0 , ∞ ) x , Q : U → ( 0 , 1 ) x,Q:U\to(0,1) x , Q : U → ( 0 , 1 ) x ( c ˉ , β ˉ ) = x ˉ x(\bar{c},\bar{\beta})=\bar{x} x ( c ˉ , β ˉ ) = x ˉ Q ( c ˉ , β ˉ ) = Q ˉ Q(\bar{c},\bar{\beta})=\bar{Q} Q ( c ˉ , β ˉ ) = Q ˉ ( c , β ) ∈ U (c,\beta)\in U ( c , β ) ∈ U c ∈ [ 0 , ∞ ) D ∖ { 0 } c\in[0,\infty)^{\mathcal{D}}\setminus\{0\} c ∈ [ 0 , ∞ ) D ∖ { 0 } ( ξ c , β ) (\xi_c,\beta) ( ξ c , β ) x ( c , β ) , Q ( c , β ) x(c,\beta),Q(c,\beta) x ( c , β ) , Q ( c , β ) Q ( c , β ) Q(c,\beta) Q ( c , β ) Proof. The proof has two steps. We first continue the two contact equations V ( Q ) = V ′ ( Q ) = 0 V(Q)=V'(Q)=0 V ( Q ) = V ′ ( Q ) = 0 V V V 0 0 0 Q Q Q 0 0 0 1 1 1
The contact equations. For ( x , Q ) ∈ ( 0 , 1 ) 2 (x,Q)\in(0,1)^2 ( x , Q ) ∈ ( 0 , 1 ) 2 c ∈ R D c\in\mathbb{R}^D c ∈ R D β > 0 \beta>0 β > 0 C = 1 − Q + x Q > 0 C=1-Q+xQ>0 C = 1 − Q + x Q > 0
H 1 = V c , β , x , Q ( Q ) = 1 x 2 ( log C 1 − Q − x Q C ) − β 2 ξ c ( Q ) , H_1=V_{c,\beta,x,Q}(Q)=\frac{1}{x^2}\left(\log\frac{C}{1-Q}-\frac{xQ}{C}\right)-\beta^2\xi_c(Q), H 1 = V c , β , x , Q ( Q ) = x 2 1 ( log 1 − Q C − C x Q ) − β 2 ξ c ( Q ) , H 2 = V c , β , x , Q ′ ( Q ) = Q ( 1 − Q ) C − β 2 ξ c ′ ( Q ) . H_2=V'_{c,\beta,x,Q}(Q)=\frac{Q}{(1-Q)C}-\beta^2\xi'_c(Q). H 2 = V c , β , x , Q ′ ( Q ) = ( 1 − Q ) C Q − β 2 ξ c ′ ( Q ) . where we used the integrals in the proof of Proposition 5.11(1). These are real-analytic functions of ( x , Q , c , β ) (x,Q,c,\beta) ( x , Q , c , β ) J j = ∫ 0 Q ( Q − v ) j ( C − x v ) − 3 d v J_j=\int_0^Q(Q-v)^j(C-xv)^{-3}\,\mathrm{d}v J j = ∫ 0 Q ( Q − v ) j ( C − xv ) − 3 d v j = 0 , 1 , 2 j=0,1,2 j = 0 , 1 , 2 H 1 = ∫ 0 Q ( Q − v ) ( C − x v ) − 2 d v − β 2 ξ c ( Q ) H_1=\int_0^Q(Q-v)(C-xv)^{-2}\,\mathrm{d}v-\beta^2\xi_c(Q) H 1 = ∫ 0 Q ( Q − v ) ( C − xv ) − 2 d v − β 2 ξ c ( Q ) H 2 = ∫ 0 Q ( C − x v ) − 2 d v − β 2 ξ c ′ ( Q ) H_2=\int_0^Q(C-xv)^{-2}\,\mathrm{d}v-\beta^2\xi'_c(Q) H 2 = ∫ 0 Q ( C − xv ) − 2 d v − β 2 ξ c ′ ( Q ) ∂ x ( C − x v ) = Q − v \partial_x(C-xv)=Q-v ∂ x ( C − xv ) = Q − v ∂ Q ( C − x v ) = − ( 1 − x ) \partial_Q(C-xv)=-(1-x) ∂ Q ( C − xv ) = − ( 1 − x )
\begin{pmatrix}
\partial_xH_1 & \partial_QH_1\\
\partial_xH_2 & \partial_QH_2
\end{pmatrix}
=
\begin{pmatrix}
-2J_2 & H_2+2(1-x)J_1\\
-2J_1 & V''(Q)+2(1-x)J_0
\end{pmatrix}. \tag*{(109)}\end{pmatrix}
where V ′ ′ ( Q ) = ( 1 − Q ) − 2 − β 2 ξ c ′ ′ ( Q ) V''(Q)=(1-Q)^{-2}-\beta^2\xi''_c(Q) V ′′ ( Q ) = ( 1 − Q ) − 2 − β 2 ξ c ′′ ( Q ) ∂ Q H 1 \partial_QH_1 ∂ Q H 1 ( Q − v ) ( C − x v ) − 2 (Q-v)(C-xv)^{-2} ( Q − v ) ( C − xv ) − 2 v = Q v=Q v = Q ∂ Q H 2 \partial_QH_2 ∂ Q H 2 ( C − x Q ) − 2 = ( 1 − Q ) − 2 (C-xQ)^{-2}=(1-Q)^{-2} ( C − x Q ) − 2 = ( 1 − Q ) − 2 ( x ˉ , Q ˉ , c ˉ , β ˉ ) (\bar{x},\bar{Q},\bar{c},\bar{\beta}) ( x ˉ , Q ˉ , c ˉ , β ˉ ) H 2 = 0 H_2=0 H 2 = 0
− 2 J 2 V ′ ′ ( Q ˉ ) − 4 ( 1 − x ˉ ) ( J 0 J 2 − J 1 2 ) < 0. -2J_2V''(\bar{Q})-4(1-\bar{x})(J_0J_2-J_1^2)<0. − 2 J 2 V ′′ ( Q ˉ ) − 4 ( 1 − x ˉ ) ( J 0 J 2 − J 1 2 ) < 0. Here J 2 > 0 J_2>0 J 2 > 0 V ′ ′ ( Q ˉ ) > 0 V''(\bar{Q})>0 V ′′ ( Q ˉ ) > 0 J 0 J 2 > J 1 2 J_0J_2>J_1^2 J 0 J 2 > J 1 2 v ↦ Q ˉ − v v\mapsto\bar{Q}-v v ↦ Q ˉ − v U 1 U_1 U 1 ( c ˉ , β ˉ ) (\bar{c},\bar{\beta}) ( c ˉ , β ˉ ) x , Q x,Q x , Q U 1 U_1 U 1 ( 0 , 1 ) (0,1) ( 0 , 1 ) H 1 = H 2 = 0 H_1=H_2=0 H 1 = H 2 = 0 ( x ( c , β ) , Q ( c , β ) , c , β ) (x(c,\beta),Q(c,\beta),c,\beta) ( x ( c , β ) , Q ( c , β ) , c , β )
Positivity. For ( c , β ) ∈ U 1 (c,\beta)\in U_1 ( c , β ) ∈ U 1 c ∈ [ 0 , ∞ ) D ∖ { 0 } c\in[0,\infty)^D\setminus\{0\} c ∈ [ 0 , ∞ ) D ∖ { 0 } V ~ = V c , β , x ( c , β ) , Q ( c , β ) \widetilde{V}=V_{c,\beta,x(c,\beta),Q(c,\beta)} V = V c , β , x ( c , β ) , Q ( c , β ) Q ~ = Q ( c , β ) \widetilde{Q}=Q(c,\beta) Q = Q ( c , β ) V 0 V_0 V 0 V V V ( ξ ˉ c , β ˉ ) (\bar{\xi}_c,\bar{\beta}) ( ξ ˉ c , β ˉ ) U 1 U_1 U 1 U U U U 1 U_1 U 1 ∣ Q ( c , β ) − Q ˉ ∣ \lvert Q(c,\beta)-\bar{Q}\rvert ∣ Q ( c , β ) − Q ˉ ∣ β 2 ∑ p ∣ c p ∣ ≤ M \beta^2\sum_p\lvert c_p\rvert\le M β 2 ∑ p ∣ c p ∣ ≤ M M < ∞ M<\infty M < ∞ ( 0 , 1 ) (0,1) ( 0 , 1 )
Near the origin: since the function x ^ \hat{x} x ^ 0 < x ^ ≤ 1 0<\hat{x}\le1 0 < x ^ ≤ 1 ξ c ( r ) ≤ r 4 ∑ p c p \xi_c(r)\le r^4\sum_p c_p ξ c ( r ) ≤ r 4 ∑ p c p 0 ≤ r ≤ 1 0\le r\le1 0 ≤ r ≤ 1 V ~ ( r ) ≥ r 2 / 2 − M r 4 ≥ r 2 / 4 \widetilde{V}(r)\ge r^2/2-Mr^4\ge r^2/4 V ( r ) ≥ r 2 /2 − M r 4 ≥ r 2 /4 0 < r ≤ r 0 = ( 4 M ) − 1 / 2 0<r\le r_0=(4M)^{-1/2} 0 < r ≤ r 0 = ( 4 M ) − 1/2
Near the contact: let λ = V 0 ′ ′ ( Q ˉ ) > 0 \lambda=V_0''(\bar{Q})>0 λ = V 0 ′′ ( Q ˉ ) > 0 V ~ ′ ′ = x ^ − 2 − β 2 ξ c ′ ′ \widetilde{V}''=\hat{x}^{-2}-\beta^2\xi''_c V ′′ = x ^ − 2 − β 2 ξ c ′′ r r r [ 0 , 1 ) [0,1) [ 0 , 1 ) x ^ \hat{x} x ^ ( v , x , Q ) (v,x,Q) ( v , x , Q ) v < 1 v<1 v < 1 U U U δ > 0 \delta>0 δ > 0 V ~ ′ ′ ≥ λ / 2 \widetilde{V}''\ge\lambda/2 V ′′ ≥ λ /2 I = [ Q ˉ − 2 δ , Q ˉ + 2 δ ] I=[\bar{Q}-2\delta,\bar{Q}+2\delta] I = [ Q ˉ − 2 δ , Q ˉ + 2 δ ] ∣ Q ~ − Q ˉ ∣ ≤ δ \lvert\widetilde{Q}-\bar{Q}\rvert\le\delta ∣ Q − Q ˉ ∣ ≤ δ V ~ ( Q ~ ) = V ~ ′ ( Q ~ ) = 0 \widetilde{V}(\widetilde{Q})=\widetilde{V}'(\widetilde{Q})=0 V ( Q ) = V ′ ( Q ) = 0 V ~ ( r ) ≥ λ ( r − Q ~ ) 2 / 4 \widetilde{V}(r)\ge\lambda(r-\widetilde{Q})^2/4 V ( r ) ≥ λ ( r − Q ) 2 /4 r ∈ I r\in I r ∈ I r ≠ Q ~ r\ne\widetilde{Q} r = Q r ≥ Q ~ r \ge\widetilde{Q} r ≥ Q Proposition 5.11 (2),
V ~ ( r ) = − log 1 − r 1 − Q ~ − r − Q ~ 1 − Q ~ − β 2 ( ξ c ( r ) − ξ c ( Q ~ ) − ( r − Q ~ ) ξ c ′ ( Q ~ ) ) . \widetilde{V}(r)=-\log\frac{1-r}{1-\widetilde{Q}}-\frac{r-\widetilde{Q}}{1-\widetilde{Q}}-\beta^2\left(\xi_c(r)-\xi_c(\widetilde{Q})-(r-\widetilde{Q})\xi_c'(\widetilde{Q})\right). V ( r ) = − log 1 − Q 1 − r − 1 − Q r − Q − β 2 ( ξ c ( r ) − ξ c ( Q ) − ( r − Q ) ξ c ′ ( Q ) ) . All terms except − log ( 1 − r ) -\log(1-r) − log ( 1 − r ) U \mathcal{U} U r ∈ [ Q ~ , 1 ) r\in[\widetilde{Q},1) r ∈ [ Q , 1 ) M 1 M_1 M 1 V ~ ( r ) ≥ − log ( 1 − r ) − M 1 > 0 \widetilde{V}(r)\ge-\log(1-r)-M_1>0 V ( r ) ≥ − log ( 1 − r ) − M 1 > 0 1 − e − M 1 < r < 1 1-e^{-M_1}<r<1 1 − e − M 1 < r < 1 1 − e − M 1 > Q ‾ + 2 δ 1-e^{-M_1}>\overline{Q}+2\delta 1 − e − M 1 > Q + 2 δ
The remaining set K = [ r 0 , 1 − e − M 1 ] ∖ ( Q ‾ − δ , Q ‾ + δ ) K=[r_0,1-e^{-M_1}]\setminus(\overline{Q}-\delta,\overline{Q}+\delta) K = [ r 0 , 1 − e − M 1 ] ∖ ( Q − δ , Q + δ ) 0 0 0 Q ‾ \overline{Q} Q 1 1 1 V 0 V_0 V 0 m 0 m_0 m 0 K K K ( c , β , r ) ↦ V ~ ( r ) (c,\beta,r)\mapsto\widetilde{V}(r) ( c , β , r ) ↦ V ( r ) U \mathcal{U} U K K K U \mathcal{U} U V ~ ≥ m 0 / 2 \widetilde{V}\ge m_0/2 V ≥ m 0 /2 K K K
Since these four regions cover ( 0 , 1 ) (0,1) ( 0 , 1 ) V ~ > 0 \widetilde{V}>0 V > 0 ( 0 , 1 ) ∖ { Q ~ } (0,1)\setminus\{\widetilde{Q}\} ( 0 , 1 ) ∖ { Q } V ~ ( Q ~ ) = V ~ ′ ( Q ~ ) = 0 \widetilde{V}(\widetilde{Q})=\widetilde{V}'(\widetilde{Q})=0 V ( Q ) = V ′ ( Q ) = 0 V ~ ′ ′ ( Q ~ ) ≥ λ / 2 \widetilde{V}''(\widetilde{Q})\ge\lambda/2 V ′′ ( Q ) ≥ λ /2 ( ξ c , β ) (\xi_c,\beta) ( ξ c , β ) x ( c , β ) x(c,\beta) x ( c , β ) Q ( c , β ) Q(c,\beta) Q ( c , β )
The proof also shows that the margins r 2 / 4 r^2/4 r 2 /4 λ ( r − Q ~ ) 2 / 4 \lambda(r-\widetilde{Q})^2/4 λ ( r − Q ) 2 /4 m 0 / 2 m_0/2 m 0 /2 U \mathcal{U} U
Corollary 5.16 (Derivative of the free energy). Let ( ξ , β 0 ) (\xi,\beta_0) ( ξ , β 0 ) x , Q x,Q x , Q Q Q Q F F F β 0 \beta_0 β 0
F ′ ( β 0 ) = β 0 ( ξ ( 1 ) − ( 1 − x ) ξ ( Q ) ) . (110) F'(\beta_0)=\beta_0\left(\xi(1)-(1-x)\xi(Q)\right). \tag*{(110)} F ′ ( β 0 ) = β 0 ( ξ ( 1 ) − ( 1 − x ) ξ ( Q ) ) . (110) Moreover, F N ′ ( β 0 ) = N − 1 E π ( H N ) F_N'(\beta_0)=N^{-1}\mathbb{E}\,\pi(H_N) F N ′ ( β 0 ) = N − 1 E π ( H N ) N N N F N ′ ( β 0 ) → F ′ ( β 0 ) F_N'(\beta_0)\to F'(\beta_0) F N ′ ( β 0 ) → F ′ ( β 0 ) N → ∞ N\to\infty N → ∞
Proof. Let D \mathcal{D} D ξ \xi ξ c c c β \beta β β 0 \beta_0 β 0 ( ξ , β ) (\xi,\beta) ( ξ , β ) x ( β ) , Q ( β ) x(\beta),Q(\beta) x ( β ) , Q ( β ) F ( β ) = F ( x ( β ) , Q ( β ) , β ) F(\beta)=\mathcal{F}(x(\beta),Q(\beta),\beta) F ( β ) = F ( x ( β ) , Q ( β ) , β )
F ( x , Q , β ) = 1 2 ( β 2 ( ξ ( 1 ) − ( 1 − x ) ξ ( Q ) ) + 1 x log 1 − Q + x Q 1 − Q + log ( 1 − Q ) ) . \mathcal{F}(x,Q,\beta)=\frac{1}{2}\left(\beta^2\left(\xi(1)-(1-x)\xi(Q)\right)+\frac{1}{x}\log\frac{1-Q+xQ}{1-Q}+\log(1-Q)\right). F ( x , Q , β ) = 2 1 ( β 2 ( ξ ( 1 ) − ( 1 − x ) ξ ( Q ) ) + x 1 log 1 − Q 1 − Q + x Q + log ( 1 − Q ) ) . This proves analyticity. By direct differentiation, we have
∂ x F = − 1 2 V x , Q ( Q ) , ∂ Q F = 1 2 ( 1 − x ) V x , Q ′ ( Q ) , \partial_x\mathcal{F}=-\frac{1}{2}V_{x,Q}(Q),\qquad\partial_Q\mathcal{F}=\frac{1}{2}(1-x)V'_{x,Q}(Q), ∂ x F = − 2 1 V x , Q ( Q ) , ∂ Q F = 2 1 ( 1 − x ) V x , Q ′ ( Q ) , where V x , Q V_{x,Q} V x , Q x , Q x,Q x , Q β \beta β H 1 H_1 H 1 H 2 H_2 H 2 F ′ ( β 0 ) = ∂ β F ( x , Q , β 0 ) F'(\beta_0)=\partial_\beta\mathcal{F}(x,Q,\beta_0) F ′ ( β 0 ) = ∂ β F ( x , Q , β 0 )
For each N N N F N F_N F N F N ′ ( β ) = N − 1 E π ( H N ) F_N'(\beta)=N^{-1}\mathbb{E}\,\pi(H_N) F N ′ ( β ) = N − 1 E π ( H N ) sup S N ∣ H N ∣ \sup_{S_N}|H_N| sup S N ∣ H N ∣ h > 0 h>0 h > 0
F N ( β 0 ) − F N ( β 0 − h ) h ≤ F N ′ ( β 0 ) ≤ F N ( β 0 + h ) − F N ( β 0 ) h . \frac{F_N(\beta_0)-F_N(\beta_0-h)}{h}\le F_N'(\beta_0)\le\frac{F_N(\beta_0+h)-F_N(\beta_0)}{h}. h F N ( β 0 ) − F N ( β 0 − h ) ≤ F N ′ ( β 0 ) ≤ h F N ( β 0 + h ) − F N ( β 0 ) . By Theorem 3.14 , F N → F F_N \to F F N → F N → ∞ N \to\infty N → ∞ h ↓ 0 h \downarrow0 h ↓ 0 F F F β 0 \beta_0 β 0 F N ′ ( β 0 ) → F ′ ( β 0 ) F_N'(\beta_0) \to F'(\beta_0) F N ′ ( β 0 ) → F ′ ( β 0 ) □ \square □
Classes of strictly 1RSB covariances
Proposition 5.17 (Strict convexity of ξ ′ ′ − 1 / 2 \xi''^{-1/2} ξ ′′ − 1/2 ξ \xi ξ r ↦ ξ ′ ′ ( r ) − 1 / 2 r \mapsto\xi''(r)^{-1/2} r ↦ ξ ′′ ( r ) − 1/2 ( 0 , 1 ] (0,1] ( 0 , 1 ] β > β s ( ξ ) \beta> \beta_s(\xi) β > β s ( ξ ) ( ξ , β ) (\xi,\beta) ( ξ , β ) Q Q Q x , Q x,Q x , Q ζ β = x δ 0 + ( 1 − x ) δ Q \zeta_\beta= x\delta_0 + (1-x)\delta_Q ζ β = x δ 0 + ( 1 − x ) δ Q
The argument that bounds the number of contacts is that of Crisanti and Sommers [52], as presented by Talagrand [146, Proposition 2.2], who treats even mixtures. The optimality criterion of Lemma 3.15 holds for all finite mixtures, and the argument applies to odd degrees as well; we include it with the strictness made explicit. The contact-count argument also appears in the rule of signs of [97, Theorem 1.3(1) and Remark 1.4] when ( ξ ′ ′ − 1 / 2 ) ′ ′ > 0 (\xi''^{-1/2})'' > 0 ( ξ ′′ − 1/2 ) ′′ > 0
Proof. Fix β > β s ( ξ ) \beta> \beta_s(\xi) β > β s ( ξ ) ζ = ζ β \zeta= \zeta_\beta ζ = ζ β x ^ = x ^ β \hat{x} = \hat{x}_\beta x ^ = x ^ β V = V β V = V_\beta V = V β Lemma 5.6 , V ≥ 0 V \ge0 V ≥ 0 [ 0 , 1 ) [0,1) [ 0 , 1 ) ζ \zeta ζ 0 0 0
D ( r ) = ( β 2 ξ ′ ′ ( r ) ) − 1 / 2 − x ^ ( r ) , 0 < r < 1. D(r) = \left(\beta^2 \xi''(r)\right)^{-1/2} - \hat{x}(r), \qquad0 < r < 1. D ( r ) = ( β 2 ξ ′′ ( r ) ) − 1/2 − x ^ ( r ) , 0 < r < 1. The function x ^ \hat{x} x ^ − ζ ( [ 0 , r ] ) -\zeta([0,r]) − ζ ([ 0 , r ]) D D D ( 0 , 1 ) (0,1) ( 0 , 1 ) (80) , for 0 < r < 1 0 < r < 1 0 < r < 1 V ′ ′ ( r ) = 0 V''(r) = 0 V ′′ ( r ) = 0 D ( r ) = 0 D(r) = 0 D ( r ) = 0
At most two contacts. Let a < b a < b a < b V ( a ) = V ( b ) = 0 V(a) = V(b) = 0 V ( a ) = V ( b ) = 0 V ′ ( a ) = V ′ ( b ) = 0 V'(a) = V'(b) = 0 V ′ ( a ) = V ′ ( b ) = 0 a = 0 a = 0 a = 0 (80) , and otherwise it holds because V ≥ 0 V \ge0 V ≥ 0 a a a b b b V V V [ a , b ] [a,b] [ a , b ] V ′ ′ = 0 V'' = 0 V ′′ = 0 ( a , b ) (a,b) ( a , b ) D D D V V V [ a , b ] [a,b] [ a , b ] c ∈ ( a , b ) c \in(a,b) c ∈ ( a , b ) V ′ ( c ) = 0 V'(c) = 0 V ′ ( c ) = 0 V ′ V' V ′ [ a , c ] [a,c] [ a , c ] [ c , b ] [c,b] [ c , b ] V ′ ′ V'' V ′′ ( a , c ) (a,c) ( a , c ) ( c , b ) (c,b) ( c , b ) D D D ( a , b ) (a,b) ( a , b ) a < b < b ′ a < b < b' a < b < b ′ D D D ( a , b ) (a,b) ( a , b ) ( b , b ′ ) (b,b') ( b , b ′ )
Two atoms. As β > β s ( ξ ) \beta> \beta_s(\xi) β > β s ( ξ ) Lemma 3.18 (1) implies that ζ ≠ δ 0 \zeta\ne\delta_0 ζ = δ 0 ζ \zeta ζ 0 0 0 { 0 } \{0\} { 0 } ζ = x δ 0 + ( 1 − x ) δ Q \zeta= x\delta_0 + (1-x)\delta_Q ζ = x δ 0 + ( 1 − x ) δ Q x , Q ∈ ( 0 , 1 ) x,Q \in(0,1) x , Q ∈ ( 0 , 1 ) { 0 , Q } \{0,Q\} { 0 , Q } x ^ \hat{x} x ^ (92) with these parameters, V V V V ( Q ) = V ′ ( Q ) = 0 V(Q)=V'(Q)=0 V ( Q ) = V ′ ( Q ) = 0 V > 0 V>0 V > 0 ( 0 , 1 ) ∖ { Q } (0,1)\setminus\{Q\} ( 0 , 1 ) ∖ { Q }
Strict curvature at Q Q Q Since Q Q Q V V V V ′ ′ ( Q ) ≥ 0 V''(Q)\geq0 V ′′ ( Q ) ≥ 0 V ′ ′ ( Q ) = 0 V''(Q)=0 V ′′ ( Q ) = 0 D ( Q ) = 0 D(Q)=0 D ( Q ) = 0 0 < Q 0<Q 0 < Q D D D ( 0 , Q ) (0,Q) ( 0 , Q ) V ′ ′ ( Q ) > 0 V''(Q)>0 V ′′ ( Q ) > 0 ( ξ , β ) (\xi,\beta) ( ξ , β )
The next proposition gives a criterion for the convexity hypothesis of Proposition 5.17 that involves only the smallest and the largest supported degrees.
Proposition 5.18 (A condition on the smallest and largest degrees). Let ξ \xi ξ d ≥ 4 d\geq4 d ≥ 4 P P P
6 ( d − 2 ) ( P − 2 ) > ( d + P − 5 ) 2 , (111) 6(d-2)(P-2)>(d+P-5)^2, \tag*{(111)} 6 ( d − 2 ) ( P − 2 ) > ( d + P − 5 ) 2 , (111) which is equivalent to P − d < d − 1 + 3 d ( d − 2 ) P-d<d-1+\sqrt{3d(d-2)} P − d < d − 1 + 3 d ( d − 2 ) ( ξ ′ ′ − 1 / 2 ) ′ ′ > 0 (\xi''^{-1/2})''>0 ( ξ ′′ − 1/2 ) ′′ > 0 ( 0 , 1 ] (0,1] ( 0 , 1 ] ξ ′ ′ − 1 / 2 \xi''^{-1/2} ξ ′′ − 1/2 ( 0 , 1 ] (0,1] ( 0 , 1 ] ( ξ , β ) (\xi,\beta) ( ξ , β ) β > β s ( ξ ) \beta>\beta_s(\xi) β > β s ( ξ ) ξ \xi ξ { 4 , 5 , … , 11 } \{4,5,\ldots,11\} { 4 , 5 , … , 11 } { 6 , 7 , … , 18 } \{6,7,\ldots,18\} { 6 , 7 , … , 18 } { 8 , 9 , … , 26 } \{8,9,\ldots,26\} { 8 , 9 , … , 26 }
The proof applies the Bhatia–Davis bound Var ( I ) ≤ ( m − n ) ( M − m ) \operatorname{Var}(I)\leq(m-n)(M-m) Var ( I ) ≤ ( m − n ) ( M − m ) I I I [ n , M ] [n,M] [ n , M ] m m m
Proof. We write v ( r ) = ξ ′ ′ ( r ) = ∑ i a i r i v(r)=\xi''(r)=\sum_i a_i r^i v ( r ) = ξ ′′ ( r ) = ∑ i a i r i i = p − 2 i=p-2 i = p − 2 a i = p ( p − 1 ) c p > 0 a_i=p(p-1)c_p>0 a i = p ( p − 1 ) c p > 0 r ∈ ( 0 , 1 ] r\in(0,1] r ∈ ( 0 , 1 ] I I I P ( I = i ) = a i r i / v ( r ) \mathbb{P}(I=i)=a_i r^i/v(r) P ( I = i ) = a i r i / v ( r ) m m m σ 2 \sigma^2 σ 2
r v ′ ( r ) v ( r ) = m , r 2 v ′ ′ ( r ) v ( r ) = E [ I ( I − 1 ) ] = m 2 + σ 2 − m , \frac{rv'(r)}{v(r)}=m,\qquad\frac{r^2v''(r)}{v(r)}=\mathbb{E}[I(I-1)]=m^2+\sigma^2-m, v ( r ) r v ′ ( r ) = m , v ( r ) r 2 v ′′ ( r ) = E [ I ( I − 1 )] = m 2 + σ 2 − m , and therefore
r 2 v ( r ) 2 ( 3 v ′ ( r ) 2 − 2 v ( r ) v ′ ′ ( r ) ) = 3 m 2 − 2 ( m 2 + σ 2 − m ) = m 2 + 2 m − 2 σ 2 . (112) \frac{r^2}{v(r)^2}\left(3v'(r)^2-2v(r)v''(r)\right)=3m^2-2(m^2+\sigma^2-m)=m^2+2m-2\sigma^2. \tag*{(112)} v ( r ) 2 r 2 ( 3 v ′ ( r ) 2 − 2 v ( r ) v ′′ ( r ) ) = 3 m 2 − 2 ( m 2 + σ 2 − m ) = m 2 + 2 m − 2 σ 2 . (112) Let n = d − 2 ≥ 2 n=d-2\geq2 n = d − 2 ≥ 2 M = P − 2 M=P-2 M = P − 2 n ≤ I ≤ M n\leq I\leq M n ≤ I ≤ M 0 ≤ E [ ( I − n ) ( M − I ) ] = ( m − n ) ( M − m ) − σ 2 0\leq\mathbb{E}[(I-n)(M-I)]=(m-n)(M-m)-\sigma^2 0 ≤ E [( I − n ) ( M − I )] = ( m − n ) ( M − m ) − σ 2
f ( m ) = m 2 + 2 m − 2 ( m − n ) ( M − m ) = 3 m 2 − 2 ( n + M − 1 ) m + 2 n M , n ≤ m ≤ M . f(m)=m^2+2m-2(m-n)(M-m)=3m^2-2(n+M-1)m+2nM,\qquad n\leq m\leq M. f ( m ) = m 2 + 2 m − 2 ( m − n ) ( M − m ) = 3 m 2 − 2 ( n + M − 1 ) m + 2 n M , n ≤ m ≤ M . The vertex of this quadratic is at m ∗ = ( n + M − 1 ) / 3 m_*=(n+M-1)/3 m ∗ = ( n + M − 1 ) /3 M ≤ 2 n + 1 M\leq2n+1 M ≤ 2 n + 1 m ∗ ≤ n m_*\leq n m ∗ ≤ n f ( m ) ≥ f ( n ) = n ( n + 2 ) > 0 f(m)\geq f(n)=n(n+2)>0 f ( m ) ≥ f ( n ) = n ( n + 2 ) > 0 [ n , M ] [n,M] [ n , M ] m ∗ ∈ ( n , M ) m_*\in(n,M) m ∗ ∈ ( n , M ) [ n , M ] [n,M] [ n , M ] f ( m ∗ ) = 2 n M − ( n + M − 1 ) 2 / 3 f(m_\ast)=2nM-(n+M-1)^2/3 f ( m ∗ ) = 2 n M − ( n + M − 1 ) 2 /3 (111) . In both cases, 3 v ′ 2 − 2 v v ′ ′ > 0 3v'^2-2vv''>0 3 v ′2 − 2 v v ′′ > 0 ( 0 , 1 ] (0,1] ( 0 , 1 ]
( v − 1 / 2 ) ′ ′ = 3 v ′ 2 − 2 v v ′ ′ 4 v 5 / 2 > 0 on ( 0 , 1 ] . \left(v^{-1/2}\right)''=\frac{3v'^2-2vv''}{4v^{5/2}}>0 \qquad\text{on }(0,1]. ( v − 1/2 ) ′′ = 4 v 5/2 3 v ′2 − 2 v v ′′ > 0 on ( 0 , 1 ] . A function with positive second derivative is strictly convex, and Proposition 5.17 implies that the 1RSB assumptions hold for every β > β s ( ξ ) \beta>\beta_s(\xi) β > β s ( ξ )
For the equivalent form, let k = P − d k=P-d k = P − d
6 ( d − 2 ) ( P − 2 ) − ( d + P − 5 ) 2 = 3 d ( d − 2 ) − ( k − ( d − 1 ) ) 2 . 6(d-2)(P-2)-(d+P-5)^2=3d(d-2)-(k-(d-1))^2. 6 ( d − 2 ) ( P − 2 ) − ( d + P − 5 ) 2 = 3 d ( d − 2 ) − ( k − ( d − 1 ) ) 2 . Since d − 1 − 3 d ( d − 2 ) < 0 ≤ k d-1-\sqrt{3d(d-2)}<0\leq k d − 1 − 3 d ( d − 2 ) < 0 ≤ k d ≥ 4 d\geq4 d ≥ 4 k < d − 1 + 3 d ( d − 2 ) k<d-1+\sqrt{3d(d-2)} k < d − 1 + 3 d ( d − 2 ) h ( d ) = d − 1 + 3 d ( d − 2 ) h(d)=d-1+\sqrt{3d(d-2)} h ( d ) = d − 1 + 3 d ( d − 2 ) d d d [ d 1 , P 1 ] [d_1,P_1] [ d 1 , P 1 ] d 1 ≤ d ≤ P ≤ P 1 d_1\leq d\leq P\leq P_1 d 1 ≤ d ≤ P ≤ P 1 P − d ≤ P 1 − d 1 P-d\leq P_1-d_1 P − d ≤ P 1 − d 1 P 1 − d 1 < h ( d 1 ) P_1-d_1<h(d_1) P 1 − d 1 < h ( d 1 ) ( d 1 , P 1 ) = ( 4 , 11 ) , ( 6 , 18 ) , ( 8 , 26 ) (d_1,P_1)=(4,11),(6,18),(8,26) ( d 1 , P 1 ) = ( 4 , 11 ) , ( 6 , 18 ) , ( 8 , 26 ) P 1 − d 1 < h ( d 1 ) P_1-d_1<h(d_1) P 1 − d 1 < h ( d 1 )
We now turn to pure models, whose static parameters Subag computed rigorously from the generalized TAP representation, without the Parisi formula (see Remark 5.21). For z > 0 z>0 z > 0
v ( z ) = ( 1 + z ) log ( 1 + z ) − z z 2 = ∫ 0 1 1 − t 1 + z t d t . (113) v(z)=\frac{(1+z)\log(1+z)-z}{z^2}=\int_0^1\frac{1-t}{1+zt}\,dt. \tag*{(113)} v ( z ) = z 2 ( 1 + z ) log ( 1 + z ) − z = ∫ 0 1 1 + z t 1 − t d t . (113) The second expression follows by computing the integral. Differentiating under the integral, we see that v v v v ( 0 + ) = 1 / 2 v(0+)=1/2 v ( 0 + ) = 1/2 v ( ∞ ) = 0 v(\infty)=0 v ( ∞ ) = 0 z z z v ( z ) = ξ ( Q ) / a Q v(z)=\xi(Q)/a_Q v ( z ) = ξ ( Q ) / a Q
Proposition 5.20 (Pure models). Let p ≥ 4 p \ge4 p ≥ 4 ξ ( r ) = r p \xi(r)=r^p ξ ( r ) = r p z p > 0 z_p>0 z p > 0 υ ( z p ) = 1 / p \upsilon(z_p)=1/p υ ( z p ) = 1/ p
q s = z p 1 + z p , e p = p + z p p ( 1 + z p ) . q_s=\frac{z_p}{1+z_p},\qquad e_p=\frac{p+z_p}{\sqrt{p(1+z_p)}}. q s = 1 + z p z p , e p = p ( 1 + z p ) p + z p . We have z p > 2 ( p − 2 ) z_p>2(p-2) z p > 2 ( p − 2 ) q s q_s q s r ↦ ( − r − log ( 1 − r ) ) / r p r\mapsto(-r-\log(1-r))/r^p r ↦ ( − r − log ( 1 − r )) / r p ( 0 , 1 ) (0,1) ( 0 , 1 )
β s ( p ) 2 = 1 p q s p − 2 ( 1 − q s ) . \beta_s(p)^2=\frac{1}{p q_s^{p-2}(1-q_s)}. β s ( p ) 2 = p q s p − 2 ( 1 − q s ) 1 . Let β > β s ( p ) \beta>\beta_s(p) β > β s ( p ) ( ξ , β ) (\xi,\beta) ( ξ , β ) Q Q Q ( q s , 1 ) (q_s,1) ( q s , 1 )
β 2 p ( 1 + z p ) Q p − 2 ( 1 − Q ) 2 = 1 , (114) \beta^2p(1+z_p)Q^{p-2}(1-Q)^2=1, \tag*{(114)} β 2 p ( 1 + z p ) Q p − 2 ( 1 − Q ) 2 = 1 , (114) and x = z p ( 1 − Q ) / Q x=z_p(1-Q)/Q x = z p ( 1 − Q ) / Q V ′ ′ ( Q ) = ( 1 − Q ) − 2 ( 1 − ( p − 1 ) / ( 1 + z p ) ) V''(Q)=(1-Q)^{-2}(1-(p-1)/(1+z_p)) V ′′ ( Q ) = ( 1 − Q ) − 2 ( 1 − ( p − 1 ) / ( 1 + z p ))
We have E ( q ) = e p q p / 2 E(q)=e_pq^{p/2} E ( q ) = e p q p /2 0 ≤ q ≤ 1 0\le q\le1 0 ≤ q ≤ 1 E 0 = e p Q p / 2 E_0=e_pQ^{p/2} E 0 = e p Q p /2 K 0 = p E 0 K_0=pE_0 K 0 = p E 0
F ( β ) = β e p Q p / 2 + 1 2 log ( 1 − Q ) + β 2 2 ( 1 − Q p − p Q p − 1 ( 1 − Q ) ) . F(\beta)=\beta e_pQ^{p/2}+\frac{1}{2}\log(1-Q)+\frac{\beta^2}{2}(1-Q^p-pQ^{p-1}(1-Q)). F ( β ) = β e p Q p /2 + 2 1 log ( 1 − Q ) + 2 β 2 ( 1 − Q p − p Q p − 1 ( 1 − Q )) . Proof. Recall that υ \upsilon υ 1 / 2 1/2 1/2 0 0 0 1 / p ∈ ( 0 , 1 / 2 ) 1/p\in(0,1/2) 1/ p ∈ ( 0 , 1/2 ) z p z_p z p
The bound on $z_p. Let m = p − 2 ≥ 2 m=p-2\ge2 m = p − 2 ≥ 2 υ \upsilon υ z p > 2 m z_p>2m z p > 2 m υ ( 2 m ) > 1 / ( m + 2 ) \upsilon(2m)>1/(m+2) υ ( 2 m ) > 1/ ( m + 2 ) h 0 ( m ) > 0 h_0(m)>0 h 0 ( m ) > 0
h 0 ( m ) = log ( 2 m + 1 ) − 2 m ( 3 m + 2 ) ( m + 2 ) ( 2 m + 1 ) . h_0(m)=\log(2m+1)-\frac{2m(3m+2)}{(m+2)(2m+1)}. h 0 ( m ) = log ( 2 m + 1 ) − ( m + 2 ) ( 2 m + 1 ) 2 m ( 3 m + 2 ) . A direct computation gives h 0 ′ ( m ) = 4 m 2 ( m − 1 ) / ( ( m + 2 ) 2 ( 2 m + 1 ) 2 ) > 0 h_0'(m)=4m^2(m-1)/((m+2)^2(2m+1)^2)>0 h 0 ′ ( m ) = 4 m 2 ( m − 1 ) / (( m + 2 ) 2 ( 2 m + 1 ) 2 ) > 0 m > 1 m>1 m > 1 m = 2 m=2 m = 2 log 5 = 2 atanh ( 2 / 3 ) \log5=2\operatorname{atanh}(2/3) log 5 = 2 atanh ( 2/3 ) atanh \operatorname{atanh} atanh
log 5 > 2 ( 2 3 + ( 2 / 3 ) 3 3 + ( 2 / 3 ) 5 5 + ( 2 / 3 ) 7 7 ) = 8 5 + 4 15309 , \log5>2\left(\frac{2}{3}+\frac{(2/3)^3}{3}+\frac{(2/3)^5}{5}+\frac{(2/3)^7}{7}\right)=\frac{8}{5}+\frac{4}{15309}, log 5 > 2 ( 3 2 + 3 ( 2/3 ) 3 + 5 ( 2/3 ) 5 + 7 ( 2/3 ) 7 ) = 5 8 + 15309 4 , that is, h 0 ( 2 ) = log 5 − 8 / 5 > 0 h_0(2)=\log5-8/5>0 h 0 ( 2 ) = log 5 − 8/5 > 0 h 0 h_0 h 0 h 0 ( m ) > 0 h_0(m)>0 h 0 ( m ) > 0 m ≥ 2 m\ge2 m ≥ 2 z p > 2 ( p − 2 ) z_p>2(p-2) z p > 2 ( p − 2 )
The static threshold. The derivative of ( − r − log ( 1 − r ) ) / r p (-r-\log(1-r))/r^p ( − r − log ( 1 − r )) / r p D p ( r ) = r 2 / ( 1 − r ) − p ( − r − log ( 1 − r ) ) D_p(r)=r^2/(1-r)-p(-r-\log(1-r)) D p ( r ) = r 2 / ( 1 − r ) − p ( − r − log ( 1 − r )) r = z / ( 1 + z ) r=z/(1+z) r = z / ( 1 + z ) ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) ( 0 , 1 ) (0,1) ( 0 , 1 ) r 2 / ( 1 − r ) = z 2 / ( 1 + z ) r^2/(1-r)=z^2/(1+z) r 2 / ( 1 − r ) = z 2 / ( 1 + z ) − r − log ( 1 − r ) = z 2 υ ( z ) / ( 1 + z ) -r-\log(1-r)=z^2\upsilon(z)/(1+z) − r − log ( 1 − r ) = z 2 υ ( z ) / ( 1 + z )
D p ( z 1 + z ) = z 2 1 + z ( 1 − p υ ( z ) ) . D_p\left(\frac{z}{1+z}\right)=\frac{z^2}{1+z}(1-p\upsilon(z)). D p ( 1 + z z ) = 1 + z z 2 ( 1 − p υ ( z )) . This is negative for z < z p z<z_p z < z p z > z p z>z_p z > z p q s q_s q s D p ( q s ) = 0 D_p(q_s)=0 D p ( q s ) = 0 q s 2 / ( p ( 1 − q s ) q s p ) q_s^2/(p(1-q_s)q_s^p) q s 2 / ( p ( 1 − q s ) q s p ) β s ( p ) 2 \beta_s(p)^2 β s ( p ) 2
The 1RSB assumptions. The function ξ ′ ′ ( r ) − 1 / 2 = ( p ( p − 1 ) ) − 1 / 2 r − ( p − 2 ) / 2 \xi^{\prime\prime}(r)^{-1/2} = (p(p-1))^{-1/2}r^{-(p-2)/2} ξ ′′ ( r ) − 1/2 = ( p ( p − 1 ) ) − 1/2 r − ( p − 2 ) /2 ( 0 , 1 ] (0,1] ( 0 , 1 ] Proposition 5.17 implies that ( ξ , β ) (\xi,\beta) ( ξ , β ) β > β s ( p ) \beta> \beta_s(p) β > β s ( p ) x , Q x,Q x , Q (100) and ξ ( Q ) / a Q = 1 / p \xi(Q)/a_Q = 1/p ξ ( Q ) / a Q = 1/ p υ ( z ) = 1 / p \upsilon(z) = 1/p υ ( z ) = 1/ p z = x Q / ( 1 − Q ) = z p z = xQ/(1-Q) = z_p z = x Q / ( 1 − Q ) = z p x = z p ( 1 − Q ) / Q x = z_p(1-Q)/Q x = z p ( 1 − Q ) / Q C = ( 1 − Q ) ( 1 + z p ) C = (1-Q)(1+z_p) C = ( 1 − Q ) ( 1 + z p ) (95) becomes (114) . We have Q > z p / ( 1 + z p ) = q s Q > z_p/(1+z_p) = q_s Q > z p / ( 1 + z p ) = q s x < 1 x < 1 x < 1 (114) has only one solution in ( q s , 1 ) (q_s,1) ( q s , 1 ) r ↦ r p − 2 ( 1 − r ) 2 r \mapsto r^{p-2}(1-r)^2 r ↦ r p − 2 ( 1 − r ) 2 ( ( p − 2 ) / p , 1 ) ((p-2)/p,1) (( p − 2 ) / p , 1 ) q s > ( p − 2 ) / p q_s > (p-2)/p q s > ( p − 2 ) / p z p > 2 ( p − 2 ) z_p > 2(p-2) z p > 2 ( p − 2 ) q s > 2 ( p − 2 ) / ( 2 p − 3 ) > ( p − 2 ) / p q_s > 2(p-2)/(2p-3) > (p-2)/p q s > 2 ( p − 2 ) / ( 2 p − 3 ) > ( p − 2 ) / p (114) , we have β 2 p ( p − 1 ) Q p − 2 = ( p − 1 ) / ( ( 1 + z p ) ( 1 − Q ) 2 ) \beta^2p(p-1)Q^{p-2} = (p-1)/((1+z_p)(1-Q)^2) β 2 p ( p − 1 ) Q p − 2 = ( p − 1 ) / (( 1 + z p ) ( 1 − Q ) 2 ) V ′ ′ ( Q ) = ( 1 − Q ) − 2 − β 2 ξ ′ ′ ( Q ) V^{\prime\prime}(Q) = (1-Q)^{-2} - \beta^2\xi^{\prime\prime}(Q) V ′′ ( Q ) = ( 1 − Q ) − 2 − β 2 ξ ′′ ( Q ) z p > p − 2 z_p > p-2 z p > p − 2
Energies and free energy. For ξ = r p \xi= r^p ξ = r p ξ ( Q ) = Q p \xi(Q) = Q^p ξ ( Q ) = Q p a Q = p Q p a_Q = pQ^p a Q = p Q p b Q = p ( p − 1 ) Q p b_Q = p(p-1)Q^p b Q = p ( p − 1 ) Q p y = ( 1 + z p ) a Q = p ( 1 + z p ) Q p / 2 y = \sqrt{(1+z_p)a_Q} = \sqrt{p(1+z_p)}Q^{p/2} y = ( 1 + z p ) a Q = p ( 1 + z p ) Q p /2 (98) , we find E 0 = ( p + z p ) Q p / y = e p Q p / 2 E_0 = (p+z_p)Q^p/y = e_pQ^{p/2} E 0 = ( p + z p ) Q p / y = e p Q p /2 K 0 = p E 0 K_0 = pE_0 K 0 = p E 0 Proposition 5.13 (2). By homogeneity, max u ∈ S N H N ( q u ) = q p / 2 max u ∈ S N H N ( u ) \max_{u\in S_N} H_N(\sqrt{q}\,u) = q^{p/2}\max_{u\in S_N}H_N(u) max u ∈ S N H N ( q u ) = q p /2 max u ∈ S N H N ( u ) q ∈ ( 0 , 1 ] q\in(0,1] q ∈ ( 0 , 1 ] N N N E ( q ) = q p / 2 E ( 1 ) E(q) = q^{p/2}E(1) E ( q ) = q p /2 E ( 1 ) q = Q q = Q q = Q E ( 1 ) = e p E(1) = e_p E ( 1 ) = e p E ( q ) = e p q p / 2 E(q) = e_pq^{p/2} E ( q ) = e p q p /2 0 ≤ q ≤ 1 0 \leq q \leq1 0 ≤ q ≤ 1 F ( β ) F(\beta) F ( β ) (102) , since ψ Q ( 1 ) = 1 − Q p − p Q p − 1 ( 1 − Q ) \psi_Q(1) = 1-Q^p-pQ^{p-1}(1-Q) ψ Q ( 1 ) = 1 − Q p − p Q p − 1 ( 1 − Q )
Remark 5.21 (Subag’s formulas for pure models). Subag obtained the formulas of Proposition 5.20 for the static overlap, the static threshold, the ground-state energy, the largest multisamplable overlap, and the free energy at all temperatures, from the generalized TAP representation and without the Parisi formula [142, Theorems 1 and 2, (1.9)–(1.13)]. The free energy, the ground-state energy, and the 1RSB parameters had been derived earlier in physics, by the replica method and the TAP approach [52, 53, 108]; see [142, Section 1.1]. His critical overlap q c q_c q c q s q_s q s D p ( q s ) = 0 D_p(q_s) = 0 D p ( q s ) = 0 1 − q s 1-q_s 1 − q s p ( 1 − q s ) log ( 1 − q s ) + p q s − ( p − 1 ) q s 2 = 0 p(1-q_s)\log(1-q_s)+pq_s-(p-1)q_s^2=0 p ( 1 − q s ) log ( 1 − q s ) + p q s − ( p − 1 ) q s 2 = 0 q s 1 − p / 2 ( p ( 1 − q s ) ) − 1 / 2 = β s ( p ) q_s^{1-p/2}(p(1-q_s))^{-1/2} = \beta_s(p) q s 1 − p /2 ( p ( 1 − q s ) ) − 1/2 = β s ( p ) e p e_p e p (114) , where z p > p − 2 z_p > p-2 z p > p − 2 F ( β ) F(\beta) F ( β ) Proposition 5.20 (3). Our derivation of these parameters from the 1RSB assumptions, through Propositions 5.11 and 5.17 , is independent of his argument. What the later sections need beyond his formulas is the strict 1RSB structure with a nondegenerate contact, in particular the strict inequalities V > 0 V > 0 V > 0 ( 0 , 1 ) ∖ { Q } (0,1)\setminus\{Q\} ( 0 , 1 ) ∖ { Q } V ′ ′ ( Q ) > 0 V^{\prime\prime}(Q) > 0 V ′′ ( Q ) > 0
The next bound is used in Section 9 to control the variance of the residual field at latitudes a ≥ Q a \geq\sqrt{Q} a ≥ Q
Lemma 5.22 (A bound on the pure residual coefficient). Let p ≥ 4 p \geq4 p ≥ 4 ξ ( r ) = r p \xi(r) = r^p ξ ( r ) = r p β > β s ( p ) \beta> \beta_s(p) β > β s ( p ) Q Q Q Proposition 5.20 . For 0 ≤ a ≤ 1 0 \leq a \leq1 0 ≤ a ≤ 1
C β ( a ) = β 2 ( 1 − a 2 p − p a 2 p − 2 ( 1 − a 2 ) ) . C_\beta(a) = \beta^2\left(1-a^{2p}-pa^{2p-2}(1-a^2)\right). C β ( a ) = β 2 ( 1 − a 2 p − p a 2 p − 2 ( 1 − a 2 ) ) . Then C β ( Q ) = β 2 ψ Q ( 1 ) C_\beta(\sqrt{Q})=\beta^2\psi_Q(1) C β ( Q ) = β 2 ψ Q ( 1 )
C β ( Q ) < 159 320 < 1 2 , C β ′ ( a ) = − 2 β 2 p ( p − 1 ) a 2 p − 3 ( 1 − a 2 ) ≤ 0. (115) C_\beta(\sqrt{Q})<\frac{159}{320}<\frac{1}{2},\qquad C_\beta'(a)=-2\beta^2p(p-1)a^{2p-3}(1-a^2)\leq0. \tag*{(115)} C β ( Q ) < 320 159 < 2 1 , C β ′ ( a ) = − 2 β 2 p ( p − 1 ) a 2 p − 3 ( 1 − a 2 ) ≤ 0. (115) Proof. The identity C β ( Q ) = β 2 ψ Q ( 1 ) C_\beta(\sqrt{Q})=\beta^2\psi_Q(1) C β ( Q ) = β 2 ψ Q ( 1 ) ξ ~ = r p \tilde{\xi}=r^p ξ ~ = r p m = p − 2 m=p-2 m = p − 2 z = z p z=z_p z = z p t = ( 1 − Q ) / Q t=(1-Q)/Q t = ( 1 − Q ) / Q ψ Q ( 1 ) = Q p ∑ k = 2 p ( p k ) t k \psi_Q(1)=Q^p\sum_{k=2}^p\binom{p}{k}t^k ψ Q ( 1 ) = Q p ∑ k = 2 p ( k p ) t k β 2 = 1 / ( p ( 1 + z ) Q p − 2 ( 1 − Q ) 2 ) \beta^2=1/(p(1+z)Q^{p-2}(1-Q)^2) β 2 = 1/ ( p ( 1 + z ) Q p − 2 ( 1 − Q ) 2 )
C β ( Q ) = Q 2 p ( 1 + z ) ( 1 − Q ) 2 ∑ k = 2 p ( p k ) t k = 1 p ( 1 + z ) ∑ k = 2 p ( p k ) t k − 2 . C_\beta(\sqrt{Q})=\frac{Q^2}{p(1+z)(1-Q)^2}\sum_{k=2}^p\binom{p}{k}t^k=\frac{1}{p(1+z)}\sum_{k=2}^p\binom{p}{k}t^{k-2}. C β ( Q ) = p ( 1 + z ) ( 1 − Q ) 2 Q 2 k = 2 ∑ p ( k p ) t k = p ( 1 + z ) 1 k = 2 ∑ p ( k p ) t k − 2 . For 2 ≤ k ≤ p 2\leq k\leq p 2 ≤ k ≤ p ( p k ) = p ( p − 1 ) k ( k − 1 ) ( p − 2 k − 2 ) ≤ p ( p − 1 ) 2 ( p − 2 k − 2 ) \binom{p}{k}=\frac{p(p-1)}{k(k-1)}\binom{p-2}{k-2}\leq\frac{p(p-1)}{2}\binom{p-2}{k-2} ( k p ) = k ( k − 1 ) p ( p − 1 ) ( k − 2 p − 2 ) ≤ 2 p ( p − 1 ) ( k − 2 p − 2 )
C β ( Q ) ≤ p − 1 2 ( 1 + z ) ∑ k = 2 p ( p − 2 k − 2 ) t k − 2 = m + 1 2 ( 1 + z ) ( 1 + t ) m . C_\beta(\sqrt{Q})\leq\frac{p-1}{2(1+z)}\sum_{k=2}^p\binom{p-2}{k-2}t^{k-2}=\frac{m+1}{2(1+z)}(1+t)^m. C β ( Q ) ≤ 2 ( 1 + z ) p − 1 k = 2 ∑ p ( k − 2 p − 2 ) t k − 2 = 2 ( 1 + z ) m + 1 ( 1 + t ) m . By Proposition 5.20(1), we have z > 2 m z>2m z > 2 m ( m + 1 ) / ( 2 ( 1 + z ) ) < ( m + 1 ) / ( 2 ( 1 + 2 m ) ) ≤ 3 / 10 (m+1)/(2(1+z))<(m+1)/(2(1+2m))\leq3/10 ( m + 1 ) / ( 2 ( 1 + z )) < ( m + 1 ) / ( 2 ( 1 + 2 m )) ≤ 3/10 m ≥ 2 m\geq2 m ≥ 2 ( m + 1 ) / ( 1 + 2 m ) (m+1)/(1+2m) ( m + 1 ) / ( 1 + 2 m ) m m m z t = x < 1 zt=x<1 z t = x < 1 t < 1 / z < 1 / ( 2 m ) t<1/z<1/(2m) t < 1/ z < 1/ ( 2 m ) ( 1 + t ) m ≤ e m t < e 1 / 2 (1+t)^m\leq e^{mt}<e^{1/2} ( 1 + t ) m ≤ e m t < e 1/2 e 1 / 2 e^{1/2} e 1/2 j ≥ 8 j\geq8 j ≥ 8 ∑ j 2 − j / j ! \sum_{j}2^{-j}/j! ∑ j 2 − j / j ! 1 / ( 2 ( j + 1 ) ) ≤ 1 / 18 1/(2(j+1))\leq1/18 1/ ( 2 ( j + 1 )) ≤ 1/18
e 1 / 2 < ∑ j = 0 7 2 − j j ! + 2 − 8 8 ! ⋅ 1 1 − 1 / 18 = 48217579 29245440 < 53 32 . e^{1/2}<\sum_{j=0}^7\frac{2^{-j}}{j!}+\frac{2^{-8}}{8!}\cdot\frac{1}{1-1/18}=\frac{48217579}{29245440}<\frac{53}{32}. e 1/2 < j = 0 ∑ 7 j ! 2 − j + 8 ! 2 − 8 ⋅ 1 − 1/18 1 = 29245440 48217579 < 32 53 . Combining these bounds, we obtain C β ( Q ) < 3 10 ⋅ 53 32 = 159 320 C_\beta(\sqrt{Q})<\frac{3}{10}\cdot\frac{53}{32}=\frac{159}{320} C β ( Q ) < 10 3 ⋅ 32 53 = 320 159
Proposition 5.23 (Small even perturbations of pure even models)
Proposition 5.23 (Small even perturbations of pure even models). Let d ≥ 4 d\geq4 d ≥ 4 P > d P>d P > d β > β s ( d ) \beta>\beta_s(d) β > β s ( d ) ε 0 = ε 0 ( d , P , β ) > 0 \varepsilon_0=\varepsilon_0(d,P,\beta)>0 ε 0 = ε 0 ( d , P , β ) > 0 ε ∈ [ 0 , ε 0 ) \varepsilon\in[0,\varepsilon_0) ε ∈ [ 0 , ε 0 ) ( b p ) (b_p) ( b p ) p p p d < p ≤ P d<p\leq P d < p ≤ P b p ≥ 0 b_p\geq0 b p ≥ 0 ∑ p b p ≤ 1 \sum_p b_p\leq1 ∑ p b p ≤ 1
ξ ~ ( r ) = r d + ε ∑ p b p r p \tilde{\xi}(r)=r^d+\varepsilon\sum_p b_pr^p ξ ~ ( r ) = r d + ε p ∑ b p r p is such that ( ξ ~ , β ) (\tilde{\xi},\beta) ( ξ ~ , β ) ( r d , β ) (r^d,\beta) ( r d , β ) ε → 0 \varepsilon\to0 ε → 0 ( b p ) (b_p) ( b p )
Proof. Recall from Proposition 5.20 that ( r d , β ) (r^d,\beta) ( r d , β ) D = { d } ∪ { p even : d < p ≤ P } D=\{d\}\cup\{p\text{ even}:d<p\leq P\} D = { d } ∪ { p even : d < p ≤ P } e d e_d e d r d r^d r d β \beta β β \beta β ε 0 > 0 \varepsilon_0>0 ε 0 > 0 B = { ( c , β ) : ∑ p ∈ D ∣ c p − ( e d ) p ∣ ≤ ε 0 } B=\{(c,\beta):\sum_{p\in D}|c_p-(e_d)_p|\leq\varepsilon_0\} B = {( c , β ) : ∑ p ∈ D ∣ c p − ( e d ) p ∣ ≤ ε 0 } β \beta β U U U lemma. The coefficient vector of r d + ε ∑ p b p r p r^d+\varepsilon\sum_p b_p r^p r d + ε ∑ p b p r p e d + ε b e_d+\varepsilon b e d + ε b ℓ 1 \ell^1 ℓ 1 ε < ε 0 \varepsilon<\varepsilon_0 ε < ε 0 e d e_d e d x ( e d + ε b , β ) x(e_d+\varepsilon b,\beta) x ( e d + ε b , β ) Q ( e d + ε b , β ) Q(e_d+\varepsilon b,\beta) Q ( e d + ε b , β ) x x x Q Q Q U \mathcal{U} U B B B ( r d , β ) (r^d,\beta) ( r d , β ) O ( ε ) O(\varepsilon) O ( ε ) b b b
The proof uses neither the parity of d d d d d d ε 0 \varepsilon_0 ε 0 β \beta β P P P β ↓ β s ( d ) \beta\downarrow\beta_s(d) β ↓ β s ( d ) P → ∞ P\to\infty P → ∞
Finally, we give a covariance that satisfies the 1RSB assumptions although ξ ′ ′ − 1 / 2 \xi^{\prime\prime-1/2} ξ ′′ − 1/2 p + s p+s p + s
Proposition 5.24 (A strictly 1RSB example without convexity). Let Q = 9 / 10 Q=9/10 Q = 9/10 x = 5 / 9 x=5/9 x = 5/9
t = 45 − 24 log 6 72 log 6 − 85 , b = t ( 10 9 ) 8 , ξ ( r ) = r 4 + b r 12 , β 2 = 15 Q 3 ( 4 + 12 t ) . (116) t=\frac{45-24\log6}{72\log6-85},\qquad b=t\left(\frac{10}{9}\right)^8,\qquad\xi(r)=r^4+br^{12},\qquad\beta^2=\frac{15}{Q^3(4+12t)}. \tag*{(116)} t = 72 log 6 − 85 45 − 24 log 6 , b = t ( 9 10 ) 8 , ξ ( r ) = r 4 + b r 12 , β 2 = Q 3 ( 4 + 12 t ) 15 . (116) Then 4539 / 100000 < t < 4541 / 100000 4539/100000<t<4541/100000 4539/100000 < t < 4541/100000 b > 0 b>0 b > 0 β > 0 \beta>0 β > 0 ( ξ , β ) (\xi,\beta) ( ξ , β ) x , Q x,Q x , Q Q Q Q β 2 ( 1 − Q ) 2 ξ ′ ′ ( Q ) < 2 / 3 \beta^2(1-Q)^2\xi^{\prime\prime}(Q)<2/3 β 2 ( 1 − Q ) 2 ξ ′′ ( Q ) < 2/3 ( ξ ′ ′ − 1 / 2 ) ′ ′ ( 5 / 6 ) < 0 (\xi^{\prime\prime-1/2})^{\prime\prime}(5/6)<0 ( ξ ′′ − 1/2 ) ′′ ( 5/6 ) < 0 ξ ′ ′ − 1 / 2 \xi^{\prime\prime-1/2} ξ ′′ − 1/2 ( 0 , 1 ] (0,1] ( 0 , 1 ]
Numerically, t ≈ 0.0453970 t\approx0.0453970 t ≈ 0.0453970 b ≈ 0.1054599 b\approx0.1054599 b ≈ 0.1054599 β ≈ 2.1277773 \beta\approx2.1277773 β ≈ 2.1277773 (111) , since 6 ⋅ 2 ⋅ 10 = 120 < 121 = 11 2 6\cdot2\cdot10=120<121=11^2 6 ⋅ 2 ⋅ 10 = 120 < 121 = 1 1 2
Proof. The proof uses two finite computations, which are carried out in Section A: the enclosure of t t t s = 1 − Q = 1 / 10 s=1-Q=1/10 s = 1 − Q = 1/10 C = s + x Q = 3 / 5 C=s+xQ=3/5 C = s + x Q = 3/5 C / s = 6 C/s=6 C / s = 6 z = x Q / s = 5 z=xQ/s=5 z = x Q / s = 5
The contact equations. By Lemma A.1, t − < t < t + t_-<t<t_+ t − < t < t + t − = 4539 / 100000 t_-=4539/100000 t − = 4539/100000 t + = 4541 / 100000 t_+=4541/100000 t + = 4541/100000 t t t b b b β 2 \beta^2 β 2 t t t log 6 \log6 log 6
1 + t 4 + 12 t = 6 log 6 − 5 25 . (117) \frac{1+t}{4+12t}=\frac{6\log6-5}{25}. \tag*{(117)} 4 + 12 t 1 + t = 25 6 log 6 − 5 . (117) We have b Q 8 = t bQ^8=t b Q 8 = t ξ ( Q ) = Q 4 ( 1 + t ) \xi(Q)=Q^4(1+t) ξ ( Q ) = Q 4 ( 1 + t ) ξ ′ ( Q ) = Q 3 ( 4 + 12 t ) \xi'(Q)=Q^3(4+12t) ξ ′ ( Q ) = Q 3 ( 4 + 12 t )
β 2 ξ ′ ( Q ) = 15 = Q s C , β 2 x 2 ξ ( Q ) = 25 ( 1 + t ) 6 ( 4 + 12 t ) = log 6 − 5 6 = log C s − x Q C , \beta^2\xi'(Q)=15=\frac{Q}{sC},\qquad\beta^2x^2\xi(Q)=\frac{25(1+t)}{6(4+12t)}=\log6-\frac{5}{6}=\log\frac{C}{s}-\frac{xQ}{C}, β 2 ξ ′ ( Q ) = 15 = s C Q , β 2 x 2 ξ ( Q ) = 6 ( 4 + 12 t ) 25 ( 1 + t ) = log 6 − 6 5 = log s C − C x Q , where we used (117) . By the integrals in the proof of Proposition 5.11(1), these are the equations V ′ ( Q ) = 0 V'(Q)=0 V ′ ( Q ) = 0 V ( Q ) = 0 V(Q)=0 V ( Q ) = 0
Below the contact. For 0 ≤ r ≤ 1 0 \le r \le1 0 ≤ r ≤ 1 U ( r ) = x 2 V ( Q r ) U(r) = x^2V(Qr) U ( r ) = x 2 V ( Q r ) 0 ≤ q ≤ Q 0 \le q \le Q 0 ≤ q ≤ Q ∫ 0 q ( q − v ) ( C − x v ) − 2 d v = x − 2 ( log ( C / ( C − x q ) ) − x q / C ) \int_0^q (q-v)(C-xv)^{-2}\,\mathrm{d}v = x^{-2}(\log(C/(C-xq))-xq/C) ∫ 0 q ( q − v ) ( C − xv ) − 2 d v = x − 2 ( log ( C / ( C − x q )) − x q / C ) C / ( C − x Q r ) = 6 / ( 6 − 5 r ) C/(C-xQr)=6/(6-5r) C / ( C − x Q r ) = 6/ ( 6 − 5 r ) x Q r / C = 5 r / 6 xQr/C=5r/6 x Q r / C = 5 r /6 x 2 β 2 Q 4 = 25 / ( 6 ( 4 + 12 t ) ) x^2\beta^2Q^4=25/(6(4+12t)) x 2 β 2 Q 4 = 25/ ( 6 ( 4 + 12 t ))
U ( r ) = − log ( 1 − 5 r 6 ) − 5 r 6 − 25 6 ( 4 + 12 t ) ( r 4 + t r 12 ) . (118) U(r)=-\log\left(1-\frac{5r}{6}\right)-\frac{5r}{6}-\frac{25}{6(4+12t)}\left(r^4+tr^{12}\right). \tag*{(118)} U ( r ) = − log ( 1 − 6 5 r ) − 6 5 r − 6 ( 4 + 12 t ) 25 ( r 4 + t r 12 ) . (118) We have U ( 0 ) = U ′ ( 0 ) = 0 U(0)=U'(0)=0 U ( 0 ) = U ′ ( 0 ) = 0 U ( 1 ) = x 2 V ( Q ) = 0 U(1)=x^2V(Q)=0 U ( 1 ) = x 2 V ( Q ) = 0 U ′ ( 1 ) = x 2 Q V ′ ( Q ) = 0 U'(1)=x^2QV'(Q)=0 U ′ ( 1 ) = x 2 Q V ′ ( Q ) = 0
U ′ ′ ( r ) = 25 P t ( r ) 6 ( 4 + 12 t ) ( 6 − 5 r ) 2 , P t ( r ) = 24 + 72 t − ( 12 r 2 + 132 t r 10 ) ( 6 − 5 r ) 2 . (119) U''(r)=\frac{25P_t(r)}{6(4+12t)(6-5r)^2},\qquad P_t(r)=24+72t-(12r^2+132tr^{10})(6-5r)^2. \tag*{(119)} U ′′ ( r ) = 6 ( 4 + 12 t ) ( 6 − 5 r ) 2 25 P t ( r ) , P t ( r ) = 24 + 72 t − ( 12 r 2 + 132 t r 10 ) ( 6 − 5 r ) 2 . (119) By Lemma A.3, there exist a 0 ∈ ( 357 / 1000 , 359 / 1000 ) a_0\in(357/1000,359/1000) a 0 ∈ ( 357/1000 , 359/1000 ) b 0 ∈ ( 890 / 1000 , 892 / 1000 ) b_0\in(890/1000,892/1000) b 0 ∈ ( 890/1000 , 892/1000 ) U ′ ′ > 0 U''>0 U ′′ > 0 ( 0 , a 0 ) (0,a_0) ( 0 , a 0 ) U ′ ′ < 0 U''<0 U ′′ < 0 ( a 0 , b 0 ) (a_0,b_0) ( a 0 , b 0 ) U ′ ′ > 0 U''>0 U ′′ > 0 ( b 0 , 1 ) (b_0,1) ( b 0 , 1 ) U ′ ( 0 ) = 0 U'(0)=0 U ′ ( 0 ) = 0 U ′ > 0 U'>0 U ′ > 0 ( 0 , a 0 ] (0,a_0] ( 0 , a 0 ] U ′ ( 1 ) = 0 U'(1)=0 U ′ ( 1 ) = 0 U ′ < 0 U'<0 U ′ < 0 [ b 0 , 1 ) [b_0,1) [ b 0 , 1 ) U ′ U' U ′ [ a 0 , b 0 ] [a_0,b_0] [ a 0 , b 0 ] c ∈ ( a 0 , b 0 ) c\in(a_0,b_0) c ∈ ( a 0 , b 0 ) U ′ > 0 U'>0 U ′ > 0 ( 0 , c ) (0,c) ( 0 , c ) U ′ < 0 U'<0 U ′ < 0 ( c , 1 ) (c,1) ( c , 1 ) U ( 0 ) = U ( 1 ) = 0 U(0)=U(1)=0 U ( 0 ) = U ( 1 ) = 0 U > 0 U>0 U > 0 ( 0 , 1 ) (0,1) ( 0 , 1 ) V > 0 V>0 V > 0 ( 0 , Q ) (0,Q) ( 0 , Q )
Above the contact. By (116) and b Q 10 = Q 2 t bQ^{10}=Q^2t b Q 10 = Q 2 t
μ Q = β 2 s 2 ζ ′ ′ ( Q ) = 15 ( 12 Q 2 + 132 t Q 2 ) 100 Q 3 ( 4 + 12 t ) = 15 ( 12 + 132 t ) 90 ( 4 + 12 t ) = 12 + 132 t 24 + 72 t . \mu_Q=\beta^2s^2\zeta''(Q)=\frac{15(12Q^2+132tQ^2)}{100Q^3(4+12t)}=\frac{15(12+132t)}{90(4+12t)}=\frac{12+132t}{24+72t}. μ Q = β 2 s 2 ζ ′′ ( Q ) = 100 Q 3 ( 4 + 12 t ) 15 ( 12 Q 2 + 132 t Q 2 ) = 90 ( 4 + 12 t ) 15 ( 12 + 132 t ) = 24 + 72 t 12 + 132 t . The right side is increasing in t t t 2304 / ( 24 + 72 t ) 2 2304/(24+72t)^2 2304/ ( 24 + 72 t ) 2 μ Q < ( 12 + 132 t + ) / ( 24 + 72 t + ) = 149951 / 227246 < 2 / 3 \mu_Q<(12+132t_+)/(24+72t_+)=149951/227246<2/3 μ Q < ( 12 + 132 t + ) / ( 24 + 72 t + ) = 149951/227246 < 2/3 p ∈ { 4 , 12 } p\in\{4,12\} p ∈ { 4 , 12 } ( 1 − q ) 2 q p − 2 (1-q)^2q^{p-2} ( 1 − q ) 2 q p − 2 ( 1 − q ) q p − 3 ( ( p − 2 ) − p q ) (1-q)q^{p-3}((p-2)-pq) ( 1 − q ) q p − 3 (( p − 2 ) − pq ) q ≥ Q q\ge Q q ≥ Q Q = 9 / 10 > 5 / 6 ≥ ( p − 2 ) / p Q=9/10>5/6\ge(p-2)/p Q = 9/10 > 5/6 ≥ ( p − 2 ) / p β 2 ( 1 − q ) 2 ζ ′ ′ ( q ) ≤ μ Q < 1 \beta^2(1-q)^2\zeta''(q)\le\mu_Q<1 β 2 ( 1 − q ) 2 ζ ′′ ( q ) ≤ μ Q < 1 Q ≤ q < 1 Q\le q<1 Q ≤ q < 1 x ^ ( q ) = 1 − q \hat{x}(q)=1-q x ^ ( q ) = 1 − q V ′ ′ ( q ) = ( 1 − q ) − 2 ( 1 − β 2 ( 1 − q ) 2 ζ ′ ′ ( q ) ) > 0 V''(q)=(1-q)^{-2}(1-\beta^2(1-q)^2\zeta''(q))>0 V ′′ ( q ) = ( 1 − q ) − 2 ( 1 − β 2 ( 1 − q ) 2 ζ ′′ ( q )) > 0 [ Q , 1 ) [Q,1) [ Q , 1 ) V ′ ′ ( Q ) > 0 V''(Q)>0 V ′′ ( Q ) > 0 V ( Q ) = V ′ ( Q ) = 0 V(Q)=V'(Q)=0 V ( Q ) = V ′ ( Q ) = 0 V > 0 V>0 V > 0 ( Q , 1 ) (Q,1) ( Q , 1 ) (93) .
Nonconvexity. Let v = ζ ′ ′ v=\zeta'' v = ζ ′′ v ( r ) = 12 r 2 ( 1 + w ( r ) ) v(r)=12r^2(1+w(r)) v ( r ) = 12 r 2 ( 1 + w ( r )) w ( r ) = 11 b r 8 w(r)=11br^8 w ( r ) = 11 b r 8 r v ′ ( r ) = 12 r 2 ( 2 + 10 w ( r ) ) rv'(r)=12r^2(2+10w(r)) r v ′ ( r ) = 12 r 2 ( 2 + 10 w ( r )) r 2 v ′ ′ ( r ) = 12 r 2 ( 2 + 90 w ( r ) ) r^2v''(r)=12r^2(2+90w(r)) r 2 v ′′ ( r ) = 12 r 2 ( 2 + 90 w ( r ))
r 2 ( 3 v ′ ( r ) 2 − 2 v ( r ) v ′ ′ ( r ) ) ( 12 r 2 ) 2 = 3 ( 2 + 10 w ) 2 − 2 ( 1 + w ) ( 2 + 90 w ) = 8 ( 1 − 3 w ) ( 1 − 5 w ) , \frac{r^2(3v'(r)^2-2v(r)v''(r))}{(12r^2)^2}=3(2+10w)^2-2(1+w)(2+90w)=8(1-3w)(1-5w), ( 12 r 2 ) 2 r 2 ( 3 v ′ ( r ) 2 − 2 v ( r ) v ′′ ( r )) = 3 ( 2 + 10 w ) 2 − 2 ( 1 + w ) ( 2 + 90 w ) = 8 ( 1 − 3 w ) ( 1 − 5 w ) , where w = w ( r ) w=w(r) w = w ( r ) r 0 = 5 / 6 r_0=5/6 r 0 = 5/6 w ( r 0 ) = 11 t ( 10 / 9 ) 8 ( 5 / 6 ) 8 = 11 t ( 25 / 27 ) 8 w(r_0)=11t(10/9)^8(5/6)^8=11t(25/27)^8 w ( r 0 ) = 11 t ( 10/9 ) 8 ( 5/6 ) 8 = 11 t ( 25/27 ) 8 1 / 4 < 11 t − ( 25 / 27 ) 8 < w ( r 0 ) < 11 t + ( 25 / 27 ) 8 < 3 / 10 1/4<11t_-(25/27)^8<w(r_0)<11t_+(25/27)^8<3/10 1/4 < 11 t − ( 25/27 ) 8 < w ( r 0 ) < 11 t + ( 25/27 ) 8 < 3/10 1 − 3 w ( r 0 ) > 0 > 1 − 5 w ( r 0 ) 1-3w(r_0)>0>1-5w(r_0) 1 − 3 w ( r 0 ) > 0 > 1 − 5 w ( r 0 ) 3 v ′ 2 − 2 v v ′ ′ < 0 3v'^2-2vv''<0 3 v ′2 − 2 v v ′′ < 0 ( v − 1 / 2 ) ′ ′ = ( 3 v ′ 2 − 2 v v ′ ′ ) / ( 4 v 5 / 2 ) < 0 (v^{-1/2})''=(3v'^2-2vv'')/(4v^{5/2})<0 ( v − 1/2 ) ′′ = ( 3 v ′2 − 2 v v ′′ ) / ( 4 v 5/2 ) < 0 r 0 r_0 r 0
The directing measure
In this section, we prove Theorem 6.3 . It concerns the overlaps of independent samples σ 1 , σ 2 , … \sigma^1,\sigma^2,\ldots σ 1 , σ 2 , … ( R ( σ i , σ j ) ) i , j ≥ 1 (R(\sigma^i,\sigma^j))_{i,j\geq1} ( R ( σ i , σ j ) ) i , j ≥ 1 G G G v 1 , v 2 , … v_1,v_2,\ldots v 1 , v 2 , … G G G R i j = v i ⋅ v j R_{ij}=v_i\cdot v_j R ij = v i ⋅ v j i ≠ j i\ne j i = j G G G G G G Theorem 6.3 states that if ( ξ , β ) (\xi,\beta) ( ξ , β ) x x x Q Q Q Q Q Q Q Q Q
This is the property of the Gibbs measure that the coarse cover by bands around critical points needs in Sections 9 and 12 (see Proposition 12.10). There, a Gibbs sample is placed in a band around a critical point of the center field H Q H_Q H Q n n n Q Q Q H Q H_Q H Q Proposition 9.12 and Lemma 12.8). In the limit, such replicas exist with probability close to one if the directing vector v 1 v_1 v 1 G G G ∥ v 1 ∥ 2 \lVert v_1\rVert^2 ∥ v 1 ∥ 2 Q Q Q v 1 v_1 v 1 Q Q Q v 1 v_1 v 1 G G G Q Q Q q q q q q q Q Q Q Theorem 6.3 excludes both possibilities.
