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The pressure transition and the proof strategy
Let d ω d\omega d ω S 2 S^2 S 2 ε > 0 \varepsilon> 0 ε > 0 x x x u u u
R ε ( x , u ) = x + [ − 1 / 2 , 1 / 2 ] u + B ( 0 , ε / 2 ) . R_\varepsilon(x,u)=x+[-1/2,1/2]u+B(0,\varepsilon/2). R ε ( x , u ) = x + [ − 1/2 , 1/2 ] u + B ( 0 , ε /2 ) . The orientation varies continuously over S 2 S^2 S 2 u u u − u -u − u Λ \Lambda Λ z > 0 z > 0 z > 0 ξ \xi ξ
Z Λ , ε ( z ∣ ξ ) = ∑ N ≥ 0 z N N ! ∫ ( Λ × S 2 ) N 1 h c ( X 1 , … , X N ; ξ ) ∏ j = 1 N d x j d ω ( u j ) . (1) Z_{\Lambda,\varepsilon}(z\mid\xi)=\sum_{N\geq0}\frac{z^N}{N!}\int_{(\Lambda\times S^2)^N}\mathbf{1}_{\mathrm{hc}}(X_1,\ldots,X_N;\xi)\prod_{j=1}^N dx_j\,d\omega(u_j). \tag*{(1)} Z Λ , ε ( z ∣ ξ ) = N ≥ 0 ∑ N ! z N ∫ ( Λ × S 2 ) N 1 hc ( X 1 , … , X N ; ξ ) j = 1 ∏ N d x j d ω ( u j ) . (1) The indicator requires disjoint rod interiors. Centres in Λ \Lambda Λ Λ s = [ − s / 2 , s / 2 ] 3 \Lambda_s=[-s/2,s/2]^3 Λ s = [ − s /2 , s /2 ] 3
P ε ( a ) = lim s → ∞ 1 ∣ Λ s ∣ log Z Λ s , ε ( ε − 1 e a ∣ ∅ ) . (2) P_\varepsilon(a)=\lim_{s\to\infty}\frac{1}{|\Lambda_s|}\log Z_{\Lambda_s,\varepsilon}(\varepsilon^{-1}e^a\mid\varnothing). \tag*{(2)} P ε ( a ) = s → ∞ lim ∣ Λ s ∣ 1 log Z Λ s , ε ( ε − 1 e a ∣ ∅ ) . (2) Theorem 1.1 (Nondifferentiability of the pressure). The pressure P ε P_\varepsilon P ε a a a a t ∈ R a_t\in\mathbb{R} a t ∈ R ε 0 > 0 \varepsilon_0>0 ε 0 > 0 Δ , c , C > 0 \Delta,c,C>0 Δ , c , C > 0 0 < ε < ε 0 0<\varepsilon<\varepsilon_0 0 < ε < ε 0 a ε a_\varepsilon a ε a ε → a t a_\varepsilon\to a_t a ε → a t ε ↓ 0 \varepsilon\downarrow0 ε ↓ 0
P ε , + ′ ( a ε ) − P ε , − ′ ( a ε ) ≥ Δ 2 ε , c ε ≤ P ε , ± ′ ( a ε ) ≤ C ε . (3) P'_{\varepsilon,+}(a_\varepsilon)-P'_{\varepsilon,-}(a_\varepsilon)\geq\frac{\Delta}{2\varepsilon},\qquad\frac{c}{\varepsilon}\leq P'_{\varepsilon,\pm}(a_\varepsilon)\leq\frac{C}{\varepsilon}. \tag*{(3)} P ε , + ′ ( a ε ) − P ε , − ′ ( a ε ) ≥ 2 ε Δ , ε c ≤ P ε , ± ′ ( a ε ) ≤ ε C . (3) The one-sided derivatives are the limiting number densities from the corresponding sides of the transition.
Thus the transition occurs at number density of order ε − 1 \varepsilon^{-1} ε − 1 ε \varepsilon ε ℓ \ell ℓ d d d ( ℓ 2 d ) − 1 (\ell^2d)^{-1} ( ℓ 2 d ) − 1 d / ℓ d/\ell d / ℓ
The strategy and its relation to LMP
Lebowitz, Mazel, and Presutti (LMP) [1 , 2 ] transferred mean-field coexistence to a continuum system with a large but finite interaction range. Their method combines a variational description of the reference phases, estimates within each phase, control of interfaces, and an adjustment of the chemical potential that makes the corrected phase pressures equal. We follow this sequence. The model-specific work replaces their Kac interactions by hard exclusion and their scalar density minima by families of angular profiles. The contour step uses Pirogov–Sinai theory [3 , 4 ] , particularly Zahradník’s truncation and stability argument [5 ] .
The conclusion here concerns the pressure. Once the physical pressure is identified with the maximum of phase pressures whose density responses are separated, their crossing proves the theorem. This avoids the further construction and analysis of distinct physical Gibbs measures carried out by LMP. The argument still needs locality of the reference gas and boundary estimates for restricted free energies; auxiliary Gibbs states are used to establish those estimates.
The following eleven steps are the subjects of the next eleven sections, in this order.
Identify the homogeneous phases. The Onsager functional has an isolated isotropic minimizer and nonisotropic minimizers at one chemical potential. Their masses are separated. Analytic coordinates divide the complete minimizing set into finitely many connected components.
Introduce an exact auxiliary field. Attach a normalized Gaussian variable to each spatial and angular cell. Integrating these variables restores the original partition function exactly. Conditional on the field, a quadratic count penalty provides the reference system used in the subsequent estimates.
Derive the variational limit. The thin-rod overlap operator and a random graph representation give the finite-volume Onsager functional for the reference gas. This supplies the microscopic approximation that plays the role of coarse graining in LMP.
Prove locality of the reference gas. The quadratic penalty makes the insertion generator coercive. Its inverse yields covariance decay uniformly in the field, the volume, and fixed exterior rods. This replaces LMP’s restricted reference-system uniqueness estimates.
Construct a positive field action. Directional averaging splits the leading free energy into nonnegative terms. A transport rigidity theorem identifies their common zeros. Minimization over density profiles then gives a local field action and quantitative separation of its minimizing components.
Restore all particle counts. A projection empties cells exceeding the reference count bound. A convergent expansion reconstructs their contribution and propagates the reference estimates to the exact rod integral.
Recover the exact field interaction. Integrate the field response to express the exact interaction as the positive leading action plus a small summable error. A common additive normalization allows pressures with different reference profiles to be compared.
Control boundary costs. Interpolate density profiles within a connected minimizing component and compare restricted field integrals. This treats the continuous families of angular minima. It gives surface bounds and a comparison of partition functions across a separating layer.
Assign contours and their costs. Blocks where a phase cannot be continued form contours. Rigidity and isoperimetry supply their positive cost. The exact regional expansion retains the boundary terms needed to combine neighboring regions.
Identify the physical pressure. Following LMP’s use of auxiliary phase pressures, truncate large contour weights and prove uniform surface estimates. An induction then shows that the physical pressure is the maximum of these phase pressures.
Locate the crossing. Differentiate the original Gaussian representation. The homogeneous mass gap gives separated increments of the complete phase pressures, while their leading values agree. Adjusting the chemical potential forces a crossing and hence the asserted density jump.
Homogeneous reference phases
Let d ω d\omega d ω S 2 S^2 S 2
( K f ) ( u ) = ∫ S 2 ∣ u × v ∣ f ( v ) d ω ( v ) , Ω a ( f ) = ∫ S 2 f ( log f − 1 − a ) d ω + ⟨ f , K f ⟩ . (Kf)(u) = \int_{S^2} |u \times v|f(v)\,d\omega(v), \qquad\Omega_a(f) = \int_{S^2} f(\log f - 1 - a)\,d\omega+ \langle f,Kf\rangle. ( K f ) ( u ) = ∫ S 2 ∣ u × v ∣ f ( v ) d ω ( v ) , Ω a ( f ) = ∫ S 2 f ( log f − 1 − a ) d ω + ⟨ f , K f ⟩ . This is the grand-canonical form of Onsager’s second-virial functional [6 ] . Profiles are nonnegative integrable functions; the entropy may be + ∞ +\infty + ∞ m a = inf Ω a m_a = \inf\Omega_a m a = inf Ω a W a = { f : Ω a ( f ) = m a } W_a = \{f : \Omega_a(f) = m_a\} W a = { f : Ω a ( f ) = m a }
For bounds on fixed-mass Onsager minimizers, see Vollmer [7 Section 7.1] . The following lemma gives the bounds needed with variable mass.
Lemma 2.1 (Bounds and compactness of the minimizers). The set W a W_a W a
log f + 2 K f = a , e a − 2 e a ≤ f ≤ e a . (4) \log f + 2Kf = a,\qquad e^{a-2e^a} \le f \le e^a. \tag*{(4)} log f + 2 K f = a , e a − 2 e a ≤ f ≤ e a . (4) The minimizers are uniformly Lipschitz on the sphere. Their union over a compact parameter interval is relatively compact in the uniform norm, and limits at a convergent sequence of parameters are minimizers at the limiting parameter.
Proof. If ρ = ∫ f d ω \rho= \int f\,d\omega ρ = ∫ f d ω
Ω a ( f ) ≥ ρ log ρ − ( 1 + a ) ρ . \Omega_a(f) \ge\rho\log\rho- (1+a)\rho. Ω a ( f ) ≥ ρ log ρ − ( 1 + a ) ρ . A minimizing sequence consequently has bounded mass and bounded positive entropy. It is uniformly integrable and has a weakly convergent subsequence in L 1 L^1 L 1 S 2 × S 2 S^2 \times S^2 S 2 × S 2 f ( u ) f(u) f ( u ) [ f ( u ) + f ( − u ) ] / 2 [f(u) + f(-u)]/2 [ f ( u ) + f ( − u )] /2 f f f
For positivity and the Euler equation, fix a minimizer f f f V = 2 K f V = 2Kf V = 2 K f L ( g ) = ∫ g ( log g − 1 − a + V ) d ω L(g) = \int g(\log g - 1 - a + V)\,d\omega L ( g ) = ∫ g ( log g − 1 − a + V ) d ω g = e a − V g = e^{a-V} g = e a − V L ( g ) < L ( f ) L(g) < L(f) L ( g ) < L ( f ) f t = ( 1 − t ) f + t g f_t = (1-t)f + tg f t = ( 1 − t ) f + t g
Ω a ( f t ) − Ω a ( f ) ≤ t [ L ( g ) − L ( f ) ] + t 2 ⟨ g − f , K ( g − f ) ⟩ < 0 \Omega_a(f_t) - \Omega_a(f) \le t[L(g) - L(f)] + t^2\langle g-f,K(g-f)\rangle< 0 Ω a ( f t ) − Ω a ( f ) ≤ t [ L ( g ) − L ( f )] + t 2 ⟨ g − f , K ( g − f )⟩ < 0 for small t > 0 t > 0 t > 0 f = e a − 2 K f f = e^{a-2Kf} f = e a − 2 K f 0 ≤ K f ≤ ∫ f d ω ≤ e a 0 \le Kf \le\int f\,d\omega\le e^a 0 ≤ K f ≤ ∫ f d ω ≤ e a f f f □ \square □
The spectrum below is the Onsager-kernel spectrum computed in [7 Corollary 10] , divided by 4 π 4\pi 4 π d ω d\omega d ω
Lemma 2.2 (Angular spectrum). On spherical harmonics of degree l l l K K K
κ 0 = π 4 , κ 2 n = − π 2 ( 2 n − 1 ) ( 2 n + 2 ) ( ( 2 n n ) 4 n ) 2 ( n ≥ 1 ) , κ 2 n + 1 = 0. (5) \kappa_0 = \frac{\pi}{4}, \qquad\kappa_{2n} = -\frac{\pi}{2(2n-1)(2n+2)}\left(\frac{\binom{2n}{n}}{4^n}\right)^2 \quad(n \ge1), \qquad\kappa_{2n+1}=0. \tag*{(5)} κ 0 = 4 π , κ 2 n = − 2 ( 2 n − 1 ) ( 2 n + 2 ) π ( 4 n ( n 2 n ) ) 2 ( n ≥ 1 ) , κ 2 n + 1 = 0. (5) In particular inf l κ l = κ 2 = − π / 32 \inf_l \kappa_l=\kappa_2=-\pi/32 inf l κ l = κ 2 = − π /32 consecutive even eigenvalues are distinct, and ∣ κ 2 n ∣ = O ( n − 3 ) |\kappa_{2n}|=O(n^{-3}) ∣ κ 2 n ∣ = O ( n − 3 )
Proof. The Funk–Hecke formula gives κ l = 1 2 ∫ − 1 1 1 − t 2 P l ( t ) d t \kappa_l=\frac{1}{2}\int_{-1}^{1}\sqrt{1-t^2}P_l(t)\,dt κ l = 2 1 ∫ − 1 1 1 − t 2 P l ( t ) d t P l P_l P l w ( t ) = 1 − t 2 w(t)=\sqrt{1-t^2} w ( t ) = 1 − t 2 w P l wP_l w P l P l / w P_l/w P l / w I l I_l I l J l J_l J l l ( l + 1 ) I l = 2 I l − J l l(l+1)I_l=2I_l-J_l l ( l + 1 ) I l = 2 I l − J l
∑ l ≥ 0 J l z l = ∫ 0 π ( 1 − z e i θ ) − 1 / 2 ( 1 − z e − i θ ) − 1 / 2 d θ = π ∑ n ≥ 0 ( ( 2 n n ) 4 n ) 2 z 2 n . \sum_{l\ge0}J_lz^l=\int_0^\pi(1-ze^{i\theta})^{-1/2}(1-ze^{-i\theta})^{-1/2}\,d\theta=\pi\sum_{n\ge0}\left(\frac{\binom{2n}{n}}{4^n}\right)^2z^{2n}. l ≥ 0 ∑ J l z l = ∫ 0 π ( 1 − z e i θ ) − 1/2 ( 1 − z e − i θ ) − 1/2 d θ = π n ≥ 0 ∑ ( 4 n ( n 2 n ) ) 2 z 2 n . Here conjugate off-diagonal terms combine into cosines with zero integral. This proves (5) . Finally,
∣ κ 2 n + 2 ∣ ∣ κ 2 n ∣ = 4 n 2 − 1 4 ( n + 1 ) ( n + 2 ) < 1 , \frac{|\kappa_{2n+2}|}{|\kappa_{2n}|}=\frac{4n^2-1}{4(n+1)(n+2)}<1, ∣ κ 2 n ∣ ∣ κ 2 n + 2 ∣ = 4 ( n + 1 ) ( n + 2 ) 4 n 2 − 1 < 1 , and the central binomial coefficient bound gives the stated decay. □
The isotropic critical point is the constant r ( a ) > 0 r(a)>0 r ( a ) > 0 log r + 2 κ 0 r = a \log r+2\kappa_0r=a log r + 2 κ 0 r = a a a a
Onsager’s degree-two trial profiles show how global minimality can be lost before local stability [6 pp. 645–646] ; see also Kayser and Raveché [8 ] . The following proof uses an explicit finite-amplitude comparison in the grand-canonical functional.
Theorem 2.3 (Homogeneous coexistence). There is a t a_t a t for which W : = W a t \mathcal{W}:=\mathcal{W}_{a_t} W := W a t contains an isolated isotropic profile r I = r ( a t ) < 5 r_I=r(a_t)<5 r I = r ( a t ) < 5 and at least one nonisotropic profile.
Proof. On the convex order interval 0 < f ≤ M 0<f\le M 0 < f ≤ M
D 2 Ω a ( f ) [ g , g ] ≥ ( M − 1 − π 16 ) ∥ g ∥ 2 2 . D^2\Omega_a(f)[g,g]\ge\left(M^{-1}-\frac{\pi}{16}\right)\lVert g\rVert_2^2. D 2 Ω a ( f ) [ g , g ] ≥ ( M − 1 − 16 π ) ∥ g ∥ 2 2 . If e a < 16 / π e^a<16/\pi e a < 16/ π r ( a ) r(a) r ( a ) r ( a ) < 16 / π r(a)<16/\pi r ( a ) < 16/ π r ( a ) < M < 16 / π r(a)<M<16/\pi r ( a ) < M < 16/ π a a a
To find loss of global minimality before this threshold, take g ( u ) = 1 + P 2 ( u ⋅ n ) g(u)=1+P_2(u\cdot n) g ( u ) = 1 + P 2 ( u ⋅ n ) 1 / 2 1/2 1/2 2 2 2 ∫ P 2 2 d ω = 1 / 5 \int P_2^2\,d\omega=1/5 ∫ P 2 2 d ω = 1/5
Ω a ( 5 g ) − Ω a ( 5 ) = 5 ( H − π 32 ) , H = ∫ 0 1 1 + 3 t 2 2 log 1 + 3 t 2 2 d t = log 2 − 1 + 2 π 9 3 . \Omega_a(5g)-\Omega_a(5)=5\left(H-\frac{\pi}{32}\right), \qquad H=\int_0^1\frac{1+3t^2}{2}\log\frac{1+3t^2}{2}\,dt=\log2-1+\frac{2\pi}{9\sqrt{3}}. Ω a ( 5 g ) − Ω a ( 5 ) = 5 ( H − 32 π ) , H = ∫ 0 1 2 1 + 3 t 2 log 2 1 + 3 t 2 d t = log 2 − 1 + 9 3 2 π . This is strictly negative. Indeed, the series log 2 = 2 ∑ n ≥ 0 3 − 2 n − 1 / ( 2 n + 1 ) \log2=2\sum_{n\ge0}3^{-2n-1}/(2n+1) log 2 = 2 ∑ n ≥ 0 3 − 2 n − 1 / ( 2 n + 1 ) log 2 < 1733 / 2500 \log2<1733/2500 log 2 < 1733/2500 3 > 19 / 11 \sqrt{3}>19/11 3 > 19/11 π < 22 / 7 \pi<22/7 π < 22/7
H − π 32 < 1733 2500 − 1 + 22 7 ( 22 171 − 1 32 ) = − 8021 11970000 < 0. (6) H-\frac{\pi}{32}<\frac{1733}{2500}-1+\frac{22}{7}\left(\frac{22}{171}-\frac{1}{32}\right)=-\frac{8021}{11970000}<0. \tag*{(6)} H − 32 π < 2500 1733 − 1 + 7 22 ( 171 22 − 32 1 ) = − 11970000 8021 < 0. (6) ::: Let a 5 = log 5 + 10 κ 0 a_5=\log5+10\kappa_0 a 5 = log 5 + 10 κ 0 a 5 a_5 a 5 r ( a 5 ) = 5 < 16 / π r(a_5)=5<16/\pi r ( a 5 ) = 5 < 16/ π
Let S = { a : r ( a ) ∈ W a } S=\{a:r(a)\in\mathcal{W}_a\} S = { a : r ( a ) ∈ W a } a t = sup { a : ( − ∞ , a ] ⊂ S } a_t=\sup\{a:(-\infty,a]\subset S\} a t = sup { a : ( − ∞ , a ] ⊂ S } m a m_a m a S S S a t ∈ S a_t\in S a t ∈ S a t < a 5 a_t<a_5 a t < a 5 a n ↓ a t a_n\downarrow a_t a n ↓ a t a n ∉ S a_n\notin S a n ∈ / S f n f_n f n f ∗ f_\ast f ∗ W \mathcal{W} W r I r_I r I f n = r ( a n ) f_n=r(a_n) f n = r ( a n )
Lemma 2.4 (Density separation). There is Δ > 0 \Delta>0 Δ > 0
∫ S 2 f d ω ≥ r I + Δ ( f ∈ W ∖ { r I } ) . (7) \int_{S^2} f\,\mathrm{d}\omega\ge r_I+\Delta\qquad(f\in\mathcal{W}\setminus\{r_I\}). \tag*{(7)} ∫ S 2 f d ω ≥ r I + Δ ( f ∈ W ∖ { r I }) . (7) Proof. We eliminate the interaction from Onsager’s equal-pressure and equal-chemical-potential equations [6 Eq. (95)] . For f ∈ W f\in\mathcal{W} f ∈ W f = ρ g f=\rho g f = ρ g ∫ g = 1 \int g=1 ∫ g = 1 t = ρ / r I t=\rho/r_I t = ρ / r I
log ρ + ∫ g log g + 2 ρ ⟨ g , K g ⟩ = a t , ρ + ρ 2 ⟨ g , K g ⟩ = r I + κ 0 r I 2 . \log\rho+\int g\log g+2\rho\langle g,Kg\rangle=a_t,\qquad\rho+\rho^2\langle g,Kg\rangle=r_I+\kappa_0r_I^2. log ρ + ∫ g log g + 2 ρ ⟨ g , K g ⟩ = a t , ρ + ρ 2 ⟨ g , K g ⟩ = r I + κ 0 r I 2 . Eliminating the interaction yields
∫ g log g = − log t + 2 ( 1 + κ 0 r I ) ( 1 − t − 1 ) . \int g\log g=-\log t+2(1+\kappa_0r_I)(1-t^{-1}). ∫ g log g = − log t + 2 ( 1 + κ 0 r I ) ( 1 − t − 1 ) . The right side is strictly increasing on ( 0 , 1 ] (0,1] ( 0 , 1 ] t ≥ 1 t\ge1 t ≥ 1 g = 1 g=1 g = 1 W ∖ { r I } \mathcal{W}\setminus\{r_I\} W ∖ { r I } r I r_I r I (7) .
The geometry of the well set
Let P N P_N P N 2 N 2N 2 N Q N = I − P N Q_N=I-P_N Q N = I − P N
∥ Q N K ∥ C → C ≤ ∑ n > N ( 4 n + 1 ) ∣ κ 2 n ∣ ⟶ 0. (8) \|Q_NK\|_{C\to C}\le\sum_{n>N}(4n+1)|\kappa_{2n}|\longrightarrow0. \tag*{(8)} ∥ Q N K ∥ C → C ≤ n > N ∑ ( 4 n + 1 ) ∣ κ 2 n ∣ ⟶ 0. (8) The reduction below is a global finite-mode version of the Lyapunov–Schmidt construction used for Onsager equilibria in [7 Section 3] : a uniform contraction on the harmonic tail covers the entire minimizing set.
Proposition 2.5 (Global analytic well coordinates). For fixed a a a E E E U ⊂ E U\subset E U ⊂ E Z ⋐ U Z\Subset U Z ⋐ U Ψ : U → C e v e n ( S 2 ) \Psi:U\to C_{\mathrm{even}}(S^2) Ψ : U → C even ( S 2 ) Ψ ( Z ) = W a \Psi(Z)=\mathcal{W}_a Ψ ( Z ) = W a
Proof. Set j − = a − 2 e a − 1 j_-=a-2e^a-1 j − = a − 2 e a − 1 j + = a + 1 j_+=a+1 j + = a + 1 T J ( t ) = exp ( max ( j − , min ( t , j + ) ) ) T_J(t)=\exp(\max(j_-,\min(t,j_+))) T J ( t ) = exp ( max ( j − , min ( t , j + ))) N N N q ∗ = 2 e j + ∥ Q N K ∥ C → C < 1 q_\ast=2e^{j_+}\|Q_NK\|_{C\to C}<1 q ∗ = 2 e j + ∥ Q N K ∥ C → C < 1 v ∈ E = P N C e v e n ( S 2 ) v\in E=P_NC_{\mathrm{even}}(S^2) v ∈ E = P N C even ( S 2 )
w ⟼ − 2 Q N K T J ( v + w ) w\longmapsto-2Q_NKT_J(v+w) w ⟼ − 2 Q N K T J ( v + w ) on Q N C e v e n ( S 2 ) Q_NC_{\mathrm{even}}(S^2) Q N C even ( S 2 ) w ( v ) w(v) w ( v ) v v v U U U v v v j − < v + w ( v ) < j + j_-<v+w(v)<j_+ j − < v + w ( v ) < j + U U U w w w w + 2 Q N K e v + w w+2Q_NKe^{v+w} w + 2 Q N K e v + w q ∗ q_\ast q ∗ w w w
For every well, (4) and the one-unit margin in J J J w ( P N log f ) = Q N log f w(\mathcal{P}_N\log f)=Q_N\log f w ( P N log f ) = Q N log f
F ( v ) = P N [ v + w ( v ) + 2 K e v + w ( v ) − a ] , G ( v ) = Ω a ( e v + w ( v ) ) − m a . F(v)=\mathcal{P}_N[v+w(v)+2Ke^{v+w(v)}-a],\qquad G(v)=\Omega_a(e^{v+w(v)})-m_a. F ( v ) = P N [ v + w ( v ) + 2 K e v + w ( v ) − a ] , G ( v ) = Ω a ( e v + w ( v ) ) − m a . Both are real analytic. Their complete common zero set in U U U Z = P N log W a Z=\mathcal{P}_N\log\mathcal{W}_a Z = P N log W a m a m_a m a Z ⋐ U Z\Subset U Z ⋐ U Ψ ( v ) = e v + w ( v ) \Psi(v)=e^{v+w(v)} Ψ ( v ) = e v + w ( v ) P N log Ψ ( v ) = v \mathcal{P}_N\log\Psi(v)=v P N log Ψ ( v ) = v
Corollary 2.6 (Finitely many well components). The compact set W a \mathcal{W}_a W a
Proof. The analytic set Z Z Z [9 Corollary 2.7] . Its continuous injection Ψ \Psi Ψ W a \mathcal{W}_a W a
Lemma 2.7 (Paths within a well component). Any two wells in one component can be joined by a Lipschitz path with values in that component, in the uniform norm.
Proof. A connected compact semianalytic set is rectifiably connected [9 Theorem 6.10] . The derivative of Ψ \Psi Ψ Z Z Z
An exact auxiliary field representation
Set β = ε − 1 \beta=\varepsilon^{-1} β = ε − 1 ζ = e a t \zeta=e^{a_t} ζ = e a t a t a_t a t Theorem 2.3 . The parameter β \beta β h = a − a t h=a-a_t h = a − a t
p ε ( h ) = P ε ( a t + h ) , z = β ζ e h . (9) p_\varepsilon(h)=P_\varepsilon(a_t+h),\qquad z=\beta\zeta e^h. \tag*{(9)} p ε ( h ) = P ε ( a t + h ) , z = β ζ e h . (9) Cells, volumes, and reference measure
Partition space into congruent fine cubes B i B_i B i v s v_s v s U a U_a U a ω a > 0 \omega_a>0 ω a > 0 C = ( i , a ) C=(i,a) C = ( i , a ) v C = v s ω a v_C=v_s\omega_a v C = v s ω a P P P L 3 L^3 L 3
V L = v s L 3 . V_L=v_sL^3. V L = v s L 3 . Volumes in pressure exponents are Euclidean volumes. Surface estimates may absorb the fixed mesh factors in their constants.
Choose a preliminary source interval ∣ h ∣ ≤ H |h|\le H ∣ h ∣ ≤ H α \alpha α
ρ > e ζ e H , R > H + 2 α ρ + 2. \rho>e\zeta e^H,\qquad R>H+2\alpha\rho+2. ρ > e ζ e H , R > H + 2 α ρ + 2. The choices of α \alpha α C C C K C = ⌈ ρ β v C ⌉ K_C=\lceil\rho\beta v_C\rceil K C = ⌈ ρβ v C ⌉ (1) remains unrestricted.
Normalized Gaussian augmentation
The completion of the square is the Gaussian auxiliary-field method of Stratonovich and Hubbard [10 , 11 ] . We use normalized conditional densities on a bounded field interval, with the particle count reset above the threshold.
For a physical configuration η \eta η n C = N C ( η ) n_C = N_C(\eta) n C = N C ( η ) n ^ C = n C 1 { n C ≤ K C } \widehat{n}_C = n_C \mathbf{1}_{\{n_C \le K_C\}} n C = n C 1 { n C ≤ K C } ϕ C ∈ [ − R , R ] \phi_C \in[-R,R] ϕ C ∈ [ − R , R ] η \eta η
1 c C , h ( n ^ C ) β v C 4 π α exp { − β v C 4 α ( ϕ C − h − 2 α n ^ C β v C ) 2 } . (10) \frac{1}{c_{C,h}(\widehat{n}_C)}\sqrt{\frac{\beta v_C}{4\pi\alpha}}\exp\left\{-\frac{\beta v_C}{4\alpha}\left(\phi_C-h-\frac{2\alpha\widehat{n}_C}{\beta v_C}\right)^2\right\}. \tag*{(10)} c C , h ( n C ) 1 4 π α β v C exp { − 4 α β v C ( ϕ C − h − β v C 2 α n C ) 2 } . (10) The normalizer c C , h ( n ) c_{C,h}(n) c C , h ( n ) [ − R , R ] [-R,R] [ − R , R ] β \beta β (10) stay a fixed positive distance from the endpoints. Thus, uniformly in 0 ≤ n ≤ K C 0 \le n \le K_C 0 ≤ n ≤ K C
c C , h ( n ) = 1 + O ( e − c β ) , c_{C,h}(n)=1+O(e^{-c\beta}), c C , h ( n ) = 1 + O ( e − c β ) , with the same conclusion for any fixed number of h h h c > 0 c>0 c > 0
Put
g C , h ( ϕ ) = β v C 4 π α e − β v C ( ϕ − h ) 2 / ( 4 α ) . g_{C,h}(\phi)=\sqrt{\frac{\beta v_C}{4\pi\alpha}}e^{-\beta v_C(\phi-h)^2/(4\alpha)}. g C , h ( ϕ ) = 4 π α β v C e − β v C ( ϕ − h ) 2 / ( 4 α ) . Expansion of the square shows that, after extracting g C , h g_{C,h} g C , h
w C , h , ϕ ( n ) = { ( β ζ ) n exp { ϕ n − α n 2 / ( β v C ) } / c C , h ( n ) , 0 ≤ n ≤ K C , ( β ζ e h ) n / c C , h ( 0 ) , n > K C . (11) w_{C,h,\phi}(n)=
\begin{cases}
(\beta\zeta)^n\exp\{\phi n-\alpha n^2/(\beta v_C)\}/c_{C,h}(n), & 0\le n\le K_C,\\
(\beta\zeta e^h)^n/c_{C,h}(0), & n>K_C.
\end{cases} \tag*{(11)} w C , h , ϕ ( n ) = { ( β ζ ) n exp { ϕ n − α n 2 / ( β v C )} / c C , h ( n ) , ( β ζ e h ) n / c C , h ( 0 ) , 0 ≤ n ≤ K C , n > K C . (11) In particular, the second branch is independent of ϕ \phi ϕ (1) .
Use the probability reference d λ C = d ϕ C / ( 2 R ) d\lambda_C=d\phi_C/(2R) d λ C = d ϕ C / ( 2 R ) R r o d , Λ ( ϕ ∣ ξ ) R_{\mathrm{rod},\Lambda}(\phi\mid\xi) R rod , Λ ( ϕ ∣ ξ ) (11) , its full field density is
F Λ ( ϕ ∣ ξ ) = ( 2 R ) ∣ I ( Λ ) ∣ ∏ C ∈ I ( Λ ) g C , h ( ϕ C ) R r o d , Λ ( ϕ ∣ ξ ) . (12) F_\Lambda(\phi\mid\xi)=(2R)^{|I(\Lambda)|}\prod_{C\in I(\Lambda)}g_{C,h}(\phi_C)R_{\mathrm{rod},\Lambda}(\phi\mid\xi). \tag*{(12)} F Λ ( ϕ ∣ ξ ) = ( 2 R ) ∣ I ( Λ ) ∣ C ∈ I ( Λ ) ∏ g C , h ( ϕ C ) R rod , Λ ( ϕ ∣ ξ ) . (12) Here I ( Λ ) I(\Lambda) I ( Λ ) Λ \Lambda Λ (10) integrates to one, Tonelli’s theorem gives the exact identity
Z Λ , ε ( β ζ e h ∣ ξ ) = ∫ F Λ ( ϕ ∣ ξ ) d λ Λ ( ϕ ) . (13) Z_{\Lambda,\varepsilon}(\beta\zeta e^h\mid\xi)=\int F_\Lambda(\phi\mid\xi)\,d\lambda_\Lambda(\phi). \tag*{(13)} Z Λ , ε ( β ζ e h ∣ ξ ) = ∫ F Λ ( ϕ ∣ ξ ) d λ Λ ( ϕ ) . (13) This equality holds for every finite volume and every admissible exterior configuration. The explicit source contribution is
log g C , h ( ϕ C ) g C , 0 ( ϕ C ) = β v C h ϕ C 2 α − β v C h 2 4 α . (14) \log\frac{g_{C,h}(\phi_C)}{g_{C,0}(\phi_C)}
=\frac{\beta v_C h\phi_C}{2\alpha}-\frac{\beta v_C h^2}{4\alpha}. \tag*{(14)} log g C , 0 ( ϕ C ) g C , h ( ϕ C ) = 2 α β v C h ϕ C − 4 α β v C h 2 . (14) Its quadratic term sums to − β h 2 ∣ Λ ∣ / ( 4 α ) -\beta h^2|\Lambda|/(4\alpha) − β h 2 ∣Λ∣/ ( 4 α ) ∑ a ω a = 1 \sum_a\omega_a=1 ∑ a ω a = 1
The finite-volume variational limit
We identify the free energy of the quadratically pinned gas. The spatial region and marked-cell partition remain fixed throughout the limit of vanishing diameter.
Let D ⊂ R 3 D \subset\mathbb{R}^{3} D ⊂ R 3 Ω = D × S 2 \Omega= D \times S^{2} Ω = D × S 2 d ν = d x d ω ( u ) \mathrm{d}\nu= \mathrm{d}x\,\mathrm{d}\omega(u) d ν = d x d ω ( u ) ω ( S 2 ) = 1 \omega(S^{2}) = 1 ω ( S 2 ) = 1 Ω \Omega Ω b ε ( X , Y ) b_{\varepsilon}(X,Y) b ε ( X , Y ) X = ( x , u ) X=(x,u) X = ( x , u ) Y = ( y , v ) Y=(y,v) Y = ( y , v )
( A f ) ( x , u ) = ∫ − 1 / 2 1 / 2 f ( x + t u , u ) d t , ( K f ) ( x , u ) = ∫ S 2 ∣ u × v ∣ f ( x , v ) d ω ( v ) , T = 2 A ∗ K A . (Af)(x,u) = \int_{-1/2}^{1/2} f(x+tu,u)\,\mathrm{d}t,\qquad(Kf)(x,u) = \int_{S^{2}} |u \times v|f(x,v)\,\mathrm{d}\omega(v),\qquad T = 2A^{*}KA. ( A f ) ( x , u ) = ∫ − 1/2 1/2 f ( x + t u , u ) d t , ( K f ) ( x , u ) = ∫ S 2 ∣ u × v ∣ f ( x , v ) d ω ( v ) , T = 2 A ∗ K A . The operator T T T π / 2 \pi/2 π /2 { C i } i = 1 m \{C_i\}_{i=1}^{m} { C i } i = 1 m Ω \Omega Ω v i = ν ( C i ) > 0 v_i=\nu(C_i)>0 v i = ν ( C i ) > 0 P P P 0 < a ≤ a max < ∞ 0<a\leq a_{\max}<\infty 0 < a ≤ a m a x < ∞
F a ( q ) = ∫ Ω q ( log q a − 1 ) d ν + 1 2 ⟨ q , T q ⟩ + α ∥ P q ∥ 2 2 . (15) \mathcal{F}_{a}(q)=\int_{\Omega}q\left(\log\frac{q}{a}-1\right)\,\mathrm{d}\nu+\frac{1}{2}\langle q,Tq\rangle+\alpha\lVert Pq\rVert_{2}^{2}. \tag*{(15)} F a ( q ) = ∫ Ω q ( log a q − 1 ) d ν + 2 1 ⟨ q , T q ⟩ + α ∥ P q ∥ 2 2 . (15) At diameter ε \varepsilon ε a / ε a/\varepsilon a / ε exp [ − α ε ∑ i N i 2 / v i ] \exp[-\alpha\varepsilon\sum_i N_i^2/v_i] exp [ − α ε ∑ i N i 2 / v i ] N i ≤ ⌈ ρ v i / ε ⌉ N_i\leq\lceil\rho v_i/\varepsilon\rceil N i ≤ ⌈ ρ v i / ε ⌉
Theorem 4.1 (Pinned finite-volume limit). Suppose T + 2 α P ≥ − κ I T+2\alpha P\geq-\kappa I T + 2 α P ≥ − κ I and κ a max < 1 \kappa a_{\max}<1 κ a m a x < 1 Then
lim ε ↓ 0 ε log Z ε , c a p = − min q ≥ 0 ∫ C i q d ν ≤ ρ v i F a ( q ) . (16) \lim_{\varepsilon\downarrow0}\varepsilon\log Z_{\varepsilon,\mathrm{cap}}=-\min_{\substack{q\geq0\\ \int_{C_i}q\,\mathrm{d}\nu\leq\rho v_i}}\mathcal{F}_{a}(q). \tag*{(16)} ε ↓ 0 lim ε log Z ε , cap = − q ≥ 0 ∫ C i q d ν ≤ ρ v i min F a ( q ) . (16) The overlap operator and common neighbors
The leading excluded volume is Onsager’s angular interaction [6 ] . The following estimate strengthens this geometric limit to convergence in operator norm.
