No replica symmetry breaking phase in random field Ginzburg-Landau model

TL;DR

This paper proves that in the infinite volume limit of the random field Ginzburg-Landau model, the variance of the spin overlap disappears, indicating no replica symmetry breaking occurs.

Contribution

It establishes the absence of replica symmetry breaking in the random field Ginzburg-Landau model by leveraging its FKG property.

Findings

Variance of spin overlap vanishes in the infinite volume limit

No replica symmetry breaking occurs in the model

Utilizes FKG property to prove the result

Abstract

It is proved that the variance of spin overlap vanishes in the infinite volume limit of random field Ginzburg-Landau model using its FKG property.

Equations

$$H(\varphi ,g):=-\sum _{|x,y\in {\Lambda }_L}{J}_{x,y}{\varphi }_x{\varphi }_y-h\sum _{x\in {\Lambda }_L}{g}_x{\varphi }_x,$$

$$Z_L(\beta ,u,h,r,g):={\int }_{{\mathbb{R}}^{|{\Lambda }_L|}}D\varphi e^{-\beta H(\varphi ,g)},$$

$$D\varphi =\prod _{x\in {\Lambda }_L}d{\varphi }_x{e}^{-u{\varphi }_x^4+r{\varphi }_x^2}$$

$$⟨f(\varphi )⟩=\frac{1}{Z_L(\beta ,h,u,r,g)}{\int }_{{\mathbb{R}}^{|{\Lambda }_L|}}D\varphi f(\varphi )e^{-\beta H(\varphi ,g)}.$$

$${\psi }_L(\beta ,h,u,r,g):=\frac{1}{|{\Lambda }_L|}\log Z_L(\beta ,h,u,r,g),$$

$$p_L(\beta ,h,u,r):=\mathbb{E}{\psi }_L(\beta ,h,u,r,g),$$

$$H({\varphi }^1,\cdots ,{\varphi }^n,g):=\sum _{a=1}^{n}H({\varphi }^a,g),$$

$$H({\varphi }^1,\cdots ,{\varphi }^n,g)=H({\varphi }^{\sigma 1},\cdots ,{\varphi }^{\sigma n},g)$$

$$R_{a,b}:=\frac{1}{|{\Lambda }_L|}\sum _{x\in {\Lambda }_L}{\varphi }_x^a{\varphi }_x^b.$$

$$\mathbb{E}H({\varphi }^a,g)H({\varphi }^b,g)-\mathbb{E}H({\varphi }^a,g)\mathbb{E}H({\varphi }^b,g)=|{\Lambda }_L|h^2{R}_{a,b}.$$

$$\underset{L\to \infty }{\lim }\mathbb{E}⟨{\Delta R_{1,2}}^2⟩\ne 0,$$

$$\underset{L\to \infty }{\lim }\mathbb{E}⟨{\Delta R_{1,2}}^2⟩=0,$$

$$\underset{L\to \infty }{\lim }[\mathbb{E}⟨{R_{1,2}}^2⟩-(\mathbb{E}⟨R_{1,2}⟩{)}^2]=0,$$

$$\underset{L\to \infty }{\lim }⟨{\delta R_{1,2}}^2⟩,$$

$$\underset{L\to \infty }{\lim }\mathbb{E}⟨{\delta R_{1,2}}^2⟩=0,$$

$$⟨{\varphi }_x{\varphi }_y⟩-⟨{\varphi }_x⟩⟨{\varphi }_y⟩,$$

$$\mathbb{E}(⟨{\varphi }_x{\varphi }_y⟩-⟨{\varphi }_x⟩⟨{\varphi }_y⟩)\le C,$$

$$\sum _{x,y\in {\Lambda }_L}\mathbb{E}(⟨{\varphi }_x{\varphi }_y⟩-⟨{\varphi }_x⟩⟨{\varphi }_y⟩{)}^2\le K|{\Lambda }_L{|}^{-\frac{1}{4}},$$

$$2\underset{L\to \infty }{\lim }\mathbb{E}⟨{\Delta R_{1,2}}^2⟩=3\underset{L\to \infty }{\lim }\mathbb{E}⟨{\delta R_{1,2}}^2⟩=6\underset{L\to \infty }{\lim }\mathbb{E}⟨\Delta R_{1,2}{⟩}^2$$