We use the following results of Section 5. Under the 1RSB assumptions of Definition 5.9, the first-variation function V V V ( 0 , 1 ) ∖ { Q } (0,1)\setminus\{Q\} ( 0 , 1 ) ∖ { Q } Theorem 5.7 and Proposition 5.11 show that 0 0 0 Q Q Q Lemma 5.8 expresses the failure of multisamplability at every other q ∈ ( 0 , 1 ) q\in(0,1) q ∈ ( 0 , 1 ) N N N q q q F F F β \beta β Corollary 5.16 ). From § 6.4 , we use the tensor representation of H N H_N H N Lemma 3.3 ), and the Parisi formula (Theorem 3.14 ).
The proof has three steps. In § 6.1.3 , we show that replica exclusion at an overlap q q q q q q q q q Lemma 6.5 ). We exclude vectors of norm one separately, by the entropy estimate of Lemma 6.4, since multisamplability concerns overlaps below one. After this step, the squared norms lie in { 0 , Q } \{0,Q\} { 0 , Q } § 6.1.4 , we derive one identity of Ghirlanda–Guerra type, with test function ξ ( R ) \xi(R) ξ ( R ) β \beta β F F F Lemma 6.7 ). In Section 6.5, we show that this identity leaves no mass at the origin. The argument uses a single identity of this type, for the original Hamiltonian without a perturbation. It uses no ultrametricity, no atomicity of the directing measure, and no Poisson–Dirichlet structure. Odd degrees are allowed throughout: ξ ( R ) \xi(R) ξ ( R ) ξ ( R ) ≤ ξ ( Q ) \xi(R) \le\xi(Q) ξ ( R ) ≤ ξ ( Q ) ∣ R ∣ ≤ Q |R| \le Q ∣ R ∣ ≤ Q
Notation. Throughout this section, ξ \xi ξ β > 0 \beta> 0 β > 0 H N H_N H N σ 1 , σ 2 , … \sigma^1,\sigma^2,\ldots σ 1 , σ 2 , … π \pi π R i j = R ( σ i , σ j ) R_{ij}=R(\sigma^i,\sigma^j) R ij = R ( σ i , σ j ) R i i = 1 R_{ii}=1 R ii = 1 ⟨ ⋅ ⟩ \langle\cdot\rangle ⟨ ⋅ ⟩ ⟨ f ⟩ = ∫ f d π ⊗ n \langle f\rangle= \int f\,d\pi^{\otimes n} ⟨ f ⟩ = ∫ f d π ⊗ n f f f σ 1 , … , σ n \sigma^1,\ldots,\sigma^n σ 1 , … , σ n F N ( β ) = N − 1 E log Z N , β F_N(\beta)=N^{-1}\mathbb{E}\log Z_{N,\beta} F N ( β ) = N − 1 E log Z N , β F ( β ) = lim N F N ( β ) F(\beta)=\lim_N F_N(\beta) F ( β ) = lim N F N ( β )
e N = E ⟨ H N ( σ 1 ) N ⟩ , ε N = E ⟨ ∣ H N ( σ 1 ) N − e N ∣ ⟩ . (120) e_N=\mathbb{E}\left\langle\frac{H_N(\sigma^1)}{N}\right\rangle,\qquad
\varepsilon_N=\mathbb{E}\left\langle\left|\frac{H_N(\sigma^1)}{N}-e_N\right|\right\rangle. \tag*{(120)} e N = E ⟨ N H N ( σ 1 ) ⟩ , ε N = E ⟨ N H N ( σ 1 ) − e N ⟩ . (120) Recall from Definition 5.9 that ( ξ , β ) (\xi,\beta) ( ξ , β ) x , Q ∈ ( 0 , 1 ) x,Q\in(0,1) x , Q ∈ ( 0 , 1 ) Q Q Q ξ \xi ξ
V ( r ) = ∫ 0 r r − v x ^ ( v ) 2 d v − β 2 ξ ( r ) , x ^ ( v ) = { 1 − Q + x ( Q − v ) , 0 ≤ v ≤ Q , 1 − v , Q ≤ v < 1 , V(r)=\int_0^r\frac{r-v}{\widehat{x}(v)^2}\,dv-\beta^2\xi(r),\qquad
\widehat{x}(v)=
\begin{cases}
1-Q+x(Q-v), & 0\le v\le Q,\\
1-v, & Q\le v<1,
\end{cases} V ( r ) = ∫ 0 r x ( v ) 2 r − v d v − β 2 ξ ( r ) , x ( v ) = { 1 − Q + x ( Q − v ) , 1 − v , 0 ≤ v ≤ Q , Q ≤ v < 1 , satisfies V ( Q ) = V ′ ( Q ) = 0 V(Q)=V'(Q)=0 V ( Q ) = V ′ ( Q ) = 0 V ′ ′ ( Q ) > 0 V''(Q)>0 V ′′ ( Q ) > 0 V ( r ) > 0 V(r)>0 V ( r ) > 0 r ∈ ( 0 , 1 ) ∖ { Q } r\in(0,1)\setminus\{Q\} r ∈ ( 0 , 1 ) ∖ { Q }
Overlap arrays and directing measures
Let M = [ − 1 , 1 ] N × N \mathcal{M}=[-1,1]^{\mathbb{N}\times\mathbb{N}} M = [ − 1 , 1 ] N × N ( r i j ) i , j ≥ 1 (r_{ij})_{i,j\ge1} ( r ij ) i , j ≥ 1 [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] R i j R_{ij} R ij M \mathcal{M} M N N N L N \mathcal{L}_N L N
L N ( A ) = E π ⊗ ∞ ( { ( R ( σ i , σ j ) ) i , j ≥ 1 ∈ A } ) \mathcal{L}_N(A)=\mathbb{E}\,\pi^{\otimes\infty}\left(\left\{\left(R(\sigma^i,\sigma^j)\right)_{i,j\ge1}\in A\right\}\right) L N ( A ) = E π ⊗ ∞ ( { ( R ( σ i , σ j ) ) i , j ≥ 1 ∈ A } ) for Borel sets A ⊂ M A\subset\mathcal{M} A ⊂ M M \mathcal{M} M ( L N ) (\mathcal{L}_N) ( L N ) M \mathcal{M} M L \mathcal{L} L E L \mathbb{E}_{\mathcal{L}} E L L \mathcal{L} L
Each L N \mathcal{L}_N L N R ( σ i , σ j ) = U ( σ i ) ⋅ U ( σ j ) R(\sigma^i,\sigma^j)=U(\sigma^i)\cdot U(\sigma^j) R ( σ i , σ j ) = U ( σ i ) ⋅ U ( σ j ) L N \mathcal{L}_N L N M \mathcal{M} M L \mathcal{L} L ( N m ) (N_m) ( N m ) L \mathcal{L} L E ⊂ M E\subset\mathcal{M} E ⊂ M L ( E ) ≤ lim inf m L N m ( E ) \mathcal{L}(E)\le\liminf_m\mathcal{L}_{N_m}(E) L ( E ) ≤ lim inf m L N m ( E ) f f f M \mathcal{M} M E L f = lim m E ⟨ f ⟩ \mathbb{E}_{\mathcal{L}}f=\lim_m\mathbb{E}\langle f\rangle E L f = lim m E ⟨ f ⟩ f f f
Let B \mathrm{B} B ℓ 2 \ell^2 ℓ 2 B ( v , r ) B(v,r) B ( v , r ) r r r v ∈ ℓ 2 v\in\ell^2 v ∈ ℓ 2 G G G B \mathrm{B} B L G \mathcal{L}^G L G v i ⋅ v j v_i\cdot v_j v i ⋅ v j i ≠ j i\ne j i = j 1 1 1 v 1 , v 2 , … v_1,v_2,\ldots v 1 , v 2 , … G G G
Definition 6.1 (Directing measure). Let L \mathcal{L} L M \mathcal{M} M directing measure of L \mathcal{L} L ( G ω ) ω ∈ Ω (G_\omega)_{\omega\in\Omega} ( G ω ) ω ∈ Ω B \mathrm{B} B ( Ω , F , Pr ) (\Omega,\mathcal{F},\Pr) ( Ω , F , Pr ) A ⊂ M A\subset\mathcal{M} A ⊂ M ω ↦ L G ω ( A ) \omega\mapsto\mathcal{L}^{G_\omega}(A) ω ↦ L G ω ( A )
L ( A ) = ∫ Ω L G ω ( A ) Pr ( d ω ) . \mathcal{L}(A)=\int_\Omega\mathcal{L}^{G_\omega}(A)\,\Pr(\mathrm{d}\omega). L ( A ) = ∫ Ω L G ω ( A ) Pr ( d ω ) . For each fixed ω \omega ω L G ω \mathcal{L}^{G_\omega} L G ω G ω G_\omega G ω L \mathcal{L} L ω ↦ G ω ( A ) \omega\mapsto G_\omega(A) ω ↦ G ω ( A ) A ⊂ B A\subset\mathrm{B} A ⊂ B ω \omega ω Pr \Pr Pr v 1 , v 2 , … v_1,v_2,\ldots v 1 , v 2 , … G ω G_\omega G ω R i j = v i ⋅ v j R_{ij}=v_i\cdot v_j R ij = v i ⋅ v j i ≠ j i\ne j i = j ω ↦ L G ω ( A ) \omega\mapsto\mathcal{L}^{G_\omega}(A) ω ↦ L G ω ( A )
Theorem 6.2 (Dovbysh–Sudakov; [123, Proposition 1]). Every limiting overlap law has a directing measure.
This is the representation theorem of Dovbysh and Sudakov [63], in the form proved by Panchenko [123, Proposition 1]. That proposition applies to the law of every symmetric, nonnegative definite, weakly exchangeable array, and we checked above that limiting overlap laws are of this kind. It states that such a law is a mixture, over a probability space ( Ω , F , Pr ) (\Omega,\mathcal{F},\Pr) ( Ω , F , Pr ) Q ω \mathcal{Q}_\omega Q ω M \mathcal{M} M ω ↦ Q ω ( A ) \omega\mapsto\mathcal{Q}_\omega(A) ω ↦ Q ω ( A ) A A A Q ω \mathcal{Q}_\omega Q ω ( h i ⋅ h j + a i 1 { i = j } ) i , j ≥ 1 (h_i\cdot h_j+a_i\mathbf{1}\{i=j\})_{i,j\ge1} ( h i ⋅ h j + a i 1 { i = j } ) i , j ≥ 1 ( h i , a i ) (h_i,a_i) ( h i , a i ) η ω \eta_\omega η ω ℓ 2 × [ 0 , ∞ ) \ell^2\times[0,\infty) ℓ 2 × [ 0 , ∞ ) L \mathcal{L} L ∫ Q ω ( { R 11 = 1 } ) Pr ( d ω ) = 1 \int Q_\omega(\{R_{11}=1\})\,\Pr(\mathrm{d}\omega)=1 ∫ Q ω ({ R 11 = 1 }) Pr ( d ω ) = 1 Ω 0 \Omega_0 Ω 0 ω \omega ω Q ω ( { R 11 = 1 } ) = 1 Q_\omega(\{R_{11}=1\})=1 Q ω ({ R 11 = 1 }) = 1 Pr ( Ω 0 ) = 1 \Pr(\Omega_0)=1 Pr ( Ω 0 ) = 1 ω ∈ Ω 0 \omega\in\Omega_0 ω ∈ Ω 0 η ω \eta_\omega η ω { ∥ h ∥ 2 + a = 1 } \{\|h\|^2+a=1\} { ∥ h ∥ 2 + a = 1 } { ∥ h ∥ ≤ 1 } \{\|h\|\le1\} { ∥ h ∥ ≤ 1 } G ω G_\omega G ω η ω \eta_\omega η ω ( h , a ) ↦ h (h,a)\mapsto h ( h , a ) ↦ h L G ω = Q ω \mathcal{L}^{G_\omega}=Q_\omega L G ω = Q ω ω ∉ Ω 0 \omega\notin\Omega_0 ω ∈ / Ω 0 G ω = δ 0 G_\omega=\delta_0 G ω = δ 0 L \mathcal{L} L ℓ 2 \ell^2 ℓ 2
Theorem 6.3 (Directing vectors have squared norm Q Q Q ( ξ , β ) (\xi,\beta) ( ξ , β ) x x x Q Q Q Q Q Q Definition 5.9 ). Let L \mathcal{L} L ( G ω ) ω ∈ Ω (G_\omega)_{\omega\in\Omega} ( G ω ) ω ∈ Ω L \mathcal{L} L Pr \Pr Pr ω \omega ω
G ω ( { v ∈ B : ∥ v ∥ 2 = Q } ) = 1. G_\omega\bigl(\{v\in B:\|v\|^2=Q\}\bigr)=1. G ω ( { v ∈ B : ∥ v ∥ 2 = Q } ) = 1. Equivalently, for Pr \Pr Pr ω \omega ω v 1 , v 2 , … v_1,v_2,\ldots v 1 , v 2 , … G ω G_\omega G ω ∥ v i ∥ 2 = Q \|v_i\|^2=Q ∥ v i ∥ 2 = Q i i i ∥ v i − v j ∥ 2 = 2 ( Q − v i ⋅ v j ) \|v_i-v_j\|^2=2(Q-v_i\cdot v_j) ∥ v i − v j ∥ 2 = 2 ( Q − v i ⋅ v j ) i ≠ j i\ne j i = j ∣ R i j ∣ ≤ Q |R_{ij}|\le Q ∣ R ij ∣ ≤ Q i ≠ j i\ne j i = j L \mathcal{L} L G G G ω ↦ G ω ( A ) \omega\mapsto G_\omega(A) ω ↦ G ω ( A ) A ⊂ B A\subset B A ⊂ B G ( { ∥ v ∥ 2 = Q } ) = 1 G(\{\|v\|^2=Q\})=1 G ({ ∥ v ∥ 2 = Q }) = 1
Pairs of samples near overlap one
The results of Section 5 concern overlaps in [ 0 , 1 ) [0,1) [ 0 , 1 ) π \pi π ν N \nu_N ν N e 2 β K N e^{2\beta K N} e 2 β K N { R ( ⋅ , u ) > 1 − δ } \{R(\mathord\cdot,u)>1-\delta\} { R ( ⋅ , u ) > 1 − δ } ν N \nu_N ν N N ( 2 δ ) ( N − 3 ) / 2 \sqrt{N}(2\delta)^{(N-3)/2} N ( 2 δ ) ( N − 3 ) /2 δ \delta δ
Lemma 6.4 (A field bound and a cap estimate). Let P = { p : c p > 0 } \mathcal{P}=\{p:c_p>0\} P = { p : c p > 0 }
a p = ( 2 p log ( 1 + 4 p ) + 2 ) 1 / 2 , K = 2 ∑ p ∈ P c p a p . a_p=(2p\log(1+4p)+2)^{1/2},\qquad K=2\sum_{p\in\mathcal{P}}\sqrt{c_p}\,a_p. a p = ( 2 p log ( 1 + 4 p ) + 2 ) 1/2 , K = 2 p ∈ P ∑ c p a p . For every N ≥ 2 N\ge2 N ≥ 2
P ( sup x ∈ S N ∣ H N ( x ) ∣ > K N ) ≤ 2 ∣ P ∣ e − N . (121) \mathbb{P}\left(\sup_{x\in S_N}|H_N(x)|>KN\right)\le2|\mathcal{P}|e^{-N}. \tag*{(121)} P ( x ∈ S N sup ∣ H N ( x ) ∣ > K N ) ≤ 2∣ P ∣ e − N . (121) For every N ≥ 3 N\ge3 N ≥ 3 u ∈ S N u\in S_N u ∈ S N δ ∈ ( 0 , 1 / 2 ) \delta\in(0,1/2) δ ∈ ( 0 , 1/2 )
ν N ( { x ∈ S N : R ( x , u ) > 1 − δ } ) ≤ N ( 2 δ ) ( N − 3 ) / 2 . (122) \nu_N\left(\{x\in S_N:R(x,u)>1-\delta\}\right)\le\sqrt{N}(2\delta)^{(N-3)/2}. \tag*{(122)} ν N ( { x ∈ S N : R ( x , u ) > 1 − δ } ) ≤ N ( 2 δ ) ( N − 3 ) /2 . (122) Let δ = e − 4 β K − 4 / 2 \delta=e^{-4\beta K-4}/2 δ = e − 4 β K − 4 /2 N ≥ 3 N\ge3 N ≥ 3
E π ⊗ 2 ( { R 12 > 1 − δ } ) ≤ 2 ∣ P ∣ e − N + e 6 β K + 6 N e − 2 N . (123) \mathbb{E}\,\pi^{\otimes2}\left(\{R_{12}>1-\delta\}\right)\le2|\mathcal{P}|e^{-N}+e^{6\beta K+6}\sqrt{N}e^{-2N}. \tag*{(123)} E π ⊗ 2 ( { R 12 > 1 − δ } ) ≤ 2∣ P ∣ e − N + e 6 β K + 6 N e − 2 N . (123) :::`
Part (1) is an explicit version of the bound on sup ∣ H N ∣ \sup|H_N| sup ∣ H N ∣ K K K N N N
Proof. For (1), let v ∈ R N v \in\mathbb{R}^N v ∈ R N (36) , we have
H N ( N v ) = N ∑ p ∈ P c p T p ( v , … , v ) , T p ( v 1 , … , v p ) = ∑ i 1 , … , i p = 1 N g i 1 … i p ( p ) v 1 , i 1 ⋯ v p , i p . H_N(\sqrt{N}v)=\sqrt{N}\sum_{p\in\mathcal{P}}\sqrt{c_p}\,T_p(v,\ldots,v),\qquad T_p(v_1,\ldots,v_p)=\sum_{i_1,\ldots,i_p=1}^N g^{(p)}_{i_1\ldots i_p}v_{1,i_1}\cdots v_{p,i_p}. H N ( N v ) = N p ∈ P ∑ c p T p ( v , … , v ) , T p ( v 1 , … , v p ) = i 1 , … , i p = 1 ∑ N g i 1 … i p ( p ) v 1 , i 1 ⋯ v p , i p . where v k , i v_{k,i} v k , i i i i v k v_k v k v 1 , … , v p v_1,\ldots,v_p v 1 , … , v p T p ( v 1 , … , v p ) T_p(v_1,\ldots,v_p) T p ( v 1 , … , v p ) ∏ k ∣ v k ∣ 2 = 1 \prod_k |v_k|^2=1 ∏ k ∣ v k ∣ 2 = 1 ∥ T p ∥ \|T_p\| ∥ T p ∥ ∣ T p ( v 1 , … , v p ) ∣ |T_p(v_1,\ldots,v_p)| ∣ T p ( v 1 , … , v p ) ∣ v 1 , … , v p v_1,\ldots,v_p v 1 , … , v p p ∈ P p\in\mathcal{P} p ∈ P ε = 1 / ( 2 p ) \varepsilon=1/(2p) ε = 1/ ( 2 p ) ε \varepsilon ε N \mathcal{N} N R N \mathbb{R}^N R N ε \varepsilon ε ε / 2 \varepsilon/2 ε /2 1 + ε / 2 1+\varepsilon/2 1 + ε /2 ∣ N ∣ ≤ ( 1 + 2 / ε ) N = ( 1 + 4 p ) N |\mathcal{N}|\leq(1+2/\varepsilon)^N=(1+4p)^N ∣ N ∣ ≤ ( 1 + 2/ ε ) N = ( 1 + 4 p ) N
Given unit vectors v 1 , … , v p v_1,\ldots,v_p v 1 , … , v p w k ∈ N w_k\in\mathcal{N} w k ∈ N ∣ v k − w k ∣ ≤ ε |v_k-w_k|\leq\varepsilon ∣ v k − w k ∣ ≤ ε
T p ( v 1 , … , v p ) − T p ( w 1 , … , w p ) = ∑ k = 1 p T p ( w 1 , … , w k − 1 , v k − w k , v k + 1 , … , v p ) . T_p(v_1,\ldots,v_p)-T_p(w_1,\ldots,w_p)=\sum_{k=1}^p T_p(w_1,\ldots,w_{k-1},v_k-w_k,v_{k+1},\ldots,v_p). T p ( v 1 , … , v p ) − T p ( w 1 , … , w p ) = k = 1 ∑ p T p ( w 1 , … , w k − 1 , v k − w k , v k + 1 , … , v p ) . and each term on the right has absolute value at most ε ∥ T p ∥ \varepsilon\|T_p\| ε ∥ T p ∥ M p M_p M p ∣ T p ∣ |T_p| ∣ T p ∣ N p \mathcal{N}^p N p v 1 , … , v p v_1,\ldots,v_p v 1 , … , v p ∥ T p ∥ ≤ M p + p ε ∥ T p ∥ = M p + ∥ T p ∥ / 2 \|T_p\|\leq M_p+p\varepsilon\|T_p\|=M_p+\|T_p\|/2 ∥ T p ∥ ≤ M p + pε ∥ T p ∥ = M p + ∥ T p ∥/2 ∥ T p ∥ ≤ 2 M p \|T_p\|\leq2M_p ∥ T p ∥ ≤ 2 M p N p \mathcal{N}^p N p ( 1 + 4 p ) p N (1+4p)^{pN} ( 1 + 4 p ) pN P ( ∣ γ ∣ > t ) ≤ 2 e − t 2 / 2 \mathbb{P}(|\gamma|>t)\leq2e^{-t^2/2} P ( ∣ γ ∣ > t ) ≤ 2 e − t 2 /2 γ \gamma γ
P ( ∥ T p ∥ > 2 a p N ) ≤ P ( M p > a p N ) ≤ 2 ( 1 + 4 p ) p N e − a p 2 N / 2 = 2 e − N , \mathbb{P}\left(\|T_p\|>2a_p\sqrt{N}\right)\leq\mathbb{P}\left(M_p>a_p\sqrt{N}\right)\leq2(1+4p)^{pN}e^{-a_p^2N/2}=2e^{-N}, P ( ∥ T p ∥ > 2 a p N ) ≤ P ( M p > a p N ) ≤ 2 ( 1 + 4 p ) pN e − a p 2 N /2 = 2 e − N , where we used a p 2 / 2 = p log ( 1 + 4 p ) + 1 a_p^2/2=p\log(1+4p)+1 a p 2 /2 = p log ( 1 + 4 p ) + 1 { ∥ T p ∥ ≤ 2 a p N } \{\|T_p\|\leq2a_p\sqrt{N}\} { ∥ T p ∥ ≤ 2 a p N } p ∈ P p\in\mathcal{P} p ∈ P ∣ H N ( x ) ∣ ≤ N ∑ p c p ∥ T p ∥ ≤ K N |H_N(x)|\leq\sqrt{N}\sum_p\sqrt{c_p}\,\|T_p\|\leq KN ∣ H N ( x ) ∣ ≤ N ∑ p c p ∥ T p ∥ ≤ K N x ∈ S N x\in S_N x ∈ S N
For (2), recall that for N ≥ 3 N\geq3 N ≥ 3 u ∈ S N u\in S_N u ∈ S N R ( x , u ) R(x,u) R ( x , u ) ν N ( d x ) \nu_N(dx) ν N ( d x ) d N ( 1 − r 2 ) ( N − 3 ) / 2 d_N(1-r^2)^{(N-3)/2} d N ( 1 − r 2 ) ( N − 3 ) /2 [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] d N − 1 = ∫ − 1 1 ( 1 − r 2 ) ( N − 3 ) / 2 d r d_N^{-1}=\int_{-1}^1(1-r^2)^{(N-3)/2}\,dr d N − 1 = ∫ − 1 1 ( 1 − r 2 ) ( N − 3 ) /2 d r ∣ r ∣ ≤ N − 1 / 2 |r|\leq N^{-1/2} ∣ r ∣ ≤ N − 1/2
d N − 1 ≥ 2 N − 1 / 2 ( 1 − 1 N ) ( N − 3 ) / 2 ≥ 2 N − 1 / 2 e − 1 / 2 , d_N^{-1}\geq2N^{-1/2}\left(1-\frac{1}{N}\right)^{(N-3)/2}\geq2N^{-1/2}e^{-1/2}, d N − 1 ≥ 2 N − 1/2 ( 1 − N 1 ) ( N − 3 ) /2 ≥ 2 N − 1/2 e − 1/2 , where we used log ( 1 − 1 / N ) ≥ − 1 / ( N − 1 ) \log(1-1/N)\geq-1/(N-1) log ( 1 − 1/ N ) ≥ − 1/ ( N − 1 ) log ( 1 − t ) ≥ − t / ( 1 − t ) \log(1-t)\geq-t/(1-t) log ( 1 − t ) ≥ − t / ( 1 − t ) ( N − 3 ) / ( N − 1 ) ≤ 1 (N-3)/(N-1)\leq1 ( N − 3 ) / ( N − 1 ) ≤ 1 d N ≤ e 1 / 2 N / 2 < N d_N\leq e^{1/2}\sqrt{N}/2<\sqrt{N} d N ≤ e 1/2 N /2 < N r ∈ ( 1 − δ , 1 ] r\in(1-\delta,1] r ∈ ( 1 − δ , 1 ] 1 − r 2 = ( 1 − r ) ( 1 + r ) ≤ 2 δ 1-r^2=(1-r)(1+r)\leq2\delta 1 − r 2 = ( 1 − r ) ( 1 + r ) ≤ 2 δ δ < 1 \delta<1 δ < 1 ν N ( { R ( ⋅ , u ) > 1 − δ } ) ≤ d N δ ( 2 δ ) ( N − 3 ) / 2 ≤ N ( 2 δ ) ( N − 3 ) / 2 \nu_N(\{R(\cdot,u)>1-\delta\})\leq d_N\delta(2\delta)^{(N-3)/2}\leq\sqrt{N}(2\delta)^{(N-3)/2} ν N ({ R ( ⋅ , u ) > 1 − δ }) ≤ d N δ ( 2 δ ) ( N − 3 ) /2 ≤ N ( 2 δ ) ( N − 3 ) /2 K \mathcal{K} K { sup S N ∣ H N ∣ ≤ K N } \{\sup_{S_N} |H_N| \le KN\} { sup S N ∣ H N ∣ ≤ K N } K \mathcal{K} K Z N , β ≥ e − β K N Z_{N,\beta} \ge e^{-\beta KN} Z N , β ≥ e − β K N π \pi π ν N \nu_N ν N e 2 β K N e^{2\beta KN} e 2 β K N K \mathcal{K} K
π ⊗ 2 { R 12 > 1 − δ } = ∫ π ( { x : R ( x , y ) > 1 − δ } ) π ( d y ) ≤ e 2 β K N N ( 2 δ ) ( N − 3 ) / 2 . \pi^{\otimes2}\{R_{12} > 1-\delta\}=\int\pi(\{x:R(x,y)>1-\delta\})\,\pi(\mathrm{d}y)\le e^{2\beta KN}\sqrt{N}(2\delta)^{(N-3)/2}. π ⊗ 2 { R 12 > 1 − δ } = ∫ π ({ x : R ( x , y ) > 1 − δ }) π ( d y ) ≤ e 2 β K N N ( 2 δ ) ( N − 3 ) /2 . The integration in y y y π \pi π log ( 2 δ ) = − 4 β K − 4 \log(2\delta)=-4\beta K-4 log ( 2 δ ) = − 4 β K − 4 N exp ( 2 β K N − ( N − 3 ) ( 2 β K + 2 ) ) = e 6 β K + 6 N e − 2 N \sqrt{N}\exp(2\beta KN-(N-3)(2\beta K+2))=e^{6\beta K+6}\sqrt{N}e^{-2N} N exp ( 2 β K N − ( N − 3 ) ( 2 β K + 2 )) = e 6 β K + 6 N e − 2 N K \mathcal{K} K π ⊗ 2 ( { R 12 > 1 − δ } ) \pi^{\otimes2}(\{R_{12}>1-\delta\}) π ⊗ 2 ({ R 12 > 1 − δ }) □ \square □
Replica exclusion and the norms of directing vectors
Recall from (5.6) the annealed probability p N , k , ρ ( q ) p_{N,k,\rho}(q) p N , k , ρ ( q ) k k k ρ \rho ρ q q q q q q N − 1 log p N , k , ρ ( q ) → 0 N^{-1}\log p_{N,k,\rho}(q)\to0 N − 1 log p N , k , ρ ( q ) → 0 k k k ρ \rho ρ q q q k k k ρ \rho ρ lim sup N N − 1 log p N , k , ρ ( q ) < 0 \limsup_N N^{-1}\log p_{N,k,\rho}(q)<0 lim sup N N − 1 log p N , k , ρ ( q ) < 0 p N , k , ρ ( q ) p_{N,k,\rho}(q) p N , k , ρ ( q ) V β = V V_\beta=V V β = V q ∈ ( 0 , 1 ) ∖ { Q } q\in(0,1)\setminus\{Q\} q ∈ ( 0 , 1 ) ∖ { Q } V ( q ) > 0 V(q)>0 V ( q ) > 0 q q q k ≥ 2 k\ge2 k ≥ 2 ρ ∈ ( 0 , 1 ) \rho\in(0,1) ρ ∈ ( 0 , 1 ) c > 0 c>0 c > 0 p N , k , ρ ( q ) ≤ e − c N p_{N,k,\rho}(q)\le e^{-cN} p N , k , ρ ( q ) ≤ e − c N N N N
The next lemma converts replica exclusion into a restriction on the directing measure. Its mechanism is the standard support argument for Dovbysh–Sudakov representations: vectors in a small ball around a point v v v ∥ v ∥ 2 \lVert v\rVert^2 ∥ v ∥ 2
Lemma 6.5 (Replica exclusion restricts the directing norms). Let A ⊂ [ 0 , 1 ) \mathcal{A}\subset[0,1) A ⊂ [ 0 , 1 ) [ 0 , 1 ) [0,1) [ 0 , 1 ) 0 ∈ A 0\in\mathcal{A} 0 ∈ A q ∈ ( 0 , 1 ) ∖ A q\in(0,1)\setminus\mathcal{A} q ∈ ( 0 , 1 ) ∖ A k q ≥ 2 k_q\ge2 k q ≥ 2 ρ q > 0 \rho_q>0 ρ q > 0 p N , k q , ρ q ( q ) → 0 p_{N,k_q,\rho_q}(q)\to0 p N , k q , ρ q ( q ) → 0 N → ∞ N\to\infty N → ∞ L \mathcal{L} L ( G ω ) ω ∈ Ω (G_\omega)_{\omega\in\Omega} ( G ω ) ω ∈ Ω L \mathcal{L} L Pr \Pr Pr ω \omega ω
G ω ( { v ∈ B : ∥ v ∥ 2 ∈ A } ) = 1. G_\omega\bigl(\{v\in B:\lVert v\rVert^2\in\mathcal{A}\}\bigr)=1. G ω ( { v ∈ B : ∥ v ∥ 2 ∈ A } ) = 1. Proof. Since A \mathcal{A} A [ 0 , 1 ) [0,1) [ 0 , 1 ) [ 0 , 1 ] [0,1] [ 0 , 1 ] { v ∈ B : ∥ v ∥ 2 ∈ A } \{v\in B:\lVert v\rVert^2\in\mathcal{A}\} { v ∈ B : ∥ v ∥ 2 ∈ A } B B B A \mathcal{A} A ( 0 , 1 ) ∖ A (0,1)\setminus\mathcal{A} ( 0 , 1 ) ∖ A G G G B B B supp G \operatorname{supp}G supp G v ∈ B v\in B v ∈ B G ( B ( v , r ) ) > 0 G(B(v,r))>0 G ( B ( v , r )) > 0 supp G \operatorname{supp} G supp G G G G ℓ 2 \ell^2 ℓ 2 G ( supp G ) = 1 G(\operatorname{supp} G)=1 G ( supp G ) = 1
Fix a subsequence ( N m ) (N_m) ( N m ) L N m \mathcal{L}_{N_m} L N m L \mathcal{L} L q ∈ ( 0 , 1 ) ∖ A q \in(0,1) \setminus\mathcal{A} q ∈ ( 0 , 1 ) ∖ A k = k q k=k_q k = k q ρ = ρ q \rho=\rho_q ρ = ρ q
E q = { max 1 ≤ i < j ≤ k ∣ R i j − q ∣ < ρ } ⊂ M E_q=\left\{\max_{1\leq i<j\leq k}\lvert R_{ij}-q\rvert<\rho\right\}\subset\mathcal{M} E q = { 1 ≤ i < j ≤ k max ∣ R ij − q ∣ < ρ } ⊂ M is open, since it is defined by strict inequalities between finitely many coordinates. By the portmanteau theorem and the hypothesis, we have L ( E q ) ≤ lim inf m p N m , k , ρ ( q ) = 0 \mathcal{L}(E_q)\leq\liminf_m p_{N_m,k,\rho}(q)=0 L ( E q ) ≤ lim inf m p N m , k , ρ ( q ) = 0 Pr \Pr Pr Ω q \Omega_q Ω q ω ∉ Ω q \omega\notin\Omega_q ω ∈ / Ω q
G ω ⊗ k ( { ( v 1 , … , v k ) : max 1 ≤ i < j ≤ k ∣ v i ⋅ v j − q ∣ < ρ } ) = L G ω ( E q ) = 0. (124) G_\omega^{\otimes k}\left(\left\{(v_1,\ldots,v_k):\max_{1\leq i<j\leq k}\lvert v_i\cdot v_j-q\rvert<\rho\right\}\right)=\mathcal{L}^{G_\omega}(E_q)=0. \tag*{(124)} G ω ⊗ k ( { ( v 1 , … , v k ) : 1 ≤ i < j ≤ k max ∣ v i ⋅ v j − q ∣ < ρ } ) = L G ω ( E q ) = 0. (124) Let ω ∉ Ω q \omega\notin\Omega_q ω ∈ / Ω q v ∈ supp G ω v\in\operatorname{supp}G_\omega v ∈ supp G ω ∣ ∥ v ∥ 2 − q ∣ < ρ / 2 \left|\|v\|^2-q\right|<\rho/2 ∥ v ∥ 2 − q < ρ /2 r = ρ / 4 r=\rho/4 r = ρ /4 v i , v j ∈ B ( v , r ) v_i,v_j\in B(v,r) v i , v j ∈ B ( v , r )
∣ v i ⋅ v j − ∥ v ∥ 2 ∣ ≤ ∣ ( v i − v ) ⋅ v j ∣ + ∣ v ⋅ ( v j − v ) ∣ < 2 r = ρ / 2 , \left|v_i\cdot v_j-\|v\|^2\right|\leq\left|(v_i-v)\cdot v_j\right|+\left|v\cdot(v_j-v)\right|<2r=\rho/2, v i ⋅ v j − ∥ v ∥ 2 ≤ ∣ ( v i − v ) ⋅ v j ∣ + ∣ v ⋅ ( v j − v ) ∣ < 2 r = ρ /2 , and therefore ∣ v i ⋅ v j − q ∣ < ρ \lvert v_i\cdot v_j-q\rvert<\rho ∣ v i ⋅ v j − q ∣ < ρ G ω ( B ( v , r ) ) k > 0 G_\omega(B(v,r))^k>0 G ω ( B ( v , r ) ) k > 0 ω ∉ Ω q \omega\notin\Omega_q ω ∈ / Ω q supp G ω \operatorname{supp}G_\omega supp G ω I q = ( q − ρ q / 2 , q + ρ q / 2 ) I_q=(q-\rho_q/2,q+\rho_q/2) I q = ( q − ρ q /2 , q + ρ q /2 )
The intervals I q I_q I q q ∈ ( 0 , 1 ) ∖ A q\in(0,1)\setminus\mathcal{A} q ∈ ( 0 , 1 ) ∖ A ( 0 , 1 ) ∖ A (0,1)\setminus\mathcal{A} ( 0 , 1 ) ∖ A R \mathbb{R} R I q 1 , I q 2 , … I_{q_1},I_{q_2},\ldots I q 1 , I q 2 , … ( 0 , 1 ) ∖ A (0,1)\setminus\mathcal{A} ( 0 , 1 ) ∖ A ω \omega ω ⋃ l Ω q l \bigcup_l\Omega_{q_l} ⋃ l Ω q l supp G ω \operatorname{supp}G_\omega supp G ω ( 0 , 1 ) ∖ A (0,1)\setminus\mathcal{A} ( 0 , 1 ) ∖ A G ω ( supp G ω ) = 1 G_\omega(\operatorname{supp}G_\omega)=1 G ω ( supp G ω ) = 1 G ω ( { ∥ v ∥ 2 ∈ ( 0 , 1 ) ∖ A } ) = 0 G_\omega(\{\|v\|^2\in(0,1)\setminus\mathcal{A}\})=0 G ω ({ ∥ v ∥ 2 ∈ ( 0 , 1 ) ∖ A }) = 0
It remains to exclude norm one. Let δ \delta δ { R 12 > 1 − δ } ⊂ M \{R_{12}>1-\delta\}\subset\mathcal{M} { R 12 > 1 − δ } ⊂ M L ( { R 12 > 1 − δ } ) = 0 \mathcal{L}(\{R_{12}>1-\delta\})=0 L ({ R 12 > 1 − δ }) = 0 Pr \Pr Pr Ω 1 \Omega_1 Ω 1 G ω ⊗ 2 ( { v 1 ⋅ v 2 > 1 − δ } ) = 0 G_\omega^{\otimes2}(\{v_1\cdot v_2>1-\delta\})=0 G ω ⊗ 2 ({ v 1 ⋅ v 2 > 1 − δ }) = 0 ω ∉ Ω 1 \omega\notin\Omega_1 ω ∈ / Ω 1 ω ∉ Ω 1 \omega\notin\Omega_1 ω ∈ / Ω 1 v ∈ supp G ω v\in\operatorname{supp}G_\omega v ∈ supp G ω ∥ v ∥ = 1 \|v\|=1 ∥ v ∥ = 1 v 1 , v 2 ∈ B ( v , δ / 2 ) v_1,v_2\in B(v,\delta/2) v 1 , v 2 ∈ B ( v , δ /2 ) r = δ / 2 r=\delta/2 r = δ /2 v 1 ⋅ v 2 > ∥ v ∥ 2 − δ = 1 − δ v_1\cdot v_2>\|v\|^2-\delta=1-\delta v 1 ⋅ v 2 > ∥ v ∥ 2 − δ = 1 − δ G ω ⊗ 2 ( { v 1 ⋅ v 2 > 1 − δ } ) ≥ G ω ( B ( v , δ / 2 ) ) 2 > 0 G_\omega^{\otimes2}(\{v_1\cdot v_2>1-\delta\})\geq G_\omega(B(v,\delta/2))^2>0 G ω ⊗ 2 ({ v 1 ⋅ v 2 > 1 − δ }) ≥ G ω ( B ( v , δ /2 ) ) 2 > 0 supp G ω \operatorname{supp}G_\omega supp G ω G ω ( { ∥ v ∥ = 1 } ) = 0 G_\omega(\{\|v\|=1\})=0 G ω ({ ∥ v ∥ = 1 }) = 0
The interval [ 0 , 1 ] [0,1] [ 0 , 1 ] A \mathcal{A} A ( 0 , 1 ) ∖ A (0,1)\setminus\mathcal{A} ( 0 , 1 ) ∖ A { 1 } \{1\} { 1 } 0 ∈ A ⊂ [ 0 , 1 ) 0\in\mathcal{A}\subset[0,1) 0 ∈ A ⊂ [ 0 , 1 ) ω \omega ω Ω 1 ∪ ⋃ l Ω q l \Omega_1\cup\bigcup_l\Omega_{q_l} Ω 1 ∪ ⋃ l Ω q l G ω G_\omega G ω
A Ghirlanda–Guerra identity for the covariance
The Ghirlanda–Guerra identities were discovered by Ghirlanda and Guerra [83], and they are closely related to the stochastic stability of Aizenman and Contucci [4]. In their full form, they hold for Hamiltonians with a small generic perturbation. We need only the single identity whose test function is the covariance ξ ( R ) \xi(R) ξ ( R ) β \beta β F F F H N / N H_N/N H N / N p p p
Proposition 6.6 (Self-averaging of the energy; [124, Remark following Theorem 1.1]). Let ξ \xi ξ β > 0 \beta> 0 β > 0 F F F β \beta β ε N → 0 \varepsilon_N \to0 ε N → 0 N → ∞ N \to\infty N → ∞
Proof. The remark following [124, Theorem 1.1] concerns probability measures ν N \nu_N ν N A N A_N A N ⟨ ⋅ ⟩ t \langle\cdot\rangle_t ⟨ ⋅ ⟩ t e t A N e^{tA_N} e t A N ν N \nu_N ν N ψ N ( t ) = N − 1 log ∫ e t A N d ν N \psi_N(t) = N^{-1}\log\int e^{tA_N}\,d\nu_N ψ N ( t ) = N − 1 log ∫ e t A N d ν N E ∣ ψ N ( t ) − E ψ N ( t ) ∣ → 0 \mathbb{E}|\psi_N(t) - \mathbb{E}\psi_N(t)| \to0 E ∣ ψ N ( t ) − E ψ N ( t ) ∣ → 0 E ψ N ( t ) \mathbb{E}\psi_N(t) E ψ N ( t ) P ( t ) P(t) P ( t ) t t t t 0 t_0 t 0 P P P t 0 t_0 t 0
lim N → ∞ 1 N E ∣ ⟨ A N − E ⟨ A N ⟩ t 0 ⟩ t 0 ∣ = 0 , \lim_{N\to\infty}\frac{1}{N}\mathbb{E}\left|\left\langle A_N-\mathbb{E}\langle A_N\rangle_{t_0}\right\rangle_{t_0}\right|=0, N → ∞ lim N 1 E ⟨ A N − E ⟨ A N ⟩ t 0 ⟩ t 0 = 0 , under measurability and integrability conditions that are implicit in its proof [124, (7), p. 190]. We take the disjoint union of the spheres S N S_N S N ν N \nu_N ν N S N S_N S N A N = H N A_N=H_N A N = H N S N S_N S N t 0 = β t_0=\beta t 0 = β P = F P=F P = F ε N → 0 \varepsilon_N\to0 ε N → 0
We verify the hypotheses. For every t > 0 t>0 t > 0 ψ N ( t ) = N − 1 log Z N , t \psi_N(t)=N^{-1}\log Z_{N,t} ψ N ( t ) = N − 1 log Z N , t B = S N B=S_N B = S N h = 0 h=0 h = 0
E ∣ ψ N ( t ) − E ψ N ( t ) ∣ = ∫ 0 ∞ P ( ∣ ψ N ( t ) − E ψ N ( t ) ∣ ≥ s ) d s ≤ ∫ 0 ∞ 2 e − s 2 N / ( 2 t 2 ξ ( 1 ) ) d s = t ( 2 π ξ ( 1 ) N ) 1 / 2 . \begin{aligned}
\mathbb{E}|\psi_N(t)-\mathbb{E}\psi_N(t)|
&=\int_0^\infty\mathbb{P}\left(|\psi_N(t)-\mathbb{E}\psi_N(t)|\ge s\right)\,ds\\
&\le\int_0^\infty2e^{-s^2N/(2t^2\xi(1))}\,ds
=t\left(\frac{2\pi\xi(1)}{N}\right)^{1/2}.
\end{aligned} E ∣ ψ N ( t ) − E ψ N ( t ) ∣ = ∫ 0 ∞ P ( ∣ ψ N ( t ) − E ψ N ( t ) ∣ ≥ s ) d s ≤ ∫ 0 ∞ 2 e − s 2 N / ( 2 t 2 ξ ( 1 )) d s = t ( N 2 π ξ ( 1 ) ) 1/2 . By Theorem 3.14, we have E ψ N ( t ) = F N ( t ) → F ( t ) \mathbb{E}\psi_N(t)=F_N(t)\to F(t) E ψ N ( t ) = F N ( t ) → F ( t ) t ≥ 0 t\ge0 t ≥ 0 F F F β \beta β H N ( x ) = g ⋅ Ψ ( x ) H_N(x)=g\cdot\Psi(x) H N ( x ) = g ⋅ Ψ ( x ) g g g ∣ Ψ ( x ) ∣ 2 = N ξ ( 1 ) |\Psi(x)|^2=N\xi(1) ∣Ψ ( x ) ∣ 2 = N ξ ( 1 ) x ∈ S N x\in S_N x ∈ S N H N H_N H N S N S_N S N sup S N ∣ H N ∣ ≤ ∣ g ∣ ( N ξ ( 1 ) ) 1 / 2 \sup_{S_N}|H_N|\le|g|(N\xi(1))^{1/2} sup S N ∣ H N ∣ ≤ ∣ g ∣ ( N ξ ( 1 ) ) 1/2 t t t
Recall from Corollary 5.16 that, under the 1RSB assumptions with parameters x x x Q Q Q F F F β \beta β F ′ ( β ) = β ( ξ ( 1 ) − ( 1 − x ) ξ ( Q ) ) F'(\beta)=\beta(\xi(1)-(1-x)\xi(Q)) F ′ ( β ) = β ( ξ ( 1 ) − ( 1 − x ) ξ ( Q )) (110) , and that e N = F N ′ ( β ) → F ′ ( β ) e_N=F_N'(\beta)\to F'(\beta) e N = F N ′ ( β ) → F ′ ( β )
Lemma 6.7 (The Ghirlanda–Guerra identity for ξ ( R ) \xi(R) ξ ( R ) ( ξ , β ) (\xi,\beta) ( ξ , β ) x x x Q Q Q Q Q Q
For every N ≥ 2 N \ge2 N ≥ 2 n ≥ 1 n \ge1 n ≥ 1 f f f S N n S_N^n S N n ( σ 1 , … , σ n ) (\sigma^1,\ldots,\sigma^n) ( σ 1 , … , σ n )
∣ E ⟨ f ξ ( R 1 , n + 1 ) ⟩ − ( 1 − x ) ξ ( Q ) n E ⟨ f ⟩ − 1 n ∑ j = 2 n E ⟨ f ξ ( R 1 j ) ⟩ ∣ ≤ ∥ f ∥ ∞ β n ( ε N + ∣ e N − F ′ ( β ) ∣ ) . (125) \left|\mathbb{E}\langle f\xi(R_{1,n+1})\rangle-\frac{(1-x)\xi(Q)}{n}\mathbb{E}\langle f\rangle-\frac{1}{n}\sum_{j=2}^{n}\mathbb{E}\langle f\xi(R_{1j})\rangle\right| \le\frac{\|f\|_\infty}{\beta n}\left(\varepsilon_N+\left|e_N-F'(\beta)\right|\right). \tag*{(125)} E ⟨ f ξ ( R 1 , n + 1 )⟩ − n ( 1 − x ) ξ ( Q ) E ⟨ f ⟩ − n 1 j = 2 ∑ n E ⟨ f ξ ( R 1 j )⟩ ≤ β n ∥ f ∥ ∞ ( ε N + ∣ e N − F ′ ( β ) ∣ ) . (125) Moreover, ε N + ∣ e N − F ′ ( β ) ∣ → 0 \varepsilon_N+\left|e_N-F'(\beta)\right|\to0 ε N + ∣ e N − F ′ ( β ) ∣ → 0 N → ∞ N\to\infty N → ∞
Let L \mathcal{L} L n ≥ 1 n\ge1 n ≥ 1 f f f ( R i j ) 1 ≤ i < j ≤ n (R_{ij})_{1\le i<j\le n} ( R ij ) 1 ≤ i < j ≤ n
E L [ f ξ ( R 1 , n + 1 ) ] = ( 1 − x ) ξ ( Q ) n E L f + 1 n ∑ j = 2 n E L [ f ξ ( R 1 j ) ] . (126) \mathbb{E}_{\mathcal{L}}\left[f\xi(R_{1,n+1})\right]=\frac{(1-x)\xi(Q)}{n}\mathbb{E}_{\mathcal{L}}f+\frac{1}{n}\sum_{j=2}^{n}\mathbb{E}_{\mathcal{L}}\left[f\xi(R_{1j})\right]. \tag*{(126)} E L [ f ξ ( R 1 , n + 1 ) ] = n ( 1 − x ) ξ ( Q ) E L f + n 1 j = 2 ∑ n E L [ f ξ ( R 1 j ) ] . (126) In particular, E L ξ ( R 12 ) = ( 1 − x ) ξ ( Q ) \mathbb{E}_{\mathcal{L}}\xi(R_{12})=(1-x)\xi(Q) E L ξ ( R 12 ) = ( 1 − x ) ξ ( Q )
For n = 1 n=1 n = 1 j j j f f f
Proof. We begin with the last assertion of (1). By Corollary 5.16 , we have e N = F N ′ ( β ) → F ′ ( β ) e_N=F'_N(\beta)\to F'(\beta) e N = F N ′ ( β ) → F ′ ( β ) F F F β \beta β Proposition 6.6 implies that ε N → 0 \varepsilon_N\to0 ε N → 0
Next, we derive an exact identity at finite N N N H N ( x ) = g ⋅ Ψ ( x ) = ∑ α g α Ψ α ( x ) H_N(x)=g\cdot\Psi(x)=\sum_\alpha g_\alpha\Psi_\alpha(x) H N ( x ) = g ⋅ Ψ ( x ) = ∑ α g α Ψ α ( x ) Proposition 6.6 . By the computation in the proof of Lemma 3.2(1), we have ∑ α Ψ α ( x ) Ψ α ( y ) = N ξ ( R ( x , y ) ) \sum_\alpha\Psi_\alpha(x)\Psi_\alpha(y)=N\xi(R(x,y)) ∑ α Ψ α ( x ) Ψ α ( y ) = N ξ ( R ( x , y )) n ≥ 1 n\ge1 n ≥ 1 f f f S N n S_N^n S N n α \alpha α Φ α ( g ) = ⟨ f Ψ α ( σ 1 ) ⟩ \Phi_\alpha(g)=\langle f\Psi_\alpha(\sigma^1)\rangle Φ α ( g ) = ⟨ f Ψ α ( σ 1 )⟩ g g g π \pi π n n n Z N , β n Z_{N,\beta}^n Z N , β n
∂ g α Φ α ( g ) = β ⟨ f Ψ α ( σ 1 ) ( ∑ l = 1 n Ψ α ( σ l ) − n Ψ α ( σ n + 1 ) ) ⟩ . \partial_{g_\alpha}\Phi_\alpha(g)=\beta\left\langle f\Psi_\alpha(\sigma^1)\left(\sum_{l=1}^{n}\Psi_\alpha(\sigma^l)-n\Psi_\alpha(\sigma^{n+1})\right)\right\rangle. ∂ g α Φ α ( g ) = β ⟨ f Ψ α ( σ 1 ) ( l = 1 ∑ n Ψ α ( σ l ) − n Ψ α ( σ n + 1 ) ) ⟩ . The function f f f g g g ∣ Ψ α ∣ ≤ ( N ξ ( 1 ) ) 1 / 2 |\Psi_\alpha|\le(N\xi(1))^{1/2} ∣ Ψ α ∣ ≤ ( N ξ ( 1 ) ) 1/2 S N S_N S N Φ α \Phi_\alpha Φ α E [ g α Φ α ( g ) ] = E [ ∂ g α Φ α ( g ) ] \mathbb{E}[g_\alpha\Phi_\alpha(g)]=\mathbb{E}[\partial_{g_\alpha}\Phi_\alpha(g)] E [ g α Φ α ( g )] = E [ ∂ g α Φ α ( g )] α \alpha α N N N
E ⟨ f H N ( σ 1 ) N ⟩ = β E ⟨ f ( ξ ( 1 ) + ∑ j = 2 n ξ ( R 1 j ) − n ξ ( R 1 , n + 1 ) ) ⟩ , (127) \mathbb{E}\left\langle f\frac{H_N(\sigma^1)}{N}\right\rangle=\beta\mathbb{E}\left\langle f\left(\xi(1)+\sum_{j=2}^{n}\xi(R_{1j})-n\xi(R_{1,n+1})\right)\right\rangle, \tag*{(127)} E ⟨ f N H N ( σ 1 ) ⟩ = β E ⟨ f ( ξ ( 1 ) + j = 2 ∑ n ξ ( R 1 j ) − n ξ ( R 1 , n + 1 ) ) ⟩ , (127) where the term l = 1 l=1 l = 1 ξ ( R 11 ) = ξ ( 1 ) \xi(R_{11})=\xi(1) ξ ( R 11 ) = ξ ( 1 )
Taking n = 1 n=1 n = 1 f = 1 f=1 f = 1 (127) , we have e N = β ( ξ ( 1 ) − E ⟨ ξ ( R 12 ) ⟩ ) e_N=\beta\bigl(\xi(1)-\mathbb{E}\langle\xi(R_{12})\rangle\bigr) e N = β ( ξ ( 1 ) − E ⟨ ξ ( R 12 )⟩ ) (127) and rearranging, we find
n E ⟨ f ξ ( R 1 , n + 1 ) ⟩ − ∑ j = 2 n E ⟨ f ξ ( R 1 j ) ⟩ − E ⟨ ξ ( R 12 ) ⟩ E ⟨ f ⟩ = − 1 β E ⟨ f ( H N ( σ 1 ) N − e N ) ⟩ . n\mathbb{E}\langle f\xi(R_{1,n+1})\rangle-\sum_{j=2}^{n}\mathbb{E}\langle f\xi(R_{1j})\rangle-\mathbb{E}\langle\xi(R_{12})\rangle\mathbb{E}\langle f\rangle=-\frac{1}{\beta}\mathbb{E}\left\langle f\left(\frac{H_N(\sigma^1)}{N}-e_N\right)\right\rangle. n E ⟨ f ξ ( R 1 , n + 1 )⟩ − j = 2 ∑ n E ⟨ f ξ ( R 1 j )⟩ − E ⟨ ξ ( R 12 )⟩ E ⟨ f ⟩ = − β 1 E ⟨ f ( N H N ( σ 1 ) − e N ) ⟩ . The right side has absolute value at most ∥ f ∥ ∞ ε N / β \lVert f\rVert_\infty\varepsilon_N/\beta ∥ f ∥ ∞ ε N / β (110) and the formula for e N e_N e N ∣ E ⟨ ξ ( R 12 ) ⟩ − ( 1 − x ) ξ ( Q ) ∣ = ∣ e N − F ′ ( β ) ∣ / β \left|\mathbb{E}\langle\xi(R_{12})\rangle-(1-x)\xi(Q)\right|=\left|e_N-F'(\beta)\right|/\beta ∣ E ⟨ ξ ( R 12 )⟩ − ( 1 − x ) ξ ( Q ) ∣ = ∣ e N − F ′ ( β ) ∣ / β ∣ E ⟨ f ⟩ ∣ ≤ ∥ f ∥ ∞ \left|\mathbb{E}\langle f\rangle\right|\leq\lVert f\rVert_\infty ∣ E ⟨ f ⟩ ∣ ≤ ∥ f ∥ ∞ n n n (125) .
For (2), let ( N m ) (N_m) ( N m ) L N m \mathcal{L}_{N_m} L N m L \mathcal{L} L R ( n ) = ( R i j ) 1 ≤ i < j ≤ n R^{(n)}=(R_{ij})_{1\leq i<j\leq n} R ( n ) = ( R ij ) 1 ≤ i < j ≤ n f f f f ( R ( n ) ) f(R^{(n)}) f ( R ( n ) ) f ( R ( n ) ) ξ ( R 1 j ) f(R^{(n)})\xi(R_{1j}) f ( R ( n ) ) ξ ( R 1 j ) 2 ≤ j ≤ n + 1 2\leq j\leq n+1 2 ≤ j ≤ n + 1 M \mathcal{M} M (125) to the bounded Borel function ( σ 1 , … , σ n ) ↦ f ( ( R ( σ i , σ j ) ) 1 ≤ i < j ≤ n ) (\sigma^1,\ldots,\sigma^n)\mapsto f((R(\sigma^i,\sigma^j))_{1\leq i<j\leq n}) ( σ 1 , … , σ n ) ↦ f (( R ( σ i , σ j ) ) 1 ≤ i < j ≤ n ) m → ∞ m\to\infty m → ∞ (125) tends to zero, which proves (126) for continuous f f f f f f (126) have the form ∫ f d μ \int f\,\mathrm{d}\mu ∫ f d μ μ \mu μ [ − 1 , 1 ] n ( n − 1 ) / 2 [-1,1]^{n(n-1)/2} [ − 1 , 1 ] n ( n − 1 ) /2 ξ \xi ξ [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] μ ( A ) = E L [ 1 A ( R ( n ) ) ξ ( R 1 , n + 1 ) ] \mu(A)=\mathbb{E}_{\mathcal{L}}[1_A(R^{(n)})\xi(R_{1,n+1})] μ ( A ) = E L [ 1 A ( R ( n ) ) ξ ( R 1 , n + 1 )] (126) extends to every bounded Borel f f f n = 1 n=1 n = 1 f = 1 f=1 f = 1 E L ξ ( R 12 ) = ( 1 − x ) ξ ( Q ) \mathbb{E}_{\mathcal{L}}\xi(R_{12})=(1-x)\xi(Q) E L ξ ( R 12 ) = ( 1 − x ) ξ ( Q )
Proof of the theorem on directing norms
Let L \mathcal{L} L ( G ω ) ω ∈ Ω (G_\omega)_{\omega\in\Omega} ( G ω ) ω ∈ Ω L \mathcal{L} L { 0 , Q } \{0,Q\} { 0 , Q }
Squared norms in { 0 , Q } \{0,Q\} { 0 , Q } Under the 1RSB assumptions, V β = V V_\beta=V V β = V V ( q ) > 0 V(q)>0 V ( q ) > 0 q ∈ ( 0 , 1 ) ∖ { Q } q\in(0,1)\setminus\{Q\} q ∈ ( 0 , 1 ) ∖ { Q } q q q k q ≥ 2 k_q\geq2 k q ≥ 2 ρ q > 0 \rho_q>0 ρ q > 0 p N , k q , ρ q ( q ) → 0 p_{N,k_q,\rho_q}(q)\to0 p N , k q , ρ q ( q ) → 0 A = { 0 , Q } \mathcal{A}=\{0,Q\} A = { 0 , Q } Pr \Pr Pr ω \omega ω G ω G_\omega G ω 0 0 0 Q Q Q ω \omega ω v 1 , v 2 v_1,v_2 v 1 , v 2 G ω G_\omega G ω ∣ v 1 ⋅ v 2 ∣ ≤ ∥ v 1 ∥ ∥ v 2 ∥ ≤ Q |v_1\cdot v_2|\leq\lVert v_1\rVert\lVert v_2\rVert\leq Q ∣ v 1 ⋅ v 2 ∣ ≤ ∥ v 1 ∥ ∥ v 2 ∥ ≤ Q ω \omega ω L \mathcal{L} L ∣ R i j ∣ ≤ Q |R_{ij}|\leq Q ∣ R ij ∣ ≤ Q i ≠ j i\neq j i = j L \mathcal{L} L ξ \xi ξ L \mathcal{L} L
ξ ( R i j ) ≤ ∑ p c p ∣ R i j ∣ p ≤ ξ ( Q ) ( i ≠ j ) . (128) \xi(R_{ij})\leq\sum_p c_p|R_{ij}|^p\leq\xi(Q)\qquad(i\neq j). \tag*{(128)} ξ ( R ij ) ≤ p ∑ c p ∣ R ij ∣ p ≤ ξ ( Q ) ( i = j ) . (128) Note that (128) holds also when ξ \xi ξ ξ ( R i j ) \xi(R_{ij}) ξ ( R ij )
A recursion. For n ≥ 1 n\geq1 n ≥ 1
A n = { ξ ( R 1 j ) = 0 for 2 ≤ j ≤ n } ⊂ M , a n = L ( A n ) , A_n=\{\xi(R_{1j})=0\text{ for }2\leq j\leq n\}\subset\mathcal{M},\qquad a_n=\mathcal{L}(A_n), A n = { ξ ( R 1 j ) = 0 for 2 ≤ j ≤ n } ⊂ M , a n = L ( A n ) , so that A 1 = M A_1=\mathcal{M} A 1 = M a 1 = 1 a_1=1 a 1 = 1 1 A n \mathbf{1}_{A_n} 1 A n ( R i j ) 1 ≤ i < j ≤ n (R_{ij})_{1\le i<j\le n} ( R ij ) 1 ≤ i < j ≤ n A n A_n A n ζ ( R 1 j ) = 0 \zeta(R_{1j})=0 ζ ( R 1 j ) = 0 2 ≤ j ≤ n 2\le j\le n 2 ≤ j ≤ n (126) vanishes for f = 1 A n f=\mathbf{1}_{A_n} f = 1 A n
E L [ 1 A n ζ ( R 1 , n + 1 ) ] = ( 1 − x ) ζ ( Q ) n a n . \mathbb{E}_{\mathcal{L}}\left[\mathbf{1}_{A_n}\zeta(R_{1,n+1})\right]=\frac{(1-x)\zeta(Q)}{n}a_n. E L [ 1 A n ζ ( R 1 , n + 1 ) ] = n ( 1 − x ) ζ ( Q ) a n . On the other hand, A n + 1 = A n ∩ { ζ ( R 1 , n + 1 ) = 0 } A_{n+1}=A_n\cap\{\zeta(R_{1,n+1})=0\} A n + 1 = A n ∩ { ζ ( R 1 , n + 1 ) = 0 } A n + 1 A_{n+1} A n + 1 A n ∖ A n + 1 A_n\setminus A_{n+1} A n ∖ A n + 1 ζ ( Q ) \zeta(Q) ζ ( Q ) (128) . The left side is therefore at most ζ ( Q ) ( a n − a n + 1 ) \zeta(Q)(a_n-a_{n+1}) ζ ( Q ) ( a n − a n + 1 ) ζ ( Q ) > 0 \zeta(Q)>0 ζ ( Q ) > 0
a n + 1 ≤ ( 1 − 1 − x n ) a n . a_{n+1}\le\left(1-\frac{1-x}{n}\right)a_n. a n + 1 ≤ ( 1 − n 1 − x ) a n . Every factor lies in ( 0 , 1 ) (0,1) ( 0 , 1 ) 0 < 1 − x < 1 0<1-x<1 0 < 1 − x < 1 a 1 = 1 a_1=1 a 1 = 1
a n ≤ ∏ j = 1 n − 1 ( 1 − 1 − x j ) ≤ exp ( − ( 1 − x ) ∑ j = 1 n − 1 1 j ) ≤ n − ( 1 − x ) . (129) a_n\le\prod_{j=1}^{n-1}\left(1-\frac{1-x}{j}\right)\le\exp\left(-(1-x)\sum_{j=1}^{n-1}\frac{1}{j}\right)\le n^{-(1-x)}. \tag*{(129)} a n ≤ j = 1 ∏ n − 1 ( 1 − j 1 − x ) ≤ exp ( − ( 1 − x ) j = 1 ∑ n − 1 j 1 ) ≤ n − ( 1 − x ) . (129) where we used log ( 1 − t ) ≤ − t \log(1-t)\le-t log ( 1 − t ) ≤ − t ∑ j = 1 n − 1 j − 1 ≥ ∫ 1 n d t / t = log n \sum_{j=1}^{n-1}j^{-1}\ge\int_1^n dt/t=\log n ∑ j = 1 n − 1 j − 1 ≥ ∫ 1 n d t / t = log n
No mass at the origin. Fix ω ∈ Ω \omega\in\Omega ω ∈ Ω v 1 , v 2 , … v_1,v_2,\ldots v 1 , v 2 , … G ω G_\omega G ω v 1 = 0 v_1=0 v 1 = 0 R 1 j = v 1 ⋅ v j = 0 R_{1j}=v_1\cdot v_j=0 R 1 j = v 1 ⋅ v j = 0 ζ ( R 1 j ) = 0 \zeta(R_{1j})=0 ζ ( R 1 j ) = 0 j ≥ 2 j\ge2 j ≥ 2 A n A_n A n n n n G ω ( { 0 } ) ≤ L G ω ( A n ) G_\omega(\{0\})\le\mathcal{L}^{G_\omega}(A_n) G ω ({ 0 }) ≤ L G ω ( A n ) n n n L G ω ( A n ) \mathcal{L}^{G_\omega}(A_n) L G ω ( A n ) n n n (129) ,
∫ Ω L G ω ( A n ) Pr ( d ω ) = a n ≤ n − ( 1 − x ) . \int_\Omega\mathcal{L}^{G_\omega}(A_n)\,\Pr(d\omega)=a_n\le n^{-(1-x)}. ∫ Ω L G ω ( A n ) Pr ( d ω ) = a n ≤ n − ( 1 − x ) . By dominated convergence, lim n L G ω ( A n ) = 0 \lim_n\mathcal{L}^{G_\omega}(A_n)=0 lim n L G ω ( A n ) = 0 Pr \Pr Pr ω \omega ω G ω ( { 0 } ) = 0 G_\omega(\{0\})=0 G ω ({ 0 }) = 0 Pr \Pr Pr ω \omega ω G ω G_\omega G ω { ∥ v ∥ 2 = Q } \{\lVert v\rVert^2=Q\} {∥ v ∥ 2 = Q } Pr \Pr Pr ω \omega ω Theorem 6.3 .
Determinant moments and the Kac–Rice formula
This section relates the critical points of the center field H Q H_Q H Q H Q H_Q H Q
The section uses only the definition of the model (Definition 3.1), the construction of the Langevin dynamics (Definition 3.4 and Lemma 3.5), and the measurability of its path laws (Lemma 4.2). Corollary 7.24 takes as hypotheses a hitting estimate for the caps under critical-point conditioning, which is proved in Section 8, and a bound on expected counts, which is proved in Sections 9 and 11. The results are used as follows: Definition 7.3 and Corollary 7.6 in Section 8; Theorem 7.17 in the criteria of Section 4 for separation and for the concentration of the Gibbs measure on bands; Theorem 7.23 and the argument of Corollary 7.24 in Proposition 13.1; Proposition 7.7, Theorem 7.17, and Corollary 7.18 in the estimates for counts and for the concentration on bands of Sections 9 to 12; and Proposition 7.16 and Lemmas 7.11 and 7.13 in the two-point estimates of Section 11. No asymptotics of expected counts are proved here.
The Kac–Rice formula is classical [3, Theorems 11.2.1 and 12.1.1], [14, Theorems 6.2 and 6.4]. Its use for spherical spin glasses goes back to Auffinger, Ben Arous, and Černý [12] and Auffinger and Ben Arous [8]. The one-point law of a mixed model at a point of a sphere of smaller radius is computed in [30, Section 4.1]. The determinantweighted Hölder argument of Lemma 7.22 is a form of the conditional Cauchy–Schwarz argument of Huang and Sellke [90, Corollary 3.16]. Two points require more care than in these references. First, the weights to which we apply the formula are probabilities of trajectory events of the diffusion in the given field. They are bounded Borel functions of the whole field, but they need not be continuous and they are not functions of finitely many derivatives at the point. The particular textbook and spin-glass statements cited above assume that these weights, which are called marks there, are continuous [14, Theorem 6.4], or that they are given by a tame auxiliary field [141, Lemma 36]. General Kac–Rice formulas allowing Borel weights that depend on the whole random field were subsequently established by Stecconi [139, Theorem 29]; see also [116, Proposition 6.1]. We retain a direct proof in coefficient space to specify the conditioning version and the observables used here. Second, the event that the gradient vanishes at a given point has probability zero, and a conditional law given this event is defined only up to a choice of version. We therefore fix an explicit version in Definition 7.3 and prove the formula in the coefficient space of the polynomial field (Proposition 7.16). The determinant bounds are proved directly from Gaussian concentration, and this yields all fixed moments uniformly in the dimension.