Lemma 4.2. If B ε B_{\varepsilon} B ε b ε b_{\varepsilon} b ε
∥ B ε / ε − T ∥ 2 → 2 ≤ C ε , sup X ∫ b ε ( X , Y ) d ν ( Y ) ≤ C ε . \lVert B_{\varepsilon}/\varepsilon-T\rVert_{2\to2}\leq C\varepsilon,\qquad\sup_X\int b_{\varepsilon}(X,Y)\,\mathrm{d}\nu(Y)\leq C\varepsilon. ∥ B ε / ε − T ∥ 2 → 2 ≤ C ε , X sup ∫ b ε ( X , Y ) d ν ( Y ) ≤ C ε . Proof. For fixed orientations u , v u,v u , v
[ − 1 / 2 , 1 / 2 ] u + [ − 1 / 2 , 1 / 2 ] v + B ‾ ( 0 , ε ) . [-1/2,1/2]u+[-1/2,1/2]v+\overline{B}(0,\varepsilon). [ − 1/2 , 1/2 ] u + [ − 1/2 , 1/2 ] v + B ( 0 , ε ) . It contains the normal prism over the parallelogram, of thickness 2 ε 2\varepsilon 2 ε
2 ε ∣ u × v ∣ + 2 π ε 2 + 4 π 3 ε 3 . 2\varepsilon|u\times v|+2\pi\varepsilon^{2}+\frac{4\pi}{3}\varepsilon^{3}. 2 ε ∣ u × v ∣ + 2 π ε 2 + 3 4 π ε 3 . This follows by integrating the planar parallel-body formula over normal heights; the parallelogram has perimeter four. Parallel orientations follow by continuity. The remainder outside the prism has volume O ( ε 2 ) O(\varepsilon^{2}) O ( ε 2 ) u , v u,v u , v C ε 2 C\varepsilon^{2} C ε 2
For k ∈ R 3 k\in\mathbb{R}^{3} k ∈ R 3 s k ( u ) = sinc ( k ⋅ u / 2 ) s_k(u)=\operatorname{sinc}(k\cdot u/2) s k ( u ) = sinc ( k ⋅ u /2 ) sinc ( t ) = sin ( t ) / t \operatorname{sinc}(t)=\sin(t)/t sinc ( t ) = sin ( t ) / t ε \varepsilon ε
2 ∣ u × v ∣ s k ( u ) s k ( v ) sinc ( ε k ⋅ n ( u , v ) ) , n ( u , v ) = u × v ∣ u × v ∣ . 2|u\times v|s_k(u)s_k(v)\operatorname{sinc}(\varepsilon k\cdot n(u,v)),\qquad n(u,v)=\frac{u\times v}{|u\times v|}. 2∣ u × v ∣ s k ( u ) s k ( v ) sinc ( ε k ⋅ n ( u , v )) , n ( u , v ) = ∣ u × v ∣ u × v . The symbol of T T T
J ( r ) = ∫ S 2 s k ( u ) 2 d ω ( u ) ≤ C 1 + r , r = ∣ k ∣ . J(r)=\int_{S^2}s_k(u)^2\,d\omega(u)\le\frac{C}{1+r},\qquad r=|k|. J ( r ) = ∫ S 2 s k ( u ) 2 d ω ( u ) ≤ 1 + r C , r = ∣ k ∣. The angular Hilbert–Schmidt norm of the difference is at most C min ( ε 2 r 2 , 1 ) J ( r ) ≤ C ε C\min(\varepsilon^2r^2,1)J(r)\le C\varepsilon C min ( ε 2 r 2 , 1 ) J ( r ) ≤ C ε
Lemma 4.3 (Compactness of the limiting interaction). The compression of T T T
Proof. Cut the symbol of T T T ∣ k ∣ ≤ R |k|\le R ∣ k ∣ ≤ R D D D
( 2 π ) − 3 ∣ D ∣ ∫ ∣ k ∣ ≤ R ∥ T ^ ( k ) ∥ H S ( L 2 ( S 2 ) ) 2 d k < ∞ . (2\pi)^{-3}|D|\int_{|k|\le R}\|\widehat{T}(k)\|_{\mathrm{HS}(L^2(S^2))}^2\,dk<\infty. ( 2 π ) − 3 ∣ D ∣ ∫ ∣ k ∣ ≤ R ∥ T ( k ) ∥ HS ( L 2 ( S 2 )) 2 d k < ∞. Right localization preserves this property, and the Fourier tail has norm at most C / ( 1 + R ) C/(1+R) C / ( 1 + R )
For d μ = w d ν d\mu=w\,d\nu d μ = w d ν w ≥ 0 w\ge0 w ≥ 0 T μ T_\mu T μ L 2 ( μ ) L^2(\mu) L 2 ( μ ) U μ f = w f U_\mu f=\sqrt{w}f U μ f = w f L 2 ( μ ) L^2(\mu) L 2 ( μ ) L 2 ( ν ) L^2(\nu) L 2 ( ν ) { w > 0 } \{w>0\} { w > 0 }
U μ T μ U μ − 1 = M w T M w . U_\mu T_\mu U_\mu^{-1}=M_{\sqrt{w}}TM_{\sqrt{w}}. U μ T μ U μ − 1 = M w T M w . The same conjugation applies to B ε / ε B_\varepsilon/\varepsilon B ε / ε T μ T_\mu T μ P → I P\to I P → I ∥ T − P T P ∥ → 0 \|T-PTP\|\to0 ∥ T − P T P ∥ → 0 P T P + 2 α P ≥ ( 2 α − ∥ T ∥ ) P PTP+2\alpha P\ge(2\alpha-\|T\|)P P T P + 2 α P ≥ ( 2 α − ∥ T ∥ ) P Theorem 4.1 holds on a sufficiently fine partition whenever 2 α > ∥ T ∥ 2\alpha>\|T\| 2 α > ∥ T ∥
Lemma 4.4 (Common-neighbor bound). Let μ \mu μ Ω \Omega Ω X = ( x , u ) X=(x,u) X = ( x , u ) Y = ( y , v ) Y=(y,v) Y = ( y , v ) s = ∣ u × v ∣ s=|u\times v| s = ∣ u × v ∣ d = ∣ ( y − x ) × u ∣ d=|(y-x)\times u| d = ∣ ( y − x ) × u ∣
c ε ( X , Y ) = ∫ b ε ( X , Z ) b ε ( Y , Z ) d μ ( Z ) . c_\varepsilon(X,Y)=\int b_\varepsilon(X,Z)b_\varepsilon(Y,Z)\,d\mu(Z). c ε ( X , Y ) = ∫ b ε ( X , Z ) b ε ( Y , Z ) d μ ( Z ) . Then
c ε ( X , Y ) ≤ { C ε , s ≤ ε , C ε 2 s − 1 ( 1 + log ( s / ε ) ) , s > ε , (17) c_\varepsilon(X,Y)\le
\begin{cases}
C\varepsilon, & s\le\varepsilon,\\
C\varepsilon^2s^{-1}(1+\log(s/\varepsilon)), & s>\varepsilon,
\end{cases} \tag*{(17)} c ε ( X , Y ) ≤ { C ε , C ε 2 s − 1 ( 1 + log ( s / ε )) , s ≤ ε , s > ε , (17) c ε ( X , Y ) ≤ C ε min { 1 , ( s + ε ) / d } . (18) c_\varepsilon(X,Y)\le C\varepsilon\min\{1,(s+\varepsilon)/d\}. \tag*{(18)} c ε ( X , Y ) ≤ C ε min { 1 , ( s + ε ) / d } . (18) The last quotient is interpreted as infinity when d = 0 d=0 d = 0
Proof. Fix the direction w w w 2 ε 2\varepsilon 2 ε
∣ det ( u , v , w ) ∣ ∣ u × w ∣ ∣ v × w ∣ ≥ ∣ det ( u , v , w ) ∣ . \frac{|\det(u,v,w)|}{|u\times w|\,|v\times w|}\ge|\det(u,v,w)|. ∣ u × w ∣ ∣ v × w ∣ ∣ det ( u , v , w ) ∣ ≥ ∣ det ( u , v , w ) ∣. Their intersection has volume at most C min { ε , ε 2 / ∣ det ( u , v , w ) ∣ } C\min\{\varepsilon,\varepsilon^2/|\det(u,v,w)|\} C min { ε , ε 2 /∣ det ( u , v , w ) ∣ } w w w det ( u , v , w ) = s n ( u , v ) ⋅ w \det(u,v,w)=sn(u,v)\cdot w det ( u , v , w ) = s n ( u , v ) ⋅ w (17) .
If both intersections occur, subtraction of the two intersection relations gives y − x = t u − t ′ v + τ w + e y-x=tu-t'v+\tau w+e y − x = t u − t ′ v + τ w + e ∣ t ∣ , ∣ t ′ ∣ ≤ 1 / 2 |t|,|t'|\le1/2 ∣ t ∣ , ∣ t ′ ∣ ≤ 1/2 ∣ e ∣ ≤ 2 ε |e|\le2\varepsilon ∣ e ∣ ≤ 2 ε u × w u\times w u × w ∣ ( ( y − x ) × u ) ⋅ w ∣ ≤ C s + C ε |((y-x)\times u)\cdot w|\le Cs+C\varepsilon ∣ (( y − x ) × u ) ⋅ w ∣ ≤ C s + C ε C min { 1 , ( s + ε ) / d } C\min\{1,(s+\varepsilon)/d\} C min { 1 , ( s + ε ) / d } C ε C\varepsilon C ε (18) .
For r > ε r > \varepsilon r > ε s = ε r s = \sqrt{\varepsilon r} s = ε r
sup max ( s , d ) ≥ r c ε ( X , Y ) ε ≤ C ε r ( 1 + log ( 1 / ε ) ) . (19) \sup_{\max(s,d)\ge r}\frac{c_\varepsilon(X,Y)}{\varepsilon} \le C\sqrt{\frac{\varepsilon}{r}}(1+\log(1/\varepsilon)). \tag*{(19)} m a x ( s , d ) ≥ r sup ε c ε ( X , Y ) ≤ C r ε ( 1 + log ( 1/ ε )) . (19) For independent marks of bounded density, the probability that both s < r s < r s < r d < r d < r d < r C r 4 Cr^4 C r 4 r r r
Sampling the interaction uniformly over trial profiles
Operator norm control makes the interaction approximation uniform over weights chosen after sampling the point cloud.
Lemma 4.5 (Covariance sampling). Let v 1 , … , v m v_1,\ldots,v_m v 1 , … , v m be independent copies of a random vector v v v in a separable Hilbert space with ∥ v ∥ 2 ≤ L \lVert v\rVert^2\le L ∥ v ∥ 2 ≤ L Set Q = m − 1 ∑ j v j ⊗ v j Q=m^{-1}\sum_j v_j\otimes v_j Q = m − 1 ∑ j v j ⊗ v j A = E ( v ⊗ v ) A=\mathbb{E}(v\otimes v) A = E ( v ⊗ v ) and k m = ( 1 + 2 ⌈ log m ⌉ ) 1 / 2 k_m=(1+2\lceil\log m\rceil)^{1/2} k m = ( 1 + 2 ⌈ log m ⌉ ) 1/2 Then, with b m = 2 k m L / m b_m=2k_m\sqrt{L/m} b m = 2 k m L / m
E ∥ Q − A ∥ ≤ b m ∥ A ∥ 1 / 2 + b m 2 , E ∥ Q ∥ ≤ ( ∥ A ∥ 1 / 2 + b m ) 2 . \mathbb{E}\lVert Q-A\rVert\le b_m\lVert A\rVert^{1/2}+b_m^2,\qquad\mathbb{E}\lVert Q\rVert\le(\lVert A\rVert^{1/2}+b_m)^2. E ∥ Q − A ∥ ≤ b m ∥ A ∥ 1/2 + b m 2 , E ∥ Q ∥ ≤ (∥ A ∥ 1/2 + b m ) 2 . Proof. Let σ j \sigma_j σ j ± 1 \pm1 ± 1 [12 Theorem 4.1] give
d : = E ∥ Q − A ∥ ≤ 2 m E E σ ∥ ∑ j σ j ( v j ⊗ v j ) ∥ d:=\mathbb{E}\lVert Q-A\rVert\le\frac{2}{m}\mathbb{E}\mathbb{E}_{\sigma}\left\lVert\sum_j\sigma_j(v_j\otimes v_j)\right\rVert d := E ∥ Q − A ∥ ≤ m 2 E E σ j ∑ σ j ( v j ⊗ v j ) ≤ 2 k m L m E ∥ Q ∥ 1 / 2 ≤ b m ∥ A ∥ + d . \le2k_m\sqrt{\frac{L}{m}}\mathbb{E}\lVert Q\rVert^{1/2}\le b_m\sqrt{\lVert A\rVert}+d. ≤ 2 k m m L E ∥ Q ∥ 1/2 ≤ b m ∥ A ∥ + d . The matrices act on the sample span, of dimension at most m m m ( v ⊗ v ) 2 ≤ L ( v ⊗ v ) (v\otimes v)^2\le L(v\otimes v) ( v ⊗ v ) 2 ≤ L ( v ⊗ v )
Lemma 4.6 (Empirical operator approximation). Suppose a real symmetric kernel R ε R_\varepsilon R ε on a probability space ( Ω , μ ) (\Omega,\mu) ( Ω , μ ) satisfies
∥ R ε ∥ ∞ ≤ C 0 / ε , sup y ∫ ∣ R ε ( x , y ) ∣ 2 d μ ( x ) ≤ C 1 / ε . \lVert R_\varepsilon\rVert_\infty\le C_0/\varepsilon,\qquad\sup_y\int\lvert R_\varepsilon(x,y)\rvert^2\,d\mu(x)\le C_1/\varepsilon. ∥ R ε ∥ ∞ ≤ C 0 / ε , y sup ∫ ∣ R ε ( x , y ) ∣ 2 d μ ( x ) ≤ C 1 / ε . For independent Z 1 , … , Z n Z_1,\ldots,Z_n Z 1 , … , Z n of law μ \mu μ let B n B_n B n have entries R ε ( Z j , Z k ) / n R_\varepsilon(Z_j,Z_k)/n R ε ( Z j , Z k ) / n off the diagonal and zero on the diagonal. Then, for even n n n
E ∥ B n ∥ ≤ ∥ R ε ∥ 2 → 2 + C log ( 2 n ) n ε + C ε e − c n ε . (20) \mathbb{E}\lVert B_n\rVert\le\lVert R_\varepsilon\rVert_{2\to2}+C\sqrt{\frac{\log(2n)}{n\varepsilon}}+\frac{C}{\varepsilon}e^{-cn\varepsilon}. \tag*{(20)} E ∥ B n ∥ ≤ ∥ R ε ∥ 2 → 2 + C n ε log ( 2 n ) + ε C e − c n ε . (20) Proof. Write n = 2 m n=2m n = 2 m X 1 , … , X m X_1,\ldots,X_m X 1 , … , X m Y 1 , … , Y m Y_1,\ldots,Y_m Y 1 , … , Y m v j = R ε ( ⋅ , Y j ) v_j=R_\varepsilon(\cdot,Y_j) v j = R ε ( ⋅ , Y j ) Σ Y = m − 1 ∑ j v j ⊗ v j \Sigma_Y=m^{-1}\sum_jv_j\otimes v_j Σ Y = m − 1 ∑ j v j ⊗ v j R ε R ε ∗ R_\varepsilon R_\varepsilon^* R ε R ε ∗ ∥ v j ∥ 2 ≤ L = C 1 / ε \lVert v_j\rVert^2\le L=C_1/\varepsilon ∥ v j ∥ 2 ≤ L = C 1 / ε
E ∥ Σ Y ∥ ≤ ( r ε + b 1 ) 2 , r ε = ∥ R ε ∥ , b 1 = 2 k m L / m . \mathbb{E}\lVert\Sigma_Y\rVert\le(r_\varepsilon+b_1)^2,\qquad r_\varepsilon=\lVert R_\varepsilon\rVert,\qquad b_1=2k_m\sqrt{L/m}. E ∥ Σ Y ∥ ≤ ( r ε + b 1 ) 2 , r ε = ∥ R ε ∥ , b 1 = 2 k m L / m . Define r Y ( x ) = m − 1 / 2 ( R ε ( x , Y 1 ) , … , R ε ( x , Y m ) ) r_Y(x)=m^{-1/2}(R_\varepsilon(x,Y_1),\ldots,R_\varepsilon(x,Y_m)) r Y ( x ) = m − 1/2 ( R ε ( x , Y 1 ) , … , R ε ( x , Y m )) G Y G_Y G Y Σ Y \Sigma_Y Σ Y G Y = J Y ∗ J Y G_Y=J_Y^*J_Y G Y = J Y ∗ J Y Σ Y = J Y J Y ∗ \Sigma_Y=J_YJ_Y^* Σ Y = J Y J Y ∗ J Y c = m − 1 / 2 ∑ j c j v j J_Yc=m^{-1/2}\sum_jc_jv_j J Y c = m − 1/2 ∑ j c j v j S Y ( x ) = ∥ r Y ( x ) ∥ 2 S_Y(x)=\|r_Y(x)\|^2 S Y ( x ) = ∥ r Y ( x ) ∥ 2 x x x L L L M = C 0 2 / ε 2 M=C_0^2/\varepsilon^2 M = C 0 2 / ε 2 M L ML M L
P Y { S Y ( x ) > 2 L } ≤ e − c m ε , E Y μ { x : S Y ( x ) > 2 L } ≤ e − c m ε . \mathbb{P}_Y\{S_Y(x)>2L\}\le e^{-cm\varepsilon},\qquad\mathbb{E}_Y\mu\{x:S_Y(x)>2L\}\le e^{-cm\varepsilon}. P Y { S Y ( x ) > 2 L } ≤ e − c m ε , E Y μ { x : S Y ( x ) > 2 L } ≤ e − c m ε . Truncate r Y r_Y r Y 2 L 2L 2 L G Y G_Y G Y Lemma 4.5 conditionally on Y Y Y X 1 , … , X m X_1,\ldots,X_m X 1 , … , X m M i j = R ε ( X i , Y j ) / m M_{ij}=R_\varepsilon(X_i,Y_j)/m M ij = R ε ( X i , Y j ) / m r ε + b 1 + b 2 r_\varepsilon+b_1+b_2 r ε + b 1 + b 2 b 2 = 2 k m 2 L / m b_2=2k_m\sqrt{2L/m} b 2 = 2 k m 2 L / m M e − c m ε Me^{-cm\varepsilon} M e − c m ε
E ∥ M ∥ ≤ r ε + C log ( 2 m ) m ε + C ε e − c m ε . \mathbb{E}\|M\|\le r_\varepsilon+C\sqrt{\frac{\log(2m)}{m\varepsilon}}+\frac{C}{\varepsilon}e^{-cm\varepsilon}. E ∥ M ∥ ≤ r ε + C m ε log ( 2 m ) + ε C e − c m ε . For a fixed full cloud, choose uniformly a set S S S m m m B n B_n B n S S S S c S^c S c C S C_S C S n / [ 2 ( n − 1 ) ] n/[2(n-1)] n / [ 2 ( n − 1 )]
B n = 2 ( n − 1 ) n E S C S . B_n=\frac{2(n-1)}{n}\mathbb{E}_S C_S. B n = n 2 ( n − 1 ) E S C S . The norm of C S C_S C S M M M E ∥ B n ∥ ≤ ( n − 1 ) E ∥ M ∥ / n \mathbb{E}\|B_n\|\le(n-1)\mathbb{E}\|M\|/n E ∥ B n ∥ ≤ ( n − 1 ) E ∥ M ∥/ n (20) .
Apply this lemma to R ε = b ε / ε − V R_\varepsilon=b_\varepsilon/\varepsilon-V R ε = b ε / ε − V V V V T μ T_\mu T μ L 2 ( μ ) L^2(\mu) L 2 ( μ ) L 2 ( μ ) L^2(\mu) L 2 ( μ ) Lemma 4.2 supplies the required row-square bound. If n ε / log n → ∞ n\varepsilon/\log n\to\infty n ε / log n → ∞
lim sup ε ↓ 0 E ∥ U n − V n ∥ ≤ ∥ T μ − V ∥ , ( U n ) j k = b ε ( Z j , Z k ) n ε ( j ≠ k ) , ( V n ) j k = V ( Z j , Z k ) n . \limsup_{\varepsilon\downarrow0}\mathbb{E}\|U_n-V_n\|\le\|T_\mu-V\|,\qquad(U_n)_{jk}=\frac{b_\varepsilon(Z_j,Z_k)}{n\varepsilon}\ (j\ne k),\qquad(V_n)_{jk}=\frac{V(Z_j,Z_k)}{n}. ε ↓ 0 lim sup E ∥ U n − V n ∥ ≤ ∥ T μ − V ∥ , ( U n ) j k = n ε b ε ( Z j , Z k ) ( j = k ) , ( V n ) j k = n V ( Z j , Z k ) . Here the diagonal correction is at most ∥ V ∥ ∞ / n \|V\|_\infty/n ∥ V ∥ ∞ / n f ∈ [ 0 , 1 ] n f\in[0,1]^n f ∈ [ 0 , 1 ] n n − 1 ∥ f ∥ 2 2 ≤ 1 n^{-1}\|f\|_2^2\le1 n − 1 ∥ f ∥ 2 2 ≤ 1
The graph representation
For a finite graph G G G Z G ( λ ) = ∑ I ⊂ V ( G ) : I independent λ ∣ I ∣ Z_G(\lambda)=\sum_{I\subset V(G):\,I\text{ independent}}\lambda^{|I|} Z G ( λ ) = ∑ I ⊂ V ( G ) : I independent λ ∣ I ∣ [13 Theorem 1.4] . If the maximum degree Δ \Delta Δ o ( Δ ) o(\Delta) o ( Δ ) p Δ p\Delta p Δ ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) λ = p / ( 1 − p ) \lambda=p/(1-p) λ = p / ( 1 − p )
log Z G ( λ ) = max 0 ≤ x j ≤ p { ∑ j ( x j − x j log x j p ) − ∑ { j , k } ∈ E ( G ) x j x k } + o ( n p ) + O ( n p 2 ) . (21) \log Z_G(\lambda)=\max_{0\le x_j\le p}\left\{\sum_j\left(x_j-x_j\log\frac{x_j}{p}\right)-\sum_{\{j,k\}\in E(G)}x_jx_k\right\}+o(np)+O(np^2). \tag*{(21)} log Z G ( λ ) = 0 ≤ x j ≤ p max ⎩ ⎨ ⎧ j ∑ ( x j − x j log p x j ) − { j , k } ∈ E ( G ) ∑ x j x k ⎭ ⎬ ⎫ + o ( n p ) + O ( n p 2 ) . (21) This follows by multiplying the independent-set probability by ( 1 + λ ) n (1+\lambda)^n ( 1 + λ ) n p p p
Fix a vector of cell tilts t = ( t i ) t=(t_i) t = ( t i ) a ( t ) = a e − t i a^{(t)}=ae^{-t_i} a ( t ) = a e − t i C i C_i C i
a ε , t = a ( t ) e − α ε / v i on C i , A ε , t = ∫ a ε , t d ν , d μ ε , t = a ε , t d ν / A ε , t . a_{\varepsilon,t}=a^{(t)}e^{-\alpha\varepsilon/v_i}\ \text{ on }C_i,\qquad A_{\varepsilon,t}=\int a_{\varepsilon,t}\,d\nu,\qquad d\mu_{\varepsilon,t}=a_{\varepsilon,t}\,d\nu/A_{\varepsilon,t}. a ε , t = a ( t ) e − α ε / v i on C i , A ε , t = ∫ a ε , t d ν , d μ ε , t = a ε , t d ν / A ε , t . Take an even sample size n ≍ ε − 1 − b n \asymp\varepsilon^{-1-b} n ≍ ε − 1 − b 1 / 2 < b < 1 1/2 < b < 1 1/2 < b < 1 p i = 1 − e − 2 α ε / v i p_i = 1 - e^{-2\alpha\varepsilon/v_i} p i = 1 − e − 2 α ε / v i λ = A ε , t / ( n ε ) \lambda= A_{\varepsilon,t}/(n\varepsilon) λ = A ε , t / ( n ε ) exp [ − α ε ∑ i N i ( N i − 1 ) / v i ] \exp[-\alpha\varepsilon\sum_i N_i(N_i-1)/v_i] exp [ − α ε ∑ i N i ( N i − 1 ) / v i ] N i 2 N_i^2 N i 2
Choose γ ∈ ( ( 1 + b ) / 2 , 1 ) \gamma\in((1+b)/2,1) γ ∈ (( 1 + b ) /2 , 1 ) r = ε γ r = \varepsilon^\gamma r = ε γ s < r s<r s < r d < r d<r d < r C n 2 r 4 = o ( 1 ) Cn^2r^4=o(1) C n 2 r 4 = o ( 1 ) o ( ε ) o(\varepsilon) o ( ε ) o ( n ε ) o(n\varepsilon) o ( n ε ) O ( n ε ) O(n\varepsilon) O ( n ε ) O ( n ε 2 + log n ) = o ( n ε ) O(n\varepsilon^2+\log n)=o(n\varepsilon) O ( n ε 2 + log n ) = o ( n ε ) O ( n ε ) O(n\varepsilon) O ( n ε ) c n ε cn\varepsilon c n ε c n ε ≤ Δ ≤ C n ε cn\varepsilon\leq\Delta\leq Cn\varepsilon c n ε ≤ Δ ≤ C n ε p Δ p\Delta p Δ (21) for graphs.
Matrix Bernstein [14 Theorem 1.4] , conditional on the marks, gives
∥ pin adjacency − conditional mean n ε ∥ = O P ( log n n ε + log n n ε ) = o P ( 1 ) . \left\|\frac{\text{pin adjacency}-\text{conditional mean}}{n\varepsilon}\right\|=O_{\mathbb{P}}\left(\sqrt{\frac{\log n}{n\varepsilon}}+\frac{\log n}{n\varepsilon}\right)=o_{\mathbb{P}}(1). n ε pin adjacency − conditional mean = O P ( n ε log n + n ε log n ) = o P ( 1 ) . Indeed each centered edge matrix has norm at most one and the variance parameter is O ( n ε ) O(n\varepsilon) O ( n ε ) 2 α ∑ i 1 C i ⊗ 1 C i / v i 2\alpha\sum_i 1_{C_i}\otimes1_{C_i}/v_i 2 α ∑ i 1 C i ⊗ 1 C i / v i O ( ε ) O(\varepsilon) O ( ε ) O ( ε ) O(\varepsilon) O ( ε ) O ( n ε ) O(n\varepsilon) O ( n ε )
Set x j = p f j x_j=pf_j x j = p f j (21) . The preceding operator estimates and finite rank approximation give
ε log Z G ( λ ) ⟶ sup 0 ≤ f ≤ 1 { A t ∫ ( f − f log f ) d μ t − 1 2 ⟨ a ( t ) f , T ( a ( t ) f ) ⟩ L 2 ( ν ) − α ∥ P ( a ( t ) f ) ∥ L 2 ( ν ) 2 } . (22) \varepsilon\log Z_G(\lambda)\longrightarrow\sup_{0\leq f\leq1}\left\{A_t\int(f-f\log f)\,d\mu_t-\frac{1}{2}\langle a^{(t)}f,T(a^{(t)}f)\rangle_{L^2(\nu)}-\alpha\|P(a^{(t)}f)\|_{L^2(\nu)}^2\right\}. \tag*{(22)} ε log Z G ( λ ) ⟶ 0 ≤ f ≤ 1 sup { A t ∫ ( f − f log f ) d μ t − 2 1 ⟨ a ( t ) f , T ( a ( t ) f ) ⟩ L 2 ( ν ) − α ∥ P ( a ( t ) f ) ∥ L 2 ( ν ) 2 } . (22) where A t = ∫ a ( t ) d ν A_t=\int a^{(t)}\,d\nu A t = ∫ a ( t ) d ν d μ t = a ( t ) d ν / A t d\mu_t=a^{(t)}\,d\nu/A_t d μ t = a ( t ) d ν / A t n p ε → A t n p\varepsilon\to A_t n pε → A t
To justify the variational maximum, express a bounded finite rank kernel in finitely many bounded features. The empirical joint laws of these features and weights in [ 0 , 1 ] [0,1] [ 0 , 1 ] Lemma 4.6 . Finally the sampling density μ ε , t \mu_{\varepsilon,t} μ ε , t μ t \mu_t μ t O ( ε ) O(\varepsilon) O ( ε )
Putting q = a ( t ) f q=a^{(t)}f q = a ( t ) f (22) gives − inf q ≥ 0 { F a ( q ) + ∑ i t i ∫ C i q } -\inf_{q\geq0}\{\mathcal{F}_a(q)+\sum_i t_i\int_{C_i}q\} − inf q ≥ 0 { F a ( q ) + ∑ i t i ∫ C i q } a ( t ) a^{(t)} a ( t ) { q > a ( t ) } \{q>a^{(t)}\} { q > a ( t ) } q ≤ a ( t ) q\leq a^{(t)} q ≤ a ( t )
Annealing and the cap constraint
Let Z ε , t Z_{\varepsilon,t} Z ε , t P ε , t \mathbb{P}_{\varepsilon,t} P ε , t a ( t ) ( x ) / ε a^{(t)}(x)/\varepsilon a ( t ) ( x ) / ε ( n ) N = n ( n − 1 ) ⋯ ( n − N + 1 ) (n)_N=n(n-1)\cdots(n-N+1) ( n ) N = n ( n − 1 ) ⋯ ( n − N + 1 ) Z G Z_G Z G N N N ( n ) N / n N (n)_N/n^N ( n ) N / n N n n n E Z G ≤ Z ε , t \mathbb{E}Z_G\leq Z_{\varepsilon,t} E Z G ≤ Z ε , t A t / ε A_t/\varepsilon A t / ε K K K
P ε , t { N > K / ε } ≤ e − c / ε . \mathbb{P}_{\varepsilon,t}\{N>K/\varepsilon\}\leq e^{-c/\varepsilon}. P ε , t { N > K / ε } ≤ e − c / ε . On N ≤ K / ϵ N \le K/\epsilon N ≤ K / ϵ exp [ − C / ( n ε 2 ) ] \exp[-C/(n\varepsilon^2)] exp [ − C / ( n ε 2 )]
0 ≤ log Z ε , t − log E Z G ≤ C / ( n ε 2 ) + o ( 1 ) = o ( ε − 1 ) . 0 \le\log Z_{\varepsilon,t}-\log\mathbb{E}Z_G \le C/(n\varepsilon^2)+o(1)=o(\varepsilon^{-1}). 0 ≤ log Z ε , t − log E Z G ≤ C / ( n ε 2 ) + o ( 1 ) = o ( ε − 1 ) . Changing one sampled point changes only edges incident to its vertex. Deleting that vertex bounds the oscillation of log Z G \log Z_G log Z G log ( 1 + λ ) \log(1+\lambda) log ( 1 + λ ) [15 ] for these 2 n 2n 2 n
0 ≤ log E Z G − E log Z G ≤ C n λ 2 = O ( ( n ε 2 ) − 1 ) = o ( ε − 1 ) . 0 \le\log\mathbb{E}Z_G-\mathbb{E}\log Z_G \le Cn\lambda^2=O((n\varepsilon^2)^{-1})=o(\varepsilon^{-1}). 0 ≤ log E Z G − E log Z G ≤ C n λ 2 = O (( n ε 2 ) − 1 ) = o ( ε − 1 ) . The variables ε log Z G \varepsilon\log Z_G ε log Z G
Proposition 4.7 (Uncapped variational limit). For every fixed bounded tilt vector,
ϕ ( t ) : = lim ε ↓ 0 ε log Z ε , t = − inf q ≥ 0 { F a ( q ) + ∑ i t i ∫ C i q d ν } . (23) \phi(t):=\lim_{\varepsilon\downarrow0}\varepsilon\log Z_{\varepsilon,t}
=-\inf_{q\ge0}\left\{\mathcal{F}_a(q)+\sum_i t_i\int_{C_i}q\,d\nu\right\}. \tag*{(23)} ϕ ( t ) := ε ↓ 0 lim ε log Z ε , t = − q ≥ 0 inf { F a ( q ) + i ∑ t i ∫ C i q d ν } . (23) Lemma 4.8 (Uniqueness of the constrained minimizer). Under the coercivity hypothesis of Theorem 4.1 , F a ( q ) + ∑ i t i ∫ C i q \mathcal{F}_a(q)+\sum_i t_i\int_{C_i}q F a ( q ) + ∑ i t i ∫ C i q t i ≥ 0 t_i\ge0 t i ≥ 0
Proof. Clipping restricts the minimization to 0 ≤ q ≤ a 0\le q\le a 0 ≤ q ≤ a 1 / a max 1/a_{\max} 1/ a m a x (15) is strongly convex under the stated coercivity margin. The order interval is weakly compact in L 2 L^2 L 2 T T T
Lemma 4.9 (Multipliers for finitely many caps). Let F F F C \mathcal{C} C m : C → [ 0 , ∞ ) k m:\mathcal{C}\to[0,\infty)^k m : C → [ 0 , ∞ ) k
V ( b ) = inf { F ( q ) : q ∈ C , m ( q ) ≤ b } V(b)=\inf\{F(q):q\in\mathcal{C},\ m(q)\le b\} V ( b ) = inf { F ( q ) : q ∈ C , m ( q ) ≤ b } is finite for every b ∈ ( 0 , ∞ ) k b\in(0,\infty)^k b ∈ ( 0 , ∞ ) k b b b q b q_b q b V ( b ) V(b) V ( b ) t ∈ [ 0 , ∞ ) k t\in[0,\infty)^k t ∈ [ 0 , ∞ ) k q b q_b q b F ( q ) + t ⋅ m ( q ) F(q)+t\cdot m(q) F ( q ) + t ⋅ m ( q ) C \mathcal{C} C t i ( m i ( q b ) − b i ) = 0 t_i(m_i(q_b)-b_i)=0 t i ( m i ( q b ) − b i ) = 0 i i i
Proof. The finite convex nonincreasing function V V V p ≤ 0 p\le0 p ≤ 0 b b b [16 ] . For q ∈ C q\in\mathcal{C} q ∈ C η > 0 \eta>0 η > 0 V ( m ( q ) + η 1 ) ≤ F ( q ) V(m(q)+\eta\mathbf{1})\le F(q) V ( m ( q ) + η 1 ) ≤ F ( q ) η ↓ 0 \eta\downarrow0 η ↓ 0 t = − p t=-p t = − p
F ( q ) + t ⋅ ( m ( q ) − b ) ≥ V ( b ) . F(q)+t\cdot(m(q)-b)\ge V(b). F ( q ) + t ⋅ ( m ( q ) − b ) ≥ V ( b ) . At q b q_b q b t ⋅ ( m ( q b ) − b ) t\cdot(m(q_b)-b) t ⋅ ( m ( q b ) − b )
Proof of Theorem 4.1. Apply Lemma 4.9 on 0 ≤ q ≤ a 0\le q\le a 0 ≤ q ≤ a F = F a F=\mathcal{F}_a F = F a m i ( q ) = ∫ C i q m_i(q)=\int_{C_i}q m i ( q ) = ∫ C i q b i = ρ v i b_i=\rho v_i b i = ρ v i t i t_i t i t i ≥ 0 t_i\ge0 t i ≥ 0 q ≥ 0 q\ge0 q ≥ 0 q ≤ a e s q\le ae^s q ≤ a e s ϕ \phi ϕ ∂ t i ϕ = − ∫ C i q \partial_{t_i}\phi=-\int_{C_i}q ∂ t i ϕ = − ∫ C i q (23) at t ± s e i t\pm se_i t ± s e i ε N i \varepsilon N_i ε N i
The cap upper bound follows by multiplying by e − ∑ i t i N i e^{-\sum_i t_iN_i} e − ∑ i t i N i N i ≤ [ ρ v i / ε ] N_i\le[\rho v_i/\varepsilon] N i ≤ [ ρ v i / ε ] O ( 1 ) O(1) O ( 1 ) ρ ′ < ρ \rho'<\rho ρ ′ < ρ − F a ( q ρ ′ ) -\mathcal{F}_a(q_{\rho'}) − F a ( q ρ ′ ) ρ ′ ↑ ρ \rho'\uparrow\rho ρ ′ ↑ ρ ( ρ ′ / ρ ) q ρ ′ (\rho'/\rho)q_{\rho'} ( ρ ′ / ρ ) q ρ ′ F a ( q ρ ′ ) → F a ( q ρ ) \mathcal{F}_a(q_{\rho'})\to\mathcal{F}_a(q_\rho) F a ( q ρ ′ ) → F a ( q ρ )
Locality of the reference gas
We now control covariances uniformly in the volume and exterior rods. This provides the locality supplied by the restricted-ensemble Dobrushin analysis in LMP [2 Sections 5–6] . For the gas conditioned on the auxiliary field, we use a coercive insertion operator. The method is the continuum Glauber coercivity argument, in particular the Mayer-kernel criterion of Kondratiev, Kuna, and Ohlerich [17 Theorem 4.2 and Corollary 4.3] . Here the cell pin makes the Mayer operator almost positive, and we estimate separately the terms produced by active count caps.
Use shape-regular marked cells with spatial side ℓ \ell ℓ ℓ \ell ℓ v c ≍ ℓ 5 v_c \asymp\ell^5 v c ≍ ℓ 5 α \alpha α ε ↓ 0 \varepsilon\downarrow0 ε ↓ 0 β = ε − 1 \beta= \varepsilon^{-1} β = ε − 1 0 ≤ z ( X ) ≤ M β 0 \le z(X) \le M\beta 0 ≤ z ( X ) ≤ M β
U ε ( X , Y ) = { + ∞ , b ε ( X , Y ) = 1 , 2 α ε / v c , X , Y ∈ C c and b ε ( X , Y ) = 0 , 0 , otherwise . U_\varepsilon(X,Y)=
\begin{cases}
+\infty, & b_\varepsilon(X,Y)=1,\\
2\alpha\varepsilon/v_c, & X,Y\in C_c\ \text{and}\ b_\varepsilon(X,Y)=0,\\
0, & \text{otherwise}.
\end{cases} U ε ( X , Y ) = ⎩ ⎨ ⎧ + ∞ , 2 α ε / v c , 0 , b ε ( X , Y ) = 1 , X , Y ∈ C c and b ε ( X , Y ) = 0 , otherwise . The self-pin e − α ε / v c e^{-\alpha\varepsilon/v_c} e − α ε / v c z z z N c ≤ K c N_c \le K_c N c ≤ K c J = ∑ A J A J=\sum_A J_A J = ∑ A J A
κ m , 1 = sup i ∑ A ∋ i ∣ A ∣ e m diam ( A ) ∥ J A ∥ ∞ . (24) \kappa_{m,1}=\sup_i\sum_{A\ni i}|A|e^{m\operatorname{diam}(A)}\lVert J_A\rVert_\infty. \tag*{(24)} κ m , 1 = i sup A ∋ i ∑ ∣ A ∣ e m diam ( A ) ∥ J A ∥ ∞ . (24) Distances are measured on the spatial cell lattice; angular cells over the same spatial cube have distance zero. Constants may depend on the fixed mesh, α \alpha α M M M μ \mu μ D X F ( η ) = F ( η + X ) − F ( η ) D_XF(\eta)=F(\eta+X)-F(\eta) D X F ( η ) = F ( η + X ) − F ( η )
r X ( η ) = z ( X ) 1 { N c ( X ) ( η ) < K c ( X ) } exp [ − ∑ Y ∈ η U ε ( X , Y ) − D X J ( η ) ] r_X(\eta)=z(X)\mathbf{1}_{\{N_{c(X)}(\eta)<K_{c(X)}\}}\exp\left[-\sum_{Y\in\eta}U_\varepsilon(X,Y)-D_XJ(\eta)\right] r X ( η ) = z ( X ) 1 { N c ( X ) ( η ) < K c ( X ) } exp − Y ∈ η ∑ U ε ( X , Y ) − D X J ( η ) for the insertion rate. Here c ( X ) c(X) c ( X ) X X X
Theorem 5.1 (Uniform covariance). Choose α > ∥ T ∥ + 1 \alpha>\lVert T\rVert+1 α > ∥ T ∥ + 1 ε \varepsilon ε β κ m , 1 \beta\kappa_{m,1} β κ m , 1 C , m > 0 C,m>0 C , m > 0
∣ Cov μ ( F , G ) ∣ ≤ C ∑ i , j e − m d ( i , j ) D i ( F ) D j ( G ) , D i ( F ) 2 = E μ ∫ C i r X ( η ) ∣ D X F ( η ) ∣ 2 d ν ( X ) . (25) \lvert\operatorname{Cov}_\mu(F,G)\rvert\le C\sum_{i,j}e^{-m d(i,j)}D_i(F)D_j(G),\qquad D_i(F)^2=\mathbb{E}_\mu\int_{C_i}r_X(\eta)\lvert D_XF(\eta)\rvert^2\,d\nu(X). \tag*{(25)} ∣ Cov μ ( F , G )∣ ≤ C i , j ∑ e − m d ( i , j ) D i ( F ) D j ( G ) , D i ( F ) 2 = E μ ∫ C i r X ( η ) ∣ D X F ( η ) ∣ 2 d ν ( X ) . (25) Here C i C_i C i i i i
A lower bound for the Mayer operator
The limiting operator maps L 2 L^2 L 2 H 1 H^1 H 1 ∥ T ^ ( k ) ∥ ≤ C / ( 1 + ∣ k ∣ ) \lVert\widehat{T}(k)\rVert\le C/(1+\lvert k\rvert) ∥ T ( k )∥ ≤ C / ( 1 + ∣ k ∣)
∫ S 2 ∣ ∇ u s k ( u ) ∣ 2 d ω ( u ) ≤ { C r , r ≥ 1 , C r 4 , r ≤ 1. \int_{S^2}\lvert\nabla_u s_k(u)\rvert^2\,d\omega(u)\le
\begin{cases}
Cr, & r\ge1,\\
Cr^4, & r\le1.
\end{cases} ∫ S 2 ∣ ∇ u s k ( u ) ∣ 2 d ω ( u ) ≤ { C r , C r 4 , r ≥ 1 , r ≤ 1. by the substitution t = r u ⋅ k ^ / 2 t=ru\cdot\hat{k}/2 t = r u ⋅ k ^ /2 J ( r ) ≤ C / ( 1 + r ) J(r)\le C/(1+r) J ( r ) ≤ C / ( 1 + r ) ∣ u × v ∣ |u\times v| ∣ u × v ∣
∥ ( I − P ℓ ) T ∥ ≤ C ℓ , ∥ T − P ℓ T P ℓ ∥ ≤ 2 C ℓ . (26) \lVert(I-P_{\ell})T\rVert\le C\ell,\qquad\lVert T-P_{\ell}TP_{\ell}\rVert\le2C\ell. \tag*{(26)} ∥( I − P ℓ ) T ∥ ≤ C ℓ , ∥ T − P ℓ T P ℓ ∥ ≤ 2 C ℓ . (26) For X ∈ C c X\in C_c X ∈ C c ε ≤ ℓ \varepsilon\le\ell ε ≤ ℓ
∫ C c b ε ( X , Y ) d ν ( Y ) ≤ C ε ℓ 4 . (27) \int_{C_c}b_{\varepsilon}(X,Y)\,d\nu(Y)\le C\varepsilon\ell^4. \tag*{(27)} ∫ C c b ε ( X , Y ) d ν ( Y ) ≤ C ε ℓ 4 . (27) Indeed, the center displacement has the form t u − s v + e tu-sv+e t u − s v + e t − s = O ( ℓ ) t-s=O(\ell) t − s = O ( ℓ ) ∣ u − v ∣ = O ( ℓ ) |u-v|=O(\ell) ∣ u − v ∣ = O ( ℓ ) O ( ℓ 2 ) O(\ell^2) O ( ℓ 2 ) O ( ε ℓ 2 + ε 2 ℓ + ε 3 ) O(\varepsilon\ell^2+\varepsilon^2\ell+\varepsilon^3) O ( ε ℓ 2 + ε 2 ℓ + ε 3 ) (27) .