$$\mathbb{E}⟨{\varphi }_x^k⟩\le C_k,$$

$${\pi }_L(s):=\frac{1}{|{\Lambda }_L|}\mathbb{E}\log \int \prod _{x\in {\Lambda }_L}d{\varphi }_x\exp [\sum _{x\in {\Lambda }_L}(-u{\varphi }_x^4+(bs+r){\varphi }_x^2+\beta h{g}_x{\varphi }_x)+\beta (1-s)\sum _{x,y\in {\Lambda }_L}{J}_{x,y}{\varphi }_x{\varphi }_y].$$

$$\frac{1}{|{\Lambda }_L|}\mathbb{E}[b\sum _{x\in {\Lambda }_L}⟨{\varphi }_x^2⟩-\beta \sum _{x,y\in {\Lambda }_L}{J}_{x,y}⟨{\varphi }_x{\varphi }_y⟩]={\pi }_L^{\prime }(0)\le {\pi }_L^{\prime }(1)$$

$$=\frac{1}{|{\Lambda }_L|}\mathbb{E}[b\sum _{x\in {\Lambda }_L}⟨{\varphi }_x^2{⟩}_0-\beta \sum _{x,y\in {\Lambda }_L}{J}_{x,y}⟨{\varphi }_x{\varphi }_y{⟩}_0].$$

$$(b-\beta d)\mathbb{E}⟨{\varphi }_x^2⟩$$

$$\le \frac{1}{|{\Lambda }_L|}\mathbb{E}[b\sum _{x\in {\Lambda }_L}⟨{\varphi }_x^2⟩-\frac{\beta }{2}\sum _{x,y\in {\Lambda }_L}{J}_{x,y}(⟨{\varphi }_x^2⟩+⟨{\varphi }_y^2⟩)]$$

$$\le \frac{1}{|{\Lambda }_L|}\mathbb{E}[b\sum _{x\in {\Lambda }_L}⟨{\varphi }_x^2{⟩}_0+\frac{\beta }{2}\sum _{x,y\in {\Lambda }_L}{J}_{x,y}(⟨{\varphi }_x^2{⟩}_0+⟨{\varphi }_y^2{⟩}_0)]$$

$$\le (b+\beta d)\mathbb{E}⟨{\varphi }_x^2{⟩}_0,$$

$$(k-3)⟨{\varphi }_x^{k-4}⟩=-⟨{\varphi }_x^{k-3}\frac{\partial }{\partial {\varphi }_x}[-u{\varphi }_x^4+r{\varphi }_x^2+\sum _{y\in {\Lambda }_L}\beta J_{x,y}{\varphi }_x{\varphi }_y+\beta h{g}_x{\varphi }_x]⟩$$

$$=⟨[4u{\varphi }_x^k-2r{\varphi }_x^{k-2}-\sum _{y\in {\Lambda }_L}\beta J_{x,y}{\varphi }_x^{k-3}{\varphi }_y-\beta h{g}_x{\varphi }_x^{k-3})]⟩$$

$$=4u⟨{\varphi }_x^k⟩-2r⟨{\varphi }_x^{k-2}⟩-\sum _{y\in {\Lambda }_L}\beta J_{x,y}⟨{\varphi }_x^{k-3}{\varphi }_y⟩-\beta h{g}_x⟨{\varphi }_x^{k-3}⟩.$$