Standing notation
Throughout this section, ξ ( r ) = ∑ p ∈ P c p r p \xi(r) = \sum_{p \in\mathcal{P}} c_p r^p ξ ( r ) = ∑ p ∈ P c p r p P = { p ≥ 2 : c p > 0 } \mathcal{P} = \{p \ge2 : c_p > 0\} P = { p ≥ 2 : c p > 0 } d = min P d = \min\mathcal{P} d = min P ξ \xi ξ pure if P = { p } \mathcal{P} = \{p\} P = { p } p p p non-pure otherwise. Part II assumes d ≥ 4 d \ge4 d ≥ 4 Lemma 7.13 and Corollary 7.24 . We fix Q ∈ ( 0 , 1 ] Q \in(0,1] Q ∈ ( 0 , 1 ] Q Q Q N N N N ≥ 2 N \ge2 N ≥ 2 n = N − 1 n = N - 1 n = N − 1 C , c > 0 C,c > 0 C , c > 0
The field and its coefficients. Let Z = ( Z p , i 1 , … , i p ) Z = (Z_{p,i_1,\ldots,i_p}) Z = ( Z p , i 1 , … , i p ) p ∈ P p \in\mathcal{P} p ∈ P i 1 , … , i p ∈ { 1 , … , N } i_1,\ldots,i_p \in\{1,\ldots,N\} i 1 , … , i p ∈ { 1 , … , N } R ℓ N \mathbb{R}^{\ell_N} R ℓ N ℓ N = ∑ p ∈ P N p \ell_N = \sum_{p \in\mathcal{P}} N^p ℓ N = ∑ p ∈ P N p z ∈ R ℓ N z \in\mathbb{R}^{\ell_N} z ∈ R ℓ N x ∈ R N x \in\mathbb{R}^N x ∈ R N
H z ( x ) = ∑ p ∈ P c p N − ( p − 1 ) / 2 ∑ i 1 , … , i p = 1 N z p , i 1 , … , i p x i 1 ⋯ x i p . (130) H_z(x) = \sum_{p \in\mathcal{P}} \sqrt{c_p}N^{-(p-1)/2}\sum_{i_1,\ldots,i_p=1}^{N} z_{p,i_1,\ldots,i_p}x_{i_1}\cdots x_{i_p}. \tag*{(130)} H z ( x ) = p ∈ P ∑ c p N − ( p − 1 ) /2 i 1 , … , i p = 1 ∑ N z p , i 1 , … , i p x i 1 ⋯ x i p . (130) Then E H Z ( x ) H Z ( y ) = ∑ p c p N 1 − p ( x ⋅ y ) p = N ξ ( x ⋅ y / N ) \mathbb{E}H_Z(x)H_Z(y) = \sum_p c_pN^{1-p}(x \cdot y)^p = N\xi(x \cdot y/N) E H Z ( x ) H Z ( y ) = ∑ p c p N 1 − p ( x ⋅ y ) p = N ξ ( x ⋅ y / N ) x , y ∈ R N x,y \in\mathbb{R}^N x , y ∈ R N H Z H_Z H Z S N S_N S N H N H_N H N H N = H Z H_N = H_Z H N = H Z H N H_N H N R N \mathbb{R}^N R N H N , p H_{N,p} H N , p p p p H N H_N H N H N \mathcal{H}_N H N R N \mathbb{R}^N R N P \mathcal{P} P z ↦ H z z \mapsto H_z z ↦ H z R ℓ N \mathbb{R}^{\ell_N} R ℓ N H N \mathcal{H}_N H N
A field-dependent weight is a Borel function φ : H N × S N → R \varphi:\mathcal{H}_N \times S_N \to\mathbb{R} φ : H N × S N → R φ ( H N , u ) \varphi(H_N,u) φ ( H N , u ) u u u ( z , u ) ↦ φ ( H z , u ) (z,u) \mapsto\varphi(H_z,u) ( z , u ) ↦ φ ( H z , u ) ( z , u ) ↦ ( H z , u ) (z,u) \mapsto(H_z,u) ( z , u ) ↦ ( H z , u )
The center field. For u ∈ R N u \in\mathbb{R}^N u ∈ R N
H Q ( u ) = H N ( Q u ) , K Q ( u ) = d d t H N ( t Q u ) ∣ t = 1 = u ⋅ ∇ H Q ( u ) . H_Q(u) = H_N(\sqrt{Q}u), \qquad K_Q(u) = \left.\frac{\mathrm{d}}{\mathrm{d}t} H_N(t\sqrt{Q}u)\right|_{t=1} = u \cdot\nabla H_Q(u). H Q ( u ) = H N ( Q u ) , K Q ( u ) = d t d H N ( t Q u ) t = 1 = u ⋅ ∇ H Q ( u ) . and H z , Q H_{z,Q} H z , Q K z , Q K_{z,Q} K z , Q H z H_z H z H Q = ∑ p Q p / 2 H N , p H_Q = \sum_p Q^{p/2}H_{N,p} H Q = ∑ p Q p /2 H N , p K Q = ∑ p p Q p / 2 H N , p K_Q = \sum_p pQ^{p/2}H_{N,p} K Q = ∑ p p Q p /2 H N , p u ∈ S N u \in S_N u ∈ S N K Q ( u ) = Q N ∂ R H N ( Q u ) K_Q(u) = \sqrt{QN}\,\partial_R H_N(\sqrt{Q}u) K Q ( u ) = QN ∂ R H N ( Q u ) ∂ R \partial_R ∂ R K Q K_Q K Q K Q ( u ) K_Q(u) K Q ( u ) radial derivative of H Q H_Q H Q u u u N N N Var K Q ( u ) = N ( a Q + b Q ) \operatorname{Var} K_Q(u) = N(a_Q+b_Q) Var K Q ( u ) = N ( a Q + b Q ) Lemma 7.1 . The covariance of H Q H_Q H Q E H Q ( x ) H Q ( y ) = N ξ ~ Q ( x ⋅ y / N ) \mathbb{E}H_Q(x)H_Q(y) = N\widetilde{\xi}_Q(x\cdot y/N) E H Q ( x ) H Q ( y ) = N ξ Q ( x ⋅ y / N ) ξ ~ Q ( t ) = ξ ~ ( Q t ) \widetilde{\xi}_Q(t)=\widetilde{\xi}(Qt) ξ Q ( t ) = ξ ( Qt )
ξ ~ Q ( 1 ) = ξ ~ ( Q ) , a Q = ξ ~ Q ′ ( 1 ) = Q ξ ~ ′ ( Q ) , b Q = ξ ~ Q ′ ′ ( 1 ) = Q 2 ξ ~ ′ ′ ( Q ) , \widetilde{\xi}_Q(1)=\widetilde{\xi}(Q), \qquad a_Q=\widetilde{\xi}_Q'(1)=Q\widetilde{\xi}'(Q), \qquad b_Q=\widetilde{\xi}_Q''(1)=Q^2\widetilde{\xi}''(Q), ξ Q ( 1 ) = ξ ( Q ) , a Q = ξ Q ′ ( 1 ) = Q ξ ′ ( Q ) , b Q = ξ Q ′′ ( 1 ) = Q 2 ξ ′′ ( Q ) , and Σ Q \Sigma_Q Σ Q 2 × 2 2\times2 2 × 2 ( ξ ~ ( Q ) , a Q ) (\widetilde{\xi}(Q),a_Q) ( ξ ( Q ) , a Q ) ( a Q , a Q + b Q ) (a_Q,a_Q+b_Q) ( a Q , a Q + b Q )
For u ∈ S N u \in S_N u ∈ S N u ⊥ = { v ∈ R N : v ⋅ u = 0 } u^\perp=\{v\in\mathbb{R}^N:v\cdot u=0\} u ⊥ = { v ∈ R N : v ⋅ u = 0 } ∇ S N H Q ( u ) ∈ u ⊥ \nabla_{S_N}H_Q(u)\in u^\perp ∇ S N H Q ( u ) ∈ u ⊥ ∇ H Q ( u ) \nabla H_Q(u) ∇ H Q ( u ) u ⊥ u^\perp u ⊥ ∇ S N 2 H Q ( u ) \nabla^2_{S_N}H_Q(u) ∇ S N 2 H Q ( u ) u ⊥ u^\perp u ⊥ ∇ 2 H Q ( u ) \nabla^2 H_Q(u) ∇ 2 H Q ( u ) u ⊥ u^\perp u ⊥ v ↦ P u ∇ 2 H Q ( u ) v v\mapsto P_u\nabla^2H_Q(u)v v ↦ P u ∇ 2 H Q ( u ) v u ⊥ u^\perp u ⊥ P u = Id − u u T / N P_u=\operatorname{Id}-uu^{T}/N P u = Id − u u T / N H Q H_Q H Q u ∈ S N u\in S_N u ∈ S N ∇ S N H Q ( u ) = 0 \nabla_{S_N}H_Q(u)=0 ∇ S N H Q ( u ) = 0 h = H Q ( u ) / N h=H_Q(u)/N h = H Q ( u ) / N k = K Q ( u ) / N k=K_Q(u)/N k = K Q ( u ) / N S N S_N S N
∣ S N ∣ = 2 π N / 2 Γ ( N / 2 ) N ( N − 1 ) / 2 , |S_N|=\frac{2\pi^{N/2}}{\Gamma(N/2)}N^{(N-1)/2}, ∣ S N ∣ = Γ ( N /2 ) 2 π N /2 N ( N − 1 ) /2 , and ν N \nu_N ν N
Random matrices. For m ≥ 1 m\geq1 m ≥ 1 G m G_m G m m × m m\times m m × m 1 / m 1/m 1/ m 2 / m 2/m 2/ m
W n , N = b Q n N G n , μ n , N ( k ) = E ∣ det ( W n , N − k Id n ) ∣ ( k ∈ R ) . W_{n,N}=\sqrt{\frac{b_Q n}{N}}G_n,\qquad\mu_{n,N}(k)=\mathbb{E}\left|\det(W_{n,N}-k\operatorname{Id}_n)\right| \qquad(k\in\mathbb{R}). W n , N = N b Q n G n , μ n , N ( k ) = E ∣ det ( W n , N − k Id n ) ∣ ( k ∈ R ) . The entries of W n , N W_{n,N} W n , N b Q / N b_Q/N b Q / N 2 b Q / N 2b_Q/N 2 b Q / N N N N n n n M M M Tr ( W n , N M ) = ∑ i W i i M i i + 2 ∑ i < j W i j M i j \operatorname{Tr}(W_{n,N}M)=\sum_i W_{ii}M_{ii}+2\sum_{i<j}W_{ij}M_{ij} Tr ( W n , N M ) = ∑ i W ii M ii + 2 ∑ i < j W ij M ij
Var Tr ( W n , N M ) = 2 b Q N ∑ i M i i 2 + 4 b Q N ∑ i < j M i j 2 = 2 b Q N Tr ( M 2 ) . (131) \operatorname{Var}\operatorname{Tr}(W_{n,N}M)=\frac{2b_Q}{N}\sum_i M_{ii}^2+\frac{4b_Q}{N}\sum_{i<j}M_{ij}^2=\frac{2b_Q}{N}\operatorname{Tr}(M^2). \tag*{(131)} Var Tr ( W n , N M ) = N 2 b Q i ∑ M ii 2 + N 4 b Q i < j ∑ M ij 2 = N 2 b Q Tr ( M 2 ) . (131) A symmetric operator on an n n n W n , N W_{n,N} W n , N W n , N W_{n,N} W n , N det ( W n , N − k Id n ) ≠ 0 \det(W_{n,N}-k\operatorname{Id}_n) \ne0 det ( W n , N − k Id n ) = 0 μ n , N ( k ) > 0 \mu_{n,N}(k)>0 μ n , N ( k ) > 0 k k k
The one-point law and the conditioned laws
Lemma 7.1 (One-point law). Let u ∈ S N u \in S_N u ∈ S N
( H Q ( u ) , K Q ( u ) ) , ∇ S N H Q ( u ) , ∇ ⊥ 2 H Q ( u ) (H_Q(u),K_Q(u)), \qquad\nabla_{S_N}H_Q(u), \qquad\nabla_{\perp}^{2}H_Q(u) ( H Q ( u ) , K Q ( u )) , ∇ S N H Q ( u ) , ∇ ⊥ 2 H Q ( u ) are centered and independent. The pair ( H Q ( u ) , K Q ( u ) ) (H_Q(u),K_Q(u)) ( H Q ( u ) , K Q ( u )) N Σ Q N\Sigma_Q N Σ Q ∇ S N H Q ( u ) \nabla_{S_N}H_Q(u) ∇ S N H Q ( u ) a Q \sqrt{a_Q} a Q u ⊥ u^\perp u ⊥ ∇ ⊥ 2 H Q ( u ) \nabla_{\perp}^{2}H_Q(u) ∇ ⊥ 2 H Q ( u ) W n , N W_{n,N} W n , N
∇ S N 2 H Q ( u ) = ∇ ⊥ 2 H Q ( u ) − K Q ( u ) N Id u ⊥ . (132) \nabla_{S_N}^{2}H_Q(u)=\nabla_{\perp}^{2}H_Q(u)-\frac{K_Q(u)}{N}\operatorname{Id}_{u^\perp}. \tag*{(132)} ∇ S N 2 H Q ( u ) = ∇ ⊥ 2 H Q ( u ) − N K Q ( u ) Id u ⊥ . (132) If ξ \xi ξ det Σ Q > 0 \det\Sigma_Q>0 det Σ Q > 0 ξ \xi ξ p p p K Q = p H Q K_Q=pH_Q K Q = p H Q R N \mathbb{R}^N R N H Q ( u ) H_Q(u) H Q ( u ) ∇ S N H Q ( u ) \nabla_{S_N}H_Q(u) ∇ S N H Q ( u ) ∇ ⊥ 2 H Q ( u ) \nabla_{\perp}^{2}H_Q(u) ∇ ⊥ 2 H Q ( u ) Var H Q ( u ) = N ξ ( Q ) \operatorname{Var}H_Q(u)=N\xi(Q) Var H Q ( u ) = N ξ ( Q )
Proof. The coefficients of the polynomial H Q H_Q H Q Z Z Z
κ ( x , y ) = E H Q ( x ) H Q ( y ) = N ξ Q ( t ) , t = x ⋅ y N . \kappa(x,y)=\mathbb{E}H_Q(x)H_Q(y)=N\xi_Q(t), \qquad t=\frac{x\cdot y}{N}. κ ( x , y ) = E H Q ( x ) H Q ( y ) = N ξ Q ( t ) , t = N x ⋅ y . The function κ \kappa κ R N \mathbb{R}^N R N H Q H_Q H Q u = N e N u=\sqrt{N}e_N u = N e N u ⊥ u^\perp u ⊥ e 1 , … , e n e_1,\ldots,e_n e 1 , … , e n u a = 0 u_a=0 u a = 0 a ≤ n a\le n a ≤ n a , b , i , j a,b,i,j a , b , i , j { 1 , … , n } \{1,\ldots,n\} { 1 , … , n } x = y = u x=y=u x = y = u t = 1 t=1 t = 1
Using ∂ x a t = y a / N \partial_{x_a}t=y_a/N ∂ x a t = y a / N ∂ y a t = x a / N \partial_{y_a}t=x_a/N ∂ y a t = x a / N
∂ x a ∂ y b κ = ξ Q ′ ′ ( t ) y a x b N + ξ Q ′ ( t ) δ a b = a Q δ a b , ∂ x i ∂ x j ∂ y a ∂ y b κ = b Q N ( δ i a δ j b + δ i b δ j a ) . \partial_{x_a}\partial_{y_b}\kappa=\xi_Q''(t)\frac{y_ax_b}{N}+\xi_Q'(t)\delta_{ab}=a_Q\delta_{ab}, \qquad\partial_{x_i}\partial_{x_j}\partial_{y_a}\partial_{y_b}\kappa=\frac{b_Q}{N}(\delta_{ia}\delta_{jb}+\delta_{ib}\delta_{ja}). ∂ x a ∂ y b κ = ξ Q ′′ ( t ) N y a x b + ξ Q ′ ( t ) δ ab = a Q δ ab , ∂ x i ∂ x j ∂ y a ∂ y b κ = N b Q ( δ ia δ j b + δ ib δ j a ) . In the second identity, every other term of the derivative contains a factor x c x_c x c y c y_c y c c ≤ n c\le n c ≤ n ∇ S N H Q ( u ) \nabla_{S_N}H_Q(u) ∇ S N H Q ( u ) a Q Id u ⊥ a_Q\operatorname{Id}_{u^\perp} a Q Id u ⊥ ∇ ⊥ 2 H Q ( u ) \nabla_{\perp}^{2}H_Q(u) ∇ ⊥ 2 H Q ( u ) b Q / N b_Q/N b Q / N 2 b Q / N 2b_Q/N 2 b Q / N W n , N W_{n,N} W n , N
Since K Q ( y ) = y ⋅ ∇ H Q ( y ) K_Q(y)=y\cdot\nabla H_Q(y) K Q ( y ) = y ⋅ ∇ H Q ( y ) y ⋅ ∇ y t = t y\cdot\nabla_y t=t y ⋅ ∇ y t = t E H Q ( x ) K Q ( y ) = y ⋅ ∇ y κ = N f ( t ) \mathbb{E}H_Q(x)K_Q(y)=y\cdot\nabla_y\kappa=Nf(t) E H Q ( x ) K Q ( y ) = y ⋅ ∇ y κ = N f ( t ) f ( t ) = t ξ Q ′ ( t ) f(t)=t\xi_Q'(t) f ( t ) = t ξ Q ′ ( t ) E K Q ( x ) K Q ( y ) = x ⋅ ∇ x ( N f ( t ) ) = N t f ′ ( t ) \mathbb{E}K_Q(x)K_Q(y)=x\cdot\nabla_x(Nf(t))=Nt f'(t) E K Q ( x ) K Q ( y ) = x ⋅ ∇ x ( N f ( t )) = N t f ′ ( t ) t = 1 t=1 t = 1
E H Q ( u ) 2 = N ξ ( Q ) , E H Q ( u ) K Q ( u ) = N a Q , E K Q ( u ) 2 = N ( a Q + b Q ) , \mathbb{E}H_Q(u)^2=N\xi(Q), \qquad\mathbb{E}H_Q(u)K_Q(u)=Na_Q, \qquad\mathbb{E}K_Q(u)^2=N(a_Q+b_Q), E H Q ( u ) 2 = N ξ ( Q ) , E H Q ( u ) K Q ( u ) = N a Q , E K Q ( u ) 2 = N ( a Q + b Q ) , so ( H Q ( u ) , K Q ( u ) ) (H_Q(u),K_Q(u)) ( H Q ( u ) , K Q ( u )) N Σ Q N\Sigma_Q N Σ Q
E H Q ( x ) ∂ y a H Q ( y ) = ξ Q ′ ( t ) x a , E K Q ( x ) ∂ y a H Q ( y ) = f ′ ( t ) x a , \mathbb{E}H_Q(x)\,\partial_{y_a}H_Q(y)=\xi'_Q(t)x_a,\qquad\mathbb{E}K_Q(x)\,\partial_{y_a}H_Q(y)=f'(t)x_a, E H Q ( x ) ∂ y a H Q ( y ) = ξ Q ′ ( t ) x a , E K Q ( x ) ∂ y a H Q ( y ) = f ′ ( t ) x a , E ∂ x i ∂ x j H Q ( x ) H Q ( y ) = ξ Q ′ ′ ( t ) y i y j N , E ∂ x i ∂ x j H Q ( x ) K Q ( y ) = f ′ ′ ( t ) y i y j N , \mathbb{E}\partial_{x_i}\partial_{x_j}H_Q(x)\,H_Q(y)=\xi''_Q(t)\frac{y_i y_j}{N},\qquad\mathbb{E}\partial_{x_i}\partial_{x_j}H_Q(x)\,K_Q(y)=f''(t)\frac{y_i y_j}{N}, E ∂ x i ∂ x j H Q ( x ) H Q ( y ) = ξ Q ′′ ( t ) N y i y j , E ∂ x i ∂ x j H Q ( x ) K Q ( y ) = f ′′ ( t ) N y i y j , E ∂ x i ∂ x j H Q ( x ) ∂ y a H Q ( y ) = ξ Q ′ ′ ′ ( t ) y i y j x a N 2 + ξ Q ′ ′ ( t ) δ i a y j + δ j a y i N . \mathbb{E}\partial_{x_i}\partial_{x_j}H_Q(x)\,\partial_{y_a}H_Q(y)=\xi'''_Q(t)\frac{y_i y_jx_a}{N^2}+\xi''_Q(t)\frac{\delta_{ia}y_j+\delta_{ja}y_i}{N}. E ∂ x i ∂ x j H Q ( x ) ∂ y a H Q ( y ) = ξ Q ′′′ ( t ) N 2 y i y j x a + ξ Q ′′ ( t ) N δ ia y j + δ j a y i . All of these vanish at x = y = u x=y=u x = y = u
For (7.3), let e ∈ u ⊥ e\in u^\perp e ∈ u ⊥ γ ( s ) = cos ( s / N ) u + N sin ( s / N ) e \gamma(s)=\cos(s/\sqrt{N})u+\sqrt{N}\sin(s/\sqrt{N})e γ ( s ) = cos ( s / N ) u + N sin ( s / N ) e S N S_N S N γ ( 0 ) = u \gamma(0)=u γ ( 0 ) = u γ ′ ( 0 ) = e \gamma'(0)=e γ ′ ( 0 ) = e γ ′ ′ ( 0 ) = − u / N \gamma''(0)=-u/N γ ′′ ( 0 ) = − u / N
∇ S N 2 H Q ( u ) ( e , e ) = d 2 d s 2 H Q ( γ ( s ) ) ∣ s = 0 = e ⋅ ∇ 2 H Q ( u ) e − u ⋅ ∇ H Q ( u ) N = e ⋅ ∇ ⊥ 2 H Q ( u ) e − K Q ( u ) N . \begin{aligned}
\nabla^2_{S_N}H_Q(u)(e,e)=\left.\frac{d^2}{ds^2}H_Q(\gamma(s))\right|_{s=0}=e\cdot\nabla^2H_Q(u)e-\frac{u\cdot\nabla H_Q(u)}{N} \\
&=e\cdot\nabla^2_{\perp}H_Q(u)e-\frac{K_Q(u)}{N}.
\end{aligned} ∇ S N 2 H Q ( u ) ( e , e ) = d s 2 d 2 H Q ( γ ( s )) s = 0 = e ⋅ ∇ 2 H Q ( u ) e − N u ⋅ ∇ H Q ( u ) = e ⋅ ∇ ⊥ 2 H Q ( u ) e − N K Q ( u ) . Polarization gives (7.3).
The weights w p = c p Q p / ξ ( Q ) w_p=c_pQ^p/\xi(Q) w p = c p Q p / ξ ( Q ) p ∈ P p\in\mathcal{P} p ∈ P ξ ( Q ) = ∑ p c p Q p \xi(Q)=\sum_p c_pQ^p ξ ( Q ) = ∑ p c p Q p a Q = ∑ p p c p Q p a_Q=\sum_p pc_pQ^p a Q = ∑ p p c p Q p a Q + b Q = ∑ p p 2 c p Q p a_Q+b_Q=\sum_p p^2c_pQ^p a Q + b Q = ∑ p p 2 c p Q p
det Σ Q = ξ ( Q ) ( a Q + b Q ) − a Q 2 = ξ ( Q ) 2 ( ∑ p w p p 2 − ( ∑ p w p p ) 2 ) . \det\Sigma_Q=\xi(Q)(a_Q+b_Q)-a_Q^2=\xi(Q)^2\left(\sum_p w_pp^2-\left(\sum_p w_pp\right)^2\right). det Σ Q = ξ ( Q ) ( a Q + b Q ) − a Q 2 = ξ ( Q ) 2 p ∑ w p p 2 − ( p ∑ w p p ) 2 . which is ξ ( Q ) 2 \xi(Q)^2 ξ ( Q ) 2 p p p w w w P \mathcal{P} P P = { p } \mathcal{P}=\{p\} P = { p } H Q H_Q H Q p p p K Q = p H Q K_Q=pH_Q K Q = p H Q
Remark 7.2 (Comparison with the literature). Lemma 7.1 is the one-point structure of [30, Section 4.1, (4.1)]. There the field lives on a sphere of radius q N q\sqrt{N} q N q = Q q=\sqrt{Q} q = Q K Q ( u ) / Q N K_Q(u)/\sqrt{QN} K Q ( u ) / QN Q u \sqrt{Q}\,u Q u 1 1 1 1 1 1 α = ξ ( 1 ) \alpha=\xi(1) α = ξ ( 1 ) ν = ξ / α \nu=\xi/\alpha ν = ξ / α H ^ N = H N / α \widehat{H}_N=H_N/\sqrt{\alpha} H N = H N / α H N H_N H N α \sqrt{\alpha} α H ^ N \widehat{H}_N H N u u u Q \sqrt{Q} Q Q Q Q u ↦ Q u u\mapsto\sqrt{Q}\,u u ↦ Q u ξ ( 1 ) > 0 \xi(1)>0 ξ ( 1 ) > 0
We now fix the conditional laws given the value, the radial derivative, and the tangent gradient at a point. For a non-pure mixture and u ∈ S N u\in S_N u ∈ S N Λ u : R ℓ N → R 2 × u ⊥ \Lambda_u:\mathbb{R}^{\ell^N}\to\mathbb{R}^2\times u^\perp Λ u : R ℓ N → R 2 × u ⊥
Λ u z = ( H z , Q ( u ) , K z , Q ( u ) , ∇ S N H z , Q ( u ) ) \Lambda_u z=\left(H_{z,Q}(u),K_{z,Q}(u),\nabla_{S_N}H_{z,Q}(u)\right) Λ u z = ( H z , Q ( u ) , K z , Q ( u ) , ∇ S N H z , Q ( u ) ) and in the pure case let Λ u Z = ( H Z , Q ( u ) , ∇ S N H Z , Q ( u ) ) ∈ R × u ⊥ \Lambda_u Z=(H_{Z,Q}(u),\nabla_{S_N}H_{Z,Q}(u))\in\mathbb{R}\times u^\perp Λ u Z = ( H Z , Q ( u ) , ∇ S N H Z , Q ( u )) ∈ R × u ⊥ T u T_u T u Λ u \Lambda_u Λ u Λ u ∗ \Lambda_u^* Λ u ∗ Λ u \Lambda_u Λ u C u = Λ u Λ u ∗ C_u=\Lambda_u\Lambda_u^* C u = Λ u Λ u ∗ Λ u Z \Lambda_uZ Λ u Z N Σ Q ⊕ a Q Id u ⊥ N\Sigma_Q\oplus a_Q\operatorname{Id}_{u^\perp} N Σ Q ⊕ a Q Id u ⊥ N ξ ~ ( Q ) ⊕ a Q Id u ⊥ N\widetilde{\xi}(Q)\oplus a_Q\operatorname{Id}_{u^\perp} N ξ ( Q ) ⊕ a Q Id u ⊥ T u T_u T u
Definition 7.3 (Conditioned laws). For u ∈ S N u\in S_N u ∈ S N a ∈ T u a\in T_u a ∈ T u
Z u , a = Λ u ∗ C u − 1 a + ( Id − Λ u ∗ C u − 1 Λ u ) Z . (133) Z^{u,a}=\Lambda_u^*C_u^{-1}a+(\operatorname{Id}-\Lambda_u^*C_u^{-1}\Lambda_u)Z. \tag*{(133)} Z u , a = Λ u ∗ C u − 1 a + ( Id − Λ u ∗ C u − 1 Λ u ) Z . (133) For a non-pure mixture and ( h , k ) ∈ R 2 (h,k)\in\mathbb{R}^2 ( h , k ) ∈ R 2 P h , k , u \mathbb{P}_{h,k,u} P h , k , u E h , k , u \mathbb{E}_{h,k,u} E h , k , u H Z u , a H_{Z^{u,a}} H Z u , a a = ( N h , N k , 0 ) a=(Nh,Nk,0) a = ( N h , N k , 0 ) E h , k , u [ Φ ( H N ) ] = E [ Φ ( H Z u , ( N h , N k , 0 ) ) ] \mathbb{E}_{h,k,u}[\Phi(H_N)]=\mathbb{E}[\Phi(H_{Z^{u,(Nh,Nk,0)}})] E h , k , u [ Φ ( H N )] = E [ Φ ( H Z u , ( N h , N k , 0 ) )] Φ ≥ 0 \Phi\geq0 Φ ≥ 0 H N \mathcal{H}_N H N P h , u \mathbb{P}_{h,u} P h , u E h , u \mathbb{E}_{h,u} E h , u a = ( N h , 0 ) a=(Nh,0) a = ( N h , 0 ) k = p h k=ph k = p h E h , k , u = E h , u \mathbb{E}_{h,k,u}=\mathbb{E}_{h,u} E h , k , u = E h , u conditioned laws at u u u ( h , k ) (h,k) ( h , k ) Γ u Z = ∇ S N H Z , Q ( u ) \Gamma_uZ=\nabla_{S_N}H_{Z,Q}(u) Γ u Z = ∇ S N H Z , Q ( u ) E u ∇ \mathbb{E}^{\nabla}_u E u ∇ H Z 0 u H_{Z^u_0} H Z 0 u Z 0 u = ( Id − a Q − 1 Γ u ∗ Γ u ) Z Z^u_0=(\operatorname{Id}-a_Q^{-1}\Gamma_u^*\Gamma_u)Z Z 0 u = ( Id − a Q − 1 Γ u ∗ Γ u ) Z
Lemma 7.4 (Properties of the conditioned laws). Let u ∈ S N u\in S_N u ∈ S N
For every a ∈ T u a\in T_u a ∈ T u Λ u Z u , a = a \Lambda_uZ^{u,a}=a Λ u Z u , a = a ( Id − Λ u ∗ C u − 1 Λ u ) Z (\operatorname{Id}-\Lambda_u^*C_u^{-1}\Lambda_u)Z ( Id − Λ u ∗ C u − 1 Λ u ) Z Λ u Z \Lambda_uZ Λ u Z Z = Z u , Λ u Z Z=Z^{u,\Lambda_uZ} Z = Z u , Λ u Z In particular , a ↦ Law ( Z u , a ) a\mapsto\operatorname{Law}(Z^{u,a}) a ↦ Law ( Z u , a ) Z Z Z Λ u Z = a \Lambda_uZ=a Λ u Z = a a a a Law ( Z 0 u ) \operatorname{Law}(Z^u_0) Law ( Z 0 u ) Z Z Z ∇ S N H Q ( u ) = 0 \nabla_{S_N}H_Q(u)=0 ∇ S N H Q ( u ) = 0
Under P h , k , u \mathbb{P}_{h,k,u} P h , k , u H Q ( u ) = N h H_Q(u)=Nh H Q ( u ) = N h K Q ( u ) = N k K_Q(u)=Nk K Q ( u ) = N k ∇ S N H Q ( u ) = 0 \nabla_{S_N}H_Q(u)=0 ∇ S N H Q ( u ) = 0 ∇ S N 2 H Q ( u ) \nabla^2_{S_N}H_Q(u) ∇ S N 2 H Q ( u ) W n , N − k Id n W_{n,N}-k\operatorname{Id}_n W n , N − k Id n k = p h k=ph k = p h E h , k , u ∣ det ∇ S N 2 H Q ( u ) ∣ = μ n , N ( k ) > 0 \mathbb{E}_{h,k,u}|\det\nabla^2_{S_N}H_Q(u)|=\mu_{n,N}(k)>0 E h , k , u ∣ det ∇ S N 2 H Q ( u ) ∣ = μ n , N ( k ) > 0
For a non-pure mixture and every field-dependent weight φ ≥ 0 \varphi\geq0 φ ≥ 0 ( u , h , k ) ↦ E h , k , u φ ( H N , u ) (u,h,k)\mapsto\mathbb{E}_{h,k,u}\varphi(H_N,u) ( u , h , k ) ↦ E h , k , u φ ( H N , u ) S N × R 2 S_N\times\mathbb{R}^2 S N × R 2 ( u , h ) ↦ E h , u φ ( H N , u ) (u,h)\mapsto\mathbb{E}_{h,u}\varphi(H_N,u) ( u , h ) ↦ E h , u φ ( H N , u ) S N × R S_N\times\mathbb{R} S N × R ( u , h , k ) ↦ E h , k , u φ ( H N , u ) (u,h,k)\mapsto\mathbb{E}_{h,k,u}\varphi(H_N,u) ( u , h , k ) ↦ E h , k , u φ ( H N , u ) S N × { ( h , k ) : k = p h } S_N\times\{(h,k):k=ph\} S N × {( h , k ) : k = p h } u ↦ E u ∇ φ ( H N , u ) u\mapsto\mathbb{E}^{\nabla}_u\varphi(H_N,u) u ↦ E u ∇ φ ( H N , u ) S N S_N S N
For a non-pure mixture and every field-dependent weight φ ≥ 0 \varphi\geq0 φ ≥ 0
E u ∇ φ ( H N , u ) = ∫ R 2 p N ( h , k ) E h , k , u φ ( H N , u ) d h d k , (134) \mathbb{E}^{\nabla}_u\varphi(H_N,u)=\int_{\mathbb{R}^2}p_N(h,k)\mathbb{E}_{h,k,u}\varphi(H_N,u)\,dh\,dk, \tag*{(134)} E u ∇ φ ( H N , u ) = ∫ R 2 p N ( h , k ) E h , k , u φ ( H N , u ) d h d k , (134) where p N ( h , k ) = N ( 2 π ) − 1 ( det Σ Q ) − 1 / 2 exp ( − N ( h , k ) Σ Q − 1 ( h , k ) T / 2 ) p_N(h,k)=N(2\pi)^{-1}(\det\Sigma_Q)^{-1/2}\exp\left(-N(h,k)\Sigma_Q^{-1}(h,k)^{\mathsf T}/2\right) p N ( h , k ) = N ( 2 π ) − 1 ( det Σ Q ) − 1/2 exp ( − N ( h , k ) Σ Q − 1 ( h , k ) T /2 ) Σ Q / N \Sigma_Q/N Σ Q / N ∫ R p N p u r e ( h ) E h , u φ ( H N , u ) d h \int_{\mathbb{R}}p_N^{\mathrm{pure}}(h)\mathbb{E}_{h,u}\varphi(H_N,u)\,dh ∫ R p N pure ( h ) E h , u φ ( H N , u ) d h p N p u r e ( h ) = ( N / ( 2 π ξ ( Q ) ) ) 1 / 2 e − N h 2 / ( 2 ξ ( Q ) ) p_N^{\mathrm{pure}}(h)=(N/(2\pi\xi(Q)))^{1/2}e^{-Nh^2/(2\xi(Q))} p N pure ( h ) = ( N / ( 2 π ξ ( Q )) ) 1/2 e − N h 2 / ( 2 ξ ( Q ))
Proof. For (1), write Π u = Λ u ∗ C u − 1 Λ u \Pi_u=\Lambda_u^*C_u^{-1}\Lambda_u Π u = Λ u ∗ C u − 1 Λ u Λ u Π u = Λ u \Lambda_u\Pi_u=\Lambda_u Λ u Π u = Λ u Λ u Λ u ∗ = C u \Lambda_u\Lambda_u^*=C_u Λ u Λ u ∗ = C u Λ u Z u , a = a \Lambda_uZ^{u,a}=a Λ u Z u , a = a ( Id − Π u ) Z (\operatorname{Id}-\Pi_u)Z ( Id − Π u ) Z Λ u Z \Lambda_uZ Λ u Z ( Id − Π u ) Λ u ∗ = Λ u ∗ − Λ u ∗ C u − 1 C u = 0 (\operatorname{Id}-\Pi_u)\Lambda_u^*=\Lambda_u^*-\Lambda_u^*C_u^{-1}C_u=0 ( Id − Π u ) Λ u ∗ = Λ u ∗ − Λ u ∗ C u − 1 C u = 0 Z = Λ u ∗ C u − 1 Λ u Z + ( Id − Π u ) Z Z=\Lambda_u^*C_u^{-1}\Lambda_uZ+(\operatorname{Id}-\Pi_u)Z Z = Λ u ∗ C u − 1 Λ u Z + ( Id − Π u ) Z Z = Z u , Λ u Z Z=Z^{u,\Lambda_uZ} Z = Z u , Λ u Z Law ( Z u , a ) \operatorname{Law}(Z^{u,a}) Law ( Z u , a ) a a a a a a (133) linearly. The same argument applies to Γ u \Gamma_u Γ u a Q Id u ⊥ a_Q\operatorname{Id}_{u^\perp} a Q Id u ⊥
For (2), the first assertion follows from Λ u Z u , a = a \Lambda_u Z^{u,a}=a Λ u Z u , a = a ∇ ⊥ 2 H z , Q ( u ) \nabla_\perp^2 H_{z,Q}(u) ∇ ⊥ 2 H z , Q ( u ) z z z Λ u Z \Lambda_u Z Λ u Z ∇ ⊥ 2 H Λ u ∗ b , Q ( u ) = 0 \nabla_\perp^2 H_{\Lambda_u^*b,Q}(u)=0 ∇ ⊥ 2 H Λ u ∗ b , Q ( u ) = 0 b ∈ T u b\in T_u b ∈ T u b = C u − 1 a b=C_u^{-1}a b = C u − 1 a b = C u − 1 Λ u Z b=C_u^{-1}\Lambda_u Z b = C u − 1 Λ u Z ∇ ⊥ 2 H Z u , a , Q ( u ) = ∇ ⊥ 2 H Z , Q ( u ) \nabla_\perp^2 H_{Z^{u,a},Q}(u)=\nabla_\perp^2 H_{Z,Q}(u) ∇ ⊥ 2 H Z u , a , Q ( u ) = ∇ ⊥ 2 H Z , Q ( u ) W n , N W_{n,N} W n , N (132) and K Q ( u ) = N k K_Q(u)=Nk K Q ( u ) = N k μ n , N \mu_{n,N} μ n , N
For (3), regard Λ u \Lambda_u Λ u R 2 × R N \mathbb{R}^2\times\mathbb{R}^N R 2 × R N u ∈ R N u\in\mathbb{R}^N u ∈ R N R 2 × R N \mathbb{R}^2\times\mathbb{R}^N R 2 × R N C u + = ( N Σ Q ) − 1 ⊕ a Q − 1 P u C_u^+=(N\Sigma_Q)^{-1}\oplus a_Q^{-1}P_u C u + = ( N Σ Q ) − 1 ⊕ a Q − 1 P u u u u C u − 1 C_u^{-1} C u − 1 T u T_u T u ( z , u , h , k ) ↦ Z u , ( N h , N k , 0 ) (z,u,h,k)\mapsto Z^{u,(Nh,Nk,0)} ( z , u , h , k ) ↦ Z u , ( N h , N k , 0 ) z z z Z Z Z ( z , u , h , k ) ↦ φ ( H Z u , ( N h , N k , 0 ) , u ) (z,u,h,k)\mapsto\varphi(H_{Z^{u,(Nh,Nk,0)}},u) ( z , u , h , k ) ↦ φ ( H Z u , ( N h , N k , 0 ) , u ) z z z ( u , h , k ) ↦ E h , k , u φ ( H N , u ) (u,h,k)\mapsto\mathbb{E}_{h,k,u}\varphi(H_N,u) ( u , h , k ) ↦ E h , k , u φ ( H N , u ) E u ∇ \mathbb{E}^{\nabla}_u E u ∇
For (4), write Λ u z = ( Λ u ′ z , Γ u z ) \Lambda_u z=(\Lambda'_u z,\Gamma_u z) Λ u z = ( Λ u ′ z , Γ u z ) Λ u ′ z = ( H z , Q ( u ) , K z , Q ( u ) ) \Lambda'_u z=(H_{z,Q}(u),K_{z,Q}(u)) Λ u ′ z = ( H z , Q ( u ) , K z , Q ( u )) Λ u ′ Γ u ∗ = 0 \Lambda'_u\Gamma_u^*=0 Λ u ′ Γ u ∗ = 0 C u − 1 C_u^{-1} C u − 1 Π u = Λ u ′ ∗ ( N Σ Q ) − 1 Λ u ′ + a Q − 1 Γ u ∗ Γ u \Pi_u=\Lambda_u^{\prime*}(N\Sigma_Q)^{-1}\Lambda'_u+a_Q^{-1}\Gamma_u^*\Gamma_u Π u = Λ u ′ ∗ ( N Σ Q ) − 1 Λ u ′ + a Q − 1 Γ u ∗ Γ u
Z 0 u = Λ u ′ ∗ ( N Σ Q ) − 1 Λ u ′ Z + ( Id − Π u ) Z = Z u , ( Λ u ′ Z , 0 ) . Z^u_0=\Lambda_u^{\prime*}(N\Sigma_Q)^{-1}\Lambda'_uZ+(\operatorname{Id}-\Pi_u)Z=Z^{u,(\Lambda'_uZ,0)}. Z 0 u = Λ u ′ ∗ ( N Σ Q ) − 1 Λ u ′ Z + ( Id − Π u ) Z = Z u , ( Λ u ′ Z , 0 ) . By (1), Λ u ′ Z \Lambda'_uZ Λ u ′ Z ( Id − Π u ) Z (\operatorname{Id}-\Pi_u)Z ( Id − Π u ) Z N ( 0 , N Σ Q ) \mathcal{N}(0,N\Sigma_Q) N ( 0 , N Σ Q ) Λ u ′ Z = ( N h , N k ) \Lambda'_uZ=(Nh,Nk) Λ u ′ Z = ( N h , N k ) ( h , k ) (h,k) ( h , k ) p N p_N p N Σ Q / N \Sigma_Q/N Σ Q / N det Σ Q / N 2 \det\Sigma_Q/N^2 det Σ Q / N 2 (134) . The pure case is the same with Λ u ′ z = H z , Q ( u ) \Lambda'_u z=H_{z,Q}(u) Λ u ′ z = H z , Q ( u )
The conditioned laws depend on no choice of frame on S N S_N S N Z Z Z Λ u Z \Lambda_uZ Λ u Z Law ( Z u , a ) \operatorname{Law}(Z^{u,a}) Law ( Z u , a ) a a a 0 0 0 z ↦ H z ( x ) z\mapsto H_z(x) z ↦ H z ( x ) (133) says that the field under P h , k , u \mathbb{P}_{h,k,u} P h , k , u
H N ( x ) + Cov ( H N ( x ) , Λ u Z ) C u − 1 ( a − Λ u Z ) , a = ( N h , N k , 0 ) , H_N(x)+\operatorname{Cov}\left(H_N(x),\Lambda_uZ\right)C_u^{-1}(a-\Lambda_uZ),\qquad a=(Nh,Nk,0), H N ( x ) + Cov ( H N ( x ) , Λ u Z ) C u − 1 ( a − Λ u Z ) , a = ( N h , N k , 0 ) , which is the explicit Gaussian regression used in Definition 8.1. The conditioned hitting estimate of Section 8 is therefore stated under the same laws.
Gaussian concentration and determinant moments
We first record two Gaussian estimates. Part (1) of the next lemma is the Gaussian concentration inequality in the form [111, Proposition 4.1 and (4.1)], which also underlies Lemma 3.3; we include its proof through Gross’s logarithmic Sobolev inequality [84] and Herbst’s argument (see [17, 112]); part (2) bounds the extreme eigenvalues of a Gaussian matrix whose covariance is dominated by that of the GOE. We give the short proofs because we need dimension-free constants and degenerate covariances.
Lemma 7.5 (Gaussian concentration and an edge bound). 1. Let Y Y Y R m \mathbb{R}^{m} R m f : R m → R f:\mathbb{R}^{m}\to\mathbb{R} f : R m → R L L L t ∈ R t\in\mathbb{R} t ∈ R
E exp ( t ( f ( Y ) − E f ( Y ) ) ) ≤ exp ( t 2 L 2 / 2 ) , (135) \mathbb{E}\exp\left(t(f(Y)-\mathbb{E}f(Y))\right)\leq\exp\left(t^{2}L^{2}/2\right), \tag*{(135)} E exp ( t ( f ( Y ) − E f ( Y )) ) ≤ exp ( t 2 L 2 /2 ) , (135) and, if L > 0 L>0 L > 0 P ( f ( Y ) ≥ E f ( Y ) + s ) ≤ exp ( − s 2 / ( 2 L 2 ) ) \mathbb{P}(f(Y)\geq\mathbb{E}f(Y)+s)\leq\exp\left(-s^{2}/(2L^{2})\right) P ( f ( Y ) ≥ E f ( Y ) + s ) ≤ exp ( − s 2 / ( 2 L 2 ) ) s > 0 s>0 s > 0 L = 0 L=0 L = 0 f f f
Let b > 0 b>0 b > 0 1 ≤ n ≤ N 1\leq n\leq N 1 ≤ n ≤ N W W W n × n n\times n n × n Var Tr ( W M ) ≤ ( 2 b / N ) Tr ( M 2 ) \operatorname{Var}\operatorname{Tr}(WM)\leq(2b/N)\operatorname{Tr}(M^{2}) Var Tr ( W M ) ≤ ( 2 b / N ) Tr ( M 2 ) M M M
E λ max ( W ) ≤ 2 b n / N , P ( ∥ W ∥ op > 2 b + s ) ≤ 2 exp ( − N s 2 / ( 4 b ) ) ( s > 0 ) . (136) \mathbb{E}\lambda_{\max}(W)\leq2\sqrt{bn/N},\qquad\mathbb{P}\left(\lVert W\rVert_{\operatorname{op}}>2\sqrt{b}+s\right)\leq2\exp\left(-Ns^{2}/(4b)\right)\quad(s>0). \tag*{(136)} E λ m a x ( W ) ≤ 2 bn / N , P ( ∥ W ∥ op > 2 b + s ) ≤ 2 exp ( − N s 2 / ( 4 b ) ) ( s > 0 ) . (136) The covariance of W W W
Proof. We begin with (1). Let γ \gamma γ R m \mathbb{R}^{m} R m Q s g ( y ) = E g ( e − s y + 1 − e − 2 s Y ) Q_{s}g(y)=\mathbb{E}g\left(e^{-s}y+\sqrt{1-e^{-2s}}Y\right) Q s g ( y ) = E g ( e − s y + 1 − e − 2 s Y ) L = Δ − y ⋅ ∇ \mathcal{L}=\Delta-y\cdot\nabla L = Δ − y ⋅ ∇ γ \gamma γ Q s Q_{s} Q s ∫ ( L ψ ) χ d γ = − ∫ ∇ ψ ⋅ ∇ χ d γ \int(\mathcal{L}\psi)\chi\,d\gamma=-\int\nabla\psi\cdot\nabla\chi\,d\gamma ∫ ( L ψ ) χ d γ = − ∫ ∇ ψ ⋅ ∇ χ d γ ∇ Q s g = e − s Q s ∇ g \nabla Q_{s}g=e^{-s}Q_{s}\nabla g ∇ Q s g = e − s Q s ∇ g g g g inf g > 0 \inf g>0 inf g > 0
d d s ∫ Q s g log Q s g d γ = ∫ ( L Q s g ) ( 1 + log Q s g ) d γ = − ∫ ∣ ∇ Q s g ∣ 2 Q s g d γ . \frac{d}{ds}\int Q_{s}g\log Q_{s}g\,d\gamma=\int(\mathcal{L}Q_{s}g)(1+\log Q_{s}g)\,d\gamma=-\int\frac{|\nabla Q_{s}g|^{2}}{Q_{s}g}\,d\gamma. d s d ∫ Q s g log Q s g d γ = ∫ ( L Q s g ) ( 1 + log Q s g ) d γ = − ∫ Q s g ∣∇ Q s g ∣ 2 d γ . Since Q s g → ∫ g d γ Q_{s}g\to\int g\,d\gamma Q s g → ∫ g d γ s → ∞ s\to\infty s → ∞ s s s
∫ g log g d γ − ∫ g d γ log ∫ g d γ = ∫ 0 ∞ ∫ ∣ ∇ Q s g ∣ 2 Q s g d γ d s . \int g\log g\,d\gamma-\int g\,d\gamma\log\int g\,d\gamma=\int_{0}^{\infty}\int\frac{|\nabla Q_{s}g|^{2}}{Q_{s}g}\,d\gamma\,ds. ∫ g log g d γ − ∫ g d γ log ∫ g d γ = ∫ 0 ∞ ∫ Q s g ∣∇ Q s g ∣ 2 d γ d s . By the Cauchy–Schwarz inequality, ∣ Q s ∇ g ∣ 2 ≤ Q s ( ∣ ∇ g ∣ 2 / g ) Q s g |Q_{s}\nabla g|^{2}\leq Q_{s}(|\nabla g|^{2}/g)Q_{s}g ∣ Q s ∇ g ∣ 2 ≤ Q s ( ∣∇ g ∣ 2 / g ) Q s g ∇ Q s g = e − s Q s ∇ g \nabla Q_{s}g=e^{-s}Q_{s}\nabla g ∇ Q s g = e − s Q s ∇ g ∣ ∇ Q s g ∣ 2 / Q s g ≤ e − 2 s Q s ( ∣ ∇ g ∣ 2 / g ) |\nabla Q_{s}g|^{2}/Q_{s}g\leq e^{-2s}Q_{s}(|\nabla g|^{2}/g) ∣∇ Q s g ∣ 2 / Q s g ≤ e − 2 s Q s ( ∣∇ g ∣ 2 / g ) γ \gamma γ
∫ g log g d γ − ∫ g d γ log ∫ g d γ ≤ 1 2 ∫ ∣ ∇ g ∣ 2 g d γ . \int g\log g\,d\gamma-\int g\,d\gamma\log\int g\,d\gamma\leq\frac{1}{2}\int\frac{|\nabla g|^{2}}{g}\,d\gamma. ∫ g log g d γ − ∫ g d γ log ∫ g d γ ≤ 2 1 ∫ g ∣∇ g ∣ 2 d γ . Now let f f f ∣ ∇ f ∣ ≤ L |\nabla f|\leq L ∣∇ f ∣ ≤ L ψ ( t ) = log E e t f ( Y ) \psi(t)=\log\mathbb{E}e^{tf(Y)} ψ ( t ) = log E e t f ( Y ) g = e t f g=e^{tf} g = e t f t > 0 t>0 t > 0 t ψ ′ ( t ) − ψ ( t ) ≤ t 2 L 2 / 2 t\psi'(t)-\psi(t)\leq t^{2}L^{2}/2 t ψ ′ ( t ) − ψ ( t ) ≤ t 2 L 2 /2 ( ψ ( t ) / t ) ′ ≤ L 2 / 2 (\psi(t)/t)'\leq L^{2}/2 ( ψ ( t ) / t ) ′ ≤ L 2 /2 ψ ( t ) / t → E f ( Y ) \psi(t)/t\to\mathbb{E}f(Y) ψ ( t ) / t → E f ( Y ) t ↓ 0 t\downarrow0 t ↓ 0 t 2 L 2 / 2 t^2L^2/2 t 2 L 2 /2 (135) . Applying this to − f -f − f t < 0 t<0 t < 0 L L L f f f ∣ ∇ f ∣ ≤ L \lvert\nabla f\rvert\le L ∣ ∇ f ∣ ≤ L f f f ∣ f ( y ) ∣ ≤ ∣ f ( 0 ) ∣ + L ∣ y ∣ \lvert f(y)\rvert\le\lvert f(0)\rvert+ L\lvert y\rvert ∣ f ( y )∣ ≤ ∣ f ( 0 )∣ + L ∣ y ∣ e c ∣ Y ∣ e^{c\lvert Y\rvert} e c ∣ Y ∣ t = s / L 2 t=s/L^2 t = s / L 2
For (2), let Γ \Gamma Γ W W W Tr ( A B ) \operatorname{Tr}(AB) Tr ( A B ) W W W Γ 1 / 2 ζ \Gamma^{1/2}\zeta Γ 1/2 ζ ζ \zeta ζ ∥ Γ 1 / 2 ∥ op ≤ 2 b / N \lVert\Gamma^{1/2}\rVert_{\operatorname{op}}\le\sqrt{2b/N} ∥ Γ 1/2 ∥ op ≤ 2 b / N v , w ∈ R n v,w\in\mathbb{R}^n v , w ∈ R n X v = v ⋅ W v = Tr ( W v v T ) X_v=v\cdot Wv=\operatorname{Tr}(Wvv^{\mathsf T}) X v = v ⋅ W v = Tr ( W v v T ) Y v = 2 b / N g ⋅ v Y_v=2\sqrt{b/N}\,g\cdot v Y v = 2 b / N g ⋅ v g g g R n \mathbb{R}^n R n
E ( X v − X w ) 2 ≤ 2 b N ∥ v v T − w w T ∥ F 2 = 4 b N ( 1 − ( v ⋅ w ) 2 ) ≤ 8 b N ( 1 − v ⋅ w ) = E ( Y v − Y w ) 2 . \mathbb{E}(X_v-X_w)^2\le\frac{2b}{N}\lVert vv^{\mathsf T}-ww^{\mathsf T}\rVert_F^2=\frac{4b}{N}\left(1-(v\cdot w)^2\right)\le\frac{8b}{N}(1-v\cdot w)=\mathbb{E}(Y_v-Y_w)^2. E ( X v − X w ) 2 ≤ N 2 b ∥ v v T − w w T ∥ F 2 = N 4 b ( 1 − ( v ⋅ w ) 2 ) ≤ N 8 b ( 1 − v ⋅ w ) = E ( Y v − Y w ) 2 . where we used 1 − ( v ⋅ w ) 2 = ( 1 − v ⋅ w ) ( 1 + v ⋅ w ) 1-(v\cdot w)^2=(1-v\cdot w)(1+v\cdot w) 1 − ( v ⋅ w ) 2 = ( 1 − v ⋅ w ) ( 1 + v ⋅ w ) ∣ v ⋅ w ∣ ≤ 1 \lvert v\cdot w\rvert\le1 ∣ v ⋅ w ∣ ≤ 1 E λ max ( W ) = E sup v X v ≤ E sup v Y v = 2 b / N E ∣ g ∣ ≤ 2 b n / N \mathbb{E}\lambda_{\max}(W)=\mathbb{E}\sup_v X_v\le\mathbb{E}\sup_v Y_v=2\sqrt{b/N}\,\mathbb{E}\lvert g\rvert\le2\sqrt{bn/N} E λ m a x ( W ) = E sup v X v ≤ E sup v Y v = 2 b / N E ∣ g ∣ ≤ 2 bn / N ζ ↦ λ max ( Γ 1 / 2 ζ ) \zeta\mapsto\lambda_{\max}(\Gamma^{1/2}\zeta) ζ ↦ λ m a x ( Γ 1/2 ζ ) 2 b / N \sqrt{2b/N} 2 b / N λ max \lambda_{\max} λ m a x 1 1 1 E λ max ( W ) ≤ 2 b \mathbb{E}\lambda_{\max}(W)\le2\sqrt b E λ m a x ( W ) ≤ 2 b P ( λ max ( W ) > 2 b + s ) ≤ exp ( − N s 2 / ( 4 b ) ) \mathbb{P}(\lambda_{\max}(W)>2\sqrt b+s)\le\exp(-Ns^2/(4b)) P ( λ m a x ( W ) > 2 b + s ) ≤ exp ( − N s 2 / ( 4 b )) − W -W − W (136) .
Combining Lemma 7.4(2) with Lemma 7.5, we record the law of the Hessian at a conditioned critical point and the meaning of a strict gap.
Corollary 7.6 (The Hessian at a conditioned critical point). Let u ∈ S N u\in S_N u ∈ S N ( h , k ) ∈ R 2 (h,k)\in\mathbb{R}^2 ( h , k ) ∈ R 2 k = p h k=ph k = p h P h , k , u \mathbb{P}_{h,k,u} P h , k , u ∇ S N 2 H Q ( u ) \nabla^2_{S_N}H_Q(u) ∇ S N 2 H Q ( u ) u ⊥ u^\perp u ⊥
b Q ( N − 1 ) / N G N − 1 − k Id N − 1 = W n , N − k Id n . \sqrt{b_Q(N-1)/N}\,G_{N-1}-k\operatorname{Id}_{N-1}=W_{n,N}-k\operatorname{Id}_n. b Q ( N − 1 ) / N G N − 1 − k Id N − 1 = W n , N − k Id n . In other words, its entries on and above the diagonal are independent Gaussians with mean − k δ i j -k\delta_{ij} − k δ ij b Q / N b_Q/N b Q / N 2 b Q / N 2b_Q/N 2 b Q / N k ≥ 2 b Q + g k\ge2\sqrt{b_Q}+g k ≥ 2 b Q + g g > 0 g>0 g > 0
P h , k , u ( ∇ S N 2 H Q ( u ) ≤ − g 2 Id u ⊥ ) ≥ 1 − 2 e − N g 2 / ( 16 b Q ) . \mathbb{P}_{h,k,u}\left(\nabla^2_{S_N}H_Q(u)\le-\frac{g}{2}\operatorname{Id}_{u^\perp}\right)\ge1-2e^{-Ng^2/(16b_Q)}. P h , k , u ( ∇ S N 2 H Q ( u ) ≤ − 2 g Id u ⊥ ) ≥ 1 − 2 e − N g 2 / ( 16 b Q ) . Proof. The first assertion is Lemma 7.4(2); it is the law of Lemma 5.14(1), stated for the conditioned laws of Definition 7.3. The shift − k Id -k\operatorname{Id} − k Id (132) and K Q ( u ) = N k K_Q(u)=Nk K Q ( u ) = N k (131) , W n , N W_{n,N} W n , N b = b Q b=b_Q b = b Q s = g / 2 s=g/2 s = g /2 P ( ∥ W n , N ∥ op > 2 b Q + g / 2 ) ≤ 2 e − N g 2 / ( 16 b Q ) \mathbb{P}(\lVert W_{n,N}\rVert_{\operatorname{op}}>2\sqrt{b_Q}+g/2)\le2e^{-Ng^2/(16b_Q)} P (∥ W n , N ∥ op > 2 b Q + g /2 ) ≤ 2 e − N g 2 / ( 16 b Q ) W n , N − k Id ≤ ( 2 b Q + g / 2 − k ) Id ≤ − ( g / 2 ) Id W_{n,N}-k\operatorname{Id}\le(2\sqrt{b_Q}+g/2-k)\operatorname{Id}\le-(g/2)\operatorname{Id} W n , N − k Id ≤ ( 2 b Q + g /2 − k ) Id ≤ − ( g /2 ) Id
We say that a set of marks has a strict gap if k ≥ 2 b Q + g k \ge2\sqrt{b_Q}+g k ≥ 2 b Q + g g > 0 g>0 g > 0 1 − O ( e − c N ) 1-O(e^{-cN}) 1 − O ( e − c N )
Proposition 7.7 (Normalized determinant moments). Let b , g > 0 b,g>0 b , g > 0 k + < ∞ k_+<\infty k + < ∞ 1 ≤ n ≤ N 1\le n\le N 1 ≤ n ≤ N W W W n × n n\times n n × n Var Tr ( W M ) ≤ ( 2 b / N ) Tr ( M 2 ) \operatorname{Var}\operatorname{Tr}(WM)\le(2b/N)\operatorname{Tr}(M^2) Var Tr ( W M ) ≤ ( 2 b / N ) Tr ( M 2 ) M M M C C C n × n n\times n n × n ∥ C ∥ o p ≤ g / 4 \lVert C\rVert_{\mathrm{op}}\le g/4 ∥ C ∥ op ≤ g /4 2 b + g ≤ k ≤ k + 2\sqrt{b}+g\le k\le k_+ 2 b + g ≤ k ≤ k +
D = ∣ det ( k Id n − C − W ) ∣ , μ = E D , Y = ∑ j = 1 n log max { ∣ k − λ j ( C + W ) ∣ , g / 2 } . D=\left|\det(k\operatorname{Id}_n-C-W)\right|,\qquad\mu=\mathbb{E}D,\qquad Y=\sum_{j=1}^{n}\log\max\{|k-\lambda_j(C+W)|,g/2\}. D = ∣ det ( k Id n − C − W ) ∣ , μ = E D , Y = j = 1 ∑ n log max { ∣ k − λ j ( C + W ) ∣ , g /2 } . Then μ > 0 \mu>0 μ > 0 r ≥ 1 r\ge1 r ≥ 1 K r = K r ( b , g , k + ) < ∞ K_r=K_r(b,g,k_+)<\infty K r = K r ( b , g , k + ) < ∞
E ( D / μ ) r ≤ E ( e Y / μ ) r ≤ K r . (137) \mathbb{E}(D/\mu)^r\le\mathbb{E}(e^Y/\mu)^r\le K_r. \tag*{(137)} E ( D / μ ) r ≤ E ( e Y / μ ) r ≤ K r . (137) The constant K r K_r K r n n n N N N k k k C C C W W W N 0 ( b , g ) N_0(b,g) N 0 ( b , g ) N ≥ N 0 N\ge N_0 N ≥ N 0 K r = 2 r exp ( 4 r 2 b / g 2 ) K_r=2^r\exp(4r^2b/g^2) K r = 2 r exp ( 4 r 2 b / g 2 ) u ∈ S N u\in S_N u ∈ S N 2 b Q + g ≤ k ≤ k + 2\sqrt{b_Q}+g\le k\le k_+ 2 b Q + g ≤ k ≤ k + P h , k , u \mathbb{P}_{h,k,u} P h , k , u D = ∣ det ∇ S N 2 H Q ( u ) ∣ D=|\det\nabla^2_{S_N}H_Q(u)| D = ∣ det ∇ S N 2 H Q ( u ) ∣ E h , k , u ( D / μ n , N ( k ) ) r ≤ K r ( b Q , g , k + ) \mathbb{E}_{h,k,u}(D/\mu_{n,N}(k))^r\le K_r(b_Q,g,k_+) E h , k , u ( D / μ n , N ( k ) ) r ≤ K r ( b Q , g , k + ) k = p h k=ph k = p h
For 0 < r < 1 0<r<1 0 < r < 1 E ( D / μ ) r ≤ 1 \mathbb{E}(D/\mu)^r\le1 E ( D / μ ) r ≤ 1 D D D C C C W W W C C C W W W
Proof. The scalar function x ↦ log max { ∣ k − x ∣ , g / 2 } x\mapsto\log\max\{|k-x|,g/2\} x ↦ log max { ∣ k − x ∣ , g /2 } ( 2 / g ) (2/g) ( 2/ g ) R \mathbb{R} R U , V U,V U , V ∑ j ( λ j ( U ) − λ j ( V ) ) 2 ≤ ∥ U − V ∥ F 2 \sum_j(\lambda_j(U)-\lambda_j(V))^2\le\lVert U-V\rVert_F^2 ∑ j ( λ j ( U ) − λ j ( V ) ) 2 ≤ ∥ U − V ∥ F 2 ∥ U − V ∥ F 2 = Tr U 2 + Tr V 2 − 2 Tr ( U V ) \lVert U-V\rVert_F^2=\operatorname{Tr}U^2+\operatorname{Tr}V^2-2\operatorname{Tr}(UV) ∥ U − V ∥ F 2 = Tr U 2 + Tr V 2 − 2 Tr ( U V ) Tr ( U V ) = ∑ i , j λ i ( U ) λ j ( V ) ∣ e i ⋅ f j ∣ 2 \operatorname{Tr}(UV)=\sum_{i,j}\lambda_i(U)\lambda_j(V)|e_i\cdot f_j|^2 Tr ( U V ) = ∑ i , j λ i ( U ) λ j ( V ) ∣ e i ⋅ f j ∣ 2 ( e i ) , ( f j ) (e_i),(f_j) ( e i ) , ( f j ) ( ∣ e i ⋅ f j ∣ 2 ) (|e_i\cdot f_j|^2) ( ∣ e i ⋅ f j ∣ 2 ) ∑ j λ j ( U ) λ j ( V ) \sum_j\lambda_j(U)\lambda_j(V) ∑ j λ j ( U ) λ j ( V ) Y Y Y ( 2 n / g ) (2\sqrt{n}/g) ( 2 n / g ) W W W W = Γ 1 / 2 ζ W=\Gamma^{1/2}\zeta W = Γ 1/2 ζ Lemma 7.5 , we find that Y Y Y ζ \zeta ζ
L = 2 g 2 b n N ≤ 2 2 b g . (138) L=\frac{2}{g}\sqrt{\frac{2bn}{N}}\le\frac{2\sqrt{2b}}{g}. \tag*{(138)} L = g 2 N 2 bn ≤ g 2 2 b . (138) We note that this concerns the clipped function Y Y Y
Let E = { ∥ W ∥ o p ≤ 2 b + g / 4 } \mathcal{E}=\{\lVert W\rVert_{\mathrm{op}}\le2\sqrt{b}+g/4\} E = {∥ W ∥ op ≤ 2 b + g /4 } P ( E c ) ≤ 2 e − N g 2 / ( 64 b ) \mathbb{P}(\mathcal{E}^{c})\le2e^{-Ng^{2}/(64b)} P ( E c ) ≤ 2 e − N g 2 / ( 64 b ) E \mathcal{E} E C + W C+W C + W 2 b + g / 2 ≤ k − g / 2 2\sqrt{b}+g/2\le k-g/2 2 b + g /2 ≤ k − g /2 D = e Y D=e^{Y} D = e Y E \mathcal{E} E D ≤ e Y D\le e^{Y} D ≤ e Y max { ∣ k − x ∣ , g / 2 } ≥ ∣ k − x ∣ \max\{|k-x|,g/2\}\ge|k-x| max { ∣ k − x ∣ , g /2 } ≥ ∣ k − x ∣
E e t Y ≤ e t E Y + t 2 L 2 / 2 ( t ≥ 0 ) , E [ e Y 1 E c ] ≤ ( E e 2 Y ) 1 / 2 P ( E c ) 1 / 2 ≤ e E Y + L 2 P ( E c ) 1 / 2 . \mathbb{E}e^{tY}\le e^{t\mathbb{E}Y+t^{2}L^{2}/2}\quad(t\ge0),\qquad\mathbb{E}[e^{Y}\mathbf{1}_{\mathcal{E}^{c}}]\le(\mathbb{E}e^{2Y})^{1/2}\mathbb{P}(\mathcal{E}^{c})^{1/2}\le e^{\mathbb{E}Y+L^{2}}\mathbb{P}(\mathcal{E}^{c})^{1/2}. E e t Y ≤ e t E Y + t 2 L 2 /2 ( t ≥ 0 ) , E [ e Y 1 E c ] ≤ ( E e 2 Y ) 1/2 P ( E c ) 1/2 ≤ e E Y + L 2 P ( E c ) 1/2 . Since E e Y ≥ e E Y \mathbb{E}e^{Y}\ge e^{\mathbb{E}Y} E e Y ≥ e E Y
μ ≥ E [ e Y 1 E ] ≥ e E Y ( 1 − e L 2 P ( E c ) 1 / 2 ) . (139) \mu\ge\mathbb{E}[e^{Y}\mathbf{1}_{\mathcal{E}}]\ge e^{\mathbb{E}Y}\left(1-e^{L^{2}}\mathbb{P}(\mathcal{E}^{c})^{1/2}\right). \tag*{(139)} μ ≥ E [ e Y 1 E ] ≥ e E Y ( 1 − e L 2 P ( E c ) 1/2 ) . (139) By (7.9), there exists N 0 ( b , g ) N_{0}(b,g) N 0 ( b , g ) 1 / 2 1/2 1/2 N ≥ N 0 N\ge N_{0} N ≥ N 0 N N N μ r ≥ 2 − r e r E Y \mu^{r}\ge2^{-r}e^{r\mathbb{E}Y} μ r ≥ 2 − r e r E Y E e r Y \mathbb{E}e^{rY} E e r Y μ r \mu^{r} μ r E ( e Y / μ ) r ≤ 2 r e r 2 L 2 / 2 ≤ 2 r exp ( 4 r 2 b / g 2 ) \mathbb{E}(e^{Y}/\mu)^{r}\le2^{r}e^{r^{2}L^{2}/2}\le2^{r}\exp(4r^{2}b/g^{2}) E ( e Y / μ ) r ≤ 2 r e r 2 L 2 /2 ≤ 2 r exp ( 4 r 2 b / g 2 ) L 2 ≤ 8 b / g 2 L^{2}\le8b/g^{2} L 2 ≤ 8 b / g 2 D ≤ e Y D\le e^{Y} D ≤ e Y N ≥ N 0 N\ge N_{0} N ≥ N 0
It remains to treat the finitely many pairs ( n , N ) (n,N) ( n , N ) n ≤ N < N 0 n\le N<N_{0} n ≤ N < N 0 Γ \Gamma Γ 0 ≤ Γ ≤ ( 2 b / N ) Id 0\le\Gamma\le(2b/N)\operatorname{Id} 0 ≤ Γ ≤ ( 2 b / N ) Id C C C k k k E D r \mathbb{E}D^{r} E D r E e r Y \mathbb{E}e^{rY} E e r Y μ \mu μ Γ ↦ Γ 1 / 2 \Gamma\mapsto\Gamma^{1/2} Γ ↦ Γ 1/2 D r D^{r} D r e r Y e^{rY} e r Y ∣ ζ ∣ |\zeta| ∣ ζ ∣ μ > 0 \mu>0 μ > 0 ζ = 0 \zeta=0 ζ = 0 k Id − C ≥ ( k − g / 4 ) Id k\operatorname{Id}-C\ge(k-g/4)\operatorname{Id} k Id − C ≥ ( k − g /4 ) Id D > 0 D>0 D > 0 0 0 0 ζ ↦ Γ 1 / 2 ζ \zeta\mapsto\Gamma^{1/2}\zeta ζ ↦ Γ 1/2 ζ E e r Y / μ r \mathbb{E}e^{rY}/\mu^{r} E e r Y / μ r N N N μ > 0 \mu>0 μ > 0
For the last assertion, recall from Lemma 7.4(2) that under P h , k , u \mathbb{P}_{h,k,u} P h , k , u − ∇ S N 2 H Q ( u ) -\nabla^{2}_{S_{N}}H_{Q}(u) − ∇ S N 2 H Q ( u ) u ⊥ u^{\perp} u ⊥ k Id n − W n , N k\operatorname{Id}_{n}-W_{n,N} k Id n − W n , N − W n , N -W_{n,N} − W n , N W n , N W_{n,N} W n , N W n , N W_{n,N} W n , N b = b Q b=b_{Q} b = b Q C = 0 C=0 C = 0
Corollary 7.8 (Removal of exceptional events). Under the hypotheses of Proposition 7.7, let A A A W W W r > 1 r>1 r > 1 s ≥ 1 s\ge1 s ≥ 1
E [ D 1 A ] μ ≤ K r 1 / r P ( A ) 1 − 1 / r , E [ D s 1 A ] μ s ≤ K 2 s 1 / 2 P ( A ) 1 / 2 . (140) \frac{\mathbb{E}[D\mathbf{1}_{A}]}{\mu}\le K_{r}^{1/r}\mathbb{P}(A)^{1-1/r},\qquad\frac{\mathbb{E}[D^{s}\mathbf{1}_{A}]}{\mu^{s}}\le K_{2s}^{1/2}\mathbb{P}(A)^{1/2}. \tag*{(140)} μ E [ D 1 A ] ≤ K r 1/ r P ( A ) 1 − 1/ r , μ s E [ D s 1 A ] ≤ K 2 s 1/2 P ( A ) 1/2 . (140) If moreover K r 1 / r P ( A ) 1 − 1 / r ≤ 1 / 2 K_{r}^{1/r}\mathbb{P}(A)^{1-1/r}\le1/2 K r 1/ r P ( A ) 1 − 1/ r ≤ 1/2 r > 1 r>1 r > 1 μ g o o d = E [ D 1 A c ] ≥ μ / 2 \mu_{\mathrm{good}}=\mathbb{E}[D\mathbf{1}_{A^{c}}]\ge\mu/2 μ good = E [ D 1 A c ] ≥ μ /2 E ( D 1 A c / μ g o o d ) s ≤ 2 s K s \mathbb{E}(D\mathbf{1}_{A^{c}}/\mu_{\mathrm{good}})^{s}\le2^{s}K_{s} E ( D 1 A c / μ good ) s ≤ 2 s K s s ≥ 1 s\ge1 s ≥ 1
Proof. The first bound in (7.11) is Hölder’s inequality with exponents r r r r / ( r − 1 ) r/(r-1) r / ( r − 1 ) D s 1 A D^{s}\mathbf{1}_{A} D s 1 A μ g o o d = μ − E [ D 1 A ] ≥ μ / 2 \mu_{\mathrm{good}}=\mu-\mathbb{E}[D\mathbf{1}_{A}]\ge\mu/2 μ good = μ − E [ D 1 A ] ≥ μ /2 E ( D 1 A c / μ g o o d ) s ≤ 2 s E ( D / μ ) s ≤ 2 s K s \mathbb{E}(D\mathbf{1}_{A^{c}}/\mu_{\mathrm{good}})^{s}\le2^{s}\mathbb{E}(D/\mu)^{s}\le2^{s}K_{s} E ( D 1 A c / μ good ) s ≤ 2 s E ( D / μ ) s ≤ 2 s K s
We note that if P ( A ) ≤ C 0 e − c 0 N \mathbb{P}(\mathcal{A}) \le C_0e^{-c_0N} P ( A ) ≤ C 0 e − c 0 N A \mathcal{A} A b , g , k + , C 0 , c 0 b,g,k_+,C_0,c_0 b , g , k + , C 0 , c 0 A \mathcal{A} A P ( A ) ≤ C 0 e − c 0 N \mathbb{P}(\mathcal{A}) \le C_0e^{-c_0N} P ( A ) ≤ C 0 e − c 0 N P ( A ) \mathbb{P}(\mathcal{A}) P ( A ) P ( A ) → 0 \mathbb{P}(\mathcal{A}) \to0 P ( A ) → 0 P ( A ) 1 − 1 / r \mathbb{P}(\mathcal{A})^{1-1/r} P ( A ) 1 − 1/ r
The semicircle law and bordered determinants
Proposition 7.7 needs no information about the size of μ \mu μ μ \mu μ
σ s c ( d x ) = 1 2 π 4 − x 2 1 [ − 2 , 2 ] ( x ) d x , Ω ( z ) = ∫ log ( z − x ) σ s c ( d x ) , \sigma_{\mathrm{sc}}(\mathrm{d}x)=\frac{1}{2\pi}\sqrt{4-x^2}\,1_{[-2,2]}(x)\,\mathrm{d}x,\qquad\Omega(z)=\int\log(z-x)\,\sigma_{\mathrm{sc}}(\mathrm{d}x), σ sc ( d x ) = 2 π 1 4 − x 2 1 [ − 2 , 2 ] ( x ) d x , Ω ( z ) = ∫ log ( z − x ) σ sc ( d x ) , and
g s c ( z ) = ∫ σ s c ( d x ) z − x = z − z 2 − 4 2 . g_{\mathrm{sc}}(z)=\int\frac{\sigma_{\mathrm{sc}}(\mathrm{d}x)}{z-x}=\frac{z-\sqrt{z^2-4}}{2}. g sc ( z ) = ∫ z − x σ sc ( d x ) = 2 z − z 2 − 4 . for z > 2 z>2 z > 2 Ω ′ = g s c \Omega'=g_{\mathrm{sc}} Ω ′ = g sc g s c g_{\mathrm{sc}} g sc x = 2 cos θ x=2\cos\theta x = 2 cos θ ∫ 0 π ( z − 2 cos θ ) − 1 d θ = π / z 2 − 4 \int_0^\pi(z-2\cos\theta)^{-1}\,\mathrm{d}\theta=\pi/\sqrt{z^2-4} ∫ 0 π ( z − 2 cos θ ) − 1 d θ = π / z 2 − 4
Proposition 7.9 (Determinants and the semicircle law). Let d 0 > 0 d_0>0 d 0 > 0 Z + < ∞ Z_+<\infty Z + < ∞ m ≥ 1 m\ge1 m ≥ 1 2 + d 0 ≤ z ≤ Z + 2+d_0\le z\le Z_+ 2 + d 0 ≤ z ≤ Z +
e m Ω ( z ) ≤ E ∣ det ( z Id m − G m ) ∣ ≤ C e m Ω ( z ) , E ∣ det ( z Id m − G m ) ∣ r ≤ C r e r m Ω ( z ) ( r ≥ 1 ) . (141) e^{m\Omega(z)}\le\mathbb{E}\left|\det(z\operatorname{Id}_m-G_m)\right|\le Ce^{m\Omega(z)},\qquad\mathbb{E}\left|\det(z\operatorname{Id}_m-G_m)\right|^r\le C_re^{rm\Omega(z)}\quad(r\ge1). \tag*{(141)} e m Ω ( z ) ≤ E ∣ det ( z Id m − G m ) ∣ ≤ C e m Ω ( z ) , E ∣ det ( z Id m − G m ) ∣ r ≤ C r e r m Ω ( z ) ( r ≥ 1 ) . (141) The same bound as the second holds for E e r Y z \mathbb{E}e^{rY_z} E e r Y z Y z = ∑ j log max { ∣ z − λ j ( G m ) ∣ , d 0 / 2 } Y_z=\sum_j\log\max\{|z-\lambda_j(G_m)|,d_0/2\} Y z = ∑ j log max { ∣ z − λ j ( G m ) ∣ , d 0 /2 } d 0 d_0 d 0 Z + Z_+ Z + r r r
Proof. The proof has four steps: we compute the signed expectation P m ( z ) = E det ( z Id m − G m ) P_m(z)=\mathbb{E}\det(z\operatorname{Id}_m-G_m) P m ( z ) = E det ( z Id m − G m ) e m Ω ( z ) e^{m\Omega(z)} e m Ω ( z )
The expected characteristic polynomial. We expand the determinant as a sum over permutations σ \sigma σ { 1 , … , m } \{1,\ldots,m\} { 1 , … , m } σ \sigma σ sgn ( σ ) ∏ i ( z δ i σ ( i ) − ( G m ) i σ ( i ) ) . \operatorname{sgn}(\sigma)\prod_i(z\delta_{i\sigma(i)}-(G_m)_{i\sigma(i)}). sgn ( σ ) ∏ i ( z δ iσ ( i ) − ( G m ) iσ ( i ) ) . z z z ( G m ) a b (G_m)_{ab} ( G m ) ab σ \sigma σ a a a b b b σ \sigma σ j j j ( − 1 ) j (-1)^j ( − 1 ) j E ( G m ) a b 2 = 1 / m \mathbb{E}(G_m)_{ab}^2=1/m E ( G m ) ab 2 = 1/ m m ! / ( 2 j j ! ( m − 2 j ) ! ) m!/(2^j j!(m-2j)!) m ! / ( 2 j j ! ( m − 2 j )!)