The exact pair Mayer kernel is
B ε p i n = b ε + ( 1 − b ε ) ∑ c ϑ c 1 C c ⊗ 1 C c , ϑ c = 1 − e − 2 α ε / v c . B_{\varepsilon}^{\mathrm{pin}}=b_{\varepsilon}+(1-b_{\varepsilon})\sum_c\vartheta_c\mathbf{1}_{C_c}\otimes\mathbf{1}_{C_c},\qquad\vartheta_c=1-e^{-2\alpha\varepsilon/v_c}. B ε pin = b ε + ( 1 − b ε ) c ∑ ϑ c 1 C c ⊗ 1 C c , ϑ c = 1 − e − 2 α ε / v c . For ε ≤ c v min / α \varepsilon\le cv_{\min}/\alpha ε ≤ c v m i n / α
∑ c v c ϑ c P c ≥ α ε P ℓ . \sum_c v_c\vartheta_cP_c\ge\alpha\varepsilon P_{\ell}. c ∑ v c ϑ c P c ≥ α ε P ℓ . By (26) , T + α P ℓ ≥ − C ℓ I T+\alpha P_{\ell}\ge-C\ell I T + α P ℓ ≥ − C ℓ I C α ε 2 / ℓ C\alpha\varepsilon^2/\ell C α ε 2 / ℓ (27) . Lemma 4.2 now gives
B ε p i n ≥ − ε δ ε I , δ ε ≤ C ( ℓ + ε + α ε / ℓ ) . (28) B_{\varepsilon}^{\mathrm{pin}}\ge-\varepsilon\delta_{\varepsilon}I,\qquad\delta_{\varepsilon}\le C(\ell+\varepsilon+\alpha\varepsilon/\ell). \tag*{(28)} B ε pin ≥ − ε δ ε I , δ ε ≤ C ( ℓ + ε + α ε / ℓ ) . (28) Its row integral is at most C ( 1 + α ) ε C(1+\alpha)\varepsilon C ( 1 + α ) ε
The insertion commutator and active caps
Since r X ≤ M β e 2 κ 0 , 1 r_X\le M\beta e^{2\kappa_{0,1}} r X ≤ M β e 2 κ 0 , 1
( L F ) ( η ) = ∫ r Y ( η ) D Y F ( η ) d ν ( Y ) + ∑ Y ∈ η [ F ( η − Y ) − F ( η ) ] (LF)(\eta)=\int r_Y(\eta)D_YF(\eta)\,d\nu(Y)+\sum_{Y\in\eta}[F(\eta-Y)-F(\eta)] ( L F ) ( η ) = ∫ r Y ( η ) D Y F ( η ) d ν ( Y ) + Y ∈ η ∑ [ F ( η − Y ) − F ( η )] has bounded birth rates. Canceling factorials in the grand-canonical density gives the Georgii–Nguyen–Zessin identity [18 Theorem 2] and hence
E ( F , G ) = E μ ∫ r X D X F D X G d ν ( X ) = ⟨ F , − L G ⟩ μ . \mathcal{E}(F,G)=\mathbb{E}_{\mu}\int r_XD_XF D_XG\,d\nu(X)=\langle F,-LG\rangle_{\mu}. E ( F , G ) = E μ ∫ r X D X F D X G d ν ( X ) = ⟨ F , − L G ⟩ μ . One-forms are measured in μ ( d η ) r X ( η ) d ν ( X ) \mu(d\eta)r_X(\eta)d\nu(X) μ ( d η ) r X ( η ) d ν ( X ) ∥ ⋅ ∥ 1 \lVert\cdot\rVert_1 ∥ ⋅ ∥ 1
Hold an inserted particle X X X L X L^X L X r Y ( η + X ) r_Y(\eta+X) r Y ( η + X ) η \eta η
D X ( − L ) F = ( − L X + I ) D X F + ∫ [ r Y ( η ) − r Y ( η + X ) ] D Y F ( η ) d ν ( Y ) , D_X(-L)F=(-L^X+I)D_XF+\int[r_Y(\eta)-r_Y(\eta+X)]D_YF(\eta)\,d\nu(Y), D X ( − L ) F = ( − L X + I ) D X F + ∫ [ r Y ( η ) − r Y ( η + X )] D Y F ( η ) d ν ( Y ) , D ( − L ) = H D , H = − L X + I + K . (29) D(-L)=\mathcal{H}D,\qquad\mathcal{H}=-L^X+I+\mathcal{K}. \tag*{(29)} D ( − L ) = H D , H = − L X + I + K . (29) The identity
r X ( η ) r Y ( η + X ) = r Y ( η ) r X ( η + Y ) r_X(\eta)r_Y(\eta+X)=r_Y(\eta)r_X(\eta+Y) r X ( η ) r Y ( η + X ) = r Y ( η ) r X ( η + Y ) makes L X L^X L X K \mathcal{K} K − L X ≥ 0 -L^X\ge0 − L X ≥ 0
First take J = 0 J = 0 J = 0 K \mathcal{K} K B ε p i n B_{\varepsilon}^{\mathrm{pin}} B ε pin (28) to the function r X g X r_X g_X r X g X r X 2 ≤ M β r X r_X^2 \le M\beta r_X r X 2 ≤ M β r X − M δ ε ∥ g ∥ 1 2 -M\delta_{\varepsilon}\lVert g\rVert_1^2 − M δ ε ∥ g ∥ 1 2 N c ( η ) = K c − 1 N_c(\eta) = K_c - 1 N c ( η ) = K c − 1
e − 2 α ε / v c 1 C c ( X ) 1 C c ( Y ) ( 1 − b ε ( X , Y ) ) r Y ( η ) . e^{-2\alpha\varepsilon/v_c}1_{C_c}(X)1_{C_c}(Y)(1-b_{\varepsilon}(X,Y))r_Y(\eta). e − 2 α ε / v c 1 C c ( X ) 1 C c ( Y ) ( 1 − b ε ( X , Y )) r Y ( η ) . In its quadratic form the term 1 1 1 C M ℓ 4 CM\ell^4 C M ℓ 4 (27) . Retaining the positive block gives
⟨ g , H g ⟩ 1 ≥ ( 1 − M δ ε − C M ℓ 4 ) ∥ g ∥ 1 2 . (30) \langle g,\mathcal{H}g\rangle_1 \ge(1-M\delta_{\varepsilon}-CM\ell^4)\lVert g\rVert_1^2. \tag*{(30)} ⟨ g , H g ⟩ 1 ≥ ( 1 − M δ ε − C M ℓ 4 ) ∥ g ∥ 1 2 . (30) Choose the mesh first so that the ℓ \ell ℓ ε \varepsilon ε
For a bounded perturbation, an allowed double birth introduces the factor e − D X D Y J e^{-D_XD_YJ} e − D X D Y J
C ∑ A ∋ c ( X ) , c ( Y ) ∥ J A ∥ ∞ , ∣ D X D Y J ∣ ≤ 4 ∑ A ∋ c ( X ) , c ( Y ) ∥ J A ∥ ∞ . C\sum_{A\ni c(X),c(Y)}\lVert J_A\rVert_{\infty},\qquad|D_XD_YJ|\le4\sum_{A\ni c(X),c(Y)}\lVert J_A\rVert_{\infty}. C A ∋ c ( X ) , c ( Y ) ∑ ∥ J A ∥ ∞ , ∣ D X D Y J ∣ ≤ 4 A ∋ c ( X ) , c ( Y ) ∑ ∥ J A ∥ ∞ . The other insertion-ratio factors lie in [ 0 , 1 ] [0,1] [ 0 , 1 ] C β κ m , 1 C\beta\kappa_{m,1} C β κ m , 1 e 2 κ 0 , 1 e^{2\kappa_{0,1}} e 2 κ 0 , 1 C β κ m , 1 C\beta\kappa_{m,1} C β κ m , 1 (30) therefore gives H ≥ c I \mathcal{H}\ge cI H ≥ c I
Weighted inverse and covariance
We obtain spatial decay by the exponential conjugation method of Combes and Thomas [19 ] . For a spatial-cell set A A A w X = e m d ( c s p a t i a l ( X ) , A ) w_X=e^{md(c_{\mathrm{spatial}}(X),A)} w X = e m d ( c spatial ( X ) , A ) L L L O ( ε ) O(\varepsilon) O ( ε ) O ( β ) O(\beta) O ( β ) O ( m ) O(m) O ( m ) (24) . For a sufficiently small m > 0 m>0 m > 0 w w w w − 1 w^{-1} w − 1 c ′ > 0 c'>0 c ′ > 0 1 / c ′ 1/c' 1/ c ′ P i P_i P i i i i
∥ P i H − 1 P j ∥ 1 → 1 ≤ C e − m d ( i , j ) . (31) \lVert P_i\mathcal{H}^{-1}P_j\rVert_{1\to1}\le Ce^{-md(i,j)}. \tag*{(31)} ∥ P i H − 1 P j ∥ 1 → 1 ≤ C e − m d ( i , j ) . (31) The caps bound the number of rods in every finite volume. Together with bounded total birth rates and reversibility, this makes the generators bounded on their respective L 2 L^2 L 2 (29) give
∥ L F ∥ 2 2 = ⟨ D F , H D F ⟩ 1 ≥ c E ( F , F ) . \lVert LF\rVert_2^2=\langle DF,\mathcal{H}DF\rangle_1\ge c\mathcal{E}(F,F). ∥ L F ∥ 2 2 = ⟨ D F , H D F ⟩ 1 ≥ c E ( F , F ) . Successive deletions reach the empty configuration. The Georgii–Nguyen–Zessin identity therefore propagates a zero insertion gradient through every particle-number sector, so the Dirichlet kernel consists of constants. The spectral theorem gives a gap at least c c c
For centered G G G ( − L ) u = G (-L)u=G ( − L ) u = G D u = H − 1 D G Du=\mathcal{H}^{-1}DG D u = H − 1 D G
Cov μ ( F , G ) = ⟨ D F , H − 1 D G ⟩ 1 . \operatorname{Cov}_{\mu}(F,G)=\langle DF,\mathcal{H}^{-1}DG\rangle_1. Cov μ ( F , G ) = ⟨ D F , H − 1 D G ⟩ 1 . This is the birth–death counterpart of the Helffer–Sjöstrand covariance representation [20 ] . Inserting (31) proves Theorem 5.1.
In particular, marked-cell counts satisfy
∣ Cov μ ( N i , N j ) ∣ ≤ C β e − m d ( i , j ) . (32) \lvert\operatorname{Cov}_{\mu}(N_i,N_j)\rvert\le C\beta e^{-m d(i,j)}. \tag*{(32)} ∣ Cov μ ( N i , N j )∣ ≤ C β e − m d ( i , j ) . (32) If the activity on any measurable set of marks is multiplied by e − t e^{-t} e − t C β e − t / 2 C\sqrt{\beta}e^{-t/2} C β e − t /2 e − t e^{-t} e − t t ≥ 0 t \ge0 t ≥ 0 V V V A A A
∣ Cov μ ( N i , V ) ∣ ≤ C β ∥ V ∥ ∞ ∑ j ∈ A e − m d ( i , j ) , (33) \lvert\operatorname{Cov}_{\mu}(N_i,V)\rvert\le C\beta\lVert V\rVert_{\infty}\sum_{j\in A}e^{-m d(i,j)}, \tag*{(33)} ∣ Cov μ ( N i , V )∣ ≤ C β ∥ V ∥ ∞ j ∈ A ∑ e − m d ( i , j ) , (33) since ∣ D X V ∣ ≤ 2 ∥ V ∥ ∞ \lvert D_XV\rvert\le2\lVert V\rVert_{\infty} ∣ D X V ∣ ≤ 2 ∥ V ∥ ∞
Spatial rigidity and a positive field action
Write D = u ⋅ ∇ x D=u\cdot\nabla_x D = u ⋅ ∇ x
The next lemma is the polynomiality property of Killing tensors in Euclidean space [21 Section IV] , stated here for distributional coefficients.
Lemma 6.1 (Polynomial transport coefficients). Let B ⊂ R 3 B\subset\mathbb{R}^3 B ⊂ R 3 H ( x , z ) H(x,z) H ( x , z ) l l l z ∈ R 3 z\in\mathbb{R}^3 z ∈ R 3 B B B ( z ⋅ ∇ x ) H = 0 (z\cdot\nabla_x)H=0 ( z ⋅ ∇ x ) H = 0 x x x l l l
Proof. For D z = z ⋅ ∇ x D_z=z\cdot\nabla_x D z = z ⋅ ∇ x [ D z , ∂ z j ] = − ∂ x j [D_z,\partial_{z_j}]=-\partial_{x_j} [ D z , ∂ z j ] = − ∂ x j
D z j ∂ z γ H = ( − 1 ) j j ! ∑ ξ ≤ γ ∣ ξ ∣ = j ( γ ξ ) ∂ x ξ ∂ z γ − ξ H ( 0 ≤ j ≤ ∣ γ ∣ ) . D_z^j\partial_z^\gamma H=(-1)^j j!\sum_{\substack{\xi\le\gamma\\\lvert\xi\rvert=j}}\binom{\gamma}{\xi}\partial_x^\xi\partial_z^{\gamma-\xi}H
\qquad(0\le j\le\lvert\gamma\rvert). D z j ∂ z γ H = ( − 1 ) j j ! ξ ≤ γ ∣ ξ ∣ = j ∑ ( ξ γ ) ∂ x ξ ∂ z γ − ξ H ( 0 ≤ j ≤ ∣ γ ∣) . Taking ∣ γ ∣ = j = l + 1 \lvert\gamma\rvert=j=l+1 ∣ γ ∣ = j = l + 1 ∂ x γ H = 0 \partial_x^\gamma H=0 ∂ x γ H = 0 l l l B B B
Theorem 6.2 (Local transport rigidity). If r r r B × S 2 B\times S^2 B × S 2 r ( x , ⋅ ) ∈ W a r(x,\cdot)\in W_a r ( x , ⋅ ) ∈ W a x x x B B B D r = 0 Dr=0 D r = 0 r ( x , u ) = f ( u ) r(x,u)=f(u) r ( x , u ) = f ( u ) f ∈ W a f\in W_a f ∈ W a
Proof. The well bounds give 0 < m ≤ r ≤ M 0<m\le r\le M 0 < m ≤ r ≤ M L 2 L^2 L 2 [ m , M ] [m,M] [ m , M ] D log r = 0 D\log r=0 D log r = 0 D K r = 0 DKr=0 D K r = 0
Let r l r_l r l l l l u j u_j u j l l l l − 1 , l + 1 l-1,l+1 l − 1 , l + 1 j j j D r = D K r = 0 Dr=DKr=0 D r = D K r = 0
A j + B j = 0 , κ j − 1 A j + κ j + 1 B j = 0 , A_j+B_j=0,\qquad\kappa_{j-1}A_j+\kappa_{j+1}B_j=0, A j + B j = 0 , κ j − 1 A j + κ j + 1 B j = 0 , where A j A_j A j B j B_j B j j j j D r j − 1 Dr_{j-1} D r j − 1 D r j + 1 Dr_{j+1} D r j + 1 A j = B j = 0 A_j=B_j=0 A j = B j = 0 D r l = 0 Dr_l=0 D r l = 0 l l l r l r_l r l H l ( x , z ) H_l(x,z) H l ( x , z ) z z z Lemma 6.1 makes its spatial coefficients polynomials of degree at most l l l
v ( x ) = P N log r ( x ) = a − 2 P N K r ( x ) v(x)=\mathcal{P}_N\log r(x)=a-2\mathcal{P}_NKr(x) v ( x ) = P N log r ( x ) = a − 2 P N K r ( x ) is consequently the restriction of a polynomial map R 3 → E \mathbb{R}^3\to E R 3 → E Z Z Z B B B
Fix x 0 ∈ B x_0\in B x 0 ∈ B e e e v e ( t ) = v ( x 0 + t e ) v_e(t)=v(x_0+te) v e ( t ) = v ( x 0 + t e ) Z Z Z I I I { t : v e ( t ) ∈ U } \{t:v_e(t)\in U\} { t : v e ( t ) ∈ U } F ( v e ( t ) ) F(v_e(t)) F ( v e ( t )) G ( v e ( t ) ) G(v_e(t)) G ( v e ( t )) v e ( I ) ⊂ Z v_e(I)\subset Z v e ( I ) ⊂ Z I I I Z ⋐ U Z\Subset U Z ⋐ U I I I I = R I=\mathbb{R} I = R e e e v v v r = Ψ ( v ) r=\Psi(v) r = Ψ ( v )
Following LMP’s averaged-density representation [2 Section 3] , we separate the action into an entropy loss under averaging and a homogeneous excess. Rods require averaging along their axes, for which Theorem 6.2 gives the equality case.
Put h ( t ) = t log t − t h(t)=t\log t-t h ( t ) = t log t − t A A A
D q ( x ) = ∫ S 2 [ ∫ − 1 / 2 1 / 2 h ( q ( x + t u , u ) ) d t − h ( A q ( x , u ) ) ] d ω ( u ) , D_q(x)=\int_{S^2}\left[\int_{-1/2}^{1/2}h(q(x+tu,u))\,dt-h(Aq(x,u))\right]\,d\omega(u), D q ( x ) = ∫ S 2 [ ∫ − 1/2 1/2 h ( q ( x + t u , u )) d t − h ( A q ( x , u )) ] d ω ( u ) , V a ( f ) = Ω a ( f ) − m a . V_a(f)=\Omega_a(f)-m_a. V a ( f ) = Ω a ( f ) − m a . Jensen’s inequality gives D q ≥ 0 D_q\geq0 D q ≥ 0 V a ≥ 0 V_a\geq0 V a ≥ 0
Corollary 6.3 (Zero local action). If a bounded nonnegative profile q q q ∫ B [ D q + V a ( A q ) ] d x = 0 \int_B[D_q+V_a(Aq)]\,dx=0 ∫ B [ D q + V a ( A q )] d x = 0 B B B q ( x , u ) = f ( u ) q(x,u)=f(u) q ( x , u ) = f ( u ) B × S 2 B\times S^2 B × S 2 f ∈ W a f\in\mathcal{W}_a f ∈ W a
Proof. Set r = A q r=Aq r = A q r ( x , ⋅ ) ∈ W a r(x,\cdot)\in\mathcal{W}_a r ( x , ⋅ ) ∈ W a q ( x + t u , u ) = r ( x , u ) q(x+tu,u)=r(x,u) q ( x + t u , u ) = r ( x , u ) ( x , u , t ) (x,u,t) ( x , u , t ) x ∈ B x\in B x ∈ B χ ∈ C c ∞ ( ( − 1 / 2 , 1 / 2 ) ) \chi\in C^\infty_c((-1/2,1/2)) χ ∈ C c ∞ (( − 1/2 , 1/2 ))
r = ∫ χ ( t ) q ( x + t u , u ) d t , D r = − ∫ χ ′ ( t ) q ( x + t u , u ) d t = 0. r=\int\chi(t)q(x+tu,u)\,dt,\qquad Dr=-\int\chi'(t)q(x+tu,u)\,dt=0. r = ∫ χ ( t ) q ( x + t u , u ) d t , D r = − ∫ χ ′ ( t ) q ( x + t u , u ) d t = 0. Theorem 6.2 gives r = f r=f r = f B ′ ⋐ B B'\Subset B B ′ ⋐ B 0 < b ≤ 1 / 2 0<b\leq1/2 0 < b ≤ 1/2 B ′ + B ( 0 , b ) ⊂ B B'+B(0,b)\subset B B ′ + B ( 0 , b ) ⊂ B
2 b ∫ B ′ × S 2 ∣ q − f ∣ 2 ≤ ∫ B × S 2 ∫ − b b ∣ q ( x + t u , u ) − f ( u ) ∣ 2 d t d x d ω = 0. 2b\int_{B'\times S^2}|q-f|^2\leq\int_{B\times S^2}\int_{-b}^{b}|q(x+tu,u)-f(u)|^2\,dt\,dx\,d\omega=0. 2 b ∫ B ′ × S 2 ∣ q − f ∣ 2 ≤ ∫ B × S 2 ∫ − b b ∣ q ( x + t u , u ) − f ( u ) ∣ 2 d t d x d ω = 0. An exhaustion proves the assertion on B B B
Minimizing the density profile at a fixed field
Fix a = a t a=a_t a = a t ζ = e a t \zeta=e^{a_t} ζ = e a t C i , b = B i × U b C_{i,b}=B_i\times U_b C i , b = B i × U b v i , b = v s ω b v_{i,b}=v_s\omega_b v i , b = v s ω b P P P [ − R , R ] [-R,R] [ − R , R ]
Set S = T + 2 α P S=T+2\alpha P S = T + 2 α P T T T
S ≥ − δ I , M = ζ e R + 1 , c = M − 1 − δ > 0. (34) S\geq-\delta I,\qquad M=\zeta e^{R+1},\qquad c=M^{-1}-\delta>0. \tag*{(34)} S ≥ − δ I , M = ζ e R + 1 , c = M − 1 − δ > 0. (34) Choose α ≥ ∥ T ∥ \alpha\ge\lVert T\rVert α ≥ ∥ T ∥ (26) gives ∥ T − P T P ∥ ≤ C ℓ \lVert T-PTP\rVert\le C\ell ∥ T − P T P ∥ ≤ C ℓ ℓ \ell ℓ P T P ≥ − ∥ T ∥ P PTP \ge-\lVert T\rVert P P T P ≥ − ∥ T ∥ P
S ≥ ( 2 α − ∥ T ∥ ) P − C ℓ I ≥ − C ℓ I . S \ge(2\alpha-\lVert T\rVert)P-C\ell I \ge-C\ell I. S ≥ ( 2 α − ∥ T ∥) P − C ℓ I ≥ − C ℓ I . After fixing α \alpha α R R R δ = C ℓ < M − 1 \delta=C\ell<M^{-1} δ = C ℓ < M − 1
ρ > sup f ∈ W ∥ f ∥ ∞ , R > 2 α sup f ∈ W ∥ f ∥ ∞ . (35) \rho>\sup_{f\in\mathcal{W}}\lVert f\rVert_\infty,\qquad R>2\alpha\sup_{f\in\mathcal{W}}\lVert f\rVert_\infty. \tag*{(35)} ρ > f ∈ W sup ∥ f ∥ ∞ , R > 2 α f ∈ W sup ∥ f ∥ ∞ . (35) Admissible density profiles satisfy q ≥ 0 q\ge0 q ≥ 0 ∫ C q ≤ ρ v C \int_C q\le\rho v_C ∫ C q ≤ ρ v C
Proposition 6.4 (The capped minimizing profile). Let D D D − R − ≤ ϕ ≤ R + 1 -R_-\le\phi\le R+1 − R − ≤ ϕ ≤ R + 1 R − < ∞ R_-<\infty R − < ∞ D D D
G D , ϕ ( q ) = ∫ D × S 2 q ( log q − 1 − a − ϕ ) + 1 2 ⟨ q , S D q ⟩ G_{D,\phi}(q)=\int_{D\times S^2}q(\log q-1-a-\phi)+\frac{1}{2}\langle q,S_Dq\rangle G D , ϕ ( q ) = ∫ D × S 2 q ( log q − 1 − a − ϕ ) + 2 1 ⟨ q , S D q ⟩ has a unique capped minimizer q D , ϕ q_{D,\phi} q D , ϕ 0 < m ∗ ≤ q D , ϕ < M 0<m_*\le q_{D,\phi}<M 0 < m ∗ ≤ q D , ϕ < M D D D λ C ≥ 0 \lambda_C\ge0 λ C ≥ 0
log ( q / ζ ) − ϕ + S q + λ C = 0 on C , λ C ( ∫ C q − ρ v C ) = 0. (36) \log(q/\zeta)-\phi+Sq+\lambda_C=0\quad\text{on }C,\qquad\lambda_C\left(\int_Cq-\rho v_C\right)=0. \tag*{(36)} log ( q / ζ ) − ϕ + S q + λ C = 0 on C , λ C ( ∫ C q − ρ v C ) = 0. (36) Proof. Replacing q q q q ∧ M q\wedge M q ∧ M log ( q / ζ ) − ϕ \log(q/\zeta)-\phi log ( q / ζ ) − ϕ M M M S S S [ 0 , M ] [0,M] [ 0 , M ] q 2 / ( 2 M ) q^2/(2M) q 2 / ( 2 M ) S ≥ − δ I S\ge-\delta I S ≥ − δ I c c c L 2 L^2 L 2
Apply Lemma 4.9 on [ 0 , M ] [0,M] [ 0 , M ] G G G λ C \lambda_C λ C q ∗ q_* q ∗ V = − a − ϕ + S q ∗ + λ V=-a-\phi+Sq_*+\lambda V = − a − ϕ + S q ∗ + λ [ h ( q ) + V q ] [h(q)+Vq] [ h ( q ) + V q ] [ 0 , M ] [0,M] [ 0 , M ] g = min ( M , e − V ) > 0 g=\min(M,e^{-V})>0 g = min ( M , e − V ) > 0 g ≠ q ∗ g\ne q_* g = q ∗ q ∗ = g q_*=g q ∗ = g q ∗ = g q_*=g q ∗ = g P q ∗ > 0 Pq_*>0 P q ∗ > 0
log M + V = R + 1 − ϕ + S q ∗ + λ ≥ 2 α P q ∗ > 0. \log M+V=R+1-\phi+Sq_*+\lambda\ge2\alpha Pq_*>0. log M + V = R + 1 − ϕ + S q ∗ + λ ≥ 2 α P q ∗ > 0. Thus q ∗ < M q_*<M q ∗ < M (36) holds. If λ C > 0 \lambda_C>0 λ C > 0 q ≤ ζ e R + 1 − λ C q\le\zeta e^{R+1-\lambda_C} q ≤ ζ e R + 1 − λ C
0 ≤ λ C ≤ ( log ζ + R + 1 − log ρ ) + . 0\le\lambda_C\le(\log\zeta+R+1-\log\rho)_+. 0 ≤ λ C ≤ ( log ζ + R + 1 − log ρ ) + . The bounded row sum of S S S
We apply the exponential conjugation method of Combes and Thomas [19 ] to the constrained Euler equation.
Lemma 6.5 (Weighted response). If W W W ∣ log W i − log W j ∣ ≤ m d ( i , j ) |\log W_i-\log W_j|\le m\,d(i,j) ∣ log W i − log W j ∣ ≤ m d ( i , j ) m > 0 m>0 m > 0
∥ W ( q D , ϕ 1 − q D , ϕ 2 ) ∥ 2 ≤ 2 c ∥ W ( ϕ 1 − ϕ 2 ) ∥ 2 . (37) \lVert W(q_{D,\phi_1}-q_{D,\phi_2})\rVert_2\le\frac{2}{c}\lVert W(\phi_1-\phi_2)\rVert_2. \tag*{(37)} ∥ W ( q D , ϕ 1 − q D , ϕ 2 ) ∥ 2 ≤ c 2 ∥ W ( ϕ 1 − ϕ 2 ) ∥ 2 . (37) Proof. Subtract (36) for two sources and test with W 2 ( q 1 − q 2 ) W^2(q_1-q_2) W 2 ( q 1 − q 2 )
∑ C W C 2 ( λ C 1 − λ C 2 ) ( ∫ C q 1 − ∫ C q 2 ) ≥ 0. \sum_C W_C^2(\lambda_C^1-\lambda_C^2)\left(\int_C q_1-\int_C q_2\right)\ge0. C ∑ W C 2 ( λ C 1 − λ C 2 ) ( ∫ C q 1 − ∫ C q 2 ) ≥ 0. The entropy difference quotient is at least M − 1 M^{-1} M − 1 W W W P P P T T T ∥ W T W − 1 − T ∥ ≤ C ( e C m − 1 ) ≤ c / 2 \lVert WTW^{-1}-T\rVert\le C(e^{Cm}-1)\le c/2 ∥ W T W − 1 − T ∥ ≤ C ( e C m − 1 ) ≤ c /2 m m m (37) .
Corollary 6.6 (The profile in the whole space). For every field in the compact product box there is a unique bounded positive solution in the whole space q ϕ q_\phi q ϕ d d d
∥ q ϕ − q ψ ∥ L 2 ( C ) ≤ C e − m d . \lVert q_\phi-q_\psi\rVert_{L^2(C)}\le Ce^{-md}. ∥ q ϕ − q ψ ∥ L 2 ( C ) ≤ C e − m d . Coordinatewise convergence of fields implies local L 2 L^2 L 2
Proof. For D ⊂ D ′ D\subset D' D ⊂ D ′ q D ′ , ϕ q_{D',\phi} q D ′ , ϕ D D D ϕ − T D , D ′ ∖ D q D ′ , ϕ \phi-T_{D,D'\setminus D}q_{D',\phi} ϕ − T D , D ′ ∖ D q D ′ , ϕ (37) with a weight decreasing away from a target cell. The squared weighted source norm is an exponentially decaying sum over the strip, bounded by C e − 2 m d Ce^{-2md} C e − 2 m d m m m
For two bounded solutions in the whole space use the same decaying weight on their difference. All terms are integrable, the cap contribution is nonnegative, and coercivity forces equality. This also proves lattice translation covariance. Weighted response proves locality and finite-coordinate continuity. To obtain product continuity, first make the exterior contribution small using locality, then use coordinate convergence in the resulting fixed finite neighborhood.
Local charges and an exact positive interaction
Define
e i ( ϕ ) = ∫ B i [ D q ϕ + V a ( A q ϕ ) ] d x + 1 4 α ∫ B i × S 2 ∣ ϕ − 2 α P q ϕ ∣ 2 . (38) e_i(\phi)=\int_{B_i}\left[Dq_\phi+V_a(Aq_\phi)\right]\,dx+\frac{1}{4\alpha}\int_{B_i\times S^2}\lvert\phi-2\alpha Pq_\phi\rvert^2. \tag*{(38)} e i ( ϕ ) = ∫ B i [ D q ϕ + V a ( A q ϕ ) ] d x + 4 α 1 ∫ B i × S 2 ∣ ϕ − 2 α P q ϕ ∣ 2 . (38) The charges are nonnegative, bounded and continuous in the product topology. Their response to a field change decays exponentially with distance, by the profile locality and the Lipschitz continuity of entropy on the uniform positive profile interval. For a homogeneous well f f f r = 2 α P f r=2\alpha Pf r = 2 α P f q r = f q_r=f q r = f Corollary 6.6 ; the caps are inactive by (35) . Consequently e i ( r ) = 0 e_i(r)=0 e i ( r ) = 0
Proposition 6.7 (Positive interactions on lattice balls). There are nonnegative continuous interactions Φ i , n 0 \Phi^0_{i,n} Φ i , n 0 B n ( i ) B_n(i) B n ( i )
e i = ∑ n ≥ 0 Φ i , n 0 , ∥ Φ i , n 0 ∥ ∞ ≤ C e − m ( n − 1 ) ( n ≥ 1 ) . e_i=\sum_{n\ge0}\Phi^0_{i,n},\qquad\lVert\Phi^0_{i,n}\rVert_\infty\le Ce^{-m(n-1)}\qquad(n\ge1). e i = n ≥ 0 ∑ Φ i , n 0 , ∥ Φ i , n 0 ∥ ∞ ≤ C e − m ( n − 1 ) ( n ≥ 1 ) . Every atom vanishes at a homogeneous well field. For every fixed p p p 0 < a ′ < m / 2 0<a'<m/2 0 < a ′ < m /2
∑ i , n : 0 ∈ B n ( i ) ∣ B n ( i ) ∣ p e a ′ diam B n ( i ) ∥ Φ i , n 0 ∥ ∞ < ∞ . \sum_{i,n:0\in B_n(i)}\lvert B_n(i)\rvert^p e^{a'\operatorname{diam} B_n(i)}\lVert\Phi^0_{i,n}\rVert_\infty<\infty. i , n : 0 ∈ B n ( i ) ∑ ∣ B n ( i ) ∣ p e a ′ diam B n ( i ) ∥ Φ i , n 0 ∥ ∞ < ∞. :::`
Proof. Set e i [ n ] ( ϕ ) = inf { e i ( ψ ) : ψ = ϕ on B n ( i ) } e_i^{[n]}(\phi)=\inf\{e_i(\psi):\psi=\phi\text{ on }B_n(i)\} e i [ n ] ( ϕ ) = inf { e i ( ψ ) : ψ = ϕ on B n ( i )} 0 ≤ e i − e i [ n ] ≤ C e − m n 0\le e_i-e_i^{[n]}\le Ce^{-mn} 0 ≤ e i − e i [ n ] ≤ C e − mn e i e_i e i n n n Φ i , 0 = e i [ 0 ] \Phi_{i,0}^{}=e_i^{[0]} Φ i , 0 = e i [ 0 ] Φ i , n = e i [ n ] − e i [ n − 1 ] \Phi_{i,n}^{}=e_i^{[n]}-e_i^{[n-1]} Φ i , n = e i [ n ] − e i [ n − 1 ] n ≥ 1 n\ge1 n ≥ 1 O ( ( n + 1 ) 3 ) O((n+1)^3) O (( n + 1 ) 3 ) 0 ∈ B n ( i ) 0\in B_n(i) 0 ∈ B n ( i ) ∣ B n ( i ) ∣ = O ( ( n + 1 ) 3 ) |B_n(i)|=O((n+1)^3) ∣ B n ( i ) ∣ = O (( n + 1 ) 3 ) 2 n 2n 2 n
Alignment with the well set
Write ∥ ⋅ ∥ i \|\cdot\|_i ∥ ⋅ ∥ i L 2 ( B i × S 2 ) L^2(B_i\times S^2) L 2 ( B i × S 2 ) f ∈ W f\in\mathcal W f ∈ W
D Q ( x ) + V a ( A Q ( x ) ) = ∫ S 2 ∫ − 1 / 2 1 / 2 [ h ( Q ( x + t u , u ) ) − h ( f ( u ) ) − log f ( u ) ( Q ( x + t u , u ) − f ( u ) ) ] d t d ω ( u ) + ⟨ A Q ( x ) − f , K ( A Q ( x ) − f ) ⟩ . D_Q(x)+V_a(A_Q(x))=\int_{S^2}\int_{-1/2}^{1/2}\left[h(Q(x+tu,u))-h(f(u))-\log f(u)(Q(x+tu,u)-f(u))\right]\,dt\,d\omega(u)+\langle A_Q(x)-f,K(A_Q(x)-f)\rangle. D Q ( x ) + V a ( A Q ( x )) = ∫ S 2 ∫ − 1/2 1/2 [ h ( Q ( x + t u , u )) − h ( f ( u )) − log f ( u ) ( Q ( x + t u , u ) − f ( u )) ] d t d ω ( u ) + ⟨ A Q ( x ) − f , K ( A Q ( x ) − f )⟩ . For z ≥ 0 z\ge0 z ≥ 0 f > 0 f>0 f > 0 ( z − f ) 2 / f (z-f)^2/f ( z − f ) 2 / f log t ≤ t − 1 \log t\le t-1 log t ≤ t − 1 m W = inf f ∈ W , u f ( u ) > 0 m_{\mathcal W}=\inf_{f\in\mathcal W,u}f(u)>0 m W = inf f ∈ W , u f ( u ) > 0
0 ≤ ∫ B i [ D Q + V a ( A Q ) ] ≤ C ∑ ∣ j − i ∣ ≤ r ∗ ∥ Q j − f ∥ L 2 ( B j × S 2 ) 2 , (39) 0\le\int_{B_i}[D_Q+V_a(A_Q)]\le C\sum_{|j-i|\le r_*}\|Q_j-f\|_{L^2(B_j\times S^2)}^2, \tag*{(39)} 0 ≤ ∫ B i [ D Q + V a ( A Q )] ≤ C ∣ j − i ∣ ≤ r ∗ ∑ ∥ Q j − f ∥ L 2 ( B j × S 2 ) 2 , (39) where r ∗ r_* r ∗
In the field coordinates
s i , b = ω b ϕ i , b , ( Π Q i ) b = 2 α ω b v s ω b ∫ B i × U b Q i , s_{i,b}=\sqrt{\omega_b}\phi_{i,b},\qquad(\Pi Q_i)_b=\frac{2\alpha\sqrt{\omega_b}}{v_s\omega_b}\int_{B_i\times U_b}Q_i, s i , b = ω b ϕ i , b , ( Π Q i ) b = v s ω b 2 α ω b ∫ B i × U b Q i , one has ∥ Π ∥ ≤ 2 α / v s \|\Pi\|\le2\alpha/\sqrt{v_s} ∥Π∥ ≤ 2 α / v s
1 4 α ∫ B i × S 2 ∣ ϕ − 2 α P Q ∣ 2 = v s 4 α ∣ s i − Π Q i ∣ 2 . (40) \frac{1}{4\alpha}\int_{B_i\times S^2}|\phi-2\alpha P Q|^2=\frac{v_s}{4\alpha}|s_i-\Pi Q_i|^2. \tag*{(40)} 4 α 1 ∫ B i × S 2 ∣ ϕ − 2 α P Q ∣ 2 = 4 α v s ∣ s i − Π Q i ∣ 2 . (40) Copying whole spatial cells preserves every marked-cell cap. The normalized Lebesgue field measure is pushed forward under this fixed diagonal coordinate change. Projected wells lie in its interior by (35) , giving uniform polynomial lower bounds for small-ball probabilities.
Lemma 6.8 (Alignment on a fixed patch). For a fixed connected inner patch I I I O O O I I I ω \omega ω ω ( t ) → 0 \omega(t)\to0 ω ( t ) → 0 t ↓ 0 t\downarrow0 t ↓ 0 E O = ∑ i ∈ O e i ( ϕ ) E_O=\sum_{i\in O}e_i(\phi) E O = ∑ i ∈ O e i ( ϕ )
min f ∈ W ∑ i ∈ I ∥ q ϕ , i − f ∥ i 2 + ∑ i , j ∈ I i ∼ j ∥ f i − f j ∥ L 2 ( S 2 ) 2 + ∑ i ∈ I ∣ s i − Π f i ∣ 2 ≤ ω ( E O ) . \min_{f\in\mathcal W}\sum_{i\in I}\|q_{\phi,i}-f\|_i^2+\sum_{\substack{i,j\in I\\i\sim j}}\|f_i-f_j\|_{L^2(S^2)}^2+\sum_{i\in I}|s_i-\Pi f_i|^2\le\omega(E_O). f ∈ W min i ∈ I ∑ ∥ q ϕ , i − f ∥ i 2 + i , j ∈ I i ∼ j ∑ ∥ f i − f j ∥ L 2 ( S 2 ) 2 + i ∈ I ∑ ∣ s i − Π f i ∣ 2 ≤ ω ( E O ) . Here i ∼ j i\sim j i ∼ j f i f_i f i q ϕ , i q_{\phi,i} q ϕ , i ∥ ⋅ ∥ i \|\cdot\|_i ∥ ⋅ ∥ i
Proof. If the outer charge sum is zero, Corollary 6.3 makes q ϕ q_\phi q ϕ ϕ = 2 α P f \phi=2\alpha P f ϕ = 2 α P f Corollary 6.6 and continuity of the charges pass zero energy and local profile convergence to the limit, a contradiction. For any nearest well f i f_i f i ∥ f i − f ∥ i ≤ 2 ∥ q ϕ , i − f ∥ i \lVert f_i-f\rVert_i\leq2\lVert q_{\phi,i}-f\rVert_i ∥ f i − f ∥ i ≤ 2 ∥ q ϕ , i − f ∥ i [22 ] .
Let ψ ( t ) \psi(t) ψ ( t ) t t t C ≥ sup ψ C\geq\sup\psi C ≥ sup ψ ω ( t ) = inf s > 0 [ ψ ( s ) + C t / s ] \omega(t)=\inf_{s>0}[\psi(s)+Ct/s] ω ( t ) = inf s > 0 [ ψ ( s ) + C t / s ] ψ \psi ψ
Lemma 6.9 (Separation after cell projection). The restriction of P P P W \mathcal{W} W
Proof. If P f = P g Pf=Pg P f = P g 2 K + 2 α P a n g 2K+2\alpha P_{\mathrm{ang}} 2 K + 2 α P ang S S S f − g f-g f − g c ∥ f − g ∥ 2 2 c\lVert f-g\rVert_2^2 c ∥ f − g ∥ 2 2 f = g f=g f = g W \mathcal{W} W
Changes of field and comparison with trial profiles
For an arbitrary trial Q Q Q ℓ i ( Q , ϕ ) \ell_i(Q,\phi) ℓ i ( Q , ϕ ) (38) with Q Q Q q ϕ q_\phi q ϕ
Proposition 6.10 (Exact change of action). If ϕ − ψ \phi-\psi ϕ − ψ
H 0 ( ϕ ; ψ ) : = ∑ i [ e i ( ϕ ) − e i ( ψ ) ] H^0(\phi;\psi):=\sum_i[e_i(\phi)-e_i(\psi)] H 0 ( ϕ ; ψ ) := i ∑ [ e i ( ϕ ) − e i ( ψ )] = ∫ 0 1 ∑ C ( ϕ C − ψ C ) [ v C { ψ C + t ( ϕ C − ψ C ) } 2 α − ∫ C q ψ + t ( ϕ − ψ ) ] d t . (41) =\int_0^1\sum_C(\phi_C-\psi_C)\left[\frac{v_C\{\psi_C+t(\phi_C-\psi_C)\}}{2\alpha}-\int_C q_{\psi+t(\phi-\psi)}\right]dt. \tag*{(41)} = ∫ 0 1 C ∑ ( ϕ C − ψ C ) [ 2 α v C { ψ C + t ( ϕ C − ψ C )} − ∫ C q ψ + t ( ϕ − ψ ) ] d t . (41) The sum is absolutely convergent. It equals the sum of changes of the interactions Φ i , n 0 \Phi^0_{i,n} Φ i , n 0
Proof. Use the same finite free box D D D
D q + V a ( A q ) = ∫ S 2 ∫ − 1 / 2 1 / 2 h ( q ( x + t u , u ) ) d t d ω − a ∫ S 2 A q d ω + ⟨ A q , K A q ⟩ − m a D_q+V_a(Aq)=\int_{S^2}\int_{-1/2}^{1/2}h(q(x+tu,u))\,dt\,d\omega-a\int_{S^2}Aq\,d\omega+\langle Aq,KAq\rangle-m_a D q + V a ( A q ) = ∫ S 2 ∫ − 1/2 1/2 h ( q ( x + t u , u )) d t d ω − a ∫ S 2 A q d ω + ⟨ A q , K A q ⟩ − m a removes the h ( A q ) h(Aq) h ( A q )
∑ i { ℓ i ( q D , ϕ , ϕ D ) − ℓ i ( q D , ψ , ψ D ) } = G D , ϕ ( q D , ϕ ) + 1 4 α ∫ D ϕ 2 − G D , ψ ( q D , ψ ) − 1 4 α ∫ D ψ 2 . \sum_i\{\ell_i(q_{D,\phi},\phi_D)-\ell_i(q_{D,\psi},\psi_D)\}
=\;G_{D,\phi}(q_{D,\phi})+\frac{1}{4\alpha}\int_D\phi^2-G_{D,\psi}(q_{D,\psi})-\frac{1}{4\alpha}\int_D\psi^2. i ∑ { ℓ i ( q D , ϕ , ϕ D ) − ℓ i ( q D , ψ , ψ D )} = G D , ϕ ( q D , ϕ ) + 4 α 1 ∫ D ϕ 2 − G D , ψ ( q D , ψ ) − 4 α 1 ∫ D ψ 2 . The common constant baseline outside D D D D D D D D D
The minimizing inequalities at two sources and the continuity of the minimizer give the envelope formula of Danskin [23 ] : the minimized full action min q G D , ϕ ( q ) + ( 4 α ) − 1 ∫ D ϕ 2 \min_q G_{D,\phi}(q) + (4\alpha)^{-1}\int_D\phi^2 min q G D , ϕ ( q ) + ( 4 α ) − 1 ∫ D ϕ 2 v C ϕ C / ( 2 α ) − ∫ C q D , ϕ v_C\phi_C/(2\alpha)-\int_Cq_{D,\phi} v C ϕ C / ( 2 α ) − ∫ C q D , ϕ (41) . For the ball interaction, atoms on balls missing the changed set have zero difference. At distant centers, only radii reaching the set contribute; their exponential norms are summable. This justifies the double sum and proves equality of the two action differences.