$$4u\mathbb{E}⟨{\varphi }_x^k⟩$$

$$p(\beta ,h,u,r)=\underset{L\to \infty }{\lim }{p}_L(\beta ,h,u,r),$$

$$H_N(\varphi ,g)=H(\varphi ,g)+H_{\mathrm{del}}(\varphi ,g),$$

Full text

1 Introduction

Replica symmetry breaking phenomena in disordered spin systems have been studied extensively, since Talagrand proved Parisi’s replica symmetry breaking formula [18] for the Sherrington-Kirkpatrick model [19] in a mathematically rigorous manner [20]. It is well known that the replica symmetry breaking appears generally in mean field disordered spin models at low temperature as a spontaneous symmetry breaking phenomenon. There have been lots of discussions whether or not, the random field Ising model has some nontrivial phases due to its randomness such as replica symmetry breaking phase or spin glass phase. Krzakala, Ricci-Tersenghi, Sherrington and Zdeborova have pointed out an evidence that spin glass phase does not exists in the random field Ising model, the random field Ginzburg-Lamdau model and random temperature Ginzburg-Landau model [16, 17], which satisfy the Fortuin-Kasteleyn-Ginibre (FKG) inequality [11]. Recently, Chatterjee has proved the absence of replica symmetry breaking in the random field Ising model in an arbitrary dimension rigorously [2]. He proved that the variance of the spin overlap vanishes almost everywhere in the coupling constant space of the random field Ising model using the FKG inequality [11] and the Ghirlanda-Guerra identities. His method has been extended to quantum systems with the weak FKG property [15]. In the present paper, we prove that the replica symmetry breaking does not occur also in the random field Ginzburg-Landau model, as well as the random field Ising model.

First, we define the model and several functions. Coupling constants in a system with quenched disorder are given by i.i.d. random variables. We can regard a given disordered sample as a system obtained by a random sampling of these variables. All physical quantities in such systems are functions of these random variables. Consider a disordered Ginzburg-Landau model on a $d$ dimensional hyper cubic lattice $\Lambda_{L}:=[1,L]^{d}\cap{\mathbb{Z}}^{d}$ whose volume is $|\Lambda_{L}|=L^{d}$. Let $J=(J_{x,y})_{x,y\in\Lambda}$ be a real symmetric matrix such that $J_{x,y}=1$, if $|x-y|=1$, otherwise $J_{x,y}=0$. Define Hamiltonian as a function of spin configuration $\phi=(\phi_{x})_{x\in\Lambda_{L}}\in{\mathbb{R}}^{\Lambda_{L}}$ by

[TABLE]

where a vector $g=(g_{x})_{x\in\Lambda_{L}}$ consists of i.i.d. standard Gaussian random variables with a real constant $h$. Here, we define Gibbs state for the Hamiltonian. For a positive $\beta$, the partition function is defined by

[TABLE]

where the measure $D\phi$ is defined by

[TABLE]

for $u>0$ and $r\in{\mathbb{R}}.$ The expectation of a function of spin configuration $f(\phi)$ in the Gibbs state is given by

[TABLE]

Define the following functions of $(\beta,h,u,r)\in[0,\infty)^{3}\times{\mathbb{R}}$ and randomness $g=(g_{x})_{x\in\Lambda_{L}}$

[TABLE]

$-\frac{|\Lambda_{L}|}{\beta}\psi_{L}(\beta,h,u,r,g)$ is called free energy in statistical physics. We define a function $p_{L}:[0,\infty)^{3}\times{\mathbb{R}}\rightarrow{\mathbb{R}}$ by

[TABLE]

where ${\mathbb{E}}$ stands for the expectation of the random variables $(g_{x})_{z\in\Lambda_{L}}$.

Next, we consider replica symmetry breaking phenomena which apparently violate self-averaging of the overlap between two replicated quantities in a replica symmetric expectation. Let $\phi^{a}(a=1,\cdots,n)$ be $n$ replicated copies of a spin configuration, and we consider the following Hamiltonian

[TABLE]

where replicated spin configurations share the same quenched disorder $g$. This Hamiltonian is invariant under an arbitrary permutation $\sigma\in S_{n}$.

[TABLE]

This permutation symmetry is the replica symmetry. The spin overlap $R_{a,b}$ between two replicated spin configurations is defined by

[TABLE]

The covariance of the Hamiltonian is written in terms of the overlap

[TABLE]

When the replica symmetry breaking occurs, broadening of the overlap distribution with a finite variance is observed. This phenomenon is well-known in several disordered systems, such as the Sherrington-Kirkpatrick model [18, 20, 21]. In the present paper, we define replica symmetry breaking by the finite variance calculated in the replica symmetric expectation in the infinite volume limit

[TABLE]

where $\Delta R_{1,2}:=R_{1,2}-{\mathbb{E}}\langle R_{1,2}\rangle$. Chatterjee has given this definition of the replica symmetry breaking and proved

[TABLE]

in the random field Ising model [2]. In the present paper, we extend his proof to the random field Ginzburg-Landau model. We prove the following theorem.