P m ( z ) = ∑ j = 0 ⌊ m / 2 ⌋ ( − 1 ) j m ! 2 j j ! ( m − 2 j ) ! m j z m − 2 j = m − m / 2 He m ( m z ) , (142) P_m(z)=\sum_{j=0}^{\lfloor m/2\rfloor}\frac{(-1)^j m!}{2^j j!(m-2j)!m^j}z^{m-2j}=m^{-m/2}\operatorname{He}_m(\sqrt{m}z), \tag*{(142)} P m ( z ) = j = 0 ∑ ⌊ m /2 ⌋ 2 j j ! ( m − 2 j )! m j ( − 1 ) j m ! z m − 2 j = m − m /2 He m ( m z ) , (142) where He m \operatorname{He}_m He m
Location of the roots. The recurrence He j + 1 ( x ) = x He j ( x ) − j He j − 1 ( x ) \operatorname{He}_{j+1}(x)=x\operatorname{He}_j(x)-j\operatorname{He}_{j-1}(x) He j + 1 ( x ) = x He j ( x ) − j He j − 1 ( x ) 1 , … , m − 1 \sqrt{1},\ldots,\sqrt{m-1} 1 , … , m − 1 λ 1 , … , λ m \lambda_1,\ldots,\lambda_m λ 1 , … , λ m P m P_m P m J m J_m J m j / m \sqrt{j/m} j / m 1 ≤ j ≤ m − 1 1\leq j\leq m-1 1 ≤ j ≤ m − 1 J m J_m J m 2 ( m − 1 ) / m < 2 2\sqrt{(m-1)/m}<2 2 ( m − 1 ) / m < 2 ( − 2 , 2 ) (-2,2) ( − 2 , 2 ) P m ( z ) > 0 P_m(z)>0 P m ( z ) > 0 z ≥ 2 z\geq2 z ≥ 2
Comparison with the semicircle law. The Hermite equation He m ′ ′ − x He m ′ + m He m = 0 \operatorname{He}_m''-x\operatorname{He}_m'+m\operatorname{He}_m=0 He m ′′ − x He m ′ + m He m = 0 P m ′ ′ − m z P m ′ + m 2 P m = 0 P_m''-mzP_m'+m^2P_m=0 P m ′′ − m z P m ′ + m 2 P m = 0 z > 2 z>2 z > 2 g m = P m ′ / ( m P m ) = m − 1 ∑ j ( z − λ j ) − 1 g_m=P_m'/(mP_m)=m^{-1}\sum_j(z-\lambda_j)^{-1} g m = P m ′ / ( m P m ) = m − 1 ∑ j ( z − λ j ) − 1 m 2 P m m^2P_m m 2 P m
g m ′ m + g m 2 − z g m + 1 = 0 , g m ′ = − 1 m ∑ j ( z − λ j ) − 2 < 0. (143) \frac{g_m'}{m}+g_m^2-zg_m+1=0,\qquad g_m'=-\frac{1}{m}\sum_j(z-\lambda_j)^{-2}<0. \tag*{(143)} m g m ′ + g m 2 − z g m + 1 = 0 , g m ′ = − m 1 j ∑ ( z − λ j ) − 2 < 0. (143) The roots of the quadratic q ( y ) = y 2 − z y + 1 q(y)=y^2-zy+1 q ( y ) = y 2 − z y + 1 g s c ( z ) g_{\mathrm{sc}}(z) g sc ( z ) g + ( z ) = ( z + z 2 − 4 ) / 2 g_+(z)=(z+\sqrt{z^2-4})/2 g + ( z ) = ( z + z 2 − 4 ) /2 q ( g m ) = − g m ′ / m > 0 q(g_m)=-g_m'/m>0 q ( g m ) = − g m ′ / m > 0 g m ( z ) g_m(z) g m ( z ) [ g s c ( z ) , g + ( z ) ] [g_{\mathrm{sc}}(z),g_+(z)] [ g sc ( z ) , g + ( z )] z > 2 z>2 z > 2 g m < g s c g_m<g_{\mathrm{sc}} g m < g sc g m > g + g_m>g_+ g m > g + ( 2 , ∞ ) (2,\infty) ( 2 , ∞ ) g m ( z ) → 0 g_m(z)\to0 g m ( z ) → 0 g + ( z ) → ∞ g_+(z)\to\infty g + ( z ) → ∞ z → ∞ z\to\infty z → ∞ ( 2 , ∞ ) (2,\infty) ( 2 , ∞ ) q ( g m ) = ( g s c − g m ) ( g + − g m ) q(g_m)=(g_{\mathrm{sc}}-g_m)(g_+-g_m) q ( g m ) = ( g sc − g m ) ( g + − g m ) g + − g m ≥ g + − g s c = z 2 − 4 g_+-g_m\geq g_+-g_{\mathrm{sc}}=\sqrt{z^2-4} g + − g m ≥ g + − g sc = z 2 − 4 z − λ j > z − 2 z-\lambda_j>z-2 z − λ j > z − 2
0 ≤ g s c ( z ) − g m ( z ) = q ( g m ) g + − g m ≤ 1 m ( z − 2 ) 2 z 2 − 4 . 0\leq g_{\mathrm{sc}}(z)-g_m(z)=\frac{q(g_m)}{g_+-g_m}\leq\frac{1}{m(z-2)^2\sqrt{z^2-4}}. 0 ≤ g sc ( z ) − g m ( z ) = g + − g m q ( g m ) ≤ m ( z − 2 ) 2 z 2 − 4 1 . Since ( log P m ) ′ = m g m (\log P_m)'=mg_m ( log P m ) ′ = m g m ( m Ω ) ′ = m g s c (m\Omega)'=mg_{\mathrm{sc}} ( m Ω ) ′ = m g sc log P m ( z ) \log P_m(z) log P m ( z ) m Ω ( z ) m\Omega(z) m Ω ( z ) m log z + O ( z − 2 ) m\log z+O(z^{-2}) m log z + O ( z − 2 ) z → ∞ z\to\infty z → ∞ m m m z z z ∞ \infty ∞
0 ≤ log P m ( z ) − m Ω ( z ) ≤ ∫ z ∞ d t ( t − 2 ) 2 t 2 − 4 ≤ C d 0 ( z ≥ 2 + d 0 ) . (144) 0\leq\log P_m(z)-m\Omega(z)\leq\int_z^\infty\frac{dt}{(t-2)^2\sqrt{t^2-4}}\leq Cd_0\qquad(z\geq2+d_0). \tag*{(144)} 0 ≤ log P m ( z ) − m Ω ( z ) ≤ ∫ z ∞ ( t − 2 ) 2 t 2 − 4 d t ≤ C d 0 ( z ≥ 2 + d 0 ) . (144) Absolute moments. Let E m = { ∥ G m ∥ o p ≤ 2 + d 0 / 2 } \mathcal{E}_m=\{\lVert G_m\rVert_{\mathrm{op}}\leq2+d_0/2\} E m = {∥ G m ∥ op ≤ 2 + d 0 /2 } Lemma 7.5 (2) with N = n = m N=n=m N = n = m b = 1 b=1 b = 1 P ( E m c ) ≤ 2 e − m d 0 2 / 16 \mathbb{P}(\mathcal{E}_m^c)\leq2e^{-md_0^2/16} P ( E m c ) ≤ 2 e − m d 0 2 /16 E m \mathcal{E}_m E m e Y z e^{Y_z} e Y z e Y z e^{Y_z} e Y z Proposition 7.7 , Y z Y_z Y z 2 2 / d 0 2\sqrt{2}/d_0 2 2 / d 0
E [ e Y z 1 E m c ] E e Y z ≤ exp ( 8 / d 0 2 ) P ( E m c ) 1 / 2 . \frac{\mathbb{E}[e^{Y_z}\mathbf{1}_{\mathcal{E}_{m}^{c}}]}{\mathbb{E}e^{Y_z}} \leq\exp(8/d_0^2)\mathbb{P}(\mathcal{E}_{m}^{c})^{1/2}. E e Y z E [ e Y z 1 E m c ] ≤ exp ( 8/ d 0 2 ) P ( E m c ) 1/2 . We have P m ≥ E e Y z − 2 E [ e Y z 1 E m c ] P_m \geq\mathbb{E}e^{Y_z}-2\mathbb{E}[e^{Y_z}\mathbf{1}_{\mathcal{E}_{m}^{c}}] P m ≥ E e Y z − 2 E [ e Y z 1 E m c ] E e Y z ≤ 2 P m ( z ) \mathbb{E}e^{Y_z}\leq2P_m(z) E e Y z ≤ 2 P m ( z ) m ≥ m 0 ( d 0 ) m\geq m_0(d_0) m ≥ m 0 ( d 0 ) E e r Y z ≤ e 4 r 2 / d 0 2 ( E e Y z ) r \mathbb{E}e^{rY_z}\leq e^{4r^2/d_0^2}(\mathbb{E}e^{Y_z})^r E e r Y z ≤ e 4 r 2 / d 0 2 ( E e Y z ) r ∣ det ∣ ≤ e Y z |\det|\leq e^{Y_z} ∣ det ∣ ≤ e Y z m ≥ m 0 m\geq m_0 m ≥ m 0 E ∣ det ( z Id m − G m ) ∣ ≥ P m ( z ) ≥ e m Ω ( z ) \mathbb{E}|\det(z\operatorname{Id}_m-G_m)|\geq P_m(z)\geq e^{m\Omega(z)} E ∣ det ( z Id m − G m ) ∣ ≥ P m ( z ) ≥ e m Ω ( z ) m < m 0 m<m_0 m < m 0 z z z [ 2 + d 0 , Z + ] [2+d_0,Z_+] [ 2 + d 0 , Z + ] e m Ω ( z ) ≥ d 0 m e^{m\Omega(z)}\geq d_0^m e m Ω ( z ) ≥ d 0 m
Corollary 7.10 (Fixed changes of dimension). Let b , g > 0 b,g>0 b , g > 0 k + < ∞ k_+<\infty k + < ∞ 2 b + g ≤ k ≤ k + 2\sqrt{b}+g\leq k\leq k_+ 2 b + g ≤ k ≤ k + m = N − j m=N-j m = N − j j j j m ≥ 1 m\geq1 m ≥ 1 b m = b m / N b_m=bm/N b m = bm / N
Λ m , N ( k ) = m ( 1 2 log b m + Ω ( k / b m ) ) . \Lambda_{m,N}(k)=m\left(\frac{1}{2}\log b_m+\Omega(k/\sqrt{b_m})\right). Λ m , N ( k ) = m ( 2 1 log b m + Ω ( k / b m ) ) . Then the first and every fixed higher moment of ∣ det ( k Id m − b m G m ) ∣ |\det(k\operatorname{Id}_m-\sqrt{b_m}G_m)| ∣ det ( k Id m − b m G m ) ∣ e m Ω e^{m\Omega} e m Ω e Λ m , N ( k ) e^{\Lambda_{m,N}(k)} e Λ m , N ( k ) Λ m , N ( k ) \Lambda_{m,N}(k) Λ m , N ( k ) N N N k k k N N N k k k
Proof. We have ∣ det ( k Id m − b m G m ) ∣ = b m m / 2 ∣ det ( z Id m − G m ) ∣ |\det(k\operatorname{Id}_m-\sqrt{b_m}G_m)|=b_m^{m/2}|\det(z\operatorname{Id}_m-G_m)| ∣ det ( k Id m − b m G m ) ∣ = b m m /2 ∣ det ( z Id m − G m ) ∣ z = k / b m ≥ 2 + g / b z=k/\sqrt{b_m}\geq2+g/\sqrt{b} z = k / b m ≥ 2 + g / b N N N z z z b m > 0 b_m>0 b m > 0 F ( t ) = t ( log ( b t / N ) / 2 + Ω ( z t ) ) F(t)=t(\log(bt/N)/2+\Omega(z_t)) F ( t ) = t ( log ( b t / N ) /2 + Ω ( z t )) z t = k N / ( b t ) z_t=k\sqrt{N/(bt)} z t = k N / ( b t ) t t t Ω ′ = g s c \Omega'=g_{\mathrm{sc}} Ω ′ = g sc d z t / d t = − z t / ( 2 t ) dz_t/dt=-z_t/(2t) d z t / d t = − z t / ( 2 t )
F ′ ( t ) = 1 2 log b t N + Ω ( z t ) + 1 2 ( 1 − z t g s c ( z t ) ) . F'(t)=\frac{1}{2}\log\frac{bt}{N}+\Omega(z_t)+\frac{1}{2}(1-z_tg_{\mathrm{sc}}(z_t)). F ′ ( t ) = 2 1 log N b t + Ω ( z t ) + 2 1 ( 1 − z t g sc ( z t )) . which is bounded uniformly for t = N − O ( 1 ) t=N-O(1) t = N − O ( 1 ) z t z_t z t
The next lemma bounds determinants of matrices obtained by adding a border of fixed size to a matrix of the form k Id − W k\operatorname{Id}-W k Id − W
Lemma 7.11 (Bordered determinants). Let b , g > 0 b,g>0 b , g > 0 k + < ∞ k_+<\infty k + < ∞ 2 b + g ≤ k ≤ k + 2\sqrt{b}+g\leq k\leq k_+ 2 b + g ≤ k ≤ k + 1 ≤ m ≤ N 1\leq m\leq N 1 ≤ m ≤ N W W W b m / N G m \sqrt{b_m/N}G_m b m / N G m T = k Id m − W T=k\operatorname{Id}_m-W T = k Id m − W j ≥ 1 j\geq1 j ≥ 1
integer, and let U U U m × j m \times j m × j C ∂ C_{\partial} C ∂ j × j j \times j j × j W W W
M = ( T U U T C ∂ ) , μ T = E ∣ det T ∣ , γ T = ∑ i = 1 m log max { ∣ k − λ i ( W ) ∣ , g / 2 } . M=\begin{pmatrix} T & U \\ U^{\mathsf T} & C_{\partial} \end{pmatrix}, \qquad\mu_T=\mathbb{E}|\det T|, \qquad\gamma_T=\sum_{i=1}^{m}\log\max\{|k-\lambda_i(W)|,g/2\}. M = ( T U T U C ∂ ) , μ T = E ∣ det T ∣ , γ T = i = 1 ∑ m log max { ∣ k − λ i ( W ) ∣ , g /2 } . There exists C j C_j C j j j j C 1 = 1 C_1=1 C 1 = 1
∣ det M ∣ ≤ C j e γ T ( ∥ C ∂ ∥ op + 2 g ∥ U ∥ F 2 ) j . (145) |\det M| \le C_j e^{\gamma_T}\left(\|C_{\partial}\|_{\operatorname{op}}+\frac{2}{g}\|U\|_{\mathrm F}^2\right)^j. \tag*{(145)} ∣ det M ∣ ≤ C j e γ T ( ∥ C ∂ ∥ op + g 2 ∥ U ∥ F 2 ) j . (145) Suppose that E ( ∥ C ∂ ∥ op + ∥ U ∥ F 2 ) q ≤ L q \mathbb{E}\left(\|C_{\partial}\|_{\operatorname{op}}+\|U\|_{\mathrm F}^2\right)^q\le L_q E ( ∥ C ∂ ∥ op + ∥ U ∥ F 2 ) q ≤ L q q ≥ 1 q\ge1 q ≥ 1 r ≥ 1 r\ge1 r ≥ 1
E ∣ det M ∣ r ≤ C r μ T r , (146) \mathbb{E}|\det M|^r\le C_r\mu_T^r, \tag*{(146)} E ∣ det M ∣ r ≤ C r μ T r , (146) where C r C_r C r b , g , k + , j , r b,g,k_+,j,r b , g , k + , j , r L 2 j r L_{2jr} L 2 j r
Suppose in addition that there exist s 0 > 0 s_0>0 s 0 > 0 G G G G G G T T T ∣ det ( C ∂ − U T T − 1 U ) ∣ ≥ s 0 |\det(C_{\partial}-U^{\mathsf T}T^{-1}U)|\ge s_0 ∣ det ( C ∂ − U T T − 1 U ) ∣ ≥ s 0 K 2 1 / 2 P ( G c ) 1 / 2 ≤ 1 / 2 K_2^{1/2}\mathbb{P}(G^{\mathrm c})^{1/2}\le1/2 K 2 1/2 P ( G c ) 1/2 ≤ 1/2 K 2 = K 2 ( b , g , k + ) K_2=K_2(b,g,k_+) K 2 = K 2 ( b , g , k + ) E ∣ det M ∣ ≥ s 0 μ T / 2 \mathbb{E}|\det M|\ge s_0\mu_T/2 E ∣ det M ∣ ≥ s 0 μ T /2 r ≥ 1 r\ge1 r ≥ 1
E ( ∣ det M ∣ E ∣ det M ∣ ) r ≤ 2 r C r s 0 r . (147) \mathbb{E}\left(\frac{|\det M|}{\mathbb{E}|\det M|}\right)^r\le\frac{2^rC_r}{s_0^r}. \tag*{(147)} E ( E ∣ det M ∣ ∣ det M ∣ ) r ≤ s 0 r 2 r C r . (147) If instead M = k Id m + j − C − W ′ M=k\operatorname{Id}_{m+j}-C-W' M = k Id m + j − C − W ′ m + j ≤ N m+j\le N m + j ≤ N C C C ∥ C ∥ op ≤ g / 4 \|C\|_{\operatorname{op}}\le g/4 ∥ C ∥ op ≤ g /4 W ′ W' W ′ Var Tr ( W ′ A ) ≤ ( 2 b / N ) Tr ( A 2 ) \operatorname{Var}\operatorname{Tr}(W'A)\le(2b/N)\operatorname{Tr}(A^2) Var Tr ( W ′ A ) ≤ ( 2 b / N ) Tr ( A 2 ) A A A ∣ det M ∣ |\det M| ∣ det M ∣ Proposition 7.7 in dimension m + j m+j m + j
Proof. For (1), conjugate M M M diag ( O , Id j ) \operatorname{diag}(O,\operatorname{Id}_j) diag ( O , Id j ) O O O O T T O = diag ( t 1 , … , t m ) O^{\mathsf T}TO=\operatorname{diag}(t_1,\ldots,t_m) O T T O = diag ( t 1 , … , t m ) U U U O T U O^{\mathsf T}U O T U det M \det M det M ∥ U ∥ F \|U\|_{\mathrm F} ∥ U ∥ F γ T \gamma_T γ T t i t_i t i t i t_i t i
det M = ∑ I ⊂ { 1 , … , m } ( ∏ i ∉ I t i ) det ( 0 ∣ I ∣ U I U I T C ∂ ) , \det M=\sum_{I\subset\{1,\ldots,m\}}\left(\prod_{i\notin I}t_i\right)\det\begin{pmatrix} 0_{|I|} & U_I \\ U_I^{\mathsf T} & C_{\partial} \end{pmatrix}, det M = I ⊂ { 1 , … , m } ∑ i ∈ / I ∏ t i det ( 0 ∣ I ∣ U I T U I C ∂ ) , where U I U_I U I U U U I I I ∣ I ∣ > j |I|>j ∣ I ∣ > j 2 j < ∣ I ∣ + j 2j<|I|+j 2 j < ∣ I ∣ + j ∣ I ∣ ≤ j |I|\le j ∣ I ∣ ≤ j ∣ I ∣ |I| ∣ I ∣ ∣ I ∣ |I| ∣ I ∣ J , J ′ ⊂ { 1 , … , j } J,J'\subset\{1,\ldots,j\} J , J ′ ⊂ { 1 , … , j } ∣ I ∣ |I| ∣ I ∣ det U I , J det U I , J ′ \det U_{I,J}\det U_{I,J'} det U I , J det U I , J ′ C ∂ C_{\partial} C ∂ j − ∣ I ∣ j-|I| j − ∣ I ∣ ∥ C ∂ ∥ op j − ∣ I ∣ \|C_{\partial}\|_{\operatorname{op}}^{j-|I|} ∥ C ∂ ∥ op j − ∣ I ∣
∑ ∣ I ∣ = ℓ ∣ det U I , J det U I , J ′ ∣ ≤ ( ∑ ∣ I ∣ = ℓ det ( U I , J ) 2 ∑ ∣ I ∣ = ℓ det ( U I , J ′ ) 2 ) 1 / 2 ≤ ∥ U ∥ F 2 ℓ , (148) \sum_{|I|=\ell}|\det U_{I,J}\det U_{I,J'}|
\le\left(\sum_{|I|=\ell}\det(U_{I,J})^2\sum_{|I|=\ell}\det(U_{I,J'})^2\right)^{1/2}
\le\|U\|_{\mathrm F}^{2\ell}, \tag*{(148)} ∣ I ∣ = ℓ ∑ ∣ det U I , J det U I , J ′ ∣ ≤ ∣ I ∣ = ℓ ∑ det ( U I , J ) 2 ∣ I ∣ = ℓ ∑ det ( U I , J ′ ) 2 1/2 ≤ ∥ U ∥ F 2 ℓ , (148) since each sum is the Gram determinant of ℓ \ell ℓ U U U ∏ i ∉ I ∣ t i ∣ ≤ e y T ( g / 2 ) − ∣ I ∣ \prod_{i\notin I}|t_i| \le e^{y_T(g/2)^{-|I|}} ∏ i ∈ / I ∣ t i ∣ ≤ e y T ( g /2 ) − ∣ I ∣ ℓ = ∣ I ∣ ≤ j \ell=|I|\le j ℓ = ∣ I ∣ ≤ j J , J ′ J,J' J , J ′
∣ det M ∣ ≤ e y T ∑ ℓ = 0 j ( j ℓ ) 2 ∥ C ∂ ∥ o p j − ℓ ( 2 g ∥ U ∥ F 2 ) ℓ , |\det M| \le e^{y_T}\sum_{\ell=0}^{j}\binom{j}{\ell}^{2}\|C_{\partial}\|_{\mathrm{op}}^{j-\ell}\left(\frac{2}{g}\|U\|_{F}^{2}\right)^{\ell}, ∣ det M ∣ ≤ e y T ℓ = 0 ∑ j ( ℓ j ) 2 ∥ C ∂ ∥ op j − ℓ ( g 2 ∥ U ∥ F 2 ) ℓ , which implies (145) with C j = max ℓ ( j ℓ ) C_j=\max_{\ell}\binom{j}{\ell} C j = max ℓ ( ℓ j ) j = 1 j=1 j = 1 e y T ( ∣ C ∂ ∣ + ( 2 / g ) ∣ U ∣ 2 ) e^{y_T}(|C_{\partial}|+(2/g)|U|^{2}) e y T ( ∣ C ∂ ∣ + ( 2/ g ) ∣ U ∣ 2 ) T T T
For (2), we apply the Cauchy–Schwarz inequality to (145) . Using (7.8) with C = 0 C=0 C = 0 E e 2 r Y T ≤ K 2 r μ T 2 r \mathbb{E}e^{2rY_T}\le K_{2r}\mu_T^{2r} E e 2 r Y T ≤ K 2 r μ T 2 r W W W 2 r 2r 2 r max { 1 , 2 / g } 2 j r L 2 j r \max\{1,2/g\}^{2jr}L_{2jr} max { 1 , 2/ g } 2 j r L 2 j r (146) .
For (3), the Schur complement formula implies that ∣ det M ∣ = ∣ det T ∣ ∣ det ( C ∂ − U T − 1 U ) ∣ ≥ s 0 ∣ det T ∣ |\det M|=|\det T||\det(C_{\partial}-UT^{-1}U)|\ge s_0|\det T| ∣ det M ∣ = ∣ det T ∣∣ det ( C ∂ − U T − 1 U ) ∣ ≥ s 0 ∣ det T ∣ G G G r = 2 r=2 r = 2
E ∣ det M ∣ ≥ s 0 E [ ∣ det T ∣ 1 G ] ≥ s 0 μ T ( 1 − K 2 1 / 2 P ( G c ) 1 / 2 ) ≥ s 0 μ T 2 . \mathbb{E}|\det M|\ge s_0\mathbb{E}[|\det T|1_G]\ge s_0\mu_T\left(1-K_2^{1/2}\mathbb{P}(G^c)^{1/2}\right)\ge\frac{s_0\mu_T}{2}. E ∣ det M ∣ ≥ s 0 E [ ∣ det T ∣ 1 G ] ≥ s 0 μ T ( 1 − K 2 1/2 P ( G c ) 1/2 ) ≥ 2 s 0 μ T . Combining this with (146) , we obtain (147) . Part (4) is Proposition 7.7 in dimension m + j ≤ N m+j\le N m + j ≤ N
Remark 7.12 (The hypothesis in part (3) is needed). Bounded moments of the border give upper bounds relative to μ T \mu_T μ T E ∣ det M ∣ \mathbb{E}|\det M| E ∣ det M ∣ j = 1 j=1 j = 1 U = 0 U=0 U = 0 C ∂ = 1 A N C_{\partial}=1_{A_N} C ∂ = 1 A N A N A_N A N W W W P ( A N ) = e − N \mathbb{P}(A_N)=e^{-N} P ( A N ) = e − N E ∣ det M ∣ = e − N μ T \mathbb{E}|\det M|=e^{-N}\mu_T E ∣ det M ∣ = e − N μ T E ∣ det M ∣ r ≥ e − N μ T r \mathbb{E}|\det M|^r\ge e^{-N}\mu_T^r E ∣ det M ∣ r ≥ e − N μ T r r r r e ( r − 1 ) N e^{(r-1)N} e ( r − 1 ) N
We turn to two points. Let u 1 , u 2 ∈ S N u_1,u_2\in S_N u 1 , u 2 ∈ S N ρ = R ( u 1 , u 2 ) \rho=R(u_1,u_2) ρ = R ( u 1 , u 2 ) ∣ ρ ∣ < 1 |\rho|<1 ∣ ρ ∣ < 1 two-point conditioned law with marks ( h i , k i ) i = 1 , 2 (h_i,k_i)_{i=1,2} ( h i , k i ) i = 1 , 2 Λ u \Lambda_u Λ u z ↦ ( Λ u 1 z , Λ u 2 z ) z\mapsto(\Lambda_{u_1}z,\Lambda_{u_2}z) z ↦ ( Λ u 1 z , Λ u 2 z ) ( N h 1 , N k 1 , 0 , N h 2 , N k 2 , 0 ) (Nh_1,Nk_1,0,Nh_2,Nk_2,0) ( N h 1 , N k 1 , 0 , N h 2 , N k 2 , 0 )
( H Q ( u 1 ) , H Q ( u 2 ) , ∇ S N H Q ( u 1 ) , ∇ S N H Q ( u 2 ) ) = ( N h 1 , N h 2 , 0 , 0 ) , (H_Q(u_1),H_Q(u_2),\nabla_{S_N}H_Q(u_1),\nabla_{S_N}H_Q(u_2))=(Nh_1,Nh_2,0,0), ( H Q ( u 1 ) , H Q ( u 2 ) , ∇ S N H Q ( u 1 ) , ∇ S N H Q ( u 2 )) = ( N h 1 , N h 2 , 0 , 0 ) , with k i = ϕ h i k_i=\phi h_i k i = ϕ h i K Q / N K_Q/\sqrt{N} K Q / N
Lemma 7.13 (Hessian determinants under two-point conditioning). Assume that d ≥ 4 d\ge4 d ≥ 4 g > 0 g>0 g > 0 K 2 ⊂ R 4 \mathcal{K}_2\subset\mathbb{R}^4 K 2 ⊂ R 4 2 b Q + g ≤ k i ≤ k + 2\sqrt{b_Q}+g\le k_i\le k_+ 2 b Q + g ≤ k i ≤ k + ( h 1 , k 1 , h 2 , k 2 ) ∈ K 2 (h_1,k_1,h_2,k_2)\in\mathcal{K}_2 ( h 1 , k 1 , h 2 , k 2 ) ∈ K 2 k i = ϕ h i k_i=\phi h_i k i = ϕ h i K 2 \mathcal{K}_2 K 2 ρ 0 ∈ ( 0 , 1 ) \rho_0\in(0,1) ρ 0 ∈ ( 0 , 1 ) ζ \zeta ζ Q Q Q g g g K 2 \mathcal{K}_2 K 2 N ≥ 2 N\ge2 N ≥ 2 u 1 , u 2 ∈ S N u_1,u_2\in S_N u 1 , u 2 ∈ S N ∣ R ( u 1 , u 2 ) ∣ ≤ ρ 0 |R(u_1,u_2)|\le\rho_0 ∣ R ( u 1 , u 2 ) ∣ ≤ ρ 0
The covariance of ( Λ u 1 Z , Λ u 2 Z ) (\Lambda_{u_1}Z,\Lambda_{u_2}Z) ( Λ u 1 Z , Λ u 2 Z )
Under the two-point conditioned law with marks in K 2 \mathcal{K}_2 K 2 i = 1 , 2 i=1,2 i = 1 , 2 r ≥ 1 r\ge1 r ≥ 1 D i = ∣ det ∇ S N 2 H Q ( u i ) ∣ D_i=\lvert\det\nabla^2_{S^N}H_Q(u_i)\rvert D i = ∣ det ∇ S N 2 H Q ( u i )∣ E ( D i / E D i ) r ≤ K r ( b Q , g , k + ) \mathbb{E}(D_i/\mathbb{E}D_i)^r\le K_r(b_Q,g,k_+) E ( D i / E D i ) r ≤ K r ( b Q , g , k + )
The determinants D 1 D_1 D 1 D 2 D_2 D 2
Proof. We first set up coordinates and record the covariances, then prove (1), and finally show that the conditional Hessian at u 1 u_1 u 1
Coordinates and covariances. We write ρ = R ( u 1 , u 2 ) \rho=R(u_1,u_2) ρ = R ( u 1 , u 2 ) ρ ˉ = 1 − ρ 2 \bar{\rho}=\sqrt{1-\rho^2} ρ ˉ = 1 − ρ 2 u 1 = N e N u_1=\sqrt{N}e_N u 1 = N e N u 2 = N ( ρ ˉ e N − 1 + ρ e N ) u_2=\sqrt{N}(\bar{\rho}e_{N-1}+\rho e_N) u 2 = N ( ρ ˉ e N − 1 + ρ e N ) τ 1 = e N − 1 ∈ u 1 ⊥ \tau_1=e_{N-1}\in u_1^\perp τ 1 = e N − 1 ∈ u 1 ⊥ τ 2 = ρ e N − 1 − ρ ˉ e N ∈ u 2 ⊥ \tau_2=\rho e_{N-1}-\bar{\rho}e_N\in u_2^\perp τ 2 = ρ e N − 1 − ρ ˉ e N ∈ u 2 ⊥ u 1 u_1 u 1 u 2 u_2 u 2 e 1 , … , e N − 2 e_1,\ldots,e_{N-2} e 1 , … , e N − 2 Y i = ( H Q ( u i ) , K Q ( u i ) ) / N Y_i=(H_Q(u_i),K_Q(u_i))/\sqrt{N} Y i = ( H Q ( u i ) , K Q ( u i )) / N η i = τ i ⋅ ∇ H Q ( u i ) \eta_i=\tau_i\cdot\nabla H_Q(u_i) η i = τ i ⋅ ∇ H Q ( u i ) ∂ c H Q ( u i ) \partial_cH_Q(u_i) ∂ c H Q ( u i ) c ≤ N − 2 c\le N-2 c ≤ N − 2 κ ( x , y ) = N ζ ~ Q ( x ⋅ y / N ) \kappa(x,y)=N\widetilde{\zeta}_Q(x\cdot y/N) κ ( x , y ) = N ζ Q ( x ⋅ y / N ) x = u 1 x=u_1 x = u 1 y = u 2 y=u_2 y = u 2 u 1 ⋅ u 2 = N ρ u_1\cdot u_2=N\rho u 1 ⋅ u 2 = N ρ u 2 ⋅ τ 1 = N ρ ˉ u_2\cdot\tau_1=\sqrt{N}\bar{\rho} u 2 ⋅ τ 1 = N ρ ˉ u 1 ⋅ τ 2 = − N ρ ˉ u_1\cdot\tau_2=-\sqrt{N}\bar{\rho} u 1 ⋅ τ 2 = − N ρ ˉ τ 1 ⋅ τ 2 = ρ \tau_1\cdot\tau_2=\rho τ 1 ⋅ τ 2 = ρ
Cov ( Y 1 , Y 2 ) = ( ζ ~ Q ( ρ ) ρ ζ ~ Q ′ ( ρ ) ρ ζ ~ Q ′ ( ρ ) ρ ζ ~ Q ′ ( ρ ) + ρ 2 ζ ~ Q ′ ′ ( ρ ) ) , Cov ( η 1 , η 2 ) = ρ ζ ~ Q ′ ( ρ ) − ρ ˉ 2 ζ ~ Q ′ ′ ( ρ ) , \operatorname{Cov}(Y_1,Y_2)=
\begin{pmatrix}
\widetilde{\zeta}_Q(\rho) & \rho\widetilde{\zeta}'_Q(\rho)\\
\rho\widetilde{\zeta}'_Q(\rho) & \rho\widetilde{\zeta}'_Q(\rho)+\rho^2\widetilde{\zeta}''_Q(\rho)
\end{pmatrix},
\qquad
\operatorname{Cov}(\eta_1,\eta_2)=\rho\widetilde{\zeta}'_Q(\rho)-\bar{\rho}^{\,2}\widetilde{\zeta}''_Q(\rho), Cov ( Y 1 , Y 2 ) = ( ζ Q ( ρ ) ρ ζ Q ′ ( ρ ) ρ ζ Q ′ ( ρ ) ρ ζ Q ′ ( ρ ) + ρ 2 ζ Q ′′ ( ρ ) ) , Cov ( η 1 , η 2 ) = ρ ζ Q ′ ( ρ ) − ρ ˉ 2 ζ Q ′′ ( ρ ) , Cov ( η 1 , Y 2 ) = − Cov ( η 2 , Y 1 ) = ρ ˉ ( ζ ~ Q ′ ( ρ ) , ζ ~ Q ′ ( ρ ) + ρ ζ ~ Q ′ ′ ( ρ ) ) , \operatorname{Cov}(\eta_1,Y_2)=-\operatorname{Cov}(\eta_2,Y_1)=\bar{\rho}\bigl(\widetilde{\zeta}'_Q(\rho),\widetilde{\zeta}'_Q(\rho)+\rho\widetilde{\zeta}''_Q(\rho)\bigr), Cov ( η 1 , Y 2 ) = − Cov ( η 2 , Y 1 ) = ρ ˉ ( ζ Q ′ ( ρ ) , ζ Q ′ ( ρ ) + ρ ζ Q ′′ ( ρ ) ) , Cov ( ∂ c H Q ( u 1 ) , ∂ c ′ H Q ( u 2 ) ) = ζ ~ Q ′ ( ρ ) δ c c ′ . (149) \operatorname{Cov}\bigl(\partial_cH_Q(u_1),\partial_{c'}H_Q(u_2)\bigr)=\widetilde{\zeta}'_Q(\rho)\delta_{cc'}. \tag*{(149)} Cov ( ∂ c H Q ( u 1 ) , ∂ c ′ H Q ( u 2 ) ) = ζ Q ′ ( ρ ) δ c c ′ . (149) For instance, Cov ( η 1 , K Q ( u 2 ) ) = τ 1 ⋅ ∇ x ( N f ( t ) ) = f ′ ( ρ ) u 2 ⋅ τ 1 \operatorname{Cov}(\eta_1,K_Q(u_2))=\tau_1\cdot\nabla_x(Nf(t))=f'(\rho)u_2\cdot\tau_1 Cov ( η 1 , K Q ( u 2 )) = τ 1 ⋅ ∇ x ( N f ( t )) = f ′ ( ρ ) u 2 ⋅ τ 1 f ( t ) = t ζ ~ Q ′ ( t ) f(t)=t\widetilde{\zeta}'_Q(t) f ( t ) = t ζ Q ′ ( t ) Y 1 , Y 2 , η 1 , η 2 Y_1,Y_2,\eta_1,\eta_2 Y 1 , Y 2 , η 1 , η 2
Invertibility. By the preceding paragraph, the covariance of all the observations is block diagonal. One block is the covariance S ( ρ ) S(\rho) S ( ρ ) ( Y 1 , Y 2 , η 1 , η 2 ) (Y_1,Y_2,\eta_1,\eta_2) ( Y 1 , Y 2 , η 1 , η 2 ) 6 × 6 6\times6 6 × 6 4 × 4 4\times4 4 × 4 ρ \rho ρ ρ ˉ \bar{\rho} ρ ˉ N N N N − 2 N-2 N − 2 2 × 2 2\times2 2 × 2 a Q a_Q a Q ζ ~ Q ′ ( ρ ) \widetilde{\zeta}'_Q(\rho) ζ Q ′ ( ρ ) ζ ~ \widetilde{\zeta} ζ d d d ∣ ζ ~ Q ( j ) ( ρ ) ∣ ≤ C ∣ ρ ∣ d − j \lvert\widetilde{\zeta}^{(j)}_Q(\rho)\rvert\le C\lvert\rho\rvert^{d-j} ∣ ζ Q ( j ) ( ρ )∣ ≤ C ∣ ρ ∣ d − j ∣ ρ ∣ ≤ 1 \lvert\rho\rvert\le1 ∣ ρ ∣ ≤ 1 j ≤ 3 j\le3 j ≤ 3 S ( 0 ) = diag ( Σ Q , Σ Q , a Q , a Q ) S(0)=\operatorname{diag}(\Sigma_Q,\Sigma_Q,a_Q,a_Q) S ( 0 ) = diag ( Σ Q , Σ Q , a Q , a Q ) ζ ~ Q ( Q ) \widetilde{\zeta}_Q(Q) ζ Q ( Q ) Σ Q \Sigma_Q Σ Q ∣ ζ ~ Q ′ ( ρ ) ∣ ≤ C ∣ ρ ∣ d − 1 \lvert\widetilde{\zeta}'_Q(\rho)\rvert\le C\lvert\rho\rvert^{d-1} ∣ ζ Q ′ ( ρ )∣ ≤ C ∣ ρ ∣ d − 1 ρ 1 > 0 \rho_1>0 ρ 1 > 0 ∣ ρ ∣ ≤ ρ 1 \lvert\rho\rvert\le\rho_1 ∣ ρ ∣ ≤ ρ 1 S ( ρ ) S(\rho) S ( ρ ) S ( 0 ) S(0) S ( 0 ) ∣ ζ ~ Q ′ ( ρ ) ∣ ≤ a Q / 2 \lvert\widetilde{\zeta}'_Q(\rho)\rvert\le a_Q/2 ∣ ζ Q ′ ( ρ )∣ ≤ a Q /2 N N N ρ 0 ≤ ρ 1 \rho_0\le\rho_1 ρ 0 ≤ ρ 1
The conditional Hessian. Consider ∇ ⊥ 2 H Q ( u 1 ) \nabla^2_{\perp}H_Q(u_1) ∇ ⊥ 2 H Q ( u 1 ) e 1 , … , e N − 2 , τ 1 e_1,\ldots,e_{N-2},\tau_1 e 1 , … , e N − 2 , τ 1 u 1 ⊥ u_1^\perp u 1 ⊥ u 1 u_1 u 1
The entries ∂ a ∂ b H Q ( u 1 ) \partial_a\partial_b H_Q(u_1) ∂ a ∂ b H Q ( u 1 ) a , b ≤ N − 2 a,b \le N-2 a , b ≤ N − 2 u 2 u_2 u 2 κ \kappa κ ( u 2 ) a (u_2)_a ( u 2 ) a ( u 2 ) b (u_2)_b ( u 2 ) b ( u 1 ) c (u_1)_c ( u 1 ) c ( τ 2 ) a (\tau_2)_a ( τ 2 ) a
The entry ∂ a ∂ τ 1 H Q ( u 1 ) \partial_a\partial_{\tau_1}H_Q(u_1) ∂ a ∂ τ 1 H Q ( u 1 ) a ≤ N − 2 a \le N-2 a ≤ N − 2 Y 2 Y_2 Y 2 η 2 \eta_2 η 2 ∂ a H Q ( u 2 ) \partial_a H_Q(u_2) ∂ a H Q ( u 2 )
The corner v 1 = τ 1 ⋅ ∇ 2 H Q ( u 1 ) τ 1 v_1=\tau_1\cdot\nabla^2H_Q(u_1)\tau_1 v 1 = τ 1 ⋅ ∇ 2 H Q ( u 1 ) τ 1
Cov ( v 1 , Y 2 ) = ρ ˉ 2 N ( ξ Q ′ ′ ( ρ ) , 2 ξ Q ′ ′ ( ρ ) + ρ ξ ˉ Q ′ ′ ′ ( ρ ) ) , \operatorname{Cov}(v_1,Y_2)=\frac{\bar{\rho}^2}{\sqrt{N}}\left(\xi_Q^{\prime\prime}(\rho),\,2\xi_Q^{\prime\prime}(\rho)+\rho\bar{\xi}_Q^{\prime\prime\prime}(\rho)\right), Cov ( v 1 , Y 2 ) = N ρ ˉ 2 ( ξ Q ′′ ( ρ ) , 2 ξ Q ′′ ( ρ ) + ρ ξ ˉ Q ′′′ ( ρ ) ) , Cov ( v 1 , η 2 ) = 1 N ( 2 ρ ρ ˉ ξ Q ′ ′ ( ρ ) − ρ 3 ξ ˉ Q ′ ′ ′ ( ρ ) ) . \operatorname{Cov}(v_1,\eta_2)=\frac{1}{\sqrt{N}}\left(2\rho\bar{\rho}\,\xi_Q^{\prime\prime}(\rho)-\rho^3\bar{\xi}_Q^{\prime\prime\prime}(\rho)\right). Cov ( v 1 , η 2 ) = N 1 ( 2 ρ ρ ˉ ξ Q ′′ ( ρ ) − ρ 3 ξ ˉ Q ′′′ ( ρ ) ) . The conditional mean of an entry is its covariance row with the observations, multiplied by the inverse covariance of the observations and by the target vector. The target vanishes on all gradient components. Since the covariance of the observations is block diagonal and the common gradient components form separate blocks with zero target, the first two items show that the conditional mean vanishes on the common block and on the cross row. Only the corner can have a nonzero conditional mean m 1 m_1 m 1
To bound m 1 m_1 m 1 6 × 6 6\times6 6 × 6 Y = ( Y 1 , Y 2 ) Y=(Y_1,Y_2) Y = ( Y 1 , Y 2 ) η = ( η 1 , η 2 ) \eta=(\eta_1,\eta_2) η = ( η 1 , η 2 ) c Y = Cov ( v 1 , Y ) c_Y=\operatorname{Cov}(v_1,Y) c Y = Cov ( v 1 , Y ) c η = Cov ( v 1 , η ) c_\eta=\operatorname{Cov}(v_1,\eta) c η = Cov ( v 1 , η ) y ∗ = N ( h 1 , k 1 , h 2 , k 2 ) y^*=\sqrt{N}(h_1,k_1,h_2,k_2) y ∗ = N ( h 1 , k 1 , h 2 , k 2 ) Y Y Y 0 0 0 η \eta η
m 1 = ( c Y − c η S η η − 1 S η Y ) ( S Y Y − S Y η S η η − 1 S η Y ) − 1 y ∗ . m_1=\left(c_Y-c_\eta S_{\eta\eta}^{-1}S_{\eta Y}\right)\left(S_{YY}-S_{Y\eta}S_{\eta\eta}^{-1}S_{\eta Y}\right)^{-1}y^*. m 1 = ( c Y − c η S η η − 1 S η Y ) ( S Y Y − S Y η S η η − 1 S η Y ) − 1 y ∗ . By the bounds on ξ Q ( j ) \xi_Q^{(j)} ξ Q ( j ) ∣ c Y ∣ ≤ C ∣ ρ ∣ d − 2 / N |c_Y|\le C|\rho|^{d-2}/\sqrt{N} ∣ c Y ∣ ≤ C ∣ ρ ∣ d − 2 / N ∣ c η ∣ ≤ C ∣ ρ ∣ d − 3 / N |c_\eta|\le C|\rho|^{d-3}/\sqrt{N} ∣ c η ∣ ≤ C ∣ ρ ∣ d − 3 / N ∣ S η Y ∣ ≤ C ∣ ρ ∣ d − 1 |S_{\eta Y}|\le C|\rho|^{d-1} ∣ S η Y ∣ ≤ C ∣ ρ ∣ d − 1 S η η − 1 S_{\eta\eta}^{-1} S η η − 1 C ( ∣ ρ ∣ d − 2 + ∣ ρ ∣ 2 d − 4 ) / N ≤ C ∣ ρ ∣ d − 2 / N C(|\rho|^{d-2}+|\rho|^{2d-4})/\sqrt{N}\le C|\rho|^{d-2}/\sqrt{N} C ( ∣ ρ ∣ d − 2 + ∣ ρ ∣ 2 d − 4 ) / N ≤ C ∣ ρ ∣ d − 2 / N Y Y Y η \eta η Y Y YY Y Y S ( ρ ) − 1 S(\rho)^{-1} S ( ρ ) − 1 ∣ y ∗ ∣ ≤ C N |y^*|\le C\sqrt{N} ∣ y ∗ ∣ ≤ C N K 2 K_2 K 2 ∣ m 1 ∣ ≤ C 1 ∣ ρ ∣ d − 2 |m_1|\le C_1|\rho|^{d-2} ∣ m 1 ∣ ≤ C 1 ∣ ρ ∣ d − 2 C 1 C_1 C 1 ξ \xi ξ Q Q Q K 2 K_2 K 2
Gaussian conditioning subtracts a positive semidefinite operator from the covariance. Then the centered part W W W ∇ ⊥ 2 H Q ( u 1 ) \nabla_{\perp}^2H_Q(u_1) ∇ ⊥ 2 H Q ( u 1 ) Var Tr ( W A ) ≤ ( 2 b Q / N ) Tr ( A 2 ) \operatorname{Var}\operatorname{Tr}(WA)\le(2b_Q/N)\operatorname{Tr}(A^2) Var Tr ( W A ) ≤ ( 2 b Q / N ) Tr ( A 2 ) (131) applied to the unconditional law. Its mean is C = m 1 τ 1 τ 1 T C=m_1\tau_1\tau_1^{\mathsf T} C = m 1 τ 1 τ 1 T ∥ C ∥ o p ≤ C 1 ∣ ρ ∣ d − 2 \|C\|_{\mathrm{op}}\le C_1|\rho|^{d-2} ∥ C ∥ op ≤ C 1 ∣ ρ ∣ d − 2 ρ 0 ≤ ρ 1 \rho_0\le\rho_1 ρ 0 ≤ ρ 1 C 1 ρ 0 d − 2 ≤ g / 4 C_1\rho_0^{d-2}\le g/4 C 1 ρ 0 d − 2 ≤ g /4 K Q ( u 1 ) = N k 1 K_Q(u_1)=Nk_1 K Q ( u 1 ) = N k 1 (132) implies that − ∇ S N 2 H Q ( u 1 ) = k 1 Id − C − W -\nabla_{S_N}^2H_Q(u_1)=k_1\operatorname{Id}-C-W − ∇ S N 2 H Q ( u 1 ) = k 1 Id − C − W i = 1 i=1 i = 1 Proposition 7.7 . The case i = 2 i=2 i = 2 u 1 u_1 u 1 u 2 u_2 u 2 Y i = H Q ( u i ) / N Y_i=H_Q(u_i)/\sqrt{N} Y i = H Q ( u i ) / N k i = ϕ i k_i=\phi_i k i = ϕ i N = 2 N=2 N = 2 u 1 ⊥ u_1^\perp u 1 ⊥ τ 1 \tau_1 τ 1
Remark 7.14 (Related determinant estimates). Subag [140, Lemma 21] proves exponentially small determinant-weighted edge tails for a fixed number of possibly dependent matrices with GOE marginals and shifts in a compact interval outside [ − 2 , 2 ] [-2,2] [ − 2 , 2 ] G G G − G -G − G
The Kac–Rice formula with field-dependent weights
We prove the Kac–Rice formula in the coefficient space of the field. The first ingredient is the equal-dimensional area formula [69], together with the corresponding elementary case of Sard’s theorem [134]; we include the short proof.
Lemma 7.15 (Equal-dimensional area formula). Let V ⊂ R n V \subset\mathbb{R}^n V ⊂ R n α : V → R n \alpha: V \to\mathbb{R}^n α : V → R n C 1 C^1 C 1 α \alpha α { t ∈ V : det D α ( t ) = 0 } \{t \in V : \det D\alpha(t) = 0\} { t ∈ V : det D α ( t ) = 0 } w ≥ 0 w \ge0 w ≥ 0 V V V
∫ R n ∑ t ∈ V : α ( t ) = y w ( t ) d y = ∫ V w ( t ) ∣ det D α ( t ) ∣ d t . (150) \int_{\mathbb{R}^n} \sum_{t \in V:\,\alpha(t)=y} w(t)\,\mathrm{d}y = \int_V w(t) \lvert\det D\alpha(t) \rvert\,\mathrm{d}t. \tag*{(150)} ∫ R n t ∈ V : α ( t ) = y ∑ w ( t ) d y = ∫ V w ( t ) ∣ det D α ( t )∣ d t . (150) The sum may be defined arbitrarily for y y y
Proof. Let K ⊂ V K \subset V K ⊂ V ϵ > 0 \epsilon> 0 ϵ > 0 L L L ∥ D α ∥ \lVert D\alpha\rVert ∥ D α ∥ K K K D α D\alpha D α δ 0 > 0 \delta_0 > 0 δ 0 > 0 Q Q Q δ ≤ δ 0 \delta\le\delta_0 δ ≤ δ 0 K K K t 0 t_0 t 0 ∣ α ( t ) − α ( t 0 ) − D α ( t 0 ) ( t − t 0 ) ∣ ≤ ϵ n δ \lvert\alpha(t) - \alpha(t_0) - D\alpha(t_0)(t - t_0) \rvert\le\epsilon\sqrt{n}\,\delta ∣ α ( t ) − α ( t 0 ) − D α ( t 0 ) ( t − t 0 )∣ ≤ ϵ n δ t ∈ Q t \in Q t ∈ Q Q Q Q t ↦ α ( t 0 ) + D α ( t 0 ) ( t − t 0 ) t \mapsto\alpha(t_0) + D\alpha(t_0)(t - t_0) t ↦ α ( t 0 ) + D α ( t 0 ) ( t − t 0 ) n − 1 n - 1 n − 1 L n δ L\sqrt{n}\,\delta L n δ α ( Q ) \alpha(Q) α ( Q ) n − 1 n - 1 n − 1 C n ( L + 1 ) δ C_n(L + 1)\delta C n ( L + 1 ) δ C n ϵ δ C_n\epsilon\delta C n ϵδ C n ( L + 1 ) n − 1 ϵ δ n C_n(L + 1)^{n-1}\epsilon\delta^n C n ( L + 1 ) n − 1 ϵ δ n δ \delta δ K K K C K δ − n C_K\delta^{-n} C K δ − n K K K K K K C K , n ( L + 1 ) n − 1 ϵ C_{K,n}(L + 1)^{n-1}\epsilon C K , n ( L + 1 ) n − 1 ϵ ϵ → 0 \epsilon\to0 ϵ → 0 V V V
Every regular point has an open neighborhood on which α \alpha α C 1 C^1 C 1 V i V_i V i P i = V i ∖ ⋃ i ′ < i V i ′ P_i = V_i \setminus\bigcup_{i'<i} V_{i'} P i = V i ∖ ⋃ i ′ < i V i ′ i i i ∫ 1 α ( P i ) ( y ) w ( ( α ∣ V i ) − 1 ( y ) ) d y = \int1_{\alpha(P_i)}(y)\,w((\alpha\vert_{V_i})^{-1}(y))\,\mathrm{d}y = ∫ 1 α ( P i ) ( y ) w (( α ∣ V i ) − 1 ( y )) d y = ∫ P i w ( t ) ∣ det D α ( t ) ∣ d t \int_{P_i} w(t)|\det D\alpha(t)|\,dt ∫ P i w ( t ) ∣ det D α ( t ) ∣ d t i i i (150) : the critical set contributes zero to the right side and only a null set of values y y y
Next we state the Kac–Rice formula for a general Gaussian field that depends linearly on finitely many coefficients. Let M M M n n n φ 1 , … , φ ℓ \varphi_1,\ldots,\varphi_\ell φ 1 , … , φ ℓ M M M f z = ∑ i z i φ i f_z=\sum_i z_i\varphi_i f z = ∑ i z i φ i z ∈ R ℓ z\in\mathbb{R}^{\ell} z ∈ R ℓ Z Z Z R ℓ \mathbb{R}^{\ell} R ℓ u ∈ M u\in M u ∈ M Γ u : R ℓ → T u M \Gamma_u:\mathbb{R}^{\ell}\to T_uM Γ u : R ℓ → T u M z ↦ ∇ f z ( u ) z\mapsto\nabla f_z(u) z ↦ ∇ f z ( u ) Γ u Γ u ∗ \Gamma_u\Gamma_u^* Γ u Γ u ∗ Z 0 u = ( Id − Γ u ∗ ( Γ u Γ u ∗ ) − 1 Γ u ) Z Z_0^u=(\operatorname{Id}-\Gamma_u^*(\Gamma_u\Gamma_u^*)^{-1}\Gamma_u)Z Z 0 u = ( Id − Γ u ∗ ( Γ u Γ u ∗ ) − 1 Γ u ) Z ker Γ u = { z : ∇ f z ( u ) = 0 } \ker\Gamma_u=\{z:\nabla f_z(u)=0\} ker Γ u = { z : ∇ f z ( u ) = 0 } p u ( 0 ) = ( 2 π ) − n / 2 det ( Γ u Γ u ∗ ) − 1 / 2 p_u(0)=(2\pi)^{-n/2}\det(\Gamma_u\Gamma_u^*)^{-1/2} p u ( 0 ) = ( 2 π ) − n /2 det ( Γ u Γ u ∗ ) − 1/2 ∇ f Z ( u ) \nabla f_Z(u) ∇ f Z ( u ) 0 0 0 T u M T_uM T u M Z 0 u Z_0^u Z 0 u Z Z Z ∇ f Z ( u ) = 0 \nabla f_Z(u)=0 ∇ f Z ( u ) = 0
Proposition 7.16 (Kac–Rice formula in coefficient space). Assume that Γ u Γ u ∗ \Gamma_u\Gamma_u^* Γ u Γ u ∗ u ∈ M u\in M u ∈ M z z z f z f_z f z f Z f_Z f Z φ ≥ 0 \varphi\geq0 φ ≥ 0 R ℓ × M \mathbb{R}^{\ell}\times M R ℓ × M z ↦ ∑ u : ∇ f z ( u ) = 0 φ ( z , u ) z\mapsto\sum_{u:\nabla f_z(u)=0}\varphi(z,u) z ↦ ∑ u : ∇ f z ( u ) = 0 φ ( z , u )
E ∑ u : ∇ f Z ( u ) = 0 φ ( Z , u ) = ∫ M p u ( 0 ) E [ ∣ det ∇ 2 f Z 0 u ( u ) ∣ φ ( Z 0 u , u ) ] vol ( d u ) . (151) \mathbb{E}\sum_{u:\nabla f_Z(u)=0}\varphi(Z,u)=\int_M p_u(0)\,\mathbb{E}\left[\left|\det\nabla^2f_{Z_0^u}(u)\right|\varphi(Z_0^u,u)\right]\operatorname{vol}(du). \tag*{(151)} E u : ∇ f Z ( u ) = 0 ∑ φ ( Z , u ) = ∫ M p u ( 0 ) E [ det ∇ 2 f Z 0 u ( u ) φ ( Z 0 u , u ) ] vol ( d u ) . (151) The same identity holds for a Borel function φ \varphi φ E ∑ u : ∇ f Z ( u ) = 0 ∣ φ ( Z , u ) ∣ < ∞ \mathbb{E}\sum_{u:\nabla f_Z(u)=0}|\varphi(Z,u)|<\infty E ∑ u : ∇ f Z ( u ) = 0 ∣ φ ( Z , u ) ∣ < ∞ (151) applied to ∣ φ ∣ |\varphi| ∣ φ ∣ (151) with ∣ φ ∣ |\varphi| ∣ φ ∣ φ \varphi φ
Proof. The proof works in local coordinates on M M M (151) .
Local coordinates. Let t ∈ V ⊂ R n t\in V\subset\mathbb{R}^n t ∈ V ⊂ R n M M M γ ( z , t ) = ( ∂ t 1 f z , … , ∂ t n f z ) ( t ) = Γ ( t ) z \gamma(z,t)=(\partial_{t_1}f_z,\ldots,\partial_{t_n}f_z)(t)=\Gamma(t)z γ ( z , t ) = ( ∂ t 1 f z , … , ∂ t n f z ) ( t ) = Γ ( t ) z Γ ( t ) \Gamma(t) Γ ( t ) n × ℓ n\times\ell n × ℓ n n n t t t V V V n n n n × n n\times n n × n Γ 1 ( t ) \Gamma_1(t) Γ 1 ( t ) Γ ( t ) \Gamma(t) Γ ( t ) V V V V V V z = ( z ′ , z ′ ′ ) ∈ R n × R ℓ − n z=(z',z'')\in\mathbb{R}^n\times\mathbb{R}^{\ell-n} z = ( z ′ , z ′′ ) ∈ R n × R ℓ − n Γ ( t ) = ( Γ 1 ( t ) , Γ 2 ( t ) ) \Gamma(t)=(\Gamma_1(t),\Gamma_2(t)) Γ ( t ) = ( Γ 1 ( t ) , Γ 2 ( t ))
γ ( z , t ) = 0 ⟺ z ′ = α z ′ ′ ( t ) , α z ′ ′ ( t ) = − Γ 1 ( t ) − 1 Γ 2 ( t ) z ′ ′ . \gamma(z,t)=0\quad\Longleftrightarrow\quad z'=\alpha_{z''}(t),\qquad\alpha_{z''}(t)=-\Gamma_1(t)^{-1}\Gamma_2(t)z''. γ ( z , t ) = 0 ⟺ z ′ = α z ′′ ( t ) , α z ′′ ( t ) = − Γ 1 ( t ) − 1 Γ 2 ( t ) z ′′ . Differentiating Γ 1 ( t ) α z ′ ′ ( t ) + Γ 2 ( t ) z ′ = 0 \Gamma_1(t)\alpha_{z''}(t)+\Gamma_2(t)z'=0 Γ 1 ( t ) α z ′′ ( t ) + Γ 2 ( t ) z ′ = 0 t t t
D α z ′ ′ ( t ) = − Γ 1 ( t ) − 1 D t γ ( ( α z ′ ′ ( t ) , z ′ ′ ) , t ) . (152) D\alpha_{z''}(t)=-\Gamma_1(t)^{-1}D_t\gamma\bigl((\alpha_{z''}(t),z''),t\bigr). \tag*{(152)} D α z ′′ ( t ) = − Γ 1 ( t ) − 1 D t γ ( ( α z ′′ ( t ) , z ′′ ) , t ) . (152) At a zero of the gradient, D t γ D_t\gamma D t γ (152) , det D α z ′ ′ ( t ) = 0 \det D\alpha_{z^{\prime\prime}}(t)=0 det D α z ′′ ( t ) = 0 t t t f ( α z ′ ′ ( t ) , z ′ ′ ) f_{(\alpha_{z^{\prime\prime}}(t),z^{\prime\prime})} f ( α z ′′ ( t ) , z ′′ ) M M M V 1 , … , V J V_1,\ldots,V_J V 1 , … , V J
Nondegeneracy. The set of pairs ( z , u ) (z,u) ( z , u ) ∇ f z ( u ) = 0 \nabla f_z(u)=0 ∇ f z ( u ) = 0 det ∇ 2 f z ( u ) = 0 \det\nabla^2 f_z(u)=0 det ∇ 2 f z ( u ) = 0 R ℓ × M \mathbb{R}^{\ell}\times M R ℓ × M ( z , u ) (z,u) ( z , u ) R ℓ \mathbb{R}^{\ell} R ℓ M M M D D D V i V_i V i z ′ ′ z^{\prime\prime} z ′′ ( z ′ , z ′ ′ ) (z',z^{\prime\prime}) ( z ′ , z ′′ ) V i V_i V i z ′ z' z ′ α z ′ ′ \alpha_{z^{\prime\prime}} α z ′′ z ′ z' z ′ D D D Z Z Z f Z f_Z f Z M M M
Measurability of the sum. At a pair ( z 0 , u 0 ) (z_0,u_0) ( z 0 , u 0 ) ∇ f z 0 ( u 0 ) = 0 \nabla f_{z_0}(u_0)=0 ∇ f z 0 ( u 0 ) = 0 det ∇ 2 f z 0 ( u 0 ) ≠ 0 \det\nabla^2 f_{z_0}(u_0)\ne0 det ∇ 2 f z 0 ( u 0 ) = 0 O ∋ z 0 O\ni z_0 O ∋ z 0 u ( ⋅ ) u(\cdot) u ( ⋅ ) O O O ( z 0 , u 0 ) (z_0,u_0) ( z 0 , u 0 ) { ( z , u ( z ) ) : z ∈ O } \{(z,u(z)):z\in O\} {( z , u ( z )) : z ∈ O } G i = { ( z , u i ( z ) ) : z ∈ O i } G_i=\{(z,u_i(z)):z\in O_i\} G i = {( z , u i ( z )) : z ∈ O i } P i = G i ∖ ⋃ i ′ < i G i ′ P_i=G_i\setminus\bigcup_{i'<i}G_{i'} P i = G i ∖ ⋃ i ′ < i G i ′ z z z G i G_i G i O i O_i O i π ( P i ) \pi(P_i) π ( P i )
∑ u : ∇ f z ( u ) = 0 φ ( z , u ) = ∑ i 1 π ( P i ) ( z ) φ ( z , u i ( z ) ) ( z ∉ D ) \sum_{u:\nabla f_z(u)=0}\varphi(z,u)=\sum_i\mathbf{1}_{\pi(P_i)}(z)\varphi(z,u_i(z))\qquad(z\notin D) u : ∇ f z ( u ) = 0 ∑ φ ( z , u ) = i ∑ 1 π ( P i ) ( z ) φ ( z , u i ( z )) ( z ∈ / D ) is Borel. The same enumeration shows that events defined in terms of the critical points, such as unions over critical points of Borel events, are Borel.
The identity in a chart. Let ( χ i ) (\chi_i) ( χ i ) ( V i ) (V_i) ( V i ) i i i V = V i V=V_i V = V i χ = χ i \chi=\chi_i χ = χ i ϕ ℓ \phi_\ell ϕ ℓ R ℓ \mathbb{R}^{\ell} R ℓ z ′ ′ z^{\prime\prime} z ′′ α z ′ ′ \alpha_{z^{\prime\prime}} α z ′′ w ( t ) = ϕ ℓ ( α z ′ ′ ( t ) , z ′ ′ ) χ ( t ) φ ( ( α z ′ ′ ( t ) , z ′ ′ ) , t ) w(t)=\phi_\ell(\alpha_{z^{\prime\prime}}(t),z^{\prime\prime})\chi(t)\varphi((\alpha_{z^{\prime\prime}}(t),z^{\prime\prime}),t) w ( t ) = ϕ ℓ ( α z ′′ ( t ) , z ′′ ) χ ( t ) φ (( α z ′′ ( t ) , z ′′ ) , t ) (152) , integrating over z ′ ′ z^{\prime\prime} z ′′
E ∑ t ∈ V : γ ( Z , t ) = 0 χ ( t ) φ ( Z , t ) = ∫ V ∫ R ℓ − n ϕ ℓ ( α z ′ ′ ( t ) , z ′ ′ ) ∣ det Γ 1 ( t ) ∣ ∣ det D t γ ( z ˉ , t ) ∣ χ ( t ) φ ( z ˉ , t ) d z ′ ′ d t , z ˉ = ( α z ′ ′ ( t ) , z ′ ′ ) . (153) \mathbb{E}\sum_{t\in V:\gamma(Z,t)=0}\chi(t)\varphi(Z,t)
=
\int_V\int_{\mathbb{R}^{\ell-n}}
\frac{\phi_\ell(\alpha_{z^{\prime\prime}}(t),z^{\prime\prime})}{\lvert\det\Gamma_1(t)\rvert}
\left\lvert\det D_t\gamma(\bar z,t)\right\rvert\chi(t)\varphi(\bar z,t)\,dz^{\prime\prime}\,dt,
\qquad\bar z=(\alpha_{z^{\prime\prime}}(t),z^{\prime\prime}). \tag*{(153)} E t ∈ V : γ ( Z , t ) = 0 ∑ χ ( t ) φ ( Z , t ) = ∫ V ∫ R ℓ − n ∣ det Γ 1 ( t )∣ ϕ ℓ ( α z ′′ ( t ) , z ′′ ) ∣ det D t γ ( z ˉ , t ) ∣ χ ( t ) φ ( z ˉ , t ) d z ′′ d t , z ˉ = ( α z ′′ ( t ) , z ′′ ) . (153) When ℓ = n \ell=n ℓ = n
Identification of the conditional law. Fix t ∈ V t\in V t ∈ V ( z ′ , z ′ ′ ) ↦ ( Γ ( t ) z , z ′ ′ ) (z',z^{\prime\prime})\mapsto(\Gamma(t)z,z^{\prime\prime}) ( z ′ , z ′′ ) ↦ ( Γ ( t ) z , z ′′ ) ∣ det Γ 1 ( t ) ∣ \lvert\det\Gamma_1(t)\rvert ∣ det Γ 1 ( t )∣ γ ( Z , t ) \gamma(Z,t) γ ( Z , t ) 0 0 0 p t c o ( 0 ) = ∫ ϕ ℓ ( α z ′ ′ ( t ) , z ′ ′ ) d z ′ ′ / ∣ det Γ 1 ( t ) ∣ p_t^{\mathrm{co}}(0)=\int\phi_\ell(\alpha_{z^{\prime\prime}}(t),z^{\prime\prime})\,dz^{\prime\prime}/\lvert\det\Gamma_1(t)\rvert p t co ( 0 ) = ∫ ϕ ℓ ( α z ′′ ( t ) , z ′′ ) d z ′′ / ∣ det Γ 1 ( t )∣ z ′ ′ ↦ ( α z ′ ′ ( t ) , z ′ ′ ) z^{\prime\prime}\mapsto(\alpha_{z^{\prime\prime}}(t),z^{\prime\prime}) z ′′ ↦ ( α z ′′ ( t ) , z ′′ ) ker Γ ( t ) \ker\Gamma(t) ker Γ ( t ) ker Γ ( t ) \ker\Gamma(t) ker Γ ( t ) ϕ ℓ \phi_\ell ϕ ℓ ker Γ ( t ) \ker\Gamma(t) ker Γ ( t ) ψ ≥ 0 \psi\ge0 ψ ≥ 0
∫ ϕ ℓ ( α z ′ ′ ( t ) , z ′ ′ ) ∣ det Γ 1 ( t ) ∣ ψ ( α z ′ ′ ( t ) , z ′ ′ ) d z ′ ′ = p t c o ( 0 ) E ψ ( Z 0 u ) . \int\frac{\phi_\ell(\alpha z^{\prime\prime}(t),z^{\prime\prime})}{\lvert\det\Gamma_1(t)\rvert}\psi(\alpha z^{\prime\prime}(t),z^{\prime\prime})\,\mathrm{d}z^{\prime\prime}=p_t^{\mathrm{co}}(0)\mathbb{E}\psi(Z_0^u). ∫ ∣ det Γ 1 ( t )∣ ϕ ℓ ( α z ′′ ( t ) , z ′′ ) ψ ( α z ′′ ( t ) , z ′′ ) d z ′′ = p t co ( 0 ) E ψ ( Z 0 u ) . where u u u t t t ψ = 1 \psi=1 ψ = 1 ker Γ ( t ) = ker Γ u \ker\Gamma(t)=\ker\Gamma_u ker Γ ( t ) = ker Γ u 0 0 0
From coordinates to orthonormal frames. Let E ( t ) E(t) E ( t ) ∂ t i \partial_{t_i} ∂ t i T u M T_uM T u M G ( t ) = E ( t ) T E ( t ) G(t)=E(t)^{\mathsf T}E(t) G ( t ) = E ( t ) T E ( t ) γ ( z , t ) \gamma(z,t) γ ( z , t ) E ( t ) T E(t)^{\mathsf T} E ( t ) T ∇ f z ( u ) \nabla f_z(u) ∇ f z ( u ) p t c o ( 0 ) = p u ( 0 ) / det G ( t ) p_t^{\mathrm{co}}(0)=p_u(0)/\sqrt{\det G(t)} p t co ( 0 ) = p u ( 0 ) / det G ( t ) D t γ = E T ∇ 2 f z ( u ) E D_t\gamma=E^{\mathsf T}\nabla^2 f_z(u)E D t γ = E T ∇ 2 f z ( u ) E ∣ det D t γ ∣ = det G ( t ) ∣ det ∇ 2 f z ( u ) ∣ \lvert\det D_t\gamma\rvert=\det G(t)\lvert\det\nabla^2 f_z(u)\rvert ∣ det D t γ ∣ = det G ( t ) ∣ det ∇ 2 f z ( u )∣ (153) , the integrand becomes p u ( 0 ) E [ ∣ det ∇ 2 f Z 0 ( u ) ∣ χ ( u ) ϕ ( Z 0 u , u ) ] det G ( t ) p_u(0)\mathbb{E}[\lvert\det\nabla^2 f_{Z_0}(u)\rvert\chi(u)\phi(Z_0^u,u)]\sqrt{\det G(t)} p u ( 0 ) E [∣ det ∇ 2 f Z 0 ( u )∣ χ ( u ) ϕ ( Z 0 u , u )] det G ( t ) det G ( t ) d t \sqrt{\det G(t)}\,\mathrm{d}t det G ( t ) d t ϕ \phi ϕ E ∑ ∣ ϕ ( Z , u ) ∣ < ∞ \mathbb{E}\sum\lvert\phi(Z,u)\rvert<\infty E ∑ ∣ ϕ ( Z , u )∣ < ∞ ϕ \phi ϕ
We now specialize to the center field on S N S_N S N
Theorem 7.17 (Kac–Rice formula with field-dependent weights). Let K ⊂ R 2 \mathcal{K}\subset\mathbb{R}^2 K ⊂ R 2 ϕ \phi ϕ H Q H_Q H Q S N S_N S N ξ \xi ξ
E ∑ u ∈ S N : ∇ S N H Q ( u ) = 0 ( H Q ( u ) , K Q ( u ) ) / N ∈ K ϕ ( H N , u ) = ∣ S N ∣ ( 2 π a Q ) n / 2 ∫ S N ∫ K p N ( h , k ) E h , k , u [ ∣ det ∇ S N 2 H Q ( u ) ∣ ϕ ( H N , u ) ] d h d k ν N ( d u ) . (154) \mathbb{E}\sum_{\substack{u\in S_N:\ \nabla_{S_N}H_Q(u)=0\\(H_Q(u),K_Q(u))/N\in\mathcal{K}}}\phi(H_N,u)
=
\frac{\lvert S_N\rvert}{(2\pi a_Q)^{n/2}}
\int_{S_N}\int_{\mathcal{K}}p_N(h,k)\mathbb{E}_{h,k,u}\left[\left\lvert\det\nabla^2_{S_N}H_Q(u)\right\rvert\phi(H_N,u)\right]\,\mathrm{d}h\,\mathrm{d}k\,\nu_N(\mathrm{d}u). \tag*{(154)} E u ∈ S N : ∇ S N H Q ( u ) = 0 ( H Q ( u ) , K Q ( u )) / N ∈ K ∑ ϕ ( H N , u ) = ( 2 π a Q ) n /2 ∣ S N ∣ ∫ S N ∫ K p N ( h , k ) E h , k , u [ det ∇ S N 2 H Q ( u ) ϕ ( H N , u ) ] d h d k ν N ( d u ) . (154) with p N p_N p N ξ \xi ξ p p p K p = { h ∈ R : ( h , p h ) ∈ K } \mathcal{K}_p=\{h\in\mathbb{R}:(h,ph)\in\mathcal{K}\} K p = { h ∈ R : ( h , p h ) ∈ K }
E ∑ u ∈ S N : ∇ S N H Q ( u ) = 0 H Q ( u ) / N ∈ K p ϕ ( H N , u ) = ∣ S N ∣ ( 2 π a Q ) n / 2 ∫ S N ∫ K p p N p u r e ( h ) E h , u [ ∣ det ∇ S N 2 H Q ( u ) ∣ ϕ ( H N , u ) ] d h ν N ( d u ) . (155) \mathbb{E}\sum_{\substack{u\in S_N:\ \nabla_{S_N}H_Q(u)=0\\H_Q(u)/N\in\mathcal{K}_p}}\phi(H_N,u)
=
\frac{\lvert S_N\rvert}{(2\pi a_Q)^{n/2}}
\int_{S_N}\int_{\mathcal{K}_p}p_N^{\mathrm{pure}}(h)\mathbb{E}_{h,u}\left[\left\lvert\det\nabla^2_{S_N}H_Q(u)\right\rvert\phi(H_N,u)\right]\,\mathrm{d}h\,\nu_N(\mathrm{d}u). \tag*{(155)} E u ∈ S N : ∇ S N H Q ( u ) = 0 H Q ( u ) / N ∈ K p ∑ ϕ ( H N , u ) = ( 2 π a Q ) n /2 ∣ S N ∣ ∫ S N ∫ K p p N pure ( h ) E h , u [ det ∇ S N 2 H Q ( u ) ϕ ( H N , u ) ] d h ν N ( d u ) . (155) For nonnegative φ \varphi φ φ \varphi φ Σ Q \Sigma_Q Σ Q
Proof. We apply Proposition 7.16 with M = S N M=S_N M = S N ℓ = ℓ N \ell=\ell_N ℓ = ℓ N f z f_z f z H z , Q H_{z,Q} H z , Q S N S_N S N z z z ∇ S N H Q ( u ) \nabla_{S_N}H_Q(u) ∇ S N H Q ( u ) a Q I d u ⊥ a_Q I_{d_{u^\perp}} a Q I d u ⊥ p u ( 0 ) = ( 2 π a Q ) − n / 2 p_u(0)=(2\pi a_Q)^{-n/2} p u ( 0 ) = ( 2 π a Q ) − n /2 Z u 0 Z_u^0 Z u 0 E ∇ \mathbb{E}_{\nabla} E ∇ S N S_N S N ∣ S N ∣ ν N |S_N|\nu_N ∣ S N ∣ ν N
Let φ ≥ 0 \varphi\geq0 φ ≥ 0
( z , u ) ↦ φ ( H z , u ) 1 { ( H z , Q ( u ) , K z , Q ( u ) ) / N ∈ K } . (z,u)\mapsto\varphi(H_{z},u)\mathbf{1}\{(H_{z,Q}(u),K_{z,Q}(u))/N\in\mathcal{K}\}. ( z , u ) ↦ φ ( H z , u ) 1 {( H z , Q ( u ) , K z , Q ( u )) / N ∈ K } . By Lemma 7.4(4) and (2), the inner expectation equals
∫ R 2 p N ( h , k ) E h , k , u [ ∣ det ∇ S N 2 H Q ( u ) ∣ φ ( H N , u ) ] 1 K ( h , k ) d h d k , \int_{\mathbb{R}^2}p_N(h,k)\mathbb{E}_{h,k,u}\left[\left|\det\nabla^2_{S_N}H_Q(u)\right|\varphi(H_N,u)\right]\mathbf{1}_{\mathcal{K}}(h,k)\,dh\,dk, ∫ R 2 p N ( h , k ) E h , k , u [ det ∇ S N 2 H Q ( u ) φ ( H N , u ) ] 1 K ( h , k ) d h d k , since ( H Q ( u ) , K Q ( u ) ) = ( N h , N k ) (H_Q(u),K_Q(u))=(Nh,Nk) ( H Q ( u ) , K Q ( u )) = ( N h , N k ) P h , k , u \mathbb{P}_{h,k,u} P h , k , u φ \varphi φ K Q = p H Q K_Q=pH_Q K Q = p H Q ( H Q ( u ) , K Q ( u ) ) / N ∈ K (H_Q(u),K_Q(u))/N\in\mathcal{K} ( H Q ( u ) , K Q ( u )) / N ∈ K H Q ( u ) / N ∈ K p H_Q(u)/N\in\mathcal{K}_p H Q ( u ) / N ∈ K p
For bounded φ \varphi φ φ = 1 \varphi=1 φ = 1 K = R 2 \mathcal{K}=\mathbb{R}^2 K = R 2 ∣ S N ∣ ( 2 π a Q ) − n / 2 ∫ R 2 p N ( h , k ) μ n , N ( k ) d h d k |S_N|(2\pi a_Q)^{-n/2}\int_{\mathbb{R}^2}p_N(h,k)\mu_{n,N}(k)\,dh\,dk ∣ S N ∣ ( 2 π a Q ) − n /2 ∫ R 2 p N ( h , k ) μ n , N ( k ) d h d k μ n , N ( k ) ≤ E ( ( ∥ W n , N ∥ o p + ∣ k ∣ ) n ) ≤ C N ( 1 + ∣ k ∣ ) n \mu_{n,N}(k)\leq\mathbb{E}((\lVert W_{n,N}\rVert_{\mathrm{op}}+|k|)^n)\leq C_N(1+|k|)^n μ n , N ( k ) ≤ E ((∥ W n , N ∥ op + ∣ k ∣ ) n ) ≤ C N ( 1 + ∣ k ∣ ) n φ \varphi φ φ \varphi φ
Corollary 7.18 (Expected numbers of critical points). Let I , U ⊂ R I,U\subset\mathbb{R} I , U ⊂ R
C N ( I , U ) = { u ∈ S N : ∇ S N H Q ( u ) = 0 , H Q ( u ) / N ∈ I , K Q ( u ) / N ∈ U } . C_N(I,U)=\{u\in S_N:\nabla_{S_N}H_Q(u)=0,\ H_Q(u)/N\in I,\ K_Q(u)/N\in U\}. C N ( I , U ) = { u ∈ S N : ∇ S N H Q ( u ) = 0 , H Q ( u ) / N ∈ I , K Q ( u ) / N ∈ U } . If ξ \xi ξ
E ∣ C N ( I , U ) ∣ = ∣ S N ∣ ( 2 π a Q ) n / 2 N 2 π det Σ Q ∫ I ∫ U e − N 2 ( h , k ) Σ Q − 1 ( h , k ) T μ n , N ( k ) d k d h . (156) \mathbb{E}|C_N(I,U)|=\frac{|S_N|}{(2\pi a_Q)^{n/2}}\frac{N}{2\pi\sqrt{\det\Sigma_Q}}\int_I\int_U e^{-\frac{N}{2}(h,k)\Sigma_Q^{-1}(h,k)^{\mathsf{T}}}\mu_{n,N}(k)\,dk\,dh. \tag*{(156)} E ∣ C N ( I , U ) ∣ = ( 2 π a Q ) n /2 ∣ S N ∣ 2 π det Σ Q N ∫ I ∫ U e − 2 N ( h , k ) Σ Q − 1 ( h , k ) T μ n , N ( k ) d k d h . (156) If ξ \xi ξ p p p
E ∣ C N ( I , U ) ∣ = ∣ S N ∣ ( 2 π a Q ) n / 2 ( N 2 π ξ ( Q ) ) 1 / 2 ∫ I 1 { p h ∈ U } e − N h 2 / ( 2 ξ ( Q ) ) μ n , N ( p h ) d h . (157) \mathbb{E}|C_N(I,U)|=\frac{|S_N|}{(2\pi a_Q)^{n/2}}\left(\frac{N}{2\pi\xi(Q)}\right)^{1/2}\int_I\mathbf{1}\{ph\in U\}e^{-Nh^2/(2\xi(Q))}\mu_{n,N}(ph)\,dh. \tag*{(157)} E ∣ C N ( I , U ) ∣ = ( 2 π a Q ) n /2 ∣ S N ∣ ( 2 π ξ ( Q ) N ) 1/2 ∫ I 1 { p h ∈ U } e − N h 2 / ( 2 ξ ( Q )) μ n , N ( p h ) d h . (157) Both quantities are finite. With I = [ ( m N − M ) / N , ( m N + M ) / N ] I=[(m_N-M)/N,(m_N+M)/N] I = [( m N − M ) / N , ( m N + M ) / N ] C N , M C_{N,M} C N , M
Proof. We take φ = 1 \varphi= 1 φ = 1 K = I × U \mathcal{K} = I \times U K = I × U Theorem 7.17 . By Lemma 7.4 (2), the conditioned expectation of the determinant is μ n , N ( k ) \mu_{n,N}(k) μ n , N ( k ) u u u h h h ν N \nu_N ν N Theorem 7.17 . □ \square □
No asymptotic estimate of these integrals is part of the statement. The counts in fixed windows are estimated in Sections 9 to 11. We record two reformulations that are used there.
Remark 7.19 (Palm form). For a non-pure mixture, let
Λ N ( d u d h d k ) = ∣ S N ∣ ( 2 π a Q ) n / 2 p N ( h , k ) μ n , N ( k ) ν N ( d u ) d h d k . \Lambda_N(du\,dh\,dk) = \frac{|S_N|}{(2\pi a_Q)^{n/2}}p_N(h,k)\mu_{n,N}(k)\nu_N(du)\,dh\,dk. Λ N ( d u d h d k ) = ( 2 π a Q ) n /2 ∣ S N ∣ p N ( h , k ) μ n , N ( k ) ν N ( d u ) d h d k . be the intensity measure of the marked critical points, so that Λ N ( S N × I × U ) = E ∣ C N ( I , U ) ∣ \Lambda_N(S_N \times I \times U) = \mathbb{E}|C_N(I,U)| Λ N ( S N × I × U ) = E ∣ C N ( I , U ) ∣ u ∈ S N u \in S_N u ∈ S N ( h , k ) ∈ R 2 (h,k) \in\mathbb{R}^2 ( h , k ) ∈ R 2 Palm law at u u u ( h , k ) (h,k) ( h , k )
E h , k , u P a l m [ Φ ] = E h , k , u [ ∣ det ∇ S N 2 H Q ( u ) ∣ Φ ] μ n , N ( k ) . \mathbb{E}^{\mathrm{Palm}}_{h,k,u}[\Phi] = \frac{\mathbb{E}_{h,k,u}[|\det\nabla^2_{S_N} H_Q(u)|\Phi]}{\mu_{n,N}(k)}. E h , k , u Palm [ Φ ] = μ n , N ( k ) E h , k , u [ ∣ det ∇ S N 2 H Q ( u ) ∣Φ ] . With this notation, (154) reads
E ∑ u : ∇ S N H Q ( u ) = 0 ( H Q ( u ) , K Q ( u ) ) / N ∈ K φ ( H N , u ) = ∫ S N × K E h , k , u P a l m [ φ ( H N , u ) ] Λ N ( d u d h d k ) . \mathbb{E}\sum_{\substack{u:\nabla_{S_N}H_Q(u)=0\\(H_Q(u),K_Q(u))/N\in\mathcal{K}}}\varphi(H_N,u)
=
\int_{S_N\times\mathcal{K}}\mathbb{E}^{\mathrm{Palm}}_{h,k,u}[\varphi(H_N,u)]\,\Lambda_N(du\,dh\,dk). E u : ∇ S N H Q ( u ) = 0 ( H Q ( u ) , K Q ( u )) / N ∈ K ∑ φ ( H N , u ) = ∫ S N × K E h , k , u Palm [ φ ( H N , u )] Λ N ( d u d h d k ) . In the pure case, Λ N \Lambda_N Λ N ∣ S N ∣ ( 2 π a Q ) − n / 2 p N p u r e ( h ) μ n , N ( p h ) |S_N|(2\pi a_Q)^{-n/2}p_N^{\mathrm{pure}}(h)\mu_{n,N}(ph) ∣ S N ∣ ( 2 π a Q ) − n /2 p N pure ( h ) μ n , N ( p h ) ν N ( d u ) d h \nu_N(du)\,dh ν N ( d u ) d h E h , u [ ∣ det ∇ S N 2 H Q ( u ) ∣ Φ ] / μ n , N ( p h ) \mathbb{E}_{h,u}[|\det\nabla^2_{S_N}H_Q(u)|\Phi]/\mu_{n,N}(ph) E h , u [ ∣ det ∇ S N 2 H Q ( u ) ∣Φ ] / μ n , N ( p h ) Lemma 7.22 .
Remark 7.20 (Antipodal points and Subag’s tameness condition). The formulas above count points of S N S_N S N u ↦ − u u \mapsto-u u ↦ − u (156) or (157) . For a general weight one must instead specify a representative of each pair, or average the two values. A weighted sum cannot be divided by two without such a convention. For pure models of degree p ≥ 3 p \ge3 p ≥ 3 Theorem 7.17 requires only Borel measurability of the weight. This matters for weights given by probabilities of trajectory events, for which the tameness hypotheses of [141, Lemma 36] have not been verified.