Proposition 6.11 (Comparison with a trial profile). Suppose ϕ ′ − ϕ \phi'-\phi ϕ ′ − ϕ Q ′ Q' Q ′ 0 ≤ Q ′ ≤ M 0\le Q'\le M 0 ≤ Q ′ ≤ M q ϕ q_\phi q ϕ
∑ i [ e i ( ϕ ′ ) − e i ( ϕ ) ] ≤ ∑ i [ ℓ i ( Q ′ , ϕ ′ ) − ℓ i ( q ϕ , ϕ ) ] . (42) \sum_i[e_i(\phi')-e_i(\phi)]\le\sum_i[\ell_i(Q',\phi')-\ell_i(q_\phi,\phi)]. \tag*{(42)} i ∑ [ e i ( ϕ ′ ) − e i ( ϕ )] ≤ i ∑ [ ℓ i ( Q ′ , ϕ ′ ) − ℓ i ( q ϕ , ϕ )] . (42) Proof. Put q ′ = q ϕ ′ q'=q_{\phi'} q ′ = q ϕ ′ Δ = Q ′ − q ′ \Delta=Q'-q' Δ = Q ′ − q ′ Δ ∈ L 1 ∩ L 2 \Delta\in L^1\cap L^2 Δ ∈ L 1 ∩ L 2 q ′ q' q ′
∑ i [ ℓ i ( Q ′ , ϕ ′ ) − ℓ i ( q ′ , ϕ ′ ) ] = ∫ [ h ( Q ′ ) − h ( q ′ ) − log q ′ ( Q ′ − q ′ ) ] + 1 2 ⟨ Δ , S Δ ⟩ − ∑ C λ C ′ ∫ C Δ ≥ c 2 ∥ Δ ∥ 2 2 ≥ 0. \begin{aligned}
\sum_i[\ell_i(Q',\phi')-\ell_i(q',\phi')]=\int[h(Q')-h(q')-\log q'(Q'-q')]+\frac{1}{2}\langle\Delta,S\Delta\rangle-\sum_C\lambda'_C\int_C\Delta\\
&\ge\frac{c}{2}\lVert\Delta\rVert_2^2\ge0.
\end{aligned} i ∑ [ ℓ i ( Q ′ , ϕ ′ ) − ℓ i ( q ′ , ϕ ′ )] = ∫ [ h ( Q ′ ) − h ( q ′ ) − log q ′ ( Q ′ − q ′ )] + 2 1 ⟨ Δ , S Δ ⟩ − C ∑ λ C ′ ∫ C Δ ≥ 2 c ∥ Δ ∥ 2 2 ≥ 0. The cap term is nonnegative, because a positive multiplier means q ′ q' q ′ Q ′ Q' Q ′ L 1 L^1 L 1 (42) .
Corollary 6.12 (A homogeneous trial profile). For a homogeneous root r = 2 α P f r=2\alpha Pf r = 2 α P f D D D
0 ≤ H D 0 ( ϕ D ∣ r ) ≤ 1 4 α ∑ C ⊂ D v C ∣ ϕ C − r C ∣ 2 , H D 0 ( ϕ D ∣ r ) = ∑ i e i ( ϕ D r D c ) . (43) 0\le H_D^0(\phi_D\mid r)\le\frac{1}{4\alpha}\sum_{C\subset D}v_C\lvert\phi_C-r_C\rvert^2,\qquad H_D^0(\phi_D\mid r)=\sum_i e_i(\phi_D r_{D^c}). \tag*{(43)} 0 ≤ H D 0 ( ϕ D ∣ r ) ≤ 4 α 1 C ⊂ D ∑ v C ∣ ϕ C − r C ∣ 2 , H D 0 ( ϕ D ∣ r ) = i ∑ e i ( ϕ D r D c ) . (43) Proof. Use Q ′ = f Q'=f Q ′ = f
Restoring all particle counts
We use the fixed spatial and angular mesh chosen above. Its marked cells are denoted by C i C_i C i v i v_i v i N i N_i N i 0 < v − ≤ v i ≤ v + < ∞ 0<v_-\le v_i\le v_+<\infty 0 < v − ≤ v i ≤ v + < ∞ G G G
Write z h = β ζ e h z_h=\beta\zeta e^h z h = β ζ e h K i = ⌈ ρ β v i ⌉ K_i=\lceil\rho\beta v_i\rceil K i = ⌈ ρβ v i ⌉ (10) –(11) . In particular,
0 ≤ − log c i , h ( n ) ≤ C e − c β , 0 ≤ n ≤ K i . (44) 0\le-\log c_{i,h}(n)\le Ce^{-c\beta},\qquad0\le n\le K_i. \tag*{(44)} 0 ≤ − log c i , h ( n ) ≤ C e − c β , 0 ≤ n ≤ K i . (44) With ν i \nu_i ν i C i C_i C i d Ω i = ∑ n ≥ 0 d v i ⊗ n / n ! d\Omega_i=\sum_{n\ge0}dv_i^{\otimes n}/n! d Ω i = ∑ n ≥ 0 d v i ⊗ n / n !
a i ϕ = ( β ζ ) N i c i , h ( N i ) e ϕ i N i − α N i 2 / ( β v i ) ( N i ≤ K i ) , b i = z h N i c i , h ( 0 ) ( N i > K i ) . (45) a_i^\phi=\frac{(\beta\zeta)^{N_i}}{c_{i,h}(N_i)}e^{\phi_iN_i-\alpha N_i^2/(\beta v_i)}\quad(N_i\le K_i),\qquad b_i=\frac{z_h^{N_i}}{c_{i,h}(0)}\quad(N_i>K_i). \tag*{(45)} a i ϕ = c i , h ( N i ) ( β ζ ) N i e ϕ i N i − α N i 2 / ( β v i ) ( N i ≤ K i ) , b i = c i , h ( 0 ) z h N i ( N i > K i ) . (45) The empty-cell weight is E i = c i , h ( 0 ) − 1 E_i=c_{i,h}(0)^{-1} E i = c i , h ( 0 ) − 1 [ 0 , 1 ] [0,1] [ 0 , 1 ]
The empty-cell projection
For a finite set D D D ξ \xi ξ R D f u l l R_D^{\mathrm{full}} R D full (45) , and let R D c a p R_D^{\mathrm{cap}} R D cap N i ≤ K i N_i\le K_i N i ≤ K i ξ \xi ξ π ω \pi\omega π ω
N i ( π ω ) = N ^ i ( ω ) , N ^ i = N i 1 { N i ≤ K i } . (46) N_i(\pi\omega)=\widehat{N}_i(\omega),\qquad\widehat{N}_i=N_i\mathbf{1}_{\{N_i\le K_i\}}. \tag*{(46)} N i ( π ω ) = N i ( ω ) , N i = N i 1 { N i ≤ K i } . (46) A nonempty G 2 G^2 G 2 S ⊂ D S\subset D S ⊂ D U S = N G [ S ] U_S=N_G[S] U S = N G [ S ] G G G σ \sigma σ
B S ( σ ; ξ ) = ∫ N i > K i , i ∈ S ∏ i ∈ S b i ( ω i ) χ S ( ω S ; σ D ∖ S , ξ ) d L S ( ω S ) , B_S(\sigma;\xi)=\int_{N_i>K_i,\ i\in S}\prod_{i\in S}b_i(\omega_i)\chi_S(\omega_S;\sigma_{D\setminus S},\xi)\,\mathrm{d}\mathcal{L}_S(\omega_S), B S ( σ ; ξ ) = ∫ N i > K i , i ∈ S i ∈ S ∏ b i ( ω i ) χ S ( ω S ; σ D ∖ S , ξ ) d L S ( ω S ) , w S ( σ ; ξ ) = 1 { σ S = ∅ } ∏ i ∈ S E i B S ( σ ; ξ ) , (47) w_S(\sigma;\xi)=\frac{\mathbf{1}_{\{\sigma_S=\varnothing\}}}{\prod_{i\in S}E_i}B_S(\sigma;\xi), \tag*{(47)} w S ( σ ; ξ ) = ∏ i ∈ S E i 1 { σ S = ∅ } B S ( σ ; ξ ) , (47) Ξ D e ( σ ; ξ ) = ∑ F c o m p a t i b l e ∏ S ∈ F w S , K D e = − log Ξ D e . \Xi_D^{e}(\sigma;\xi)=\sum_{\mathcal{F}\ \mathrm{compatible}}\prod_{S\in\mathcal{F}}w_S,\qquad K_D^{e}=-\log\Xi_D^{e}. Ξ D e ( σ ; ξ ) = F compatible ∑ S ∈ F ∏ w S , K D e = − log Ξ D e . Here χ S \chi_S χ S S S S
Proposition 7.1 (Exact projection). For every D , ξ , ϕ D,\xi,\phi D , ξ , ϕ
R D f u l l ( ϕ ∣ ξ ) = R D c a p ( ϕ ∣ ξ ) Ξ D , ϕ , ξ c a p e − K D e . (48) R_D^{\mathrm{full}}(\phi\mid\xi)=R_D^{\mathrm{cap}}(\phi\mid\xi)\Xi_{D,\phi,\xi}^{\mathrm{cap}}e^{-K_D^{e}}. \tag*{(48)} R D full ( ϕ ∣ ξ ) = R D cap ( ϕ ∣ ξ ) Ξ D , ϕ , ξ cap e − K D e . (48) The pushforward of the full law by π \pi π e − K D e e^{-K_D^{e}} e − K D e K D e K_D^{e} K D e ϕ \phi ϕ
Proof. Decompose the full integral according to its actual bad set B = { i : N i > K i } B=\{i:N_i>K_i\} B = { i : N i > K i } G 2 G^2 G 2 B B B G G G (48) , the indicators force the cells in B B B ∏ i ∈ B E i \prod_{i\in B}E_i ∏ i ∈ B E i (47) . All remaining factors agree with the original sector. This proves the identity sector by sector. Inserting any bounded function of π ω \pi\omega π ω b i / E i = z h N i b_i/E_i=z_h^{N_i} b i / E i = z h N i
Lemma 7.2 (Exceptional-count tails). There is q β = e − c β q_\beta=e^{-c\beta} q β = e − c β c > 0 c>0 c > 0
0 ≤ w S ≤ q β ∣ S ∣ , ∑ n > K i n m ( β ζ e H v i ) n n ! ≤ C m β m q β (49) 0\le w_S\le q_\beta^{|S|},\qquad\sum_{n>K_i}\frac{n^m(\beta\zeta e^{H}v_i)^n}{n!}\le C_m\beta^m q_\beta \tag*{(49)} 0 ≤ w S ≤ q β ∣ S ∣ , n > K i ∑ n ! n m ( β ζ e H v i ) n ≤ C m β m q β (49) for every fixed integer m ≥ 0 m\ge0 m ≥ 0
Proof. Drop the hard constraints and accessibility multipliers. The empty weights cancel the factors c i , h ( 0 ) − 1 c_{i,h}(0)^{-1} c i , h ( 0 ) − 1 r = e ζ e H / ρ < 1 r=e^{\zeta e^{H}/\rho}<1 r = e ζ e H / ρ < 1 n ! ≥ ( n / e ) n n!\ge(n/e)^n n ! ≥ ( n / e ) n
∑ n > K i ( β ζ e H v i ) n n ! ≤ r K i + 1 1 − r . \sum_{n>K_i}\frac{(\beta\zeta e^{H}v_i)^n}{n!}\le\frac{r^{K_i+1}}{1-r}. n > K i ∑ n ! ( β ζ e H v i ) n ≤ 1 − r r K i + 1 . Since v i ≥ v − v_i \ge v_- v i ≥ v − β \beta β ζ e H / ρ < 1 \zeta e^H/\rho< 1 ζ e H / ρ < 1 n m n^m n m
The expansion with a weak interaction
We shall allow a bounded measurable interaction J = ∑ A J A J = \sum_A J_A J = ∑ A J A G G G a > 0 a > 0 a > 0
∥ J ∥ a = sup x ∑ A ∋ x e a ∣ A ∣ ∥ J A ∥ ∞ = κ . (50) \lVert J\rVert_a = \sup_x \sum_{A\ni x} e^{a|A|}\lVert J_A\rVert_\infty= \kappa. \tag*{(50)} ∥ J ∥ a = x sup A ∋ x ∑ e a ∣ A ∣ ∥ J A ∥ ∞ = κ . (50) The exponent a a a e − J A = 1 + f A e^{-J_A}=1+f_A e − J A = 1 + f A η A = e ∥ J A ∥ ∞ − 1 \eta_A=e^{\lVert J_A\rVert_\infty}-1 η A = e ∥ J A ∥ ∞ − 1 i i i i + = N G [ i ] i^+=N_G[i] i + = N G [ i ] q β q_\beta q β A A A A A A η A \eta_A η A γ \gamma γ
U γ = ⋃ a ∈ γ B a , ℓ γ = ∑ a ∈ γ ∣ B a ∣ , z γ # = ∏ a ∈ γ w a . (51) U_\gamma=\bigcup_{a\in\gamma}B_a,\qquad\ell_\gamma=\sum_{a\in\gamma}|B_a|,\qquad z_\gamma^\#=\prod_{a\in\gamma}w_a. \tag*{(51)} U γ = a ∈ γ ⋃ B a , ℓ γ = a ∈ γ ∑ ∣ B a ∣ , z γ # = a ∈ γ ∏ w a . (51) For γ = ( S , F ) \gamma=(S,\mathcal F) γ = ( S , F )
z γ ( σ ; ξ ) = 1 { σ S = ∅ } ∏ i ∈ S E i 1 { N i > K i , i ∈ S } ∫ ∏ i ∈ S b i ( ω i ) χ S ( ω S ; σ D ∖ S , ξ ) ∏ A ∈ F f A ( ω S σ D ∖ S ξ , ϕ ) d L S ( ω S ) . (52) z_\gamma(\sigma;\xi)=\frac{\mathbf{1}_{\{\sigma_S=\varnothing\}}}{\prod_{i\in S}E_i\mathbf{1}_{\{N_i>K_i,\ i\in S\}}}\int\prod_{i\in S}b_i(\omega_i)\chi_S(\omega_S;\sigma_{D\setminus S},\xi)\prod_{A\in\mathcal F}f_A(\omega_S\sigma_{D\setminus S}\xi,\phi)\,d\mathcal L_S(\omega_S). \tag*{(52)} z γ ( σ ; ξ ) = ∏ i ∈ S E i 1 { N i > K i , i ∈ S } 1 { σ S = ∅ } ∫ i ∈ S ∏ b i ( ω i ) χ S ( ω S ; σ D ∖ S , ξ ) A ∈ F ∏ f A ( ω S σ D ∖ S ξ , ϕ ) d L S ( ω S ) . (52) When S S S (52) are evaluated on the reconstructed actual configuration. The bound
∣ z γ ∣ ≤ z γ # = q β ∣ S ∣ ∏ A ∈ F η A (53) |z_\gamma|\le z_\gamma^\#=q_\beta^{|S|}\prod_{A\in\mathcal F}\eta_A \tag*{(53)} ∣ z γ ∣ ≤ z γ # = q β ∣ S ∣ A ∈ F ∏ η A (53) follows from Lemma 7.2.
Lemma 7.3 (Projection with a weak interaction). Let K D ( J ) K_D(J) K D ( J ) (52) , with disjoint dependency supports. Then
R D f u l l , J = R D c a p E D c a p e − K D ( J ) . (54) \mathcal R_D^{\mathrm{full},J}=\mathcal R_D^{\mathrm{cap}}\mathbb E_D^{\mathrm{cap}}e^{-K_D(J)}. \tag*{(54)} R D full , J = R D cap E D cap e − K D ( J ) . (54) The full law projects to the capped law tilted by e − K D ( J ) e^{-K_D(J)} e − K D ( J )
Proof. Expand the weak factors and partition each selected atom set into its overlap components. Distinct components have disjoint dependence and no hard factor connecting their bad rods, so their integrals factor. The empty weights cancel as in Proposition 7.1. Inserting a bounded function of the projected configuration proves the assertion about its law.
The logarithmic expansion uses Penrose’s partition identity [24 ] and the Kotecký–Preiss convergence criterion [25 ] , in the formulation by iteration over trees in Fernández and Procacci [26 Section 4] .
Lemma 7.4 (Mixed cluster expansion). Suppose that, for the required fixed s ≥ 0 s \ge0 s ≥ 0
ρ s : = sup x ∑ a : x ∈ B a w a e ( s + 2 ) ∣ B a ∣ ≤ 1 4 . (55) \rho_s := \sup_x \sum_{a:x\in B_a} w_a e^{(s+2)|B_a|} \le\frac{1}{4}. \tag*{(55)} ρ s := x sup a : x ∈ B a ∑ w a e ( s + 2 ) ∣ B a ∣ ≤ 4 1 . (55) The gas has an absolutely convergent logarithm. Grouping the logarithm by connected union supports gives
K D ( J ) = ∑ U K D , U ( J ) , sup x ∑ U ∋ x e b ∣ U ∣ ( 1 + ∣ U ∣ ) p ∥ K D , U ( J ) ∥ ∞ ≤ C b , p ( q β + κ ) (56) K_D(J) = \sum_U K_{D,U}(J), \qquad\sup_x \sum_{U\ni x} e^{b|U|}(1+|U|)^p \lVert K_{D,U}(J)\rVert_\infty\le C_{b,p}(q_\beta+\kappa) \tag*{(56)} K D ( J ) = U ∑ K D , U ( J ) , x sup U ∋ x ∑ e b ∣ U ∣ ( 1 + ∣ U ∣ ) p ∥ K D , U ( J ) ∥ ∞ ≤ C b , p ( q β + κ ) (56) whenever (55) holds with sufficient reserve beyond b b b q β q_\beta q β κ \kappa κ q β κ q_\beta\kappa q β κ
Proof. For an ordered tuple C = ( γ 1 , … , γ n ) C=(\gamma_1,\ldots,\gamma_n) C = ( γ 1 , … , γ n ) φ T ( C ) \varphi^T(C) φ T ( C ) ℓ ( C ) = ∑ j ℓ γ j \ell(C)=\sum_j\ell_{\gamma_j} ℓ ( C ) = ∑ j ℓ γ j U ( C ) = ⋃ j U γ j U(C)=\bigcup_j U_{\gamma_j} U ( C ) = ⋃ j U γ j
∣ φ T ( C ) ∣ ≤ ∑ T t r e e ∏ { i , j } ∈ E ( T ) 1 { U γ i ∩ U γ j ≠ ∅ } . (57) |\varphi^T(C)| \le\sum_{T\ \mathrm{tree}} \prod_{\{i,j\}\in E(T)} \mathbf{1}_{\{U_{\gamma_i}\cap U_{\gamma_j}\ne\varnothing\}}. \tag*{(57)} ∣ φ T ( C ) ∣ ≤ T tree ∑ { i , j } ∈ E ( T ) ∏ 1 { U γ i ∩ U γ j = ∅ } . (57) To prove it, order edges and group connected graphs by their minimum spanning tree. Each group is a Boolean interval, whose alternating sum is zero or has absolute value one.
For atom trees, iteration from zero in
T a = w a e u ∣ B a ∣ exp { ∑ b : B b ∩ B a ≠ ∅ T b } T_a = w_a e^{u|B_a|}\exp\left\{\sum_{b:B_b\cap B_a\ne\varnothing} T_b\right\} T a = w a e u ∣ B a ∣ exp ⎩ ⎨ ⎧ b : B b ∩ B a = ∅ ∑ T b ⎭ ⎬ ⎫ is bounded by w a e ( u + 1 ) ∣ B a ∣ w_a e^{(u+1)|B_a|} w a e ( u + 1 ) ∣ B a ∣ sup x ∑ a : x ∈ B a w a e ( u + 1 ) ∣ B a ∣ ≤ 1 \sup_x\sum_{a:x\in B_a}w_a e^{(u+1)|B_a|}\le1 sup x ∑ a : x ∈ B a w a e ( u + 1 ) ∣ B a ∣ ≤ 1
∑ γ : a ∈ γ z γ # e u ℓ γ ≤ w a e ( u + 1 ) ∣ B a ∣ , ∑ γ : x ∈ U γ z γ # e u ℓ γ ≤ ∑ a : x ∈ B a w a e ( u + 1 ) ∣ B a ∣ . (58) \sum_{\gamma:a\in\gamma} z_\gamma^\# e^{u\ell_\gamma}\le w_a e^{(u+1)|B_a|}, \qquad\sum_{\gamma:x\in U_\gamma} z_\gamma^\# e^{u\ell_\gamma}\le\sum_{a:x\in B_a}w_a e^{(u+1)|B_a|}. \tag*{(58)} γ : a ∈ γ ∑ z γ # e u ℓ γ ≤ w a e ( u + 1 ) ∣ B a ∣ , γ : x ∈ U γ ∑ z γ # e u ℓ γ ≤ a : x ∈ B a ∑ w a e ( u + 1 ) ∣ B a ∣ . (58) Apply the same recursion to object trees. Its supersolution is z γ # e s ℓ γ + ∣ U γ ∣ z_\gamma^\#e^{s\ell_\gamma+|U_\gamma|} z γ # e s ℓ γ + ∣ U γ ∣
∑ γ ′ : U γ ′ ∩ U γ ≠ ∅ z γ ′ # e s ℓ γ ′ + ∣ U γ ′ ∣ ≤ ∣ U γ ∣ ρ s . \sum_{\gamma':U_{\gamma'}\cap U_\gamma\ne\varnothing} z_{\gamma'}^\# e^{s\ell_{\gamma'}+|U_{\gamma'}|}\le|U_\gamma|\rho_s. γ ′ : U γ ′ ∩ U γ = ∅ ∑ z γ ′ # e s ℓ γ ′ + ∣ U γ ′ ∣ ≤ ∣ U γ ∣ ρ s . Rooting at an object containing x x x (57) yields
∑ n ≥ 1 1 n ! ∑ C ∣ φ T ( C ) ∣ ∏ j z γ j # e s ℓ ( C ) 1 { x ∈ U ( C ) } ≤ ρ s . (59) \sum_{n\ge1}\frac{1}{n!}\sum_C|\varphi^T(C)|\prod_j z_{\gamma_j}^\#e^{s\ell(C)}\mathbf{1}_{\{x\in U(C)\}}\le\rho_s. \tag*{(59)} n ≥ 1 ∑ n ! 1 C ∑ ∣ φ T ( C ) ∣ j ∏ z γ j # e s ℓ ( C ) 1 { x ∈ U ( C )} ≤ ρ s . (59) If n i ( C ) n_i(C) n i ( C ) i i i
∑ n ≥ 1 1 n ! ∑ C ∣ φ T ( C ) ∣ n i ( C ) ∏ j z γ j # e s ℓ ( C ) ≤ q β e ( s + 2 ) ∣ B i ∣ . (60) \sum_{n\ge1}\frac{1}{n!}\sum_C|\varphi^T(C)|n_i(C)\prod_j z_{\gamma_j}^\#e^{s\ell(C)}\le q_\beta e^{(s+2)|B_i|}. \tag*{(60)} n ≥ 1 ∑ n ! 1 C ∑ ∣ φ T ( C ) ∣ n i ( C ) j ∏ z γ j # e s ℓ ( C ) ≤ q β e ( s + 2 ) ∣ B i ∣ . (60) Extra occurrence marks cost powers of ℓ ( C ) \ell(C) ℓ ( C ) κ s \kappa_s κ s ρ s \rho_s ρ s λ 0 = 1 + ( 4 κ s ) − 1 \lambda_0=1+(4\kappa_s)^{-1} λ 0 = 1 + ( 4 κ s ) − 1 3 / 4 3/4 3/4 k ≤ ( λ 0 k − 1 ) / ( λ 0 − 1 ) k\le(\lambda_0^k-1)/(\lambda_0-1) k ≤ ( λ 0 k − 1 ) / ( λ 0 − 1 ) C κ s C\kappa_s C κ s (60) , C q β κ s Cq_\beta\kappa_s C q β κ s κ s = 0 \kappa_s=0 κ s = 0
Because ∣ U ( C ) ∣ ≤ ℓ ( C ) |U(C)|\leq\ell(C) ∣ U ( C ) ∣ ≤ ℓ ( C ) (56) . A cluster joining two sites has occurrence size at least a fixed multiple of their distance, so the reserved exponent also gives spatial decay. For finitely many atoms the series is the formal logarithm of the gas polynomial. The same bounds hold in a complex activity disk larger than the unit disk; analytic continuation identifies its exponential with that polynomial. Positivity of the original fiber integral selects the real logarithm. Absolute convergence then permits countable atom families and marked derivatives.
For fixed norm exponents, (55) follows from q β → 0 q_\beta\to0 q β → 0 κ \kappa κ (50) . For J = 0 J=0 J = 0 K D ( 0 ) = K D e K_D(0)=K_D^{e} K D ( 0 ) = K D e
Evaluate both expansions at η = σ D ξ \eta=\sigma_D\xi η = σ D ξ D D D D D D i i i i i i D c D^c D c Δ U \Delta U Δ U
∑ U ∋ i osc i Δ U ≤ C q β e − m d ( i , D c ) . (61) \sum_{U\ni i}\operatorname{osc}_i\Delta U\leq Cq_\beta e^{-m d(i,D^c)}. \tag*{(61)} U ∋ i ∑ osc i Δ U ≤ C q β e − m d ( i , D c ) . (61) Uniform response and reconstruction
We use Theorem 5.1 in the following form. If a capped reference is perturbed by Q Q Q β ∥ Q ∥ a \beta\lVert Q\rVert_a β ∥ Q ∥ a
∣ Cov c a p , Q ( F , G ) ∣ ≤ C ∑ i , j e − m d ( i , j ) D i ( F ) D j ( G ) , D i ( F ) 2 = E ∫ C i r X ∣ D X F ∣ 2 d X . (62) \left|\operatorname{Cov}_{\mathrm{cap},Q}(F,G)\right|\leq C\sum_{i,j}e^{-m d(i,j)}D_i(F)D_j(G),\qquad D_i(F)^2=\mathbb{E}\int_{C_i}r_X|D_XF|^2\,\mathrm{d}X. \tag*{(62)} ∣ Cov cap , Q ( F , G ) ∣ ≤ C i , j ∑ e − m d ( i , j ) D i ( F ) D j ( G ) , D i ( F ) 2 = E ∫ C i r X ∣ D X F ∣ 2 d X . (62) Insertion rates satisfy r X ≤ C β r_X\leq C\beta r X ≤ C β e − t e^{-t} e − t (44) and (56) , they apply to the exact projected law whenever
β ( q β + κ ) ≤ c 0 (63) \beta(q_\beta+\kappa)\leq c_0 \tag*{(63)} β ( q β + κ ) ≤ c 0 (63) with the fixed constant c 0 c_0 c 0
Lemma 7.5 (Reset-count covariance). In the bare full reference,
∣ Cov ( N ^ i , N ^ j ) ∣ ≤ C β e − m d ( i , j ) . (64) \left|\operatorname{Cov}(\widehat N_i,\widehat N_j)\right|\leq C\beta e^{-m d(i,j)}. \tag*{(64)} Cov ( N i , N j ) ≤ C β e − m d ( i , j ) . (64) Proof. The projected reset counts are the capped counts. Their insertion gradients equal one in their own cells and zero elsewhere, so (62) applies.
Lemma 7.6 (Response to deleting particle states). Let A A A (45) by e − t N A e^{-tN_A} e − t N A h h h I A = { j : ν j ( A ∩ C j ) > 0 } I_A=\{j:\nu_j(A\cap C_j)>0\} I A = { j : ν j ( A ∩ C j ) > 0 }
∣ d d t β − 1 E t N ^ i ∣ ≤ C ( 1 + β q β ) e − t / 2 ∑ j ∈ I A e − m d ( i , j ) . (65) \left|\frac{\mathrm{d}}{\mathrm{d}t}\beta^{-1}\mathbb{E}_t\widehat N_i\right|\leq C(1+\beta q_\beta)e^{-t/2}\sum_{j\in I_A}e^{-m d(i,j)}. \tag*{(65)} d t d β − 1 E t N i ≤ C ( 1 + β q β ) e − t /2 j ∈ I A ∑ e − m d ( i , j ) . (65) Proof. Under deletion, the same projection formula holds with correction K t K_t K t − N A -N_A − N A N A ≥ 1 N_A \ge1 N A ≥ 1 N A e − t N A ≤ e − t / 2 N A N_A e^{-tN_A} \le e^{-t/2}N_A N A e − t N A ≤ e − t /2 N A
∂ t K t = ∑ j ∈ I A ∑ U ∋ j D t , U ( j ) , ∑ U ∋ j e b ∣ U ∣ ( 1 + ∣ U ∣ ) p ∥ D t , U ( j ) ∥ ∞ ≤ C b , p β q β e − t / 2 . (66) \partial_t K_t = \sum_{j\in I_A}\sum_{U\ni j}D_{t,U}^{(j)}, \qquad\sum_{U\ni j}e^{b|U|}(1+|U|)^p\lVert D_{t,U}^{(j)}\rVert_\infty\le C_{b,p}\beta q_\beta e^{-t/2}. \tag*{(66)} ∂ t K t = j ∈ I A ∑ U ∋ j ∑ D t , U ( j ) , U ∋ j ∑ e b ∣ U ∣ ( 1 + ∣ U ∣ ) p ∥ D t , U ( j ) ∥ ∞ ≤ C b , p β q β e − t /2 . (66) If T A T_A T A A A A
d d t E t N ^ i = − Cov c a p , K t ( N i , T A + ∂ t K t ) . \frac{d}{dt}\mathbb{E}_t\widehat{N}_i=-\operatorname{Cov}_{\mathrm{cap},K_t}\left(N_i,T_A+\partial_tK_t\right). d t d E t N i = − Cov cap , K t ( N i , T A + ∂ t K t ) . The insertion norm of a deleted count in cell j j j C β e − t / 2 C\sqrt{\beta}e^{-t/2} C β e − t /2 V U V_U V U 2 ∥ V U ∥ ∞ 2\lVert V_U\rVert_\infty 2 ∥ V U ∥ ∞
∣ Cov ( N i , V U ) ∣ ≤ C β ∥ V U ∥ ∞ ∑ x ∈ U e − m d ( i , x ) . \left|\operatorname{Cov}(N_i,V_U)\right|\le C\beta\lVert V_U\rVert_\infty\sum_{x\in U}e^{-m d(i,x)}. ∣ Cov ( N i , V U ) ∣ ≤ C β ∥ V U ∥ ∞ x ∈ U ∑ e − m d ( i , x ) . Apply this to (66) ; connected support and its exponential reserve move x x x j j j C β e − t / 2 C\beta e^{-t/2} C β e − t /2 C β 2 q β e − t / 2 C\beta^2q_\beta e^{-t/2} C β 2 q β e − t /2 (65) . □
For an exterior configuration ξ \xi ξ ξ \xi ξ (65) from zero to infinity shows that
∣ β − 1 E D ξ N ^ i − β − 1 E D f r e e N ^ i ∣ ≤ C e − m d ( i , D c ) . (67) \left|\beta^{-1}\mathbb{E}^{\xi}_{D}\widehat{N}_i-\beta^{-1}\mathbb{E}^{\mathrm{free}}_{D}\widehat{N}_i\right|\le Ce^{-m d(i,D^c)}. \tag*{(67)} β − 1 E D ξ N i − β − 1 E D free N i ≤ C e − m d ( i , D c ) . (67) The same argument works for every bounded local observable of the projected configuration: its insertion differences are bounded by twice its supremum norm. Its unscaled boundary error has a constant finite for each fixed β \beta β
To recover the actual configuration, condition on the projected configuration σ \sigma σ Ω i b a d = ⨆ n > K i C i n / S n \Omega_i^{\mathrm{bad}}=\bigsqcup_{n>K_i}\mathcal{C}_i^n/\mathfrak{S}_n Ω i bad = ⨆ n > K i C i n / S n
d ν i b a d ( ω i ) = ∑ n > K i z i n n ! d ν i ⊗ n ( ω i ) , ν i b a d ( Ω i b a d ) ≤ q β . (68) d\nu_i^{\mathrm{bad}}(\omega_i)=\sum_{n>K_i}\frac{z_i^n}{n!}\,d\nu_i^{\otimes n}(\omega_i),\qquad\nu_i^{\mathrm{bad}}(\Omega_i^{\mathrm{bad}})\le q_\beta. \tag*{(68)} d ν i bad ( ω i ) = n > K i ∑ n ! z i n d ν i ⊗ n ( ω i ) , ν i bad ( Ω i bad ) ≤ q β . (68) The hard constraints are imposed separately and only reduce this measure. Under a weak interaction B B B κ \kappa κ 2 κ 2\kappa 2 κ q β e 2 κ q_\beta e^{2\kappa} q β e 2 κ
The following is Dobrushin’s comparison argument [27 Theorem 3] applied to the dilute conditional system.
Lemma 7.7 (Fiber comparison). The conditional bad-cell system has a unique infinite-volume law. If c i j c_{ij} c ij
c i j ≤ C q β [ 1 { i ∼ G j } + ∑ A ∋ i , j ∥ B A ∥ ∞ ] , sup i ∑ j e m d ( i , j ) c i j < 1 2 . (69) c_{ij}\le Cq_\beta\left[\mathbf{1}_{\{i\sim_G j\}}+\sum_{A\ni i,j}\lVert B_A\rVert_\infty\right],\qquad\sup_i\sum_j e^{m d(i,j)}c_{ij}<\frac{1}{2}. \tag*{(69)} c ij ≤ C q β 1 { i ∼ G j } + A ∋ i , j ∑ ∥ B A ∥ ∞ , i sup j ∑ e m d ( i , j ) c ij < 2 1 . (69) For bounded F F F U U U r r r U U U C ∣ U ∣ ∥ F ∥ ∞ e − m r C|U|\lVert F\rVert_\infty e^{-mr} C ∣ U ∣ ∥ F ∥ ∞ e − m r
Proof. Changing a hard neighbor changes only the nonvacuum part of the one-cell conditional law, of mass O ( q β ) O(q^\beta) O ( q β ) i i i j j j 4 ∥ B A ∥ ∞ 4\lVert B_A\rVert_\infty 4 ∥ B A ∥ ∞ C q β Cq^\beta C q β (50) give the second for large β \beta β
For the comparison statement, couple single-cell heat-bath updates by maximal coupling. If two specifications differ by conditional discrepancies b i b_i b i d i ≤ b i + ∑ j c i j d j d_i \le b_i + \sum_j c_{ij}d_j d i ≤ b i + ∑ j c ij d j [ 0 , 1 ] [0,1] [ 0 , 1 ] d ≤ ( I − C ) − 1 b d \le(I-C)^{-1}b d ≤ ( I − C ) − 1 b (69) makes the resolvent converge and gives exponential decay of its entries. A boundary change, or a change of retained rods outside a large ball, supplies discrepancies only near that change, together with the summable weak tails. Thus local marginals are uniformly Cauchy in total variation as boundaries recede. Their consistent limits give the asserted law and quasilocality.
Lemma 7.8 (The bare reference law). For every field with ∣ ϕ i ∣ ≤ R |\phi_i| \le R ∣ ϕ i ∣ ≤ R ν ϕ \nu_\phi ν ϕ β \beta β
Proof. The deletion estimate gives convergence in total variation of the projected local marginals. Approximate a conditional fiber expectation by retaining its arguments in a finite neighborhood of the observable. Lemma 7.7 bounds the approximation error exponentially; projected convergence then constructs the actual local limits. Consistency gives a law on configurations. Local finite-range conditional distributions pass to the limit in total variation, so this law satisfies the DLR equations. Conditioning any Gibbs law on larger boxes and using the uniform boundary estimate proves uniqueness.
Covariance for actual rods
Proposition 7.9. Let B B B (63) , including any coefficientwise partial sum or interpolation of such an interaction. For bounded actual-rod observables F U F_U F U G V G_V G V U , V U,V U , V
∣ Cov f u l l , B ( F U , G V ) ∣ ≤ C β ∥ F U ∥ ∞ ∥ G V ∥ ∞ ∑ u ∈ U , v ∈ V e − m d ( u , v ) , (70) \left|\operatorname{Cov}_{\mathrm{full},B}(F_U,G_V)\right| \le C\beta\lVert F_U\rVert_\infty\lVert G_V\rVert_\infty\sum_{u\in U,v\in V} e^{-m d(u,v)}, \tag*{(70)} ∣ Cov full , B ( F U , G V ) ∣ ≤ C β ∥ F U ∥ ∞ ∥ G V ∥ ∞ u ∈ U , v ∈ V ∑ e − m d ( u , v ) , (70) ∣ Cov f u l l , B ( F U , N ^ j ) ∣ ≤ C β ∥ F U ∥ ∞ ∑ u ∈ U e − m d ( u , j ) . (71) \left|\operatorname{Cov}_{\mathrm{full},B}(F_U,\widehat{N}_j)\right| \le C\beta\lVert F_U\rVert_\infty\sum_{u\in U} e^{-m d(u,j)}. \tag*{(71)} Cov full , B ( F U , N j ) ≤ C β ∥ F U ∥ ∞ u ∈ U ∑ e − m d ( u , j ) . (71) The estimates hold in finite volumes uniformly in frozen admissible exteriors and accessibility masks.
Proof. Let K ( s , t ) K(s,t) K ( s , t ) s F U + t G V sF_U+tG_V s F U + t G V
Cov f u l l , B ( F U , G V ) = Cov c a p , K ( K s , K t ) − E c a p , K K s t . (72) \operatorname{Cov}_{\mathrm{full},B}(F_U,G_V)=\operatorname{Cov}_{\mathrm{cap},K}(K_s,K_t)-\mathbb{E}_{\mathrm{cap},K}K_{st}. \tag*{(72)} Cov full , B ( F U , G V ) = Cov cap , K ( K s , K t ) − E cap , K K s t . (72) Conditional differentiation of the positive fiber integral yields
K s = E f i b e r , B F U , K t = E f i b e r , B G V , K s t = − Cov f i b e r , B ( F U , G V ) . K_s=\mathbb{E}_{\mathrm{fiber},B}F_U,\qquad K_t=\mathbb{E}_{\mathrm{fiber},B}G_V,\qquad K_{st}=-\operatorname{Cov}_{\mathrm{fiber},B}(F_U,G_V). K s = E fiber , B F U , K t = E fiber , B G V , K s t = − Cov fiber , B ( F U , G V ) . Perturbing the fiber by t G V tG_V t G V V V V C q β ∣ t ∣ ∥ G V ∥ ∞ Cq_\beta|t|\|G_V\|_\infty C q β ∣ t ∣∥ G V ∥ ∞ Lemma 7.7 by ∣ t ∣ |t| ∣ t ∣ t → 0 t \to0 t → 0
∣ K s t ∣ ≤ C q β ∥ F U ∥ ∞ ∥ G V ∥ ∞ ∑ u ∈ U , v ∈ V e − m d ( u , v ) . (73) |K_{st}| \le Cq_\beta\|F_U\|_\infty\|G_V\|_\infty\sum_{u\in U,v\in V} e^{-m d(u,v)}. \tag*{(73)} ∣ K s t ∣ ≤ C q β ∥ F U ∥ ∞ ∥ G V ∥ ∞ u ∈ U , v ∈ V ∑ e − m d ( u , v ) . (73) An allowed retained insertion in cell x x x F U F_U F U x ∈ U x \in U x ∈ U x x x x x x O ( q β ) O(q_\beta) O ( q β )
∣ D X K s ∣ ≤ C ∥ F U ∥ ∞ [ 1 { x ∈ U } + q β ∑ u ∈ U e − m d ( x , u ) ] . (74) |D_XK_s| \le C\|F_U\|_\infty\left[\mathbf{1}_{\{x\in U\}}+q_\beta\sum_{u\in U}e^{-m d(x,u)}\right]. \tag*{(74)} ∣ D X K s ∣ ≤ C ∥ F U ∥ ∞ [ 1 { x ∈ U } + q β u ∈ U ∑ e − m d ( x , u ) ] . (74) Applying (62) to the two gradients in (74) , and convolving exponential kernels, bounds the first term of (72) by the right side of (70) , with a smaller exponent. Equation (73) bounds its second term. If G = N ^ j G=\widehat{N}_j G = N j K t = N j ( σ ) K_t=N_j(\sigma) K t = N j ( σ ) K s t = 0 K_{st}=0 K s t = 0 j j j (71) .