Theorem 1.1

In the random field ferromagnetic Ginzburg-Landau model, the following variance vanishes

[TABLE]

for almost all coupling constants $(\beta,u,h,r)\in[0,\infty)^{3}\times{\mathbb{R}}^{2}$ in the infinite volume limit.

Here, we roughly sketch Chatterjee’s proof for the random field Ising model and explain a key to extend it to the random field Ginzburg-Landau model.

From the view point of detecting the spontaneous symmetry breaking, observe another variance

[TABLE]

calculated in the replica symmetric Gibbs state, where $\delta R_{1,2}:=R_{1,2}-\langle R_{1,2}\rangle.$ If this variance does not vanish in a sample, a strong fluctuation should yield an instability of the replica symmetric Gibbs state and one can expect spontaneous replica symmetry breaking. This phenomenon corresponds to a long range order in systems without disorder as discussed by Griffiths [13]. First, Chatterjee prove

[TABLE]

by the FKG inequality and another inequality obtained from the boundedness of $R_{1,2}$ in the random field Ising model. In the random field Ginzburg-Landau model, however, there is no simple bound for the truncated correlation function

[TABLE]

unlike its sample expectation

[TABLE]

for a positive number $C$. To show the limit (7) in the random field Ginzburg-Landau model, we prove a bound

[TABLE]

for a positive number $K$. Next, Chatterjee points out a relation between two variances ${\mathbb{E}}\langle\Delta R_{1,2}^{2}\rangle$ and ${\mathbb{E}}\langle\delta R_{1,2}^{2}\rangle$. For the disordered Ising systems,

[TABLE]

are obtained from the Aizenman-Contucci [1, 5] or the Ghirlanda-Guerra identities [2, 3, 6, 7, 10, 12]. Therefore, the identity (7) implies stronger identity (5), then the proof has been completed for the random field Ising model. The naive extension of this argument to the random field Ginzburg-Landau model derives a kind of the Ghirlanda-Guerra type identities including the self-overlap $R_{1,1}$ because of $\phi_{x}^{2}\neq 1$, they are not useful to prove the identities (8). We use a new method to obtain the variance bound on $R_{1,1}$, and then we obtain the Ghirlanda-Guerra identities as a useful form to prove the identities (8). In the present paper, we prove that all variances of the spin overlap vanish in the replica symmetric Gibbs state in another way for the random field ferromagnetic Ginzburg-Landau model.

2

Proof

2.1 Properties of the free energy density in the infinite volume limit

Lemma 2.1

The expectation of function of spin variable at a single site has an upper bound

[TABLE]

*where $k$ is an arbitrary even integer and $C_{k}$ is independent of the system size $L$.

Proof.*

This is proved by an inductivity for even integer $k$. First, we consider this bound in the case $k=2$. Define the following interpolating function $\pi_{L}(s)$ of a parameter $s\in[0,1]$ for a positive number $b$

[TABLE]

Note that $\pi_{L}(0)=p_{L}(\beta,u,h,r)$ and $\pi_{L}(1)$ is defined in an independent spin model. Since $\pi_{L}$ is convex in $s$, we have a bound

[TABLE]

If we use $2\phi_{x}\phi_{y}\leq\phi_{x}^{2}+\phi_{y}^{2}$, the above inequality and the translational symmetry, we have

[TABLE]

Note that the right hand side in the above denotes the Gibbs expectation in the independent spin model. Then the left hand side ${\mathbb{E}}\langle\phi_{x}^{2}\rangle$ is finite for a sufficiently large $b$, since the one point function ${\mathbb{E}}\langle\phi_{x}^{2}\rangle_{0}$ in the independent model has a simple upper bound.