Hitting probabilities as weights
We now show that hitting probabilities of the Langevin diffusion are field-dependent weights. For H ∈ H N H \in\mathcal{H}_N H ∈ H N β ≥ 0 \beta\ge0 β ≥ 0 x ∈ S N x \in S_N x ∈ S N P x H \mathbb{P}^H_x P x H C ( [ 0 , T ] , S N ) C([0,T],S_N) C ([ 0 , T ] , S N ) Definition 3.4 started at x x x (40) , which has noise coefficient 2 \sqrt{2} 2 S N S_N S N P ν N H = ∫ P x H ν N ( d x ) \mathbb{P}^H_{\nu_N} = \int\mathbb{P}^H_x \nu_N(\mathrm{d}x) P ν N H = ∫ P x H ν N ( d x ) ν N \nu_N ν N
Lemma 7.21 (Hitting probabilities are field-dependent weights). Let N ≥ 2 N \ge2 N ≥ 2 β ≥ 0 \beta\ge0 β ≥ 0 T ∈ ( 0 , ∞ ) T \in(0,\infty) T ∈ ( 0 , ∞ ) A ⊂ C ( [ 0 , T ] , S N ) A \subset C([0,T],S_N) A ⊂ C ([ 0 , T ] , S N ) H ↦ P ν N H ( A ) H \mapsto\mathbb{P}^H_{\nu_N}(A) H ↦ P ν N H ( A ) H N \mathcal{H}_N H N A ⊂ C ( [ 0 , T ] , S N ) × S N A \subset C([0,T],S_N) \times S_N A ⊂ C ([ 0 , T ] , S N ) × S N
φ A ( H , u ) = P ν N H ( { ω : ( ω , u ) ∈ A } ) \varphi_A(H,u) = \mathbb{P}^H_{\nu_N}\left(\{\omega: (\omega,u) \in A\}\right) φ A ( H , u ) = P ν N H ( { ω : ( ω , u ) ∈ A } ) is a field-dependent weight with values in [ 0 , 1 ] [0,1] [ 0 , 1 ] a ∈ [ 0 , 1 ] a \in[0,1] a ∈ [ 0 , 1 ]
φ T , a ( H , u ) = P ν N H ( sup 0 ≤ t ≤ T ∣ R ( X t , u ) ∣ ≥ a ) (158) \varphi_{T,a}(H,u) = \mathbb{P}^H_{\nu_N}\left(\sup_{0 \le t \le T} \lvert R(X_t,u)\rvert\ge a\right) \tag*{(158)} φ T , a ( H , u ) = P ν N H ( 0 ≤ t ≤ T sup ∣ R ( X t , u )∣ ≥ a ) (158) is a field-dependent weight. This applies to T = e γ N T = e^{\gamma N} T = e γ N N N N
Proof. By Lemma 4.2 (a), ( H , x ) ↦ P x H (H,x) \mapsto\mathbb{P}^H_x ( H , x ) ↦ P x H H N × S N \mathcal{H}_N \times S_N H N × S N C ( [ 0 , T ] , S N ) C([0,T],S_N) C ([ 0 , T ] , S N ) ( H , x ) (H,x) ( H , x ) F F F C ( [ 0 , T ] , S N ) C([0,T],S_N) C ([ 0 , T ] , S N ) ( H , x ) ↦ ∫ F d P x H (H,x) \mapsto\int F\,\mathrm{d}\mathbb{P}^H_x ( H , x ) ↦ ∫ F d P x H H ↦ ∫ F d P ν N H H \mapsto\int F\,\mathrm{d}\mathbb{P}^H_{\nu_N} H ↦ ∫ F d P ν N H A A A ( H , u ) ↦ ∫ 1 A ( ω , u ) P ν N H ( d ω ) (H,u) \mapsto\int\mathbf{1}_A(\omega,u)\,\mathbb{P}^H_{\nu_N}(\mathrm{d}\omega) ( H , u ) ↦ ∫ 1 A ( ω , u ) P ν N H ( d ω ) A = A 1 × E A = A_1 \times E A = A 1 × E A A A π \pi π λ \lambda λ A A A ( ω , u ) ↦ sup t ≤ T ∣ R ( ω ( t ) , u ) ∣ (\omega,u) \mapsto\sup_{t \le T}\lvert R(\omega(t),u)\rvert ( ω , u ) ↦ sup t ≤ T ∣ R ( ω ( t ) , u )∣ ∣ R ( x , u ) − R ( x ′ , u ′ ) ∣ ≤ ( ∣ x − x ′ ∣ + ∣ u − u ′ ∣ ) / N \lvert R(x,u) - R(x',u')\rvert\le(\lvert x-x'\rvert+ \lvert u-u'\rvert)/\sqrt{N} ∣ R ( x , u ) − R ( x ′ , u ′ )∣ ≤ (∣ x − x ′ ∣ + ∣ u − u ′ ∣) / N S N S_N S N (158) is a Borel subset of C ( [ 0 , T ] , S N ) × S N C([0,T],S_N) \times S_N C ([ 0 , T ] , S N ) × S N T T T N N N T T T □ \square □
Lemma 4.3(b) states the special case of Lemma 7.21 that is used in Sections 4 and 13.
Determinant-weighted transfer
The estimate of Section 8 bounds a hitting probability under the conditioned law P h , k , u \mathbb{P}_{h,k,u} P h , k , u
Lemma 7.22 (Hölder bound for determinant-weighted expectations). Let D ≥ 0 D \ge0 D ≥ 0 random variable with 0 < μ = E D < ∞ 0 < \mu= \mathbb{E}D < \infty 0 < μ = E D < ∞ E ( D / μ ) r ≤ K \mathbb{E}(D/\mu)^r \le K E ( D / μ ) r ≤ K r > 1 r > 1 r > 1 φ \varphi φ 0 ≤ φ ≤ 1 0 \le\varphi\le1 0 ≤ φ ≤ 1 E φ ≤ q \mathbb{E}\varphi\le q E φ ≤ q
E [ D φ ] μ ≤ K 1 / r q 1 − 1 / r . (159) \frac{\mathbb{E}[D\varphi]}{\mu} \le K^{1/r}q^{1-1/r}. \tag*{(159)} μ E [ D φ ] ≤ K 1/ r q 1 − 1/ r . (159) No assumption on the dependence between φ \varphi φ D D D
Proof. Let r ′ = r / ( r − 1 ) r' = r/(r-1) r ′ = r / ( r − 1 ) E [ ( D / μ ) φ ] ≤ ( E ( D / μ ) r ) 1 / r ( E φ r ′ ) 1 / r ′ \mathbb{E}[(D/\mu)\varphi] \le(\mathbb{E}(D/\mu)^r)^{1/r}(\mathbb{E}\varphi^{r'})^{1/r'} E [( D / μ ) φ ] ≤ ( E ( D / μ ) r ) 1/ r ( E φ r ′ ) 1/ r ′ 0 ≤ φ ≤ 1 0 \le\varphi\le1 0 ≤ φ ≤ 1 φ r ′ ≤ φ \varphi^{r'} \le\varphi φ r ′ ≤ φ q 1 / r ′ = q 1 − 1 / r q^{1/r'} = q^{1-1/r} q 1/ r ′ = q 1 − 1/ r
For r = 2 r = 2 r = 2 K 2 1 / 2 q 1 / 2 K_2^{1/2}q^{1/2} K 2 1/2 q 1/2 φ \varphi φ D D D D / μ D/\mu D / μ φ \varphi φ
Theorem 7.23 (Determinant-weighted transfer). Let g > 0 g > 0 g > 0 K ⊂ R 2 \mathcal{K} \subset\mathbb{R}^2 K ⊂ R 2 k ≥ 2 b Q + g k \ge2\sqrt{b_Q}+g k ≥ 2 b Q + g ( h , k ) ∈ K (h,k) \in\mathcal{K} ( h , k ) ∈ K k = p h k = ph k = p h ( h , k ) ∈ K (h,k) \in\mathcal{K} ( h , k ) ∈ K K ≠ ∅ \mathcal{K} \ne\varnothing K = ∅ k + = max { k : ( h , k ) ∈ K } k_+ = \max\{k : (h,k) \in\mathcal{K}\} k + = max { k : ( h , k ) ∈ K } K = ∅ \mathcal{K} = \varnothing K = ∅ k + = 2 b Q + g k_+ = 2\sqrt{b_Q}+g k + = 2 b Q + g N N N W N ⊂ K \mathcal{W}_N \subset\mathcal{K} W N ⊂ K
C N = { u ∈ S N : ∇ S N H Q ( u ) = 0 , ( H Q ( u ) , K Q ( u ) ) / N ∈ W N } , C_N = \{u \in S_N : \nabla_{S_N} H_Q(u) = 0,\ (H_Q(u),K_Q(u))/N \in\mathcal{W}_N\}, C N = { u ∈ S N : ∇ S N H Q ( u ) = 0 , ( H Q ( u ) , K Q ( u )) / N ∈ W N } , and let φ N \varphi_N φ N [ 0 , 1 ] [0,1] [ 0 , 1 ] q N ≥ 0 q_N \ge0 q N ≥ 0
E h , k , u φ N ( H N , u ) ≤ q N for all u ∈ S N and ( h , k ) ∈ W N . (160) \mathbb{E}_{h,k,u}\varphi_N(H_N,u) \le q_N \qquad\text{for all }u \in S_N\text{ and }(h,k) \in\mathcal{W}_N. \tag*{(160)} E h , k , u φ N ( H N , u ) ≤ q N for all u ∈ S N and ( h , k ) ∈ W N . (160) Then for every r > 1 r > 1 r > 1 N ≥ 2 N \ge2 N ≥ 2
E ∑ u ∈ C N φ N ( H N , u ) ≤ K r 1 / r q N 1 − 1 / r E ∣ C N ∣ , (161) \mathbb{E}\sum_{u\in C_N}\varphi_N(H_N,u) \le K_r^{1/r}q_N^{1-1/r}\mathbb{E}|C_N|, \tag*{(161)} E u ∈ C N ∑ φ N ( H N , u ) ≤ K r 1/ r q N 1 − 1/ r E ∣ C N ∣ , (161) where K r = K r ( b Q , g , k + ) K_r = K_r(b_Q,g,k_+) K r = K r ( b Q , g , k + ) φ N \varphi_N φ N θ φ N \theta\varphi_N θ φ N θ \theta θ [ 0 , 1 ] [0,1] [ 0 , 1 ]
Proof. If K = ∅ \mathcal{K} = \varnothing K = ∅ W N = C N = ∅ \mathcal{W}_N = C_N = \varnothing W N = C N = ∅ u ∈ S N u \in S_N u ∈ S N ( h , k ) ∈ W N (h,k) \in\mathcal{W}_N ( h , k ) ∈ W N D = ∣ det ∇ S N 2 H Q ( u ) ∣ D = |\det\nabla^2_{S_N}H_Q(u)| D = ∣ det ∇ S N 2 H Q ( u ) ∣ P h , k , u \mathbb{P}_{h,k,u} P h , k , u E h , k , u D = μ n , N ( k ) \mathbb{E}_{h,k,u}D = \mu_{n,N}(k) E h , k , u D = μ n , N ( k ) E h , k , u ( D / μ n , N ( k ) ) r ≤ K r \mathbb{E}_{h,k,u}(D/\mu_{n,N}(k))^r \le K_r E h , k , u ( D / μ n , N ( k ) ) r ≤ K r
E h , k , u [ D φ N ( H N , u ) ] ≤ K r 1 / r q N 1 − 1 / r μ n , N ( k ) . \mathbb{E}_{h,k,u}[D\varphi_N(H_N,u)] \le K_r^{1/r}q_N^{1-1/r}\mu_{n,N}(k). E h , k , u [ D φ N ( H N , u )] ≤ K r 1/ r q N 1 − 1/ r μ n , N ( k ) . We insert this bound into (7.24) with K \mathcal{K} K W N \mathcal{W}_N W N φ = 1 \varphi= 1 φ = 1 E ∣ C N ∣ \mathbb{E}|C_N| E ∣ C N ∣
Theorem 7.17 with φ = 1 \varphi= 1 φ = 1 (161) . For the last assertion, D θ φ N ≤ D φ N D\theta\varphi_N \le D\varphi_N D θ φ N ≤ D φ N (161) remains the expected number of all critical points in C N C_N C N θ \theta θ
We now apply Theorem 7.23 to hitting probabilities and to the critical points in a fixed window. For a deterministic sequence ( m N ) (m_N) ( m N ) U ⊂ R U \subset\mathbb{R} U ⊂ R M > 0 M > 0 M > 0
C N , M = { u ∈ S N : ∇ S N H Q ( u ) = 0 , ∣ H Q ( u ) − m N ∣ ≤ M , K Q ( u ) / N ∈ U } . (162) C_{N,M}=\left\{u\in S_N:\nabla_{S_N}H_Q(u)=0,\ \lvert H_Q(u)-m_N\rvert\le M,\ K_Q(u)/N\in U\right\}. \tag*{(162)} C N , M = { u ∈ S N : ∇ S N H Q ( u ) = 0 , ∣ H Q ( u ) − m N ∣ ≤ M , K Q ( u ) / N ∈ U } . (162) We call this set the critical points in a fixed window , since its half-width M M M N N N W N , M = [ ( m N − M ) / N , ( m N + M ) / N ] × U W_{N,M}=[(m_N-M)/N,(m_N+M)/N]\times U W N , M = [( m N − M ) / N , ( m N + M ) / N ] × U { k = p h } \{k=ph\} { k = p h }
Corollary 7.24 (Critical points in a fixed window). Assume that d ≥ 4 d\ge4 d ≥ 4 β ≥ 0 \beta\ge0 β ≥ 0 g g g K K K k + k_+ k +
There exist a ∈ ( 0 , 1 ] a\in(0,1] a ∈ ( 0 , 1 ] γ > 0 \gamma>0 γ > 0 q N q_N q N q N → 0 q_N\to0 q N → 0 N N N u ∈ S N u\in S_N u ∈ S N ( h , k ) ∈ K (h,k)\in K ( h , k ) ∈ K
E h , k , u P v N H N ( sup 0 ≤ t ≤ e γ N ∣ R ( X t , u ) ∣ ≥ a ) ≤ q N . \mathbb{E}_{h,k,u}\mathbb{P}_{v_N}^{H_N}\left(\sup_{0\le t\le e^{\gamma N}}\lvert R(X_t,u)\rvert\ge a\right)\le q_N. E h , k , u P v N H N ( 0 ≤ t ≤ e γ N sup ∣ R ( X t , u )∣ ≥ a ) ≤ q N . For every M > 0 M>0 M > 0 N M N_M N M W N , M ⊂ K W_{N,M}\subset K W N , M ⊂ K N ≥ N M N\ge N_M N ≥ N M
For every M > 0 M>0 M > 0 C M = sup N E ∣ C N , M ∣ < ∞ C_M=\sup_N\mathbb{E}\lvert C_{N,M}\rvert<\infty C M = sup N E ∣ C N , M ∣ < ∞
Then for every r > 1 r>1 r > 1 M > 0 M>0 M > 0 N ≥ N M N\ge N_M N ≥ N M
E P v N H N ( ∃ u ∈ C N , M , ∃ t ≤ e γ N : ∣ R ( X t , u ) ∣ ≥ a ) ≤ C M K r 1 / r q N 1 − 1 / r . (163) \mathbb{E}\mathbb{P}_{v_N}^{H_N}\left(\exists u\in C_{N,M},\ \exists t\le e^{\gamma N}:\lvert R(X_t,u)\rvert\ge a\right)\le C_MK_r^{1/r}q_N^{1-1/r}. \tag*{(163)} E P v N H N ( ∃ u ∈ C N , M , ∃ t ≤ e γ N : ∣ R ( X t , u )∣ ≥ a ) ≤ C M K r 1/ r q N 1 − 1/ r . (163) The same bound holds for every subfamily of C N , M C_{N,M} C N , M { 0 , 1 } \{0,1\} { 0 , 1 } γ \gamma γ M M M
Proof. Let T N = e γ N T_N=e^{\gamma N} T N = e γ N φ N = φ T N , a \varphi_N=\varphi_{T_N,a} φ N = φ T N , a φ N \varphi_N φ N [ 0 , 1 ] [0,1] [ 0 , 1 ] M M M N ≥ N M N\ge N_M N ≥ N M (160) holds on W N = W N , M ⊂ K W_N=W_{N,M}\subset K W N = W N , M ⊂ K
E ∑ u ∈ C N , M φ N ( H N , u ) ≤ K r 1 / r q N 1 − 1 / r E ∣ C N , M ∣ ≤ C M K r 1 / r q N 1 − 1 / r . \mathbb{E}\sum_{u\in C_{N,M}}\varphi_N(H_N,u)\le K_r^{1/r}q_N^{1-1/r}\mathbb{E}\lvert C_{N,M}\rvert\le C_MK_r^{1/r}q_N^{1-1/r}. E u ∈ C N , M ∑ φ N ( H N , u ) ≤ K r 1/ r q N 1 − 1/ r E ∣ C N , M ∣ ≤ C M K r 1/ r q N 1 − 1/ r . By Theorem 7.17, C N , M C_{N,M} C N , M P v N H N \mathbb{P}_{v_N}^{H_N} P v N H N (163) by ∑ u ∈ C N , M φ N ( H N , u ) \sum_{u\in C_{N,M}}\varphi_N(H_N,u) ∑ u ∈ C N , M φ N ( H N , u ) (163) . For a subfamily selected by a weight θ \theta θ Theorem 7.23 . The constant K r \mathrm{K}_r K r b Q b_Q b Q g g g k + k_+ k + r r r q N q_N q N K \mathcal{K} K a a a γ \gamma γ M M M M M M C M C_M C M N M N_M N M
Note that E ∣ C N , M ∣ < ∞ \mathbb{E}|C_{N,M}| < \infty E ∣ C N , M ∣ < ∞ N N N Corollary 7.18 , and hypothesis (3) is equivalent to lim sup N E ∣ C N , M ∣ < ∞ \limsup_N \mathbb{E}|C_{N,M}| < \infty lim sup N E ∣ C N , M ∣ < ∞ K = [ h − , h + ] × [ k − , k + ] \mathcal{K} = [h_-,h_+] \times[k_-,k_+] K = [ h − , h + ] × [ k − , k + ] U ⊂ [ k − , k + ] U \subset[k_-,k_+] U ⊂ [ k − , k + ] m N / N m_N/N m N / N ( h − , h + ) (h_-,h_+) ( h − , h + ) { ( h , k ) : k = p h } \{(h,k): k = ph\} {( h , k ) : k = p h } W N , M \mathcal{W}_{N,M} W N , M (163) implies that for each fixed M M M ν N \nu_N ν N C a ( u ) = { x ∈ S N : ∣ R ( x , u ) ∣ ≥ a } C_a(u) = \{x \in S_N : |R(x,u)| \ge a\} C a ( u ) = { x ∈ S N : ∣ R ( x , u ) ∣ ≥ a } Definition 4.1 for some u ∈ C N , M u \in C_{N,M} u ∈ C N , M e γ N e^{\gamma N} e γ N
Remark 7.25 (Polynomial rates). Let a ∗ > 0 a_* > 0 a ∗ > 0 ξ \xi ξ Q Q Q β \beta β K \mathcal{K} K a ∈ ( 0 , min { 1 , a ∗ } ] a \in(0,\min\{1,a_*\}] a ∈ ( 0 , min { 1 , a ∗ }] γ \gamma γ c c c C C C N 0 N_0 N 0 C C C 2 ≤ N < N 0 2 \le N < N_0 2 ≤ N < N 0 Corollary 7.24 holds with
q N = C ( N 1 − d / 2 + N 3 − d ) + e − c N . q_N = C(N^{1-d/2} + N^{3-d}) + e^{-cN}. q N = C ( N 1 − d /2 + N 3 − d ) + e − c N . Since d ≥ 4 d \ge4 d ≥ 4 N 3 − d ≤ N 1 − d / 2 N^{3-d} \le N^{1-d/2} N 3 − d ≤ N 1 − d /2 q N ≤ C ′ N − ( d − 2 ) / 2 q_N \le C'N^{-(d-2)/2} q N ≤ C ′ N − ( d − 2 ) /2 M > 0 M > 0 M > 0 r > 1 r > 1 r > 1 (163) is at most C M , r N − ( d − 2 ) ( 1 − 1 / r ) / 2 C_{M,r}N^{-(d-2)(1-1/r)/2} C M , r N − ( d − 2 ) ( 1 − 1/ r ) /2 N ≥ N M N \ge N_M N ≥ N M r = 2 r = 2 r = 2 ( d − 2 ) / 4 (d-2)/4 ( d − 2 ) /4 Proposition 7.7 holds for every fixed r r r ( d − 2 ) / 2 (d-2)/2 ( d − 2 ) /2 γ \gamma γ M M M r r r
Remark 7.26 (Order of the parameters). Table 7.1 lists the parameters of Theorem 7.23 and Corollary 7.24 in the order in which they are fixed. All of them are fixed before N → ∞ N \to\infty N → ∞ C M C_M C M N M N_M N M M M M γ \gamma γ M M M
Table 7.1. Parameters in the determinant-weighted transfer, in the order in which they are fixed.
Cap hitting estimates under critical-point conditioning
In this section, we prove Theorem 8.2 , the hitting estimate for the caps under critical-point conditioning. It concerns the original Langevin diffusion started from the uniform law ν N \nu_N ν N H Q H_Q H Q u u u { x : ∣ R ( x , u ) ∣ ≥ a } \{x : \lvert R(x,u)\rvert\ge a\} { x : ∣ R ( x , u )∣ ≥ a } e γ N e^{\gamma N} e γ N C ( N 1 − d / 2 + N 3 − d ) + e − c N C(N^{1-d/2}+N^{3-d})+e^{-cN} C ( N 1 − d /2 + N 3 − d ) + e − c N d ≥ 4 d\ge4 d ≥ 4 Theorem 7.23 and the separation criterion of Proposition 4.5 .
The section is self-contained. From earlier sections, it uses only the model (Definition 3.1), the construction of the dynamics (Lemma 3.5), and the rotation covariance of the path kernels (Lemma 4.2 ). It does not use the static results of Section 5, the counts of critical points and the concentration of the Gibbs measure on bands in Sections 9 to 12, or the Kac–Rice formula of § 6.2 , and it needs no lower bound on the Hessian at the critical point. The spherical harmonic computations, the concentration estimate on the sphere, and the comparison of resolvents that we need are proved here.
Standing notation. Throughout this section, ξ ( r ) = ∑ p ∈ P c p r p \xi(r)=\sum_{p\in\mathcal{P}}c_pr^p ξ ( r ) = ∑ p ∈ P c p r p c p > 0 c_p>0 c p > 0 p ∈ P p\in\mathcal{P} p ∈ P d = min P ≥ 4 d=\min\mathcal{P}\ge4 d = min P ≥ 4 p ˉ = max P \bar p=\max\mathcal{P} p ˉ = max P H N = ∑ p ∈ P H N , p H_N=\sum_{p\in\mathcal{P}}H_{N,p} H N = ∑ p ∈ P H N , p H N , p H_{N,p} H N , p p p p R N \mathbb{R}^N R N E H N , p ( x ) H N , p ( y ) = N c p R ( x , y ) p \mathbb{E}H_{N,p}(x)H_{N,p}(y)=Nc_pR(x,y)^p E H N , p ( x ) H N , p ( y ) = N c p R ( x , y ) p N N N J J J H J H_J H J E H N ( x ) H N ( y ) = N ξ ( R ( x , y ) ) \mathbb{E}H_N(x)H_N(y)=N\xi(R(x,y)) E H N ( x ) H N ( y ) = N ξ ( R ( x , y )) R ( x , y ) = x ⋅ y / N R(x,y)=x\cdot y/N R ( x , y ) = x ⋅ y / N x , y ∈ R N x,y\in\mathbb{R}^N x , y ∈ R N Q ∈ ( 0 , 1 ] Q\in(0,1] Q ∈ ( 0 , 1 ] β ≥ 0 \beta\ge0 β ≥ 0
H Q ( u ) = H N ( Q u ) = ∑ p ∈ P Q p / 2 H N , p ( u ) , K Q ( u ) = d d t H N ( t Q u ) ∣ t = 1 = ∑ p ∈ P p Q p / 2 H N , p ( u ) . (164) \begin{aligned}
H_Q(u)&=H_N(\sqrt{Q}\,u)=\sum_{p\in\mathcal{P}}Q^{p/2}H_{N,p}(u),\\
K_Q(u)&=\left.\frac{\mathrm{d}}{\mathrm{d}t}H_N(t\sqrt{Q}\,u)\right|_{t=1}=\sum_{p\in\mathcal{P}}pQ^{p/2}H_{N,p}(u). \tag*{(164)}
\end{aligned} H Q ( u ) K Q ( u ) = H N ( Q u ) = p ∈ P ∑ Q p /2 H N , p ( u ) , = d t d H N ( t Q u ) t = 1 = p ∈ P ∑ p Q p /2 H N , p ( u ) . (164) and the parameters ζ ( Q ) \zeta(Q) ζ ( Q ) a Q = Q ζ ′ ( Q ) a_Q = Q\zeta'(Q) a Q = Q ζ ′ ( Q ) b Q = Q 2 ζ ′ ′ ( Q ) b_Q = Q^2\zeta''(Q) b Q = Q 2 ζ ′′ ( Q )
Σ Q = ( ζ ( Q ) a Q a Q a Q + b Q ) . \Sigma_Q =
\begin{pmatrix}
\zeta(Q) & a_Q \\
a_Q & a_Q+b_Q
\end{pmatrix}. Σ Q = ( ζ ( Q ) a Q a Q a Q + b Q ) . For u ∈ S N u \in S_N u ∈ S N G u = ∇ S N H Q ( u ) G_u = \nabla_{S_N}H_Q(u) G u = ∇ S N H Q ( u ) u ⊥ u^\perp u ⊥ ∇ E \nabla_E ∇ E ∇ = ∇ S N \nabla= \nabla_{S_N} ∇ = ∇ S N S N S_N S N Π x = I − x x T / N \Pi_x = I - xx^{\mathsf T}/N Π x = I − x x T / N x ⊥ x^\perp x ⊥ ∇ H = Π x ∇ E H \nabla H = \Pi_x\nabla_E H ∇ H = Π x ∇ E H x ∈ S N x \in S_N x ∈ S N H H H L β , H = Δ S N + β ∇ H ⋅ ∇ L_{\beta,H} = \Delta_{S_N} + \beta\nabla H \cdot\nabla L β , H = Δ S N + β ∇ H ⋅ ∇ π H \pi^H π H P ν N H \mathbb{P}^H_{\nu_N} P ν N H ν N \nu_N ν N 0 < a < 1 0 < a < 1 0 < a < 1 u ∈ S N u \in S_N u ∈ S N
C a ( u ) = { x ∈ S N : ∣ R ( x , u ) ∣ ≥ a } C_a(u) = \{x \in S_N : |R(x,u)| \ge a\} C a ( u ) = { x ∈ S N : ∣ R ( x , u ) ∣ ≥ a } for the union of the two antipodal caps of latitude a a a ± u \pm u ± u I ( a ) = − 1 2 log ( 1 − a 2 ) I(a) = -\frac{1}{2}\log(1-a^2) I ( a ) = − 2 1 log ( 1 − a 2 )
We prescribe the values H Q ( u ) = N h H_Q(u) = Nh H Q ( u ) = N h K Q ( u ) = N k K_Q(u) = Nk K Q ( u ) = N k ( h , k ) (h,k) ( h , k ) h ˉ , k ˉ > 0 \bar h,\bar k > 0 h ˉ , k ˉ > 0 ( h , k ) (h,k) ( h , k ) ∣ h ∣ ≤ h ˉ |h| \le\bar h ∣ h ∣ ≤ h ˉ ∣ k ∣ ≤ k ˉ |k| \le\bar k ∣ k ∣ ≤ k ˉ ζ \zeta ζ p p p k = p h k = ph k = p h C C C c c c ζ \zeta ζ Q Q Q β \beta β h ˉ \bar h h ˉ k ˉ \bar k k ˉ N N N u u u ( h , k ) (h,k) ( h , k ) S N S_N S N Lemma 8.6 .
The main estimate
We first define the conditioned field. When ζ \zeta ζ Σ Q \Sigma_Q Σ Q Proposition 8.7 ), and we define polynomials θ 1 , θ 2 , θ 3 \theta_1,\theta_2,\theta_3 θ 1 , θ 2 , θ 3
( θ 1 ( r ) , θ 2 ( r ) ) = ( ζ ( Q r ) , Q r ζ ′ ( Q r ) ) Σ Q − 1 , θ 3 ( r ) = Q ζ ′ ( Q r ) a Q . (165) (\theta_1(r),\theta_2(r)) = (\zeta(\sqrt{Q}r),\sqrt{Q}r\zeta'(\sqrt{Q}r))\Sigma_Q^{-1}, \qquad\theta_3(r) = \frac{\sqrt{Q}\zeta'(\sqrt{Q}r)}{a_Q}. \tag*{(165)} ( θ 1 ( r ) , θ 2 ( r )) = ( ζ ( Q r ) , Q r ζ ′ ( Q r )) Σ Q − 1 , θ 3 ( r ) = a Q Q ζ ′ ( Q r ) . (165) When ζ = c p r p \zeta= c_p r^p ζ = c p r p θ 1 ( r ) = Q − p / 2 r p \theta_1(r) = Q^{-p/2}r^p θ 1 ( r ) = Q − p /2 r p θ 2 = 0 \theta_2 = 0 θ 2 = 0 θ 3 ( r ) = Q − p / 2 r p − 1 \theta_3(r) = Q^{-p/2}r^{p-1} θ 3 ( r ) = Q − p /2 r p − 1
Definition 8.1 (The conditioned field). Let u ∈ S N u \in S_N u ∈ S N x ∈ R N x \in\mathbb{R}^N x ∈ R N r = x ⋅ u / N r = x \cdot u/N r = x ⋅ u / N Gaussian regression projection of H N H_N H N u u u
D u ( x ) = θ 1 ( r ) H Q ( u ) + θ 2 ( r ) K Q ( u ) + θ 3 ( r ) G u ⋅ x . (166) D_u(x) = \theta_1(r)H_Q(u) + \theta_2(r)K_Q(u) + \theta_3(r)G_u \cdot x. \tag*{(166)} D u ( x ) = θ 1 ( r ) H Q ( u ) + θ 2 ( r ) K Q ( u ) + θ 3 ( r ) G u ⋅ x . (166) For an admissible pair ( h , k ) (h,k) ( h , k ) conditioned field is
H h , k , u = H N + δ h , k , u , δ h , k , u ( x ) = N ( θ 1 ( r ) h + θ 2 ( r ) k ) − D u ( x ) . (167) H^{h,k,u} = H_N + \delta_{h,k,u}, \qquad\delta_{h,k,u}(x) = N\bigl(\theta_1(r)h + \theta_2(r)k\bigr) - D_u(x). \tag*{(167)} H h , k , u = H N + δ h , k , u , δ h , k , u ( x ) = N ( θ 1 ( r ) h + θ 2 ( r ) k ) − D u ( x ) . (167) We write E h , k , u \mathbb{E}_{h,k,u} E h , k , u H h , k , u H^{h,k,u} H h , k , u
We show in Proposition 8.7 below that the law of H h , k , u H^{h,k,u} H h , k , u H N H_N H N H Q ( u ) = N h H_Q(u)=Nh H Q ( u ) = N h K Q ( u ) = N k K_Q(u)=Nk K Q ( u ) = N k G u = 0 G_u=0 G u = 0 K Q ( u ) = p H Q ( u ) K_Q(u)=pH_Q(u) K Q ( u ) = p H Q ( u ) H Q ( u ) = N h H_Q(u)=Nh H Q ( u ) = N h G u = 0 G_u=0 G u = 0 ( h , k , u ) (h,k,u) ( h , k , u )
Theorem 8.2 (Cap hitting bound from the uniform initial law). Let Z \mathfrak{Z} Z d ≥ 4 d\ge4 d ≥ 4 Q ∈ ( 0 , 1 ] Q\in(0,1] Q ∈ ( 0 , 1 ] β ≥ 0 \beta\ge0 β ≥ 0 h ˉ , k ˉ > 0 \bar{h},\bar{k}>0 h ˉ , k ˉ > 0 a ∗ > 0 a_*>0 a ∗ > 0 a ∈ ( 0 , a ∗ ] a\in(0,a_*] a ∈ ( 0 , a ∗ ] γ , c , C > 0 \gamma,c,C>0 γ , c , C > 0 N 0 N_0 N 0 N ≥ N 0 N\ge N_0 N ≥ N 0 u ∈ S N u\in S_N u ∈ S N ( h , k ) (h,k) ( h , k )
E h , k , u [ P ν N H ( sup 0 ≤ t ≤ e γ N ∣ R ( X t , u ) ∣ ≥ a ) ] ≤ C ( N 1 − d / 2 + N 3 − d ) + e − c N . (168) \mathbb{E}_{h,k,u}\left[\mathbb{P}^{H}_{\nu_N}\left(\sup_{0\le t\le e^{\gamma N}}|R(X_t,u)|\ge a\right)\right]\le C\left(N^{1-d/2}+N^{3-d}\right)+e^{-cN}. \tag*{(168)} E h , k , u [ P ν N H ( 0 ≤ t ≤ e γ N sup ∣ R ( X t , u ) ∣ ≥ a ) ] ≤ C ( N 1 − d /2 + N 3 − d ) + e − c N . (168) The constant a ∗ a_* a ∗ Z \mathfrak{Z} Z Q Q Q β \beta β h ˉ , k ˉ \bar{h},\bar{k} h ˉ , k ˉ γ , c , C , N 0 \gamma,c,C,N_0 γ , c , C , N 0 a a a
In (8.5), the field H H H H h , k , u H^{h,k,u} H h , k , u X X X L β , H L_{\beta,H} L β , H H H H C a ( u ) C_a(u) C a ( u ) e γ N e^{\gamma N} e γ N d ≥ 4 d\ge4 d ≥ 4
Corollary 8.3 (Uniformity over a rectangle of marks). Let Z \mathfrak{Z} Z Q Q Q β \beta β K ⊂ R 2 \mathcal{K}\subset\mathbb{R}^2 K ⊂ R 2 a ∗ > 0 a_*>0 a ∗ > 0 a ∈ ( 0 , a ∗ ] a\in(0,a_*] a ∈ ( 0 , a ∗ ] γ , c , C > 0 \gamma,c,C>0 γ , c , C > 0 N 0 N_0 N 0 Z , Q , β , K , a \mathfrak{Z},Q,\beta,\mathcal{K},a Z , Q , β , K , a N ≥ N 0 N\ge N_0 N ≥ N 0
sup u ∈ S N sup ( h , k ) ∈ K E h , k , u [ P ν N H ( sup 0 ≤ t ≤ e γ N ∣ R ( X t , u ) ∣ ≥ a ) ] ≤ C ( N 1 − d / 2 + N 3 − d ) + e − c N . (169) \sup_{u\in S_N}\ \sup_{(h,k)\in\mathcal{K}}\mathbb{E}_{h,k,u}\left[\mathbb{P}^{H}_{\nu_N}\left(\sup_{0\le t\le e^{\gamma N}}|R(X_t,u)|\ge a\right)\right]\le C\left(N^{1-d/2}+N^{3-d}\right)+e^{-cN}. \tag*{(169)} u ∈ S N sup ( h , k ) ∈ K sup E h , k , u [ P ν N H ( 0 ≤ t ≤ e γ N sup ∣ R ( X t , u ) ∣ ≥ a ) ] ≤ C ( N 1 − d /2 + N 3 − d ) + e − c N . (169) When Z \mathfrak{Z} Z p p p ( h , k ) ∈ K (h,k)\in\mathcal{K} ( h , k ) ∈ K k = p h k=ph k = p h
Proof. Fix h ˉ , k ˉ \bar{h},\bar{k} h ˉ , k ˉ K ⊂ [ − h ˉ , h ˉ ] × [ − k ˉ , k ˉ ] \mathcal{K}\subset[-\bar{h},\bar{h}]\times[-\bar{k},\bar{k}] K ⊂ [ − h ˉ , h ˉ ] × [ − k ˉ , k ˉ ] h ˉ , k ˉ \bar{h},\bar{k} h ˉ , k ˉ ∣ H Q ( u ) ∣ ≤ N h ˉ |H_Q(u)|\le N\bar{h} ∣ H Q ( u ) ∣ ≤ N h ˉ ∣ K Q ( u ) ∣ ≤ N k ˉ |K_Q(u)|\le N\bar{k} ∣ K Q ( u ) ∣ ≤ N k ˉ Z , Q , β , a \mathfrak{Z},Q,\beta,a Z , Q , β , a
Remark 8.4 (Scope). In Proposition 13.1, Corollary 8.3 is applied to a compact rectangle of marks that contains the marks of every fixed energy window [ m N − M , m N + M ] [mN-M,mN+M] [ m N − M , m N + M ] γ \gamma γ M M M K \mathcal{K} K k k k 2 b ˉ Q 2\sqrt{\bar{b}_Q} 2 b ˉ Q ∇ 2 H Q ( u ) \nabla^2H_Q(u) ∇ 2 H Q ( u ) E h , k , u \mathbb{E}_{h,k,u} E h , k , u Q ↓ 0 Q\downarrow0 Q ↓ 0 Z \mathfrak{Z} Z a a a N N N d = 3 d=3 d = 3 N 3 − d N^{3-d} N 3 − d
The mechanism and the origin of the errors
We now explain why the dynamics started from ν N \nu_N ν N H Q H_Q H Q e γ N e^{\gamma N} e γ N (168) come from. The discussion in this section is informal; the precise statements and proofs follow in the next sections.
The unconditioned field. We begin with the case where the field is not conditioned and the axis u u u S N S_N S N H N H_N H N N − 1 / 2 N^{-1/2} N − 1/2 u u u ν N ( C a ( u ) ) ≤ e − N I ( a ) / 2 \nu_N(C_a(u)) \le e^{-N I(a)/2} ν N ( C a ( u )) ≤ e − N I ( a ) /2 N N N C a ( u ) C_a(u) C a ( u ) [ 0 , e γ N ] [0,e^{\gamma N}] [ 0 , e γ N ] κ ≥ 0 \kappa\ge0 κ ≥ 0 C a ( u ) C_a(u) C a ( u ) τ \tau τ ∫ 0 t κ ( X v ) d v \int_0^t \kappa(X_v)\,\mathrm{d}v ∫ 0 t κ ( X v ) d v E e − λ τ \mathbb{E}e^{-\lambda\tau} E e − λ τ λ − 1 ∫ κ d π H \lambda^{-1}\int\kappa\,\mathrm{d}\pi^H λ − 1 ∫ κ d π H ν N \nu_N ν N π H \pi^H π H N N N λ = e − γ N \lambda=e^{-\gamma N} λ = e − γ N λ − 1 ∫ κ d π H \lambda^{-1}\int\kappa\,\mathrm{d}\pi^H λ − 1 ∫ κ d π H e γ N − N I ( a ) / 2 e^{\gamma N-NI(a)/2} e γ N − N I ( a ) /2 γ \gamma γ
The conditioning and the axis average. Conditioning u u u H Q H_Q H Q H h , k , u = H N + δ h , k , u H^{h,k,u}=H_N+\delta_{h,k,u} H h , k , u = H N + δ h , k , u N N N u u u H N H_N H N u u u θ 1 \theta_1 θ 1 θ 2 \theta_2 θ 2 ∣ r ∣ d |r|^d ∣ r ∣ d θ 3 \theta_3 θ 3 ∣ r ∣ d − 1 |r|^{d-1} ∣ r ∣ d − 1
∣ δ h , k , u ( x ) ∣ ≤ C N ∣ r ∣ d + C N ∣ r ∣ d − 1 ∣ V x ( u ) ∣ , V x ( u ) = N − 1 / 2 G u ⋅ x , |\delta_{h,k,u}(x)| \le CN|r|^d+C\sqrt{N}|r|^{d-1}|V_x(u)|,\qquad V_x(u)=N^{-1/2}G_u\cdot x, ∣ δ h , k , u ( x ) ∣ ≤ C N ∣ r ∣ d + C N ∣ r ∣ d − 1 ∣ V x ( u ) ∣ , V x ( u ) = N − 1/2 G u ⋅ x , where r = R ( x , u ) r=R(x,u) r = R ( x , u ) x x x N − 1 / 2 N^{-1/2} N − 1/2 u u u V x ( u ) V_x(u) V x ( u ) N 1 − d / 2 N^{1-d/2} N 1 − d /2 ( x , u ) (x,u) ( x , u ) δ α \delta_\alpha δ α δ h , k , u \delta_{h,k,u} δ h , k , u α \alpha α (190) and the last paragraph of this section), through the density e w e^w e w w = β δ α w=\beta\delta_\alpha w = β δ α u u u N N N e β δ h , k , u e^{\beta\delta_{h,k,u}} e β δ h , k , u α \alpha α e w e^w e w x x x
η = E u [ ∣ e w − 1 ∣ ] = O ( N 1 − d / 2 ) , ϵ = E u [ e w ( 1 − e − w ) 2 ] = O ( N 2 − d ) . \eta=\mathbb{E}_u[|e^w-1|]=O(N^{1-d/2}),\qquad\epsilon=\mathbb{E}_u[e^w(1-e^{-w})^2]=O(N^{2-d}). η = E u [ ∣ e w − 1∣ ] = O ( N 1 − d /2 ) , ϵ = E u [ e w ( 1 − e − w ) 2 ] = O ( N 2 − d ) . The correlated gradient. This raises a difficulty, since the point x x x G u G_u G u G u G_u G u x x x x x x u u u { u : R ( x , u ) = r } \{u : R(x,u) = r\} { u : R ( x , u ) = r } V x V_x V x H N , p H_{N,p} H N , p N ∣ r ∣ p − 1 \sqrt{N}|r|^{p-1} N ∣ r ∣ p − 1 ω p , j , N = O ( N − j ) \omega_{p,j,N} = O(N^{-j}) ω p , j , N = O ( N − j ) N 1 − j N^{1-j} N 1 − j Lemma 8.5 ); they must be kept.
The time horizon. The main obstacle is the length e γ N e^{\gamma N} e γ N Lemma 8.11 . It compares the Laplace transform E e − λ τ \mathbb{E}e^{-\lambda\tau} E e − λ τ
η + ϵ ⏟ O ( N 1 − d / 2 ) + ϵ ( 1 + H 0 ) ⏟ O ( N 3 − d ) . (170) \underbrace{\eta+\sqrt{\epsilon}}_{O(N^{1-d/2})}
+
\underbrace{\epsilon(1+\mathcal{H}_0)}_{O(N^{3-d})}. \tag*{(170)} O ( N 1 − d /2 ) η + ϵ + O ( N 3 − d ) ϵ ( 1 + H 0 ) . (170) where H 0 = ∫ ( 1 + f 0 ) log ( 1 + f 0 ) d μ \mathcal{H}_0 = \int(1+f_0)\log(1+f_0)\,d\mu H 0 = ∫ ( 1 + f 0 ) log ( 1 + f 0 ) d μ f 0 f_0 f 0 μ \mu μ Lemma 8.11 . In our application, μ \mu μ f 0 μ f_0\mu f 0 μ H 0 = O ( N ) \mathcal{H}_0 = O(N) H 0 = O ( N ) (214) . Since ∫ f 0 d μ = 1 \int f_0\,d\mu= 1 ∫ f 0 d μ = 1 a ↦ ( 1 + a ) log ( 1 + a ) − a log a a \mapsto(1+a)\log(1+a) - a\log a a ↦ ( 1 + a ) log ( 1 + a ) − a log a H 0 \mathcal{H}_0 H 0 ∫ f 0 log f 0 d μ \int f_0\log f_0\,d\mu ∫ f 0 log f 0 d μ ν N \nu_N ν N 2 log 2 2\log2 2 log 2 e γ N e^{\gamma N} e γ N d = 4 d=4 d = 4 (170) are O ( N − 1 ) O(N^{-1}) O ( N − 1 ) d = 3 d=3 d = 3
Truncation and the return to the true process. The comparison lemma requires a bounded perturbation whose size is small compared with the entropy cost of the caps. For this reason, we truncate δ h k , u \delta h_{k,u} δ h k , u α \alpha α r = R ( x , u ) r=R(x,u) r = R ( x , u ) s = G u ⋅ x / N s=G_u\cdot x/N s = G u ⋅ x / N C N α d CN\alpha^d C N α d α / 4 \alpha/4 α /4 N I ( α / 4 ) ≥ N α 2 / 32 NI(\alpha/4)\ge N\alpha^2/32 N I ( α /4 ) ≥ N α 2 /32 d > 2 d>2 d > 2 α \alpha α γ \gamma γ C α / 2 ( u ) C_{\alpha/2}(u) C α /2 ( u ) ∣ r ∣ |r| ∣ r ∣ ∣ s ∣ |s| ∣ s ∣ α \alpha α ∣ r ∣ = α |r| = \alpha ∣ r ∣ = α C a ( u ) C_a(u) C a ( u ) a = α / 2 a = \alpha/2 a = α /2 ∣ s ∣ = α |s| = \alpha ∣ s ∣ = α G u G_u G u s t s_t s t a Q / N a_Q/N a Q / N
Organization. In § 6.3.3 , we compute the harmonic decomposition of the field and prove regularity bounds with exponential tails. § 6.3.4 constructs the conditioned field by regression and bounds the correction. § 6.4.1 proves the latitude concentration and the two moment bounds for e w e^w e w § 6.4.2 states and proves the entropy–resolvent comparison for abstract diffusions. § 6.4.3 applies it to the truncated process, and § 6.4.4 shows that the true process does not reach the cutoff boundary { ∣ G u ⋅ x ∣ / N = α } \{|G_u \cdot x|/N = \alpha\} { ∣ G u ⋅ x ∣/ N = α } Theorem 8.2 in § 6.4.5 . Table 8.1 lists the parameters in the order in which they are fixed; all of them are fixed before N → ∞ N \to\infty N → ∞
Table 8.1. Parameters in the proof of Theorem 8.2, in the order in which they are chosen.
Harmonic coefficients and regularity of the field
The latitude mean of the tangent gradient term G u ⋅ x G_u \cdot x G u ⋅ x H N , p H_{N,p} H N , p
Let N ≥ 3 N \ge3 N ≥ 3 α N = ( N − 2 ) / 2 \alpha_N = (N - 2)/2 α N = ( N − 2 ) /2 C ℓ ( α N ) C_\ell^{(\alpha_N)} C ℓ ( α N )
( 1 − 2 r t + t 2 ) − α N = ∑ ℓ ≥ 0 C ℓ ( α N ) ( r ) t ℓ . (1 - 2rt + t^2)^{-\alpha_N} = \sum_{\ell\ge0} C_\ell^{(\alpha_N)}(r)t^\ell. ( 1 − 2 r t + t 2 ) − α N = ℓ ≥ 0 ∑ C ℓ ( α N ) ( r ) t ℓ . Expanding the left side by the binomial series gives
C ℓ ( α N ) ( r ) = ∑ j = 0 ⌊ ℓ / 2 ⌋ ( − 1 ) j ( α N ) ℓ − j j ! ( ℓ − 2 j ) ! ( 2 r ) ℓ − 2 j , C ℓ ( α N ) ( 1 ) = ( 2 α N ) ℓ ℓ ! . C_\ell^{(\alpha_N)}(r) = \sum_{j=0}^{\lfloor\ell/2 \rfloor} \frac{(-1)^j(\alpha_N)_{\ell-j}}{j!(\ell- 2j)!}(2r)^{\ell- 2j}, \qquad C_\ell^{(\alpha_N)}(1) = \frac{(2\alpha_N)_\ell}{\ell!}. C ℓ ( α N ) ( r ) = j = 0 ∑ ⌊ ℓ /2 ⌋ j ! ( ℓ − 2 j )! ( − 1 ) j ( α N ) ℓ − j ( 2 r ) ℓ − 2 j , C ℓ ( α N ) ( 1 ) = ℓ ! ( 2 α N ) ℓ . where ( t ) j = t ( t + 1 ) ⋯ ( t + j − 1 ) (t)_j = t(t + 1)\cdots(t + j - 1) ( t ) j = t ( t + 1 ) ⋯ ( t + j − 1 ) ( t ) 0 = 1 (t)_0 = 1 ( t ) 0 = 1
Z ℓ , N ( r ) = C ℓ ( α N ) ( r ) C ℓ ( α N ) ( 1 ) = ℓ ! ( 2 α N ) ℓ ∑ j = 0 ⌊ ℓ / 2 ⌋ ( − 1 ) j ( α N ) ℓ − j j ! ( ℓ − 2 j ) ! ( 2 r ) ℓ − 2 j , (171) Z_{\ell,N}(r) = \frac{C_\ell^{(\alpha_N)}(r)}{C_\ell^{(\alpha_N)}(1)} = \frac{\ell!}{(2\alpha_N)_\ell} \sum_{j=0}^{\lfloor\ell/2 \rfloor} \frac{(-1)^j(\alpha_N)_{\ell-j}}{j!(\ell- 2j)!}(2r)^{\ell- 2j}, \tag*{(171)} Z ℓ , N ( r ) = C ℓ ( α N ) ( 1 ) C ℓ ( α N ) ( r ) = ( 2 α N ) ℓ ℓ ! j = 0 ∑ ⌊ ℓ /2 ⌋ j ! ( ℓ − 2 j )! ( − 1 ) j ( α N ) ℓ − j ( 2 r ) ℓ − 2 j , (171) so that Z 0 , N = 1 Z_{0,N}=1 Z 0 , N = 1 Z ℓ , N ( 1 ) = 1 Z_{\ell,N}(1)=1 Z ℓ , N ( 1 ) = 1
( 1 − r 2 ) Z ′ ′ − ( N − 1 ) r Z ′ + ℓ ( ℓ + N − 2 ) Z = 0 , Z = Z ℓ , N . (172) (1-r^2)Z''-(N-1)rZ'+\ell(\ell+N-2)Z=0,\qquad Z=Z_{\ell,N}. \tag*{(172)} ( 1 − r 2 ) Z ′′ − ( N − 1 ) r Z ′ + ℓ ( ℓ + N − 2 ) Z = 0 , Z = Z ℓ , N . (172) Lemma 8.5 (Harmonic coefficients). Let N ≥ 3 N\ge3 N ≥ 3
For every integer p ≥ 0 p\ge0 p ≥ 0
r p = ∑ j = 0 ⌊ p / 2 ⌋ ω p , j , N Z p − 2 j , N ( r ) , ω p , j , N = p ! Γ ( α N ) ( p − 2 j + α N ) 2 j j ! Γ ( p − j + α N + 1 ) ⋅ ( 2 α N ) p − 2 j ( p − 2 j ) ! . (173) r^p=\sum_{j=0}^{\lfloor p/2\rfloor}\omega_{p,j,N}Z_{p-2j,N}(r),\qquad
\omega_{p,j,N}=\frac{p!\,\Gamma(\alpha_N)(p-2j+\alpha_N)}{2^j j!\,\Gamma(p-j+\alpha_N+1)}\cdot\frac{(2\alpha_N)_{p-2j}}{(p-2j)!}. \tag*{(173)} r p = j = 0 ∑ ⌊ p /2 ⌋ ω p , j , N Z p − 2 j , N ( r ) , ω p , j , N = 2 j j ! Γ ( p − j + α N + 1 ) p ! Γ ( α N ) ( p − 2 j + α N ) ⋅ ( p − 2 j )! ( 2 α N ) p − 2 j . (173) The coefficients ω p , j , N \omega_{p,j,N} ω p , j , N
For fixed p p p 0 ≤ j ≤ p / 2 0\le j\le p/2 0 ≤ j ≤ p /2 N → ∞ N\to\infty N → ∞
ω p , j , N = p ! 2 j j ! ( p − 2 j ) ! N − j ( 1 + O p ( N − 1 ) ) ; in particular ω p , j , N ≤ C p N − j . (174) \omega_{p,j,N}=\frac{p!}{2^j j!(p-2j)!}N^{-j}(1+O_p(N^{-1}));\qquad\text{in particular }\omega_{p,j,N}\le C_pN^{-j}. \tag*{(174)} ω p , j , N = 2 j j ! ( p − 2 j )! p ! N − j ( 1 + O p ( N − 1 )) ; in particular ω p , j , N ≤ C p N − j . (174) For fixed ℓ \ell ℓ m m m Z ℓ , N ( r ) − r ℓ Z_{\ell,N}(r)-r^\ell Z ℓ , N ( r ) − r ℓ m m m C ℓ , m / N C_{\ell,m}/N C ℓ , m / N [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] ℓ ≥ 1 \ell\ge1 ℓ ≥ 1 ∣ r ∣ ≤ 1 |r|\le1 ∣ r ∣ ≤ 1
∣ Z ℓ , N ′ ( r ) ∣ ≤ C ℓ ( ∣ r ∣ ℓ − 1 + N − 1 ) . (175) |Z'_{\ell,N}(r)|\le C_\ell(|r|^{\ell-1}+N^{-1}). \tag*{(175)} ∣ Z ℓ , N ′ ( r ) ∣ ≤ C ℓ ( ∣ r ∣ ℓ − 1 + N − 1 ) . (175) Let H N , p ∣ S N = ∑ j H p , p − 2 j H_{N,p}|_{S_N}=\sum_j H_{p,p-2j} H N , p ∣ S N = ∑ j H p , p − 2 j H p , ℓ H_{p,\ell} H p , ℓ ℓ \ell ℓ H p , p − 2 j H_{p,p-2j} H p , p − 2 j p ∈ P p\in\mathcal{P} p ∈ P 0 ≤ j ≤ p / 2 0\le j\le p/2 0 ≤ j ≤ p /2 S N S_N S N
E H p , p − 2 j ( x ) H p , p − 2 j ( y ) = N c p ω p , j , N Z p − 2 j , N ( R ( x , y ) ) , x , y ∈ S N . (176) \mathbb{E}H_{p,p-2j}(x)H_{p,p-2j}(y)=Nc_p\omega_{p,j,N}Z_{p-2j,N}(R(x,y)),\qquad x,y\in S_N. \tag*{(176)} E H p , p − 2 j ( x ) H p , p − 2 j ( y ) = N c p ω p , j , N Z p − 2 j , N ( R ( x , y )) , x , y ∈ S N . (176) For example, at p = 4 p=4 p = 4
r 4 = ( N − 1 ) ( N + 1 ) ( N + 2 ) ( N + 4 ) Z 4 , N ( r ) + 6 ( N − 1 ) N ( N + 4 ) Z 2 , N ( r ) + 3 N ( N + 2 ) . r^4=\frac{(N-1)(N+1)}{(N+2)(N+4)}Z_{4,N}(r)+\frac{6(N-1)}{N(N+4)}Z_{2,N}(r)+\frac{3}{N(N+2)}. r 4 = ( N + 2 ) ( N + 4 ) ( N − 1 ) ( N + 1 ) Z 4 , N ( r ) + N ( N + 4 ) 6 ( N − 1 ) Z 2 , N ( r ) + N ( N + 2 ) 3 . In particular, the component of degree two of a quartic field has a variance of order N − 1 N^{-1} N − 1
Proof. We first recall the uniqueness of zonal harmonics. Fix z ∈ S N z\in S_N z ∈ S N Δ S N \Delta_{S_N} Δ S N r = R ( x , z ) r=R(x,z) r = R ( x , z ) g ↦ N − 1 [ ( 1 − r 2 ) g ′ ′ − ( N − 1 ) r g ′ ] g\mapsto N^{-1}[(1-r^2)g''-(N-1)rg'] g ↦ N − 1 [( 1 − r 2 ) g ′′ − ( N − 1 ) r g ′ ] x ↦ Z ℓ , N ( R ( x , z ) ) x\mapsto Z_{\ell,N}(R(x,z)) x ↦ Z ℓ , N ( R ( x , z )) Δ S N \Delta_{S_N} Δ S N − ℓ ( ℓ + N − 2 ) / N -\ell(\ell+N-2)/N − ℓ ( ℓ + N − 2 ) / N ℓ \ell ℓ ℓ \ell ℓ z z z ℓ \ell ℓ R ( x , z ) R(x,z) R ( x , z ) ℓ \ell ℓ ℓ \ell ℓ Z ℓ , N ( R ( ⋅ , z ) ) Z_{\ell,N}(R(\cdot,z)) Z ℓ , N ( R ( ⋅ , z ))
The expansion (173) . By (171) , ω p , j , N Z p − 2 j , N = p ! Γ ( α N ) ( p − 2 j + α N ) ( 2 p j ! Γ ( p − j + α N + 1 ) ) − 1 C p − 2 j ( α N ) \omega_{p,j,N}Z_{p-2j,N}=p!\,\Gamma(\alpha_N)(p-2j+\alpha_N)(2^{p}j!\,\Gamma(p-j+\alpha_N+1))^{-1}C_{p-2j}^{(\alpha_N)} ω p , j , N Z p − 2 j , N = p ! Γ ( α N ) ( p − 2 j + α N ) ( 2 p j ! Γ ( p − j + α N + 1 ) ) − 1 C p − 2 j ( α N ) C p − 2 j ( α N ) C_{p-2j}^{(\alpha_N)} C p − 2 j ( α N ) r p − 2 s r^{p-2s} r p − 2 s (173) is
p ! 2 2 s ( p − 2 s ) ! ∑ j = 0 s ( − 1 ) s − j ( p − 2 j + α N ) Γ ( p − s − j + α N ) j ! ( s − j ) ! Γ ( p − j + α N + 1 ) \frac{p!}{2^{2s}(p-2s)!}\sum_{j=0}^{s}\frac{(-1)^{s-j}(p-2j+\alpha_N)\Gamma(p-s-j+\alpha_N)}{j!(s-j)!\Gamma(p-j+\alpha_N+1)} 2 2 s ( p − 2 s )! p ! j = 0 ∑ s j ! ( s − j )! Γ ( p − j + α N + 1 ) ( − 1 ) s − j ( p − 2 j + α N ) Γ ( p − s − j + α N ) For s = 0 s=0 s = 0 s ≥ 1 s\ge1 s ≥ 1
Γ ( p − s − j + α N ) Γ ( p − j + α N + 1 ) = 1 s ! ∫ 0 1 t p − s − j + α N − 1 ( 1 − t ) s d t . \frac{\Gamma(p-s-j+\alpha_N)}{\Gamma(p-j+\alpha_N+1)}=\frac{1}{s!}\int_0^1 t^{p-s-j+\alpha_N-1}(1-t)^s\,\mathrm{d}t. Γ ( p − j + α N + 1 ) Γ ( p − s − j + α N ) = s ! 1 ∫ 0 1 t p − s − j + α N − 1 ( 1 − t ) s d t . Writing y = 1 / t y=1/t y = 1/ t y y y
∑ j = 0 s ( − 1 ) s − j ( p + α N − 2 j ) t − j j ! ( s − j ) ! = t − s ( 1 − t ) s − 1 s ! [ ( p + α N ) ( 1 − t ) − 2 s ] . \sum_{j=0}^{s}\frac{(-1)^{s-j}(p+\alpha_N-2j)t^{-j}}{j!(s-j)!}=\frac{t^{-s}(1-t)^{s-1}}{s!}\left[(p+\alpha_N)(1-t)-2s\right]. j = 0 ∑ s j ! ( s − j )! ( − 1 ) s − j ( p + α N − 2 j ) t − j = s ! t − s ( 1 − t ) s − 1 [ ( p + α N ) ( 1 − t ) − 2 s ] . Inserting these two identities, we find that the sum over j j j ( s ! ) − 2 (s!)^{-2} ( s ! ) − 2 [ 0 , 1 ] [0,1] [ 0 , 1 ]
t p − 2 s + α N − 1 ( 1 − t ) 2 s − 1 [ ( p + α N ) ( 1 − t ) − 2 s ] = d d t [ t p − 2 s + α N ( 1 − t ) 2 s ] . t^{p-2s+\alpha_N-1}(1-t)^{2s-1}\left[(p+\alpha_N)(1-t)-2s\right]=\frac{\mathrm{d}}{\mathrm{d}t}\left[t^{p-2s+\alpha_N}(1-t)^{2s}\right]. t p − 2 s + α N − 1 ( 1 − t ) 2 s − 1 [ ( p + α N ) ( 1 − t ) − 2 s ] = d t d [ t p − 2 s + α N ( 1 − t ) 2 s ] . Both boundary values vanish, because p − 2 s + α N > 0 p-2s+\alpha_N>0 p − 2 s + α N > 0 2 s > 0 2s>0 2 s > 0 (173) . The coefficients are positive by their formula, and evaluating (173) at r = 1 r=1 r = 1
Asymptotics. For fixed p p p j j j ω p , j , N \omega_{p,j,N} ω p , j , N α N + c \alpha_N+c α N + c 2 α N + c 2\alpha_N+c 2 α N + c c c c α N + c = α N ( 1 + c / α N ) \alpha_N+c=\alpha_N(1+c/\alpha_N) α N + c = α N ( 1 + c / α N )
ω p , j , N = p ! 2 2 j j ! ( p − 2 j ) ! α N − j ( 1 + O p ( α N − 1 ) ) , \omega_{p,j,N}=\frac{p!}{2^{2j}j!(p-2j)!}\alpha_N^{-j}(1+O_p(\alpha_N^{-1})), ω p , j , N = 2 2 j j ! ( p − 2 j )! p ! α N − j ( 1 + O p ( α N − 1 )) , which is (174) , since α N − j = 2 j N − j ( 1 + O j ( N − 1 ) ) \alpha_N^{-j}=2^jN^{-j}(1+O_j(N^{-1})) α N − j = 2 j N − j ( 1 + O j ( N − 1 )) Z ℓ , N Z_{\ell,N} Z ℓ , N 2 ℓ ( α N ) ℓ / ( 2 α N ) ℓ = 1 + O ℓ ( N − 1 ) 2^\ell(\alpha_N)_\ell/(2\alpha_N)_\ell=1+O_\ell(N^{-1}) 2 ℓ ( α N ) ℓ / ( 2 α N ) ℓ = 1 + O ℓ ( N − 1 ) r ℓ − 2 j r^{\ell-2j} r ℓ − 2 j O ℓ ( N − j ) O_\ell(N^{-j}) O ℓ ( N − j ) j ≥ 1 j\ge1 j ≥ 1
The harmonic components. The decomposition of H N , p ∣ S N H_{N,p}|_{S_N} H N , p ∣ S N N c p R ( x , y ) p = N c p ∑ j ω p , j , N Z p − 2 j , N ( R ( x , y ) ) N c_p R(x,y)^p=N c_p\sum_j\omega_{p,j,N}Z_{p-2j,N}(R(x,y)) N c p R ( x , y ) p = N c p ∑ j ω p , j , N Z p − 2 j , N ( R ( x , y )) Z ℓ , N ( R ( x , y ) ) Z_{\ell,N}(R(x,y)) Z ℓ , N ( R ( x , y )) ℓ \ell ℓ L 2 ( ν N ) L^2(\nu_N) L 2 ( ν N ) Δ S N \Delta_{S_N} Δ S N − ℓ ( ℓ + N − 2 ) / N -\ell(\ell+N-2)/N − ℓ ( ℓ + N − 2 ) / N ℓ ′ \ell' ℓ ′ ℓ = ℓ ′ \ell=\ell' ℓ = ℓ ′ (176) , and it shows that components of different degrees, or of different p p p
The next lemma is a refinement of Lemma 3.2 for the finite mixtures of this section: it also controls the lower harmonic components, at their own scale. For a component H p , ℓ H_{p,\ell} H p , ℓ ℓ \ell ℓ R N \mathbb{R}^{N} R N
Lemma 8.6 (Regularity with exponential tails). There exist C 0 C_0 C 0 N 0 N_0 N 0 ξ \xi ξ Q Q Q Ω N \Omega_N Ω N J J J P ( Ω N c ) ≤ e − N \mathbb{P}(\Omega_N^c)\le e^{-N} P ( Ω N c ) ≤ e − N N ≥ N 0 N\ge N_0 N ≥ N 0 Ω N \Omega_N Ω N H ↦ H ∘ g − 1 H\mapsto H\circ g^{-1} H ↦ H ∘ g − 1 g ∈ O ( N ) g\in O(N) g ∈ O ( N ) Ω N \Omega_N Ω N
For every p ∈ P p\in\mathcal{P} p ∈ P m ∈ { 0 , 1 , 2 } m\in\{0,1,2\} m ∈ { 0 , 1 , 2 } sup x ∈ S N ∥ ∇ E m H N , p ( x ) ∥ o p ≤ C 0 N 1 − m / 2 \sup_{x\in S_N}\lVert\nabla_E^m H_{N,p}(x)\rVert_{\mathrm{op}}\le C_0N^{1-m/2} sup x ∈ S N ∥ ∇ E m H N , p ( x ) ∥ op ≤ C 0 N 1 − m /2
For every p ∈ P p\in\mathcal{P} p ∈ P 0 ≤ j ≤ p / 2 0\le j\le p/2 0 ≤ j ≤ p /2 sup x ∈ S N ∣ H p , p − 2 j ( x ) ∣ ≤ C 0 N 1 − j / 2 \sup_{x\in S_N}|H_{p,p-2j}(x)|\le C_0N^{1-j/2} sup x ∈ S N ∣ H p , p − 2 j ( x ) ∣ ≤ C 0 N 1 − j /2
With ∥ ⋅ ∥ ∞ \lVert\cdot\rVert_\infty ∥ ⋅ ∥ ∞ S N S_N S N
∥ H N ∥ ∞ + ∥ H Q ∥ ∞ + ∥ K Q ∥ ∞ ≤ C 0 N , \lVert H_N\rVert_\infty+\lVert H_Q\rVert_\infty+\lVert K_Q\rVert_\infty\le C_0N, ∥ H N ∥ ∞ + ∥ H Q ∥ ∞ + ∥ K Q ∥ ∞ ≤ C 0 N , ∥ ∇ E H N ∥ ∞ + ∥ ∇ E H Q ∥ ∞ + ∥ ∇ H N ∥ ∞ + ∥ ∇ H Q ∥ ∞ ≤ C 0 N , \lVert\nabla_EH_N\rVert_\infty+\lVert\nabla_EH_Q\rVert_\infty+\lVert\nabla H_N\rVert_\infty+\lVert\nabla H_Q\rVert_\infty\le C_0\sqrt{N}, ∥ ∇ E H N ∥ ∞ + ∥ ∇ E H Q ∥ ∞ + ∥ ∇ H N ∥ ∞ + ∥ ∇ H Q ∥ ∞ ≤ C 0 N , ∥ ∇ E 2 H N ∥ ∞ + ∥ ∇ E 2 H Q ∥ ∞ + ∥ ∇ 2 H N ∥ ∞ + ∥ ∇ 2 H Q ∥ ∞ ≤ C 0 . (177) \lVert\nabla_E^2H_N\rVert_\infty+\lVert\nabla_E^2H_Q\rVert_\infty+\lVert\nabla^2H_N\rVert_\infty+\lVert\nabla^2H_Q\rVert_\infty\le C_0. \tag*{(177)} ∥ ∇ E 2 H N ∥ ∞ + ∥ ∇ E 2 H Q ∥ ∞ + ∥ ∇ 2 H N ∥ ∞ + ∥ ∇ 2 H Q ∥ ∞ ≤ C 0 . (177) where the norms of second derivatives are operator norms.