For an infinite weak background B B B B ( R ) = ∑ A ⊂ B R ( 0 ) B A B^{(R)}=\sum_{A\subset B_R(0)}B_A B ( R ) = ∑ A ⊂ B R ( 0 ) B A ν ϕ \nu_\phi ν ϕ e − B ( R ) e^{-B^{(R)}} e − B ( R ) S > R S>R S > R (70) , gives for local F U F_U F U
∣ E ν ϕ B ( S ) F U − E ν ϕ B ( R ) F U ∣ ≤ C β , U κ ∥ F U ∥ ∞ e − c dist ( U , B R ( 0 ) c ) . (75) \left|\mathbb{E}_{\nu_\phi^{B^{(S)}}}F_U-\mathbb{E}_{\nu_\phi^{B^{(R)}}}F_U\right|
\le C_{\beta,U}\kappa\|F_U\|_\infty e^{-c\operatorname{dist}(U,B_R(0)^c)}. \tag*{(75)} E ν ϕ B ( S ) F U − E ν ϕ B ( R ) F U ≤ C β , U κ ∥ F U ∥ ∞ e − c dist ( U , B R ( 0 ) c ) . (75) Taking the supremum over ∥ F U ∥ ∞ ≤ 1 \|F_U\|_\infty\le1 ∥ F U ∥ ∞ ≤ 1 ν ϕ B \nu_\phi^B ν ϕ B (70) –(71) .
Weak free energies at a frozen boundary
Lemma 7.10 (Deletion response for actual observables). Let F U F_U F U U U U N B N_B N B B B B e − t e^{-t} e − t t ≥ 0 t\ge0 t ≥ 0 0 ≤ s ≤ 1 0\le s\le1 0 ≤ s ≤ 1 s J D sJ_D s J D
∣ Cov f u l l , t , s J D ( F U , N B ) ∣ ≤ C β ( 1 + β q β ) e − t / 2 ∥ F U ∥ ∞ ∑ x ∈ U , y ∈ B e − m d ( x , y ) . (76) |\operatorname{Cov}_{\mathrm{full},t,sJ_D}(F_U,N_B)|
\le C\beta(1+\beta q_\beta)e^{-t/2}\|F_U\|_\infty\sum_{x\in U,y\in B}e^{-m d(x,y)}. \tag*{(76)} ∣ Cov full , t , s J D ( F U , N B ) ∣ ≤ C β ( 1 + β q β ) e − t /2 ∥ F U ∥ ∞ x ∈ U , y ∈ B ∑ e − m d ( x , y ) . (76) Proof. Use a scalar source u F U uF_U u F U S = ∂ u K t ( 0 ) = E f i b e r F U S=\partial_uK_t(0)=\mathbb{E}_{\mathrm{fiber}}F_U S = ∂ u K t ( 0 ) = E fiber F U T B T_B T B B B B N B b a d N_B^{\mathrm{bad}} N B bad
Cov f u l l ( F U , N B ) = Cov c a p ( S , T B + ∂ t K t ) + E c a p Cov f i b e r ( F U , N B b a d ) . (77) \operatorname{Cov}_{\mathrm{full}}(F_U,N_B)
=\operatorname{Cov}_{\mathrm{cap}}(S,T_B+\partial_tK_t)
+\mathbb{E}_{\mathrm{cap}}\operatorname{Cov}_{\mathrm{fiber}}(F_U,N_B^{\mathrm{bad}}). \tag*{(77)} Cov full ( F U , N B ) = Cov cap ( S , T B + ∂ t K t ) + E cap Cov fiber ( F U , N B bad ) . (77) The insertion norm of T B T_B T B e − t / 2 e^{-t/2} e − t /2 ∂ t K t \partial_tK_t ∂ t K t (49) supplies one count factor C β C\beta C β (66) with the same anchors in B B B (62) and (74) bound the first term of (77) by the right side of (76) . For the fiber term, the derivative of a deleted cell’s conditional law has total mass at most C β q β e − t / 2 C\beta q_\beta e^{-t/2} C β q β e − t /2 C β q β C\beta q_\beta C β q β
Lemma 7.11 (Response of a logarithmic tilt). Let μ \mu μ A , B A,B A , B A A A μ t , s \mu_{t,s} μ t , s μ \mu μ e − t B − s A e^{-tB-sA} e − tB − s A E μ e − ( t ± η ) B < ∞ \mathbb{E}_{\mu}e^{-(t\pm\eta)B}<\infty E μ e − ( t ± η ) B < ∞ η > 0 \eta>0 η > 0
d d t [ − log E μ t , 0 e − A ] = − ∫ 0 1 Cov μ t , s ( A , B ) d s (78) \frac{d}{dt}\left[-\log\mathbb{E}_{\mu_{t,0}}e^{-A}\right]=-\int_{0}^{1}\operatorname{Cov}_{\mu_{t,s}}(A,B)\,\mathrm{d}s \tag*{(78)} d t d [ − log E μ t , 0 e − A ] = − ∫ 0 1 Cov μ t , s ( A , B ) d s (78) Proof. Differentiating the numerator and denominator gives E μ t , 1 B − E μ t , 0 B \mathbb{E}_{\mu_{t,1}}B-\mathbb{E}_{\mu_{t,0}}B E μ t , 1 B − E μ t , 0 B ∂ s E μ t , s B = − Cov μ t , s ( A , B ) \partial_s\mathbb{E}_{\mu_{t,s}}B=-\operatorname{Cov}_{\mu_{t,s}}(A,B) ∂ s E μ t , s B = − Cov μ t , s ( A , B ) s s s
Proposition 7.12. Let J D = ∑ A ⊂ D J A J_D=\sum_{A\subset D}J_A J D = ∑ A ⊂ D J A μ D ξ \mu_D^\xi μ D ξ ξ \xi ξ D D D ν ϕ \nu_\phi ν ϕ
∣ − log E μ D ξ e − J D + log E ν ϕ e − J D ∣ ≤ C β ( 1 + β q β ) κ A ( D ) (79) \left|-\log\mathbb{E}_{\mu_D^\xi}e^{-J_D}+\log\mathbb{E}_{\nu_\phi}e^{-J_D}\right|\le C\beta(1+\beta q_\beta)\kappa A(D) \tag*{(79)} − log E μ D ξ e − J D + log E ν ϕ e − J D ≤ C β ( 1 + β q β ) κ A ( D ) (79) where A ( D ) A(D) A ( D )
Proof. Choose a strip B ⊂ D B\subset D B ⊂ D ∣ B ∣ ≤ C A ( D ) |B|\le CA(D) ∣ B ∣ ≤ C A ( D ) B B B e − t e^{-t} e − t h h h J D J_D J D V ξ ( t ) = − log E μ D , t ξ e − J D V_\xi(t)=-\log\mathbb{E}_{\mu_{D,t}^\xi}e^{-J_D} V ξ ( t ) = − log E μ D , t ξ e − J D A = J D A=J_D A = J D B = N B B=N_B B = N B (76) over the terms of J D J_D J D ∣ V ξ ′ ( t ) ∣ ≤ C β ( 1 + β q β ) κ e − t / 2 ∣ B ∣ |V_\xi'(t)|\le C\beta(1+\beta q_\beta)\kappa e^{-t/2}|B| ∣ V ξ ′ ( t ) ∣ ≤ C β ( 1 + β q β ) κ e − t /2 ∣ B ∣ t = ∞ t=\infty t = ∞ ξ \xi ξ J D J_D J D D D D V ξ ( ∞ ) V_\xi(\infty) V ξ ( ∞ ) ξ \xi ξ V ξ ( 0 ) V_\xi(0) V ξ ( 0 ) (79) . Finally the bare DLR identity gives
E ν ϕ e − J D = E ν ϕ ∣ D c E μ D ξ e − J D . \mathbb{E}_{\nu_\phi}e^{-J_D}=\mathbb{E}_{\nu_\phi\mid D^c}\mathbb{E}_{\mu_D^\xi}e^{-J_D}. E ν ϕ e − J D = E ν ϕ ∣ D c E μ D ξ e − J D . The negative logarithm of this average lies between the extreme values of V ξ ( 0 ) V_\xi(0) V ξ ( 0 )
A common field interaction for all regional weak gases
Let J J J
J D = ∑ U ⊂ D J U . (80) J_D=\sum_{U\subset D}J_U. \tag*{(80)} J D = U ⊂ D ∑ J U . (80) If J J J D D D (80) .
Choose a translation-covariant lexicographic order and anchor each support at its least site. Define
J i = ∑ anchor ( U ) = i J U , J < i = ∑ anchor ( U ) < i J U , v i ( ϕ ) = − log E ν ϕ e − J i . (81) J_i=\sum_{\operatorname{anchor}(U)=i}J_U,\qquad J_{<i}=\sum_{\operatorname{anchor}(U)<i}J_U,\qquad v_i(\phi)=-\log\mathbb{E}_{\nu_\phi}e^{-J_i}. \tag*{(81)} J i = anchor ( U ) = i ∑ J U , J < i = anchor ( U ) < i ∑ J U , v i ( ϕ ) = − log E ν ϕ e − J i . (81) The state for the interaction J < i J_{<i} J < i ∥ J i ∥ ∞ ≤ κ \lVert J_i\rVert_\infty\le\kappa ∥ J i ∥ ∞ ≤ κ ∣ v i ∣ ≤ κ |v_i|\le\kappa ∣ v i ∣ ≤ κ D D D J i , D J_{i,D} J i , D J < i , D J_{<i,D} J < i , D U ⊂ D U\subset D U ⊂ D v i , D v_{i,D} v i , D
V D ( ϕ ) : = − log E ν ϕ e − J D = ∑ i ∈ D v i , D ( ϕ ) . (82) V_D(\phi):=-\log\mathbb{E}_{\nu_\phi}e^{-J_D}=\sum_{i\in D}v_{i,D}(\phi). \tag*{(82)} V D ( ϕ ) := − log E ν ϕ e − J D = i ∈ D ∑ v i , D ( ϕ ) . (82) Put t β = C β κ t_\beta=C\beta\kappa t β = C β κ C C C
Lemma 7.13 (Comparison of regional increments). For d i = dist ( i , D c ) d_i=\operatorname{dist}(i,D^c) d i = dist ( i , D c )
∣ v i , D − v i ∣ ≤ C ( κ + β κ 2 ) e − c d i ≤ t β e − c d i . (83) |v_{i,D}-v_i|\le C(\kappa+\beta\kappa^2)e^{-cd_i}\le t_\beta e^{-cd_i}. \tag*{(83)} ∣ v i , D − v i ∣ ≤ C ( κ + β κ 2 ) e − c d i ≤ t β e − c d i . (83) Proof. For background changes, use Lemma 7.11 with μ = ν ϕ B 0 \mu=\nu_\phi^{B_0} μ = ν ϕ B 0 B = Δ B B=\Delta B B = Δ B (70) . If d i = dist ( i , D c ) d_i=\operatorname{dist}(i,D^c) d i = dist ( i , D c )
∥ J i − J i , D ∥ ∞ ≤ C κ e − c d i , ∑ U ∋ x , U ⊄ D e b ∣ U ∣ ∥ J U ∥ ∞ ≤ C κ e − c dist ( x , D c ) ( x ∈ D ) . \|J_i-J_{i,D}\|_\infty\le C\kappa e^{-cd_i},\qquad
\sum_{\substack{U\ni x,\,U\not\subset D}}e^{b|U|}\|J_U\|_\infty\le C\kappa e^{-c\operatorname{dist}(x,D^c)}\quad(x\in D). ∥ J i − J i , D ∥ ∞ ≤ C κ e − c d i , U ∋ x , U ⊂ D ∑ e b ∣ U ∣ ∥ J U ∥ ∞ ≤ C κ e − c dist ( x , D c ) ( x ∈ D ) . with bound C κ C\kappa C κ D D D J i J_i J i C κ e − c d ( x , i ) C\kappa e^{-c d(x,i)} C κ e − c d ( x , i ) (78) and convolving the exponential profiles yields (83) . For every finite cubical union, ∑ i ∈ D e − c d i ≤ C A ( D ) \sum_{i\in D}e^{-cd_i}\le CA(D) ∑ i ∈ D e − c d i ≤ C A ( D ) (82) and (83) therefore show
∣ V D − ∑ i ∈ D v i ∣ ≤ C t β A ( D ) . (84) \left|V_D-\sum_{i\in D}v_i\right|\le Ct_\beta A(D). \tag*{(84)} V D − i ∈ D ∑ v i ≤ C t β A ( D ) . (84) Lemma 7.14 (Local dependence of the increments). If ϕ \phi ϕ ϕ ′ \phi' ϕ ′ B r ( i ) B_r(i) B r ( i )
∣ v i ( ϕ ) − v i ( ϕ ′ ) ∣ ≤ C t β e − c r . (85) |v_i(\phi)-v_i(\phi')|\le Ct_\beta e^{-cr}. \tag*{(85)} ∣ v i ( ϕ ) − v i ( ϕ ′ ) ∣ ≤ C t β e − cr . (85) Proof. First hold the weak functions fixed. Changing ϕ j \phi_j ϕ j t t t e t N ^ j e^{t\widehat N_j} e t N j A = J i A=J_i A = J i B = − N ^ j B=-\widehat N_j B = − N j
∂ ϕ j v i = ∫ 0 1 Cov ϕ J i + t J i ( J i , N ^ j ) d t . \partial_{\phi_j}v_i=\int_0^1\operatorname{Cov}_{\phi}^{J_i+tJ_i}(J_i,\widehat N_j)\,\mathrm{d}t. ∂ ϕ j v i = ∫ 0 1 Cov ϕ J i + t J i ( J i , N j ) d t . The mixed covariance estimate bounds the sum over changed fields by C t β e − c r Ct_\beta e^{-cr} C t β e − cr ϕ ′ \phi' ϕ ′ B r ( i ) B_r(i) B r ( i ) J i J_i J i C κ e − c r C\kappa e^{-cr} C κ e − cr ∣ log E e − A − log E e − A ′ ∣ ≤ ∥ A − A ′ ∥ ∞ |\log\mathbb{E}e^{-A}-\log\mathbb{E}e^{-A'}|\le\|A-A'\|_\infty ∣ log E e − A − log E e − A ′ ∣ ≤ ∥ A − A ′ ∥ ∞ (78) with the same profiles as above. This proves (85) .
We now convert the local increments into an interaction by shell telescoping, following Kozlov’s reconstruction method [28 Sections 2 and 4] .
Proposition 7.15 (Regional interaction). There are bounded measurable field potentials Ψ i , r J \Psi^J_{i,r} Ψ i , r J r r r B r ( i ) B_r(i) B r ( i )
sup x ∑ i , r : x ∈ B r ( i ) e c ′ r ( 1 + r ) p ∥ Ψ i , r J ∥ ∞ ≤ C p , c ′ t β , t β = C β κ , (86) \sup_x\sum_{i,r:x\in B_r(i)}e^{c'r}(1+r)^p\|\Psi^J_{i,r}\|_\infty\le C_{p,c'}t_\beta,\qquad t_\beta=C\beta\kappa, \tag*{(86)} x sup i , r : x ∈ B r ( i ) ∑ e c ′ r ( 1 + r ) p ∥ Ψ i , r J ∥ ∞ ≤ C p , c ′ t β , t β = C β κ , (86) for fixed p p p c ′ > 0 c'>0 c ′ > 0 D D D
∣ V D ( ϕ D r ϕ D c ) − H D J ( ϕ D ∣ r ) ∣ ≤ C t β A ( D ) . (87) \left|V_D(\phi_{D^r} \phi_{D^c})-H_D^J(\phi_D\mid r)\right|\le Ct_\beta A(D). \tag*{(87)} V D ( ϕ D r ϕ D c ) − H D J ( ϕ D ∣ r ) ≤ C t β A ( D ) . (87) Proof. Fix a reference field r ∗ r_\ast r ∗
v i [ r ] ( ϕ ) = v i ( ϕ B r ( i ) r ∗ , B r ( i ) c ) , Ψ i , 0 J = v i [ 0 ] , Ψ i , r J = v i [ r ] − v i [ r − 1 ] ( r ≥ 1 ) . v_i^{[r]}(\phi)=v_i(\phi_{B_r(i)^{r_\ast},B_r(i)^c}),\qquad\Psi^J_{i,0}=v_i^{[0]},\qquad\Psi^J_{i,r}=v_i^{[r]}-v_i^{[r-1]}\quad(r\ge1). v i [ r ] ( ϕ ) = v i ( ϕ B r ( i ) r ∗ , B r ( i ) c ) , Ψ i , 0 J = v i [ 0 ] , Ψ i , r J = v i [ r ] − v i [ r − 1 ] ( r ≥ 1 ) . Their exact sum is v i v_i v i (85) bounds the r r r C t β e − c r Ct_\beta e^{-cr} C t β e − cr (86) . In an ordinary conditional sum, every term anchored in D D D D D D D D D D D D C t β A ( D ) Ct_\beta A(D) C t β A ( D ) (84) to obtain (87) .
The weak potentials above are contributions to the exponent. If the field action is written β H \beta H β H Ψ J / β \Psi^J/\beta Ψ J / β
The effective field interaction
The auxiliary field lives on the fine spatial lattice. A site carries the angular-cell vector ϕ i ∈ K = [ − R , R ] n ω \phi_i\in K=[-R,R]^{n_\omega} ϕ i ∈ K = [ − R , R ] n ω λ \lambda λ n ω n_\omega n ω v s v_s v s v a v_a v a ∑ a v a = v s \sum_a v_a=v_s ∑ a v a = v s β = ε − 1 \beta=\varepsilon^{-1} β = ε − 1 Φ 0 \Phi^0 Φ 0 s i , a = v a ϕ i , a s_{i,a}=\sqrt{v_a}\phi_{i,a} s i , a = v a ϕ i , a
From finite boxes to an exact field interaction
We follow LMP in separating a leading variational functional from a local remainder [2 Sections 3–5] . Let R β , D h ( ϕ ∣ ξ ) \mathcal{R}^h_{\beta,D}(\phi\mid\xi) R β , D h ( ϕ ∣ ξ )
R β , D h ( ϕ ∣ ξ ) = ∫ w β , h , D ( η ∣ ξ ) exp { ∑ i ∈ D , a ϕ i , a N ^ i , a ( η ) } d η , N ^ i , a = N i , a 1 { N i , a ≤ K a } , (88) \mathcal{R}^h_{\beta,D}(\phi\mid\xi)=\int w_{\beta,h,D}(\eta\mid\xi)\exp\left\{\sum_{i\in D,a}\phi_{i,a}\widehat{N}_{i,a}(\eta)\right\}\,d\eta,\qquad\widehat{N}_{i,a}=N_{i,a}\mathbf{1}_{\{N_{i,a}\le K_a\}}, \tag*{(88)} R β , D h ( ϕ ∣ ξ ) = ∫ w β , h , D ( η ∣ ξ ) exp ⎩ ⎨ ⎧ i ∈ D , a ∑ ϕ i , a N i , a ( η ) ⎭ ⎬ ⎫ d η , N i , a = N i , a 1 { N i , a ≤ K a } , (88) where w w w ϕ \phi ϕ log R \log\mathcal{R} log R K + = [ − R − 1 , R + 1 ] n ω K^+=[-R-1,R+1]^{n_\omega} K + = [ − R − 1 , R + 1 ] n ω K K K
The finite-box variational limit, the microscopic locality estimate, and the response estimate for the minimizing profile give the following facts. First, in each fixed free box,
P β , D h ( ϕ ) : = β − 1 log R β , D h ( ϕ ) ⟶ P 0 , D ( ϕ ) , (89) P^h_{\beta,D}(\phi):=\beta^{-1}\log\mathcal{R}^h_{\beta,D}(\phi)\longrightarrow P_{0,D}(\phi), \tag*{(89)} P β , D h ( ϕ ) := β − 1 log R β , D h ( ϕ ) ⟶ P 0 , D ( ϕ ) , (89) P 0 , D ( ϕ ) = − min q { ∫ D q ( log ( q / ζ ) − 1 ) + 1 2 ⟨ q , T D q ⟩ + α ∥ P D q ∥ 2 2 − ∫ D ϕ q } , P_{0,D}(\phi)=-\min_q\left\{\int_D q(\log(q/\zeta)-1)+\frac{1}{2}\langle q,T_Dq\rangle+\alpha\lVert P_Dq\rVert_2^2-\int_D\phi q\right\}, P 0 , D ( ϕ ) = − q min { ∫ D q ( log ( q / ζ ) − 1 ) + 2 1 ⟨ q , T D q ⟩ + α ∥ P D q ∥ 2 2 − ∫ D ϕq } , with the marked-cell mass caps imposed on q q q ϕ \phi ϕ
∣ β − 1 E D , ϕ , ξ N ^ i , a − β − 1 E ϕ N ^ i , a ∣ ≤ C e − m 0 d ( i , D c ) , (90) \left|\beta^{-1}E_{D,\phi,\xi}\widehat{N}_{i,a}-\beta^{-1}E_{\phi}\widehat{N}_{i,a}\right|\le Ce^{-m_0d(i,D^c)}, \tag*{(90)} β − 1 E D , ϕ , ξ N i , a − β − 1 E ϕ N i , a ≤ C e − m 0 d ( i , D c ) , (90) β − 1 ∣ Cov ϕ ( N ^ i , a , N ^ j , b ) ∣ ≤ C e − m 0 d ( i , j ) . \beta^{-1}\left|\operatorname{Cov}_{\phi}(\widehat{N}_{i,a},\widehat{N}_{j,b})\right|\le Ce^{-m_0d(i,j)}. β − 1 Cov ϕ ( N i , a , N j , b ) ≤ C e − m 0 d ( i , j ) . These bounds are uniform in the original admissible exterior rods. Third, the density minimizers in free boxes converge exponentially locally to the unique bounded all-space minimizer q ϕ q_\phi q ϕ (90) use the unique whole-space bare reference of Lemma 7.8.
Lemma 8.1 (Uniform force convergence). Put
g β , h ; i , a ( ϕ ) = v a 2 α ( ϕ i , a − h ) − β − 1 E ϕ N ^ i , a , g_{\beta,h;i,a}(\phi)=\frac{v_a}{2\alpha}(\phi_{i,a}-h)-\beta^{-1}\mathbb{E}_{\phi}\widehat{N}_{i,a}, g β , h ; i , a ( ϕ ) = 2 α v a ( ϕ i , a − h ) − β − 1 E ϕ N i , a , g 0 ; i , a ( ϕ ) = v a 2 α ϕ i , a − ∫ C i , a q ϕ . g_{0;i,a}(\phi)=\frac{v_a}{2\alpha}\phi_{i,a}-\int_{C_{i,a}}q\phi. g 0 ; i , a ( ϕ ) = 2 α v a ϕ i , a − ∫ C i , a q ϕ . Uniformly over all fields in K Z 3 K^{\mathbb{Z}^3} K Z 3
d β : = sup ϕ , h , i , a ∣ g β , h ; i , a ( ϕ ) − g 0 ; i , a ( ϕ ) + h v a 2 α ∣ ⟶ 0. (91) d_\beta:=\sup_{\phi,h,i,a}\left|g_{\beta,h;i,a}(\phi)-g_{0;i,a}(\phi)+\frac{h v_a}{2\alpha}\right|\longrightarrow0. \tag*{(91)} d β := ϕ , h , i , a sup g β , h ; i , a ( ϕ ) − g 0 ; i , a ( ϕ ) + 2 α h v a ⟶ 0. (91) Proof. In a fixed box, we use the convergence theorem for gradients of convex functions [16 Theorem 25.7] , with the following uniform estimate. The capped counts make the functions in (89) equi-Lipschitz on K + K^+ K + η β , D \eta_{\beta,D} η β , D ω D \omega_D ω D 0 < t < 1 0<t<1 0 < t < 1 ϕ ∈ K \phi\in K ϕ ∈ K
∣ ∂ i , a P β , D h ( ϕ ) − ∂ i , a P 0 , D ( ϕ ) ∣ ≤ 2 η β , D / t + ω D ( t ) . \left|\partial_{i,a}P_{\beta,D}^{h}(\phi)-\partial_{i,a}P_{0,D}(\phi)\right|\leq2\eta_{\beta,D}/t+\omega_D(t). ∂ i , a P β , D h ( ϕ ) − ∂ i , a P 0 , D ( ϕ ) ≤ 2 η β , D / t + ω D ( t ) . The two derivative bounds are the forward and backward difference quotients. Let β → ∞ \beta\to\infty β → ∞ t ↓ 0 t\downarrow0 t ↓ 0 ℓ \ell ℓ 2 C e − m ℓ 2Ce^{-m\ell} 2 C e − m ℓ ℓ \ell ℓ β \beta β (91) . □
Lemma 8.2 (Locality of the force error). The error force E β , h E_{\beta,h} E β , h in this formula is quasilocal: if ϕ = ψ \phi=\psi ϕ = ψ on B r ( i ) B_r(i) B r ( i ) then
∣ E β , h ; i ( ϕ ) − E β , h ; i ( ψ ) ∣ ≤ C e − m r . (92) \left|E_{\beta,h;i}(\phi)-E_{\beta,h;i}(\psi)\right|\leq Ce^{-mr}. \tag*{(92)} ∣ E β , h ; i ( ϕ ) − E β , h ; i ( ψ ) ∣ ≤ C e − m r . (92) Proof. Integrating the covariance bounds and variational response bounds along successive field coordinates proves (92) for finitely supported changes. Summability and exhaustion extend the bound to arbitrary exterior changes. □
For finite-volume fields the force is the gradient of minus β − 1 \beta^{-1} β − 1 E β , h E_{\beta,h} E β , h
C E ( ϕ ; ψ ) = ∫ 0 1 ∑ i ∈ A E i ( ψ + t ( ϕ − ψ ) ) ⋅ ( ϕ i − ψ i ) d t , A = { i : ϕ i ≠ ψ i } , (93) C_E(\phi;\psi)=\int_0^1\sum_{i\in A}E_i\bigl(\psi+t(\phi-\psi)\bigr)\cdot(\phi_i-\psi_i)\,\mathrm{d}t,\qquad A=\{i:\phi_i\ne\psi_i\}, \tag*{(93)} C E ( ϕ ; ψ ) = ∫ 0 1 i ∈ A ∑ E i ( ψ + t ( ϕ − ψ ) ) ⋅ ( ϕ i − ψ i ) d t , A = { i : ϕ i = ψ i } , (93) whenever A A A
Reconstructing potentials from quasilocal conditional energies goes back to Sullivan [29 ] and Kozlov [28 ] . The construction below adapts Kozlov’s lexicographic regrouping [28 Section 4] ; exponential locality gives the weighted norm.
Proposition 8.3 (Small interaction representing the error). Fix a homogeneous well field r ∗ r_* r ∗ There is a translation-covariant interaction Δ β , h \Delta_{\beta,h} Δ β , h on complete lattice balls whose cocycle is (93) and for which, whenever a < m a<m a < m and p p p is fixed,
∥ Δ β , h ∥ a , p : = sup i , r : x ∈ B r ( i ) e a r ( 1 + r ) p ∥ Δ i , r ∥ ∞ ⟶ 0. (94) \|\Delta_{\beta,h}\|_{a,p}:=\sup_{i,r:\,x\in B_r(i)}e^{ar}(1+r)^p\|\Delta_{i,r}\|_\infty\longrightarrow0. \tag*{(94)} ∥ Δ β , h ∥ a , p := i , r : x ∈ B r ( i ) sup e a r ( 1 + r ) p ∥ Δ i , r ∥ ∞ ⟶ 0. (94) Every term vanishes on r ∗ . r_*. r ∗ .
Proof. Order the lattice lexicographically and set
U i ( ϕ ) = ∫ 0 1 E i ( ϕ j ≺ i , r ∗ + t ( ϕ i − r ∗ ) , r ∗ , j ≻ i ) ⋅ ( ϕ i − r ∗ ) d t . U_i(\phi)=\int_0^1 E_i(\phi_{j\prec i},r_*+t(\phi_i-r_*),r_{*,j\succ i})\cdot(\phi_i-r_*)\,\mathrm{d}t. U i ( ϕ ) = ∫ 0 1 E i ( ϕ j ≺ i , r ∗ + t ( ϕ i − r ∗ ) , r ∗ , j ≻ i ) ⋅ ( ϕ i − r ∗ ) d t . This is well defined on the infinite past by quasilocality. Let U i ( r ) U_i^{(r)} U i ( r ) B r ( i ) B_r(i) B r ( i ) r ∗ r_* r ∗ Δ i , 0 = U i ( 0 ) \Delta_{i,0}=U_i^{(0)} Δ i , 0 = U i ( 0 ) Δ i , r = U i ( r ) − U i ( r − 1 ) \Delta_{i,r}=U_i^{(r)}-U_i^{(r-1)} Δ i , r = U i ( r ) − U i ( r − 1 ) r ≥ 1 r\ge1 r ≥ 1
∥ Δ i , r ∥ ∞ ≤ C min { d β , e − m r } . \lVert\Delta_{i,r}\rVert_\infty\le C\min\{d_\beta,e^{-mr}\}. ∥ Δ i , r ∥ ∞ ≤ C min { d β , e − m r } . Finite lexicographic telescoping proves the cocycle identity for fields equal to r ∗ r_* r ∗
C E ( ϕ ; ψ ) = ∑ i , r [ Δ i , r ( ϕ ) − Δ i , r ( ψ ) ] C_E(\phi;\psi)=\sum_{i,r}\left[\Delta_{i,r}(\phi)-\Delta_{i,r}(\psi)\right] C E ( ϕ ; ψ ) = i , r ∑ [ Δ i , r ( ϕ ) − Δ i , r ( ψ ) ] for arbitrary finite changes, since centers far from the changed set have an exponentially summable contribution. Finally,
∥ Δ β , h ∥ a , p ≤ C ∑ r ≥ 0 ( 1 + r ) p + 3 e a r min { d β , e − m r } ⟶ 0. \lVert\Delta_{\beta,h}\rVert_{a,p}\le C\sum_{r\ge0}(1+r)^{p+3}e^{ar}\min\{d_\beta,e^{-mr}\}\longrightarrow0. ∥ Δ β , h ∥ a , p ≤ C r ≥ 0 ∑ ( 1 + r ) p + 3 e a r min { d β , e − m r } ⟶ 0. Dominated convergence applies because a < m a<m a < m m − 1 log ( 1 / d β ) m^{-1}\log(1/d_\beta) m − 1 log ( 1/ d β ) C d β 1 − a / m ( 1 + ∣ log d β ∣ ) p + 4 Cd_\beta^{1-a/m}(1+|\log d_\beta|)^{p+4} C d β 1 − a / m ( 1 + ∣ log d β ∣ ) p + 4
Retain the positive leading interaction Φ 0 \Phi^0 Φ 0
Ψ β , h = Φ 0 + Δ β , h − ∑ i h 2 α ∑ a v a ( ϕ i , a − r ∗ , a ) . (95) \Psi_{\beta,h}=\Phi^0+\Delta_{\beta,h}-\sum_i\frac{h}{2\alpha}\sum_a v_a(\phi_{i,a}-r_{*,a}). \tag*{(95)} Ψ β , h = Φ 0 + Δ β , h − i ∑ 2 α h a ∑ v a ( ϕ i , a − r ∗ , a ) . (95) Its force is exactly g β , h g_{\beta,h} g β , h E β , h = Ψ β , h − Φ 0 E_{\beta,h}=\Psi_{\beta,h}-\Phi^0 E β , h = Ψ β , h − Φ 0
∥ E β , h ∥ a , p ≤ C ∣ h ∣ + δ m i c r o , β , δ m i c r o , β ⟶ 0. (96) \lVert E_{\beta,h}\rVert_{a,p}\le C|h|+\delta_{\mathrm{micro},\beta},\qquad\delta_{\mathrm{micro},\beta}\longrightarrow0. \tag*{(96)} ∥ E β , h ∥ a , p ≤ C ∣ h ∣ + δ micro , β , δ micro , β ⟶ 0. (96) Let F D F_D F D λ \lambda λ D D D A ( D ) A(D) A ( D ) L 2 L^2 L 2
Lemma 8.4 (Pressure at a fixed homogeneous field). For every homogeneous field r ∈ K r\in K r ∈ K
b r = lim D ( v s # D ) − 1 log F D ( r ∣ f r e e ) b_r=\lim_D(v_s\#D)^{-1}\log F_D(r\mid\mathrm{free}) b r = D lim ( v s # D ) − 1 log F D ( r ∣ free ) exists along aligned boxes. Changing the admissible exterior rods changes the logarithm by at most C β A ( D ) C_\beta A(D) C β A ( D ) r r r
Proof. We use the argument with separated boxes for the thermodynamic limit [30 Section 2] . Divide each cell weight by its empty weight and remove the Gaussian prefactor. The normalized empty weight is one. Dropping exclusions, (11) bounds the one-cell sum by 2 exp [ C β v a ] 2\exp[C\beta v_a] 2 exp [ C β v a ] r r r ∣ h ∣ ≤ H |h|\le H ∣ h ∣ ≤ H β \beta β
For a finite region D D D H D Ψ H_D^{\Psi} H D Ψ D D D e ( r ) = v s − 1 ∑ A ∋ 0 Ψ A ( r ) / ∣ A ∣ e(r)=v_s^{-1}\sum_{A\ni0}\Psi_A(r)/|A| e ( r ) = v s − 1 ∑ A ∋ 0 Ψ A ( r ) /∣ A ∣
Lemma 8.5 (Common pressure normalization). With b ⋆ = b r ⋆ b_{\star}=b_{r_{\star}} b ⋆ = b r ⋆
b r + β e ( r ) = b ⋆ = : b 0 . (97) b_r+\beta e(r)=b_{\star}=:b_0. \tag*{(97)} b r + β e ( r ) = b ⋆ =: b 0 . (97) Proof. Integrate the difference between the finite-box and global forces along r ⋆ + t ( r − r ⋆ ) r_{\star}+t(r-r_{\star}) r ⋆ + t ( r − r ⋆ ) D D D (90) sums to C A ( D ) CA(D) C A ( D )
log F D ( r ∣ f r e e ) F D ( r ⋆ ∣ f r e e ) = − β [ H D Ψ ( r D ∣ r ⋆ ) − H D Ψ ( r ⋆ ∣ r ⋆ ) ] + O β ( A ( D ) ) . \log\frac{F_D(r\mid\mathrm{free})}{F_D(r_{\star}\mid\mathrm{free})}
=-\beta\left[H_D^{\Psi}(r_D\mid r_{\star})-H_D^{\Psi}(r_{\star}\mid r_{\star})\right]+O_{\beta}(A(D)). log F D ( r ⋆ ∣ free ) F D ( r ∣ free ) = − β [ H D Ψ ( r D ∣ r ⋆ ) − H D Ψ ( r ⋆ ∣ r ⋆ ) ] + O β ( A ( D )) . The first spatial moment of the interaction is summable, so the energy difference equals ( e ( r ) − e ( r ⋆ ) ) v s # D + O ( A ( D ) ) (e(r)-e(r_{\star}))v_s\#D+O(A(D)) ( e ( r ) − e ( r ⋆ )) v s # D + O ( A ( D )) r ⋆ r_{\star} r ⋆ e ( r ⋆ ) = 0 e(r_{\star})=0 e ( r ⋆ ) = 0
Define
σ D ( r ) = H D Ψ ( r ∣ r ) − e ( r ) v s # D . (98) \sigma_D(r)=H_D^{\Psi}(r\mid r)-e(r)v_s\#D. \tag*{(98)} σ D ( r ) = H D Ψ ( r ∣ r ) − e ( r ) v s # D . (98) We retain σ D ( r ) \sigma_D(r) σ D ( r )
For frozen admissible rods ξ \xi ξ − β − 1 log F D ( ϕ ∣ ξ ) -\beta^{-1}\log F_D(\phi\mid\xi) − β − 1 log F D ( ϕ ∣ ξ )
∣ ∂ i , a B D ξ ( ϕ ) ∣ ≤ C e − m d ( i , D c ) . \left|\partial_{i,a}B_D^{\xi}(\phi)\right|\le Ce^{-m d(i,D^{\mathrm c})}. ∂ i , a B D ξ ( ϕ ) ≤ C e − m d ( i , D c ) . If O O O c L cL c L
sup ϕ ∣ B D ξ ( ϕ ) − B D ξ ( ϕ O ∩ D ∖ O ) ∣ ≤ C e − m ′ L A ( D ) . (99) \sup_{\phi}\left|B_D^{\xi}(\phi)-B_D^{\xi}(\phi_{O\cap D\setminus O})\right|
\le Ce^{-m' L}A(D). \tag*{(99)} ϕ sup B D ξ ( ϕ ) − B D ξ ( ϕ O ∩ D ∖ O ) ≤ C e − m ′ L A ( D ) . (99) The retained function of the outer fields is part of the boundary weight.
Surface estimates for restricted fields
Local interpolation
Write Q ( ϕ ) = q ϕ Q(\phi)=q_{\phi} Q ( ϕ ) = q ϕ ℓ i \ell_i ℓ i (38) . In the original angular field coordinates, set ∣ z ∣ ω 2 = ∑ a ω a z a 2 |z|_{\omega}^{2}=\sum_a\omega_a z_a^2 ∣ z ∣ ω 2 = ∑ a ω a z a 2 ( Π 0 Q i ) a = 2 α ( v s ω a ) − 1 ∫ B i × U a Q i (\Pi_0Q_i)_a=2\alpha(v_s\omega_a)^{-1}\int_{B_i\times U_a}Q_i ( Π 0 Q i ) a = 2 α ( v s ω a ) − 1 ∫ B i × U a Q i ( v s / 4 α ) ∣ ϕ i − Π 0 Q i ∣ ω 2 (v_s/4\alpha)|\phi_i-\Pi_0Q_i|_{\omega}^{2} ( v s /4 α ) ∣ ϕ i − Π 0 Q i ∣ ω 2 (39) and the global comparison (42) ; write r 0 r_0 r 0
Fix a component W q W_q W q M q = Π 0 W q M_q=\Pi_0W_q M q = Π 0 W q f q f_q f q r q = Π 0 f q r_q=\Pi_0f_q r q = Π 0 f q ∣ ⋅ ∣ ω |\cdot|_{\omega} ∣ ⋅ ∣ ω Lemma 6.8 gives measurable nearest wells f i f_i f i ω 0 \omega_0 ω 0
ω ( t ) = C { ω 0 ( t ) + ω 0 ( t ) } . \omega(t)=C\{\omega_0(t)+\sqrt{\omega_0(t)}\}. ω ( t ) = C { ω 0 ( t ) + ω 0 ( t ) } . This bounded concave modulus controls both the squared errors and the linear neighboring-well differences used in randomized rounding below. The results 2.6, 2.7, and 6.9 give finitely many separated projected components and rectifiable paths within each component.
Let E E E δ \delta δ β \beta β L L L E E E E E E L Z 3 L\mathbb{Z}^{3} L Z 3 Proposition 7.15 . The phase restriction allows at most θ L 3 \theta L^{3} θ L 3 M q M_q M q s s s λ \lambda λ c s n ω cs^{n\omega} c s nω s s s
For a finite vector X X X λ X \lambda_X λ X H ( X ∣ W ) = − E D ( L ( X ∣ W ) ∥ λ X ) \mathcal{H}(X \mid W)=-\mathbb{E}\mathcal{D}(\mathcal{L}(X \mid W)\mathbin\Vert\lambda_X) H ( X ∣ W ) = − E D ( L ( X ∣ W ) ∥ λ X ) W W W
Lemma 9.1 (Entropy after replacing coordinates). Let X X X S S S U U U V V V H ( V ∣ S , W ) ≥ − a \mathcal{H}(V \mid S,W)\ge-a H ( V ∣ S , W ) ≥ − a
H ( U , V ∣ W ) ≥ H ( X ∣ W ) − a . \mathcal{H}(U,V \mid W)\ge\mathcal{H}(X \mid W)-a. H ( U , V ∣ W ) ≥ H ( X ∣ W ) − a . Proof. The chain rule and H ( ⋅ ∣ ⋅ ) ≤ 0 \mathcal{H}(\cdot\mid\cdot)\le0 H ( ⋅ ∣ ⋅ ) ≤ 0 H ( U ∣ W ) ≥ H ( X ∣ W ) \mathcal{H}(U\mid W)\ge\mathcal{H}(X\mid W) H ( U ∣ W ) ≥ H ( X ∣ W ) U U U S S S H ( V ∣ U , W ) ≥ H ( V ∣ S , W ) \mathcal{H}(V\mid U,W)\ge\mathcal{H}(V\mid S,W) H ( V ∣ U , W ) ≥ H ( V ∣ S , W ) [31 Proposition 1] .