Next, we consider the case for an even integer $k\geq 4$. An integration by parts with respect to a spin variable $\phi_{x}$ at a fixed site $x\in\Lambda_{L}$ gives

[TABLE]

This equality (10), Hölder’s and the Jensen’s inequalities give the following recursive inequality for a bound on ${\mathbb{E}}\langle\phi_{x}^{k}\rangle$ for an even integer $k>4$

[TABLE]

Therefore, ${\mathbb{E}}\langle\phi_{x}^{k}\rangle$ for an even integer $k$ is bounded from the above by ${\mathbb{E}}\langle\phi_{x}^{k-2}\rangle$. This and the bound (2.1) for $k=2$ complete the proof. $\Box$

Lemma 2.2

The following infinite volume limit exists

[TABLE]

*for arbitrary coupling constants.

Proof.*

This is proved by a standard argument based on the decomposition of the lattice into disjoint blocks [8, 9, 13]. Let $L,M$ be positive integers and denote $N=LM$, then we divide the lattice $\Lambda_{N}$ into $M^{d}$ disjoint translated blocks of $\Lambda_{L}$. Define a new Hamiltonian $H$ on $\Lambda_{N}$ by deleting interaction bonds near the boundaries of blocks, such that $M^{d}$ spin systems on the block $\Lambda_{L}$ and its $M^{d}-1$ translations have no interaction with each other. The original Hamiltonian $H_{N}$ has the following two terms

[TABLE]

where $H_{\rm del}$ is deleted interaction Hamiltonian, and $H$ denotes the summation of Hamiltonians on the $M^{d}$ disjoint blocks. Define the following function of $s\in[0,1]$ by

[TABLE]

Note that $\pi_{N}(1)=p_{N}(\beta,u,h,r)$ and $\pi_{N}(0)=p_{L}(\beta,u,h,r).$ The derivative functions of $\pi_{N}$ are given by

[TABLE]

where $\langle f(\phi)\rangle_{s}$ is the Gibbs and expectations with the Hamiltonian $H+sH_{\rm del}$ for function $f$ of a spin configuration. Since the function $\pi_{N}(s)$ is convex, we have

[TABLE]

then

[TABLE]

Since the expectation

[TABLE]

is bounded as shown in Lemma 2.1, $|{\mathbb{E}}\langle H_{\rm del}\rangle_{s}|$ is bounded by the number of deleted bonds. Then, there exist positive numbers $K$ independent of $L$ and $N$, such that the function $p_{N}$ and $p_{L}$ obey

[TABLE]

In the same argument for $M$ instead of $L$, we have

[TABLE]

and therefore

[TABLE]

The sequence $p_{L}$ is Cauchy. $\Box$

Note 2.3

The functions $p$, $p_{L}$ $\psi_{L}$ are convex functions of each argument for arbitrarily fixed others.

Note 2.4

The function $p$ is differentiable almost everywhere in the coupling constant space $[0,\infty)^{3}\times{\mathbb{R}}$ because of its convexity.

Hereafter, we use a lighter notation $\psi_{L}(g)$ for $\psi_{L}(\beta,h,u,r,g)$. Define the function of $s\in[0,1]$

[TABLE]

where $g=(g_{x})_{x\in\Lambda_{L}}$ and $g^{\prime}=(g^{\prime}_{x})_{x\in\Lambda_{L}}$ consist of i.i.d. standard Gaussian variables. Define a generating function $\gamma_{L}(s)$ of a parameter $s\in[0,1]$ by

[TABLE]

where ${\mathbb{E}}$ and ${\mathbb{E}}^{\prime}$ denote expectation over $g$ and $g^{\prime}$, respectively. This generating function $\gamma_{L}$ is introduced by Chatterjee [4]. We denote the Gibbs expectation of a function of spin configuration $f(\phi)$ with Hamiltonian $H(\phi,G(s))$

[TABLE]

Lemma 2.5

For any $(\beta,h,u,r)\in[0,\infty)^{3}\times{\mathbb{R}}$, any positive integer $L$, any positive integer $k$ and any $s_{0}\in[0,1]$, an upper bound on the $k$-th order partial derivative of the function $\gamma_{L}$ is given by

[TABLE]

The $k$-th order derivative of $\gamma_{L}$ is represented in the following

[TABLE]