Proof. We first treat a component of degree zero, which occurs as H p , 0 H_{p,0} H p , 0 p p p Z Z Z Var Z = N σ 2 \operatorname{Var} Z=N\sigma^2 Var Z = N σ 2 σ 2 = c p ω p , p / 2 , N \sigma^2=c_p\omega_{p,p/2,N} σ 2 = c p ω p , p /2 , N P ( ∣ Z ∣ > 2 L σ N ) ≤ 2 e − 2 L 2 N \mathbb{P}(|Z|>2L\sigma N)\le2e^{-2L^2N} P ( ∣ Z ∣ > 2 L σ N ) ≤ 2 e − 2 L 2 N L > 0 L>0 L > 0 ∣ Z ∣ ≤ 2 L σ N ≤ C L N 1 − p / 4 |Z|\le2L\sigma N\le CLN^{1-p/4} ∣ Z ∣ ≤ 2 L σ N ≤ C L N 1 − p /4 j = p / 2 j=p/2 j = p /2
Now let T T T 1 ≤ ℓ ≤ p ˉ 1\le\ell\le\bar p 1 ≤ ℓ ≤ p ˉ R N \mathbb{R}^{N} R N E T ( x ) 2 = N σ 2 \mathbb{E}T(x)^2=N\sigma^2 E T ( x ) 2 = N σ 2 ∣ x ∣ 2 = N |x|^2=N ∣ x ∣ 2 = N E T ( y ) 2 = N σ 2 ( ∣ y ∣ 2 / N ) ℓ \mathbb{E}T(y)^2=N\sigma^2(|y|^2/N)^\ell E T ( y ) 2 = N σ 2 ( ∣ y ∣ 2 / N ) ℓ y y y T \mathcal{T} T ℓ \ell ℓ T ( y ) = T ( y , … , y ) T(y)=\mathcal{T}(y,\ldots,y) T ( y ) = T ( y , … , y ) v 1 , … , v ℓ v_1,\ldots,v_\ell v 1 , … , v ℓ
T ( v 1 , … , v ℓ ) = 1 2 ℓ ℓ ! ∑ ε ∈ { ± 1 } ℓ ( ∏ i ε i ) T ( ∑ i ε i v i ) \mathcal{T}(v_1,\ldots,v_\ell)=\frac{1}{2^\ell\ell!}\sum_{\varepsilon\in\{\pm1\}^\ell}\left(\prod_i\varepsilon_i\right)T\left(\sum_i\varepsilon_i v_i\right) T ( v 1 , … , v ℓ ) = 2 ℓ ℓ ! 1 ε ∈ { ± 1 } ℓ ∑ ( i ∏ ε i ) T ( i ∑ ε i v i ) and ∣ ∑ i ε i v i ∣ ≤ ℓ |\sum_i\varepsilon_i v_i|\le\ell ∣ ∑ i ε i v i ∣ ≤ ℓ E T ( v 1 , … , v ℓ ) 2 ≤ D ℓ 2 σ 2 N 1 − ℓ \mathbb{E}\mathcal{T}(v_1,\ldots,v_\ell)^2\le D_\ell^2\sigma^2N^{1-\ell} E T ( v 1 , … , v ℓ ) 2 ≤ D ℓ 2 σ 2 N 1 − ℓ D ℓ = ℓ ℓ / ℓ ! D_\ell=\ell^\ell/\ell! D ℓ = ℓ ℓ / ℓ ! N \mathcal{N} N ( 4 p ˉ ) − 1 (4\bar p)^{-1} ( 4 p ˉ ) − 1 ∣ N ∣ ≤ ( 1 + 8 p ˉ ) N |\mathcal{N}|\le(1+8\bar p)^N ∣ N ∣ ≤ ( 1 + 8 p ˉ ) N ( 4 p ˉ ) − 1 ∥ T ∥ o p (4\bar{p})^{-1}\lVert T\rVert_{\mathrm{op}} ( 4 p ˉ ) − 1 ∥ T ∥ op ∥ T ∥ o p ≤ 4 3 max v i ∈ N ∣ T ( v 1 , … , v ℓ ) ∣ \lVert T\rVert_{\mathrm{op}} \le\frac{4}{3}\max_{v_i\in\mathcal{N}}|T(v_1,\ldots,v_\ell)| ∥ T ∥ op ≤ 3 4 max v i ∈ N ∣ T ( v 1 , … , v ℓ ) ∣ ( 1 + 8 p ˉ ) ℓ N (1+8\bar{p})^{\ell N} ( 1 + 8 p ˉ ) ℓ N L > 0 L>0 L > 0
P ( ∥ T ∥ o p > 2 L D ℓ σ N 1 − ℓ / 2 ) ≤ 2 exp ( − N [ L 2 / 2 − p ˉ log ( 1 + 8 p ˉ ) ] ) . (178) \mathbb{P}\left(\lVert T\rVert_{\mathrm{op}}>2LD_\ell\sigma N^{1-\ell/2}\right)\le2\exp\left(-N\left[L^2/2-\bar{p}\log(1+8\bar{p})\right]\right). \tag*{(178)} P ( ∥ T ∥ op > 2 L D ℓ σ N 1 − ℓ /2 ) ≤ 2 exp ( − N [ L 2 /2 − p ˉ log ( 1 + 8 p ˉ ) ] ) . (178) For ∣ x ∣ = N |x|=\sqrt{N} ∣ x ∣ = N 0 ≤ m ≤ ℓ 0\le m\le\ell 0 ≤ m ≤ ℓ m m m T T T x x x ( ℓ m ) T ( x , … , x , ⋅ , … , ⋅ ) \binom{\ell}{m}T(x,\ldots,x,\cdot,\ldots,\cdot) ( m ℓ ) T ( x , … , x , ⋅ , … , ⋅ ) ( ℓ m ) = ℓ ( ℓ − 1 ) ⋯ ( ℓ − m + 1 ) \binom{\ell}{m}=\ell(\ell-1)\cdots(\ell-m+1) ( m ℓ ) = ℓ ( ℓ − 1 ) ⋯ ( ℓ − m + 1 ) ( ℓ m ) ∥ T ∥ o p N ( ℓ − m ) / 2 \binom{\ell}{m}\lVert T\rVert_{\mathrm{op}}N^{(\ell-m)/2} ( m ℓ ) ∥ T ∥ op N ( ℓ − m ) /2 (178) , it is at most 2 L D ℓ ( ℓ m ) σ N 1 − m / 2 2LD_\ell\binom{\ell}{m}\sigma N^{1-m/2} 2 L D ℓ ( m ℓ ) σ N 1 − m /2 m > ℓ m>\ell m > ℓ
We use these bounds for each H N , p H_{N,p} H N , p σ 2 = c p \sigma^2=c_p σ 2 = c p H p , p − 2 j H_{p,p-2j} H p , p − 2 j σ 2 = c p ω p , j , N ≤ C N − j \sigma^2=c_p\omega_{p,j,N}\le CN^{-j} σ 2 = c p ω p , j , N ≤ C N − j (176) and (174) . This is a fixed finite number M 0 M_0 M 0 L L L L 2 / 2 ≥ p ˉ log ( 1 + 8 p ˉ ) + log ( 2 M 0 ) + 2 L^2/2\ge\bar{p}\log(1+8\bar{p})+\log(2M_0)+2 L 2 /2 ≥ p ˉ log ( 1 + 8 p ˉ ) + log ( 2 M 0 ) + 2 M 0 M_0 M 0 e − N e^{-N} e − N N N N L L L Ω N \Omega_N Ω N C 0 C_0 C 0 J J J Ω N \Omega_N Ω N
Part (3) follows by summing the components with the weights 1 1 1 Q p / 2 Q^{p/2} Q p /2 p Q p / 2 pQ^{p/2} p Q p /2 (164) . For the spherical derivatives, recall that ∇ T = Π x ∇ E T \nabla T=\Pi_x\nabla_E T ∇ T = Π x ∇ E T
∇ 2 T ( x ) = Π x ∇ E 2 T ( x ) Π x − x ⋅ ∇ E T ( x ) N Π x ⊥ , \nabla^2T(x)=\Pi_x\nabla_E^2T(x)\Pi_x-\frac{x\cdot\nabla_E T(x)}{N}\Pi_{x^\perp}, ∇ 2 T ( x ) = Π x ∇ E 2 T ( x ) Π x − N x ⋅ ∇ E T ( x ) Π x ⊥ , and that x ⋅ ∇ E T = ℓ T x\cdot\nabla_E T=\ell T x ⋅ ∇ E T = ℓ T ℓ \ell ℓ C 0 C_0 C 0
Exact regression of the whole field
In this section, we show that the field of Definition 8.1 has the conditional law, and we bound the correction δ h , k , u \delta_{h,k,u} δ h , k , u H N H_N H N Q \sqrt{Q} Q S N S_N S N
Proposition 8.7 (Whole-field regression). Let u ∈ S N u\in S_N u ∈ S N r = x ⋅ u / N r=x\cdot u/N r = x ⋅ u / N x ∈ R N x\in\mathbb{R}^N x ∈ R N
If ξ \xi ξ det Σ Q > 0 \det\Sigma_Q>0 det Σ Q > 0 H N − D u H_N-D_u H N − D u ( H Q ( u ) , K Q ( u ) , G u ) (H_Q(u),K_Q(u),G_u) ( H Q ( u ) , K Q ( u ) , G u ) ξ \xi ξ p p p K Q ( u ) = p H Q ( u ) K_Q(u)=pH_Q(u) K Q ( u ) = p H Q ( u ) H N − D u H_N-D_u H N − D u ( H Q ( u ) , G u ) (H_Q(u),G_u) ( H Q ( u ) , G u ) H h , k , u H^{h,k,u} H h , k , u H N H_N H N H Q ( u ) = N h H_Q(u)=Nh H Q ( u ) = N h K Q ( u ) = N k K_Q(u)=Nk K Q ( u ) = N k G u = 0 G_u=0 G u = 0
For ∣ r ∣ ≤ 1 |r|\le1 ∣ r ∣ ≤ 1
∣ θ 1 ( r ) ∣ + ∣ θ 2 ( r ) ∣ ≤ C ∣ r ∣ d , ∣ θ 3 ′ ( r ) ∣ ≤ C ∣ r ∣ d − 2 , |\theta_1(r)|+|\theta_2(r)|\le C|r|^d,\qquad|\theta_3'(r)|\le C|r|^{d-2}, ∣ θ 1 ( r ) ∣ + ∣ θ 2 ( r ) ∣ ≤ C ∣ r ∣ d , ∣ θ 3 ′ ( r ) ∣ ≤ C ∣ r ∣ d − 2 , ∣ θ 1 ′ ( r ) ∣ + ∣ θ 2 ′ ( r ) ∣ + ∣ θ 3 ( r ) ∣ ≤ C ∣ r ∣ d − 1 . (179) |\theta_1'(r)|+|\theta_2'(r)|+|\theta_3(r)|\le C|r|^{d-1}. \tag*{(179)} ∣ θ 1 ′ ( r ) ∣ + ∣ θ 2 ′ ( r ) ∣ + ∣ θ 3 ( r ) ∣ ≤ C ∣ r ∣ d − 1 . (179) The correction has spherical gradient
∇ δ h , k , u ( x ) = [ θ 1 ′ ( r ) ( N h − H Q ( u ) ) + θ 2 ′ ( r ) ( N k − K Q ( u ) ) − θ 3 ′ ( r ) G u ⋅ x ] Π x u N − θ 3 ( r ) Π x G u (180) \nabla\delta_{h,k,u}(x)=\left[\theta'_1(r)(Nh-H_Q(u))+\theta'_2(r)(Nk-K_Q(u))-\theta'_3(r)G_u\cdot x\right]\frac{\Pi_xu}{N}-\theta_3(r)\Pi_xG_u \tag*{(180)} ∇ δ h , k , u ( x ) = [ θ 1 ′ ( r ) ( N h − H Q ( u )) + θ 2 ′ ( r ) ( N k − K Q ( u )) − θ 3 ′ ( r ) G u ⋅ x ] N Π x u − θ 3 ( r ) Π x G u (180) at x ∈ S N x\in S_N x ∈ S N Ω N \Omega_N Ω N ( h , k ) (h,k) ( h , k ) x ∈ S N x\in S_N x ∈ S N
∣ δ h , k , u ( x ) ∣ ≤ C N ∣ r ∣ d + C ∣ r ∣ d − 1 ∣ G u ⋅ x ∣ , ∥ ∇ δ h , k , u ∥ ∞ ≤ C N . (181) |\delta_{h,k,u}(x)|\le CN|r|^d+C|r|^{d-1}|G_u\cdot x|,\qquad\|\nabla\delta_{h,k,u}\|_\infty\le C\sqrt{N}. \tag*{(181)} ∣ δ h , k , u ( x ) ∣ ≤ C N ∣ r ∣ d + C ∣ r ∣ d − 1 ∣ G u ⋅ x ∣ , ∥∇ δ h , k , u ∥ ∞ ≤ C N . (181) Proof. We begin with the covariances. Recall that E H N ( x ) H N ( y ) = N ξ ( x ⋅ y / N ) \mathbb{E}H_N(x)H_N(y)=N\xi(x\cdot y/N) E H N ( x ) H N ( y ) = N ξ ( x ⋅ y / N ) x , y ∈ R N x,y\in\mathbb{R}^N x , y ∈ R N y y y t Q u t\sqrt{Q}\,u t Q u e ⊥ u e\perp u e ⊥ u x ∈ R N x\in\mathbb{R}^N x ∈ R N
Cov ( ( H Q ( u ) , K Q ( u ) ) ) = N Σ Q , Cov ( G u ) = a Q I u ⊥ , Cov ( ( H Q ( u ) , K Q ( u ) ) , G u ) = 0 , Cov ( H N ( x ) , G u ) = Q ξ ′ ( Q r ) Π u x , Cov ( H N ( x ) , ( H Q ( u ) , K Q ( u ) ) ) = N ( ξ ( Q r ) , Q r ξ ′ ( Q r ) ) . (182) \begin{aligned}
\operatorname{Cov}((H_Q(u),K_Q(u)))&=N\Sigma_Q,\qquad\operatorname{Cov}(G_u)=a_QI_{u^\perp},\\
\operatorname{Cov}((H_Q(u),K_Q(u)),G_u)&=0,\qquad\operatorname{Cov}(H_N(x),G_u)=\sqrt{Q}\,\xi'(\sqrt{Q}r)\Pi_{u}x,\\
\operatorname{Cov}(H_N(x),(H_Q(u),K_Q(u)))&=N\bigl(\xi(\sqrt{Q}r),\sqrt{Q}r\xi'(\sqrt{Q}r)\bigr). \tag*{(182)}
\end{aligned} Cov (( H Q ( u ) , K Q ( u ))) Cov (( H Q ( u ) , K Q ( u )) , G u ) Cov ( H N ( x ) , ( H Q ( u ) , K Q ( u ))) = N Σ Q , Cov ( G u ) = a Q I u ⊥ , = 0 , Cov ( H N ( x ) , G u ) = Q ξ ′ ( Q r ) Π u x , = N ( ξ ( Q r ) , Q r ξ ′ ( Q r ) ) . (182) For example, the identity for Cov ( H N ( x ) , G u ) \operatorname{Cov}(H_N(x),G_u) Cov ( H N ( x ) , G u ) N ξ ( Q x ⋅ v / N ) N\xi(\sqrt{Q}\,x\cdot v/N) N ξ ( Q x ⋅ v / N ) v v v v = u v=u v = u e ⊥ u e\perp u e ⊥ u Q ξ ′ ( Q r ) x ⋅ e \sqrt{Q}\,\xi'(\sqrt{Q}r)\,x\cdot e Q ξ ′ ( Q r ) x ⋅ e G u G_u G u ∂ e ∂ e ′ N ξ ( Q v ⋅ v ′ / N ) = Q ξ ′ ( Q ) e ⋅ e ′ \partial_e\partial_{e'}N\xi(Qv\cdot v'/N)=Q\xi'(Q)e\cdot e' ∂ e ∂ e ′ N ξ ( Q v ⋅ v ′ / N ) = Q ξ ′ ( Q ) e ⋅ e ′ v = v ′ = u v=v'=u v = v ′ = u u ⋅ e = u ⋅ e ′ = 0 u\cdot e=u\cdot e'=0 u ⋅ e = u ⋅ e ′ = 0 ( H Q ( u ) , K Q ( u ) ) (H_Q(u),K_Q(u)) ( H Q ( u ) , K Q ( u )) G u G_u G u w p = c p Q p w_p=c_pQ^p w p = c p Q p ξ ( Q ) = ∑ p w p \xi(Q)=\sum_p w_p ξ ( Q ) = ∑ p w p a Q = ∑ p p w p a_Q=\sum_p pw_p a Q = ∑ p p w p a Q + b Q = ∑ p p 2 w p a_Q+b_Q=\sum_p p^2w_p a Q + b Q = ∑ p p 2 w p
det Σ Q = ( ∑ p w p ) ( ∑ p p 2 w p ) − ( ∑ p p w p ) 2 > 0 \det\Sigma_Q=\left(\sum_p w_p\right)\left(\sum_p p^2w_p\right)-\left(\sum_p pw_p\right)^2>0 det Σ Q = ( p ∑ w p ) ( p ∑ p 2 w p ) − ( p ∑ p w p ) 2 > 0 by the Cauchy–Schwarz inequality, with equality exactly when only one degree occurs. Consider a non-pure mixture. By (182) , D u ( x ) D_u(x) D u ( x ) H N ( x ) H_N(x) H N ( x ) ( H Q ( u ) , K Q ( u ) , G u ) (H_Q(u),K_Q(u),G_u) ( H Q ( u ) , K Q ( u ) , G u ) H Q ( u ) H_Q(u) H Q ( u ) K Q ( u ) K_Q(u) K Q ( u ) N ( ξ ( Q r ) , Q r ξ ′ ( Q r ) ) ( N Σ Q ) − 1 = ( θ 1 ( r ) , θ 2 ( r ) ) N(\xi(\sqrt{Q}r),\sqrt{Q}r\xi'(\sqrt{Q}r))(N\Sigma_Q)^{-1}=(\theta_1(r),\theta_2(r)) N ( ξ ( Q r ) , Q r ξ ′ ( Q r )) ( N Σ Q ) − 1 = ( θ 1 ( r ) , θ 2 ( r )) G u G_u G u Q ξ ′ ( Q r ) Π u x / a Q \sqrt{Q}\,\xi'(\sqrt{Q}r)\Pi_ux/a_Q Q ξ ′ ( Q r ) Π u x / a Q Π u x ⋅ G u = x ⋅ G u \Pi_ux\cdot G_u=x\cdot G_u Π u x ⋅ G u = x ⋅ G u { ( H N − D u ) ( x ) : x ∈ R N } \{(H_N-D_u)(x):x\in\mathbb{R}^N\} {( H N − D u ) ( x ) : x ∈ R N } ( H Q ( u ) , K Q ( u ) , G u ) (H_Q(u),K_Q(u),G_u) ( H Q ( u ) , K Q ( u ) , G u ) p ˉ \bar p p ˉ H N = ( H N − D u ) + D u H_N=(H_N-D_u)+D_u H N = ( H N − D u ) + D u D u D_u D u ( H Q ( u ) , K Q ( u ) , G u ) (H_Q(u),K_Q(u),G_u) ( H Q ( u ) , K Q ( u ) , G u ) ( N h , N k , 0 ) (Nh,Nk,0) ( N h , N k , 0 ) D u D_u D u H N = H N , p H_N=H_{N,p} H N = H N , p p p p K Q ( u ) = p H Q ( u ) K_Q(u)=pH_Q(u) K Q ( u ) = p H Q ( u ) Var H Q ( u ) = N c p Q p \operatorname{Var}H_Q(u)=Nc_pQ^p Var H Q ( u ) = N c p Q p Cov ( H N ( x ) , H Q ( u ) ) = N c p Q p / 2 r p \operatorname{Cov}(H_N(x),H_Q(u))=Nc_pQ^p/2r_p Cov ( H N ( x ) , H Q ( u )) = N c p Q p /2 r p (182) , the regression coefficients on H Q ( u ) H_Q(u) H Q ( u ) G u G_u G u Q − p / 2 r p Q^{-p/2}r^p Q − p /2 r p Q − p / 2 r p − 1 Q^{-p/2}r^{p-1} Q − p /2 r p − 1
For (2), note that θ 1 , θ 2 \theta_1,\theta_2 θ 1 , θ 2 ξ ( Q r ) \xi(\sqrt{Q}\,r) ξ ( Q r ) Q r ξ ′ ( Q r ) \sqrt{Q}\,r\,\xi'(\sqrt{Q}\,r) Q r ξ ′ ( Q r ) r r r r d r^d r d θ 3 \theta_3 θ 3 ξ ′ ( Q r ) \xi'(\sqrt{Q}\,r) ξ ′ ( Q r ) r d − 1 r^{d-1} r d − 1 p ˉ \bar p p ˉ
For (3), the identity (8.17) is the product rule, with ∇ r = Π x u / N \nabla r=\Pi_xu/N ∇ r = Π x u / N ∇ ( G u ⋅ x ) = Π x G u \nabla(G_u\cdot x)=\Pi_xG_u ∇ ( G u ⋅ x ) = Π x G u Ω N \Omega_N Ω N ∣ N h − H Q ( u ) ∣ ≤ ( h ˉ + C 0 ) N |Nh-H_Q(u)|\le(\bar h+C_0)N ∣ N h − H Q ( u ) ∣ ≤ ( h ˉ + C 0 ) N ∣ N k − K Q ( u ) ∣ ≤ ( k ˉ + C 0 ) N |Nk-K_Q(u)|\le(\bar k+C_0)N ∣ N k − K Q ( u ) ∣ ≤ ( k ˉ + C 0 ) N ∣ G u ∣ ≤ C 0 N |G_u|\le C_0\sqrt{N} ∣ G u ∣ ≤ C 0 N ∣ Π x u ∣ / N ≤ N − 1 / 2 |\Pi_xu|/N\le N^{-1/2} ∣ Π x u ∣/ N ≤ N − 1/2 ∣ G u ⋅ x ∣ ≤ C 0 N |G_u\cdot x|\le C_0N ∣ G u ⋅ x ∣ ≤ C 0 N N ( θ 1 h + θ 2 k ) N(\theta_1h+\theta_2k) N ( θ 1 h + θ 2 k ) N N N u u u
Latitude concentration and moments of the density ratio
In this section, we bound two moments of the density e w e^w e w Ω N \Omega_N Ω N x x x
Lemma 8.8 (Concentration on the sphere). Let m ≥ 2 m\ge2 m ≥ 2 σ m \sigma_m σ m S m ⊂ R m + 1 S^m\subset\mathbb{R}^{m+1} S m ⊂ R m + 1 g : S m → R g:S^m\to\mathbb{R} g : S m → R L L L Then
∫ e t ( g − ∫ g d σ m ) d σ m ≤ exp ( L 2 t 2 2 ( m − 1 ) ) , t ∈ R . (183) \int e^{t(g-\int g\,d\sigma_m)}\,d\sigma_m\le\exp\left(\frac{L^2t^2}{2(m-1)}\right),\qquad t\in\mathbb{R}. \tag*{(183)} ∫ e t ( g − ∫ g d σ m ) d σ m ≤ exp ( 2 ( m − 1 ) L 2 t 2 ) , t ∈ R . (183) Proof. For a function q > 0 q>0 q > 0 Ent ( q ) = ∫ q log q d σ m − ∫ q d σ m log ∫ q d σ m \operatorname{Ent}(q)=\int q\log q\,d\sigma_m-\int q\,d\sigma_m\log\int q\,d\sigma_m Ent ( q ) = ∫ q log q d σ m − ∫ q d σ m log ∫ q d σ m I ( q ) = ∫ ∣ ∇ q ∣ 2 / q d σ m \mathcal{I}(q)=\int|\nabla q|^2/q\,d\sigma_m I ( q ) = ∫ ∣∇ q ∣ 2 / q d σ m
Ent ( q ) ≤ I ( q ) 2 ( m − 1 ) (184) \operatorname{Ent}(q)\le\frac{\mathcal{I}(q)}{2(m-1)} \tag*{(184)} Ent ( q ) ≤ 2 ( m − 1 ) I ( q ) (184) for smooth q > 0 q>0 q > 0 q t = e t Δ q q_t=e^{t\Delta}q q t = e t Δ q S m S^m S m d d t Ent ( q t ) = − I ( q t ) \frac{d}{dt}\operatorname{Ent}(q_t)=-\mathcal{I}(q_t) d t d Ent ( q t ) = − I ( q t ) v = log q t v=\log q_t v = log q t ∂ t v = Δ v + ∣ ∇ v ∣ 2 \partial_t v=\Delta v+|\nabla v|^2 ∂ t v = Δ v + ∣∇ v ∣ 2 I ( q t ) = ∫ q t ∣ ∇ v ∣ 2 \mathcal{I}(q_t)=\int q_t|\nabla v|^2 I ( q t ) = ∫ q t ∣∇ v ∣ 2
d d t I ( q t ) = ∫ q t Δ ∣ ∇ v ∣ 2 + 2 ∫ q t ∇ v ⋅ ∇ Δ v + 2 ∫ q t ∇ v ⋅ ∇ ∣ ∇ v ∣ 2 . \frac{d}{dt}\mathcal{I}(q_t)=\int q_t\Delta|\nabla v|^2+2\int q_t\nabla v\cdot\nabla\Delta v+2\int q_t\nabla v\cdot\nabla|\nabla v|^2. d t d I ( q t ) = ∫ q t Δ∣∇ v ∣ 2 + 2 ∫ q t ∇ v ⋅ ∇Δ v + 2 ∫ q t ∇ v ⋅ ∇∣∇ v ∣ 2 . where we moved the Laplacian in the first term. The divergence theorem for q t ∇ ∣ ∇ v ∣ 2 q_t\nabla|\nabla v|^2 q t ∇∣∇ v ∣ 2 ∇ q t = q t ∇ v \nabla q_t=q_t\nabla v ∇ q t = q t ∇ v − 2 ∫ q t Δ ∣ ∇ v ∣ 2 -2\int q_t\Delta|\nabla v|^2 − 2 ∫ q t Δ∣∇ v ∣ 2 1 2 Δ ∣ ∇ v ∣ 2 = ∇ v ⋅ ∇ Δ v + ∣ ∇ 2 v ∣ 2 + Ric ( ∇ v , ∇ v ) \frac{1}{2}\Delta|\nabla v|^2=\nabla v\cdot\nabla\Delta v+|\nabla^2v|^2+\operatorname{Ric}(\nabla v,\nabla v) 2 1 Δ∣∇ v ∣ 2 = ∇ v ⋅ ∇Δ v + ∣ ∇ 2 v ∣ 2 + Ric ( ∇ v , ∇ v ) Ric = ( m − 1 ) g S m \operatorname{Ric}=(m-1)g_{S^m} Ric = ( m − 1 ) g S m
d d t I ( q t ) = − 2 ∫ q t ( ∣ ∇ 2 v ∣ 2 + ( m − 1 ) ∣ ∇ v ∣ 2 ) ≤ − 2 ( m − 1 ) I ( q t ) . \frac{\mathrm{d}}{\mathrm{d}t}\mathcal{I}(q_t)=-2\int q_t\left(|\nabla^2v|^2+(m-1)|\nabla v|^2\right)\leq-2(m-1)\mathcal{I}(q_t). d t d I ( q t ) = − 2 ∫ q t ( ∣ ∇ 2 v ∣ 2 + ( m − 1 ) ∣∇ v ∣ 2 ) ≤ − 2 ( m − 1 ) I ( q t ) . Hence I ( q t ) ≤ e − 2 ( m − 1 ) t I ( q ) \mathcal{I}(q_t)\leq e^{-2(m-1)t}\mathcal{I}(q) I ( q t ) ≤ e − 2 ( m − 1 ) t I ( q ) q t q_t q t ∫ q d σ m \int q\,\mathrm{d}\sigma_m ∫ q d σ m Ent ( q t ) → 0 \operatorname{Ent}(q_t)\to0 Ent ( q t ) → 0 t t t (184) .
Now let g g g ∣ ∇ g ∣ ≤ L |\nabla g|\leq L ∣∇ g ∣ ≤ L ψ ( t ) = log ∫ e t g d σ m \psi(t)=\log\int e^{tg}\,\mathrm{d}\sigma_m ψ ( t ) = log ∫ e t g d σ m (184) to q = e t g q=e^{tg} q = e t g I ( q ) ≤ L 2 t 2 ∫ e t g \mathcal{I}(q)\leq L^2t^2\int e^{tg} I ( q ) ≤ L 2 t 2 ∫ e t g t ψ ′ ( t ) − ψ ( t ) ≤ L 2 t 2 / ( 2 ( m − 1 ) ) t\psi'(t)-\psi(t)\leq L^2t^2/(2(m-1)) t ψ ′ ( t ) − ψ ( t ) ≤ L 2 t 2 / ( 2 ( m − 1 )) ψ ( t ) / t \psi(t)/t ψ ( t ) / t L 2 / ( 2 ( m − 1 ) ) L^2/(2(m-1)) L 2 / ( 2 ( m − 1 )) ψ ( t ) / t → ∫ g d σ m \psi(t)/t\to\int g\,\mathrm{d}\sigma_m ψ ( t ) / t → ∫ g d σ m t ↓ 0 t\downarrow0 t ↓ 0 (183) for t > 0 t>0 t > 0 t < 0 t<0 t < 0 − g -g − g
We now fix x ∈ S N x\in S_N x ∈ S N u u u S N S_N S N r = R ( x , u ) r=R(x,u) r = R ( x , u )
p N ( r ) = c N ( 1 − r 2 ) ( N − 3 ) / 2 , − 1 < r < 1 , c N = Γ ( N / 2 ) π Γ ( ( N − 1 ) / 2 ) ≤ N , (185) p_N(r)=c_N(1-r^2)^{(N-3)/2},\qquad-1<r<1,\qquad c_N=\frac{\Gamma(N/2)}{\sqrt{\pi}\,\Gamma((N-1)/2)}\leq\sqrt{N}, \tag*{(185)} p N ( r ) = c N ( 1 − r 2 ) ( N − 3 ) /2 , − 1 < r < 1 , c N = π Γ (( N − 1 ) /2 ) Γ ( N /2 ) ≤ N , (185) by spherical coordinates and the beta integral; the bound on c N c_N c N Γ ( z + 1 2 ) ≤ z Γ ( z ) \Gamma(z+\frac{1}{2})\leq\sqrt{z}\,\Gamma(z) Γ ( z + 2 1 ) ≤ z Γ ( z ) r r r u = r x + 1 − r 2 y u=rx+\sqrt{1-r^2}\,y u = r x + 1 − r 2 y y y y S N ∩ x ⊥ S_N\cap x^\perp S N ∩ x ⊥ N − 2 N-2 N − 2 N \sqrt{N} N E l a t \mathbb{E}_{\mathrm{lat}} E lat y y y r r r
V x ( u ) = N − 1 / 2 G u ⋅ x = N − 1 / 2 ∇ H Q ( u ) ⋅ x . (186) V_x(u)=N^{-1/2}G_u\cdot x=N^{-1/2}\nabla H_Q(u)\cdot x. \tag*{(186)} V x ( u ) = N − 1/2 G u ⋅ x = N − 1/2 ∇ H Q ( u ) ⋅ x . (186) Lemma 8.9 (The tangent gradient on a latitude). Let N ≥ 6 N\geq6 N ≥ 6 Ω N \Omega_N Ω N x ∈ S N x\in S_N x ∈ S N − 1 < r < 1 -1<r<1 − 1 < r < 1 m x ( r ) = E l a t V x ( u ) m_x(r)=\mathbb{E}_{\mathrm{lat}}V_x(u) m x ( r ) = E lat V x ( u ) Z = V x ( u ) − m x ( r ) Z=V_x(u)-m_x(r) Z = V x ( u ) − m x ( r )
∣ m x ( r ) ∣ ≤ C ( N ∣ r ∣ d − 1 + 1 ) , E l a t e t Z ≤ e C t 2 , t ∈ R . (187) |m_x(r)|\leq C(\sqrt{N}|r|^{d-1}+1),\qquad\mathbb{E}_{\mathrm{lat}}e^{tZ}\leq e^{Ct^2},\qquad t\in\mathbb{R}. \tag*{(187)} ∣ m x ( r ) ∣ ≤ C ( N ∣ r ∣ d − 1 + 1 ) , E lat e tZ ≤ e C t 2 , t ∈ R . (187) The point x x x x x x
Proof. We first prove the concentration. Along a unit-speed curve in S N S_N S N Π u ∇ E H Q ( u ) \Pi_u\nabla E H_Q(u) Π u ∇ E H Q ( u ) 2 ∣ ∇ E H Q ( u ) ∣ / N + ∥ ∇ 2 E H Q ( u ) ∥ o p ≤ 3 C 0 2|\nabla E H_Q(u)|/\sqrt{N}+\|\nabla^2 E H_Q(u)\|_{\mathrm{op}}\leq3C_0 2∣∇ E H Q ( u ) ∣/ N + ∥ ∇ 2 E H Q ( u ) ∥ op ≤ 3 C 0 (177) . Because ∣ x ∣ = N |x|=\sqrt{N} ∣ x ∣ = N V x V_x V x 3 C 0 3C_0 3 C 0 S N S_N S N y ↦ r x + 1 − r 2 y y\mapsto rx+\sqrt{1-r^2}y y ↦ r x + 1 − r 2 y 1 1 1 S N − 2 S^{N-2} S N − 2 y ↦ V x ( u ) y\mapsto V_x(u) y ↦ V x ( u ) 3 C 0 N 3C_0\sqrt{N} 3 C 0 N m = N − 2 m=N-2 m = N − 2 E l a t e t Z ≤ exp ( 9 C 0 2 N t 2 / ( 2 ( N − 3 ) ) ) ≤ e C t 2 \mathbb{E}_{\mathrm{lat}}e^{tZ}\leq\exp(9C_0^2Nt^2/(2(N-3)))\leq e^{Ct^2} E lat e tZ ≤ exp ( 9 C 0 2 N t 2 / ( 2 ( N − 3 ))) ≤ e C t 2
We next compute the mean, keeping all harmonic components. Let Y Y Y ℓ \ell ℓ S N S_N S N x x x ℓ \ell ℓ Y ( x ) Y(x) Y ( x ) x x x Y ( x ) Z ℓ , N ( R ( x , ⋅ ) ) Y(x)Z_{\ell,N}(R(x,\cdot)) Y ( x ) Z ℓ , N ( R ( x , ⋅ )) r r r E l a t Y ( u ) = Z ℓ , N ( r ) Y ( x ) \mathbb{E}_{\mathrm{lat}}Y(u)=Z_{\ell,N}(r)Y(x) E lat Y ( u ) = Z ℓ , N ( r ) Y ( x )
E l a t H Q ( u ) = ∑ p ∈ P Q p / 2 ∑ j = 0 ⌊ p / 2 ⌋ Z p − 2 j , N ( r ) H p , p − 2 j ( x ) . (188) \mathbb{E}_{\mathrm{lat}}H_Q(u)=\sum_{p\in\mathcal{P}}Q^{p/2}\sum_{j=0}^{\lfloor p/2\rfloor}Z_{p-2j,N}(r)H_{p,p-2j}(x). \tag*{(188)} E lat H Q ( u ) = p ∈ P ∑ Q p /2 j = 0 ∑ ⌊ p /2 ⌋ Z p − 2 j , N ( r ) H p , p − 2 j ( x ) . (188) For fixed y y y ∂ r u = ( x − r y ) / 1 − r 2 = ( x − r u ) / ( 1 − r 2 ) \partial_r u=(x-ry)/\sqrt{1-r^2}=(x-ru)/(1-r^2) ∂ r u = ( x − r y ) / 1 − r 2 = ( x − r u ) / ( 1 − r 2 ) ( x − r u ) ⋅ u = 0 (x-ru)\cdot u=0 ( x − r u ) ⋅ u = 0 G u G_u G u u u u
d d r H Q ( u ) = ∇ E H Q ( u ) ⋅ x − r u 1 − r 2 = G u ⋅ x 1 − r 2 = N V x ( u ) 1 − r 2 . \frac{\mathrm{d}}{\mathrm{d}r}H_Q(u)=\nabla_EH_Q(u)\cdot\frac{x-ru}{1-r^2}=\frac{G_u\cdot x}{1-r^2}=\frac{\sqrt{N}V_x(u)}{1-r^2}. d r d H Q ( u ) = ∇ E H Q ( u ) ⋅ 1 − r 2 x − r u = 1 − r 2 G u ⋅ x = 1 − r 2 N V x ( u ) . Since the law of y y y r r r
m x ( r ) = 1 − r 2 N ∑ p ∈ P Q p / 2 ∑ j = 0 ⌊ p / 2 ⌋ Z p − 2 j , N ′ ( r ) H p , p − 2 j ( x ) . (189) m_x(r)=\frac{1-r^2}{\sqrt{N}}\sum_{p\in\mathcal{P}}Q^{p/2}\sum_{j=0}^{\lfloor p/2\rfloor}Z'_{p-2j,N}(r)H_{p,p-2j}(x). \tag*{(189)} m x ( r ) = N 1 − r 2 p ∈ P ∑ Q p /2 j = 0 ∑ ⌊ p /2 ⌋ Z p − 2 j , N ′ ( r ) H p , p − 2 j ( x ) . (189) For j = 0 j=0 j = 0 C N ∣ r ∣ p − 1 + C N − 1 / 2 C\sqrt{N}|r|^{p-1}+CN^{-1/2} C N ∣ r ∣ p − 1 + C N − 1/2 j ≥ 1 j\geq1 j ≥ 1 ∣ H p , p − 2 j ( x ) ∣ ≤ C 0 N |H_{p,p-2j}(x)|\leq C_0\sqrt{N} ∣ H p , p − 2 j ( x ) ∣ ≤ C 0 N ∣ Z p − 2 j , N ′ ∣ ≤ C |Z'_{p-2j,N}|\leq C ∣ Z p − 2 j , N ′ ∣ ≤ C [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] ∣ r ∣ p − 1 ≤ ∣ r ∣ d − 1 |r|^{p-1}\leq|r|^{d-1} ∣ r ∣ p − 1 ≤ ∣ r ∣ d − 1 p ≥ d p\geq d p ≥ d m x ( r ) m_x(r) m x ( r )
We now define the truncation. We fix once and for all an even C ∞ C^\infty C ∞ χ : R → [ 0 , 1 ] \chi:\mathbb{R}\to[0,1] χ : R → [ 0 , 1 ] χ = 1 \chi=1 χ = 1 [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] χ = 0 \chi=0 χ = 0 [ − 2 , 2 ] [-2,2] [ − 2 , 2 ] α > 0 \alpha>0 α > 0 u , x ∈ S N u,x\in S_N u , x ∈ S N ( h , k ) (h,k) ( h , k )
r = R ( x , u ) , s = G u ⋅ x N , r=R(x,u),\qquad s=\frac{G_u\cdot x}{N}, r = R ( x , u ) , s = N G u ⋅ x , δ α ( u , x ) = χ ( r / α ) χ ( s / α ) δ h , k , u ( x ) , w ( u , x ) = β δ α ( u , x ) . (190) \delta_\alpha(u,x)=\chi(r/\alpha)\chi(s/\alpha)\delta_{h,k,u}(x),\qquad w(u,x)=\beta\delta_\alpha(u,x). \tag*{(190)} δ α ( u , x ) = χ ( r / α ) χ ( s / α ) δ h , k , u ( x ) , w ( u , x ) = β δ α ( u , x ) . (190) The function δ α ( u , ⋅ ) \delta_\alpha(u,\cdot) δ α ( u , ⋅ ) R N \mathbb{R}^N R N H Q ( u ) H_Q(u) H Q ( u ) K Q ( u ) K_Q(u) K Q ( u ) G u G_u G u
Proposition 8.10 (Axis-averaged moments of the density ratio). There exist α 0 ∈ ( 0 , 1 / 8 ] \alpha_0\in(0,1/8] α 0 ∈ ( 0 , 1/8 ] C 1 C_1 C 1 N 0 N_0 N 0 α ∈ ( 0 , α 0 ] \alpha\in(0,\alpha_0] α ∈ ( 0 , α 0 ] N ≥ N 0 N\geq N_0 N ≥ N 0 Ω N \Omega_N Ω N ( h , k ) (h,k) ( h , k ) x ∈ S N x\in S_N x ∈ S N u u u S N S_N S N
E u ∣ e w ( u , x ) − 1 ∣ ≤ C 1 N 1 − d / 2 , (191) \mathbb{E}_u\left|e^{w(u,x)}-1\right|\leq C_1N^{1-d/2}, \tag*{(191)} E u e w ( u , x ) − 1 ≤ C 1 N 1 − d /2 , (191) E u [ e w ( u , x ) ( 1 − e − w ( u , x ) ) 2 ] ≤ C 1 N 2 − d . (192) \mathbb{E}_u\left[e^{w(u,x)}\left(1-e^{-w(u,x)}\right)^2\right]\leq C_1N^{2-d}. \tag*{(192)} E u [ e w ( u , x ) ( 1 − e − w ( u , x ) ) 2 ] ≤ C 1 N 2 − d . (192) Moreover, for all u ∈ S N u \in S_N u ∈ S N
sup x ∈ S N ∣ δ α ( u , x ) ∣ ≤ C 1 N α d , ∥ ∇ δ α ( u , ⋅ ) ∥ ∞ ≤ C 1 N α d − 1 . (193) \sup_{x \in S_N} |\delta_\alpha(u,x)| \le C_1N\alpha^d,\qquad\|\nabla\delta_\alpha(u,\cdot)\|_\infty\le C_1\sqrt{N}\alpha^{d-1}. \tag*{(193)} x ∈ S N sup ∣ δ α ( u , x ) ∣ ≤ C 1 N α d , ∥∇ δ α ( u , ⋅ ) ∥ ∞ ≤ C 1 N α d − 1 . (193) The constants α 0 \alpha_0 α 0 C 1 C_1 C 1 N 0 N_0 N 0 α \alpha α
Proof. The proof has three steps. We first bound exponential moments on each latitude, then integrate over the overlap r r r
Moments on a latitude. Fix ζ ≥ 0 \zeta\ge0 ζ ≥ 0 ζ = 2 β \zeta= 2\beta ζ = 2 β ∣ G u ⋅ x ∣ = N ∣ V x ( u ) ∣ |G_u \cdot x| = \sqrt{N}|V_x(u)| ∣ G u ⋅ x ∣ = N ∣ V x ( u ) ∣ (181) , we find
∣ δ h , k , u ( x ) ∣ ≤ C 2 N ∣ r ∣ d + C 2 N ∣ r ∣ d − 1 ∣ V x ( u ) ∣ . (194) |\delta_{h,k,u}(x)| \le C_2N|r|^d + C_2\sqrt{N}|r|^{d-1}|V_x(u)|. \tag*{(194)} ∣ δ h , k , u ( x ) ∣ ≤ C 2 N ∣ r ∣ d + C 2 N ∣ r ∣ d − 1 ∣ V x ( u ) ∣. (194) Here V x V_x V x (187) satisfies P ( ∣ Z ∣ > v ) ≤ 2 e − v 2 / ( 4 C ) \mathbb{P}(|Z| > v) \le2e^{-v^2/(4C)} P ( ∣ Z ∣ > v ) ≤ 2 e − v 2 / ( 4 C ) E l a t Z 4 ≤ C \mathbb{E}_{\mathrm{lat}}Z^4 \le C E lat Z 4 ≤ C E l a t e t ∣ Z ∣ ≤ 2 e C t 2 \mathbb{E}_{\mathrm{lat}}e^{t|Z|} \le2e^{Ct^2} E lat e t ∣ Z ∣ ≤ 2 e C t 2 t ≥ 0 t \ge0 t ≥ 0 E l a t [ Z 2 e t ∣ Z ∣ ] ≤ C e 2 C t 2 \mathbb{E}_{\mathrm{lat}}[Z^2e^{t|Z|}] \le Ce^{2Ct^2} E lat [ Z 2 e t ∣ Z ∣ ] ≤ C e 2 C t 2 V x = m x + Z V_x=m_x+Z V x = m x + Z m x m_x m x (187) into (194) , and apply these estimates with t = ζ C 2 N ∣ r ∣ d − 1 t=\zeta C_2\sqrt{N}|r|^{d-1} t = ζ C 2 N ∣ r ∣ d − 1
E l a t e ζ ∣ δ ∣ ≤ C ζ e Ψ ζ ( r ) , E l a t [ V x 2 e ζ ∣ δ ∣ ] ≤ C ζ ( 1 + N ∣ r ∣ 2 d − 2 ) e Ψ ζ ( r ) , (195) \mathbb{E}_{\mathrm{lat}}e^{\zeta|\delta|} \le C_\zeta e^{\Psi_\zeta(r)},\qquad\mathbb{E}_{\mathrm{lat}}\left[V_x^2e^{\zeta|\delta|}\right] \le C_\zeta(1+N|r|^{2d-2})e^{\Psi_\zeta(r)}, \tag*{(195)} E lat e ζ ∣ δ ∣ ≤ C ζ e Ψ ζ ( r ) , E lat [ V x 2 e ζ ∣ δ ∣ ] ≤ C ζ ( 1 + N ∣ r ∣ 2 d − 2 ) e Ψ ζ ( r ) , (195) where δ = δ h , k , u ( x ) \delta=\delta_{h,k,u}(x) δ = δ h , k , u ( x )
Ψ ζ ( r ) = C ζ ( N ∣ r ∣ d + N ∣ r ∣ 2 d − 2 + N ∣ r ∣ d − 1 ) . \Psi_\zeta(r)=C_\zeta\left(N|r|^d+N|r|^{2d-2}+\sqrt{N}|r|^{d-1}\right). Ψ ζ ( r ) = C ζ ( N ∣ r ∣ d + N ∣ r ∣ 2 d − 2 + N ∣ r ∣ d − 1 ) . In Ψ ζ \Psi_\zeta Ψ ζ t t t m x m_x m x m x m_x m x
Integration over the overlap. For N ≥ 6 N \ge6 N ≥ 6 (185) satisfies p N ( r ) ≤ N e − N r 2 / 4 p_N(r) \le\sqrt{N}e^{-Nr^2/4} p N ( r ) ≤ N e − N r 2 /4 α 0 ≤ 1 / 8 \alpha_0 \le1/8 α 0 ≤ 1/8
C ζ ( ( 2 α 0 ) d − 2 + ( 2 α 0 ) 2 d − 4 + ( 2 α 0 ) d − 3 ) ≤ 1 / 8. C_\zeta\left((2\alpha_0)^{d-2}+(2\alpha_0)^{2d-4}+(2\alpha_0)^{d-3}\right)\le1/8. C ζ ( ( 2 α 0 ) d − 2 + ( 2 α 0 ) 2 d − 4 + ( 2 α 0 ) d − 3 ) ≤ 1/8. Since N ≤ N \sqrt{N}\le N N ≤ N d ≥ 4 d\ge4 d ≥ 4 Ψ ζ ( r ) ≤ N r 2 / 8 \Psi_\zeta(r)\le Nr^2/8 Ψ ζ ( r ) ≤ N r 2 /8 ∣ r ∣ ≤ 2 α 0 |r|\le2\alpha_0 ∣ r ∣ ≤ 2 α 0 j ≥ 0 j\ge0 j ≥ 0 α ≤ α 0 \alpha\le\alpha_0 α ≤ α 0
∫ ∣ r ∣ ≤ 2 α p N ( r ) ∣ r ∣ j e Ψ ζ ( r ) d r ≤ N ∫ R ∣ r ∣ j e − N r 2 / 8 d r ≤ C j N − j / 2 . (196) \int_{|r|\le2\alpha}p_N(r)|r|^j e^{\Psi_\zeta(r)}\,dr\le\sqrt{N}\int_{\mathbb{R}}|r|^j e^{-Nr^2/8}\,dr\le C_jN^{-j/2}. \tag*{(196)} ∫ ∣ r ∣ ≤ 2 α p N ( r ) ∣ r ∣ j e Ψ ζ ( r ) d r ≤ N ∫ R ∣ r ∣ j e − N r 2 /8 d r ≤ C j N − j /2 . (196) Note that δ α 2 e ζ ∣ δ α ∣ ≤ 1 { ∣ r ∣ ≤ 2 α } δ 2 e ζ ∣ δ ∣ \delta_\alpha^2e^{\zeta|\delta_\alpha|}\le1_{\{|r|\le2\alpha\}}\delta^2e^{\zeta|\delta|} δ α 2 e ζ ∣ δ α ∣ ≤ 1 { ∣ r ∣ ≤ 2 α } δ 2 e ζ ∣ δ ∣ [ 0 , 1 ] [0,1] [ 0 , 1 ] ∣ r ∣ > 2 α |r|>2\alpha ∣ r ∣ > 2 α (194) , taking the latitude average with (195) , and integrating with (196) , we find
E u [ δ α 2 e ζ ∣ δ α ∣ ] ≤ C ( N 2 N − d + N N − ( d − 1 ) + N 2 N − ( 2 d − 2 ) ) ≤ C N 2 − d . (197) \mathbb{E}_u\left[\delta_\alpha^2e^{\zeta|\delta_\alpha|}\right]\le C\left(N^2N^{-d}+NN^{-(d-1)}+N^2N^{-(2d-2)}\right)\le CN^{2-d}. \tag*{(197)} E u [ δ α 2 e ζ ∣ δ α ∣ ] ≤ C ( N 2 N − d + N N − ( d − 1 ) + N 2 N − ( 2 d − 2 ) ) ≤ C N 2 − d . (197) The three terms come from N 2 r 2 d N^2r^{2d} N 2 r 2 d N r 2 d − 2 Nr^{2d-2} N r 2 d − 2 V x V_x V x N r 2 d − 2 Nr^{2d-2} N r 2 d − 2 N r 2 d − 2 Nr^{2d-2} N r 2 d − 2
We now take ζ = 2 β \zeta= 2\beta ζ = 2 β ∣ e t − 1 ∣ ≤ ∣ t ∣ e ∣ t ∣ |e^t-1| \le|t|e^{|t|} ∣ e t − 1∣ ≤ ∣ t ∣ e ∣ t ∣ e t ( 1 − e − t ) 2 = e t + e − t − 2 ≤ t 2 e ∣ t ∣ e^t(1-e^{-t})^2=e^t+e^{-t}-2\le t^2e^{|t|} e t ( 1 − e − t ) 2 = e t + e − t − 2 ≤ t 2 e ∣ t ∣
E u ∣ e w − 1 ∣ ≤ β ( E u [ δ 2 e 2 β ∣ δ ∣ ] ) 1 / 2 , E u [ e w ( 1 − e − w ) 2 ] ≤ β 2 E u [ δ 2 e 2 β ∣ δ ∣ ] . \mathbb{E}_{u}|e^w-1|\le\beta\left(\mathbb{E}_{u}[\delta^2e^{2\beta|\delta|}]\right)^{1/2},\qquad\mathbb{E}_{u}[e^w(1-e^{-w})^2]\le\beta^2\mathbb{E}_{u}[\delta^2e^{2\beta|\delta|}]. E u ∣ e w − 1∣ ≤ β ( E u [ δ 2 e 2 β ∣ δ ∣ ] ) 1/2 , E u [ e w ( 1 − e − w ) 2 ] ≤ β 2 E u [ δ 2 e 2 β ∣ δ ∣ ] . Together with (8.34), this proves (8.28) and (8.29).
Deterministic bounds. Wherever the cutoff product or one of its derivatives is nonzero, we have ∣ r ∣ ≤ 2 α |r|\le2\alpha ∣ r ∣ ≤ 2 α ∣ s ∣ ≤ 2 α |s|\le2\alpha ∣ s ∣ ≤ 2 α G u ⋅ x = N s G_u\cdot x=Ns G u ⋅ x = N s Ω N \Omega_N Ω N ∣ δ h , k , u ∣ ≤ C N α d |\delta_{h,k,u}|\le CN\alpha^d ∣ δ h , k , u ∣ ≤ C N α d θ 1 ′ , θ 2 ′ \theta'_1,\theta'_2 θ 1 ′ , θ 2 ′ C N α d − 1 C\sqrt{N}\alpha^{d-1} C N α d − 1 θ 3 ′ \theta'_3 θ 3 ′ C α d − 2 N ∣ s ∣ N − 1 / 2 ≤ C N α d − 1 C\alpha^{d-2}N|s|N^{-1/2}\le C\sqrt{N}\alpha^{d-1} C α d − 2 N ∣ s ∣ N − 1/2 ≤ C N α d − 1 C N α d − 1 C\sqrt{N}\alpha^{d-1} C N α d − 1 ∣ ∇ r ∣ ≤ N − 1 / 2 |\nabla r|\le N^{-1/2} ∣∇ r ∣ ≤ N − 1/2 ∣ ∇ s ∣ = ∣ Π x G u ∣ / N ≤ C 0 N − 1 / 2 |\nabla s|=|\Pi_xG_u|/N\le C_0N^{-1/2} ∣∇ s ∣ = ∣ Π x G u ∣/ N ≤ C 0 N − 1/2 C α − 1 N − 1 / 2 ⋅ N α d C\alpha^{-1}N^{-1/2}\cdot N\alpha^d C α − 1 N − 1/2 ⋅ N α d
An entropy–resolvent comparison
The next lemma does not involve the Gaussian model. It compares the killing time of a perturbed reversible diffusion, killed at a rate κ \kappa κ
Setting. Let ( U , ϑ ) (\mathcal{U},\vartheta) ( U , ϑ ) ( E , μ ) (E,\mu) ( E , μ ) μ ˉ = ϑ ⊗ μ \bar{\mu}=\vartheta\otimes\mu μ ˉ = ϑ ⊗ μ u ∈ U u\in\mathcal{U} u ∈ U E 0 \mathcal{E}_0 E 0 L 2 ( μ ˉ ) L^2(\bar{\mu}) L 2 ( μ ˉ ) V \mathcal{V} V A 0 ≥ 0 A_0\ge0 A 0 ≥ 0 E 0 ( g , v ) = ⟨ A 0 g , v ⟩ μ ˉ \mathcal{E}_0(g,v)=\langle A_0g,v\rangle_{\bar{\mu}} E 0 ( g , v ) = ⟨ A 0 g , v ⟩ μ ˉ g ∈ Dom ( A 0 ) g\in\operatorname{Dom}(A_0) g ∈ Dom ( A 0 ) v ∈ V v\in\mathcal{V} v ∈ V
(D1) (Conservativeness.) 1 ∈ V 1\in\mathcal{V} 1 ∈ V E 0 ( 1 , 1 ) = 0 \mathcal{E}_0(1,1)=0 E 0 ( 1 , 1 ) = 0
(D2) (Carré du champ.) There exists a symmetric bilinear map ( g , v ) ↦ D g ⋅ D v (g,v)\mapsto Dg\cdot Dv ( g , v ) ↦ D g ⋅ D v V × V \mathcal{V}\times\mathcal{V} V × V L 1 ( μ ˉ ) L^1(\bar{\mu}) L 1 ( μ ˉ ) D g ⋅ D g ≥ 0 Dg\cdot Dg\ge0 D g ⋅ D g ≥ 0 E 0 ( g , v ) = ∫ D g ⋅ D v d μ ˉ \mathcal{E}_0(g,v)=\int Dg\cdot Dv\,d\bar{\mu} E 0 ( g , v ) = ∫ D g ⋅ D v d μ ˉ
(D3) (Diffusion property.) If g , v ∈ V ∩ L ∞ g,v\in\mathcal{V}\cap L^\infty g , v ∈ V ∩ L ∞ g v ∈ V gv\in\mathcal{V} g v ∈ V D ( g v ) ⋅ D ψ = g D v ⋅ D ψ + v D g ⋅ D ψ D(gv)\cdot D\psi=gDv\cdot D\psi+vDg\cdot D\psi D ( g v ) ⋅ D ψ = g D v ⋅ D ψ + v D g ⋅ D ψ ψ ∈ V \psi\in\mathcal{V} ψ ∈ V g ∈ V ∩ L ∞ g\in\mathcal{V}\cap L^\infty g ∈ V ∩ L ∞ φ ∈ C 1 ( R ) \varphi\in C^1(\mathbb{R}) φ ∈ C 1 ( R ) φ ( g ) ∈ V \varphi(g)\in\mathcal{V} φ ( g ) ∈ V D φ ( g ) ⋅ D ψ = φ ′ ( g ) D g ⋅ D ψ D\varphi(g)\cdot D\psi=\varphi'(g)Dg\cdot D\psi D φ ( g ) ⋅ D ψ = φ ′ ( g ) D g ⋅ D ψ
(D4) (Frozen label.) The resolvents ( λ + A 0 ) − 1 (\lambda+A_0)^{-1} ( λ + A 0 ) − 1 λ > 0 \lambda>0 λ > 0 ( g , v ) ↦ D g ⋅ D v (g,v)\mapsto Dg\cdot Dv ( g , v ) ↦ D g ⋅ D v u u u u u u
By (D2), ∣ D g ⋅ D v ∣ ≤ ( D g ⋅ D g ) 1 / 2 ( D v ⋅ D v ) 1 / 2 |Dg\cdot Dv|\le(Dg\cdot Dg)^{1/2}(Dv\cdot Dv)^{1/2} ∣ D g ⋅ D v ∣ ≤ ( D g ⋅ D g ) 1/2 ( D v ⋅ D v ) 1/2 ∣ D g ∣ 2 = D g ⋅ D g |Dg|^2=Dg\cdot Dg ∣ D g ∣ 2 = D g ⋅ D g E E E U × E \mathcal{U}\times E U × E u u u E E E S N S_N S N D g ⋅ D v = ∇ g ⋅ ∇ v Dg \cdot Dv = \nabla_g \cdot\nabla_v D g ⋅ D v = ∇ g ⋅ ∇ v S N S_N S N
Let w w w U × E \mathcal{U} \times E U × E
M = ∫ e w d μ ˉ , μ w = M − 1 e w μ ˉ , E w ( g , v ) = ∫ D g ⋅ D v d μ w , g , v ∈ V . M = \int e^w \,d\bar{\mu}, \qquad\mu_w = M^{-1}e^w\bar{\mu}, \qquad\mathcal{E}_w(g,v) = \int Dg \cdot Dv \,d\mu_w,\quad g,v\in\mathcal{V}. M = ∫ e w d μ ˉ , μ w = M − 1 e w μ ˉ , E w ( g , v ) = ∫ D g ⋅ D v d μ w , g , v ∈ V . Since w w w L 2 ( μ w ) L^2(\mu_w) L 2 ( μ w ) L 2 ( μ ˉ ) L^2(\bar{\mu}) L 2 ( μ ˉ ) E w \mathcal{E}_w E w L 2 ( μ w ) L^2(\mu_w) L 2 ( μ w ) V \mathcal{V} V A w ≥ 0 A_w \ge0 A w ≥ 0 w w w − A w = − A 0 + D w ⋅ D -A_w=-A_0+Dw\cdot D − A w = − A 0 + D w ⋅ D ( u , X t ) t ≥ 0 (u,X_t)_{t\ge0} ( u , X t ) t ≥ 0 U × E \mathcal{U}\times E U × E u u u e − t A w e^{-tA_w} e − t A w μ w \mu_w μ w
Lemma 8.11 (Entropy–resolvent comparison). In the setting above, suppose that for μ \mu μ x ∈ E x\in E x ∈ E
∫ ∣ e w ( u , x ) − 1 ∣ d ϑ ( u ) ≤ η ≤ 1 2 , ∫ e w ( u , x ) ( 1 − e − w ( u , x ) ) 2 d ϑ ( u ) ≤ ϵ . (198) \int\left|e^{w(u,x)}-1\right|\,d\vartheta(u)\le\eta\le\frac{1}{2},\qquad\int e^{w(u,x)}\left(1-e^{-w(u,x)}\right)^2\,d\vartheta(u)\le\epsilon. \tag*{(198)} ∫ e w ( u , x ) − 1 d ϑ ( u ) ≤ η ≤ 2 1 , ∫ e w ( u , x ) ( 1 − e − w ( u , x ) ) 2 d ϑ ( u ) ≤ ϵ . (198) Let f 0 ≥ 0 f_0\ge0 f 0 ≥ 0 E E E ∫ f 0 d μ = 1 \int f_0\,d\mu=1 ∫ f 0 d μ = 1 H 0 = ∫ ( 1 + f 0 ) log ( 1 + f 0 ) d μ < ∞ \mathcal{H}_0=\int(1+f_0)\log(1+f_0)\,d\mu<\infty H 0 = ∫ ( 1 + f 0 ) log ( 1 + f 0 ) d μ < ∞ κ ≥ 0 \kappa\ge0 κ ≥ 0 U × E \mathcal{U}\times E U × E ϑ ⊗ ( f 0 μ ) \vartheta\otimes(f_0\mu) ϑ ⊗ ( f 0 μ ) e be a unit exponential variable independent of it, and let τ κ = inf { t ≥ 0 : ∫ 0 t κ ( u , X v ) d v ≥ e } \tau_\kappa=\inf\{t\ge0:\int_0^t\kappa(u,X_v)\,dv\ge\mathbf{e}\} τ κ = inf { t ≥ 0 : ∫ 0 t κ ( u , X v ) d v ≥ e } inf ∅ = ∞ \inf\varnothing=\infty inf ∅ = ∞ κ \kappa κ λ > 0 \lambda>0 λ > 0
f = λ ( λ + A 0 ) − 1 f 0 , F = 1 + f , R λ = λ − 1 ∫ κ F d μ w . (199) f=\lambda(\lambda+A_0)^{-1}f_0,\qquad F=1+f,\qquad\mathcal{R}_\lambda=\lambda^{-1}\int\kappa F\,d\mu_w. \tag*{(199)} f = λ ( λ + A 0 ) − 1 f 0 , F = 1 + f , R λ = λ − 1 ∫ κ F d μ w . (199) Then
E e − λ τ κ ≤ 4 η + 5 ϵ + R λ + 2 H 0 [ R λ + ϵ ( 1 + H 0 ) ] , (200) \mathbb{E}e^{-\lambda\tau_\kappa}\le4\eta+5\sqrt{\epsilon}+\mathcal{R}_\lambda+2\sqrt{\mathcal{H}_0\left[\mathcal{R}_\lambda+\epsilon(1+\mathcal{H}_0)\right]}, \tag*{(200)} E e − λ τ κ ≤ 4 η + 5 ϵ + R λ + 2 H 0 [ R λ + ϵ ( 1 + H 0 ) ] , (200) with e − λ ⋅ ∞ = 0 e^{-\lambda\cdot\infty}=0 e − λ ⋅ ∞ = 0 T > 0 T>0 T > 0
P ( τ κ ≤ T ) ≤ 14 ( η + ϵ + ϵ ( 1 + H 0 ) + R 1 / T ) . (201) \mathbb{P}(\tau_\kappa\le T)\le14\left(\eta+\sqrt{\epsilon}+\epsilon(1+\mathcal{H}_0)+\mathcal{R}_{1/T}\right). \tag*{(201)} P ( τ κ ≤ T ) ≤ 14 ( η + ϵ + ϵ ( 1 + H 0 ) + R 1/ T ) . (201) The first resolvent in (199) acts on the unperturbed form; for unbounded f 0 f_0 f 0 L 1 ( μ ) L^1(\mu) L 1 ( μ ) (200) depends on ∥ w ∥ ∞ \lVert w\rVert_\infty ∥ w ∥ ∞ λ \lambda λ w w w E w \mathcal{E}_w E w A w A_w A w
Proof. We first treat bounded f 0 f_0 f 0
Two consequences of the moment bounds. By the first bound in (8.35), we have 1 − η ≤ M ≤ 1 + η 1-\eta\le M \le1+\eta 1 − η ≤ M ≤ 1 + η k = 1 − e − w k=1-e^{-w} k = 1 − e − w Γ \Gamma Γ E E E
∫ k 2 Γ d μ w ≤ 2 ϵ ∫ Γ d μ , ∫ Γ d μ w ≤ 3 ∫ Γ d μ . (202) \int k^{2}\Gamma\,\mathrm{d}\mu_w \le2\epsilon\int\Gamma\,\mathrm{d}\mu,\qquad\int\Gamma\,\mathrm{d}\mu_w \le3\int\Gamma\,\mathrm{d}\mu. \tag*{(202)} ∫ k 2 Γ d μ w ≤ 2 ϵ ∫ Γ d μ , ∫ Γ d μ w ≤ 3 ∫ Γ d μ . (202) Indeed, e w k 2 = e w ( 1 − e − w ) 2 e^{w}k^{2}=e^{w}(1-e^{-w})^{2} e w k 2 = e w ( 1 − e − w ) 2 M ≥ 1 / 2 M\ge1/2 M ≥ 1/2 ( 1 + η ) / ( 1 − η ) ≤ 3 (1+\eta)/(1-\eta)\le3 ( 1 + η ) / ( 1 − η ) ≤ 3 e w e^{w} e w
The resolvent of the unperturbed generator. For bounded f 0 f_0 f 0 f = λ ∫ 0 ∞ e − λ t e − t A 0 f 0 d t f=\lambda\int_0^\infty e^{-\lambda t}e^{-tA_0}f_0\,\mathrm{d}t f = λ ∫ 0 ∞ e − λ t e − t A 0 f 0 d t f ≥ 0 f\ge0 f ≥ 0 ∫ f d μ = 1 \int f\,\mathrm{d}\mu=1 ∫ f d μ = 1 f ∈ Dom ( A 0 ) ∩ L ∞ f\in\operatorname{Dom}(A_0)\cap L^\infty f ∈ Dom ( A 0 ) ∩ L ∞ f f f u u u F 0 = 1 + f 0 F_0=1+f_0 F 0 = 1 + f 0 A 0 F = A 0 f = λ ( F 0 − F ) A_0F=A_0f=\lambda(F_0-F) A 0 F = A 0 f = λ ( F 0 − F ) F ≥ 1 F\ge1 F ≥ 1 log F ∈ V \log F\in\mathcal{V} log F ∈ V D log F ⋅ D ψ = F − 1 D F ⋅ D ψ D\log F\cdot D\psi=F^{-1}DF\cdot D\psi D log F ⋅ D ψ = F − 1 D F ⋅ D ψ J = ∫ ∣ D F ∣ 2 F − 1 d μ \mathcal{J}=\int|DF|^2F^{-1}\,\mathrm{d}\mu J = ∫ ∣ D F ∣ 2 F − 1 d μ log F \log F log F
J = ⟨ A 0 F , log F ⟩ μ = λ ⟨ F 0 − F , log F ⟩ μ ≤ λ [ ∫ F 0 log F 0 d μ − ∫ F log F d μ ] ≤ λ H 0 . (203) \mathcal{J}=\langle A_0F,\log F\rangle_\mu=\lambda\langle F_0-F,\log F\rangle_\mu
\le\lambda\left[\int F_0\log F_0\,\mathrm{d}\mu-\int F\log F\,\mathrm{d}\mu\right]\le\lambda\mathcal{H}_0. \tag*{(203)} J = ⟨ A 0 F , log F ⟩ μ = λ ⟨ F 0 − F , log F ⟩ μ ≤ λ [ ∫ F 0 log F 0 d μ − ∫ F log F d μ ] ≤ λ H 0 . (203) In the first inequality, we used the tangent line inequality ( a − b ) log b ≤ a log a − b log b − ( a − b ) (a-b)\log b\le a\log a-b\log b-(a-b) ( a − b ) log b ≤ a log a − b log b − ( a − b ) a , b > 0 a,b>0 a , b > 0 F 0 F_0 F 0 F F F F log F ≥ 0 F\log F\ge0 F log F ≥ 0
The resolvent of the killed diffusion. The form E w ( g , v ) + ∫ κ g v d μ w \mathcal{E}_w(g,v)+\int\kappa gv\,\mathrm{d}\mu_w E w ( g , v ) + ∫ κ g v d μ w V \mathcal{V} V
h = ( λ + A w + κ ) − 1 κ ∈ Dom ( A w ) ⊂ V . h=(\lambda+A_w+\kappa)^{-1}\kappa\in\operatorname{Dom}(A_w)\subset\mathcal{V}. h = ( λ + A w + κ ) − 1 κ ∈ Dom ( A w ) ⊂ V . The Duhamel formula expresses e − t ( A w + κ ) e^{-t(A_w+\kappa)} e − t ( A w + κ )
e − t ( A w + κ ) = ∑ n ≥ 0 ( − 1 ) n ∫ 0 < v 1 < ⋯ < v n < t e − v 1 A w κ e − ( v 2 − v 1 ) A w κ ⋯ κ e − ( t − v n ) A w d v . e^{-t(A_w+\kappa)}=\sum_{n\ge0}(-1)^n\int_{0<v_1<\cdots<v_n<t}e^{-v_1A_w}\kappa e^{-(v_2-v_1)A_w}\kappa\cdots\kappa e^{-(t-v_n)A_w}\,\mathrm{d}v. e − t ( A w + κ ) = n ≥ 0 ∑ ( − 1 ) n ∫ 0 < v 1 < ⋯ < v n < t e − v 1 A w κ e − ( v 2 − v 1 ) A w κ ⋯ κ e − ( t − v n ) A w d v . whose n n n ∥ κ ∥ ∞ n t n / n ! \lVert\kappa\rVert_\infty^n t^n/n! ∥ κ ∥ ∞ n t n / n ! E u , x [ e − ∫ 0 t κ ( u , X v ) d v g ( X t ) ] \mathbb{E}_{u,x}\left[e^{-\int_0^t\kappa(u,X_v)\,\mathrm{d}v}g(X_t)\right] E u , x [ e − ∫ 0 t κ ( u , X v ) d v g ( X t ) ] e − t ( A w + κ ) g ( u , x ) = E u , x [ e − ∫ 0 t κ ( u , X v ) d v g ( X t ) ] e^{-t(A_w+\kappa)}g(u,x)=\mathbb{E}_{u,x}\left[e^{-\int_0^t\kappa(u,X_v)\,\mathrm{d}v}g(X_t)\right] e − t ( A w + κ ) g ( u , x ) = E u , x [ e − ∫ 0 t κ ( u , X v ) d v g ( X t ) ] g g g τ κ \tau_\kappa τ κ κ ( u , X t ) e − ∫ 0 t κ ( u , X v ) d v \kappa(u,X_t)e^{-\int_0^t\kappa(u,X_v)\,\mathrm{d}v} κ ( u , X t ) e − ∫ 0 t κ ( u , X v ) d v [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) g = κ g=\kappa g = κ e − λ t d t e^{-\lambda t}\,\mathrm{d}t e − λ t d t
h ( u , x ) = E u , x e − λ τ κ . (204) h(u,x)=\mathbb{E}_{u,x}e^{-\lambda\tau_\kappa}. \tag*{(204)} h ( u , x ) = E u , x e − λ τ κ . (204) Note that 0 ≤ h ≤ 1 0\le h\le1 0 ≤ h ≤ 1
The commutator. Let g ∈ Dom ( A 0 ) g \in\operatorname{Dom}(A_0) g ∈ Dom ( A 0 ) u u u v ∈ V v \in\mathcal{V} v ∈ V
E w ( g , v ) = ⟨ ( 1 − k ) A 0 g , v ⟩ μ w + ∫ k D g ⋅ D v d μ w . (205) \mathcal{E}_w(g,v)=\langle(1-k)A_0g,v\rangle_{\mu_w}+\int k\,Dg\cdot Dv\,\mathrm{d}\mu_w. \tag*{(205)} E w ( g , v ) = ⟨( 1 − k ) A 0 g , v ⟩ μ w + ∫ k D g ⋅ D v d μ w . (205) Indeed, since ( 1 − k ) e w = 1 (1-k)e^{w}=1 ( 1 − k ) e w = 1 M − 1 ∫ A 0 g v d μ ˉ = M − 1 ∫ D g ⋅ D v d μ ˉ M^{-1}\int A_0gv\,\mathrm{d}\bar{\mu}=M^{-1}\int Dg\cdot Dv\,\mathrm{d}\bar{\mu} M − 1 ∫ A 0 g v d μ ˉ = M − 1 ∫ D g ⋅ D v d μ ˉ 1 + ( e w − 1 ) = e w 1+(e^{w}-1)=e^{w} 1 + ( e w − 1 ) = e w A w g = ( 1 − k ) A 0 g + D w ∗ ( k D g ) A_wg=(1-k)A_0g+D_w^*(kDg) A w g = ( 1 − k ) A 0 g + D w ∗ ( k D g ) D w ∗ D_w^* D w ∗ L 2 ( μ w ) L^2(\mu_w) L 2 ( μ w ) w w w k k k
The weighted energy. Let B h = ∫ F ∣ D h ∣ 2 d μ w B_h=\int F|Dh|^2\,\mathrm{d}\mu_w B h = ∫ F ∣ D h ∣ 2 d μ w A h = ∫ F h 2 d μ w A_h=\int Fh^2\,\mathrm{d}\mu_w A h = ∫ F h 2 d μ w F h ∈ V Fh\in\mathcal{V} F h ∈ V D ( F h ) ⋅ D h = F ∣ D h ∣ 2 + h D F ⋅ D h D(Fh)\cdot Dh=F|Dh|^2+h\,DF\cdot Dh D ( F h ) ⋅ D h = F ∣ D h ∣ 2 + h D F ⋅ D h ( λ + A w + κ ) h = κ (\lambda+A_w+\kappa)h=\kappa ( λ + A w + κ ) h = κ F h Fh F h h ≤ 1 h\leq1 h ≤ 1
B h + ∫ h D F ⋅ D h d μ w + ∫ ( λ + κ ) F h 2 d μ w = ∫ κ F h d μ w ≤ ∫ κ F d μ w = λ R λ . (206) B_h+\int h\,DF\cdot Dh\,\mathrm{d}\mu_w+\int(\lambda+\kappa)Fh^2\,\mathrm{d}\mu_w=\int\kappa Fh\,\mathrm{d}\mu_w\leq\int\kappa F\,\mathrm{d}\mu_w=\lambda R_\lambda. \tag*{(206)} B h + ∫ h D F ⋅ D h d μ w + ∫ ( λ + κ ) F h 2 d μ w = ∫ κ F h d μ w ≤ ∫ κ F d μ w = λ R λ . (206) The cross term is the delicate one, since ∣ D F ∣ |DF| ∣ D F ∣ h D F ⋅ D h = 1 2 D F ⋅ D ( h 2 ) h\,DF\cdot Dh=\frac{1}{2}DF\cdot D(h^2) h D F ⋅ D h = 2 1 D F ⋅ D ( h 2 ) g = F g=F g = F v = h 2 v=h^2 v = h 2 A 0 F = λ ( f 0 − f ) A_0F=\lambda(f_0-f) A 0 F = λ ( f 0 − f ) ( 1 − k ) e w = 1 (1-k)e^w=1 ( 1 − k ) e w = 1
∫ h D F ⋅ D h d μ w = λ 2 M ∫ f 0 h 2 d μ ˉ − λ 2 ∫ f h 2 d μ w + λ 2 ∫ k f h 2 d μ w + ∫ k h D F ⋅ D h d μ w . (207) \int h\,DF\cdot Dh\,\mathrm{d}\mu_w=\frac{\lambda}{2M}\int f_0h^2\,\mathrm{d}\bar{\mu}-\frac{\lambda}{2}\int fh^2\,\mathrm{d}\mu_w+\frac{\lambda}{2}\int kfh^2\,\mathrm{d}\mu_w+\int kh\,DF\cdot Dh\,\mathrm{d}\mu_w. \tag*{(207)} ∫ h D F ⋅ D h d μ w = 2 M λ ∫ f 0 h 2 d μ ˉ − 2 λ ∫ f h 2 d μ w + 2 λ ∫ k f h 2 d μ w + ∫ k h D F ⋅ D h d μ w . (207) The first term on the right is nonnegative. Since F − f / 2 ≥ F / 2 F-f/2\geq F/2 F − f /2 ≥ F /2 λ A h \lambda A_h λ A h λ A h / 2 \lambda A_h/2 λ A h /2 Γ = f \Gamma=f Γ = f Γ = ∣ D F ∣ 2 / F \Gamma=|DF|^2/F Γ = ∣ D F ∣ 2 / F u u u h ≤ 1 h\leq1 h ≤ 1 ∫ f d μ = 1 \int f\,\mathrm{d}\mu=1 ∫ f d μ = 1
∣ ∫ k f h 2 d μ w ∣ ≤ 2 ϵ A h , ∣ ∫ k h D F ⋅ D h d μ w ∣ ≤ 2 ϵ J B h . (208) \left|\int kfh^2\,\mathrm{d}\mu_w\right|\leq\sqrt{2\epsilon A_h},\qquad\left|\int kh\,DF\cdot Dh\,\mathrm{d}\mu_w\right|\leq\sqrt{2\epsilon\mathcal{J}B_h}. \tag*{(208)} ∫ k f h 2 d μ w ≤ 2 ϵ A h , ∫ k h D F ⋅ D h d μ w ≤ 2 ϵ J B h . (208) For the first bound, we used ∫ f h 4 d μ w ≤ A h \int fh^4\,\mathrm{d}\mu_w\leq A_h ∫ f h 4 d μ w ≤ A h
B h + λ 2 A h ≤ λ R λ + λ 2 2 ϵ A h + 2 ϵ J B h . B_h+\frac{\lambda}{2}A_h\leq\lambda R_\lambda+\frac{\lambda}{2}\sqrt{2\epsilon A_h}+\sqrt{2\epsilon\mathcal{J}B_h}. B h + 2 λ A h ≤ λ R λ + 2 λ 2 ϵ A h + 2 ϵ J B h . Using Young’s inequality in the forms λ 2 2 ϵ A h ≤ λ 4 A h + λ 2 ϵ \frac{\lambda}{2}\sqrt{2\epsilon A_h}\leq\frac{\lambda}{4}A_h+\frac{\lambda}{2}\epsilon 2 λ 2 ϵ A h ≤ 4 λ A h + 2 λ ϵ 2 ϵ J B h ≤ 1 2 B h + ϵ J \sqrt{2\epsilon\mathcal{J}B_h}\leq\frac{1}{2}B_h+\epsilon\mathcal{J} 2 ϵ J B h ≤ 2 1 B h + ϵ J
B h λ ≤ 2 R λ + ϵ + 2 ϵ H 0 . (209) \frac{B_h}{\lambda}\leq2R_\lambda+\epsilon+2\epsilon\mathcal{H}_0. \tag*{(209)} λ B h ≤ 2 R λ + ϵ + 2 ϵ H 0 . (209) The term λ A h \lambda A_h λ A h B h / λ B_h/\lambda B h / λ ϵ H 0 \sqrt{\epsilon\mathcal{H}_0} ϵ H 0 N ( 3 − d ) / 2 N^{(3-d)/2} N ( 3 − d ) /2
Comparison of initial laws. The actual initial law ϑ ⊗ ( f 0 μ ) \vartheta\otimes(f_0\mu) ϑ ⊗ ( f 0 μ ) q 0 = M e − w f 0 q_0 = M e^{-w}f_0 q 0 = M e − w f 0 μ w \mu_w μ w x x x
∥ q 0 − f 0 ∥ L 1 ( μ w ) = M − 1 ∫ f 0 ∣ M − e w ∣ d μ ˉ ≤ 2 ∫ f 0 ( ∣ M − 1 ∣ + ∫ ∣ 1 − e w ∣ d ϑ ) d μ ≤ 4 η . (210) \lVert q_0-f_0\rVert_{L^1(\mu_w)}=M^{-1}\int f_0\lvert M-e^w\rvert\,d\bar{\mu}\le2\int f_0\left(\lvert M-1\rvert+\int\lvert1-e^w\rvert\,d\vartheta\right)d\mu\le4\eta. \tag*{(210)} ∥ q 0 − f 0 ∥ L 1 ( μ w ) = M − 1 ∫ f 0 ∣ M − e w ∣ d μ ˉ ≤ 2 ∫ f 0 ( ∣ M − 1 ∣ + ∫ ∣ 1 − e w ∣ d ϑ ) d μ ≤ 4 η . (210) By (8.41), E e − λ τ κ = ⟨ q 0 , h ⟩ μ w ≤ ⟨ f 0 , h ⟩ μ w + 4 η \mathbb{E}e^{-\lambda\tau_\kappa}=\langle q_0,h\rangle_{\mu_w}\le\langle f_0,h\rangle_{\mu_w}+4\eta E e − λ τ κ = ⟨ q 0 , h ⟩ μ w ≤ ⟨ f 0 , h ⟩ μ w + 4 η h h h f f f g = f g=f g = f v = h v=h v = h E w ( f , h ) = λ ⟨ f 0 − f , h ⟩ μ w − ⟨ k A 0 f , h ⟩ μ w + ∫ k D f ⋅ D h d μ w \mathcal{E}_w(f,h)=\lambda\langle f_0-f,h\rangle_{\mu_w}-\langle kA_0f,h\rangle_{\mu_w}+\int k\,Df\cdot Dh\,d\mu_w E w ( f , h ) = λ ⟨ f 0 − f , h ⟩ μ w − ⟨ k A 0 f , h ⟩ μ w + ∫ k D f ⋅ D h d μ w λ ⟨ h , f ⟩ μ w + E w ( h , f ) + ⟨ κ h , f ⟩ μ w = ⟨ κ , f ⟩ μ w \lambda\langle h,f\rangle_{\mu_w}+\mathcal{E}_w(h,f)+\langle\kappa h,f\rangle_{\mu_w}=\langle\kappa,f\rangle_{\mu_w} λ ⟨ h , f ⟩ μ w + E w ( h , f ) + ⟨ κh , f ⟩ μ w = ⟨ κ , f ⟩ μ w
⟨ f 0 , h ⟩ μ w = λ − 1 ⟨ f , κ ( 1 − h ) ⟩ μ w + ⟨ k ( f 0 − f ) , h ⟩ μ w − λ − 1 ∫ k D f ⋅ D h d μ w . (211) \langle f_0,h\rangle_{\mu_w}=\lambda^{-1}\langle f,\kappa(1-h)\rangle_{\mu_w}+\langle k(f_0-f),h\rangle_{\mu_w}-\lambda^{-1}\int k\,Df\cdot Dh\,d\mu_w. \tag*{(211)} ⟨ f 0 , h ⟩ μ w = λ − 1 ⟨ f , κ ( 1 − h ) ⟩ μ w + ⟨ k ( f 0 − f ) , h ⟩ μ w − λ − 1 ∫ k D f ⋅ D h d μ w . (211) The first term lies in [ 0 , R λ ] [0,R_\lambda] [ 0 , R λ ] Γ = f 0 \Gamma=f_0 Γ = f 0 Γ = f \Gamma=f Γ = f ∫ ∣ k ∣ ( f 0 + f ) d μ w ≤ 2 6 ϵ ≤ 5 ϵ \int\lvert k\rvert(f_0+f)\,d\mu_w\le2\sqrt{6}\epsilon\le5\sqrt{\epsilon} ∫ ∣ k ∣ ( f 0 + f ) d μ w ≤ 2 6 ϵ ≤ 5 ϵ D f = D F Df=DF D f = D F
λ − 1 ∣ ∫ k D f ⋅ D h d μ w ∣ ≤ λ − 1 2 ϵ J B h ≤ 2 ϵ H 0 B h / λ ≤ 2 ϵ H 0 [ R λ + ϵ ( 1 + H 0 ) ] . \lambda^{-1}\left\lvert\int k\,Df\cdot Dh\,d\mu_w\right\rvert\le\lambda^{-1}\sqrt{2\epsilon\mathcal{J}B_h}\le\sqrt{2\epsilon\mathcal{H}_0B_h/\lambda}\le2\sqrt{\epsilon\mathcal{H}_0\left[R_\lambda+\epsilon(1+\mathcal{H}_0)\right]}. λ − 1 ∫ k D f ⋅ D h d μ w ≤ λ − 1 2 ϵ J B h ≤ 2 ϵ H 0 B h / λ ≤ 2 ϵ H 0 [ R λ + ϵ ( 1 + H 0 ) ] . Combining these bounds proves (8.37) for bounded f 0 f_0 f 0 F F F h h h F h Fh F h h 2 h^2 h 2 log F \log F log F V \mathcal{V} V w w w V \mathcal{V} V