For m m m S , W S,W S , W λ \lambda λ β − 2 \beta^{-2} β − 2 a = C m log β a=Cm\log\beta a = C m log β β ≥ 2 \beta\ge2 β ≥ 2
The elementary cubical bound used repeatedly is
# { i : d ∞ ( i , ∂ D ) ≤ t } ≤ C ( t + 1 ) A ( D ) , 1 ≤ t ≤ c L . (100) \#\{i:d_{\infty}(i,\partial D)\le t\}\le C(t+1)A(D),\qquad1\le t\le cL. \tag*{(100)} # { i : d ∞ ( i , ∂ D ) ≤ t } ≤ C ( t + 1 ) A ( D ) , 1 ≤ t ≤ c L . (100) For one coarse face the relevant prism, edge, and corner volumes are bounded by C ( t L 2 + t 2 L + t 3 ) ≤ C t L 2 C(tL^{2}+t^{2}L+t^{3})\le CtL^{2} C ( t L 2 + t 2 L + t 3 ) ≤ C t L 2
Lemma 9.2 (Buffer energy). Let A > 0 A>0 A > 0 B B B L L L B B B r 0 r_{0} r 0 J J J # J ≤ C L A \#J\le CLA # J ≤ C L A C A CA C A
E ∑ i ∈ B e i ≤ C A + C L A ( δ + log β β + β − 4 ) . (101) \mathbb{E}\sum_{i\in B}e_i\le CA+CLA\left(\delta+\frac{\log\beta}{\beta}+\beta^{-4}\right). \tag*{(101)} E i ∈ B ∑ e i ≤ C A + C L A ( δ + β log β + β − 4 ) . (101) Consequently a fine moving cut through a distance interval of length c L cL c L
η L = C ( L − 1 + δ + log β β + β − 4 ) . (102) \eta_L=C\left(L^{-1}+\delta+\frac{\log\beta}{\beta}+\beta^{-4}\right). \tag*{(102)} η L = C ( L − 1 + δ + β log β + β − 4 ) . (102) Proof. Replace the fields in J J J β − 2 \beta^{-2} β − 2 r q r_q r q f q f_q f q J J J A A A B B B (42) and drop the other negative terms. The weak change costs at most C δ L A C\delta LA C δ L A C L A log β CLA\log\beta C L A log β (101) . Whole-block replacement preserves the phase restriction. A fixed-width patch belongs to only a bounded number of the c L cL c L (100) give (102) .
Lemma 9.3 (Interpolation through a well component). For source laws satisfying Lemma 9.2 , choose a ν \nu ν net in W q W_q W q of cardinality N ν N_\nu N ν and paths to f q f_q f q of length at most D ν D_\nu D ν A moving interface can be joined to a root plateau through R 1 R_1 R 1 fine layers with expected energy cost at most A κ E A\kappa_E A κ E where κ E ≤ κ \kappa_E \le\kappa κ E ≤ κ and
κ ≤ C [ θ + ν 2 + D ν 2 R 1 + ( 1 + R 1 N ν ν ) ω ( η L ) + L δ + L log β β + L β − 4 ] (103) \kappa\le C\left[\theta+\nu^2+\frac{D_\nu^2}{R_1}+\left(1+\frac{R_1N_\nu}{\nu}\right)\omega(\eta_L)
+L\delta+\frac{L\log\beta}{\beta}+L\beta^{-4}\right] \tag*{(103)} κ ≤ C [ θ + ν 2 + R 1 D ν 2 + ( 1 + ν R 1 N ν ) ω ( η L ) + L δ + β L log β + L β − 4 ] (103) also includes its entropy cost divided by β A \beta A β A The coefficient of θ \theta θ is independent of the subsequent net, paths, and R 1 R_1 R 1
Proof. Fix a separator made of coarse-block faces. Assign the blocks on each side to the source that will be retained there. On either side choose a cut at fine distance between L / 4 L/4 L /4 L / 3 L/3 L /3 R 1 + r 0 < L / 6 R_1+r_0<L/6 R 1 + r 0 < L /6 Lemma 9.2 is imposed on the available bands on both sides. For a single source the other side is already at the root.
First map a nearest well f i f_i f i W q W_q W q f q f_q f q W q W_q W q W W W h i h_i h i
Let the net be v 1 , … , v N ν v_1,\ldots,v_{N_\nu} v 1 , … , v N ν [32 ] ; see also the construction in [33 Lemma 3.1] . Choose u u u [ ν , 2 ν ] [\nu,2\nu] [ ν , 2 ν ] h h h v j v_j v j ∥ h − v j ∥ ≤ u \lVert h-v_j\rVert\le u ∥ h − v j ∥ ≤ u
P u ( n ( h ) ≠ n ( h ′ ) ) ≤ C N ν ν ∥ h − h ′ ∥ . \mathbb{P}_u\bigl(n(h)\ne n(h')\bigr)\le\frac{CN_\nu}{\nu}\lVert h-h'\rVert. P u ( n ( h ) = n ( h ′ ) ) ≤ ν C N ν ∥ h − h ′ ∥ . Indeed, a different first successful index requires u u u N ν ∥ h − h ′ ∥ N_\nu\lVert h-h'\rVert N ν ∥ h − h ′ ∥ [ 0 , 1 ] [0,1] [ 0 , 1 ] D ν D_\nu D ν Q i ′ = γ n ( h i ) ( s i ) Q'_i=\gamma_{n(h_i)}(s_i) Q i ′ = γ n ( h i ) ( s i ) s i s_i s i R 1 R_1 R 1 C D ν / R 1 CD_\nu/R_1 C D ν / R 1 (39) , patch rigidity, and concavity of ω \omega ω
C A [ D ν 2 R 1 + R 1 N ν ν ω ( η L ) ] . CA\left[\frac{D_\nu^2}{R_1}+\frac{R_1N_\nu}{\nu}\omega(\eta_L)\right]. C A [ R 1 D ν 2 + ν R 1 N ν ω ( η L ) ] . The initial interface costs C A [ ν 2 + ω ( η L ) + θ ] CA[\nu^2+\omega(\eta_L)+\theta] C A [ ν 2 + ω ( η L ) + θ ] E i p a t c h E_i^{\mathrm{patch}} E i patch i i i
1 { f i ∉ W q } ≤ 1 { ϕ i ∉ N ( M q ) } + C ω ( E i p a t c h ) . \mathbf{1}_{\{f_i\notin W_q\}}\le\mathbf{1}_{\{\phi_i\notin N(\mathcal{M}_q)\}}+C\omega(E_i^{\mathrm{patch}}). 1 { f i ∈ / W q } ≤ 1 { ϕ i ∈ / N ( M q )} + C ω ( E i patch ) . Only the fixed-width initial interface pays this cost. Each minority site is charged O ( L − 1 ) O(L^{-1}) O ( L − 1 ) O ( L A ) O(LA) O ( L A ) C θ A C\theta A C θ A h i h_i h i W q W_q W q R 1 θ R_1\theta R 1 θ
Conditional on the source fields, sample the fresh spins independently from λ \lambda λ β − 2 \beta^{-2} β − 2 O ( L A ) O(LA) O ( L A ) O ( L A β − 4 ) O(LA\beta^{-4}) O ( L A β − 4 ) − C L A log β -CLA\log\beta − C L A log β O ( δ L A ) O(\delta LA) O ( δ L A ) u u u
One first fixes θ \theta θ ν \nu ν R 1 R_1 R 1 L ≫ R 1 + r 0 L \gg R_1+r_0 L ≫ R 1 + r 0 δ \delta δ β \beta β (103) can be made arbitrarily small.
Rooted restricted partition functions
Let Z D q , r q Z_D^{q,r_q} Z D q , r q D D D r q r_q r q
H D 0 ( ϕ D ∣ r q ) = ∑ i e i ( ϕ D r q , D c ) ≥ 0. H_D^0(\phi_D \mid r_q)=\sum_i e_i(\phi_D r_q,D^c)\ge0. H D 0 ( ϕ D ∣ r q ) = i ∑ e i ( ϕ D r q , D c ) ≥ 0. The weak part is the ordinary conditional Hamiltonian H D E = ∑ A : A ∩ D ≠ ∅ E A H_D^E=\sum_{A:A\cap D\ne\varnothing}E_A H D E = ∑ A : A ∩ D = ∅ E A
Lemma 9.4 (Partition bound near a homogeneous well). For sufficiently large β \beta β D D D
− C ( log β + β δ ) # D ≤ log Z D q , r q ≤ C β δ # D . -C(\log\beta+\beta\delta)\#D\le\log Z_D^{q,r_q}\le C\beta\delta\#D. − C ( log β + β δ ) # D ≤ log Z D q , r q ≤ C β δ # D . The same bounds hold after omitting nonnegative leading terms or weak interaction terms.
Proof. The leading energy is nonnegative, the weak energy has absolute value at most C δ # D C\delta\#D C δ # D β − 2 \beta^{-2} β − 2 r q r_q r q exp [ − C # D log β ] \exp[-C\#D\log\beta] exp [ − C # D log β ] C β − 4 # D C\beta^{-4}\#D C β − 4 # D (43) . Omitting nonnegative leading terms preserves both energy bounds; omitting weak terms cannot increase their absolute norm.
For a signed interaction, allocate every term E A E_A E A ∣ A ∣ |A| ∣ A ∣ B L B_L B L μ \mu μ
e 0 = 1 L 3 ∑ i ∈ B L E μ e i , e E = 1 L 3 ∑ i ∈ B L ∑ A ∋ i E μ E A ∣ A ∣ . e_0=\frac{1}{L^3}\sum_{i\in B_L}\mathbb{E}_{\mu}e_i,\qquad
e_E=\frac{1}{L^3}\sum_{i\in B_L}\sum_{A\ni i}\frac{\mathbb{E}_{\mu}E_A}{|A|}. e 0 = L 3 1 i ∈ B L ∑ E μ e i , e E = L 3 1 i ∈ B L ∑ A ∋ i ∑ ∣ A ∣ E μ E A . Absolute summability gives ∣ e E ∣ ≤ C δ |e_E|\le C\delta ∣ e E ∣ ≤ C δ
Lemma 9.5 (Restricted pressure). Along complete-block cubes D D D
π q = lim D log Z D q , r q # D \pi_q=\lim_D\frac{\log Z_D^{q,r_q}}{\#D} π q = D lim # D log Z D q , r q exists and is independent of the bounded exterior field.
Proof. We again use the proof of the thermodynamic limit by division into boxes [30 Section 2] . Normalize the reference on the allowed configurations of each block. For any coarse union D D D D D D D c D^c D c
b ( D ) = ∑ A : A ∩ D ≠ ∅ , A ∩ D c ≠ ∅ ∥ Φ A 0 + E A ∥ ∞ = O ( A ( D ) ) . (104) b(D)=\sum_{\substack{A:A\cap D\ne\varnothing,\,A\cap D^c\ne\varnothing}}\left\|\Phi_A^0+E_A\right\|_\infty=O(A(D)). \tag*{(104)} b ( D ) = A : A ∩ D = ∅ , A ∩ D c = ∅ ∑ Φ A 0 + E A ∞ = O ( A ( D )) . (104) Deleting those terms changes log Z D \log Z_D log Z D β b ( D ) \beta b(D) β b ( D )
Lemma 9.6 (Restricted Gibbs state). There is a restricted Gibbs state μ \mu μ
Proof. This is the finite-volume compactness construction of Gibbs states, adapted to the allowed-block reference; compare [34 Appendix A] . Relative to the normalized allowed-block reference, summability bounds each fixed marginal density in L ∞ L^\infty L ∞
Lemma 9.7 (Entropy bounds). The state μ \mu μ s s s C C C
H μ ( X C ) ≥ s # C , H μ ( X C ∣ X C c ) ≤ s # C . (105) \mathcal{H}_{\mu}(X_C) \ge s\#C,\qquad\mathcal{H}_{\mu}(X_C \mid X_{C^c}) \le s\#C. \tag*{(105)} H μ ( X C ) ≥ s # C , H μ ( X C ∣ X C c ) ≤ s # C . (105) Proof. We use the entropy inequalities and lexicographic construction of Robinson and Ruelle [31 Propositions 1–2] . In a lexicographic block order, conditioning on larger finite pasts converges to conditioning on the infinite past. The chain rule on large cubes gives its entropy per site as s s s C C C C c C^c C c
The next identity is the restricted-block form of the Gibbs variational principle [35 Theorem 3] ; the passage from Gibbs conditional laws to this identity follows the boundary comparison in [34 Appendix A] .
Lemma 9.8 (Pressure identity). The entropy and mean energies of μ \mu μ
π q = s − β ( e 0 + e E ) . (106) \pi_q = s - \beta(e_0 + e_E). \tag*{(106)} π q = s − β ( e 0 + e E ) . (106) Proof. For a complete-block box D D D e − 4 β b ( D ) e^{-4\beta b(D)} e − 4 β b ( D ) e 4 β b ( D ) e^{4\beta b(D)} e 4 β b ( D ) X D X_D X D X D c X_{D^c} X D c 4 β b ( D ) 4\beta b(D) 4 β b ( D )
H μ ( X D ∣ X D c ) = β E μ H D ( X D ∣ X D c ) + E μ log Z D ( X D c ) \mathcal{H}_{\mu}(X_D \mid X_{D^c}) = \beta E_{\mu}H_D(X_D \mid X_{D^c}) + E_{\mu}\log Z_D(X_{D^c}) H μ ( X D ∣ X D c ) = β E μ H D ( X D ∣ X D c ) + E μ log Z D ( X D c ) then gives (106) after division by # D \#D # D O ( b ( D ) ) O(b(D)) O ( b ( D ))
Lemma 9.4 gives the pressure bounds below; (106) , s ≤ 0 s \le0 s ≤ 0 ∣ e E ∣ ≤ C δ \lvert e_E\rvert\le C\delta ∣ e E ∣ ≤ C δ
− C ( log β + β δ ) ≤ π q ≤ C β δ , 0 ≤ e 0 ≤ C ( δ + log β β ) . (107) -C(\log\beta+ \beta\delta) \le\pi_q \le C\beta\delta,\qquad0 \le e_0 \le C\left(\delta+ \frac{\log\beta}{\beta}\right). \tag*{(107)} − C ( log β + β δ ) ≤ π q ≤ C β δ , 0 ≤ e 0 ≤ C ( δ + β log β ) . (107) For a finite fine-site set J J J C C C
H μ ( X J ∣ X J c ) = H μ ( X J ∖ C ∣ X J c ) + H μ ( X C ∣ X C c ) ≤ s # C . (108) \mathcal{H}_{\mu}(X_J \mid X_{J^c}) = \mathcal{H}_{\mu}(X_{J\setminus C} \mid X_{J^c}) + \mathcal{H}_{\mu}(X_C \mid X_{C^c}) \le s\#C. \tag*{(108)} H μ ( X J ∣ X J c ) = H μ ( X J ∖ C ∣ X J c ) + H μ ( X C ∣ X C c ) ≤ s # C . (108) Gibbs optimality on J J J
The next bound has the role of LMP’s surface error in comparing restricted partition functions [2 Statement 3.5 and Section 6] . Here the boundary modification interpolates within an entire connected component of minimizing profiles.
Proposition 9.9 (Rooted surface bound). Under the preceding hypotheses and scale choices,
∣ log Z D q , r q − π q # D ∣ ≤ β κ A ( D ) (109) \left|\log Z_D^{q,r_q}-\pi_q\#D\right|\le\beta\kappa A(D) \tag*{(109)} log Z D q , r q − π q # D ≤ β κ A ( D ) (109) uniformly over finite unions of complete coarse blocks, with κ \kappa κ (103) .
Proof. For the lower bound take X ∼ μ X\sim\mu X ∼ μ D D D D D D (42) relative to the global root, using a trial density profile equal to f q f_q f q D D D D D D
E H D 0 ( Z D ∣ r q ) ≤ e 0 # D + κ E A ( D ) . \mathbb{E}H_D^0(Z_D\mid r_q)\le e_0\#D+\kappa_E A(D). E H D 0 ( Z D ∣ r q ) ≤ e 0 # D + κ E A ( D ) . The equal site allocation of signed weak terms gives e E # D + C δ L A ( D ) e_E\#D+C\delta LA(D) e E # D + C δ L A ( D ) O ( δ L A ( D ) ) O(\delta LA(D)) O ( δ L A ( D )) Lemma 9.1 to X D X_D X D X X X
H ( Z D ) ≥ s # D − C L A ( D ) log β . \mathcal{H}(Z_D)\ge s\#D-CLA(D)\log\beta. H ( Z D ) ≥ s # D − C L A ( D ) log β . The finite-volume variational principle proves the lower estimate.
For the upper bound let Y Y Y D D D X ∼ μ X\sim\mu X ∼ μ X X X J J J C ⊂ J C\subset J C ⊂ J r 0 r_0 r 0 C C C
∣ # D − # C ∣ ≤ C L A ( D ) . \left|\#D-\#C\right|\le CLA(D). ∣ # D − # C ∣ ≤ C L A ( D ) . Every cell of J J J Y Y Y X X X J J J Y Y Y (42) relative to X X X C C C Y Y Y
E H 0 ( Z ; X ) ≤ E H D 0 ( Y D ∣ r q ) − e 0 # C + κ E A ( D ) , \mathbb{E}H^0(Z;X)\le\mathbb{E}H_D^0(Y_D\mid r_q)-e_0\#C+\kappa_EA(D), E H 0 ( Z ; X ) ≤ E H D 0 ( Y D ∣ r q ) − e 0 # C + κ E A ( D ) , E H E ( Z ; X ) ≤ E H D E ( Y D ∣ r q ) − e E # C + C δ L A ( D ) . \mathbb{E}H^E(Z;X)\le\mathbb{E}H_D^E(Y_D\mid r_q)-e_E\#C+C\delta LA(D). E H E ( Z ; X ) ≤ E H D E ( Y D ∣ r q ) − e E # C + C δ L A ( D ) . Apply Lemma 9.1 to Y D Y_D Y D ( X , Y ) (X,Y) ( X , Y ) X J c X_{J^c} X J c
H ( Z J ∣ X J c ) ≥ H ( Y D ) − C L A ( D ) log β . \mathcal{H}(Z_J\mid X_{J^c})\ge\mathcal{H}(Y_D)-CLA(D)\log\beta. H ( Z J ∣ X J c ) ≥ H ( Y D ) − C L A ( D ) log β . Conditional Gibbs optimality of μ \mu μ (108) now imply
log Z D q , r q = H ( Y D ) − β E H D ( Y D ∣ r q ) ≤ π q # C + β κ A ( D ) . \log Z_D^{q,r_q}=\mathcal{H}(Y_D)-\beta\mathbb{E}H_D(Y_D\mid r_q)\le\pi_q\#C+\beta\kappa A(D). log Z D q , r q = H ( Y D ) − β E H D ( Y D ∣ r q ) ≤ π q # C + β κ A ( D ) . Replacing # C \#C # C # D \#D # D C β L A ( D ) ( δ + log β / β ) C\beta LA(D)(\delta+\log\beta/\beta) C β L A ( D ) ( δ + log β / β ) (107) , already included in (103) . If an eroded core is empty, the affected region has at most C L A ( D ) CLA(D) C L A ( D ) Lemma 9.4 gives both bounds directly.
For the chosen perturbation E E E
p q ( E ) = b 0 + π q / v s . (110) p_q(E)=b_0+\pi_q/v_s. \tag*{(110)} p q ( E ) = b 0 + π q / v s . (110) The root boundary normalization is (98) .
For a regional weak gas, add the potentials of Proposition 7.15 divided by β \beta β C t β / β Ct_\beta/\beta C t β / β (87) compares the actual regional correction with its ordinary conditional Hamiltonian at a logarithmic cost C t β A ( D ) Ct_\beta A(D) C t β A ( D ) Proposition 9.9 with this enlarged weak interaction therefore gives the same rooted surface estimate for the actual auxiliary field integral, on every coarse union, including punctured regions. The homogeneous value of the correction is included in its branch pressure.
Collar kernels and arbitrary caps
Let S S S S S S a , b a,b a , b q q q S S S T T T
If P P P ∂ T \partial T ∂ T S S S Lemma 9.2 . Writing A = L 2 A=L^2 A = L 2 O ( L A ) O(LA) O ( L A ) L / 4 L/4 L /4 L / 3 L/3 L /3 O ( A ) O(A) O ( A ) R 1 R_1 R 1 C R 1 CR_1 C R 1
Proposition 9.10 (Collar cross ratio). Let K ( a , b ) > 0 K(a,b)>0 K ( a , b ) > 0
∣ log K ( a , b ) + log K ( a ′ , b ′ ) − log K ( a , b ′ ) − log K ( a ′ , b ) ∣ ≤ β A κ . (111) \lvert\log K(a,b)+\log K(a',b')-\log K(a,b')-\log K(a',b)\rvert\le\beta A\kappa. \tag*{(111)} ∣ log K ( a , b ) + log K ( a ′ , b ′ ) − log K ( a , b ′ ) − log K ( a ′ , b )∣ ≤ β A κ . (111) with κ \kappa κ (103) , uniformly in the caps and in the collar’s size and topology.
Proof. Take independent source fields X , Y X,Y X , Y ( a , b ) (a,b) ( a , b ) ( a ′ , b ′ ) (a',b') ( a ′ , b ′ ) T T T r 0 r_0 r 0 T T T
For the first output use a global trial density equal to Q ( X ) Q(X) Q ( X ) Q ( Y ) Q(Y) Q ( Y ) (42) . Each trial agrees with its own source profile outside a finite set, and every stencil copied entirely from a source cancels across the pair. The plateau has only the small field-pinning cost. Dropping the remaining negative old terms and applying Lemmas 9.2 –9.3 bounds the leading change by A κ E A\kappa_E A κ E
The weak changes are also added before they are bounded. Terms wholly in a copied side cancel; every remaining connected support meets the erased band. Their total norm is at most C δ L A C\delta LA C δ L A
The retained collar coordinates are a deterministic permutation of a marginal of ( X S , Y S ) (X_S,Y_S) ( X S , Y S ) Lemma 9.1 , with the complete source pair as randomness, bounds the joint output entropy below by H ( X S ) + H ( Y S ) − C L A log β \mathcal{H}(X_S)+\mathcal{H}(Y_S)-CLA\log\beta H ( X S ) + H ( Y S ) − C L A log β
An interaction missing S S S
log K ( a ′ , b ) + log K ( a , b ′ ) ≥ log K ( a , b ) + log K ( a ′ , b ′ ) − β A κ . \log K(a',b)+\log K(a,b')\ge\log K(a,b)+\log K(a',b')-\beta A\kappa. log K ( a ′ , b ) + log K ( a , b ′ ) ≥ log K ( a , b ) + log K ( a ′ , b ′ ) − β A κ . Interchanging the source and target cap pairings gives the reverse inequality. Positivity follows from bounded action and the positive-mass small-ball trial. ∎
Contours and interface costs
We use fine spatial lattice units in this section. A fine cell has physical volume v s v_s v s L 3 L^3 L 3
V L = v s L 3 , Vol ( D ) = V L n ( D ) , (112) V_L = v_sL^3,\qquad\operatorname{Vol}(D) = V_L n(D), \tag*{(112)} V L = v s L 3 , Vol ( D ) = V L n ( D ) , (112) where n ( D ) n(D) n ( D ) A ( D ) A(D) A ( D ) V L V_L V L
Let Q v = L ( v + [ 0 , 1 ] 3 ) Q_v = L(v + [0,1]^3) Q v = L ( v + [ 0 , 1 ] 3 ) v ∈ Z 3 v \in\mathbb{Z}^3 v ∈ Z 3
N k ( B ) = { v : dist ∞ ( v , B ) ≤ k } . N_k(B) = \{v : \operatorname{dist}_{\infty}(v,B) \le k\}. N k ( B ) = { v : dist ∞ ( v , B ) ≤ k } . Every spatial variable includes the entire angular fiber and all rod and field variables assigned to that spatial cell. We choose the physical coarse side longer than the microscopic interaction range.
Use the compact projected sets M 1 , … , M m \mathcal{M}_1,\ldots,\mathcal{M}_m M 1 , … , M m ∣ ⋅ ∣ ω |\cdot|_\omega ∣ ⋅ ∣ ω ρ ∗ > 0 \rho_* > 0 ρ ∗ > 0 q q q ρ ∗ \rho_* ρ ∗ M q \mathcal{M}_q M q τ v = q \tau_v=q τ v = q ( 1 − θ ) L 3 (1-\theta)L^3 ( 1 − θ ) L 3 q q q 0 < θ < 1 / 2 0<\theta<1/2 0 < θ < 1/2
Set τ v = 0 \tau_v=0 τ v = 0
Θ v = q ⟺ τ w = q for every ∥ w − v ∥ ∞ ≤ 1 ; Θ v = 0 otherwise . \Theta_v=q \quad\Longleftrightarrow\quad\tau_w=q\ \text{for every }\|w-v\|_\infty\le1;\qquad\Theta_v=0\ \text{otherwise}. Θ v = q ⟺ τ w = q for every ∥ w − v ∥ ∞ ≤ 1 ; Θ v = 0 otherwise . Two star-adjacent nonzero Θ \Theta Θ B = { v : Θ v = 0 } B=\{v:\Theta_v=0\} B = { v : Θ v = 0 } N k ( B ) N_k(B) N k ( B ) B Γ B_\Gamma B Γ k k k A Γ = N k ( B Γ ) A_\Gamma=N_k(B_\Gamma) A Γ = N k ( B Γ )
Complementary components and regional domains
Lemma 10.1 (Connected complementary boundaries). For a finite connected union A A A
Proof. A path in the open complement can be perturbed away from grid edges and vertices, so it crosses only open faces; the converse follows by joining face neighbors. Compactify space to S 3 S^3 S 3 D D D F = S 3 ∖ D F=S^3\setminus D F = S 3 ∖ D G = D ‾ G=\overline{D} G = D F F F A A A A A A x , y x,y x , y F ∩ G F\cap G F ∩ G a ⊂ F a\subset F a ⊂ F b ⊂ G b\subset G b ⊂ G y − x y-x y − x [36 Section 2.2, pp. 149–150] . Since H 1 ( S 3 ) = 0 H_1(S^3)=0 H 1 ( S 3 ) = 0 a − b = ∂ ( z F + z G ) a-b=\partial(z_F+z_G) a − b = ∂ ( z F + z G ) F F F G G G a − ∂ z F = b + ∂ z G a-\partial z_F=b+\partial z_G a − ∂ z F = b + ∂ z G F ∩ G F\cap G F ∩ G x x x y y y ∂ D = F ∩ G \partial D=F\cap G ∂ D = F ∩ G D D D A A A D D D
Lemma 10.2 (Containment of inherited regions). If D D D A ′ ⊆ D A' \subseteq D A ′ ⊆ D A ′ A' A ′ D D D
Proof. The connected set S 3 ∖ D \mathbb{S}^3 \setminus D S 3 ∖ D A ′ A' A ′ S 3 ∖ A ′ \mathbb{S}^3 \setminus A' S 3 ∖ A ′
A physical recursion domain is a box or a bounded complementary component of a connected closed support.
Lemma 10.3. Every complementary boundary of a contour support has a single nonzero facing label. Components with different facing labels have index distance at least 2 k + 2 2k+2 2 k + 2
Proof. A cube outside the support and touching it has distance at least k + 1 k+1 k + 1 Lemma 10.1 connects these facing cubes by star steps, so their nonzero labels coincide. A star path between differently labeled facing cubes contains a bad cube, and hence has length at least 2 k + 2 2k+2 2 k + 2
Lemma 10.4 (Regional gap). If D D D A A A v ∈ D v \in D v ∈ D
dist ∞ ( v , D c ) = dist ∞ ( v , A ) . \operatorname{dist}_{\infty}(v,D^{c})=\operatorname{dist}_{\infty}(v,A). dist ∞ ( v , D c ) = dist ∞ ( v , A ) . For a child support A ′ = N k ( B ′ ) ⊂ D A'=N_k(B')\subset D A ′ = N k ( B ′ ) ⊂ D
dist ∞ ( A ′ , D c ) ≥ 2 ⟺ dist ∞ ( B ′ , D c ) ≥ k + 2. (113) \operatorname{dist}_{\infty}(A',D^{c})\ge2 \quad\Longleftrightarrow\quad\operatorname{dist}_{\infty}(B',D^{c})\ge k+2. \tag*{(113)} dist ∞ ( A ′ , D c ) ≥ 2 ⟺ dist ∞ ( B ′ , D c ) ≥ k + 2. (113) Proof. For the distance identity, choose a nearest cube of D c D^c D c D D D A A A A ⊂ D c A \subset D^c A ⊂ D c C C C
dist ∞ ( N k ( B ′ ) , C ) = max { dist ∞ ( B ′ , C ) − k , 0 } , \operatorname{dist}_{\infty}(N_k(B'),C)=\max\{\operatorname{dist}_{\infty}(B',C)-k,0\}, dist ∞ ( N k ( B ′ ) , C ) = max { dist ∞ ( B ′ , C ) − k , 0 } , by the triangle inequality and a shortest star path. This proves (113) .
Write U q ( D ) U_q(D) U q ( D ) τ = q \tau=q τ = q D D D R q ( D ) R_q(D) R q ( D ) Θ = q \Theta=q Θ = q k + 1 k+1 k + 1 q q q (113) is equivalent to regional admissibility. Indeed, the connected complement of D D D k + 1 k+1 k + 1 q q q
On arbitrary finite unions, including punctured sets, we will also use an auxiliary gas indexed by support inclusion.
Removed regions, shielding, and the contour identity
A contour datum records B Γ B_{\Gamma} B Γ A Γ A_{\Gamma} A Γ τ \tau τ B Γ B_\Gamma B Γ q q q D j D_j D j p j ≠ q p_j \ne q p j = q
V Γ = A Γ ∪ ⋃ j D j , K Γ = { support cubes touching a q component } , U Γ = V Γ ∖ K Γ . V_\Gamma= A_\Gamma\cup\bigcup_j D_j,\qquad K_\Gamma= \{\text{support cubes touching a }q\text{ component}\},\qquad U_\Gamma= V_\Gamma\setminus K_\Gamma. V Γ = A Γ ∪ j ⋃ D j , K Γ = { support cubes touching a q component } , U Γ = V Γ ∖ K Γ . The set K Γ K_\Gamma K Γ q q q
Lemma 10.5 (Nesting of removed regions). The removed regions of strongly compatible q q q
Proof. A connected support avoiding another lies in one complementary component. If this component is bounded, Lemma 10.2 keeps all its interiors there: the removed regions are nested in a non-q q q q q q H 1 , H 2 H_1,H_2 H 1 , H 2 H 1 H_1 H 1 H 1 ⊂ H 2 H_1 \subset H_2 H 1 ⊂ H 2 H 2 H_2 H 2
Lemma 10.6 (Microscopic separation). Every microscopic factor meeting U Γ U_\Gamma U Γ K Γ K_\Gamma K Γ
Proof. Lemma 10.3 separates the inherited non-q q q q q q K Γ K_\Gamma K Γ
Let x v x_v x v ρ U \rho_U ρ U q q q x K x_K x K
N Γ ( x K ) = ∫ ρ U ( x U ∣ x K ) 1 { recorded pattern on A } ∏ j 1 R p j ( D j ) d x U , (114) N_\Gamma(x_K)=\int\rho_U(x_U\mid x_K)1_{\{\text{recorded pattern on }A\}}\prod_j1_{R_{p_j}(D_j)}\,dx_U, \tag*{(114)} N Γ ( x K ) = ∫ ρ U ( x U ∣ x K ) 1 { recorded pattern on A } j ∏ 1 R p j ( D j ) d x U , (114) D Γ ( x K ) = ∫ ρ U ( x U ∣ x K ) 1 U q ( A ∖ K ) ∏ j 1 R q ( D j ) d x U , W Γ = N Γ / D Γ . (115) D_\Gamma(x_K)=\int\rho_U(x_U\mid x_K)1_{U_q(A\setminus K)}\prod_j1_{R_q(D_j)}\,dx_U,\qquad W_\Gamma=N_\Gamma/D_\Gamma. \tag*{(115)} D Γ ( x K ) = ∫ ρ U ( x U ∣ x K ) 1 U q ( A ∖ K ) j ∏ 1 R q ( D j ) d x U , W Γ = N Γ / D Γ . (115) The empty rod configuration and fields near the phase root give D Γ > 0 D_\Gamma>0 D Γ > 0 W Γ W_\Gamma W Γ x K x_K x K
Lemma 10.7 (Interior contour indices). After a maximal q q q D j D_j D j q q q D j D_j D j
Proof. Compatibility with the parent is equivalent to (113) . This gap also ensures compatibility with siblings and exterior contours and admissibility in the parent domain. The denominator in (115) prescribes phase q q q D j D_j D j
The cancellation of interior partition functions is the contour reweighting used by Disertori–Giuliani [37 Section 4.3, Lemma 1] and Disertori–Giuliani–Jauslin [38 Section 4.2, Lemma 2] . Here the quotients remain conditional on the collar variables, and the identity is established for each exterior configuration.
Proposition 10.8 (Exact regional identity). Let C q ( D ) C_q(D) C q ( D ) q q q
G q ( D ; x ) = ∑ F ⊂ C q ( D ) c o m p a t i b l e ∏ Γ ∈ F W Γ ( x K Γ ) , G_q(D;x)=\sum_{\mathcal{F}\subset C_q(D)\ \mathrm{compatible}}\prod_{\Gamma\in\mathcal{F}}W_\Gamma(x_{K_\Gamma}), G q ( D ; x ) = F ⊂ C q ( D ) compatible ∑ Γ ∈ F ∏ W Γ ( x K Γ ) , allowing nested removed regions. For every compatible exterior configuration ξ \xi ξ
∫ U q ( D ) ρ D ( x D ∣ ξ ) G q ( D ; x D ) d x D = ∫ R q ( D ) ρ D ( x D ∣ ξ ) d x D . (116) \int_{U_q(D)}\rho_D(x_D\mid\xi)G_q(D;x_D)\,\mathrm{d}x_D=\int_{R_q(D)}\rho_D(x_D\mid\xi)\,\mathrm{d}x_D. \tag*{(116)} ∫ U q ( D ) ρ D ( x D ∣ ξ ) G q ( D ; x D ) d x D = ∫ R q ( D ) ρ D ( x D ∣ ξ ) d x D . (116) Proof. Physical configurations have a unique outside decomposition. Extract the components of the full thickened bad set, start in the exterior q q q q q q q q q q q q p ≠ q p\ne q p = q p p p τ = q \tau=q τ = q q q q k k k
Induct on the number of cubes of D D D Lemma 10.7 , the residual family indices are a product of regional q q q q q q U U U D Γ D_\Gamma D Γ W Γ W_\Gamma W Γ N Γ N_\Gamma N Γ
Following Zahradník’s truncation construction [5 ] , used in the continuum setting by LMP [2 Section 3] , we clip contour weights to define controlled phase pressures before proving stability. For fixed thresholds t Γ t_\Gamma t Γ W Γ > t Γ W_\Gamma>t_\Gamma W Γ > t Γ E = D ∖ ⋃ Γ ∈ F V Γ E=D\setminus\bigcup_{\Gamma\in\mathcal{F}}V_\Gamma E = D ∖ ⋃ Γ ∈ F V Γ
∫ U q ( D ) ρ D ∏ Γ ∈ F ( W Γ 1 { W Γ > t Γ } ) G q , E s t a b l e , F ∏ Γ ∈ F , j G q ( D j ) d x D . (117) \int_{U_q(D)}\rho_D\prod_{\Gamma\in\mathcal{F}}\left(W_\Gamma\mathbf{1}_{\{W_\Gamma>t_\Gamma\}}\right)G^{\mathrm{stable},\mathcal{F}}_{q,E}\prod_{\Gamma\in\mathcal{F},j}G_q(D_j)\,\mathrm{d}x_D. \tag*{(117)} ∫ U q ( D ) ρ D Γ ∈ F ∏ ( W Γ 1 { W Γ > t Γ } ) G q , E stable , F Γ ∈ F , j ∏ G q ( D j ) d x D . (117) The stable gas retains the original parent admissibility and compatibility conditions. Its actual supports lie in E E E E E E K Γ K_\Gamma K Γ E E E (117) . Dropping the active indicators and enlarging the stable gas gives the positive bound
selected contribution ≤ ∫ ρ D 1 R F ( ϕ ) G q , E a u x ( x E ) d x D . (118) \text{selected contribution}\le\int\rho_D\mathbf{1}_{R_{\mathcal{F}}(\phi)}G^{\mathrm{aux}}_{q,E}(x_E)\,\mathrm{d}x_D. \tag*{(118)} selected contribution ≤ ∫ ρ D 1 R F ( ϕ ) G q , E aux ( x E ) d x D . (118) where the auxiliary gas contains all compatible q q q A Γ ⊂ E A_\Gamma\subset E A Γ ⊂ E w Γ = min { W Γ , t Γ } w_\Gamma=\min\{W_\Gamma,t_\Gamma\} w Γ = min { W Γ , t Γ }
A positive charge on each contour
The contour cost supplies the energy bound in Peierls’ energy–entropy argument [39 ] ; here it follows from local rigidity and lattice isoperimetry.
Call a fine site defective when its label is zero or it has a nearest neighbor with a different nonzero label. The fixed-patch rigidity estimate supplies a finite patch P i P_i P i e ∗ > 0 e_*>0 e ∗ > 0
i defective ⟹ ∑ x ∈ P i e x ( ϕ ) ≥ e ∗ . (119) i\ \text{defective}\quad\Longrightarrow\quad\sum_{x\in P_i} e_x(\phi)\ge e_*. \tag*{(119)} i defective ⟹ x ∈ P i ∑ e x ( ϕ ) ≥ e ∗ . (119) Here e x e_x e x e ∗ e_* e ∗ ≥ ρ ∗ \ge\rho_* ≥ ρ ∗ ≤ ρ ∗ \le\rho_* ≤ ρ ∗
Use the positive ball atoms of Proposition 6.7, writing Φ x , r = Φ x , r 0 \Phi_{x,r}=\Phi^0_{x,r} Φ x , r = Φ x , r 0 C e − c r Ce^{-cr} C e − cr R 0 R_0 R 0 ∣ P i ∣ C e − c R 0 ≤ e ∗ / 2 |P_i|Ce^{-cR_0}\le e_*/2 ∣ P i ∣ C e − c R 0 ≤ e ∗ /2
F i = ∑ x ∈ P i ∑ r = 0 R 0 Φ x , r ≥ e ∗ / 2 F_i=\sum_{x\in P_i}\sum_{r=0}^{R_0}\Phi_{x,r}\ge e_*/2 F i = x ∈ P i ∑ r = 0 ∑ R 0 Φ x , r ≥ e ∗ /2 at each defect, and its support lies in a fixed ball B S ( i ) B_S(i) B S ( i )
Lemma 10.9 (Relative lattice isoperimetry). For H ⊂ { 1 , … , M } 3 H\subset\{1,\ldots,M\}^3 H ⊂ { 1 , … , M } 3 ∂ Q H \partial_QH ∂ Q H
∣ ∂ Q H ∣ ≥ ( 1 − 2 − 1 / 3 ) min { ∣ H ∣ , ∣ Q ∖ H ∣ } 2 / 3 . |\partial_QH|\ge(1-2^{-1/3})\min\{|H|,|Q\setminus H|\}^{2/3}. ∣ ∂ Q H ∣ ≥ ( 1 − 2 − 1/3 ) min { ∣ H ∣ , ∣ Q ∖ H ∣ } 2/3 . Proof. Replace H H H a = ∣ H ∣ ≤ M 3 / 2 a=|H|\le M^3/2 a = ∣ H ∣ ≤ M 3 /2 P j P_j P j j j j [40 Theorem 2] a 2 ≤ P 1 P 2 P 3 a^{2}\le P_1P_2P_3 a 2 ≤ P 1 P 2 P 3 P j ≥ a 2 / 3 P_j\ge a^{2/3} P j ≥ a 2/3 ∑ x , y , z a x y b y z c z x \sum_{x,y,z}a_{xy}b_{yz}c_{zx} ∑ x , y , z a x y b y z c z x ( ∑ a x y ) ( ∑ b y z ) ( ∑ c z x ) (\sum a_{xy})(\sum b_{yz})(\sum c_{zx}) ( ∑ a x y ) ( ∑ b y z ) ( ∑ c z x ) a / M a/M a / M
∣ ∂ Q H ∣ ≥ a 2 / 3 − a / M ≥ ( 1 − 2 − 1 / 3 ) a 2 / 3 . |\partial_QH|\ge a^{2/3}-a/M\ge(1-2^{-1/3})a^{2/3}. ∣ ∂ Q H ∣ ≥ a 2/3 − a / M ≥ ( 1 − 2 − 1/3 ) a 2/3 . Proposition 10.10 (Protected Peierls charge). There is c 0 > 0 c_0>0 c 0 > 0 L , θ , β L,\theta,\beta L , θ , β n = ∣ A Γ ∣ n=|A_\Gamma| n = ∣ A Γ ∣ B Γ B_\Gamma B Γ P B Γ P_{B_\Gamma} P B Γ
supp P B Γ ⊂ N 2 ( B Γ ) , P B Γ ≥ τ L 2 n , τ = c 0 θ 2 / 3 ( 2 k + 1 ) 3 . (120) \operatorname{supp}P_{B_\Gamma}\subset N_2(B_\Gamma),\qquad P_{B_\Gamma}\ge\tau L^2n,\qquad\tau=\frac{c_0\theta^{2/3}}{(2k+1)^3}. \tag*{(120)} supp P B Γ ⊂ N 2 ( B Γ ) , P B Γ ≥ τ L 2 n , τ = ( 2 k + 1 ) 3 c 0 θ 2/3 . (120) For strongly compatible contours these protected atom sets are disjoint.