*for an arbitrary $s\in[0,1].$

Proof.*

The first derivative of $\gamma_{L}$ is calculated in integration by parts

[TABLE]

The bound ${\mathbb{E}}\langle\phi_{x}^{2}\rangle\leq C_{2}$ given by Lemma 2.1 has been used. The formula (14) for the $k$-th derivative is proved by inductivity. The positive semi-definiteness of arbitrary order derivative $\gamma_{L}^{(k)}(s)$ and Taylor’s theorem

[TABLE]

for $s_{1}\in(s_{0},1)$ give the first inequality (13). $\Box$

Lemma 2.5 gives the following lemma

Lemma 2.6

The variance of $\psi_{L}$ is bounded from the above as follows

[TABLE]

Proof.

The left hand side is given by

[TABLE]

This completes the proof. $\Box$

2.2 Variance inequalities for the Hamiltonian density

Next we evaluate the following variance of the random Hamiltonian density defined by

[TABLE]

Lemma 2.7

For any coupling constants, we have

[TABLE]

*where $\delta\xi_{L}:=\xi_{L}-\langle\xi_{L}\rangle$.

Proof.*

This bound on the variance ${\mathbb{E}}\langle\delta\xi_{L}^{2}\rangle$ is obtained as follows:

[TABLE]

The boundedness of the spins and Lemma 2.5 have been used. $\Box$

Next lemma can be proved by a standard argument of the continuous differentiability of the function $p(\beta,h)$ for almost all $h$ because of the convexity [2, 21].

Lemma 2.8

For almost all coupling constants,

[TABLE]

and

[TABLE]

Proof.

Regard $p_{L}(h)$ $p(h)$ and $\psi_{L}(h)$ as functions of $h$ for lighter notation. Define the following functions

[TABLE]

for $\epsilon>0$. Lemma 2.2 gives

[TABLE]

and Lemma 2.6 and the Schwarz inequality give also

[TABLE]

for any $\epsilon>0$. Since $\psi_{L}$, $p_{L}$ and $p$ are convex functions of $h$, we have

[TABLE]

As in the same calculation, we have

[TABLE]

Then,

[TABLE]

Convergence of ${\mathbb{E}}w_{L},$ ${\mathbb{E}}w_{L}^{2}$ and $e_{L}$ in the infinite volume limit implies

[TABLE]

The right hand side vanishes, since the convex function $p(h)$ is continuously differentiable almost everywhere and $\epsilon>0$ is arbitrary. Therefore

[TABLE]

for almost all $h$. Jensen’s inequality gives

[TABLE]

This implies the first equality (16). These equalities imply also

[TABLE]

This completes the proof. $\Box$

Lemma 2.9

The following limit vanishes

[TABLE]

*almost everywhere in coupling constant space, where $\Delta\xi_{L}:=\xi_{L}-{\mathbb{E}}\langle\xi_{L}\rangle$

Proof.*

[TABLE]

Lemma 2.7 and Lemma 2.8 complete the proof. $\Box$

2.3 Variance inequalities for spin overlap functions

Notation An expression $\langle A\ ;B\rangle$ denotes a two point truncated correlation function

[TABLE]

*for functions $A,B$ of spin configurations $(\phi_{x})_{x\in\Lambda_{L}}$, *

Lemma 2.10

Let $f$ and $g$ be monotonically increasing functions of spin configuration, then $\langle f;g\rangle\geq 0.$ Therefore, $\langle\phi_{x};\phi_{y}\rangle\geq 0,$ for any $x,y\in\Lambda_{L}.$

This inequality proved by Fortuin, Kasteleyn and Ginibre is called the FKG inequality [11].

Lemma 2.11

There exists a positive number $K$ independent of the system size $L$, such that the expectation of square of truncated two point function is bounded from the above

[TABLE]

Proof.

Define an indicator defined $I$ by $I[true]=1,$ and $I[false]=0$. Let $C$ be a positive number and evaluate

[TABLE]

The FKG inequality and Lemma 2.5 have been used. Next we evaluate expectation of summation of squared two point correlation functions over the lattice

[TABLE]

where an upper bound ${\mathbb{E}}\langle\phi_{x};\phi_{y}\rangle^{4}\leq 16C_{8}$ is guaranteed by the bound on one point functions (9) and the Schwarz inequality. This implies the upper bound. $\Box$

Lemma 2.12

For any coupling constants

[TABLE]

Proof.