Unbounded initial densities. For general f 0 f_0 f 0 f 0 ( n ) = ( f 0 ∧ n ) / ∫ ( f 0 ∧ n ) d μ f_0^{(n)}=(f_0\wedge n)/\int(f_0\wedge n)\,d\mu f 0 ( n ) = ( f 0 ∧ n ) / ∫ ( f 0 ∧ n ) d μ f 0 ( n ) → f 0 f_0^{(n)}\to f_0 f 0 ( n ) → f 0 L 1 ( μ ) L^1(\mu) L 1 ( μ ) n n n f 0 ( n ) ≤ 2 f 0 f_0^{(n)}\le2f_0 f 0 ( n ) ≤ 2 f 0 ( 1 + 2 a ) log ( 1 + 2 a ) ≤ 2 ( 1 + a ) log ( 1 + a ) + 2 a (1+2a)\log(1+2a)\le2(1+a)\log(1+a)+2a ( 1 + 2 a ) log ( 1 + 2 a ) ≤ 2 ( 1 + a ) log ( 1 + a ) + 2 a a ≥ 0 a\ge0 a ≥ 0 H 0 ( n ) → H 0 \mathcal{H}_0^{(n)}\to\mathcal{H}_0 H 0 ( n ) → H 0 λ ( λ + A 0 ) − 1 \lambda(\lambda+A_0)^{-1} λ ( λ + A 0 ) − 1 L 1 ( μ ) L^1(\mu) L 1 ( μ ) f ( n ) → f f^{(n)}\to f f ( n ) → f L 1 ( μ ) L^1(\mu) L 1 ( μ ) κ \kappa κ e w e^w e w R λ ( n ) → R λ \mathcal{R}_\lambda^{(n)}\to\mathcal{R}_\lambda R λ ( n ) → R λ h h h f 0 f_0 f 0
The time horizon. Finally, 1 { τ κ ≤ T } ≤ e ⋅ e − τ κ / T \mathbf{1}_{\{\tau_\kappa\le T\}}\le e\cdot e^{-\tau_\kappa/T} 1 { τ κ ≤ T } ≤ e ⋅ e − τ κ / T λ = 1 / T \lambda=1/T λ = 1/ T 2 ϵ H 0 R ≤ R + ϵ H 0 2\sqrt{\epsilon\mathcal{H}_0}\mathcal{R}\le\mathcal{R}+\epsilon\mathcal{H}_0 2 ϵ H 0 R ≤ R + ϵ H 0 2 ϵ H 0 ( 1 + H 0 ) ≤ 2 ϵ ( 1 + H 0 ) 2\epsilon\sqrt{\mathcal{H}_0}(1+\mathcal{H}_0)\le2\epsilon(1+\mathcal{H}_0) 2 ϵ H 0 ( 1 + H 0 ) ≤ 2 ϵ ( 1 + H 0 ) P ( τ κ ≤ T ) ≤ e ( 4 η + 5 ϵ + 2 R 1 / T / T + 3 ϵ ( 1 + H 0 ) ) \mathbb{P}(\tau_\kappa\le T)\le e(4\eta+5\sqrt{\epsilon}+2\mathcal{R}_{1/T}/T+3\epsilon(1+\mathcal{H}_0)) P ( τ κ ≤ T ) ≤ e ( 4 η + 5 ϵ + 2 R 1/ T / T + 3 ϵ ( 1 + H 0 ))
Killing-time bound for the truncated process
We now apply Lemma 8.11 to the diffusion whose Hamiltonian is H N + δ α ( u , ⋅ ) H_N+\delta_\alpha(u,\cdot) H N + δ α ( u , ⋅ ) u u u α 0 \alpha_0 α 0 C 1 C_1 C 1
For a smooth function H ~ \widetilde{H} H R N \mathbb{R}^N R N L β , H ~ L_{\beta,\widetilde{H}} L β , H
d X t = 2 Π X t d B t + [ β Π X t ∇ E H ~ ( X t ) − N − 1 N X t ] d t (212) dX_t=\sqrt{2}\,\Pi_{X_t}\,dB_t+\left[\beta\Pi_{X_t}\nabla_E\widetilde{H}(X_t)-\frac{N-1}{N}X_t\right]dt \tag*{(212)} d X t = 2 Π X t d B t + [ β Π X t ∇ E H ( X t ) − N N − 1 X t ] d t (212) in R N \mathbb{R}^N R N B B B R N \mathbb{R}^N R N (63) . The coefficients are smooth and in particular locally Lipschitz. Let ρ t = ∣ X t ∣ 2 − N \rho_t = |X_t|^2-N ρ t = ∣ X t ∣ 2 − N x ⋅ Π x v = − ρ x ⋅ v / N x\cdot\Pi_xv=-\rho\,x\cdot v/N x ⋅ Π x v = − ρ x ⋅ v / N tr ( Π x Π x ) = N − 1 + ρ 2 / N 2 \operatorname{tr}(\Pi_x\Pi_x)=N-1+\rho^2/N^2 tr ( Π x Π x ) = N − 1 + ρ 2 / N 2 ρ = ∣ x ∣ 2 − N \rho=|x|^2-N ρ = ∣ x ∣ 2 − N d ρ t = ρ t ( a t d t + b t ⋅ d B t ) d\rho_t=\rho_t(a_t\,dt+b_t\cdot dB_t) d ρ t = ρ t ( a t d t + b t ⋅ d B t ) a t a_t a t b t b_t b t ρ t = 0 \rho_t=0 ρ t = 0 t t t ρ 0 = 0 \rho_0=0 ρ 0 = 0 S N S_N S N H ~ \widetilde H H B B B
Proposition 8.12 (The truncated process avoids the caps). There exists α 1 ∈ ( 0 , α 0 ] \alpha_1\in(0,\alpha_0] α 1 ∈ ( 0 , α 0 ] α ∈ ( 0 , α 1 ] \alpha\in(0,\alpha_1] α ∈ ( 0 , α 1 ] γ 0 , c , C > 0 \gamma_0,c,C>0 γ 0 , c , C > 0 N 0 N_0 N 0 Q α Q_\alpha Q α J J J H N H_N H N Ω N \Omega_N Ω N u ∈ S N u\in S_N u ∈ S N X 0 X_0 X 0 ν N \nu_N ν N X X X H ~ = H J + δ α ( u , ⋅ ) \widetilde H=H_J+\delta_\alpha(u,\cdot) H = H J + δ α ( u , ⋅ ) N ≥ N 0 N\ge N_0 N ≥ N 0 ( h , k ) (h,k) ( h , k )
Q α ( sup 0 ≤ t ≤ e γ 0 N ∣ R ( X t , u ) ∣ ≥ α 2 ) ≤ C ( N 1 − d / 2 + N 3 − d ) + e − c N . (213) Q_\alpha\left(\sup_{0\le t\le e^{\gamma_0N}}|R(X_t,u)|\ge\frac{\alpha}{2}\right)\le C\left(N^{1-d/2}+N^{3-d}\right)+e^{-cN}. \tag*{(213)} Q α ( 0 ≤ t ≤ e γ 0 N sup ∣ R ( X t , u ) ∣ ≥ 2 α ) ≤ C ( N 1 − d /2 + N 3 − d ) + e − c N . (213) The same bound holds with γ 0 \gamma_0 γ 0 γ ∈ ( 0 , γ 0 ] \gamma\in(0,\gamma_0] γ ∈ ( 0 , γ 0 ]
Proof. The proof has three steps: we verify the hypotheses of Lemma 8.11, bound the integrated killing rate, and convert the killing at a smooth rate into a hit of the cap C α / 2 ( u ) C_{\alpha/2}(u) C α /2 ( u )
The setting of the comparison lemma. Let Θ \Theta Θ P Ω ( d J ) = P ( d J ∣ Ω N ) \mathbb{P}_{\Omega}(dJ)=\mathbb{P}(dJ\mid\Omega_N) P Ω ( dJ ) = P ( dJ ∣ Ω N ) E = Θ × S N E=\Theta\times S_N E = Θ × S N
μ ( d J , d x ) = P Ω ( d J ) π H J ( d x ) , f 0 ( J , x ) = Z H J e − β H J ( x ) , \mu(dJ,dx)=\mathbb{P}_{\Omega}(dJ)\,\pi^{H_J}(dx),\qquad f_0(J,x)=Z_{H_J}e^{-\beta H_J(x)}, μ ( dJ , d x ) = P Ω ( dJ ) π H J ( d x ) , f 0 ( J , x ) = Z H J e − β H J ( x ) , where Z H J = ∫ e β H J d ν N Z_{H_J}=\int e^{\beta H_J}\,d\nu_N Z H J = ∫ e β H J d ν N ( U , ϑ ) = ( S N , ν N ) (\mathcal{U},\vartheta)=(S_N,\nu_N) ( U , ϑ ) = ( S N , ν N ) f 0 μ f_0\mu f 0 μ P Ω ⊗ ν N \mathbb{P}_{\Omega}\otimes\nu_N P Ω ⊗ ν N Ω N \Omega_N Ω N ∣ H J ∣ ≤ C 0 N |H_J|\le C_0N ∣ H J ∣ ≤ C 0 N f 0 ≤ e 2 β C 0 N f_0\le e^{2\beta C_0N} f 0 ≤ e 2 β C 0 N
H 0 ≤ log ( 1 + e 2 β C 0 N ) ∫ ( 1 + f 0 ) d μ ≤ C ( 1 + N ) . (214) \mathcal{H}_0\le\log(1+e^{2\beta C_0N})\int(1+f_0)\,d\mu\le C(1+N). \tag*{(214)} H 0 ≤ log ( 1 + e 2 β C 0 N ) ∫ ( 1 + f 0 ) d μ ≤ C ( 1 + N ) . (214) The base form is E 0 ( g , v ) = ∫ ∇ x g ⋅ ∇ x v d μ ˉ \mathcal{E}_0(g,v)=\int\nabla_xg\cdot\nabla_xv\,d\bar\mu E 0 ( g , v ) = ∫ ∇ x g ⋅ ∇ x v d μ ˉ μ ˉ = ν N ⊗ μ \bar\mu=\nu_N\otimes\mu μ ˉ = ν N ⊗ μ ∇ x \nabla_x ∇ x S N S_N S N V = L 2 ( ν N ⊗ P Ω ; H 1 ( S N , ν N ) ) \mathcal{V}=L^2(\nu_N\otimes\mathbb{P}_{\Omega};H^1(S_N,\nu_N)) V = L 2 ( ν N ⊗ P Ω ; H 1 ( S N , ν N )) e − 2 β C 0 N ≤ d μ / d ( P Ω ⊗ ν N ) ≤ e 2 β C 0 N e^{-2\beta C_0N}\le d\mu/d(\mathbb{P}_{\Omega}\otimes\nu_N)\le e^{2\beta C_0N} e − 2 β C 0 N ≤ d μ / d ( P Ω ⊗ ν N ) ≤ e 2 β C 0 N ( ∥ g ∥ L 2 ( μ ˉ ) 2 + E 0 ( g , g ) ) 1 / 2 (\|g\|_{L^2(\bar\mu)}^2+\mathcal{E}_0(g,g))^{1/2} ( ∥ g ∥ L 2 ( μ ˉ ) 2 + E 0 ( g , g ) ) 1/2 V \mathcal{V} V E 0 \mathcal{E}_0 E 0 V \mathcal{V} V ( u , J ) (u,J) ( u , J ) N N N ( u , J ) (u,J) ( u , J ) A 0 A_0 A 0 − L β , H J -L_{\beta,H_J} − L β , H J π H J \pi^{H_J} π H J
We take w = β δ α ( u , x ) w=\beta\delta_\alpha(u,x) w = β δ α ( u , x ) ( u , J , x ) (u,J,x) ( u , J , x ) x x x μ w \mu_w μ w ( u , J ) (u,J) ( u , J ) H J + δ α ( u , ⋅ ) H_J+\delta_\alpha(u,\cdot) H J + δ α ( u , ⋅ ) ( u , J ) (u,J) ( u , J ) E w \mathcal{E}_w E w L β , H J + δ α ( u , ⋅ ) L_{\beta,H_J+\delta_\alpha(u,\cdot)} L β , H J + δ α ( u , ⋅ ) A w A_w A w − L β , H J + δ α ( u , ⋅ ) -L_{\beta,H_J+\delta_\alpha(u,\cdot)} − L β , H J + δ α ( u , ⋅ ) (212) , constructed for each fixed ( u , J ) (u,J) ( u , J ) u u u J J J ϑ ⊗ ( f 0 μ ) \vartheta\otimes(f_0\mu) ϑ ⊗ ( f 0 μ ) Q α Q_\alpha Q α J J J x x x
For α ≤ α 0 \alpha\le\alpha_0 α ≤ α 0 (198) for every ( J , x ) (J,x) ( J , x ) J ∈ Ω N J\in\Omega_N J ∈ Ω N
η = C 1 N 1 − d / 2 , ϵ = C 1 N 2 − d , (215) \eta=C_1N^{1-d/2},\qquad\epsilon=C_1N^{2-d}, \tag*{(215)} η = C 1 N 1 − d /2 , ϵ = C 1 N 2 − d , (215) and η ≤ 1 / 2 \eta\le1/2 η ≤ 1/2 N N N
The integrated killing rate. Let ϕ : R → [ 0 , 1 ] \phi:\mathbb{R}\to[0,1] ϕ : R → [ 0 , 1 ] ϕ ( r ) = 0 \phi(r)=0 ϕ ( r ) = 0 ∣ r ∣ ≤ α / 4 |r|\le\alpha/4 ∣ r ∣ ≤ α /4 ϕ ( r ) = 1 \phi(r)=1 ϕ ( r ) = 1 ∣ r ∣ ≥ α / 3 |r|\ge\alpha/3 ∣ r ∣ ≥ α /3 κ ( u , J , x ) = ϕ ( R ( x , u ) ) \kappa(u,J,x)=\phi(R(x,u)) κ ( u , J , x ) = ϕ ( R ( x , u )) f f f u u u ∫ F d μ = 2 \int F\,d\mu=2 ∫ F d μ = 2 e w e^w e w (193) , and use M ≥ 1 / 2 M\ge1/2 M ≥ 1/2 x x x u u u κ ( u , J , x ) ≠ 0 \kappa(u,J,x)\ne0 κ ( u , J , x ) = 0 C α / 4 ( x ) C_{\alpha/4}(x) C α /4 ( x ) ν N ( C b ( x ) ) \nu_N(C_b(x)) ν N ( C b ( x )) x ∈ S N x\in S_N x ∈ S N ν N ( C b ) \nu_N(C_b) ν N ( C b )
R λ ≤ 4 λ − 1 e C 1 β N α d ν N ( C α / 4 ) . (216) R_\lambda\le4\lambda^{-1}e^{C_1\beta N\alpha^d}\nu_N(C_{\alpha/4}). \tag*{(216)} R λ ≤ 4 λ − 1 e C 1 β N α d ν N ( C α /4 ) . (216) This step uses that (193) holds for every u u u x x x (185) , for every fixed b ∈ ( 0 , 1 ) b\in(0,1) b ∈ ( 0 , 1 ) N N N
ν N ( C b ) = 2 ∫ b 1 p N ( r ) d r ≤ 2 N e − ( N − 3 ) I ( b ) ≤ e − N I ( b ) / 2 . (217) \nu_N(C_b)=2\int_b^1p_N(r)\,dr\le2\sqrt{N}e^{-(N-3)I(b)}\le e^{-NI(b)/2}. \tag*{(217)} ν N ( C b ) = 2 ∫ b 1 p N ( r ) d r ≤ 2 N e − ( N − 3 ) I ( b ) ≤ e − N I ( b ) /2 . (217) Since I ( b ) ≥ b 2 / 2 I(b)\ge b^2/2 I ( b ) ≥ b 2 /2 d > 2 d>2 d > 2 α 1 ∈ ( 0 , α 0 ] \alpha_1\in(0,\alpha_0] α 1 ∈ ( 0 , α 0 ] C 1 β α d ≤ I ( α / 4 ) / 8 C_1\beta\alpha^d\le I(\alpha/4)/8 C 1 β α d ≤ I ( α /4 ) /8 α ≤ α 1 \alpha\le\alpha_1 α ≤ α 1 α ≤ α 1 \alpha\le\alpha_1 α ≤ α 1 γ 0 = I ( α / 4 ) / 8 \gamma_0=I(\alpha/4)/8 γ 0 = I ( α /4 ) /8 λ = ( e γ 0 N + 1 ) − 1 \lambda=(e^{\gamma_0N}+1)^{-1} λ = ( e γ 0 N + 1 ) − 1 (216) and (217) imply that R λ ≤ 8 e − N I ( α / 4 ) / 4 R_\lambda\le8e^{-NI(\alpha/4)/4} R λ ≤ 8 e − N I ( α /4 ) /4 N N N (201) , (214) , and (215) , we obtain
Q α ( τ κ ≤ e γ 0 N + 1 ) ≤ C ( N 1 − d / 2 + N 3 − d ) + e − c N , (218) Q_\alpha\left(\tau_\kappa\le e^{\gamma_0N}+1\right)\le C\left(N^{1-d/2}+N^{3-d}\right)+e^{-cN}, \tag*{(218)} Q α ( τ κ ≤ e γ 0 N + 1 ) ≤ C ( N 1 − d /2 + N 3 − d ) + e − c N , (218) where τ κ \tau_\kappa τ κ κ \kappa κ (218) we used N 2 − d ≤ N 1 − d / 2 N^{2-d}\le N^{1-d/2} N 2 − d ≤ N 1 − d /2
From the killing time to the cap. Let r t = R ( X t , u ) r_t=R(X_t,u) r t = R ( X t , u ) (212) , r t r_t r t − N − 1 N r t + β N − 1 u ⋅ ∇ H ~ ( X t ) -\frac{N-1}{N}r_t+\beta N^{-1}u\cdot\nabla\widetilde{H}(X_t) − N N − 1 r t + β N − 1 u ⋅ ∇ H ( X t ) 2 ∣ Π X t u ∣ 2 / N 2 = 2 ( 1 − r t 2 ) / N ≤ 2 / N 2|\Pi_{X_t}u|^2/N^2=2(1-r_t^2)/N\le2/N 2∣ Π X t u ∣ 2 / N 2 = 2 ( 1 − r t 2 ) / N ≤ 2/ N Ω N \Omega_N Ω N ∥ ∇ H ~ ∥ ∞ ≤ C N \lVert\nabla\widetilde{H}\rVert_\infty\le C\sqrt{N} ∥ ∇ H ∥ ∞ ≤ C N (177) and (193) , and the drift is bounded by a constant d ˉ r \bar d_r d ˉ r τ α / 2 = inf { t ≥ 0 : ∣ r t ∣ ≥ α / 2 } \tau_{\alpha/2}=\inf\{t\ge0: |r_t|\ge\alpha/2\} τ α /2 = inf { t ≥ 0 : ∣ r t ∣ ≥ α /2 } Δ = α / 2 − α / 3 = α / 6 \Delta=\alpha/2-\alpha/3=\alpha/6 Δ = α /2 − α /3 = α /6 t 0 = Δ / ( 4 ( d ˉ r + 1 ) ) < 1 t_0=\Delta/(4(\bar d_r+1))<1 t 0 = Δ/ ( 4 ( d ˉ r + 1 )) < 1 T > 0 T>0 T > 0
Q α ( { τ α / 2 ≤ T } ∩ { inf τ α / 2 ≤ t ≤ τ α / 2 + t 0 ∣ r t ∣ < α 3 } ) ≤ 2 exp ( − N Δ 2 16 t 0 ) . (219) Q_\alpha\left(\{\tau_{\alpha/2}\le T\}\cap\left\{\inf_{\tau_{\alpha/2}\le t\le\tau_{\alpha/2}+t_0}|r_t|<\frac{\alpha}{3}\right\}\right)\le2\exp\left(-\frac{N\Delta^2}{16t_0}\right). \tag*{(219)} Q α ( { τ α /2 ≤ T } ∩ { τ α /2 ≤ t ≤ τ α /2 + t 0 inf ∣ r t ∣ < 3 α } ) ≤ 2 exp ( − 16 t 0 N Δ 2 ) . (219) Indeed, after τ α / 2 \tau_{\alpha/2} τ α /2 r t r_t r t d ˉ r t 0 ≤ Δ / 4 \bar d_r t_0\le\Delta/4 d ˉ r t 0 ≤ Δ/4 t 0 t_0 t 0 ∣ r t ∣ |r_t| ∣ r t ∣ α / 2 \alpha/2 α /2 α / 3 \alpha/3 α /3 Δ / 2 \Delta/2 Δ/2 2 t 0 / N 2t_0/N 2 t 0 / N v v v b b b 2 e − b 2 / ( 2 v ) 2e^{-b^2/(2v)} 2 e − b 2 / ( 2 v ) τ α / 2 \tau_{\alpha/2} τ α /2 (219) holds. This also covers the case τ α / 2 = 0 \tau_{\alpha/2}=0 τ α /2 = 0
On the complement of the event in (219) , and on { τ α / 2 ≤ T } \{\tau_{\alpha/2}\le T\} { τ α /2 ≤ T } [ τ α / 2 , τ α / 2 + t 0 ] [\tau_{\alpha/2},\tau_{\alpha/2}+t_0] [ τ α /2 , τ α /2 + t 0 ] t 0 t_0 t 0 T + t 0 T+t_0 T + t 0 1 − e − t 0 1-e^{-t_0} 1 − e − t 0 (219) , we find
Q α ( τ α / 2 ≤ T ) ≤ ( 1 − e − t 0 ) − 1 Q α ( τ κ ≤ T + t 0 ) + 2 e − N Δ 2 / ( 16 t 0 ) . Q_\alpha(\tau_{\alpha/2}\le T)\le(1-e^{-t_0})^{-1}Q_\alpha(\tau_\kappa\le T+t_0)+2e^{-N\Delta^2/(16t_0)}. Q α ( τ α /2 ≤ T ) ≤ ( 1 − e − t 0 ) − 1 Q α ( τ κ ≤ T + t 0 ) + 2 e − N Δ 2 / ( 16 t 0 ) . We take T = e γ 0 N T=e^{\gamma_0N} T = e γ 0 N (218) with t 0 < 1 t_0<1 t 0 < 1 C C C c c c N 0 N_0 N 0 (213) . The left side of (213) decreases when γ 0 \gamma_0 γ 0
The cutoff in the gradient variable
The truncated and true processes coincide until ∣ r ∣ |r| ∣ r ∣ ∣ s ∣ |s| ∣ s ∣ ∣ s ∣ = α |s|=\alpha ∣ s ∣ = α G u G_u G u
Lemma 8.13 (The gradient cutoff is not reached). Let ( h , k ) (h,k) ( h , k ) u u u S N S_N S N H N H_N H N X 0 X_0 X 0 ν N \nu_N ν N X X X (212) with H ~ = H h , k , u \widetilde H=H^{h,k,u} H = H h , k , u X 0 X_0 X 0 ( H N , u ) (H_N,u) ( H N , u ) s t = G u ⋅ X t / N s_t=G_u\cdot X_t/N s t = G u ⋅ X t / N b > 0 b>0 b > 0 γ b , c b > 0 \gamma_b,c_b>0 γ b , c b > 0 N 0 N_0 N 0 N ≥ N 0 N\ge N_0 N ≥ N 0
P ( sup 0 ≤ t ≤ e γ b N ∣ s t ∣ ≥ b ) ≤ e − c b N . (220) \mathbb{P}\left(\sup_{0\le t\le e^{\gamma_bN}}|s_t|\ge b\right)\le e^{-c_bN}. \tag*{(220)} P ( 0 ≤ t ≤ e γ b N sup ∣ s t ∣ ≥ b ) ≤ e − c b N . (220) The same bound, with a smaller c b c_b c b H N H_N H N Ω N \Omega_N Ω N ( h , k ) (h,k) ( h , k )
Proof. We first work under the unrestricted law. Conditionally on u u u H N − D u H_N-D_u H N − D u G u G_u G u Proposition 8.7 , and G u G_u G u u ⊥ u^\perp u ⊥ a Q I u ⊥ a_Q I_{u^\perp} a Q I u ⊥ (182) . The coefficients of H h , k , u = ( H N − D u ) + N ( θ 1 h + θ 2 k ) H^{h,k,u}=(H_N-D_u)+N(\theta_1h+\theta_2k) H h , k , u = ( H N − D u ) + N ( θ 1 h + θ 2 k ) H N − D u H_N-D_u H N − D u u u u (212) , the whole trajectory ( X t ) t ≥ 0 (X_t)_{t\geq0} ( X t ) t ≥ 0 u u u X 0 X_0 X 0 u u u G u G_u G u t t t u u u s t s_t s t a Q ∣ Π u X t ∣ 2 / N 2 ≤ a Q / N a_Q|\Pi_uX_t|^2/N^2\leq a_Q/N a Q ∣ Π u X t ∣ 2 / N 2 ≤ a Q / N
P ( ∣ s t ∣ ≥ b / 2 ) ≤ 2 e − N b 2 / ( 8 a Q ) . (221) \mathbb{P}\left(|s_t|\geq b/2\right)\leq2e^{-Nb^2/(8a_Q)}. \tag*{(221)} P ( ∣ s t ∣ ≥ b /2 ) ≤ 2 e − N b 2 / ( 8 a Q ) . (221) We choose 0 < γ b < b 2 / ( 32 a Q ) 0<\gamma_b<b^2/(32a_Q) 0 < γ b < b 2 / ( 32 a Q ) j / N j/N j / N 0 ≤ j ≤ ⌈ N e γ b N ⌉ 0\leq j\leq\lceil Ne^{\gamma_bN}\rceil 0 ≤ j ≤ ⌈ N e γ b N ⌉ (221) shows that some grid value satisfies ∣ s j / N ∣ ≥ b / 2 |s_{j/N}|\geq b/2 ∣ s j / N ∣ ≥ b /2 e − c N e^{-cN} e − c N Ω N \Omega_N Ω N ∣ G u ∣ ≤ C 0 N |G_u|\leq C_0\sqrt{N} ∣ G u ∣ ≤ C 0 N ∥ ∇ H h , k , u ∥ ∞ ≤ C N \|\nabla H^{h,k,u}\|_\infty\leq C\sqrt{N} ∥∇ H h , k , u ∥ ∞ ≤ C N (177) and (181) . By (212) , s t s_t s t − N − 1 N s t + β N − 1 G u ⋅ ∇ H h , k , u ( X t ) -\frac{N-1}{N}s_t+\beta N^{-1}G_u\cdot\nabla H^{h,k,u}(X_t) − N N − 1 s t + β N − 1 G u ⋅ ∇ H h , k , u ( X t ) d ˉ s \bar d_s d ˉ s ∣ s t ∣ ≤ C 0 |s_t|\leq C_0 ∣ s t ∣ ≤ C 0 2 ∣ Π X t G u ∣ 2 / N 2 ≤ q ˉ s / N 2|\Pi_{X_t}G_u|^2/N^2\leq\bar q_s/N 2∣ Π X t G u ∣ 2 / N 2 ≤ q ˉ s / N 1 / N 1/N 1/ N s t s_t s t b / 4 b/4 b /4 N N N ∣ s ∣ < b / 2 |s|<b/2 ∣ s ∣ < b /2 ∣ s ∣ |s| ∣ s ∣ b b b b / 4 b/4 b /4 q ˉ s / N 2 \bar q_s/N^2 q ˉ s / N 2 Proposition 8.12 shows that this has probability at most 2 e − b 2 N 2 / ( 32 q ˉ s ) 2e^{-b^2N^2/(32\bar q_s)} 2 e − b 2 N 2 / ( 32 q ˉ s ) Ω N \Omega_N Ω N O ( N e γ b N ) O(Ne^{\gamma_bN}) O ( N e γ b N ) e − c N 2 e^{-cN^2} e − c N 2 P ( Ω N c ) ≤ e − N \mathbb{P}(\Omega_N^c)\leq e^{-N} P ( Ω N c ) ≤ e − N (220) holds.
Under the law conditioned on Ω N \Omega_N Ω N (220) is at most its unrestricted probability divided by P ( Ω N ) ≥ 1 / 2 \mathbb{P}(\Omega_N)\geq1/2 P ( Ω N ) ≥ 1/2 G u G_u G u
Proof of the main estimate
Proof of Theorem 8.2. We fix the parameters, couple the true and truncated processes, remove the regularity restriction, and finally pass from a uniform axis to a deterministic one.
Choice of parameters. We fix ξ \xi ξ Q Q Q β \beta β h ˉ \bar h h ˉ k ˉ \bar k k ˉ C 0 C_0 C 0 Ω N \Omega_N Ω N α 1 \alpha_1 α 1 a ∗ = α 1 / 2 a_*=\alpha_1/2 a ∗ = α 1 /2 a ∈ ( 0 , a ∗ ] a\in(0,a_*] a ∈ ( 0 , a ∗ ] α = 2 a \alpha=2a α = 2 a C α / 2 ( u ) C_{\alpha/2}(u) C α /2 ( u ) (213) is C a ( u ) C_a(u) C a ( u )
γ = min { γ 0 , γ α } . \gamma=\min\{\gamma_0,\gamma_\alpha\}. γ = min { γ 0 , γ α } . where γ 0 \gamma_0 γ 0 γ α \gamma_\alpha γ α b = α b=\alpha b = α N → ∞ N\to\infty N → ∞ u u u ( h , k ) (h,k) ( h , k ) Table 8.1 .
The coupling. Let the coefficient vector J J J H N H_N H N Ω N \Omega_N Ω N u u u X 0 X_0 X 0 ν N \nu_N ν N B B B X t r X^{\mathrm{tr}} X tr (212) with H ~ = H h , k , u = H N + δ h , k , u \widetilde H=H^{h,k,u}=H_N+\delta_{h,k,u} H = H h , k , u = H N + δ h , k , u X c u t X^{\mathrm{cut}} X cut H ~ = H N + δ α ( u , ⋅ ) \widetilde H=H_N+\delta_\alpha(u,\cdot) H = H N + δ α ( u , ⋅ ) X 0 X_0 X 0 B B B r t r_t r t s t s_t s t (190) along each path. On the open set { x : ∣ R ( x , u ) ∣ < α , ∣ G u ⋅ x ∣ / N < α } \{x:|R(x,u)|<\alpha,\ |G_u\cdot x|/N<\alpha\} { x : ∣ R ( x , u ) ∣ < α , ∣ G u ⋅ x ∣/ N < α }
Let T = e γ N T=e^{\gamma N} T = e γ N τ a t r \tau_a^{\mathrm{tr}} τ a tr τ a c u t \tau_a^{\mathrm{cut}} τ a cut C a ( u ) C_a(u) C a ( u ) ζ t r = inf { t ≥ 0 : ∣ s t t r ∣ ≥ α } \zeta^{\mathrm{tr}}=\inf\{t\ge0:|s_t^{\mathrm{tr}}|\ge\alpha\} ζ tr = inf { t ≥ 0 : ∣ s t tr ∣ ≥ α }
{ τ a t r ≤ T } ⊂ { τ a c u t ≤ T } ∪ { ζ t r ≤ T } . (222) \{\tau_a^{\mathrm{tr}}\le T\}\subset\{\tau_a^{\mathrm{cut}}\le T\}\cup\{\zeta^{\mathrm{tr}}\le T\}. \tag*{(222)} { τ a tr ≤ T } ⊂ { τ a cut ≤ T } ∪ { ζ tr ≤ T } . (222) If ∣ R ( X 0 , u ) ∣ ≥ a |R(X_0,u)|\ge a ∣ R ( X 0 , u ) ∣ ≥ a ∣ s 0 ∣ ≥ α |s_0|\ge\alpha ∣ s 0 ∣ ≥ α ζ t r = 0 \zeta^{\mathrm{tr}}=0 ζ tr = 0 τ a t r ≤ T \tau_a^{\mathrm{tr}}\le T τ a tr ≤ T ζ t r > T \zeta^{\mathrm{tr}}>T ζ tr > T a < α a<\alpha a < α ∣ r ∣ = a |r|=a ∣ r ∣ = a ∣ r ∣ = α |r|=\alpha ∣ r ∣ = α ∣ s ∣ = α |s|=\alpha ∣ s ∣ = α T T T [ 0 , τ a t r ] [0,\tau_a^{\mathrm{tr}}] [ 0 , τ a tr ] τ a c u t = τ a t r ≤ T \tau_a^{\mathrm{cut}}=\tau_a^{\mathrm{tr}}\le T τ a cut = τ a tr ≤ T (222) .
The law of ( J , u , X c u t ) (J,u,X^{\mathrm{cut}}) ( J , u , X cut ) Q α Q_\alpha Q α γ ≤ γ 0 \gamma\le\gamma_0 γ ≤ γ 0 (222) by C ( N 1 − d / 2 + N 3 − d ) + e − c N C(N^{1-d/2}+N^{3-d})+e^{-cN} C ( N 1 − d /2 + N 3 − d ) + e − c N γ ≤ γ α \gamma\le\gamma_\alpha γ ≤ γ α e − c α N e^{-c_\alpha N} e − c α N
Removing the restriction. Let now J J J A A A P ( A ) ≤ P ( A ∣ Ω N ) + P ( Ω N c ) \mathbb{P}(A)\le\mathbb{P}(A\mid\Omega_N)+\mathbb{P}(\Omega_N^c) P ( A ) ≤ P ( A ∣ Ω N ) + P ( Ω N c ) P ( Ω N c ) ≤ e − N \mathbb{P}(\Omega_N^c)\le e^{-N} P ( Ω N c ) ≤ e − N u u u H h , k , u H^{h,k,u} H h , k , u E h , k , u \mathbb{E}_{h,k,u} E h , k , u X 0 X_0 X 0 B B B (212) , the probability of { τ a t r ≤ T } \{\tau_a^{\mathrm{tr}}\le T\} { τ a tr ≤ T } u u u u u u u u u
∫ E h , k , u [ P ν N H ( sup 0 ≤ t ≤ e γ N ∣ R ( X t , u ) ∣ ≥ a ) ] ν N ( d u ) ≤ C ( N 1 − d / 2 + N 3 − d ) + e − c N . (223) \int\mathbb{E}_{h,k,u}\left[\mathbb{P}_{\nu_N}^{H}\left(\sup_{0\le t\le e^{\gamma N}}|R(X_t,u)|\ge a\right)\right]\nu_N(du)
\le C(N^{1-d/2}+N^{3-d})+e^{-cN}. \tag*{(223)} ∫ E h , k , u [ P ν N H ( 0 ≤ t ≤ e γ N sup ∣ R ( X t , u ) ∣ ≥ a ) ] ν N ( d u ) ≤ C ( N 1 − d /2 + N 3 − d ) + e − c N . (223) Deterministic axes. Let g ∈ O ( N ) g\in O(N) g ∈ O ( N ) H N H_N H N H ↦ H ∘ g − 1 H\mapsto H\circ g^{-1} H ↦ H ∘ g − 1 H ∘ g − 1 H\circ g^{-1} H ∘ g − 1 H Q ( g u ) H_Q(gu) H Q ( g u ) K Q ( g u ) K_Q(gu) K Q ( g u ) G g u G_{gu} G g u H Q ( u ) H_Q(u) H Q ( u ) K Q ( u ) K_Q(u) K Q ( u ) g G u gG_u g G u H H H (166) and (167) , the conditioned field at g u gu g u H ∘ g − 1 H\circ g^{-1} H ∘ g − 1 H h , k , u ∘ g − 1 H^{h,k,u}\circ g^{-1} H h , k , u ∘ g − 1 H h , k , g u H^{h,k,gu} H h , k , g u H h , k , u H^{h,k,u} H h , k , u H ↦ H ∘ g − 1 H\mapsto H\circ g^{-1} H ↦ H ∘ g − 1 X X X (212) for a Hamiltonian H ~ \widetilde H H B B B g X gX g X H ~ ∘ g − 1 \widetilde H\circ g^{-1} H ∘ g − 1 g B gB g B ν N \nu_N ν N R ( g x , g u ) = R ( x , u ) R(gx,gu)=R(x,u) R ( g x , g u ) = R ( x , u ) (223) takes the same value at u u u g u gu g u O ( N ) O(N) O ( N ) S N S_N S N (223) implies (168) for every u ∈ S N u \in S_N u ∈ S N ( H Q ( u ) , G u ) (H_Q(u), G_u) ( H Q ( u ) , G u ) k = p h k = ph k = p h e − c N e^{-cN} e − c N c c c N 0 N_0 N 0
Pure models: critical-point counts and Gibbs concentration on bands
In this section, we prove the hypotheses of the reduction in Section 4 on critical-point counts and on the concentration of the Gibbs measure on bands, for the pure model ξ ( r ) = r p \xi(r)=r^p ξ ( r ) = r p p ≥ 4 p \ge4 p ≥ 4 β > β s ( p ) \beta> \beta_s(p) β > β s ( p ) H Q H_Q H Q m N = N E 0 − ( log N ) / ( 2 c 0 ) m_N = NE_0 - (\log N)/(2c_0) m N = N E 0 − ( log N ) / ( 2 c 0 ) N − 1 / 2 N^{-1/2} N − 1/2
We use the following results from other sections. From § 6.4 , we use the representation and regularity of the field (Lemma 3.2) and the concentration of free energies (Lemma 3.3). From Section 5, we use that the pure model is strictly 1RSB with a nondegenerate contact at Q Q Q E 0 E_0 E 0 K 0 K_0 K 0 Q Q Q § 6.2 , we use the conditioned laws and the law of the Hessian at a conditioned critical point (Definition 7.3 and Corollary 7.6), and Gaussian concentration (Lemma 7.5). We also use the moment bounds for conditioned determinants, the removal of exceptional events, and the bound on expected determinants by the logarithmic potential of the semicircle law (Propositions 7.7 and 7.9 and Corollary 7.8), as well as the marked Kac–Rice formula, the exact count formula, and the Hölder bound for determinant-weighted expectations (Theorem 7.17, Corollary 7.18, and Lemma 7.22).
The pure model has more explicit structure than a general mixture. We use several results of Subag and of Subag and Zeitouni about it. The parameters Q Q Q x x x β \beta β β \beta β β > β s ( p ) \beta> \beta_{\mathrm{s}}(p) β > β s ( p ) Theorem 6.3 , together with Subag’s fixed-parameter estimates for free energies of replicated bands [143]. Since ξ \xi ξ K Q = p H Q K_Q = pH_Q K Q = p H Q
The proof proceeds in stages. In § 6.5.3 , we bound the density of critical values by an exponential with a bounded prefactor on every window of sublinear width (Lemma 9.6 ). In § 6.5.4 , we show that all critical points with energy close to the ground energy are nondegenerate maxima, and that every point with such energy is close to a critical maximum (Proposition 9.12 ); we also prove Proposition 9.1 . In § 6.5.5 , we show that the empirical mean of a fixed number of Gibbs samples with pairwise overlaps close to Q Q Q Proposition 9.15 ). In § 6.5.6 , we combine these two facts with the directing measure and obtain a cover of the Gibbs measure by caps around critical points of near-ground energy (Proposition 9.16). In Sections 9.7 to 9.9, we pass from this cover to critical points in a fixed window and to microscopic bands, of width of order N − 1 / 2 N^{-1/2} N − 1/2 L L L L L L
Notation. Throughout this section, p ≥ 4 p \ge4 p ≥ 4 ξ ( r ) = r p \xi(r) = r^p ξ ( r ) = r p β > β s ( p ) \beta> \beta_{\mathrm{s}}(p) β > β s ( p ) H N H_N H N p p p R N \mathbb{R}^N R N π \pi π Z N , β = ∫ e β H N d ν N Z_{N,\beta} = \int e^{\beta H_N}\,d\nu_N Z N , β = ∫ e β H N d ν N P N = N − 1 log Z N , β P_N = N^{-1}\log Z_{N,\beta} P N = N − 1 log Z N , β ∇ \nabla ∇ ∇ 2 \nabla^2 ∇ 2 ∇ S N \nabla_{S_N} ∇ S N ∇ S N 2 \nabla^2_{S_N} ∇ S N 2 ∇ S N 3 \nabla^3_{S_N} ∇ S N 3 d S N d_{S_N} d S N S N S_N S N ∣ S N ∣ = 2 π N / 2 N ( N − 1 ) / 2 / Γ ( N / 2 ) |S_N| = 2\pi^{N/2}N^{(N-1)/2}/\Gamma(N/2) ∣ S N ∣ = 2 π N /2 N ( N − 1 ) /2 /Γ ( N /2 ) C , c > 0 C,c > 0 C , c > 0 p p p β \beta β N N N N → ∞ N \to\infty N → ∞ Table 9.1 summarizes this order.
Table 9.1. Parameters in the proofs of Proposition 9.1 and Theorem 9.2, in the order in which they are fixed. The window M M M b b b U U U N → ∞ N \to\infty N → ∞
Parameters and statements
Thermodynamic parameters. By Proposition 5.20 , the pair ( ξ , β ) (\xi,\beta) ( ξ , β ) Q Q Q Definition 5.9 ), and the parameters x x x Q Q Q z p > 0 z_p > 0 z p > 0
p [ ( 1 + z p ) log ( 1 + z p ) − z p ] = z p 2 (224) p\left[(1+z_p)\log(1+z_p)-z_p\right]=z_p^2 \tag*{(224)} p [ ( 1 + z p ) log ( 1 + z p ) − z p ] = z p 2 (224) which is the equation υ ( z p ) = 1 / p \upsilon(z_p)=1/p υ ( z p ) = 1/ p
q s = z p 1 + z p , e p = p + z p p ( 1 + z p ) , e ∞ = 2 p − 1 p , q_s=\frac{z_p}{1+z_p},\qquad e_p=\frac{p+z_p}{\sqrt{p(1+z_p)}},\qquad e_\infty=2\sqrt{\frac{p-1}{p}}, q s = 1 + z p z p , e p = p ( 1 + z p ) p + z p , e ∞ = 2 p p − 1 , c p = z p p ( 1 + z p ) , w 0 = p − 1 1 + z p . (225) c_p=\frac{z_p}{\sqrt{p(1+z_p)}},\qquad w_0=\sqrt{\frac{p-1}{1+z_p}}. \tag*{(225)} c p = p ( 1 + z p ) z p , w 0 = 1 + z p p − 1 . (225) By Proposition 5.20, we have z p > 2 ( p − 2 ) z_p>2(p-2) z p > 2 ( p − 2 ) Q Q Q ( q s , 1 ) (q_s,1) ( q s , 1 )
β 2 p ( 1 + z p ) Q p − 2 ( 1 − Q ) 2 = 1 ; (226) \beta^2p(1+z_p)Q^{p-2}(1-Q)^2=1; \tag*{(226)} β 2 p ( 1 + z p ) Q p − 2 ( 1 − Q ) 2 = 1 ; (226) the other parameter is x = z p s / Q x=z_ps/Q x = z p s / Q s = 1 − Q s=1-Q s = 1 − Q H N H_N H N E ( 1 ) = lim N N − 1 E max S N H N = e p E(1)=\lim_{N}N^{-1} \mathbb{E}\max_{S_N}H_N=e_p E ( 1 ) = lim N N − 1 E max S N H N = e p E 0 = e p Q p / 2 E_0=e_pQ^{p/2} E 0 = e p Q p /2 K 0 = p E 0 K_0=pE_0 K 0 = p E 0 ψ Q ( t ) = ( Q + s t ) p − Q p − p Q p − 1 s t \psi_Q(t)=(Q+st)^p-Q^p-pQ^{p-1}st ψ Q ( t ) = ( Q + s t ) p − Q p − p Q p − 1 s t
F ( β ) = β e p Q p / 2 + 1 2 log s + β 2 2 ψ Q ( 1 ) . (227) F(\beta)=\beta e_pQ^{p/2}+\frac{1}{2}\log s+\frac{\beta^2}{2}\psi_Q(1). \tag*{(227)} F ( β ) = β e p Q p /2 + 2 1 log s + 2 β 2 ψ Q ( 1 ) . (227) These formulas were obtained by Subag [142, Theorems 1 and 2] from the generalized TAP representation; see Remark 5.21 for the dictionary.
We record some consequences. Since z p > p − 2 z_p>p-2 z p > p − 2 0 < w 0 < 1 0<w_0<1 0 < w 0 < 1 q s > ( p − 2 ) / ( p − 1 ) ≥ 2 / 3 q_s>(p-2)/(p-1)\ge2/3 q s > ( p − 2 ) / ( p − 1 ) ≥ 2/3
e p = p − 1 p ( w 0 + w 0 − 1 ) , e p 2 − e ∞ 2 = ( z p − p + 2 ) 2 p ( 1 + z p ) > 0. (228) e_p=\sqrt{\frac{p-1}{p}}\left(w_0+w_0^{-1}\right),\qquad e_p^2-e_\infty^2=\frac{(z_p-p+2)^2}{p(1+z_p)}>0. \tag*{(228)} e p = p p − 1 ( w 0 + w 0 − 1 ) , e p 2 − e ∞ 2 = p ( 1 + z p ) ( z p − p + 2 ) 2 > 0. (228) By (9.3) and the definition of x x x
Q > q s > 2 3 , 0 < x < 1 , β s Q ( p − 2 ) / 2 = 1 p ( 1 + z p ) , c p = β x Q p / 2 . (229) Q>q_s>\frac{2}{3},\qquad0<x<1,\qquad\beta sQ^{(p-2)/2}=\frac{1}{\sqrt{p(1+z_p)}},\qquad c_p=\beta xQ^{p/2}. \tag*{(229)} Q > q s > 3 2 , 0 < x < 1 , β s Q ( p − 2 ) /2 = p ( 1 + z p ) 1 , c p = β x Q p /2 . (229) From the last identity, we find that the slope c 0 = β x c_0=\beta x c 0 = β x c 0 = c p Q − p / 2 c_0=c_pQ^{-p/2} c 0 = c p Q − p /2 ξ ( Q ) = Q p \xi(Q)=Q^p ξ ( Q ) = Q p a Q = p Q p a_Q=pQ^p a Q = p Q p b Q = p ( p − 1 ) Q p b_Q=p(p-1)Q^p b Q = p ( p − 1 ) Q p
Center field and windows. The center field and its radial derivative are
H Q ( u ) = H N ( Q u ) = Q p / 2 H N ( u ) , K Q ( u ) = p H Q ( u ) , u ∈ S N , H_Q(u)=H_N(\sqrt{Q}\,u)=Q^{p/2}H_N(u),\qquad K_Q(u)=pH_Q(u),\qquad u\in S_N, H Q ( u ) = H N ( Q u ) = Q p /2 H N ( u ) , K Q ( u ) = p H Q ( u ) , u ∈ S N , where the second identity is Euler’s identity for homogeneous polynomials. We have K 0 = p Q p / 2 e p > p Q p / 2 e ∞ = 2 b Q K_0 = pQ^{p/2}e_p > pQ^{p/2}e_\infty= 2\sqrt{b_Q} K 0 = p Q p /2 e p > p Q p /2 e ∞ = 2 b Q U U U K 0 K_0 K 0 min U > 2 b Q \min U > 2\sqrt{b_Q} min U > 2 b Q
M N = N e p − log N 2 c p , m N = Q p / 2 M N = N E 0 − log N 2 c 0 , (230) M_N = Ne_p - \frac{\log N}{2c_p}, \qquad m_N = Q^{p/2}M_N = NE_0 - \frac{\log N}{2c_0}, \tag*{(230)} M N = N e p − 2 c p log N , m N = Q p /2 M N = N E 0 − 2 c 0 log N , (230) and we let C N , M C_{N,M} C N , M B N , M B_{N,M} B N , M Q Q Q U U U m N m_N m N
C N , M = { u ∈ S N : ∇ S N H Q ( u ) = 0 , ∣ H Q ( u ) − m N ∣ ≤ M , K Q ( u ) / N ∈ U } . C_{N,M} = \left\{u \in S_N : \nabla_{S_N} H_Q(u) = 0,\ \lvert H_Q(u) - m_N\rvert\le M,\ K_Q(u)/N \in U\right\}. C N , M = { u ∈ S N : ∇ S N H Q ( u ) = 0 , ∣ H Q ( u ) − m N ∣ ≤ M , K Q ( u ) / N ∈ U } . We also use the critical points of H N H_N H N M N M_N M N
D N , L = { u ∈ S N : ∇ S N H N ( u ) = 0 , ∣ H N ( u ) − M N ∣ ≤ L } . D_{N,L} = \left\{u \in S_N : \nabla_{S_N} H_N(u) = 0,\ \lvert H_N(u) - M_N\rvert\le L\right\}. D N , L = { u ∈ S N : ∇ S N H N ( u ) = 0 , ∣ H N ( u ) − M N ∣ ≤ L } . The critical points of H Q H_Q H Q H N H_N H N u ∈ D N , L u \in D_{N,L} u ∈ D N , L K Q ( u ) / N = p Q p / 2 H N ( u ) / N = K 0 + O ( ( L + log N ) / N ) K_Q(u)/N = pQ^{p/2}H_N(u)/N = K_0 + O((L + \log N)/N) K Q ( u ) / N = p Q p /2 H N ( u ) / N = K 0 + O (( L + log N ) / N ) K 0 K_0 K 0 U U U L > 0 L > 0 L > 0
C N , Q p / 2 L ⊂ D N , L for every N , C N , Q p / 2 L = D N , L for all sufficiently large N . (231) C_{N,Q^{p/2}L} \subset D_{N,L}\ \text{for every }N,\qquad C_{N,Q^{p/2}L} = D_{N,L}\ \text{for all sufficiently large }N. \tag*{(231)} C N , Q p /2 L ⊂ D N , L for every N , C N , Q p /2 L = D N , L for all sufficiently large N . (231) We count points of S N S_N S N p p p u u u − u -u − u
We now state the two main results of this section.
Proposition 9.1 (Critical points in fixed windows). Let p ≥ 4 p \ge4 p ≥ 4 β > β s ( p ) \beta> \beta_s(p) β > β s ( p ) U U U m N m_N m N C N , M C_{N,M} C N , M M > 0 M > 0 M > 0
sup N ≥ 2 E ∣ C N , M ∣ < ∞ . (232) \sup_{N \ge2} \mathbb{E}\lvert C_{N,M}\rvert< \infty. \tag*{(232)} N ≥ 2 sup E ∣ C N , M ∣ < ∞. (232) Further,
lim M → ∞ lim inf N → ∞ P { C N , M ≠ ∅ and max S N H Q ≤ m N + M } = 1. (233) \lim_{M \to\infty}\liminf_{N \to\infty}\mathbb{P}\left\{C_{N,M} \ne\varnothing\ \text{and}\ \max_{S_N} H_Q \le m_N + M\right\} = 1. \tag*{(233)} M → ∞ lim N → ∞ lim inf P { C N , M = ∅ and S N max H Q ≤ m N + M } = 1. (233) *Finally, for every fixed M > 0 M > 0 M > 0 u ∈ C N , M u \in C_{N,M} u ∈ C N , M H Q H_Q H Q
For b > 0 b > 0 b > 0 u ∈ S N u \in S_N u ∈ S N B u ( b ) = { x ∈ S N : ∣ R ( x , u ) − Q ∣ ≤ b } B_u(b) = \{x \in S_N : \lvert R(x,u) - \sqrt{Q}\rvert\le b\} B u ( b ) = { x ∈ S N : ∣ R ( x , u ) − Q ∣ ≤ b } B N , M ( b ) = ⋃ u ∈ C N , M B u ( b ) B_{N,M}(b) = \bigcup_{u \in C_{N,M}} B_u(b) B N , M ( b ) = ⋃ u ∈ C N , M B u ( b ) b b b
Theorem 9.2 (Gibbs concentration on bands). Let p ≥ 4 p \ge4 p ≥ 4 β > β s ( p ) \beta> \beta_s(p) β > β s ( p ) U U U m N m_N m N C N , M C_{N,M} C N , M
There exists b 0 = b 0 ( p , β ) ∈ ( 0 , Q ) b_0 = b_0(p,\beta) \in(0,\sqrt{Q}) b 0 = b 0 ( p , β ) ∈ ( 0 , Q ) b ∈ ( 0 , b 0 ] b \in(0,b_0] b ∈ ( 0 , b 0 ]
lim M → ∞ lim sup N → ∞ E π ( B N , M ( b ) c ) = 0. (234) \lim_{M \to\infty}\limsup_{N \to\infty}\mathbb{E}\,\pi\left(B_{N,M}(b)^c\right) = 0. \tag*{(234)} M → ∞ lim N → ∞ lim sup E π ( B N , M ( b ) c ) = 0. (234) :::`
For every ε > 0 \varepsilon> 0 ε > 0 M 0 , w ∗ > 0 M_0, w_\ast> 0 M 0 , w ∗ > 0 M ≥ M 0 M \ge M_0 M ≥ M 0 w ≥ w ∗ w \ge w_\ast w ≥ w ∗
lim sup N → ∞ P ( B N , M ( w N − 1 / 2 ) c ) ≤ ε . (235) \limsup_{N \to\infty} \mathbb{P}\left(B_{N,M}(wN^{-1/2})^{c}\right) \le\varepsilon. \tag*{(235)} N → ∞ lim sup P ( B N , M ( w N − 1/2 ) c ) ≤ ε . (235) Part (1) is the concentration property (4.2) for every band width b ∈ ( 0 , b 0 ] b \in(0,b_0] b ∈ ( 0 , b 0 ] § 6.5.4 and Section 9.9, respectively. Table 9.1 lists the parameters that appear in these proofs, in the order in which they are fixed.
The exponent of the latitude integral
The bands around a critical point are weighed by a one-dimensional integral over latitudes. In this section, we choose the latitude a a a 0 ≤ q ≤ 1 0 \le q \le1 0 ≤ q ≤ 1
C β ( q ) = β 2 [ 1 − q 2 p − p q 2 p − 2 ( 1 − q 2 ) ] , C β ′ ( q ) = − 2 β 2 p ( p − 1 ) q 2 p − 3 ( 1 − q 2 ) ≤ 0. (236) C_\beta(q) = \beta^2\left[1-q^{2p}-pq^{2p-2}(1-q^2)\right], \qquad C'_\beta(q) = -2\beta^2p(p-1)q^{2p-3}(1-q^2) \le0. \tag*{(236)} C β ( q ) = β 2 [ 1 − q 2 p − p q 2 p − 2 ( 1 − q 2 ) ] , C β ′ ( q ) = − 2 β 2 p ( p − 1 ) q 2 p − 3 ( 1 − q 2 ) ≤ 0. (236) The quantity N C β ( q ) / β 2 NC_\beta(q)/\beta^2 N C β ( q ) / β 2 q q q
C β ( Q ) = β 2 ψ Q ( 1 ) < 159 320 < 1 2 (237) C_\beta(\sqrt{Q}) = \beta^2\psi_Q(1) < \frac{159}{320} < \frac{1}{2} \tag*{(237)} C β ( Q ) = β 2 ψ Q ( 1 ) < 320 159 < 2 1 (237) for every p ≥ 4 p \ge4 p ≥ 4 β > β s ( p ) \beta> \beta_s(p) β > β s ( p )
Lemma 9.3 (Choice of the cap latitude). For 0 < q < 1 0 < q < 1 0 < q < 1
Λ ( q ) = β e p q p + 1 2 log ( 1 − q 2 ) + 1 2 C β ( q ) . \Lambda(q) = \beta e_p q^{p} + \frac{1}{2}\log(1-q^2) + \frac{1}{2}C_\beta(q). Λ ( q ) = β e p q p + 2 1 log ( 1 − q 2 ) + 2 1 C β ( q ) . There exists a ∈ ( 1 / 2 , Q ) a \in(1/\sqrt{2},\sqrt{Q}) a ∈ ( 1/ 2 , Q )
C β ( a ) < 1 2 , β a p > c p , Λ ( Q ) = F ( β ) , Λ ′ ′ ( Q ) < 0 , (238) C_\beta(a) < \frac{1}{2}, \qquad\beta a^p > c_p, \qquad\Lambda(\sqrt{Q}) = F(\beta), \qquad\Lambda''(\sqrt{Q}) < 0, \tag*{(238)} C β ( a ) < 2 1 , β a p > c p , Λ ( Q ) = F ( β ) , Λ ′′ ( Q ) < 0 , (238) Λ \Lambda Λ [ a , Q ] [a,\sqrt{Q}] [ a , Q ] Λ \Lambda Λ [ Q , 1 ) [\sqrt{Q},1) [ Q , 1 ) Q \sqrt{Q} Q Λ \Lambda Λ [ a , 1 ) [a,1) [ a , 1 ) a a a k ∗ = − Λ ′ ′ ( Q ) > 0 k_*=-\Lambda''(\sqrt{Q})>0 k ∗ = − Λ ′′ ( Q ) > 0
Proof. Let f ( q ) = β q p − 2 ( 1 − q 2 ) f(q)=\beta q^{p-2}(1-q^2) f ( q ) = β q p − 2 ( 1 − q 2 ) Λ \Lambda Λ (236) , we find
Λ ′ ( q ) = p β e p q p − 1 − q 1 − q 2 − β 2 p ( p − 1 ) q 2 p − 3 ( 1 − q 2 ) \Lambda'(q)=p\beta e_p q^{p-1}-\frac{q}{1-q^2}-\beta^2p(p-1)q^{2p-3}(1-q^2) Λ ′ ( q ) = pβ e p q p − 1 − 1 − q 2 q − β 2 p ( p − 1 ) q 2 p − 3 ( 1 − q 2 ) = q 1 − q 2 ( p e p f ( q ) − 1 − p ( p − 1 ) f ( q ) 2 ) . =\frac{q}{1-q^2}\left(p e_p f(q)-1-p(p-1)f(q)^2\right). = 1 − q 2 q ( p e p f ( q ) − 1 − p ( p − 1 ) f ( q ) 2 ) . The quadratic p ( p − 1 ) f 2 − p e p f + 1 p(p-1)f^2-p e_p f+1 p ( p − 1 ) f 2 − p e p f + 1
a ± = e p ± e p 2 − e ∞ 2 2 ( p − 1 ) , a − = 1 p ( 1 + z p ) , a + = 1 + z p p ( p − 1 ) . a_\pm=\frac{e_p\pm\sqrt{e_p^2-e_\infty^2}}{2(p-1)},\qquad
a_-=\frac{1}{\sqrt{p(1+z_p)}},\qquad
a_+=\frac{\sqrt{1+z_p}}{\sqrt{p}(p-1)}. a ± = 2 ( p − 1 ) e p ± e p 2 − e ∞ 2 , a − = p ( 1 + z p ) 1 , a + = p ( p − 1 ) 1 + z p . where we used (9.5) and z p > p − 2 z_p>p-2 z p > p − 2 ( z p − p + 2 ) / p ( 1 + z p ) (z_p-p+2)/\sqrt{p(1+z_p)} ( z p − p + 2 ) / p ( 1 + z p ) a + > a − a_+>a_- a + > a − z p > p − 2 z_p>p-2 z p > p − 2
Λ ′ ( q ) = − p ( p − 1 ) q 1 − q 2 ( f ( q ) − a − ) ( f ( q ) − a + ) . (239) \Lambda'(q)=-\frac{p(p-1)q}{1-q^2}(f(q)-a_-)(f(q)-a_+). \tag*{(239)} Λ ′ ( q ) = − 1 − q 2 p ( p − 1 ) q ( f ( q ) − a − ) ( f ( q ) − a + ) . (239) Note that f ( Q ) = a − f(\sqrt{Q})=a_- f ( Q ) = a − f ′ ( q ) = β q p − 3 [ ( p − 2 ) − p q 2 ] f'(q)=\beta q^{p-3}[(p-2)-pq^2] f ′ ( q ) = β q p − 3 [( p − 2 ) − p q 2 ] q 2 > ( p − 2 ) / p q^2>(p-2)/p q 2 > ( p − 2 ) / p Q > ( p − 2 ) / p Q>(p-2)/p Q > ( p − 2 ) / p f f f ( q 1 , 1 ) (q_1,1) ( q 1 , 1 ) Q \sqrt{Q} Q q ∈ ( Q , 1 ) q\in(\sqrt{Q},1) q ∈ ( Q , 1 ) f ( q ) < a − < a + f(q)<a_-<a_+ f ( q ) < a − < a + (239) implies that Λ ′ ( q ) < 0 \Lambda'(q)<0 Λ ′ ( q ) < 0 q ∈ ( q 1 , Q ) q\in(q_1,\sqrt{Q}) q ∈ ( q 1 , Q ) Q \sqrt{Q} Q a − < f ( q ) < a + a_-<f(q)<a_+ a − < f ( q ) < a + Λ ′ ( q ) > 0 \Lambda'(q)>0 Λ ′ ( q ) > 0 (239) at Q \sqrt{Q} Q
Λ ′ ′ ( Q ) = − p ( p − 1 ) Q 1 − Q f ′ ( Q ) ( a − − a + ) < 0 , \Lambda''(\sqrt{Q})=-\frac{p(p-1)\sqrt{Q}}{1-Q}f'(\sqrt{Q})(a_--a_+)<0, Λ ′′ ( Q ) = − 1 − Q p ( p − 1 ) Q f ′ ( Q ) ( a − − a + ) < 0 , since f ′ ( Q ) < 0 f'(\sqrt{Q})<0 f ′ ( Q ) < 0 a − − a + < 0 a_- - a_+<0 a − − a + < 0 Λ ( Q ) = F ( β ) \Lambda(\sqrt{Q})=F(\beta) Λ ( Q ) = F ( β ) C β ( Q ) = β 2 ψ Q ( 1 ) C_\beta(\sqrt{Q})=\beta^2\psi_Q(1) C β ( Q ) = β 2 ψ Q ( 1 )
Recall from (9.14) that C β ( Q ) < 1 / 2 C_\beta(\sqrt{Q})<1/2 C β ( Q ) < 1/2 β Q p / 2 = c p / χ > c p \beta Q^{p/2}=c_p/\chi>c_p β Q p /2 = c p / χ > c p Q > 2 / 3 > 1 / 2 \sqrt{Q}>\sqrt{2/3}>1/\sqrt{2} Q > 2/3 > 1/ 2 Q \sqrt{Q} Q q 2 q_2 q 2 { 0 } ∪ { q ∈ ( 0 , Q ) : Λ ′ ( q ) ≤ 0 } \{0\}\cup\{q\in(0,\sqrt{Q}):\Lambda'(q)\leq0\} { 0 } ∪ { q ∈ ( 0 , Q ) : Λ ′ ( q ) ≤ 0 } Q \sqrt{Q} Q Λ ′ \Lambda' Λ ′ a < Q a<\sqrt{Q} a < Q q 2 q_2 q 2 1 / 2 1/\sqrt{2} 1/ 2 Q \sqrt{Q} Q
The conditional field and the density of critical values
Lemma 9.4 (Regularity of the pure field). There exist C , c > 0 C,c>0 C , c > 0 N ≥ 2 N\geq2 N ≥ 2 1 − e − c N 1-e^{-cN} 1 − e − c N
sup x ∈ S N ∥ ∇ S N 3 H N ( x ) ∥ o p ≤ C N . (240) \sup_{x\in S_N}\left\|\nabla^3_{S_N}H_N(x)\right\|_{\mathrm{op}}\leq\frac{C}{\sqrt{N}}. \tag*{(240)} x ∈ S N sup ∇ S N 3 H N ( x ) op ≤ N C . (240) Moreover, E sup ∣ x ∣ ≤ N ∣ ∇ H N ( x ) ∣ ≤ C N \mathbb{E}\sup_{|x|\leq\sqrt{N}}|\nabla H_N(x)|\leq C\sqrt{N} E sup ∣ x ∣ ≤ N ∣∇ H N ( x ) ∣ ≤ C N for all N ≥ 2 , ∗ a n d f o r e v e r y f i x e d ∗ N\geq2, *and for every fixed* N ≥ 2 , ∗ an df or e v er y f i x e d ∗ and all sufficiently large N N N
P { ∣ 1 N max S N H N − e p ∣ > r } ≤ 2 e − N r 2 / 8 . (241) \mathbb{P}\left\{\left|\frac{1}{N}\max_{S_N}H_N-e_p\right|>r\right\}\leq2e^{-Nr^2/8}. \tag*{(241)} P { N 1 S N max H N − e p > r } ≤ 2 e − N r 2 /8 . (241) Proof. Let K , c K,c K , c x ∈ S N x\in S_N x ∈ S N z , v , w z,v,w z , v , w x x x A = ∇ 2 H N − N − 1 ( x ⋅ ∇ H N ) I = ∇ 2 H N − ( p / N ) H N I A=\nabla^2H_N-N^{-1}(x\cdot\nabla H_N)I=\nabla^2H_N-(p/N)H_NI A = ∇ 2 H N − N − 1 ( x ⋅ ∇ H N ) I = ∇ 2 H N − ( p / N ) H N I x ⋅ ∇ H N = p H N x\cdot\nabla H_N=pH_N x ⋅ ∇ H N = p H N V V V S N S_N S N ∇ z V + N − 1 ( z ⋅ V ) x \nabla_zV+N^{-1}(z\cdot V)x ∇ z V + N − 1 ( z ⋅ V ) x A ( V , W ) A(V,W) A ( V , W ) z z z V V V W W W
∇ S N 3 H N ( x ) [ z , v , w ] = ∇ 3 H N ( x ) [ z , v , w ] − p N ( ∇ H N ( x ) ⋅ z ) ( v ⋅ w ) − p − 1 N [ ( z ⋅ v ) ∇ H N ( x ) ⋅ w + ( z ⋅ w ) ∇ H N ( x ) ⋅ v ] . \nabla^3_{S_N}H_N(x)[z,v,w]=\nabla^3H_N(x)[z,v,w]-\frac{p}{N}(\nabla H_N(x)\cdot z)(v\cdot w)
-\frac{p-1}{N}\left[(z\cdot v)\nabla H_N(x)\cdot w+(z\cdot w)\nabla H_N(x)\cdot v\right]. ∇ S N 3 H N ( x ) [ z , v , w ] = ∇ 3 H N ( x ) [ z , v , w ] − N p ( ∇ H N ( x ) ⋅ z ) ( v ⋅ w ) − N p − 1 [ ( z ⋅ v ) ∇ H N ( x ) ⋅ w + ( z ⋅ w ) ∇ H N ( x ) ⋅ v ] . Here we used A ( x , w ) = ∇ 2 H N ( x ) [ x , w ] = ( p − 1 ) ∇ H N ( x ) ⋅ w A(x,w)=\nabla^2H_N(x)[x,w]=(p-1)\nabla H_N(x)\cdot w A ( x , w ) = ∇ 2 H N ( x ) [ x , w ] = ( p − 1 ) ∇ H N ( x ) ⋅ w w w w ∇ H N ⋅ w \nabla H_N\cdot w ∇ H N ⋅ w p − 1 p-1 p − 1 K / N + 3 p K / N K/\sqrt{N}+3pK/\sqrt{N} K / N + 3 p K / N
For the second assertion, let Y = sup ∣ x ∣ ≤ N , ∣ v ∣ ≤ 1 ∇ H N ( x ) ⋅ v Y=\sup_{|x|\leq\sqrt{N},|v|\leq1}\nabla H_N(x)\cdot v Y = sup ∣ x ∣ ≤ N , ∣ v ∣ ≤ 1 ∇ H N ( x ) ⋅ v H N ( x ) = g ⋅ Ψ ( x ) H_N(x)=g\cdot\Psi(x) H N ( x ) = g ⋅ Ψ ( x ) ∇ H N ( x ) ⋅ v \nabla H_N(x)\cdot v ∇ H N ( x ) ⋅ v g g g p ∣ x ∣ p − 1 ∣ v ∣ N − ( p − 1 ) / 2 ≤ p p|x|^{p-1}|v|N^{-(p-1)/2}\leq p p ∣ x ∣ p − 1 ∣ v ∣ N − ( p − 1 ) /2 ≤ p Y Y Y p p p g g g P { Y ≤ E Y − t } ≤ e − t 2 / ( 2 p 2 ) \mathbb{P}\{Y\leq\mathbb{E}Y-t\}\leq e^{-t^2/(2p^2)} P { Y ≤ E Y − t } ≤ e − t 2 / ( 2 p 2 ) t > 0 t>0 t > 0 P { Y > K N } ≤ e − c N \mathbb{P}\{Y>K\sqrt{N}\}\leq e^{-cN} P { Y > K N } ≤ e − c N E Y > K N + 2 p c N \mathbb{E}Y>K\sqrt{N}+2p\sqrt{cN} E Y > K N + 2 p c N 1 − e − 2 c N ≤ P { Y > K N } ≤ e − c N 1-e^{-2cN}\leq\mathbb{P}\{Y>K\sqrt{N}\}\leq e^{-cN} 1 − e − 2 c N ≤ P { Y > K N } ≤ e − c N N N N E Y ≤ ( K + 2 p C ) N \mathbb{E}Y \le(K+2p\sqrt{C})\sqrt{N} E Y ≤ ( K + 2 p C ) N N N N N N N E Y ≤ p E ∣ g ∣ < ∞ \mathbb{E}Y \le p\mathbb{E}|g| < \infty E Y ≤ p E ∣ g ∣ < ∞
For (9.18), note that max S N H N \max_{S_N} H_N max S N H N N \sqrt{N} N g g g g g g ∣ Ψ ( x ) ∣ = N |\Psi(x)|=\sqrt{N} ∣Ψ ( x ) ∣ = N E ( 1 ) E(1) E ( 1 ) ∣ E max S N H N / N − e p ∣ ≤ r / 2 |\mathbb{E}\max_{S_N} H_N/N-e_p| \le r/2 ∣ E max S N H N / N − e p ∣ ≤ r /2 N N N N N N s = N r / 2 s=Nr/2 s = N r /2 □ \square □
Conditioned laws and the Kac–Rice formula. We use the results of Section 7 with the center radius equal to one, so that the center field is H N H_N H N ξ ( 1 ) = 1 \xi(1)=1 ξ ( 1 ) = 1 a 1 = p a_1=p a 1 = p b 1 = p ( p − 1 ) b_1=p(p-1) b 1 = p ( p − 1 ) p H N pH_N p H N u ∈ S N u\in S_N u ∈ S N h ∈ R h\in\mathbb{R} h ∈ R P u ; h \mathbb{P}_{u;h} P u ; h E u ; h \mathbb{E}_{u;h} E u ; h u u u h / N h/N h / N H N ( u ) = h H_N(u)=h H N ( u ) = h ∇ S N H N ( u ) = 0 \nabla_{S_N}H_N(u)=0 ∇ S N H N ( u ) = 0 n = N − 1 n=N-1 n = N − 1 G n G_n G n n n n
ζ N = ( p ( p − 1 ) ( N − 1 ) N ) 1 / 2 , D ( u ) = ∣ det ∇ S N 2 H N ( u ) ∣ . \zeta_N=\left(\frac{p(p-1)(N-1)}{N}\right)^{1/2},\qquad D(u)=\left|\det\nabla^2_{S_N}H_N(u)\right|. ζ N = ( N p ( p − 1 ) ( N − 1 ) ) 1/2 , D ( u ) = det ∇ S N 2 H N ( u ) . Recall from Corollary 7.6 that, under P u ; h \mathbb{P}_{u;h} P u ; h ∇ S N 2 H N ( u ) \nabla^2_{S_N}H_N(u) ∇ S N 2 H N ( u ) u ⊥ u^\perp u ⊥
ζ N G n − p h N I d n . (242) \zeta_NG_n-\frac{ph}{N}I d_n. \tag*{(242)} ζ N G n − N p h I d n . (242) It follows that μ N ( h ) = E u ; h D ( u ) = μ n , N ( p h / N ) \mu_N(h)=\mathbb{E}_{u;h}D(u)=\mu_{n,N}(ph/N) μ N ( h ) = E u ; h D ( u ) = μ n , N ( p h / N ) u u u h h h ∣ det ( ζ N G n − ( p h / N ) I d n ) ∣ |\det(\zeta_NG_n-(ph/N)I d_n)| ∣ det ( ζ N G n − ( p h / N ) I d n ) ∣ h h h h h h ( ζ N ∥ G n ∥ op + C ) n (\zeta_N\|G_n\|_{\operatorname{op}}+C)^n ( ζ N ∥ G n ∥ op + C ) n h h h J ⊂ R J\subset\mathbb{R} J ⊂ R φ ( H , u ) ≥ 0 \varphi(H,u)\ge0 φ ( H , u ) ≥ 0
E ∑ u : ∇ S N H N ( u ) = 0 H N ( u ) ∈ J φ ( H , u ) = ∫ J ρ N ( h ) ∫ S N E u ; h [ D ( u ) φ ( H , u ) ] μ N ( h ) ν N ( d u ) d h . (243) \mathbb{E}\sum_{\substack{u:\nabla_{S_N}H_N(u)=0\\H_N(u)\in J}}\varphi(H,u)
=\int_J\rho_N(h)\int_{S_N}\frac{\mathbb{E}_{u;h}[D(u)\varphi(H,u)]}{\mu_N(h)}\,\nu_N(du)\,dh. \tag*{(243)} E u : ∇ S N H N ( u ) = 0 H N ( u ) ∈ J ∑ φ ( H , u ) = ∫ J ρ N ( h ) ∫ S N μ N ( h ) E u ; h [ D ( u ) φ ( H , u )] ν N ( d u ) d h . (243) where
ρ N ( h ) = ∣ S N ∣ ( 2 π p ) − ( N − 1 ) / 2 ( 2 π N ) − 1 / 2 e − h 2 / ( 2 N ) μ N ( h ) . (244) \rho_N(h)=|S_N|(2\pi p)^{-(N-1)/2}(2\pi N)^{-1/2}e^{-h^2/(2N)}\mu_N(h). \tag*{(244)} ρ N ( h ) = ∣ S N ∣ ( 2 π p ) − ( N − 1 ) /2 ( 2 π N ) − 1/2 e − h 2 / ( 2 N ) μ N ( h ) . (244) With φ = 1 \varphi=1 φ = 1 ρ N \rho_N ρ N { H N ( u ) : ∇ S N H N ( u ) = 0 } \{H_N(u):\nabla_{S_N}H_N(u)=0\} { H N ( u ) : ∇ S N H N ( u ) = 0 } S N S_N S N
Let Δ = e p − e ∞ > 0 \Delta=e_p-e_\infty>0 Δ = e p − e ∞ > 0 k = p h / N k=ph/N k = p h / N g = p Δ / 2 g=p\Delta/2 g = p Δ/2 k + = p ( e p + Δ / 2 ) k_+=p(e_p+\Delta/2) k + = p ( e p + Δ/2 ) K 2 K_2 K 2 p p p
E u ; h [ D ( u ) 2 ] ≤ K 2 μ N ( h ) 2 for all N ≥ 2 , u ∈ S N and ∣ h N − e p ∣ ≤ Δ 2 . (245) \mathbb{E}_{u;h}\left[D(u)^2\right]\le K_2\mu_N(h)^2\qquad\text{for all }N\ge2,\ u\in S_N\text{ and }\left|\frac{h}{N}-e_p\right|\le\frac{\Delta}{2}. \tag*{(245)} E u ; h [ D ( u ) 2 ] ≤ K 2 μ N ( h ) 2 for all N ≥ 2 , u ∈ S N and N h − e p ≤ 2 Δ . (245) Indeed, for such h h h p h / N ≥ p e ∞ + p Δ / 2 = 2 b ˉ 1 + p Δ / 2 ph/N\ge pe_\infty+p\Delta/2=2\sqrt{\bar b_1}+p\Delta/2 p h / N ≥ p e ∞ + p Δ/2 = 2 b ˉ 1 + p Δ/2
The next lemma describes the whole field under P u ; h \mathbb{P}_{u;h} P u ; h u u u
Lemma 9.5 (The field conditioned on a critical point). Let u ∈ S N u\in S_N u ∈ S N h ∈ R h\in\mathbb{R} h ∈ R x ∈ S N ∖ { ± u } x\in S_N\setminus\{\pm u\} x ∈ S N ∖ { ± u } x = q u + N ( 1 − q 2 ) ω x=qu+\sqrt{N(1-q^2)}\,\omega x = q u + N ( 1 − q 2 ) ω q = R ( x , u ) q=R(x,u) q = R ( x , u ) ω \omega ω u u u
α k ( q ) = ( p k ) 1 / 2 q p − k ( 1 − q 2 ) k / 2 , 0 ≤ k ≤ p . \alpha_k(q)=\binom{p}{k}^{1/2}q^{p-k}(1-q^2)^{k/2},\qquad0\le k\le p. α k ( q ) = ( k p ) 1/2 q p − k ( 1 − q 2 ) k /2 , 0 ≤ k ≤ p . Under P u ; h \mathbb{P}_{u;h} P u ; h
H N ( q u + N ( 1 − q 2 ) ω ) = h q p + N ∑ k = 2 p α k ( q ) X k ( ω ) , (246) H_N\left(qu+\sqrt{N(1-q^2)}\,\omega\right)=hq^p+\sqrt{N}\sum_{k=2}^{p}\alpha_k(q)X_k(\omega), \tag*{(246)} H N ( q u + N ( 1 − q 2 ) ω ) = h q p + N k = 2 ∑ p α k ( q ) X k ( ω ) , (246) where X 2 , … , X p X_2,\ldots,X_p X 2 , … , X p u ⊥ u^\perp u ⊥ E X k ( ω ) X k ( ω ′ ) = ( ω ⋅ ω ′ ) k \mathbb{E}X_k(\omega)X_k(\omega')=(\omega\cdot\omega')^k E X k ( ω ) X k ( ω ′ ) = ( ω ⋅ ω ′ ) k ∇ S N 2 H N ( u ) + ( p h / N ) Id \nabla^2_{S_N}H_N(u)+(ph/N)\operatorname{Id} ∇ S N 2 H N ( u ) + ( p h / N ) Id X 2 X_2 X 2 (246) .