Proof. First suppose τ v = 0 \tau_v=0 τ v = 0 M = L 3 M=L^3 M = L 3 a 0 a_0 a 0 θ M / 2 \theta M/2 θ M /2
a 0 ≥ max { 1 , θ M / 2 } ≥ 2 − 2 / 3 ( θ M ) 2 / 3 . a_0\ge\max\{1,\theta M/2\}\ge2^{-2/3}(\theta M)^{2/3}. a 0 ≥ max { 1 , θ M /2 } ≥ 2 − 2/3 ( θ M ) 2/3 . Otherwise at least 3 M / 4 3M/4 3 M /4 a q ≥ 3 M / ( 4 m ) a_q\ge3M/(4m) a q ≥ 3 M / ( 4 m ) θ M \theta M θ M min ( a q , M − a q ) ≥ c m θ M \min(a_q,M-a_q)\ge c_m\theta M min ( a q , M − a q ) ≥ c m θ M
applied to that phase gives at least c θ 2 / 3 L 2 c\theta^{2/3}L^2 c θ 2/3 L 2
If Θ b = 0 \Theta_b=0 Θ b = 0 τ = 0 \tau=0 τ = 0 N 1 ( b ) N_1(b) N 1 ( b ) L 3 / 2 L^3/2 L 3 /2 c θ 2 / 3 L 2 c\theta^{2/3}L^2 c θ 2/3 L 2 B Γ B_\Gamma B Γ
∣ { defects in N 1 ( B Γ ) } ∣ ≥ c θ 2 / 3 L 2 ∣ B Γ ∣ / 27. |\{\text{defects in }N_1(B_\Gamma)\}| \ge c\theta^{2/3}L^2|B_\Gamma|/27. ∣ { defects in N 1 ( B Γ )} ∣ ≥ c θ 2/3 L 2 ∣ B Γ ∣/27. Color fine sites by coordinate residues modulo 2 S + 1 2S+1 2 S + 1 ( 2 S + 1 ) − 3 (2S+1)^{-3} ( 2 S + 1 ) − 3 B S ( i ) B_S(i) B S ( i ) F i F_i F i n ≤ ( 2 k + 1 ) 3 ∣ B Γ ∣ n\le(2k+1)^3|B_\Gamma| n ≤ ( 2 k + 1 ) 3 ∣ B Γ ∣
To make the protected Hamiltonian independent of the field, include, once per label, every Φ x , r \Phi_{x,r} Φ x , r r ≤ R 0 r\le R_0 r ≤ R 0 N 1 ( B Γ ) N_1(B_\Gamma) N 1 ( B Γ ) L ≥ S + 2 L\ge S+2 L ≥ S + 2 N 2 ( B Γ ) N_2(B_\Gamma) N 2 ( B Γ ) k k k 2 k + 2 2k+2 2 k + 2 N 2 N_2 N 2 k ≥ 24 k\ge24 k ≥ 24
Seams, restoration, and separated collars
Let D j D_j D j N N N
A i n : = ∑ j A ( D j ) ≤ 6 L 2 N . (121) \mathcal{A}_{\mathrm{in}}:=\sum_j\mathcal{A}(D_j)\le6L^2N. \tag*{(121)} A in := j ∑ A ( D j ) ≤ 6 L 2 N . (121) Suppose ∥ Φ i , r ∥ ∞ ≤ C 0 e − c 0 r \lVert\Phi_{i,r}\rVert_\infty\le C_0e^{-c_0r} ∥ Φ i , r ∥ ∞ ≤ C 0 e − c 0 r m ( i , r ) m(i,r) m ( i , r )
Lemma 10.11 (Seam estimate). For every fixed $p\ge 0,
∑ m ( i , r ) ≥ 2 ( 1 + m ( i , r ) ) p ∥ Φ i , r ∥ ∞ ≤ C p A i n / L . (122) \sum_{m(i,r)\ge2}(1+m(i,r))^p\lVert\Phi_{i,r}\rVert_\infty\le C_p\mathcal{A}_{\mathrm{in}}/L. \tag*{(122)} m ( i , r ) ≥ 2 ∑ ( 1 + m ( i , r ) ) p ∥ Φ i , r ∥ ∞ ≤ C p A in / L . (122) If a signed interaction ψ \psi ψ δ \delta δ
∑ X : m ( X ) ≥ 2 m ( X ) ∥ ψ X ∥ ∞ ≤ C δ A i n . \sum_{X:m(X)\ge2}m(X)\lVert\psi_X\rVert_\infty\le C\delta\mathcal{A}_{\mathrm{in}}. X : m ( X ) ≥ 2 ∑ m ( X ) ∥ ψ X ∥ ∞ ≤ C δ A in . Proof. For r < L / 4 r<L/4 r < L /4 r r r r r r
C ∑ j ∣ ∂ D j ∣ { L ( r + 1 ) 2 + ( r + 1 ) 3 } ≤ C A i n L ( r + 1 ) 2 . C\sum_j|\partial D_j|\{L(r+1)^2+(r+1)^3\}\le C\frac{\mathcal{A}_{\mathrm{in}}}{L}(r+1)^2. C j ∑ ∣ ∂ D j ∣ { L ( r + 1 ) 2 + ( r + 1 ) 3 } ≤ C L A in ( r + 1 ) 2 . Here ∣ ∂ D j ∣ |\partial D_j| ∣ ∂ D j ∣ e − c 0 r e^{-c_0r} e − c 0 r m ≤ 8 m\le8 m ≤ 8 r ≥ L / 4 r\ge L/4 r ≥ L /4 C L 2 ( r + 1 ) ( 1 + ( r + 1 ) / L ) 2 CL^{2}(r+1)(1+(r+1)/L)^{2} C L 2 ( r + 1 ) ( 1 + ( r + 1 ) / L ) 2 m ≤ C ( 1 + r / L ) 3 m \le C(1+r/L)^{3} m ≤ C ( 1 + r / L ) 3
C p A i n ∑ r ≥ L / 4 ( r + 1 ) ( 1 + ( r + 1 ) / L ) 2 + 3 p e − c 0 r ≤ C p e − c L A i n . CpA_{\mathrm{in}}\sum_{r\ge L/4}(r+1)(1+(r+1)/L)^{2+3p}e^{-c_{0}r}\le Cp e^{-cL}A_{\mathrm{in}}. C p A in r ≥ L /4 ∑ ( r + 1 ) ( 1 + ( r + 1 ) / L ) 2 + 3 p e − c 0 r ≤ C p e − c L A in . A connected weak support crossing ∂ D j \partial D_j ∂ D j j j j m m m m m m
□ \square □
Deleting or restoring terms of total norm M M M β M \beta M β M m m m m m m m m m C ( 1 + m ) C(1+m) C ( 1 + m ) C β ( L − 1 + δ ) A i n C\beta(L^{-1}+\delta)A_{\mathrm{in}} C β ( L − 1 + δ ) A in
Take k ≥ 24 k\ge24 k ≥ 24 D j D_j D j
ℓ j ( v ) = dist ∞ ( v , D j c ) , S j = { 1 ≤ ℓ j ≤ 8 } , I j = { ℓ j ≥ 9 } . \ell_j(v)=\operatorname{dist}_{\infty}(v,D_j^{c}),\qquad S_j=\{1\le\ell_j\le8\},\qquad I_j=\{\ell_j\ge9\}. ℓ j ( v ) = dist ∞ ( v , D j c ) , S j = { 1 ≤ ℓ j ≤ 8 } , I j = { ℓ j ≥ 9 } . Use the level-four/five faces as separator Σ j \Sigma_j Σ j S j S_j S j p j p_j p j ℓ j \ell_j ℓ j I j I_j I j D j c D_j^{c} D j c
∣ Σ j ∣ ≤ C A ( D j ) , ∣ S j ∣ + ∣ N t ( Σ j ) ∣ ≤ C L A ( D j ) , (123) |\Sigma_j|\le C\mathcal{A}(D_j),\qquad|S_j|+|N_t(\Sigma_j)|\le CL\mathcal{A}(D_j), \tag*{(123)} ∣ Σ j ∣ ≤ C A ( D j ) , ∣ S j ∣ + ∣ N t ( Σ j ) ∣ ≤ C L A ( D j ) , (123) for L / 4 ≤ t ≤ L / 3 L/4\le t\le L/3 L /4 ≤ t ≤ L /3
∣ N t + R ( Σ j ) ∖ N t ( Σ j ) ∣ ≤ C ( R + 1 ) A ( D j ) . |N_{t+R}(\Sigma_j)\setminus N_t(\Sigma_j)|\le C(R+1)\mathcal{A}(D_j). ∣ N t + R ( Σ j ) ∖ N t ( Σ j ) ∣ ≤ C ( R + 1 ) A ( D j ) . Indeed a single coarse face contributes O ( L 3 ) O(L^3) O ( L 3 ) O ( ( R + 1 ) L 2 ) O((R+1)L^2) O (( R + 1 ) L 2 ) L L L c L cL c L
For a selected family let V V V E E E q q q
S 0 = { v ∈ A : 1 ≤ dist ∞ ( v , E ) ≤ 8 } . S_0=\{v\in A:1\le\operatorname{dist}_{\infty}(v,E)\le8\}. S 0 = { v ∈ A : 1 ≤ dist ∞ ( v , E ) ≤ 8 } . The same level-four/five construction gives its outer separator. Every inherited non-q q q q q q S 0 S_0 S 0 q q q
dist ∞ ( S 0 , D j ) ≥ 2 k + 2 − 8. \operatorname{dist}_{\infty}(S_0,D_j)\ge2k+2-8. dist ∞ ( S 0 , D j ) ≥ 2 k + 2 − 8. Leading balls meeting both outer and inner collars therefore have radius at least c L cL c L C e − c L A i n Ce^{-cL}A_{\mathrm{in}} C e − c L A in C δ A i n C\delta A_{\mathrm{in}} C δ A in
The collar switch exchanges the complete bounded sides { ℓ j ≥ 5 } \{\ell_j\ge5\} { ℓ j ≥ 5 } { v ∈ V : dist ∞ ( v , E ) ≥ 5 } \{v\in V:\operatorname{dist}_{\infty}(v,E)\ge5\} { v ∈ V : dist ∞ ( v , E ) ≥ 5 } q q q I j = ∅ I_j=\varnothing I j = ∅
Collar kernels impose τ = q \tau=q τ = q Θ \Theta Θ Lemma 10.3 .
For a conditional single-contour quotient, freeze K = K Γ K = K_{\Gamma} K = K Γ U = V Γ ∖ K U = V_{\Gamma} \setminus K U = V Γ ∖ K ℓ ( v ) = dist ∞ ( v , K ) \ell(v) = \operatorname{dist}_{\infty}(v,K) ℓ ( v ) = dist ∞ ( v , K )
O = { ℓ ≤ 4 } , S = { 5 ≤ ℓ ≤ 12 } , V ′ = U ∖ ( O ∪ S ) . O = \{\ell\le4\}, \qquad S = \{5 \le\ell\le12\}, \qquad V' = U \setminus(O \cup S). O = { ℓ ≤ 4 } , S = { 5 ≤ ℓ ≤ 12 } , V ′ = U ∖ ( O ∪ S ) . The level-eight/nine separator has the same tube bounds. The collar lies in the q q q D j D_j D j V ′ V' V ′ P P P 4 , 5 4,5 4 , 5 8 , 9 8,9 8 , 9 N 3 ( P ) ⊂ S N_3(P) \subset S N 3 ( P ) ⊂ S [ 1 , 8 ] [1,8] [ 1 , 8 ] [ 5 , 12 ] [5,12] [ 5 , 12 ] A A A E E E D j D_j D j N 2 ( P ) N_2(P) N 2 ( P ) L / 4 L/4 L /4 L / 3 L/3 L /3 R 1 + r 0 < L / 6 R_1 + r_0 < L/6 R 1 + r 0 < L /6
The boundary-cocycle estimate ∣ ∂ i B ξ ∣ ≤ C e − c dist ( i , U c ) |\partial_i B^{\xi}| \le Ce^{-c\operatorname{dist}(i,U^c)} ∣ ∂ i B ξ ∣ ≤ C e − c dist ( i , U c )
∥ B ξ ( ϕ ) − B ξ ( ϕ O ∩ O c ) ∥ ∞ ≤ C e − c ′ L A ( U ) . (124) \lVert B^{\xi}(\phi) - B^{\xi}(\phi_{O\cap O^c})\rVert_{\infty} \le Ce^{-c' L} A(U). \tag*{(124)} ∥ B ξ ( ϕ ) − B ξ ( ϕ O ∩ O c ) ∥ ∞ ≤ C e − c ′ L A ( U ) . (124) To sum the gradient bound, assign each fine site to a nearest boundary face; at distance d d d C A ( U ) ( d + 1 ) 2 CA(U)(d+1)^2 C A ( U ) ( d + 1 ) 2 d ≥ c L d \ge cL d ≥ c L V a u g = S ∪ V ′ V_{\mathrm{aug}} = S \cup V' V aug = S ∪ V ′ S j S_j S j V a u g c V_{\mathrm{aug}}^c V aug c C e − c L ∑ j A ( D j ) Ce^{-cL}\sum_j A(D_j) C e − c L ∑ j A ( D j ) C δ ∑ j A ( D j ) C\delta\sum_j A(D_j) C δ ∑ j A ( D j ) N 2 ( B Γ ) N_2(B_{\Gamma}) N 2 ( B Γ )
Removed volume and contour counting
Lemma 10.12. For a contour of support size n n n
n ( V Γ ) ≤ C n 3 / 2 . n(V_{\Gamma}) \le Cn^{3/2}. n ( V Γ ) ≤ C n 3/2 . The number of size-n n n C 1 n C_1^n C 1 n
Proof. By the same Loomis–Whitney inequality, a finite lattice set d d d ∣ d ∣ 2 ≤ P 1 P 2 P 3 |d|^2 \le P_1P_2P_3 ∣ d ∣ 2 ≤ P 1 P 2 P 3 ∣ ∂ d ∣ ≥ 2 ( P 1 + P 2 + P 3 ) ≥ 6 ∣ d ∣ 2 / 3 |\partial d| \ge2(P_1 + P_2 + P_3) \ge6|d|^{2/3} ∣ ∂ d ∣ ≥ 2 ( P 1 + P 2 + P 3 ) ≥ 6∣ d ∣ 2/3
∑ j ∣ D j ∣ ≤ C ∑ j ∣ ∂ D j ∣ 3 / 2 ≤ C ( ∑ j ∣ ∂ D j ∣ ) 3 / 2 ≤ C n 3 / 2 . \sum_j |D_j| \le C\sum_j |\partial D_j|^{3/2} \le C\left(\sum_j |\partial D_j|\right)^{3/2} \le Cn^{3/2}. j ∑ ∣ D j ∣ ≤ C j ∑ ∣ ∂ D j ∣ 3/2 ≤ C ( j ∑ ∣ ∂ D j ∣ ) 3/2 ≤ C n 3/2 . Add the support. Every bounded complementary component lies in its support’s coordinate bounding box, since a ray beyond an extreme coordinate reaches infinity without meeting the support. A star-connected support of size n n n n − 1 n-1 n − 1 O ( n 3 ) O(n^3) O ( n 3 ) 2 ( n − 1 ) 2(n-1) 2 ( n − 1 ) 6 n 6n 6 n
For a contour put n Γ = ∣ A Γ ∣ n_{\Gamma}=|A_{\Gamma}| n Γ = ∣ A Γ ∣ S Γ S_{\Gamma} S Γ A Γ A_{\Gamma} A Γ ∣ S Γ ∣ ≤ d L n Γ |S_{\Gamma}|\le d_Ln_{\Gamma} ∣ S Γ ∣ ≤ d L n Γ C = ( Γ 1 , … , Γ m ) C=(\Gamma_1,\ldots,\Gamma_m) C = ( Γ 1 , … , Γ m ) s ( C ) = ∑ j n Γ j s(C)=\sum_j n_{\Gamma_j} s ( C ) = ∑ j n Γ j S ( C ) = ⋃ j S Γ j S(C)=\bigcup_j S_{\Gamma_j} S ( C ) = ⋃ j S Γ j φ T ( C ) \varphi^{\mathcal{T}}(C) φ T ( C ) (57) .
Lemma 10.13 (Anchored contour clusters). For every fixed b ≥ 0 b\ge0 b ≥ 0 T 0 ( L , b ) T_0(L,b) T 0 ( L , b )
sup x ∑ m ≥ 1 1 m ! ∑ C ∣ φ T ( C ) ∣ e b s ( C ) ∏ j = 1 m e − T n Γ j 1 { x ∈ S ( C ) } ≤ C L e − T / 2 (125) \sup_x\sum_{m\ge1}\frac{1}{m!}\sum_C|\varphi^{\mathcal{T}}(C)|e^{bs(C)}\prod_{j=1}^{m}e^{-Tn_{\Gamma_j}}\mathbf{1}_{\{x\in S(C)\}}\le C_Le^{-T/2} \tag*{(125)} x sup m ≥ 1 ∑ m ! 1 C ∑ ∣ φ T ( C ) ∣ e b s ( C ) j = 1 ∏ m e − T n Γ j 1 { x ∈ S ( C )} ≤ C L e − T /2 (125) for T ≥ T 0 ( L , b ) T\ge T_0(L,b) T ≥ T 0 ( L , b )
Proof. The counting bound gives at most C 1 n C_1^n C 1 n n n n 27 n Γ 27n_{\Gamma} 27 n Γ a Γ = e − ( T − b ) n Γ a_{\Gamma}=e^{-(T-b)n_{\Gamma}} a Γ = e − ( T − b ) n Γ
∑ Γ ′ ≁ Γ a Γ ′ e n Γ ′ ≤ 27 n Γ ∑ n ≥ 1 C 1 n e − ( T − b − 1 ) n ≤ n Γ \sum_{\Gamma'\not\sim\Gamma}a_{\Gamma'}e^{n_{\Gamma'}}\le27n_{\Gamma}\sum_{n\ge1}C_1^ne^{-(T-b-1)n}\le n_{\Gamma} Γ ′ ∼ Γ ∑ a Γ ′ e n Γ ′ ≤ 27 n Γ n ≥ 1 ∑ C 1 n e − ( T − b − 1 ) n ≤ n Γ when T T T [25 Section 2, Theorem] . The rooted tree bound from Lemma 7.4 bounds the left side of (125) by C L ∑ n ≥ 1 C 1 n e − ( T − b − 1 ) n C_L\sum_{n\ge1}C_1^ne^{-(T-b-1)n} C L ∑ n ≥ 1 C 1 n e − ( T − b − 1 ) n T 0 ( L , b ) T_0(L,b) T 0 ( L , b ) C L e − T / 2 C_Le^{-T/2} C L e − T /2
For ∣ w Γ ∣ ≤ e − T n Γ |w_{\Gamma}|\le e^{-Tn_{\Gamma}} ∣ w Γ ∣ ≤ e − T n Γ ∣ S ( C ) ∣ ≤ d L s ( C ) |S(C)|\le d_Ls(C) ∣ S ( C ) ∣ ≤ d L s ( C ) diam S ( C ) ≤ C L s ( C ) \operatorname{diam}S(C)\le C_Ls(C) diam S ( C ) ≤ C L s ( C ) b b b κ = C L e − T / 2 \kappa=C_Le^{-T/2} κ = C L e − T /2
Lemma 10.14 (Localization of additional contour species). Let D j D_j D j E E E C = E ∖ ⋃ j D j C=E\setminus\bigcup_jD_j C = E ∖ ⋃ j D j q q q E E E q q q D j D_j D j
S = C ∪ ⋃ j { v ∈ D j : dist ∞ ( v , D j c ) < 2 } . S=C\cup\bigcup_j\{v\in D_j:\operatorname{dist}_{\infty}(v,D_j^c)<2\}. S = C ∪ j ⋃ { v ∈ D j : dist ∞ ( v , D j c ) < 2 } . Proof. A connected support avoiding S S S D j D_j D j D j c D_j^c D j c (113) makes it regionally admissible there; Lemma 10.2 keeps all its inherited regions inside D j D_j D j
Lemma 10.15 (Cost of adding species). Suppose two clipped contour gases have nested index sets, identical activities on common indices, and every additional species has support meeting a coarse set S S S
∥ log G 2 − log G 1 ∥ ∞ ≤ C L κ # S . \|\log G_2-\log G_1\|_{\infty}\le C_L\kappa\,\#S. ∥ log G 2 − log G 1 ∥ ∞ ≤ C L κ # S . Proof. This is the local cluster comparison underlying [25 Section 2, estimate (4)] . Their logarithmic cluster series differ only in clusters containing an additional species. Mark one such species and a site of its support in S S S Lemma 10.13 , summed over these sites, gives the bound. It is pointwise in the rods and fields and hence also bounds the ratio of the corresponding joint partition functions.
The physical pressure as a maximum of phase pressures
Following LMP [2 Sections 3 and 6] , we first define auxiliary pressures using clipped contours and then bound the physical partition functions in terms of those pressures. Roots, phase neighborhoods, regional restrictions, and the threshold T \mathcal{T} T
Normalization and quantitative inputs
Fix the source h h h ν r \nu_r ν r r r r Lemma 7.8 ), and ν r , o u t \nu_{r,\mathrm{out}} ν r , out D D D ν r , ϕ ; D \nu_{r,\phi;D} ν r , ϕ ; D ϕ D r D c \phi_D r_{D^c} ϕ D r D c F D ( r ∣ ξ ) F_D(r\mid\xi) F D ( r ∣ ξ ) b r b_r b r R R R J J J
C D R , J ( r ) = e b r Vol ( D ) E ν r , o u t Z D R , J ( ξ ) F D ( r ∣ ξ ) , V D J ( ϕ ) = − log E ν r , ϕ ; D e − J . C_D^{R,J}(r)=e^{b_r\operatorname{Vol}(D)}\mathbb{E}_{\nu_{r,\mathrm{out}}}\frac{Z_D^{R,J}(\xi)}{F_D(r\mid\xi)},\qquad V_D^J(\phi)=-\log\mathbb{E}_{\nu_{r,\phi;D}}e^{-J}. C D R , J ( r ) = e b r Vol ( D ) E ν r , out F D ( r ∣ ξ ) Z D R , J ( ξ ) , V D J ( ϕ ) = − log E ν r , ϕ ; D e − J . The conditional Hamiltonian H D Ψ H_D^\Psi H D Ψ D D D r r r
b r + β e ( r ) = b 0 , σ D ( r ) = H D Ψ ( r ∣ r ) − e ( r ) Vol ( D ) . b_r+\beta e(r)=b_0,\qquad\sigma_D(r)=H_D^\Psi(r\mid r)-e(r)\operatorname{Vol}(D). b r + β e ( r ) = b 0 , σ D ( r ) = H D Ψ ( r ∣ r ) − e ( r ) Vol ( D ) . Lemma 11.1 (Exact normalized field integral). *With the probability reference measure λ D \lambda_D λ D
J D R , J ( r ) : = e − β σ D ( r ) C D R , J ( r ) = e b 0 Vol ( D ) ∫ R e − β H D Ψ ( ϕ ∣ r ) − V D J ( ϕ ) d λ D . (126) \mathcal{J}_D^{R,J}(r):=e^{-\beta\sigma_D(r)}C_D^{R,J}(r)=e^{b_0\operatorname{Vol}(D)}\int_R e^{-\beta H_D^\Psi(\phi\mid r)-V_D^J(\phi)}\,d\lambda_D. \tag*{(126)} J D R , J ( r ) := e − β σ D ( r ) C D R , J ( r ) = e b 0 Vol ( D ) ∫ R e − β H D Ψ ( ϕ ∣ r ) − V D J ( ϕ ) d λ D . (126) Proof. Let G D \mathcal{G}_D G D (12) and set L D = ∑ i ∈ I ( D ) ( ϕ i − r i ) N ^ i \mathcal{L}_D=\sum_{i\in I(D)}(\phi_i-r_i)\widehat{N}_i L D = ∑ i ∈ I ( D ) ( ϕ i − r i ) N i h h h (11) cancel, giving
F D ( ϕ ∣ ξ ) F D ( r ∣ ξ ) = G D ( ϕ ) G D ( r ) E ν r ( ⋅ ∣ ξ ) e L D . \frac{F_D(\phi\mid\xi)}{F_D(r\mid\xi)}=\frac{\mathcal{G}_D(\phi)}{\mathcal{G}_D(r)}\mathbb{E}_{\nu_r(\cdot\mid\xi)}e^{\mathcal{L}_D}. F D ( r ∣ ξ ) F D ( ϕ ∣ ξ ) = G D ( r ) G D ( ϕ ) E ν r ( ⋅ ∣ ξ ) e L D . Average over ν r , o u t \nu_{r,\mathrm{out}} ν r , out e L D ν r / E ν r e L D e^{\mathcal{L}_D}\nu_r/\mathbb{E}_{\nu_r}e^{\mathcal{L}_D} e L D ν r / E ν r e L D ϕ D r D c \phi_Dr_{D^c} ϕ D r D c ν r , ϕ ; D \nu_{r,\phi;D} ν r , ϕ ; D e − V D J ( ϕ ) e^{-V_D^J(\phi)} e − V D J ( ϕ ) G D ( ϕ ) E ν r e L D / G D ( r ) \mathcal{G}_D(\phi)\mathbb{E}_{\nu_r}e^{\mathcal{L}_D}/\mathcal{G}_D(r) G D ( ϕ ) E ν r e L D / G D ( r ) − β g β , h -\beta g_{\beta,h} − β g β , h − β H D Ψ -\beta H_D^\Psi − β H D Ψ r r r ϕ \phi ϕ
G D ( ϕ ) G D ( r ) E ν r e L D = e − β [ H D Ψ ( ϕ ∣ r ) − H D Ψ ( r ∣ r ) ] . \frac{\mathcal{G}_D(\phi)}{\mathcal{G}_D(r)}\mathbb{E}_{\nu_r}e^{\mathcal{L}_D}=e^{-\beta[H_D^\Psi(\phi\mid r)-H_D^\Psi(r\mid r)]}. G D ( r ) G D ( ϕ ) E ν r e L D = e − β [ H D Ψ ( ϕ ∣ r ) − H D Ψ ( r ∣ r )] . Insert this identity into C D R , J C_D^{R,J} C D R , J e − β σ D ( r ) e^{-\beta\sigma_D(r)} e − β σ D ( r ) e ( b r + β e ( r ) ) Vol ( D ) e^{(b_r+\beta e(r))\operatorname{Vol}(D)} e ( b r + β e ( r )) Vol ( D ) □ \square □
Set J q p h y s ( D ) = J D R q ( D ) , 0 ( r q ) \mathcal{J}_q^{\mathrm{phys}}(D)=\mathcal{J}_D^{R_q(D),0}(r_q) J q phys ( D ) = J D R q ( D ) , 0 ( r q ) R = U q ( D ) R=U_q(D) R = U q ( D ) J = − log G q , D a u x J=-\log G_{q,D}^{\mathrm{aux}} J = − log G q , D aux I I I J = e b 0 Vol I \mathcal{J}=e^{b_0\operatorname{Vol}}I J = e b 0 Vol I Z D , F ( ξ ) Z_{D,F}(\xi) Z D , F ( ξ )
J F = e b r q Vol ( D ) − β σ D ( r q ) E ν r q , o u t Z D , F ( ξ ) F D ( r q ∣ ξ ) . \mathcal{J}_F=e^{b_{r_q}\operatorname{Vol}(D)-\beta\sigma_D(r_q)}\mathbb{E}_{\nu_{r_q,\mathrm{out}}}\frac{Z_{D,F}(\xi)}{F_D(r_q\mid\xi)}. J F = e b r q Vol ( D ) − β σ D ( r q ) E ν r q , out F D ( r q ∣ ξ ) Z D , F ( ξ ) . The interaction Ψ = Φ 0 + ψ \Psi=\Phi^0+\psi Ψ = Φ 0 + ψ ∥ ψ ∥ ≤ δ β \lVert\psi\rVert\leq\delta_\beta ∥ ψ ∥ ≤ δ β (124) ; its homogeneous trial bound is (43) .
Apply the restricted-pressure construction to the field interaction obtained from the clipped contour gas, and denote its physical pressure by p q p_q p q G G G
∣ log I a u x ( G ) − ( p q − b 0 ) Vol ( G ) ∣ ≤ σ 0 A ( G ) , ∣ p q − b 0 ∣ ≤ r β . (127) \left|\log I^{\mathrm{aux}}(G)-(p_q-b_0)\operatorname{Vol}(G)\right|\leq\sigma_0A(G),\qquad|p_q-b_0|\leq r_\beta. \tag*{(127)} ∣ log I aux ( G ) − ( p q − b 0 ) Vol ( G ) ∣ ≤ σ 0 A ( G ) , ∣ p q − b 0 ∣ ≤ r β . (127) Here r β = O ( log β + β δ β + t β ) r_\beta=O(\log\beta+\beta\delta_\beta+t_\beta) r β = O ( log β + β δ β + t β ) π q \pi_q π q p q = b 0 + π q / v s p_q=b_0+\pi_q/v_s p q = b 0 + π q / v s
The positive bare collar kernels satisfy, by 9.10, for arbitrary bounded caps,
K S ( v , w ) = K S ( v , r q ) K S ( r q , w ) k S e U S ( v , w ) , k S = K S ( r q , r q ) , ∣ U S ∣ ≤ β ϵ K A ( S ) . (128) K_S(v,w)=\frac{K_S(v,r_q)K_S(r_q,w)}{k_S}e^{U_S(v,w)},\qquad k_S=K_S(r_q,r_q),\qquad|U_S|\leq\beta\epsilon K_A(S). \tag*{(128)} K S ( v , w ) = k S K S ( v , r q ) K S ( r q , w ) e U S ( v , w ) , k S = K S ( r q , r q ) , ∣ U S ∣ ≤ β ϵ K A ( S ) . (128) The marked estimates 7.12 and 7.15 bound changes outside the source region E E E V D J E V_D^{JE} V D J E C t β A ( E ) Ct_\beta A(E) C t β A ( E ) H H H C κ C\kappa C κ
V D J E ( ϕ E r D ∖ E ) = V E J E ( ϕ E ) . (129) V_D^{JE}(\phi_{E r_D\setminus E})=V_E^{JE}(\phi_E). \tag*{(129)} V D J E ( ϕ E r D ∖ E ) = V E J E ( ϕ E ) . (129) At fixed L L L t β t_\beta t β L L L β \beta β κ \kappa κ
Put P = max q p q P=\max_q p_q P = max q p q σ = C σ ( σ 0 + t β + β δ β ) \sigma=C_\sigma(\sigma_0+t_\beta+\beta\delta_\beta) σ = C σ ( σ 0 + t β + β δ β ) C σ C_\sigma C σ
η c u t = C / L + C δ β + C e − c L , (130) \eta_{\mathrm{cut}}=C/L+C\delta_\beta+Ce^{-cL}, \tag*{(130)} η cut = C / L + C δ β + C e − c L , (130) E β = C [ σ + β ϵ K + β η c u t + β e − c L + t β ] L 2 + C ( r β + t β ) V L , (131) \mathcal{E}_\beta=C[\sigma+\beta\epsilon_K+\beta\eta_{\mathrm{cut}}+\beta e^{-cL}+t_\beta]L^2+C(r_\beta+t_\beta)V_L, \tag*{(131)} E β = C [ σ + β ϵ K + β η cut + β e − c L + t β ] L 2 + C ( r β + t β ) V L , (131) B ∗ = β τ L 2 − E β . (132) B_*=\beta\tau L^2-\mathcal{E}_\beta. \tag*{(132)} B ∗ = β τ L 2 − E β . (132) The induction requires B ∗ ≥ 3 T B_*\geq3\mathcal{T} B ∗ ≥ 3 T B ∗ B_* B ∗ log C 1 \log C_1 log C 1 T = β τ L 2 / 16 \mathcal{T}=\beta\tau L^2/16 T = β τ L 2 /16 E β ≤ β τ L 2 / 2 \mathcal{E}_\beta\leq\beta\tau L^2/2 E β ≤ β τ L 2 /2 B ∗ ≥ 8 T B_*\geq8\mathcal{T} B ∗ ≥ 8 T
The root divisors satisfy
log k S ≥ − C ( log β + β δ β ) Vol ( S ) . (133) \log k_S\geq-C(\log\beta+\beta\delta_\beta)\operatorname{Vol}(S). \tag*{(133)} log k S ≥ − C ( log β + β δ β ) Vol ( S ) . (133) Apply Lemma 9.4 on S S S r q r_q r q
Single-contour and simultaneous-family estimates
Lemma 11.2 (Allocation of interactions to collars). Partition the integrated coordinates into an exterior cap E E E C C C I j I_j I j S 0 , S j S_0,S_j S 0 , S j S 0 S_0 S 0 E E E S j S_j S j I j I_j I j § 10.3 , so that a remaining declared support meets at most one collar. A support missing all collars is then assigned to its unique cap; a support meeting a collar is assigned to that collar. Assign by full declared supports, even when a term is constant at the fields in question. With the fixed exterior coordinates included in the appropriate cap, this gives the exact decomposition
H c u t = H E + H C + ∑ j H I j + h 0 ( S 0 ; E , C ) + ∑ j h j ( S j ; C , I j ) . H^{\mathrm{cut}}=H_E+H_C+\sum_j H_{I_j}+h_0(S_0;E,C)+\sum_j h_j(S_j;C,I_j). H cut = H E + H C + j ∑ H I j + h 0 ( S 0 ; E , C ) + j ∑ h j ( S j ; C , I j ) . Suppose the restrictions, after fixing the prescribed majority label on every collar, are products of collar-majority indicators and cap-only indicators. Allow any additional nonnegative cap weights, and absorb these and e − β H α e^{-\beta H_\alpha} e − β H α a α a_\alpha a α K 0 c u t ( c , e ) K_0^{\mathrm{cut}}(c,e) K 0 cut ( c , e ) K j c u t ( i , c ) K_j^{\mathrm{cut}}(i,c) K j cut ( i , c ) e − β h 0 e^{-\beta h_0} e − β h 0 e − β h j e^{-\beta h_j} e − β h j
Proof. A connected support meeting distinct caps crosses a collar. If it meets only S j S_j S j C , I j C,I_j C , I j S 0 S_0 S 0 C , E C,E C , E Lemma 10.3 .
Lemma 11.3 (Factorization through collar kernels). With the preceding allocation, let I c u t I^{\mathrm{cut}} I cut r e r_e r e M e M_e M e ϵ e \epsilon_e ϵ e
K e c u t ( v , w ) ≤ e ϵ e + 4 β M e K e c u t ( v , r e ) K e c u t ( r e , w ) k e c u t , k e c u t = K e c u t ( r e , r e ) . K_e^{\mathrm{cut}}(v,w) \le e^{\epsilon_e+4\beta M_e}\frac{K_e^{\mathrm{cut}}(v,r_e)K_e^{\mathrm{cut}}(r_e,w)}{k_e^{\mathrm{cut}}}, \qquad k_e^{\mathrm{cut}}=K_e^{\mathrm{cut}}(r_e,r_e). K e cut ( v , w ) ≤ e ϵ e + 4 β M e k e cut K e cut ( v , r e ) K e cut ( r e , w ) , k e cut = K e cut ( r e , r e ) . Consequently the cut integral obeys
I c u t ≤ e ∑ e ( ϵ e + 4 β M e ) F E c u t F C c u t ∏ j F j c u t k 0 c u t ∏ j k j c u t , (134) I^{\mathrm{cut}}\le e^{\sum_e(\epsilon_e+4\beta M_e)}\frac{F_E^{\mathrm{cut}}F_C^{\mathrm{cut}}\prod_jF_j^{\mathrm{cut}}}{k_0^{\mathrm{cut}}\prod_j k_j^{\mathrm{cut}}}, \tag*{(134)} I cut ≤ e ∑ e ( ϵ e + 4 β M e ) k 0 cut ∏ j k j cut F E cut F C cut ∏ j F j cut , (134) where
F E c u t = ∫ a E ( e ) K 0 c u t ( r q , e ) d λ E ( e ) , F_E^{\mathrm{cut}}=\int a_E(e)K_0^{\mathrm{cut}}(r_q,e)\,d\lambda_E(e), F E cut = ∫ a E ( e ) K 0 cut ( r q , e ) d λ E ( e ) , F C c u t = ∫ a C ( c ) K 0 c u t ( c , r q ) ∏ j K j c u t ( r p j , c ) d λ C ( c ) , F_C^{\mathrm{cut}}=\int a_C(c)K_0^{\mathrm{cut}}(c,r_q)\prod_jK_j^{\mathrm{cut}}(r_{p_j},c)\,d\lambda_C(c), F C cut = ∫ a C ( c ) K 0 cut ( c , r q ) j ∏ K j cut ( r p j , c ) d λ C ( c ) , F j c u t = ∫ a j ( i ) K j c u t ( i , r p j ) d λ I j ( i ) . F_j^{\mathrm{cut}}=\int a_j(i)K_j^{\mathrm{cut}}(i,r_{p_j})\,d\lambda_{I_j}(i). F j cut = ∫ a j ( i ) K j cut ( i , r p j ) d λ I j ( i ) . Proof. The energy comparison gives ∣ log K e c u t − log K e ∣ ≤ β M e \lvert\log K_e^{\mathrm{cut}}-\log K_e\rvert\le\beta M_e ∣ log K e cut − log K e ∣ ≤ β M e 4 β M e 4\beta M_e 4 β M e
∫ a E ( e ) a C ( c ) ∏ j a j ( i j ) K 0 c u t ( c , e ) ∏ j K j c u t ( i j , c ) d λ E d λ C ∏ j d λ I j . \int a_E(e)a_C(c)\prod_j a_j(i_j)K_0^{\mathrm{cut}}(c,e)\prod_jK_j^{\mathrm{cut}}(i_j,c)\,d\lambda_E\,d\lambda_C\prod_jd\lambda_{I_j}. ∫ a E ( e ) a C ( c ) j ∏ a j ( i j ) K 0 cut ( c , e ) j ∏ K j cut ( i j , c ) d λ E d λ C j ∏ d λ I j . Apply the cross-ratio inequality pointwise. The remaining factors depend on separate caps, and Tonelli gives (134) . The cap weights may include boundary cocycles. For an empty inner cap the root factor cancels its divisor.
In each factor, terms assigned to an incident collar are integrated with that collar’s copied variables. Its root divisor retains the same terms at the root. A term meeting m m m C ( 1 + m ) C(1+m) C ( 1 + m ) Lemma 10.11 bounds the total cost.