Proof. This is [141, Lemmas 14 and 15 and Corollary 17], written in our notation: Subag’s pure field of degree k k k N − 1 \sqrt{N-1} N − 1 u ⊥ u^\perp u ⊥ N / ( N − 1 ) \sqrt{N/(N-1)} N / ( N − 1 ) N X k ( ω ) \sqrt{N}X_k(\omega) N X k ( ω ) u = N e N u=\sqrt{N}e_N u = N e N k k k N N N k k k N α k ( q ) X k ( ω ) \sqrt{N}\alpha_k(q)X_k(\omega) N α k ( q ) X k ( ω ) X k X_k X k k k k ω \omega ω E H N ( x ) H N ( x ′ ) = N ( q q ′ + ( 1 − q 2 ) ( 1 − q ′ 2 ) ω ⋅ ω ′ ) p \mathbb{E}H_N(x)H_N(x')=N\left(qq'+\sqrt{(1-q^2)(1-q'^2)}\,\omega\cdot\omega'\right)^p E H N ( x ) H N ( x ′ ) = N ( q q ′ + ( 1 − q 2 ) ( 1 − q ′2 ) ω ⋅ ω ′ ) p k = 0 k=0 k = 0 q p H N ( u ) q^pH_N(u) q p H N ( u ) k = 1 k=1 k = 1 N α 1 ( q ) ζ ⋅ ω \sqrt{N}\alpha_1(q)\,\zeta\cdot\omega N α 1 ( q ) ζ ⋅ ω ζ \zeta ζ u ⊥ u^\perp u ⊥ ∇ S N H N ( u ) = p ζ \nabla_{S_N}H_N(u)=\sqrt{p}\,\zeta ∇ S N H N ( u ) = p ζ u u u ( H N ( u ) , ∇ S N H N ( u ) ) (H_N(u),\nabla_{S_N}H_N(u)) ( H N ( u ) , ∇ S N H N ( u )) h q p hq^p h q p 0 0 0 y = N ( 1 − q 2 ) ω y=\sqrt{N(1-q^2)}\,\omega y = N ( 1 − q 2 ) ω u u u k ≥ 3 k\ge3 k ≥ 3 ∣ y ∣ 3 |y|^3 ∣ y ∣ 3 h q p = h − ( p h / 2 N ) ∣ y ∣ 2 + O ( ∣ y ∣ 4 ) hq^p=h-(ph/2N)|y|^2+O(|y|^4) h q p = h − ( p h /2 N ) ∣ y ∣ 2 + O ( ∣ y ∣ 4 ) u u u − ( p h / N ) Id -(ph/N)\operatorname{Id} − ( p h / N ) Id y = 0 y=0 y = 0 k = 2 k=2 k = 2 X 2 X_2 X 2
We now bound ρ N \rho_N ρ N t > 2 t > 2 t > 2 w ( t ) = ( t − t 2 − 4 ) / 2 w(t) = (t - \sqrt{t^2 - 4})/2 w ( t ) = ( t − t 2 − 4 ) /2 g s c ( t ) g_{\mathrm{sc}}(t) g sc ( t ) σ s c \sigma_{\mathrm{sc}} σ sc § 6.2.4 , let Ω ( t ) = ∫ log ( t − λ ) σ s c ( d λ ) \Omega(t) = \int\log(t-\lambda)\,\sigma_{\mathrm{sc}}(\mathrm{d}\lambda) Ω ( t ) = ∫ log ( t − λ ) σ sc ( d λ )
γ p = p p − 1 , θ ( e ) = 1 2 + 1 2 log ( p − 1 ) − e 2 2 + Ω ( γ p e ) , e > e ∞ . \gamma_p = \sqrt{\frac{p}{p-1}}, \qquad\theta(e) = \frac{1}{2} + \frac{1}{2}\log(p-1) - \frac{e^2}{2} + \Omega(\gamma_p e), \qquad e > e_\infty. γ p = p − 1 p , θ ( e ) = 2 1 + 2 1 log ( p − 1 ) − 2 e 2 + Ω ( γ p e ) , e > e ∞ . Note that γ p e > 2 \gamma_p e > 2 γ p e > 2 e > e ∞ e > e_\infty e > e ∞ θ ( e ) \theta(e) θ ( e ) Θ p ( − e ) \Theta_p(-e) Θ p ( − e ) N e Ne N e
Lemma 9.6 (Density of critical values). 1. The function θ \theta θ ( e ∞ , ∞ ) (e_\infty,\infty) ( e ∞ , ∞ ) θ ( e p ) = 0 \theta(e_p) = 0 θ ( e p ) = 0 θ ′ ( e p ) = − c p \theta'(e_p) = -c_p θ ′ ( e p ) = − c p
Let J J J ( e ∞ , ∞ ) (e_\infty,\infty) ( e ∞ , ∞ ) C C C p p p J J J N ≥ 2 N \ge2 N ≥ 2 h / N ( N − 1 ) ∈ J h/\sqrt{N(N-1)} \in J h / N ( N − 1 ) ∈ J
ρ N ( h ) ≤ C N − 1 / 2 exp [ ( N − 1 ) θ ( h N ( N − 1 ) ) ] . (247) \rho_N(h) \le C N^{-1/2}\exp\left[(N-1)\theta\left(\frac{h}{\sqrt{N(N-1)}}\right)\right]. \tag*{(247)} ρ N ( h ) ≤ C N − 1/2 exp [ ( N − 1 ) θ ( N ( N − 1 ) h ) ] . (247) Let a N , a N ′ ≥ 0 a_N,a'_N \ge0 a N , a N ′ ≥ 0 a N + a N ′ = o ( N ) a_N+a'_N=o(N) a N + a N ′ = o ( N ) C C C p p p N N N h ∈ [ M N − a N , M N + a N ′ ] h \in[M_N-a_N,M_N+a'_N] h ∈ [ M N − a N , M N + a N ′ ]
ρ N ( h ) ≤ C e c p ( M N − h ) . (248) \rho_N(h) \le C e^{c_p(M_N-h)}. \tag*{(248)} ρ N ( h ) ≤ C e c p ( M N − h ) . (248) The bound (248) has a bounded prefactor on every window of sublinear width. This is what the summable bounds in Section 9.9, on the bands around critical points far below the centering, require.
Proof. For (1), we have Ω ′ ( t ) = w ( t ) \Omega'(t) = w(t) Ω ′ ( t ) = w ( t ) Ω ′ ′ ( t ) = w ′ ( t ) = ( 1 − t / t 2 − 4 ) / 2 < 0 \Omega''(t) = w'(t) = (1-t/\sqrt{t^2-4})/2 < 0 Ω ′′ ( t ) = w ′ ( t ) = ( 1 − t / t 2 − 4 ) /2 < 0 t > 2 t > 2 t > 2 θ ′ ′ ( e ) = − 1 + γ p 2 Ω ′ ′ ( γ p e ) < 0 \theta''(e) = -1+\gamma_p^2\Omega''(\gamma_p e)<0 θ ′′ ( e ) = − 1 + γ p 2 Ω ′′ ( γ p e ) < 0 v ∈ ( 0 , 1 ] v \in(0,1] v ∈ ( 0 , 1 ] w ( v + v − 1 ) = v w(v+v^{-1})=v w ( v + v − 1 ) = v Ω ( 2 ) = 1 / 2 \Omega(2)=1/2 Ω ( 2 ) = 1/2 Ω \Omega Ω Ω ′ = w \Omega'=w Ω ′ = w t = v ′ + 1 / v ′ t=v'+1/v' t = v ′ + 1/ v ′ v ′ ∈ [ v , 1 ] v'\in[v,1] v ′ ∈ [ v , 1 ]
Ω ( v + v − 1 ) = 1 2 + ∫ 1 v ( v ′ − 1 v ′ ) d v ′ = v 2 2 − log v . \Omega(v+v^{-1}) = \frac{1}{2}+\int_1^v\left(v'-\frac{1}{v'}\right)\,\mathrm{d}v'=\frac{v^2}{2}-\log v. Ω ( v + v − 1 ) = 2 1 + ∫ 1 v ( v ′ − v ′ 1 ) d v ′ = 2 v 2 − log v . By (9.5), γ p e p = w 0 + w 0 − 1 \gamma_p e_p=w_0+w_0^{-1} γ p e p = w 0 + w 0 − 1 Ω ( γ p e p ) = w 0 2 / 2 − log w 0 \Omega(\gamma_p e_p)=w_0^2/2-\log w_0 Ω ( γ p e p ) = w 0 2 /2 − log w 0 Ω ′ ( γ p e p ) = w 0 \Omega'(\gamma_p e_p)=w_0 Ω ′ ( γ p e p ) = w 0 w 0 2 = ( p − 1 ) / ( 1 + z p ) w_0^2=(p-1)/(1+z_p) w 0 2 = ( p − 1 ) / ( 1 + z p ) e p 2 = ( p + z p ) 2 / ( p ( 1 + z p ) ) e_p^2=(p+z_p)^2/(p(1+z_p)) e p 2 = ( p + z p ) 2 / ( p ( 1 + z p ))
θ ( e p ) = p ( 1 + z p ) log ( 1 + z p ) − p z p − z p 2 2 p ( 1 + z p ) = 0 , θ ′ ( e p ) = − e p + γ p w 0 = − c p , \theta(e_p)=\frac{p(1+z_p)\log(1+z_p)-p z_p-z_p^2}{2p(1+z_p)}=0,\qquad\theta'(e_p)=-e_p+\gamma_p w_0=-c_p, θ ( e p ) = 2 p ( 1 + z p ) p ( 1 + z p ) log ( 1 + z p ) − p z p − z p 2 = 0 , θ ′ ( e p ) = − e p + γ p w 0 = − c p , where the first identity is (9.1), and the second follows from γ p w 0 = p / ( 1 + z p ) \gamma_p w_0=\sqrt{p/(1+z_p)} γ p w 0 = p / ( 1 + z p ) e = h / N ( N − 1 ) e = h/\sqrt{N(N-1)} e = h / N ( N − 1 ) v = γ p e v = \gamma_p e v = γ p e v = ( p h / N ) / ζ N v = (ph/N)/\zeta_N v = ( p h / N ) / ζ N (242) , μ N ( h ) = ζ N − 1 E ∣ det ( v Id n − G n ) ∣ \mu_N(h) = \zeta_N^{-1}\mathbb{E}|\det(v\operatorname{Id}_n-G_n)| μ N ( h ) = ζ N − 1 E ∣ det ( v Id n − G n ) ∣ − G n -G_n − G n G n G_n G n e 2 ( N − 1 ) = h 2 / N e^2(N-1)=h^2/N e 2 ( N − 1 ) = h 2 / N (244) becomes
ρ N ( h ) = ∣ S N ∣ N − ( N − 1 ) / 2 ( ( p − 1 ) ( N − 1 ) 2 π ) ( N − 1 ) / 2 ( 2 π N ) − 1 / 2 e − ( N − 1 ) e 2 / 2 E ∣ det ( v Id n − G n ) ∣ . (249) \rho_N(h)=|S_N|N^{-(N-1)/2}\left(\frac{(p-1)(N-1)}{2\pi}\right)^{(N-1)/2}(2\pi N)^{-1/2}e^{-(N-1)e^2/2}\mathbb{E}|\det(v\operatorname{Id}_n-G_n)|. \tag*{(249)} ρ N ( h ) = ∣ S N ∣ N − ( N − 1 ) /2 ( 2 π ( p − 1 ) ( N − 1 ) ) ( N − 1 ) /2 ( 2 π N ) − 1/2 e − ( N − 1 ) e 2 /2 E ∣ det ( v Id n − G n ) ∣. (249) The interval γ p J \gamma_p\mathcal{J} γ p J ( 2 , ∞ ) (2,\infty) ( 2 , ∞ ) E ∣ det ( v Id n − G n ) ∣ ≤ C e ( N − 1 ) Ω ( v ) \mathbb{E}|\det(v\operatorname{Id}_n-G_n)|\le Ce^{(N-1)\Omega(v)} E ∣ det ( v Id n − G n ) ∣ ≤ C e ( N − 1 ) Ω ( v ) C C C p p p J \mathcal{J} J ∣ S N ∣ N − ( N − 1 ) / 2 ( ( N − 1 ) / ( 2 π ) ) ( N − 1 ) / 2 ( 2 π N ) − 1 / 2 e − ( N − 1 ) / 2 = ( 1 + o ( 1 ) ) ( π N ) − 1 / 2 |S_N|N^{-(N-1)/2}((N-1)/(2\pi))^{(N-1)/2}(2\pi N)^{-1/2}e^{-(N-1)/2}=(1+o(1))(\pi N)^{-1/2} ∣ S N ∣ N − ( N − 1 ) /2 (( N − 1 ) / ( 2 π ) ) ( N − 1 ) /2 ( 2 π N ) − 1/2 e − ( N − 1 ) /2 = ( 1 + o ( 1 )) ( π N ) − 1/2 (249) and using ( N − 1 ) [ log ( p − 1 ) + 1 − e 2 ] / 2 + ( N − 1 ) Ω ( γ p e ) = ( N − 1 ) θ ( e ) (N-1)[\log(p-1)+1-e^2]/2+(N-1)\Omega(\gamma_p e)=(N-1)\theta(e) ( N − 1 ) [ log ( p − 1 ) + 1 − e 2 ] /2 + ( N − 1 ) Ω ( γ p e ) = ( N − 1 ) θ ( e ) (247) .
For (3), let J = [ e p − Δ / 2 , e p + Δ / 2 ] \mathcal{J}=[e_p-\Delta/2,e_p+\Delta/2] J = [ e p − Δ/2 , e p + Δ/2 ] C C C h ∈ [ M N − a N ′ , M N + a N ′ ] h\in[M_N-a'_N,M_N+a'_N] h ∈ [ M N − a N ′ , M N + a N ′ ] N N N h / N ( N − 1 ) ∈ J h/\sqrt{N(N-1)}\in\mathcal{J} h / N ( N − 1 ) ∈ J θ ( e ) ≤ − c p ( e − e p ) \theta(e)\le-c_p(e-e_p) θ ( e ) ≤ − c p ( e − e p ) e > e ∞ e>e_\infty e > e ∞ r N = ( N − 1 ) / N r_N=\sqrt{(N-1)/N} r N = ( N − 1 ) / N v = M N − h v=M_N-h v = M N − h (247) , we find
log ρ N ( h ) ≤ log C − 1 2 log N − c p r N h + c p ( N − 1 ) e p = log C + c p r N v + c p e p [ ( N − 1 ) − N r N ] + r N − 1 2 log N , (250) \begin{aligned}
\log\rho_N(h)\le\log C-\frac{1}{2}\log N-c_p r_N h+c_p(N-1)e_p \\
&=\log C+c_p r_Nv+c_p e_p\left[(N-1)-Nr_N\right]+\frac{r_N-1}{2}\log N, \tag*{(250)}
\end{aligned} log ρ N ( h ) ≤ log C − 2 1 log N − c p r N h + c p ( N − 1 ) e p = log C + c p r N v + c p e p [ ( N − 1 ) − N r N ] + 2 r N − 1 log N , (250) where we used r N M N = r N N e p − r N ( log N ) / ( 2 c p ) r_NM_N=r_NNe_p-r_N(\log N)/(2c_p) r N M N = r N N e p − r N ( log N ) / ( 2 c p ) ( N − 1 ) − N r N → − 1 / 2 (N-1)-Nr_N\to-1/2 ( N − 1 ) − N r N → − 1/2 ( 1 − r N ) log N → 0 (1-r_N)\log N\to0 ( 1 − r N ) log N → 0 v ≥ 0 v\ge0 v ≥ 0 r N v ≤ v r_Nv\le v r N v ≤ v v < 0 v<0 v < 0 c p ( r N − 1 ) v ≤ c p ∣ v ∣ / N = o ( 1 ) c_p(r_N-1)v\le c_p|v|/N=o(1) c p ( r N − 1 ) v ≤ c p ∣ v ∣/ N = o ( 1 ) ∣ v ∣ ≤ a N ′ = o ( N ) |v|\le a'_N=o(N) ∣ v ∣ ≤ a N ′ = o ( N ) (248) .
The proof of (2) is the computation of [144, Section 4], with the Hermite asymptotics replaced by the upper bound of Proposition 7.9; only an upper bound with a bounded prefactor is needed below.
Remark 9.7 (Windows of sublinear width). For windows of fixed width, (248) also follows from the case of compact windows in [144, Proposition 3]. On windows of width much larger than N \sqrt{N} N (247) differs from its linearization − c p ( h − M N ) -c_p(h-M_N) − c p ( h − M N ) ( h − N e p ) 2 / N (h-Ne_p)^2/N ( h − N e p ) 2 / N θ ′ ′ ( e p ) < 0 \theta''(e_p)<0 θ ′′ ( e p ) < 0 (248) , which is what the arguments below need, and not a relative asymptotic for ρ N \rho_N ρ N
Lemma 9.8 (Tightness of the maximum). The sequence max S N H N − M N \max_{S_N}H_N-M_N max S N H N − M N is tight. Moreover, there exists C C C such that lim sup N → ∞ P { max S N H N > M N + L } ≤ C e − c p L / 2 \limsup_{N\to\infty}\mathbb{P}\{\max_{S_N}H_N>M_N+L\}\le Ce^{-c_pL/2} lim sup N → ∞ P { max S N H N > M N + L } ≤ C e − c p L /2 for every L ≥ 0 L\ge0 L ≥ 0
Proof. We begin with the upper tail. Let L ≥ 0 L\ge0 L ≥ 0 J = [ M N + L , N ( e p + Δ / 4 ) ] \mathcal{J}=[M_N+L,N(e_p+\Delta/4)] J = [ M N + L , N ( e p + Δ/4 )] { M N + L < max H N ≤ N ( e p + Δ / 4 ) } \{M_N+L<\max H_N\le N(e_p+\Delta/4)\} { M N + L < max H N ≤ N ( e p + Δ/4 )} H N H_N H N
with value in J J J (243) with φ = 1 \varphi=1 φ = 1 ∫ J ρ N ( h ) d h \int_J \rho_N(h)\,dh ∫ J ρ N ( h ) d h h ∈ J h\in J h ∈ J N N N h / N ( N − 1 ) ∈ [ e p − Δ / 2 , e p + Δ / 2 ] h/\sqrt{N(N-1)}\in[e_p-\Delta/2,e_p+\Delta/2] h / N ( N − 1 ) ∈ [ e p − Δ/2 , e p + Δ/2 ] (250) , which uses only (247) and the tangent bound, implies that ρ N ( h ) ≤ C exp ( − c p r N ( h − M N ) ) ≤ C e − c p ( h − M N ) / 2 \rho_N(h)\le C\exp(-c_p r_N(h-M_N))\le Ce^{-c_p(h-M_N)/2} ρ N ( h ) ≤ C exp ( − c p r N ( h − M N )) ≤ C e − c p ( h − M N ) /2 ∫ J ρ N ≤ ( 2 C / c p ) e − c p L / 2 \int_J\rho_N\le(2C/c_p)e^{-c_pL/2} ∫ J ρ N ≤ ( 2 C / c p ) e − c p L /2 { max H N > N ( e p + Δ / 4 ) } \{\max H_N>N(e_p+\Delta/4)\} { max H N > N ( e p + Δ/4 )} (241) , and the bound on the upper tail follows.
The lower tail is the main content of the work of Subag and Zeitouni. They identified the extremal process of the critical values and the limiting law of the minimum of H N H_N H N − N e p + ( log N ) / ( 2 c p ) -Ne_p+(\log N)/(2c_p) − N e p + ( log N ) / ( 2 c p ) p p p p ≥ 3 p\ge3 p ≥ 3 E 0 E_0 E 0 Θ p ( − ⋅ ) \Theta_p(-\cdot) Θ p ( − ⋅ ) E ∞ E_\infty E ∞ e p e_p e p c p = Θ p ′ ( − E 0 ) c_p=\Theta_p'(-E_0) c p = Θ p ′ ( − E 0 ) c p c_p c p Lemma 9.6 (1). Applied to the field − H N -H_N − H N H N H_N H N max S N H N − M N \max_{S_N}H_N-M_N max S N H N − M N
Lemma 9.9 (Marked first moments in windows of sublinear width). Let a N , a N ′ ≥ 0 a_N,a'_N\ge0 a N , a N ′ ≥ 0 be deterministic with a N + a N ′ = o ( N ) a_N+a'_N=o(N) a N + a N ′ = o ( N ) and let J N ⊂ [ M N − a N , M N + a N ′ ] J_N\subset[M_N-a_N,M_N+a'_N] J N ⊂ [ M N − a N , M N + a N ′ ] be deterministic Borel sets. Let Y ( H , u ) ≥ 0 Y(H,u)\ge0 Y ( H , u ) ≥ 0 be a Borel function of the field and the point, and let y N : J N → [ 0 , ∞ ] y_N:J_N\to[0,\infty] y N : J N → [ 0 , ∞ ] be a Borel function such that E u ; h [ Y ( H , u ) 2 ] ≤ y N ( h ) 2 \mathbb{E}_{u;h}[Y(H,u)^2]\le y_N(h)^2 E u ; h [ Y ( H , u ) 2 ] ≤ y N ( h ) 2 for all u ∈ S N u\in S_N u ∈ S N and h ∈ J N h\in J_N h ∈ J N There exists C C C depending only on p p p such that for all sufficiently large N N N
E ∑ u : ∇ S N H N ( u ) = 0 H N ( u ) ∈ J N Y ( H , u ) ≤ C ∫ J N e c p ( M N − h ) y N ( h ) d h . (251) \mathbb{E}\sum_{\substack{u:\nabla_{S_N}H_N(u)=0\\ H_N(u)\in J_N}}Y(H,u)\le C\int_{J_N}e^{c_p(M_N-h)}y_N(h)\,dh. \tag*{(251)} E u : ∇ S N H N ( u ) = 0 H N ( u ) ∈ J N ∑ Y ( H , u ) ≤ C ∫ J N e c p ( M N − h ) y N ( h ) d h . (251) Proof. For large N N N h ∈ J N h\in J_N h ∈ J N ∣ h / N − e p ∣ ≤ Δ / 2 |h/N-e_p|\le\Delta/2 ∣ h / N − e p ∣ ≤ Δ/2 P u ; h \mathbb{P}_{u;h} P u ; h (245) , we have E u ; h [ D ( u ) Y ( H , u ) ] ≤ K 2 1 / 2 μ N ( h ) y N ( h ) \mathbb{E}_{u;h}[D(u)Y(H,u)]\le K_2^{1/2}\mu_N(h)y_N(h) E u ; h [ D ( u ) Y ( H , u )] ≤ K 2 1/2 μ N ( h ) y N ( h ) (243) and using (248) , we obtain (251) .
For an indicator Y = 1 A Y=1_A Y = 1 A Lemma 7.22 for determinant-weighted expectations, with r = 2 r=2 r = 2 Y Y Y Lemma 9.6 (3) directly and apply the estimates to arbitrary Borel weights, including indicators of bands with small partition function and partition functions of disconnected sets of latitudes.
Localization at critical maxima
In this section, we show that every point with energy close to the ground energy lies close to a critical maximum, and complete the proof of Proposition 9.1 . We write Δ = e p − e ∞ \Delta= e_p-e_\infty Δ = e p − e ∞
Lemma 9.10 (A Hessian bound at all high critical points). Let κ = p Δ / 4 \kappa= p\Delta/4 κ = p Δ/4 τ 0 ∈ ( 0 , Δ / 4 ] \tau_0 \in(0,\Delta/4] τ 0 ∈ ( 0 , Δ/4 ] c > 0 c > 0 c > 0 p p p N N N 1 − e − c N 1-e^{-cN} 1 − e − c N u u u H N H_N H N H N ( u ) ≥ N ( e p − τ 0 ) H_N(u) \ge N(e_p-\tau_0) H N ( u ) ≥ N ( e p − τ 0 ) ∇ S N 2 H N ( u ) ≤ − κ Id \nabla^2_{S_N} H_N(u) \le-\kappa\operatorname{Id} ∇ S N 2 H N ( u ) ≤ − κ Id
Proof. Let h ∈ [ N ( e p − Δ / 4 ) , N ( e p + Δ / 4 ) ] h \in[N(e_p-\Delta/4),N(e_p+\Delta/4)] h ∈ [ N ( e p − Δ/4 ) , N ( e p + Δ/4 )] u ∈ S N u \in S_N u ∈ S N A A A ( H , u ) (H,u) ( H , u ) ∇ S N 2 H N ( u ) \nabla^2_{S_N}H_N(u) ∇ S N 2 H N ( u ) − κ -\kappa − κ (242) , under P u ; h \mathbb{P}_{u;h} P u ; h A A A λ max ( ς N G n ) > p h / N − κ \lambda_{\max}(\varsigma_NG_n)>ph/N-\kappa λ m a x ( ς N G n ) > p h / N − κ p h / N − κ ≥ p ( e p − Δ / 2 ) = 2 b 1 + p Δ / 2 ph/N-\kappa\ge p(e_p-\Delta/2)=2\sqrt{b_1}+p\Delta/2 p h / N − κ ≥ p ( e p − Δ/2 ) = 2 b 1 + p Δ/2 ς N G n \varsigma_NG_n ς N G n W n , N W_{n,N} W n , N § 6.2 with b Q b_Q b Q b 1 b_1 b 1 Lemma 7.5 (2) with b = b 1 b=b_1 b = b 1 (136) with s = p Δ / 2 s=p\Delta/2 s = p Δ/2 P u ; h ( A ) ≤ 2 e − c ′ N \mathbb{P}_{u;h}(A)\le2e^{-c'N} P u ; h ( A ) ≤ 2 e − c ′ N c ′ = p 2 Δ 2 / ( 16 b 1 ) c'=p^2\Delta^2/(16b_1) c ′ = p 2 Δ 2 / ( 16 b 1 ) p h / N ph/N p h / N [ 2 b 1 + 3 p Δ / 4 , p ( e p + Δ / 4 ) ] [2\sqrt{b_1}+3p\Delta/4,p(e_p+\Delta/4)] [ 2 b 1 + 3 p Δ/4 , p ( e p + Δ/4 )] Corollary 7.8 with r = 2 r=2 r = 2
E u ; h [ D ( u ) 1 A ( H , u ) ] ≤ K 2 1 / 2 ( 2 e − c ′ N ) 1 / 2 μ N ( h ) ≤ e − c 1 N μ N ( h ) \mathbb{E}_{u;h}\left[D(u)1_A(H,u)\right]\le K_2^{1/2}(2e^{-c'N})^{1/2}\mu_N(h)\le e^{-c_1N}\mu_N(h) E u ; h [ D ( u ) 1 A ( H , u ) ] ≤ K 2 1/2 ( 2 e − c ′ N ) 1/2 μ N ( h ) ≤ e − c 1 N μ N ( h ) for large N N N c 1 = c ′ / 3 c_1=c'/3 c 1 = c ′ /3 K 2 K_2 K 2 p p p c 1 c_1 c 1
Let τ 0 ≤ Δ / 4 \tau_0\le\Delta/4 τ 0 ≤ Δ/4 (243) with φ = 1 A \varphi=1_A φ = 1 A u u u ( H , u ) ∈ A (H,u)\in A ( H , u ) ∈ A H N ( u ) ∈ [ N ( e p − τ 0 ) , N ( e p + τ 0 ) ] H_N(u)\in[N(e_p-\tau_0),N(e_p+\tau_0)] H N ( u ) ∈ [ N ( e p − τ 0 ) , N ( e p + τ 0 )] e − c 1 N e^{-c_1N} e − c 1 N ∫ ρ N ( h ) d h \int\rho_N(h)\,dh ∫ ρ N ( h ) d h h h h h / N ( N − 1 ) ≥ e p − τ 0 h/\sqrt{N(N-1)}\ge e_p-\tau_0 h / N ( N − 1 ) ≥ e p − τ 0 Lemma 9.6 (2) with J = [ e p − Δ / 2 , e p + Δ / 2 ] J=[e_p-\Delta/2,e_p+\Delta/2] J = [ e p − Δ/2 , e p + Δ/2 ] θ ( e ) ≤ − c p ( e − e p ) \theta(e)\le-c_p(e-e_p) θ ( e ) ≤ − c p ( e − e p ) ρ N ( h ) ≤ C N − 1 / 2 e ( N − 1 ) c p τ 0 \rho_N(h)\le CN^{-1/2}e^{(N-1)c_p\tau_0} ρ N ( h ) ≤ C N − 1/2 e ( N − 1 ) c p τ 0 2 C τ 0 N 1 / 2 e c p τ 0 N 2C\tau_0N^{1/2}e^{c_p\tau_0N} 2 C τ 0 N 1/2 e c p τ 0 N τ 0 \tau_0 τ 0 c p τ 0 ≤ c 1 / 2 c_p\tau_0\le c_1/2 c p τ 0 ≤ c 1 /2 2 C τ 0 N 1 / 2 e − c 1 N / 2 2C\tau_0N^{1/2}e^{-c_1N/2} 2 C τ 0 N 1/2 e − c 1 N /2 N ( e p + τ 0 ) N(e_p+\tau_0) N ( e p + τ 0 ) max H N > N ( e p + τ 0 ) \max H_N>N(e_p+\tau_0) max H N > N ( e p + τ 0 ) 2 e − N τ 0 2 / 8 2e^{-N\tau_0^2/8} 2 e − N τ 0 2 /8 (241) . This proves the lemma. □ \square □
The determinant weight enters through Corollary 7.8 . The probability of the event A A A D ( u ) D(u) D ( u )
Lemma 9.11 (Deterministic localization in superlevel sets). Let f ∈ C 3 ( S N ) f\in C^3(S_N) f ∈ C 3 ( S N ) f ∗ = max S N f f_*=\max_{S_N}f f ∗ = max S N f sup S N ∥ ∇ 3 f ∥ o p ≤ C 3 / N \sup_{S_N}\lVert\nabla^3f\rVert_{\mathrm{op}}\le C_3/\sqrt{N} sup S N ∥ ∇ 3 f ∥ op ≤ C 3 / N δ 0 , κ > 0 \delta_0,\kappa>0 δ 0 , κ > 0 c c c f f f f ( c ) > f ∗ − δ 0 N f(c)>f_*-\delta_0N f ( c ) > f ∗ − δ 0 N ∇ S N 2 f ( c ) ≤ − κ I \nabla^2_{S_N}f(c)\le-\kappa I ∇ S N 2 f ( c ) ≤ − κ I r 0 = min ( 1 , κ / ( 2 C 3 ) ) r_0=\min(1,\kappa/(2C_3)) r 0 = min ( 1 , κ / ( 2 C 3 )) r 0 = 1 r_0=1 r 0 = 1 C 3 = 0 C_3=0 C 3 = 0 0 < δ < min ( δ 0 , κ r 0 2 / 4 ) 0<\delta<\min(\delta_0,\kappa r_0^2/4) 0 < δ < min ( δ 0 , κ r 0 2 /4 ) V \mathcal{V} V
{ f > f ∗ − δ N } \{f>f_*-\delta N\} { f > f ∗ − δ N } c c c f f f f f f V \mathcal V V V \mathcal V V r 0 N r_0\sqrt{N} r 0 N c c c x ∈ V x\in\mathcal V x ∈ V
d S N ( x , c ) < 2 δ / κ N , f ( c ) ≥ f ( x ) . (252) d_{S_N}(x,c)<2\sqrt{\delta/\kappa}\sqrt{N},\qquad f(c)\ge f(x). \tag*{(252)} d S N ( x , c ) < 2 δ / κ N , f ( c ) ≥ f ( x ) . (252) Proof. The function f f f V \mathcal V V c c c f ( c ) > f ∗ − δ N f(c)>f_*-\delta N f ( c ) > f ∗ − δ N V \mathcal V V c c c f > f ∗ − δ N f>f_*-\delta N f > f ∗ − δ N V \mathcal V V V \mathcal V V c ∈ V c\in\mathcal V c ∈ V c c c f f f ∇ S N 2 f ( c ) ≤ − κ I \nabla_{S_N}^{2}f(c)\le-\kappa I ∇ S N 2 f ( c ) ≤ − κ I
Let γ \gamma γ c c c γ \gamma γ C 3 t / N C_3t/\sqrt{N} C 3 t / N t t t t ≤ r 0 N t\le r_0\sqrt{N} t ≤ r 0 N − κ I + C 3 r 0 I ≤ − ( κ / 2 ) I -\kappa I+C_3r_0I\le-(\kappa/2)I − κ I + C 3 r 0 I ≤ − ( κ /2 ) I γ \gamma γ ∇ S N f ( c ) = 0 \nabla_{S_N}f(c)=0 ∇ S N f ( c ) = 0 y y y r 0 N r_0\sqrt{N} r 0 N c c c
f ( y ) ≤ f ( c ) − κ 4 d S N ( y , c ) 2 ≤ f ∗ − κ 4 d S N ( y , c ) 2 . (253) f(y)\le f(c)-\frac{\kappa}{4}d_{S_N}(y,c)^2\le f_*-\frac{\kappa}{4}d_{S_N}(y,c)^2. \tag*{(253)} f ( y ) ≤ f ( c ) − 4 κ d S N ( y , c ) 2 ≤ f ∗ − 4 κ d S N ( y , c ) 2 . (253) The geodesics in this argument are unique, since the radius r 0 N ≤ N r_0\sqrt{N}\le\sqrt{N} r 0 N ≤ N π N \pi\sqrt{N} π N S N S_N S N f ≤ f ∗ − κ r 0 2 N / 4 < f ∗ − δ N f\le f_*-\kappa r_0^2N/4<f_*-\delta N f ≤ f ∗ − κ r 0 2 N /4 < f ∗ − δ N V \mathcal V V c c c c ′ ≠ c c'\ne c c ′ = c c c c c ′ c' c ′ t 1 > 0 t_1>0 t 1 > 0 f f f t 1 t_1 t 1 − κ t 1 / 2 < 0 -\kappa t_1/2<0 − κ t 1 /2 < 0 ∇ S N f ( c ′ ) = 0 \nabla_{S_N}f(c')=0 ∇ S N f ( c ′ ) = 0 f ( x ) > f ∗ − δ N f(x)>f_*-\delta N f ( x ) > f ∗ − δ N □ \square □
Proposition 9.12 (Proximity of high-energy points to local maxima). For every fixed r > 0 r>0 r > 0 ϵ 0 , c > 0 \epsilon_0,c>0 ϵ 0 , c > 0 N N N 1 − e − c N 1-e^{-cN} 1 − e − c N y ∈ S N y\in S_N y ∈ S N H N ( y ) ≥ N ( e p − ϵ 0 ) H_N(y)\ge N(e_p-\epsilon_0) H N ( y ) ≥ N ( e p − ϵ 0 ) u u u H N H_N H N
∣ u − y ∣ ≤ r N , H N ( u ) ≥ H N ( y ) . (254) |u-y|\le r\sqrt{N},\qquad H_N(u)\ge H_N(y). \tag*{(254)} ∣ u − y ∣ ≤ r N , H N ( u ) ≥ H N ( y ) . (254) The same holds with ϵ 0 \epsilon_0 ϵ 0
Proof. Let τ 0 , κ \tau_0,\kappa τ 0 , κ C 3 C_3 C 3 r 0 = min ( 1 , κ / ( 2 C 3 ) ) r_0=\min(1,\kappa/(2C_3)) r 0 = min ( 1 , κ / ( 2 C 3 )) ϵ 0 ∈ ( 0 , τ 0 / 6 ) \epsilon_0\in(0,\tau_0/6) ϵ 0 ∈ ( 0 , τ 0 /6 ) 3 ϵ 0 < κ r 0 2 / 4 3\epsilon_0<\kappa r_0^2/4 3 ϵ 0 < κ r 0 2 /4 2 3 ϵ 0 / κ < r 2\sqrt{3\epsilon_0/\kappa}<r 2 3 ϵ 0 / κ < r ϵ 0 \epsilon_0 ϵ 0 E \mathcal E E { ∣ max H N / N − e p ∣ ≤ ϵ 0 } \{|\max H_N/N-e_p|\le\epsilon_0\} { ∣ max H N / N − e p ∣ ≤ ϵ 0 } P ( E c ) ≤ e − c N \mathbb P(\mathcal E^c)\le e^{-cN} P ( E c ) ≤ e − c N c > 0 c>0 c > 0
On E \mathcal E E f = H N f=H_N f = H N f ∗ = max H N f_*=\max H_N f ∗ = max H N c c c f ( c ) > f ∗ − τ 0 N / 2 f(c)>f_*-\tau_0N/2 f ( c ) > f ∗ − τ 0 N /2 f ( c ) > N ( e p − ϵ 0 − τ 0 / 2 ) ≥ N ( e p − τ 0 ) f(c)>N(e_p-\epsilon_0-\tau_0/2)\ge N(e_p-\tau_0) f ( c ) > N ( e p − ϵ 0 − τ 0 /2 ) ≥ N ( e p − τ 0 ) ∇ S N 2 f ( c ) ≤ − κ I \nabla_{S_N}^{2}f(c)\le-\kappa I ∇ S N 2 f ( c ) ≤ − κ I δ 0 = τ 0 / 2 \delta_0=\tau_0/2 δ 0 = τ 0 /2 δ = 3 ϵ 0 \delta=3\epsilon_0 δ = 3 ϵ 0 3 ϵ 0 < τ 0 / 2 3\epsilon_0<\tau_0/2 3 ϵ 0 < τ 0 /2 H N ( y ) ≥ N ( e p − ϵ 0 ) H_N(y)\ge N(e_p-\epsilon_0) H N ( y ) ≥ N ( e p − ϵ 0 ) f ∗ − f ( y ) ≤ 2 ϵ 0 N < 3 ϵ 0 N f_*-f(y)\le2\epsilon_0N<3\epsilon_0N f ∗ − f ( y ) ≤ 2 ϵ 0 N < 3 ϵ 0 N y y y { f > f ∗ − 3 ϵ 0 N } \{f>f_*-3\epsilon_0N\} { f > f ∗ − 3 ϵ 0 N } u u u H N ( u ) ≥ H N ( y ) H_N(u)\ge H_N(y) H N ( u ) ≥ H N ( y ) ∣ u − y ∣ ≤ d S N ( u , y ) < 2 3 ϵ 0 / κ N < r N |u-y|\le d_{S_N}(u,y)<2\sqrt{3\epsilon_0/\kappa}\sqrt{N}<r\sqrt{N} ∣ u − y ∣ ≤ d S N ( u , y ) < 2 3 ϵ 0 / κ N < r N u u u ∇ S N 2 H N ( u ) ≤ − κ I \nabla_{S_N}^{2}H_N(u)\le-\kappa I ∇ S N 2 H N ( u ) ≤ − κ I □ \square □
Proof of Proposition 9.1. Let M > 0 M>0 M > 0 L = Q − p / 2 M L=Q^{-p/2}M L = Q − p /2 M (231) , C N , M ⊂ D N , L C_{N,M}\subset D_{N,L} C N , M ⊂ D N , L N N N (243) with φ = 1 \varphi=1 φ = 1 (248) with a N = a N ′ = L a_N=a'_N=L a N = a N ′ = L N ≥ N 0 N\ge N_0 N ≥ N 0
E ∣ D N , L ∣ = ∫ M N − L M N + L ρ N ( h ) d h ≤ C ∫ − L L e − c p t d t ≤ 2 C L e c p L . \mathbb{E}|D_{N,L}|=\int_{M_N-L}^{M_N+L}\rho_N(h)\,\mathrm{d}h\le C\int_{-L}^{L}e^{-c_pt}\,\mathrm{d}t\le2CL e^{c_pL}. E ∣ D N , L ∣ = ∫ M N − L M N + L ρ N ( h ) d h ≤ C ∫ − L L e − c p t d t ≤ 2 C L e c p L . For each of the finitely many N < N 0 N<N_0 N < N 0 ρ N \rho_N ρ N (232) .
For (233) , let ε > 0 \varepsilon>0 ε > 0 Lemma 9.8 , there exists L L L lim inf N P { ∣ max H N − M N ∣ ≤ L } ≥ 1 − ε \liminf_N\mathbb{P}\{|\max H_N-M_N|\le L\}\ge1-\varepsilon lim inf N P { ∣ max H N − M N ∣ ≤ L } ≥ 1 − ε u ∗ u_* u ∗ H N H_N H N ∣ H N ( u ∗ ) − M N ∣ ≤ L |H_N(u_*)-M_N|\le L ∣ H N ( u ∗ ) − M N ∣ ≤ L u ∗ ∈ D N , L u_*\in D_{N,L} u ∗ ∈ D N , L u ∗ ∈ C N , Q p / 2 L u_*\in C_{N,Q^{p/2}L} u ∗ ∈ C N , Q p /2 L N N N (231) . Further, max H Q = Q p / 2 max H N ≤ m N + Q p / 2 L \max H_Q=Q^{p/2}\max H_N\le m_N+Q^{p/2}L max H Q = Q p /2 max H N ≤ m N + Q p /2 L (233) increases with M M M lim inf \liminf lim inf 1 − ε 1-\varepsilon 1 − ε M ≥ Q p / 2 L M\ge Q^{p/2}L M ≥ Q p /2 L (233) .
Finally, let M M M N N N u ∈ C N , M u\in C_{N,M} u ∈ C N , M H N ( u ) ≥ M N − Q − p / 2 M ≥ N ( e p − τ 0 ) H_N(u)\ge M_N-Q^{-p/2}M\ge N(e_p-\tau_0) H N ( u ) ≥ M N − Q − p /2 M ≥ N ( e p − τ 0 ) Lemma 9.10 , with probability at least 1 − e − c N 1-e^{-cN} 1 − e − c N u u u ∇ S N 2 H Q ( u ) = Q p / 2 ∇ S N 2 H N ( u ) ≤ − Q p / 2 κ I \nabla^2_{S_N}H_Q(u)=Q^{p/2}\nabla^2_{S_N}H_N(u)\le-Q^{p/2}\kappa I ∇ S N 2 H Q ( u ) = Q p /2 ∇ S N 2 H N ( u ) ≤ − Q p /2 κ I H Q H_Q H Q
Empirical means of fixed replica groups
In this section, we show that if n n n n n n Q Q Q N Q NQ N Q N E 0 NE_0 N E 0
Let m ∈ R N m\in\mathbb{R}^N m ∈ R N ∣ m ∣ 2 = N Q |m|^2=NQ ∣ m ∣ 2 = N Q m = Q u m=\sqrt{Q}\,u m = Q u u ∈ S N u\in S_N u ∈ S N n ≥ 1 n\ge1 n ≥ 1 ρ , δ > 0 \rho,\delta>0 ρ , δ > 0
T N ( m , n , ρ , δ ) = { ( σ 1 , … , σ n ) ∈ S N n : ∣ R ( σ i , m ) − Q ∣ ≤ δ Q for all i , ∣ R ( σ i , σ j ) − Q ∣ < ρ for all i ≠ j } , T_N(m,n,\rho,\delta)=\left\{(\sigma^1,\ldots,\sigma^n)\in S_N^n:\left|R(\sigma^i,m)-Q\right|\le\delta\sqrt{Q}\text{ for all }i,\ \left|R(\sigma^i,\sigma^j)-Q\right|<\rho\text{ for all }i\ne j\right\}, T N ( m , n , ρ , δ ) = { ( σ 1 , … , σ n ) ∈ S N n : R ( σ i , m ) − Q ≤ δ Q for all i , R ( σ i , σ j ) − Q < ρ for all i = j } , and
P ~ N ( m , n , ρ , δ ) = 1 N n log ∫ T N ( m , n , ρ , δ ) exp ( β ∑ i = 1 n [ H N ( σ i ) − H N ( m ) ] ) ν N ⊗ n ( d σ ) . (255) \widetilde{P}_N(m,n,\rho,\delta)=\frac{1}{Nn}\log\int_{T_N(m,n,\rho,\delta)}\exp\left(\beta\sum_{i=1}^{n}[H_N(\sigma^i)-H_N(m)]\right)\nu_N^{\otimes n}(\mathrm{d}\sigma). \tag*{(255)} P N ( m , n , ρ , δ ) = N n 1 log ∫ T N ( m , n , ρ , δ ) exp ( β i = 1 ∑ n [ H N ( σ i ) − H N ( m )] ) ν N ⊗ n ( d σ ) . (255) The condition on R ( σ i , m ) R(\sigma^i,m) R ( σ i , m ) σ i ∈ B u ( δ ) \sigma^i\in B_u(\delta) σ i ∈ B u ( δ ) R ( σ , m ) = Q R ( σ , u ) R(\sigma,m)=\sqrt{Q}R(\sigma,u) R ( σ , m ) = Q R ( σ , u ) q = Q q=Q q = Q N ≥ n + 1 N\ge n+1 N ≥ n + 1 T N ( m , n , ρ , δ ) T_N(m,n,\rho,\delta) T N ( m , n , ρ , δ ) σ i = m + N s ω i \sigma^i=m+\sqrt{N}s\,\omega_i σ i = m + N s ω i ω 1 , … , ω n \omega_1,\ldots,\omega_n ω 1 , … , ω n m m m R ( σ i , m ) = Q R(\sigma^i,m)=Q R ( σ i , m ) = Q R ( σ i , σ j ) = Q R(\sigma^i,\sigma^j)=Q R ( σ i , σ j ) = Q T N ( m , n , ρ , δ ) T_N(m,n,\rho,\delta) T N ( m , n , ρ , δ ) P ~ N \widetilde{P}_N P N E P ~ N ( m , n , ρ , δ ) \mathbb{E}\widetilde{P}_N(m,n,\rho,\delta) E P N ( m , n , ρ , δ ) m m m
Proposition 9.13 (Uniform concentration; [143, Proposition 24]). Let t , c > 0 t,c>0 t , c > 0 δ 0 , ρ 0 > 0 \delta_0,\rho_0>0 δ 0 , ρ 0 > 0 n 0 n_0 n 0 δ ∈ ( 0 , δ 0 ] \delta\in(0,\delta_0] δ ∈ ( 0 , δ 0 ] ρ ∈ ( 0 , ρ 0 ] \rho\in(0,\rho_0] ρ ∈ ( 0 , ρ 0 ] n ≥ n 0 n\ge n_0 n ≥ n 0 N ≥ n + 1 N\ge n+1 N ≥ n + 1
P { sup ∣ m ∣ 2 = N Q [ P ~ N ( m , n , ρ , δ ) − E P ~ N ( m , n , ρ , δ ) ] < t } > 1 − e − c N . \mathbb{P}\left\{\sup_{|m|^2=NQ}\left[\widetilde{P}_N(m,n,\rho,\delta)-\mathbb{E}\widetilde{P}_N(m,n,\rho,\delta)\right]<t\right\}>1-e^{-cN}. P { ∣ m ∣ 2 = N Q sup [ P N ( m , n , ρ , δ ) − E P N ( m , n , ρ , δ ) ] < t } > 1 − e − c N . This is [143, Proposition 24, (3.1)], which holds for every mixture with exponentially decaying coefficients, in particular for the pure covariance of either parity, and for every q ∈ [ 0 , 1 ) q\in[0,1) q ∈ [ 0 , 1 ) Q Q Q N ≥ n + 1 N\ge n+1 N ≥ n + 1 n n n δ , ρ , n \delta,\rho,n δ , ρ , n N → ∞ N\to\infty N → ∞ P ~ N \widetilde{P}_N P N L N L\sqrt{N} L N H N H_N H N N \sqrt{N} N 1 − e − c ′ N 1-e^{-c'N} 1 − e − c ′ N L L L δ \delta δ ρ \rho ρ ρ + 8 δ \rho+\sqrt{8\delta} ρ + 8 δ N − 1 log ν N ( B u ( δ ) ) + β L 2 δ N^{-1}\log\nu_N(B_u(\delta))+\beta L\sqrt{2\delta} N − 1 log ν N ( B u ( δ )) + β L 2 δ β γ 1 2 1 / n + ρ \beta\gamma_1^2\sqrt{1/n+\rho} β γ 1 2 1/ n + ρ γ 1 2 \gamma_1^2 γ 1 2 n n n ρ \rho ρ δ \delta δ
Lemma 9.14 (Expected free energy of a replicated band). Let T Q = ( log s ) / 2 + β 2 ψ Q ( 1 ) / 2 T_Q=(\log s)/2+\beta^2\psi_Q(1)/2 T Q = ( log s ) /2 + β 2 ψ Q ( 1 ) /2 C C C δ ∈ ( 0 , Q / 2 ] \delta\in(0,\sqrt{Q}/2] δ ∈ ( 0 , Q /2 ] ρ > 0 \rho>0 ρ > 0 n ≥ 1 n\ge1 n ≥ 1
lim sup N → ∞ E P ~ N ( m , n , ρ , δ ) ≤ T Q + C δ + C ( 1 n + ρ + 8 δ s ) 1 / 2 . (256) \limsup_{N\to\infty}\mathbb{E}\widetilde{P}_N(m,n,\rho,\delta)\le T_Q+C\sqrt{\delta}+C\left(\frac{1}{n}+\frac{\rho+\sqrt{8\delta}}{s}\right)^{1/2}. \tag*{(256)} N → ∞ lim sup E P N ( m , n , ρ , δ ) ≤ T Q + C δ + C ( n 1 + s ρ + 8 δ ) 1/2 . (256) Note that F ( β ) = β E 0 + T Q F(\beta)=\beta E_0+T_Q F ( β ) = β E 0 + T Q E 0 = e p Q p / 2 E_0=e_pQ^{p/2} E 0 = e p Q p /2
Proof. Let u = m / Q u=m/\sqrt{Q} u = m / Q (255) in three steps: we project the replicas to the exact latitude R ( ⋅ , u ) = Q R(\cdot,u)=\sqrt{Q} R ( ⋅ , u ) = Q u u u
Projection to the exact latitude. We write σ ∈ B u ( δ ) \sigma\in B_u(\delta) σ ∈ B u ( δ ) σ = r u + N ( 1 − r 2 ) ω \sigma=ru+\sqrt{N(1-r^2)}\,\omega σ = r u + N ( 1 − r 2 ) ω ∣ r − Q ∣ ≤ δ |r-\sqrt{Q}|\le\delta ∣ r − Q ∣ ≤ δ ω \omega ω u u u σ ′ = Q u + N s ω \sigma'=\sqrt{Q}\,u+\sqrt{N}s\,\omega σ ′ = Q u + N s ω r = cos ϕ r=\cos\phi r = cos ϕ Q = cos ϕ ′ \sqrt{Q}=\cos\phi' Q = cos ϕ ′ ϕ , ϕ ′ ∈ [ 0 , π ] \phi,\phi'\in[0,\pi] ϕ , ϕ ′ ∈ [ 0 , π ]
∣ σ − σ ′ ∣ 2 = 4 N sin 2 ϕ − ϕ ′ 2 ≤ 4 N ∣ sin ϕ + ϕ ′ 2 sin ϕ − ϕ ′ 2 ∣ = 2 N ∣ cos ϕ − cos ϕ ′ ∣ ≤ 2 N δ , |\sigma-\sigma'|^2=4N\sin^2\frac{\phi-\phi'}{2}\le4N\left|\sin\frac{\phi+\phi'}{2}\sin\frac{\phi-\phi'}{2}\right|=2N|\cos\phi-\cos\phi'|\le2N\delta, ∣ σ − σ ′ ∣ 2 = 4 N sin 2 2 ϕ − ϕ ′ ≤ 4 N sin 2 ϕ + ϕ ′ sin 2 ϕ − ϕ ′ = 2 N ∣ cos ϕ − cos ϕ ′ ∣ ≤ 2 N δ , where we used ∣ ϕ − ϕ ′ ∣ / 2 ≤ ( ϕ + ϕ ′ ) / 2 ≤ π − ∣ ϕ − ϕ ′ ∣ / 2 |\phi-\phi'|/2\le(\phi+\phi')/2\le\pi-|\phi-\phi'|/2 ∣ ϕ − ϕ ′ ∣/2 ≤ ( ϕ + ϕ ′ ) /2 ≤ π − ∣ ϕ − ϕ ′ ∣/2 ∣ σ − σ ′ ∣ ≤ 2 N δ |\sigma-\sigma'|\le\sqrt{2N\delta} ∣ σ − σ ′ ∣ ≤ 2 N δ ∣ R ( σ i , σ j ) − R ( σ ′ i , σ ′ j ) ∣ ≤ 2 2 δ = 8 δ |R(\sigma^i,\sigma^j)-R(\sigma'^i,\sigma'^j)|\le2\sqrt{2\delta}=\sqrt{8\delta} ∣ R ( σ i , σ j ) − R ( σ ′ i , σ ′ j ) ∣ ≤ 2 2 δ = 8 δ Lip \operatorname{Lip} Lip H N H_N H N N \sqrt{N} N σ \sigma σ σ ′ \sigma' σ ′
∑ i = 1 n [ H N ( σ ′ i ) − H N ( m ) ] ≤ ∑ i = 1 n [ H N ( σ i ) − H N ( m ) ] + n Lip 2 N δ . (257) \sum_{i=1}^{n}\left[H_N(\sigma'^i)-H_N(m)\right]\le\sum_{i=1}^{n}\left[H_N(\sigma^i)-H_N(m)\right]+n\operatorname{Lip}\sqrt{2N\delta}. \tag*{(257)} i = 1 ∑ n [ H N ( σ ′ i ) − H N ( m ) ] ≤ i = 1 ∑ n [ H N ( σ i ) − H N ( m ) ] + n Lip 2 N δ . (257) Latitude and direction. Under ν N \nu_N ν N r = R ( σ , u ) r=R(\sigma,u) r = R ( σ , u ) ℓ N ( 1 − r 2 ) ( N − 3 ) / 2 \ell_N(1-r^2)^{(N-3)/2} ℓ N ( 1 − r 2 ) ( N − 3 ) /2 ( − 1 , 1 ) (-1,1) ( − 1 , 1 )
ℓ N = Γ ( N / 2 ) π Γ ( ( N − 1 ) / 2 ) ≤ N , (258) \ell_N=\frac{\Gamma(N/2)}{\sqrt{\pi}\Gamma((N-1)/2)}\le\sqrt{N}, \tag*{(258)} ℓ N = π Γ (( N − 1 ) /2 ) Γ ( N /2 ) ≤ N , (258) and, given r r r ω \omega ω u ⊥ u^\perp u ⊥ ( σ i ) ∈ T N ( m , n , ρ , δ ) (\sigma^i)\in T_N(m,n,\rho,\delta) ( σ i ) ∈ T N ( m , n , ρ , δ ) ∣ R ( σ i , σ j ) − Q ∣ < ρ + 8 δ \lvert R(\sigma^i,\sigma^j)-Q\rvert<\rho+\sqrt{8\delta} ∣ R ( σ i , σ j ) − Q ∣ < ρ + 8 δ R ( σ i , σ j ) = Q + s ω i ⋅ ω j R(\sigma^i,\sigma^j)=Q+s\,\omega_i\cdot\omega_j R ( σ i , σ j ) = Q + s ω i ⋅ ω j ∣ ω i ⋅ ω j ∣ < ρ ′ = ( ρ + 8 δ ) / s \lvert\omega_i\cdot\omega_j\rvert<\rho'=(\rho+\sqrt{8\delta})/s ∣ ω i ⋅ ω j ∣ < ρ ′ = ( ρ + 8 δ ) / s
∫ T N exp ( β ∑ i [ H N ( σ i ) − H N ( m ) ] ) d ν N ⊗ n ≤ ν N ( B u ( δ ) ) n e n β Lip 2 N δ \int_{T_N}\exp\left(\beta\sum_i[H_N(\sigma^i)-H_N(m)]\right)\,d\nu_N^{\otimes n}\le\nu_N(B_u(\delta))^n e^{n\beta\operatorname{Lip}\sqrt{2N\delta}} ∫ T N exp ( β i ∑ [ H N ( σ i ) − H N ( m )] ) d ν N ⊗ n ≤ ν N ( B u ( δ ) ) n e n β Lip 2 N δ × ∫ { ∣ ω i ⋅ ω j ∣ < ρ ′ } e β ∑ i X ( ω i ) d ω , \qquad\times\int_{\{\lvert\omega_i\cdot\omega_j\rvert<\rho'\}}e^{\beta\sum_i X(\omega_i)}\,d\omega, × ∫ {∣ ω i ⋅ ω j ∣ < ρ ′ } e β ∑ i X ( ω i ) d ω , where X ( ω ) = H N ( Q u + N s ω ) − H N ( m ) X(\omega)=H_N(\sqrt{Q}\,u+\sqrt{N}s\,\omega)-H_N(m) X ( ω ) = H N ( Q u + N s ω ) − H N ( m ) d ω d\omega d ω u ⊥ u^\perp u ⊥ 1 − r 2 ≤ 1 − ( Q − δ ) 2 1-r^2\le1-(\sqrt{Q}-\delta)^2 1 − r 2 ≤ 1 − ( Q − δ ) 2 δ ≤ Q / 2 \delta\le\sqrt{Q}/2 δ ≤ Q /2
1 N log ν N ( B u ( δ ) ) ≤ 1 N log ( 2 δ N ) + N − 3 2 N log ( s + 2 Q δ ) ≤ 1 2 log s + 2 δ s + o ( 1 ) . \frac{1}{N}\log\nu_N(B_u(\delta))\le\frac{1}{N}\log(2\delta\sqrt{N})+\frac{N-3}{2N}\log(s+2\sqrt{Q}\delta)\le\frac{1}{2}\log s+\frac{2\delta}{s}+o(1). N 1 log ν N ( B u ( δ )) ≤ N 1 log ( 2 δ N ) + 2 N N − 3 log ( s + 2 Q δ ) ≤ 2 1 log s + s 2 δ + o ( 1 ) . The field on the exact latitude. Since R ( Q u + N s ω , Q u + N s ω ′ ) = Q + s ω ⋅ ω ′ R(\sqrt{Q}u+\sqrt{N}s\,\omega,\sqrt{Q}u+\sqrt{N}s\,\omega')=Q+s\,\omega\cdot\omega' R ( Q u + N s ω , Q u + N s ω ′ ) = Q + s ω ⋅ ω ′ R ( Q u + N s ω , m ) = Q R(\sqrt{Q}u+\sqrt{N}s\,\omega,m)=Q R ( Q u + N s ω , m ) = Q X X X
E X ( ω ) X ( ω ′ ) = N [ ( Q + s ω ⋅ ω ′ ) p − Q p ] = N [ p Q p − 1 s ω ⋅ ω ′ + ψ Q ( ω ⋅ ω ′ ) ] . \mathbb{E}X(\omega)X(\omega')=N\left[(Q+s\,\omega\cdot\omega')^p-Q^p\right]=N\left[pQ^{p-1}s\,\omega\cdot\omega'+\psi_Q(\omega\cdot\omega')\right]. E X ( ω ) X ( ω ′ ) = N [ ( Q + s ω ⋅ ω ′ ) p − Q p ] = N [ p Q p − 1 s ω ⋅ ω ′ + ψ Q ( ω ⋅ ω ′ ) ] . Since the polynomial ψ Q \psi_Q ψ Q X ( ω ) = N γ 1 g ⋅ ω + Y ( ω ) X(\omega)=\sqrt{N}\gamma_1g\cdot\omega+Y(\omega) X ( ω ) = N γ 1 g ⋅ ω + Y ( ω ) γ 1 = ( p Q p − 1 s ) 1 / 2 \gamma_1=(pQ^{p-1}s)^{1/2} γ 1 = ( p Q p − 1 s ) 1/2 g g g u ⊥ u^\perp u ⊥ Y Y Y N ψ Q ( ω ⋅ ω ′ ) N\psi_Q(\omega\cdot\omega') N ψ Q ( ω ⋅ ω ′ ) { ∣ ω i ⋅ ω j ∣ < ρ ′ } \{\lvert\omega_i\cdot\omega_j\rvert<\rho'\} {∣ ω i ⋅ ω j ∣ < ρ ′ }
∣ g ⋅ ∑ i = 1 n ω i ∣ ≤ ∣ g ∣ ∣ ∑ i = 1 n ω i ∣ ≤ ∣ g ∣ ( n + n ( n − 1 ) ρ ′ ) 1 / 2 ≤ n ∣ g ∣ ( 1 n + ρ ′ ) 1 / 2 . \left\lvert g\cdot\sum_{i=1}^n\omega_i\right\rvert\le\lvert g\rvert\left\lvert\sum_{i=1}^n\omega_i\right\rvert\le\lvert g\rvert\left(n+n(n-1)\rho'\right)^{1/2}\le n\lvert g\rvert\left(\frac{1}{n}+\rho'\right)^{1/2}. g ⋅ i = 1 ∑ n ω i ≤ ∣ g ∣ i = 1 ∑ n ω i ≤ ∣ g ∣ ( n + n ( n − 1 ) ρ ′ ) 1/2 ≤ n ∣ g ∣ ( n 1 + ρ ′ ) 1/2 . After this bound, we drop the constraint on the directions, so that the remaining integral factorizes into n n n
1 N E log ∫ e β Y ( ω ) d ω ≤ 1 N log ∫ E e β Y ( ω ) d ω = β 2 2 ψ Q ( 1 ) . \frac{1}{N}\mathbb{E}\log\int e^{\beta Y(\omega)}\,d\omega\le\frac{1}{N}\log\int\mathbb{E}e^{\beta Y(\omega)}\,d\omega=\frac{\beta^2}{2}\psi_Q(1). N 1 E log ∫ e β Y ( ω ) d ω ≤ N 1 log ∫ E e β Y ( ω ) d ω = 2 β 2 ψ Q ( 1 ) . Conclusion. Combining the three steps, taking expectations, and using E ∣ g ∣ ≤ N \mathbb{E}\lvert g\rvert\le\sqrt{N} E ∣ g ∣ ≤ N E Lip ≤ C N \mathbb{E}\operatorname{Lip}\le C\sqrt{N} E Lip ≤ C N
E P ~ N ( m , n , ρ , δ ) ≤ 1 2 log s + 2 δ s + o ( 1 ) + C β 2 δ + β γ 1 ( 1 n + ρ ′ ) 1 / 2 + β 2 2 ψ Q ( 1 ) . \mathbb{E}\widetilde{P}_N(m,n,\rho,\delta)\le\frac{1}{2}\log s+\frac{2\delta}{s}+o(1)+C\beta\sqrt{2\delta}+\beta\gamma_1\left(\frac{1}{n}+\rho'\right)^{1/2}+\frac{\beta^2}{2}\psi_Q(1). E P N ( m , n , ρ , δ ) ≤ 2 1 log s + s 2 δ + o ( 1 ) + C β 2 δ + β γ 1 ( n 1 + ρ ′ ) 1/2 + 2 β 2 ψ Q ( 1 ) . Since δ ≤ δ \delta\le\sqrt{\delta} δ ≤ δ (256) . The bound uses the expected Lipschitz constant; no contribution from the complement of a regularity event is lost.
Proposition 9.15 (Empirical means of Gibbs replica groups). Let ε > 0 \varepsilon> 0 ε > 0 n 0 n_0 n 0 η , c > 0 \eta,c > 0 η , c > 0 n ≥ n 0 n \ge n_0 n ≥ n 0 N N N n n n
E π ⊗ n { ∣ R ( σ i , σ j ) − Q ∣ < η for all i ≠ j , H Q ( u ^ ) ≤ N ( E 0 − ε ) } ≤ e − c N , (259) \mathbb{E}\,\pi^{\otimes n}\left\{ \lvert R(\sigma^i,\sigma^j)-Q\rvert< \eta\ \text{for all }i\ne j,\ H_Q(\hat{u})\le N(E_0-\varepsilon) \right\}\le e^{-cN}, \tag*{(259)} E π ⊗ n { ∣ R ( σ i , σ j ) − Q ∣ < η for all i = j , H Q ( u ^ ) ≤ N ( E 0 − ε ) } ≤ e − c N , (259) where σ ˉ = n − 1 ∑ i = 1 n σ i \bar{\sigma}=n^{-1}\sum_{i=1}^{n}\sigma^i σ ˉ = n − 1 ∑ i = 1 n σ i q ˉ = ∣ σ ˉ ∣ 2 / N \bar{q}=\lvert\bar{\sigma}\rvert^2/N q ˉ = ∣ σ ˉ ∣ 2 / N u ^ = σ ˉ / q ˉ ∈ S N \hat{u}=\bar{\sigma}/\sqrt{\bar{q}}\in S_N u ^ = σ ˉ / q ˉ ∈ S N (259) , we have q ˉ > 0 \bar{q}>0 q ˉ > 0 u ^ \hat{u} u ^
The diagonal of the overlap matrix enters the mean: since R ( σ i , σ i ) = 1 R(\sigma^i,\sigma^i)=1 R ( σ i , σ i ) = 1 σ ˉ \bar{\sigma} σ ˉ Q + ( 1 − Q ) / n Q+(1-Q)/n Q + ( 1 − Q ) / n Q Q Q n n n N → ∞ N\to\infty N → ∞
Proof. We first relate u ^ \hat{u} u ^
Geometry of the mean. Let q n = Q + ( 1 − Q ) / n q_n=Q+(1-Q)/n q n = Q + ( 1 − Q ) / n ∣ R ( σ i , σ j ) − Q ∣ < η \lvert R(\sigma^i,\sigma^j)-Q\rvert<\eta ∣ R ( σ i , σ j ) − Q ∣ < η i ≠ j i\ne j i = j η < Q / 2 \eta<Q/2 η < Q /2 q ˉ = n − 2 ∑ i , j R ( σ i , σ j ) \bar{q}=n^{-2}\sum_{i,j}R(\sigma^i,\sigma^j) q ˉ = n − 2 ∑ i , j R ( σ i , σ j ) R ( σ i , σ ˉ ) = n − 1 ∑ j R ( σ i , σ j ) R(\sigma^i,\bar{\sigma})=n^{-1}\sum_jR(\sigma^i,\sigma^j) R ( σ i , σ ˉ ) = n − 1 ∑ j R ( σ i , σ j )
∣ q ˉ − q n ∣ ≤ η , ∣ R ( σ i , σ ˉ ) − q ˉ ∣ ≤ 2 η . \lvert\bar{q}-q_n\rvert\le\eta,\qquad\lvert R(\sigma^i,\bar{\sigma})-\bar{q}\rvert\le2\eta. ∣ q ˉ − q n ∣ ≤ η , ∣ R ( σ i , σ ˉ ) − q ˉ ∣ ≤ 2 η . In particular, q ˉ ≥ Q − η > Q / 2 \bar{q}\ge Q-\eta>Q/2 q ˉ ≥ Q − η > Q /2 R ( σ i , Q u ^ ) − Q = Q / q ˉ [ R ( σ i , σ ˉ ) − q ˉ ] + Q q ˉ − Q R(\sigma^i,\sqrt{Q}\hat{u})-Q=\sqrt{Q/\bar{q}}\,[R(\sigma^i,\bar{\sigma})-\bar{q}]+\sqrt{Q\bar{q}}-Q R ( σ i , Q u ^ ) − Q = Q / q ˉ [ R ( σ i , σ ˉ ) − q ˉ ] + Q q ˉ − Q ∣ Q q ˉ − Q ∣ ≤ ∣ q ˉ − Q ∣ ≤ η + 1 / n \lvert\sqrt{Q\bar{q}}-Q\rvert\le\lvert\bar{q}-Q\rvert\le\eta+1/n ∣ Q q ˉ − Q ∣ ≤ ∣ q ˉ − Q ∣ ≤ η + 1/ n Q / q ˉ ≤ 2 \sqrt{Q/\bar{q}}\le\sqrt{2} Q / q ˉ ≤ 2
∣ R ( σ i , Q u ^ ) − Q ∣ ≤ 1 n + 4 η . (260) \lvert R(\sigma^i,\sqrt{Q}\hat{u})-Q\rvert\le\frac{1}{n}+4\eta. \tag*{(260)} ∣ R ( σ i , Q u ^ ) − Q ∣ ≤ n 1 + 4 η . (260) Choice of the parameters. Let L L L { Lip ≤ L N } \{\operatorname{Lip}\le L\sqrt{N}\} { Lip ≤ L N } Lip \operatorname{Lip} Lip 1 − e − c 2 N 1-e^{-c_2N} 1 − e − c 2 N c 2 > 0 c_2>0 c 2 > 0 L L L δ 0 , ρ 0 , n 0 ′ \delta_0,\rho_0,n'_0 δ 0 , ρ 0 , n 0 ′ t = β ε / 8 t=\beta\varepsilon/8 t = β ε /8 c = 1 c=1 c = 1 δ ∈ ( 0 , min ( δ 0 , Q / 2 ) ] \delta\in(0,\min(\delta_0,\sqrt{Q}/2)] δ ∈ ( 0 , min ( δ 0 , Q /2 )] ρ ∈ ( 0 , ρ 0 ] \rho\in(0,\rho_0] ρ ∈ ( 0 , ρ 0 ] n 1 n_1 n 1 (256) are at most β ε / 16 \beta\varepsilon/16 β ε /16 n ≥ n 1 n\ge n_1 n ≥ n 1 n ≥ n 1 n\ge n_1 n ≥ n 1 N N N
E P ~ N ( m , n , ρ , δ ) ≤ T Q + β ε 8 . (261) \mathbb{E}\widetilde{P}_N(m,n,\rho,\delta)\le T_Q+\frac{\beta\varepsilon}{8}. \tag*{(261)} E P N ( m , n , ρ , δ ) ≤ T Q + 8 β ε . (261) Let ϑ > 0 \vartheta>0 ϑ > 0 ϑ ≤ δ Q / 4 \vartheta\le\delta\sqrt{Q}/4 ϑ ≤ δ Q /4 L ϑ ≤ ε / 4 L\vartheta\le\varepsilon/4 L ϑ ≤ ε /4 { ∣ m ∣ 2 = N Q } \{\lvert m\rvert^2=NQ\} {∣ m ∣ 2 = N Q } ϑ N \vartheta\sqrt{N} ϑ N N \mathcal{N} N e A ϑ N e^{A_\vartheta N} e A ϑ N A ϑ = log ( 1 + 2 Q / ϑ ) A_\vartheta=\log(1+2\sqrt{Q}/\vartheta) A ϑ = log ( 1 + 2 Q / ϑ ) n 0 ≥ max ( n 0 ′ , n 1 ) n_0\ge\max(n'_0,n_1) n 0 ≥ max ( n 0 ′ , n 1 ) 1 / n 0 ≤ δ Q / 4 1/n_0\le\delta\sqrt{Q}/4 1/ n 0 ≤ δ Q /4 n 0 β ε / 4 ≥ 2 A ϑ n_0\beta\varepsilon/4\ge2A_\vartheta n 0 β ε /4 ≥ 2 A ϑ η ∈ ( 0 , min ( Q / 2 , ρ , δ Q / 16 ) ) \eta\in(0,\min(Q/2,\rho,\delta\sqrt{Q}/16)) η ∈ ( 0 , min ( Q /2 , ρ , δ Q /16 )) N → ∞ N\to\infty N → ∞ n ≥ n 0 n\ge n_0 n ≥ n 0 A union over the net. Let n ≥ n 0 n \ge n_0 n ≥ n 0 E \mathcal{E} E { Lip ≤ L N } \{\operatorname{Lip} \le L\sqrt{N}\} { Lip ≤ L N } { P N ≥ F ( β ) − β ε / 8 } \{P_N \ge F(\beta)-\beta\varepsilon/8\} { P N ≥ F ( β ) − β ε /8 } E P N → F ( β ) \mathbb{E}P_N \to F(\beta) E P N → F ( β ) 1 − 2 e − c N 1-2e^{-cN} 1 − 2 e − c N N N N P ( E c ) ≤ e − c 3 N \mathbb{P}(\mathcal{E}^{c}) \le e^{-c_3N} P ( E c ) ≤ e − c 3 N c 3 > 0 c_3>0 c 3 > 0 N N N ( σ i ) (\sigma^i) ( σ i ) (259) , and let m j ∈ N m_j \in\mathbb{N} m j ∈ N ∣ m j − Q u ^ ∣ ≤ ϑ N |m_j-\sqrt{Q}\hat{u}| \le\vartheta\sqrt{N} ∣ m j − Q u ^ ∣ ≤ ϑ N ∣ R ( σ i , m j ) − R ( σ i , Q u ^ ) ∣ ≤ ϑ |R(\sigma^i,m_j)-R(\sigma^i,\sqrt{Q}\hat{u})| \le\vartheta ∣ R ( σ i , m j ) − R ( σ i , Q u ^ ) ∣ ≤ ϑ (260) and the choice of the parameters give ∣ R ( σ i , m j ) − Q ∣ ≤ 3 δ Q / 4 |R(\sigma^i,m_j)-Q| \le3\delta\sqrt{Q}/4 ∣ R ( σ i , m j ) − Q ∣ ≤ 3 δ Q /4 ∣ R ( σ i , σ k ) − Q ∣ < η ≤ ρ |R(\sigma^i,\sigma^k)-Q|<\eta\le\rho ∣ R ( σ i , σ k ) − Q ∣ < η ≤ ρ i ≠ k i\ne k i = k ( σ i ) ∈ T N ( m j , n , ρ , δ ) (\sigma^i)\in T_N(m_j,n,\rho,\delta) ( σ i ) ∈ T N ( m j , n , ρ , δ )
On E \mathcal{E} E
H N ( m j ) ≤ H N ( Q u ^ ) + L ϑ N = H Q ( u ^ ) + L ϑ N ≤ N ( E 0 − 3 ε / 4 ) , H_N(m_j) \le H_N(\sqrt{Q}\hat{u})+L\vartheta N=H_Q(\hat{u})+L\vartheta N\le N(E_0-3\varepsilon/4), H N ( m j ) ≤ H N ( Q u ^ ) + L ϑN = H Q ( u ^ ) + L ϑN ≤ N ( E 0 − 3 ε /4 ) , where we used L ϑ ≤ ε / 4 L\vartheta\le\varepsilon/4 L ϑ ≤ ε /4 E \mathcal{E} E m j m_j m j
π ⊗ n ( T N ( m j , n , ρ , δ ) ) = exp { N n [ β N H N ( m j ) + P ~ N ( m j , n , ρ , δ ) − P N ] } ≤ e − 3 N n β ε / 8 . \pi^{\otimes n}(T_N(m_j,n,\rho,\delta))=\exp\left\{Nn\left[\frac{\beta}{N}H_N(m_j)+\widetilde{P}_N(m_j,n,\rho,\delta)-P_N\right]\right\}\le e^{-3Nn\beta\varepsilon/8}. π ⊗ n ( T N ( m j , n , ρ , δ )) = exp { N n [ N β H N ( m j ) + P N ( m j , n , ρ , δ ) − P N ] } ≤ e − 3 N n β ε /8 . Indeed, on E \mathcal{E} E (261) give P ~ N ( m j , n , ρ , δ ) ≤ T Q + β ε / 4 \widetilde{P}_N(m_j,n,\rho,\delta)\le T_Q+\beta\varepsilon/4 P N ( m j , n , ρ , δ ) ≤ T Q + β ε /4 P N ≥ F ( β ) − β ε / 8 = β E 0 + T Q − β ε / 8 P_N\ge F(\beta)-\beta\varepsilon/8=\beta E_0+T_Q-\beta\varepsilon/8 P N ≥ F ( β ) − β ε /8 = β E 0 + T Q − β ε /8 N n [ − 3 β ε / 4 + 3 β ε / 8 ] Nn[-3\beta\varepsilon/4+3\beta\varepsilon/8] N n [ − 3 β ε /4 + 3 β ε /8 ] E \mathcal{E} E (259) is at most e A ϑ N − 3 N n β ε / 8 ≤ e − N n β ε / 8 e^{A_\vartheta N-3Nn\beta\varepsilon/8}\le e^{-Nn\beta\varepsilon/8} e A ϑ N − 3 N n β ε /8 ≤ e − N n β ε /8 n β ε / 4 ≥ 2 A ϑ n\beta\varepsilon/4\ge2A_\vartheta n β ε /4 ≥ 2 A ϑ (259) is at most e − c 3 N + e − N n β ε / 8 e^{-c_3N}+e^{-Nn\beta\varepsilon/8} e − c 3 N + e − N n β ε /8 e − c N e^{-cN} e − c N c = min ( c 3 , β ε / 8 ) / 2 c=\min(c_3,\beta\varepsilon/8)/2 c = min ( c 3 , β ε /8 ) /2 N N N
A coarse cover from the directing measure
For τ > 0 \tau>0 τ > 0
U N ( τ ) = ⋃ u : ∇ S N H N ( u ) = 0 H N ( u ) ≥ N ( e p − τ ) { x ∈ S N : R ( x , u ) ≥ a } , \mathcal{U}_N(\tau)=\bigcup_{\substack{u:\nabla S_NH_N(u)=0\\H_N(u)\ge N(e_p-\tau)}}\{x\in S_N:R(x,u)\ge a\}, U N ( τ ) = u : ∇ S N H N ( u ) = 0 H N ( u ) ≥ N ( e p − τ ) ⋃ { x ∈ S N : R ( x , u ) ≥ a } , where a a a ( H , x ) (H,x) ( H , x ) x ∈ U N ( τ ) x\in\mathcal{U}_N(\tau) x ∈ U N ( τ ) π ( U N ( τ ) ) \pi(\mathcal{U}_N(\tau)) π ( U N ( τ )) U N ( τ ) \mathcal{U}_N(\tau) U N ( τ ) τ > 0 \tau>0 τ > 0
We use the notation of Section 6.1. Let σ 1 , σ 2 , … \sigma^1,\sigma^2,\ldots σ 1 , σ 2 , … π \pi π L N \mathcal{L}_N L N ( R ( σ i , σ j ) ) i , j ≥ 1 (R(\sigma^i,\sigma^j))_{i,j\ge1} ( R ( σ i , σ j ) ) i , j ≥ 1 M = [ − 1 , 1 ] N × N \mathcal{M}=[-1,1]^{\mathbb{N}\times\mathbb{N}} M = [ − 1 , 1 ] N × N L \mathcal{L} L ( L N ) (\mathcal{L}_N) ( L N ) Q Q Q L ∈ L \mathcal{L}\in\mathscr{L} L ∈ L ( G ω ) ω ∈ Ω (G_\omega)_{\omega\in\Omega} ( G ω ) ω ∈ Ω L \mathcal{L} L ω \omega ω v 1 , v 2 , … v_1,v_2,\ldots v 1 , v 2 , … B B B ℓ 2 \ell^2 ℓ 2 G ω G_\omega G ω R i i = 1 R_{ii}=1 R ii = 1 R i j = v i ⋅ v j R_{ij}=v_i\cdot v_j R ij = v i ⋅ v j i ≠ j i\ne j i = j ω \omega ω G ω { v : ∥ v ∥ 2 = Q } = 1 G_\omega\{v:\|v\|^2=Q\}=1 G ω { v : ∥ v ∥ 2 = Q } = 1