Lemma 11.4. Suppose, in every strictly smaller physical inherited domain H H H p p p
Z p p h y s ( H ) ≤ e P Vol ( H ) + σ A ( H ) . \mathcal{Z}_p^{\mathrm{phys}}(H)\le e^{P\operatorname{Vol}(H)+\sigma A(H)}. Z p phys ( H ) ≤ e P Vol ( H ) + σ A ( H ) . Then every q q q D D D
log W Γ ≤ − B ∗ n Γ + ( P − p q ) Vol ( V Γ ) . (135) \log W_\Gamma\le-B_*n_\Gamma+(P-p_q)\operatorname{Vol}(V_\Gamma). \tag*{(135)} log W Γ ≤ − B ∗ n Γ + ( P − p q ) Vol ( V Γ ) . (135) Proof. Keep numerator and denominator of (115) as joint microscopic integrals on the same U = ( A ∖ K ) ∪ ⋃ j D j U=(A\setminus K)\cup\bigcup_jD_j U = ( A ∖ K ) ∪ ⋃ j D j D j D_j D j Lemma 10.14 , every added auxiliary species meets the core or the first inherited boundary layer. The core and these layers contain at most C L n Γ C_Ln_\Gamma C L n Γ Lemma 10.15 therefore bounds the comparison cost by C t β V L n Γ C_t\beta V_Ln_\Gamma C t β V L n Γ
Use the shifted collar O , S , V ′ O,S,V' O , S , V ′ V ′ V' V ′ U U U R R R J J J
Z U R , J ( ξ ) = F U ( r q ∣ ξ ) e β H U Ψ ( r q ∣ r q ) ∫ R e − β H U Ψ ( ϕ ∣ r q ) − β B U ξ , r q ( ϕ ) − V U J , ξ ( ϕ ) d λ U . Z_U^{R,J}(\xi)=F_U(r_q\mid\xi)e^{\beta H_U^\Psi(r_q\mid r_q)}
\int_R e^{-\beta H_U^\Psi(\phi\mid r_q)-\beta B_U^{\xi,r_q}(\phi)-V_U^{J,\xi}(\phi)}\,d\lambda_U. Z U R , J ( ξ ) = F U ( r q ∣ ξ ) e β H U Ψ ( r q ∣ r q ) ∫ R e − β H U Ψ ( ϕ ∣ r q ) − β B U ξ , r q ( ϕ ) − V U J , ξ ( ϕ ) d λ U . The prefactor is the same for numerator and denominator and cancels. The frozen-boundary weak comparison replaces V U J , ξ V_U^{J,\xi} V U J , ξ B U ξ , r q B_U^{\xi,r_q} B U ξ , r q O O O C β e − c L L 2 n Γ C\beta e^{-cL}L^{2n_\Gamma} C β e − c L L 2 n Γ O , S O,S O , S C t β A ( V ′ ) C_{t\beta}\mathcal{A}(V') C tβ A ( V ′ ) (129) leaves exactly V V ′ J V ′ V_{V'}^{J V'} V V ′ J V ′
Let a ξ ( w ) a_\xi(w) a ξ ( w ) (128) gives
∫ a ξ ( w ) K S ( v , w ) d λ O ( w ) = A ξ K S ( v , r q ) e ϵ ( v ) , A ξ = ∫ a ξ ( w ) K S ( r q , w ) d λ O ( w ) k S , ∣ ϵ ∣ ≤ β ϵ K A ( S ) . \int a_\xi(w)K_S(v,w)\,d\lambda_O(w)=A_\xi K_S(v,r_q)e^{\epsilon(v)},\qquad A_\xi=\frac{\int a_\xi(w)K_S(r_q,w)\,d\lambda_O(w)}{k_S},\qquad|\epsilon|\le\beta\epsilon_K\mathcal{A}(S). ∫ a ξ ( w ) K S ( v , w ) d λ O ( w ) = A ξ K S ( v , r q ) e ϵ ( v ) , A ξ = k S ∫ a ξ ( w ) K S ( r q , w ) d λ O ( w ) , ∣ ϵ ∣ ≤ β ϵ K A ( S ) . The same A ξ A_\xi A ξ V a u g = S ∪ V ′ V_{\mathrm{aug}}=S\cup V' V aug = S ∪ V ′ S S S S S S V a u g V_{\mathrm{aug}} V aug (127) applies also when this domain is punctured.
Choose the inner collars S j ⊂ D j S_j\subset D_j S j ⊂ D j Lemma 10.11 counts every restoration; the protected charge is untouched. Allocate terms by full declared supports and apply Lemma 11.3 . Keep both cap factors and each root divisor, then restore the deleted terms. If A 0 = V a u g ∖ ⋃ j D j A_0=V_{\mathrm{aug}}\setminus\bigcup_jD_j A 0 = V aug ∖ ⋃ j D j
I N ( V a u g ) ≤ e E i n I c o r e + ∏ j I p j p h y s ( D j ) ∏ j k j , Vol ( c o r e + ) = Vol ( A 0 ) + ∑ j Vol ( S j ) . I_N(V_{\mathrm{aug}})\le e^{E_{\mathrm{in}}}\frac{I_{\mathrm{core}^+}\prod_j I_{p_j}^{\mathrm{phys}}(D_j)}{\prod_j k_j},\qquad\operatorname{Vol}(\mathrm{core}^+)=\operatorname{Vol}(A_0)+\sum_j\operatorname{Vol}(S_j). I N ( V aug ) ≤ e E in ∏ j k j I core + ∏ j I p j phys ( D j ) , Vol ( core + ) = Vol ( A 0 ) + j ∑ Vol ( S j ) . The core integral is at most exp [ − β τ L 2 n Γ + C r β Vol ( c o r e + ) ] \exp[-\beta\tau L^{2n_\Gamma}+C r_\beta\operatorname{Vol}(\mathrm{core}^+)] exp [ − β τ L 2 n Γ + C r β Vol ( core + )] Proposition 10.10 . Apply the induction hypothesis to the unchanged D j D_j D j
( P − b 0 ) ∑ j Vol ( D j ) − ( p q − b 0 ) Vol ( V a u g ) = ( P − p q ) Vol ( V a u g ) − ( P − b 0 ) Vol ( A 0 ) . \begin{aligned}
(P-b_0)\sum_j\operatorname{Vol}(D_j)-(p_q-b_0)\operatorname{Vol}(V_{\mathrm{aug}}) \\
&=(P-p_q)\operatorname{Vol}(V_{\mathrm{aug}})-(P-b_0)\operatorname{Vol}(A_0).
\end{aligned} ( P − b 0 ) j ∑ Vol ( D j ) − ( p q − b 0 ) Vol ( V aug ) = ( P − p q ) Vol ( V aug ) − ( P − b 0 ) Vol ( A 0 ) . The last term is a core error, and the first is at most ( P − p q ) Vol ( V Γ ) (P-p_q)\operatorname{Vol}(V_\Gamma) ( P − p q ) Vol ( V Γ ) C L 2 n Γ CL^{2n_\Gamma} C L 2 n Γ C V L n Γ CVL^{n_\Gamma} C V L n Γ
Lemma 11.5. Under the same smaller-domain hypothesis, a selected maximal active q q q F \mathcal{F} F N = ∑ Γ n Γ N=\sum_\Gamma n_\Gamma N = ∑ Γ n Γ E = D ∖ ⋃ Γ V Γ E=D\setminus\bigcup_\Gamma V_\Gamma E = D ∖ ⋃ Γ V Γ
I F ≤ exp { P Vol ( D ) + σ A ( D ) − B ∗ N − ( P − p q ) Vol ( E ) } . (136) I_{\mathcal{F}}\le\exp\{P\operatorname{Vol}(D)+\sigma\mathcal{A}(D)-B_*N-(P-p_q)\operatorname{Vol}(E)\}. \tag*{(136)} I F ≤ exp { P Vol ( D ) + σ A ( D ) − B ∗ N − ( P − p q ) Vol ( E )} . (136) Proof. Apply (10.7), integrate all rods together, and use the common canonical normalization once. The result is bounded by
e b 0 Vol ( D ) ∫ R F e − β H D Ψ ( ϕ ∣ r q ) − V D J E ( ϕ ) d λ D . e^{b_0\operatorname{Vol}(D)}\int_{R_{\mathcal{F}}}e^{-\beta H_D^\Psi(\phi\mid r_q)-V_D^{JE}(\phi)}\,d\lambda_D. e b 0 Vol ( D ) ∫ R F e − β H D Ψ ( ϕ ∣ r q ) − V D J E ( ϕ ) d λ D . Set fields outside E E E C t β A ( E ) C t_{\beta}\mathcal{A}(E) C t β A ( E ) (129) . Let S 0 ⊂ A = ⋃ Γ A Γ S_0 \subset A = \bigcup_{\Gamma} A_{\Gamma} S 0 ⊂ A = ⋃ Γ A Γ S j ⊂ D j S_j \subset D_j S j ⊂ D j C = A ∖ S 0 C = A \setminus S_0 C = A ∖ S 0 I j = D j ∖ S j I_j = D_j \setminus S_j I j = D j ∖ S j Lemma 11.3 factors the collar integrations and applies the cross ratios with complete root divisors. Restore all terms with their multiplicities. The exterior acquires a copy of S 0 S_0 S 0 S j S_j S j
Write E + = E ∪ S 0 E^+ = E \cup S_0 E + = E ∪ S 0 S 0 S_0 S 0 J E J_E J E S 0 S_0 S 0 C t β A ( E ) + C κ Vol ( S 0 ) C t_{\beta}\mathcal{A}(E) + C_{\kappa}\operatorname{Vol}(S_0) C t β A ( E ) + C κ Vol ( S 0 )
e − b 0 Vol ( D ) J Γ ≤ e E F I q a u x ( E + ) ( ∏ j I p j p h y s ( D j ) ) I c o r e + k 0 ∏ j k j . e^{-b_0\operatorname{Vol}(D)}\mathcal{J}_{\Gamma} \le e^{E^F}\frac{I^{\mathrm{aux}}_q(E^+)\left(\prod_j I^{\mathrm{phys}}_{p_j}(D_j)\right)I_{\mathrm{core}^+}}{k_0\prod_j k_j}. e − b 0 Vol ( D ) J Γ ≤ e E F k 0 ∏ j k j I q aux ( E + ) ( ∏ j I p j phys ( D j ) ) I core + . where
Vol ( E + ) = Vol ( E ) + Vol ( S 0 ) , Vol ( c o r e + ) = Vol ( A ) + ∑ j Vol ( S j ) . \operatorname{Vol}(E^+) = \operatorname{Vol}(E) + \operatorname{Vol}(S_0), \qquad\operatorname{Vol}(\mathrm{core}^+) = \operatorname{Vol}(A) + \sum_j \operatorname{Vol}(S_j). Vol ( E + ) = Vol ( E ) + Vol ( S 0 ) , Vol ( core + ) = Vol ( A ) + j ∑ Vol ( S j ) . The copied field volumes obey
Vol ( E + ) + ∑ j Vol ( D j ) + Vol ( c o r e + ) − Vol ( S 0 ) − ∑ j Vol ( S j ) = Vol ( D ) . \operatorname{Vol}(E^+) + \sum_j\operatorname{Vol}(D_j) + \operatorname{Vol}(\mathrm{core}^+) - \operatorname{Vol}(S_0) - \sum_j\operatorname{Vol}(S_j) = \operatorname{Vol}(D). Vol ( E + ) + j ∑ Vol ( D j ) + Vol ( core + ) − Vol ( S 0 ) − j ∑ Vol ( S j ) = Vol ( D ) . Apply the auxiliary surface estimate, the physical induction, the root-divisor trial, and the protected charge. The bulk expression is
b 0 Vol ( D ) + ( p q − b 0 ) [ Vol ( E ) + Vol ( S 0 ) ] + ( P − b 0 ) ∑ j Vol ( D j ) = P Vol ( D ) − ( P − p q ) Vol ( E ) + ( p q − b 0 ) Vol ( S 0 ) − ( P − b 0 ) Vol ( A ) . \begin{aligned}
b_0\operatorname{Vol}(D) + (p_q-b_0)[\operatorname{Vol}(E)+\operatorname{Vol}(S_0)] + (P-b_0)\sum_j\operatorname{Vol}(D_j) \\
&= P\operatorname{Vol}(D) - (P-p_q)\operatorname{Vol}(E) + (p_q-b_0)\operatorname{Vol}(S_0) - (P-b_0)\operatorname{Vol}(A).
\end{aligned} b 0 Vol ( D ) + ( p q − b 0 ) [ Vol ( E ) + Vol ( S 0 )] + ( P − b 0 ) j ∑ Vol ( D j ) = P Vol ( D ) − ( P − p q ) Vol ( E ) + ( p q − b 0 ) Vol ( S 0 ) − ( P − b 0 ) Vol ( A ) . The last two terms have magnitude at most C r β V L N C r_{\beta} V L N C r β V L N A ( E + ) ≤ A ( D ) + C L 2 N \mathcal{A}(E^+) \le\mathcal{A}(D) + C L^2 N A ( E + ) ≤ A ( D ) + C L 2 N ∑ j A ( D j ) ≤ C L 2 N \sum_j\mathcal{A}(D_j) \le C L^2 N ∑ j A ( D j ) ≤ C L 2 N σ \sigma σ (132) , leaving − B ∗ N -B_*N − B ∗ N
The numerical tree and the physical pressure
The simultaneous induction below adapts Zahradník’s stability argument [5 Theorem 1.7 and Section 1.8] : an active contour must be large relative to the pressure deficit, and summing such contours yields the physical pressure bound.
Proposition 11.6 (Finite-volume pressure bounds). For the canonical model constructed above, let the error budget (132) satisfy B ∗ ≥ 3 T B_* \ge3T B ∗ ≥ 3 T B ∗ B_* B ∗ D D D q q q
e p q Vol ( D ) − σ A ( D ) ≤ Z q p h y s ( D ) ≤ e P Vol ( D ) + σ A ( D ) . (137) e^{p_q\operatorname{Vol}(D)-\sigma\mathcal{A}(D)} \le\mathcal{Z}^{\mathrm{phys}}_q(D) \le e^{P\operatorname{Vol}(D)+\sigma\mathcal{A}(D)}. \tag*{(137)} e p q Vol ( D ) − σ A ( D ) ≤ Z q phys ( D ) ≤ e P Vol ( D ) + σ A ( D ) . (137) Proof. Induct simultaneously over all phases and physical domains, ordered by their number of coarse cubes. The cutoff pressures and P P P a = ( P − p q ) V L ≥ 0 a = (P-p_q)V_L \ge0 a = ( P − p q ) V L ≥ 0
− T n < log W Γ ≤ − B ∗ n + C a n 3 / 2 , -Tn < \log W_{\Gamma} \le-B_*n + Can^{3/2}, − T n < log W Γ ≤ − B ∗ n + C a n 3/2 , so for a > 0 a>0 a > 0
N ( a ) = max { 1 , ⌈ c ( B ∗ − T a ) 2 ⌉ } . N(a)=\max\left\{1,\left\lceil c\left(\frac{B_*-\mathcal{T}}{a}\right)^2\right\rceil\right\}. N ( a ) = max { 1 , ⌈ c ( a B ∗ − T ) 2 ⌉ } . At a = 0 a=0 a = 0 e P Vol ( D ) + σ A ( D ) e^{P\operatorname{Vol}(D)+\sigma A(D)} e P Vol ( D ) + σ A ( D ) (136) leaves a sum of weights e − B ∗ ∑ n Γ e − a n ( E ) e^{-B_* \sum n_\Gamma}e^{-an(E)} e − B ∗ ∑ n Γ e − an ( E ) E E E e − a e^{-a} e − a n ≥ N ( a ) n\ge N(a) n ≥ N ( a ) C 1 n C_1^n C 1 n q q q
e − a + ∑ n ≥ N ( a ) C 1 n e − B ∗ n ≤ 1. e^{-a}+\sum_{n\ge N(a)}C_1^n e^{-B_*n}\le1. e − a + n ≥ N ( a ) ∑ C 1 n e − B ∗ n ≤ 1. For a ≥ 1 a\ge1 a ≥ 1 1 − e − 1 1-e^{-1} 1 − e − 1 B ∗ B_* B ∗ 0 < a ≤ 1 0<a\le1 0 < a ≤ 1 C exp [ − c B ∗ 3 / a 2 ] ≤ a / 2 ≤ 1 − e − a C\exp[-cB_*^3/a^2]\le a/2\le1-e^{-a} C exp [ − c B ∗ 3 / a 2 ] ≤ a /2 ≤ 1 − e − a a = 0 a=0 a = 0
For the lower bound, clip the exact regional gas before enlarging it. Positivity makes the physical partition at least this original regional cutoff partition. Apply Lemma 10.14 with one domain and empty core. Added species meet its first boundary layer, whose size is bounded by C A ( D ) / L 2 CA(D)/L^2 C A ( D ) / L 2 σ \sigma σ
Theorem 11.7 (Pressure maximum). For the parameter range in Proposition 11.6,
p ε ( h ) = max q p q . (138) p_\varepsilon(h)=\max_q p_q. \tag*{(138)} p ε ( h ) = q max p q . (138) Proof. At fixed β \beta β e − C β e^{-C\beta} e − C β q q q F D ( r ∣ ξ ) F_D(r\mid\xi) F D ( r ∣ ξ ) e b r Vol ( D ) e^{b_r\operatorname{Vol}(D)} e b r Vol ( D ) ξ \xi ξ σ D ( r ) \sigma_D(r) σ D ( r ) (137) by volume along boxes. The upper bound gives a pressure limsup at most P P P P P P (138) .
The crossing and the density jump
We first record the elementary pressure facts, then compare the source responses of the phase pressures. All pressures are per Euclidean volume. The mesh, count caps, field interval, phase neighborhoods, and contour index sets remain fixed as h h h
Lemma 12.1 (Thermodynamic pressure). For each ε > 0 \varepsilon>0 ε > 0 p ε ( h ) p_\varepsilon(h) p ε ( h ) h h h
Proof. We use the separated-box construction for the thermodynamic limit [30 Section 2] . Dropping exclusions between boxes gives a product upper bound. Boxes separated by an empty strip wider than 1 + ε 1+\varepsilon 1 + ε S S S log Z Λ \log Z_\Lambda log Z Λ z ∣ S ∣ z|S| z ∣ S ∣
the lower bound. Tile large cubes with boxes of fixed side, then let that side increase. The upper and lower bounds converge to the same limit; the strip estimate also removes exterior boundary effects. Finally, 0 ≤ log Z Λ ≤ z ∣ Λ ∣ 0 \le\log Z_{\Lambda} \le z|\Lambda| 0 ≤ log Z Λ ≤ z ∣Λ∣ h h h
Lemma 12.2 (Density response). At every differentiability point of p ε p_{\varepsilon} p ε p ε ′ p_{\varepsilon}' p ε ′ h h h
Proof. The first two derivatives of log Z Λ \log Z_{\Lambda} log Z Λ E N Λ \mathbb{E}N_{\Lambda} E N Λ Var N Λ \operatorname{Var}N_{\Lambda} Var N Λ [16 Theorems 24.1 and 24.5] .
Source response of the joint model
Let D D D E E E J h = ∑ A J A , h J_h = \sum_A J_{A,h} J h = ∑ A J A , h Z D E , J ( h ∣ ξ ) Z_D^{E,J}(h \mid\xi) Z D E , J ( h ∣ ξ ) (12) over E E E e − J h e^{-J_h} e − J h ξ \xi ξ E E E h h h
sup i ∑ A ∋ i ∥ J A , h ∥ ∞ ≤ κ , sup i ∑ A ∋ i Lip h ( J A , h ) < ∞ . (139) \sup_{i}\sum_{A\ni i}\|J_{A,h}\|_{\infty}\le\kappa,\qquad\sup_{i}\sum_{A\ni i}\operatorname{Lip}_{h}(J_{A,h})<\infty. \tag*{(139)} i sup A ∋ i ∑ ∥ J A , h ∥ ∞ ≤ κ , i sup A ∋ i ∑ Lip h ( J A , h ) < ∞. (139) Fix 0 < h ∗ ≤ H 0<h_{*}\le H 0 < h ∗ ≤ H ∣ h ∣ ≤ h ∗ |h|\le h_{*} ∣ h ∣ ≤ h ∗
Lemma 12.3 (Source score). For almost every h h h
S h = ∑ i ∈ I ( D ) [ β v i 2 α ( ϕ i − h ) + N i − N ^ i − ∂ h log c i , h ( N ^ i ) ] − ∂ h J h . (140) S_h=\sum_{i\in I(D)}\left[\frac{\beta v_i}{2\alpha}(\phi_i-h)+N_i-\widehat N_i-\partial_h\log c_{i,h}(\widehat N_i)\right]-\partial_hJ_h. \tag*{(140)} S h = i ∈ I ( D ) ∑ [ 2 α β v i ( ϕ i − h ) + N i − N i − ∂ h log c i , h ( N i ) ] − ∂ h J h . (140) Proof. Differentiate the physical activity factor and the attached Gaussian in (10) . Their count terms combine to N i − N ^ i N_i-\widehat N_i N i − N i h h h J h J_h J h
Lemma 12.4 (Exceptional counts). If ρ > e ζ e h ∗ \rho>e^{\zeta}e^{h_*} ρ > e ζ e h ∗ E E E Z D E , J > 0 Z_D^{E,J}>0 Z D E , J > 0
E h E , J ∑ i ∈ I ( D ) ( N i − N ^ i ) ≤ C β ∣ D ∣ e − c β . (141) \mathbb{E}_h^{E,J}\sum_{i\in I(D)}(N_i-\widehat N_i)\le C\beta|D|e^{-c\beta}. \tag*{(141)} E h E , J i ∈ I ( D ) ∑ ( N i − N i ) ≤ C β ∣ D ∣ e − c β . (141) Proof. Condition on the fields and the rods outside cell i i i J h J_h J h 2 κ 2\kappa 2 κ (11) , the empty term and every term with n > K i n>K_i n > K i F ≥ 0 F\ge0 F ≥ 0
E [ F ( N i ) 1 { N i > K i } ∣ ϕ , η i c ] ≤ e 2 κ ∑ n > K i F ( n ) ( β ζ e h ∗ v i ) n n ! . (142) \mathbb{E}\left[F(N_i)1_{\{N_i>K_i\}}\mid\phi,\eta_i^c\right]\le e^{2\kappa}\sum_{n>K_i}F(n)\frac{(\beta\zeta e^{h_*}v_i)^n}{n!}. \tag*{(142)} E [ F ( N i ) 1 { N i > K i } ∣ ϕ , η i c ] ≤ e 2 κ n > K i ∑ F ( n ) n ! ( β ζ e h ∗ v i ) n . (142) Apply the scalar tail bound in Lemma 7.2 with m = 1 m=1 m = 1 H H H h ∗ h_* h ∗ C β e − c β C\beta e^{-c\beta} C β e − c β ∣ D ∣ / v − |D|/v_- ∣ D ∣/ v −
The Gaussian means stay at distance at least d 0 > 0 d_0>0 d 0 > 0
1 − c i , h ( m ) ≤ 2 e − β v i d 0 2 / ( 4 α ) , ∣ ∂ h c i , h ( m ) ∣ ≤ C β v i e − β v i d 0 2 / ( 4 α ) . 1-c_{i,h}(m)\le2e^{-\beta v_i d_0^2/(4\alpha)},\qquad|\partial_h c_{i,h}(m)|\le C\sqrt{\beta v_i}\,e^{-\beta v_i d_0^2/(4\alpha)}. 1 − c i , h ( m ) ≤ 2 e − β v i d 0 2 / ( 4 α ) , ∣ ∂ h c i , h ( m ) ∣ ≤ C β v i e − β v i d 0 2 / ( 4 α ) . In particular, c i , h ( m ) ≥ 1 / 2 c_{i,h}(m) \ge1/2 c i , h ( m ) ≥ 1/2 β \beta β
∑ i ∈ I ( D ) ∣ ∂ h log c i , h ( N ^ i ) ∣ ≤ C β ∣ D ∣ e − c β . (143) \sum_{i\in I(D)} \left\lvert\partial_h \log c_{i,h}(\widehat{N}_i) \right\rvert\le C\beta|D|e^{-c\beta}. \tag*{(143)} i ∈ I ( D ) ∑ ∂ h log c i , h ( N i ) ≤ C β ∣ D ∣ e − c β . (143) These bounds justify differentiation under the finite-volume integral, so ∂ h log Z D E , J = E h E , J S h \partial_h \log Z_D^{E,J} = \mathbb{E}_h^{E,J} S_h ∂ h log Z D E , J = E h E , J S h
Differentiating the cutoff gas
The numerator and denominator of W Γ W_\Gamma W Γ h h h n Γ n_\Gamma n Γ V Γ V_\Gamma V Γ ∣ V Γ ∣ ≤ C V L n Γ 3 / 2 \lvert V_\Gamma\rvert\le C V L n_\Gamma^{3/2} ∣ V Γ ∣ ≤ C V L n Γ 3/2 J = 0 J=0 J = 0
∣ ∂ h log W Γ ∣ ≤ C β V L n Γ 3 / 2 . (144) \left\lvert\partial_h \log W_\Gamma\right\rvert\le C\beta V L n_\Gamma^{3/2}. \tag*{(144)} ∣ ∂ h log W Γ ∣ ≤ C β V L n Γ 3/2 . (144) A zero numerator has an allowed set of zero reference measure and vanishes for every h h h h h h T \mathcal{T} T
w Γ = min { W Γ , e − T n Γ } , ∣ ∂ h w Γ ∣ ≤ C β V L n Γ 3 / 2 e − T n Γ (145) w_\Gamma= \min\{W_\Gamma,e^{-\mathcal{T}n_\Gamma}\}, \qquad\left\lvert\partial_h w_\Gamma\right\rvert\le C\beta V L n_\Gamma^{3/2}e^{-\mathcal{T}n_\Gamma} \tag*{(145)} w Γ = min { W Γ , e − T n Γ } , ∣ ∂ h w Γ ∣ ≤ C β V L n Γ 3/2 e − T n Γ (145) almost everywhere. Its declared marked-cell support is unchanged, with ∣ S Γ ∣ ≤ d L n Γ \lvert S_\Gamma\rvert\le d_L n_\Gamma ∣ S Γ ∣ ≤ d L n Γ
Lemma 12.5 (Differentiated cutoff expansion). For every fixed a > 0 a>0 a > 0 J h = − log G h J_h=-\log G_h J h = − log G h
sup i ∑ A ⊃ i e a ∣ A ∣ ∥ J A , h ∥ ∞ ≤ C L e − T / 2 , (146) \sup_i \sum_{A\supset i} e^{a|A|}\lVert J_{A,h}\rVert_\infty\le C_L e^{-\mathcal{T}/2}, \tag*{(146)} i sup A ⊃ i ∑ e a ∣ A ∣ ∥ J A , h ∥ ∞ ≤ C L e − T /2 , (146) sup i ∑ A ⊃ i e a ∣ A ∣ Lip h ( J A , h ) ≤ C L β e − T / 4 , (147) \sup_i \sum_{A\supset i} e^{a|A|}\operatorname{Lip}_h(J_{A,h}) \le C_L\beta e^{-\mathcal{T}/4}, \tag*{(147)} i sup A ⊃ i ∑ e a ∣ A ∣ Lip h ( J A , h ) ≤ C L β e − T /4 , (147) when T \mathcal{T} T a a a L L L
Proof. Since ∣ S ( C ) ∣ ≤ d L s ( C ) \lvert S(C)\rvert\le d_Ls(C) ∣ S ( C )∣ ≤ d L s ( C ) (146) follows from (125) with b = a d L b=ad_L b = a d L (145) bounds its absolute majorant by C β V L ∑ j n j 3 / 2 ≤ C β V L s ( C ) 3 / 2 C\beta V L\sum_j n_j^{3/2}\le C\beta V Ls(C)^{3/2} C β V L ∑ j n j 3/2 ≤ C β V L s ( C ) 3/2 s 3 / 2 ≤ C e T s / 4 s^{3/2}\le Ce^{\mathcal{T}s/4} s 3/2 ≤ C e T s /4 3 T / 4 3\mathcal{T}/4 3 T /4 C L β e − 3 T / 8 ≤ C L β e − T / 4 C_L\beta e^{-3\mathcal{T}/8}\le C_L\beta e^{-\mathcal{T}/4} C L β e − 3 T /8 ≤ C L β e − T /4 (147) . This is the rooted derivative expansion of [26 Section 2, equations (2.6)–(2.7)] . Uniform absolute convergence permits integration between source values and gives the stated Lipschitz norm. □
Increment bounds for the phase pressures
Let [ m q − , m q + ] [m_q^-,m_q^+] [ m q − , m q + ] q q q q q q
m q − − ω 0 ≤ M D ( ϕ ) : = 1 2 α ∣ D ∣ ∑ i ∈ I ( D ) v i ϕ i ≤ m q + + ω 0 . (148) m_q^- - \omega_0 \le M_D(\phi) := \frac{1}{2\alpha|D|}\sum_{i\in I(D)}v_i\phi_i \le m_q^+ + \omega_0. \tag*{(148)} m q − − ω 0 ≤ M D ( ϕ ) := 2 α ∣ D ∣ 1 i ∈ I ( D ) ∑ v i ϕ i ≤ m q + + ω 0 . (148) Indeed, if a good spatial cell has mass error at most η / ( 2 α ) \eta/(2\alpha) η / ( 2 α ) θ \theta θ ω 0 = η / ( 2 α ) + R θ / α \omega_0=\eta/(2\alpha)+R\theta/\alpha ω 0 = η / ( 2 α ) + R θ / α
Lemma 12.6 (Microscopic realization of a branch). Let Z D , q c u t ( h ) Z^{\mathrm{cut}}_{D,q}(h) Z D , q cut ( h ) complete joint partition function restricted to q q q J q , D J_{q,D} J q , D
lim D ↑ R 3 ∣ D ∣ − 1 log Z D , q c u t ( h ) = p q ( h ) . \lim_{D\uparrow\mathbb{R}^{3}} |D|^{-1}\log Z^{\mathrm{cut}}_{D,q}(h)=p_q(h). D ↑ R 3 lim ∣ D ∣ − 1 log Z D , q cut ( h ) = p q ( h ) . Proof. The bare boundary cocycle, the frozen weak comparison (87) , and the rooted surface estimate give
log Z D , q c u t ( h ) = ( b 0 ( h ) + π q ( h ) v s ) ∣ D ∣ + O β ( A ( D ) ) . \log Z^{\mathrm{cut}}_{D,q}(h)=\left(b_0(h)+\frac{\pi_q(h)}{v_s}\right)|D|+O_\beta(A(D)). log Z D , q cut ( h ) = ( b 0 ( h ) + v s π q ( h ) ) ∣ D ∣ + O β ( A ( D )) . Divide by the volume of aligned boxes and use (110) .
Lemma 12.7 (Branch increments). Put Q q ( h ) = β − 1 p q ( h ) + h 2 / ( 4 α ) Q_q(h)=\beta^{-1}p_q(h)+h^2/(4\alpha) Q q ( h ) = β − 1 p q ( h ) + h 2 / ( 4 α ) σ β = C e − c β + C e − T / 4 → 0 \sigma_\beta=Ce^{-c\beta}+Ce^{-T/4}\to0 σ β = C e − c β + C e − T /4 → 0 h 1 < h 2 h_1<h_2 h 1 < h 2
( m ‾ q − ω 0 − σ β ) ( h 2 − h 1 ) ≤ Q q ( h 2 ) − Q q ( h 1 ) ≤ ( m q + + ω 0 + σ β ) ( h 2 − h 1 ) . (149) (\overline{m}_q-\omega_0-\sigma_\beta)(h_2-h_1)\le Q_q(h_2)-Q_q(h_1)\le(m_q^++\omega_0+\sigma_\beta)(h_2-h_1). \tag*{(149)} ( m q − ω 0 − σ β ) ( h 2 − h 1 ) ≤ Q q ( h 2 ) − Q q ( h 1 ) ≤ ( m q + + ω 0 + σ β ) ( h 2 − h 1 ) . (149) Proof. The source score, exceptional-count bound, and cutoff estimate imply
∂ h log Z D , q c u t β ∣ D ∣ = E D , q , h M D − h 2 α + ε D , q ( h ) , ∣ ε D , q ( h ) ∣ ≤ σ β . (150) \frac{\partial_h\log Z^{\mathrm{cut}}_{D,q}}{\beta|D|}=\mathbb{E}_{D,q,h}M_D-\frac{h}{2\alpha}+\varepsilon_{D,q}(h),\qquad|\varepsilon_{D,q}(h)|\le\sigma_\beta. \tag*{(150)} β ∣ D ∣ ∂ h log Z D , q cut = E D , q , h M D − 2 α h + ε D , q ( h ) , ∣ ε D , q ( h ) ∣ ≤ σ β . (150) Conversion from the marked-cell norm uses the positive minimum of v i v_i v i (148) , and pass to the pressure limit supplied by the preceding lemma.
Choosing the scales
Reduce the homogeneous mass gap so that 0 < Δ ≤ r I 0<\Delta\le r_I 0 < Δ ≤ r I
Fix H > 0 H>0 H > 0 α > ∥ T ∥ + 1 \alpha>\lVert T\rVert+1 α > ∥ T ∥ + 1 ρ > ζ e H \rho>\zeta e^H ρ > ζ e H R > H + 2 α ρ + 2 R>H+2\alpha\rho+2 R > H + 2 α ρ + 2 M = ζ e R + 1 M=\zeta e^{R+1} M = ζ e R + 1 K i K_i K i β \beta β
Choose disjoint phase neighborhoods preserving the mass gap, with good-cell mass error at most Δ / 16 \Delta/16 Δ/16 k ≥ 24 k\ge24 k ≥ 24
The protected charge has coefficient τ = c P θ 2 / 3 \tau=c_P\theta^{2/3} τ = c P θ 2/3 θ \theta θ C m θ / τ C_m\theta/\tau C m θ / τ C m C_m C m
Choose a well net of radius ν ≤ 1 \nu\le1 ν ≤ 1 R i n t R_{\mathrm{int}} R int C ν C\nu C ν C D ν 2 / R i n t CD_\nu^2/R_{\mathrm{int}} C D ν 2 / R int τ \tau τ η 0 > 0 \eta_0>0 η 0 > 0
C ( 1 + R i n t N ν ν ) ω ( η 0 ) ≤ c 0 τ . C\left(1+\frac{R_{\mathrm{int}}N_\nu}{\nu}\right)\omega(\eta_0)\le c_0\tau. C ( 1 + ν R int N ν ) ω ( η 0 ) ≤ c 0 τ . where c 0 c_0 c 0 L L L C / L ≤ η 0 / 3 C/L\le\eta_0/3 C / L ≤ η 0 /3 C / L + C e − c L ≪ τ C/L+Ce^{-cL}\ll\tau C / L + C e − c L ≪ τ
The weak interaction obeys δ β ( h ) ≤ C ∣ h ∣ + δ m i c r o , β \delta_\beta(h)\le C|h|+\delta_{\mathrm{micro},\beta} δ β ( h ) ≤ C ∣ h ∣ + δ micro , β δ m i c r o , β → 0 \delta_{\mathrm{micro},\beta}\to0 δ micro , β → 0 0 < h ∗ ≤ H 0<h_*\le H 0 < h ∗ ≤ H C L h ∗ CLh_* C L h ∗ C h ∗ Ch_* C h ∗ h ∗ / ( 2 α ) ≤ r I / 4 h_*/(2\alpha)\le r_I/4 h ∗ / ( 2 α ) ≤ r I /4 β \beta β L log β / β L\log\beta/\beta L log β / β
All choices except β \beta β
T = β τ L 2 16 . (151) \mathcal{T} = \frac{\beta\tau L^2}{16}. \tag*{(151)} T = 16 β τ L 2 . (151) This threshold satisfies Lemma 12.5 for large β \beta β (49) and (146) give β ( q β + κ ) → 0 \beta(q_\beta+\kappa)\to0 β ( q β + κ ) → 0 (63) . For the error E β E_\beta E β (132) , r β / β ≤ C h ∗ + o ( 1 ) r_\beta/\beta\le Ch_\ast+o(1) r β / β ≤ C h ∗ + o ( 1 ) t β / β → 0 t_\beta/\beta\to0 t β / β → 0 β L 2 \beta L^2 β L 2 O ( L h ∗ ) + o ( 1 ) O(Lh_\ast)+o(1) O ( L h ∗ ) + o ( 1 )
B ∗ = β τ L 2 − E β ≥ 1 2 β τ L 2 = 8 T . (152) B_\ast=\beta\tau L^2-E_\beta\ge\frac{1}{2}\beta\tau L^2=8\mathcal{T}. \tag*{(152)} B ∗ = β τ L 2 − E β ≥ 2 1 β τ L 2 = 8 T . (152) Lemma 12.8 (Equal limiting intercepts). At zero source.
max q ∣ p q ( 0 ) − b 0 ( 0 ) β ∣ ≤ C ( log β β + δ m i c r o , β + t β β + β − 4 ) ⟶ 0. (153) \max_q\left|\frac{p_q(0)-b_0(0)}{\beta}\right|\le C\left(\frac{\log\beta}{\beta}+\delta_{\mathrm{micro},\beta}+\frac{t_\beta}{\beta}+\beta^{-4}\right)\longrightarrow0. \tag*{(153)} q max β p q ( 0 ) − b 0 ( 0 ) ≤ C ( β log β + δ micro , β + β t β + β − 4 ) ⟶ 0. (153) Proof. At h = 0 h=0 h = 0 C ( δ m i c r o , β + t β / β ) C(\delta_{\mathrm{micro},\beta}+t_\beta/\beta) C ( δ micro , β + t β / β ) (96) and Proposition 7.15 . Apply (107) with this norm and use p q = b 0 + π q / v s p_q=b_0+\pi_q/v_s p q = b 0 + π q / v s β \beta β
Proof of the main theorem
The source adjustment is the LMP coexistence mechanism [2 Section 6, proof of Statement 3.3] : separated density responses force a crossing of the corrected phase pressures.
Proof of Theorem 1.1. The reserve (152) satisfies the quantitative conditions of Proposition 11.6 . Theorem 11.7 therefore gives
p ε ( h ) = max q p q ( h ) , ∣ h ∣ ≤ h ∗ . (154) p_\varepsilon(h)=\max_q p_q(h),\qquad|h|\le h_\ast. \tag*{(154)} p ε ( h ) = q max p q ( h ) , ∣ h ∣ ≤ h ∗ . (154) The isotropic component has mass r I r_I r I r I + Δ r_I+\Delta r I + Δ ω 0 ≤ Δ / 8 \omega_0\le\Delta/8 ω 0 ≤ Δ/8 β \beta β σ β ≤ Δ / 16 \sigma_\beta\le\Delta/16 σ β ≤ Δ/16 (149) gives
( r I − 3 Δ 16 ) ( h 2 − h 1 ) ≤ Q I ( h 2 ) − Q I ( h 1 ) ≤ ( r I + 3 Δ 16 ) ( h 2 − h 1 ) , (155) \left(r_I-\frac{3\Delta}{16}\right)(h_2-h_1)\le Q_I(h_2)-Q_I(h_1)\le\left(r_I+\frac{3\Delta}{16}\right)(h_2-h_1), \tag*{(155)} ( r I − 16 3Δ ) ( h 2 − h 1 ) ≤ Q I ( h 2 ) − Q I ( h 1 ) ≤ ( r I + 16 3Δ ) ( h 2 − h 1 ) , (155) Q q ( h 2 ) − Q q ( h 1 ) ≥ ( r I + 13 Δ 16 ) ( h 2 − h 1 ) , q ≠ I . (156) Q_q(h_2)-Q_q(h_1)\ge\left(r_I+\frac{13\Delta}{16}\right)(h_2-h_1),\qquad q\ne I. \tag*{(156)} Q q ( h 2 ) − Q q ( h 1 ) ≥ ( r I + 16 13Δ ) ( h 2 − h 1 ) , q = I . (156) There are finitely many components, and their increment bounds imply continuity. Thus D β ( h ) = max q ≠ I Q q ( h ) − Q I ( h ) D_\beta(h)=\max_{q\ne I}Q_q(h)-Q_I(h) D β ( h ) = max q = I Q q ( h ) − Q I ( h ) h 1 h_1 h 1
D β ( h 2 ) − D β ( h 1 ) ≥ 5 Δ 8 ( h 2 − h 1 ) . (157) D_\beta(h_2)-D_\beta(h_1)\ge\frac{5\Delta}{8}(h_2-h_1). \tag*{(157)} D β ( h 2 ) − D β ( h 1 ) ≥ 8 5Δ ( h 2 − h 1 ) . (157) By (153) , D β ( 0 ) → 0 D_\beta(0)\to0 D β ( 0 ) → 0 D β D_\beta D β h ε ∈ ( − h ∗ , h ∗ ) h_\varepsilon\in(-h_\ast,h_\ast) h ε ∈ ( − h ∗ , h ∗ ) β \beta β ∣ h ε ∣ ≤ 8 ∣ D β ( 0 ) ∣ / ( 5 Δ ) → 0 |h_\varepsilon|\le8|D_\beta(0)|/(5\Delta)\to0 ∣ h ε ∣ ≤ 8∣ D β ( 0 ) ∣/ ( 5Δ ) → 0
To the left of the zero the physical pressure equals the isotropic branch; to the right it equals the maximum of the other branches. Convexity gives one-sided derivatives. Their secant bounds are
p ε , − ′ ( h ε ) ≤ β ( r I + 3 Δ 16 − h ε 2 α ) , p'_{\varepsilon,-}(h_\varepsilon)\le\beta\left(r_I+\frac{3\Delta}{16}-\frac{h_\varepsilon}{2\alpha}\right), p ε , − ′ ( h ε ) ≤ β ( r I + 16 3Δ − 2 α h ε ) , p ε , + ′ ( h ε ) ≥ β ( r I + 13 Δ 16 − h ε 2 α ) . p'_{\varepsilon,+}(h_\varepsilon)\ge\beta\left(r_I+\frac{13\Delta}{16}-\frac{h_\varepsilon}{2\alpha}\right). p ε , + ′ ( h ε ) ≥ β ( r I + 16 13Δ − 2 α h ε ) . Therefore
p ε , + ′ ( h ε ) − p ε , − ′ ( h ε ) ≥ 5 Δ 8 ε ≥ Δ 2 ε . (158) p'_{\varepsilon,+}(h_{\varepsilon}) - p'_{\varepsilon,-}(h_{\varepsilon}) \ge\frac{5\Delta}{8\varepsilon} \ge\frac{\Delta}{2\varepsilon}. \tag*{(158)} p ε , + ′ ( h ε ) − p ε , − ′ ( h ε ) ≥ 8 ε 5Δ ≥ 2 ε Δ . (158) The isotropic lower increment bound, Δ ≤ r I \Delta\le r_I Δ ≤ r I h ∗ / ( 2 α ) ≤ r I / 4 h_*/(2\alpha) \le r_I/4 h ∗ / ( 2 α ) ≤ r I /4 β \beta β a ε = a t + h ε a_{\varepsilon} = a_t + h_{\varepsilon} a ε = a t + h ε P ε ( a ) = p ε ( a − a t ) P_{\varepsilon}(a) = p_{\varepsilon}(a - a_t) P ε ( a ) = p ε ( a − a t )