This can be proved using the FKG inequality $\langle\phi_{x};\phi_{y}\rangle\geq 0$ as proved for the random field Ising model [2].

[TABLE]

We have used Lemma 2.11. $\Box$

The following two lemmas show that two kinds of variance of self-overlap $R_{1,1}$ vanish. Proving these two lemmas is necessary to obtain the Ghirlanda-Guerra identities for the random field Ginzburg-Landau model, although they are automatically valid in the random field Ising model because of $R_{1,1}=1$.

Lemma 2.13

For all coupling constants,

[TABLE]

Prrof.

To prove this lemma we have to evaluate the upper bound of the following Gibbs expectation

[TABLE]

To obtain the bound, represent the following derivative function in terms of correlation functions

[TABLE]

Consider the following function to estimate the first and the last terms

[TABLE]

where $(G_{x}(t))_{x\in\Lambda_{L}}:=(\sqrt{t}g_{x}+\sqrt{1-t}g_{x}^{\prime})_{x\in\Lambda_{L}}$ for $t\in[0,1]$. The $k$-th derivative of $\chi_{L}$ is given by

[TABLE]

Note that any order derivative is monotonically increasing in $t$. The second derivative function of $\chi_{L}$ is represented in the following summation of three point correlation functions which gives us the bound on the summation of two point functions.

[TABLE]

A bound for $\chi_{L}^{\prime\prime}(0)$ is given in terms of $\chi_{L}(1)$ using Taylor’s theorem as for $\gamma_{L}^{(k)}(0)$

[TABLE]

The following correlation is estimated using the above bound

[TABLE]

We obtain the following bound

[TABLE]

The FKG inequality for $\langle\phi_{x};\phi_{y}\rangle$, bounds on $\chi_{L}^{\prime\prime}(0)$, $\gamma_{L}^{\prime\prime\prime\prime}(0)$ and Lemma 2.11 have been used. These estimates conclude that

[TABLE]

Then the variance of $R_{1,1}$ vanishes in the infinite volume limit. $\Box$

Next lemma can be proved by the continuous differentiability of the function $p(\beta,r)$ in $r$ for almost all $r$ because of its convexity as in the proof of Lemma 2.8.

Lemma 2.14

For almost all coupling constants,

[TABLE]

and

[TABLE]

2.4 The Ghirlanda-Guerra identities

Lemma 2.8, 2.12 and 2.14 enable us to derive the Ghirlanda-Guerra identities for the Ginzburg-Landau model in the useful form as well as those for Ising systems [1, 12] .

Lemma 2.15

Let $f(\phi^{1},\cdots,\phi^{n})$ be an arbitrary function of $n$ replicated spins, satisfying a bound ${\mathbb{E}}\langle f^{2}\rangle\leq C_{f}^{2}$. For $\beta h\neq 0$, almost everywhere in the coupling constant space, the following identity is valid

[TABLE]

Proof.

Lemma 2.8, the boundedness of $f$ and the Schwarz inequality imply

[TABLE]

in the infinite volume limit. The left hand side can be calculated using integration by parts.

[TABLE]

Substituting $f=1$ to the above, we have

[TABLE]

From the above two identities, we have

[TABLE]

Therefore, Lemma 2.14 and Lemma 2.13 enable us to obtain the identity (27) $\Box$

2.5 Concluding the proof of Theorem 1.1

As proved by Chatterjee for the random field Ising model [2], we use the Ghirlanda-Guerra identities. Lemma 2.15 for $n=2$ and $f=R_{1,2}$ implies

[TABLE]

and it for $n=3$ and $f=R_{2,3}$ implies

[TABLE]

Both are valid almost everywhere in the coupling constant space. The replica symmetric Gibbs state gives

[TABLE]

Then the following relation between two kinds of variance

[TABLE]

is obtained. Lemma 2.12 implies

[TABLE]

This completes the proof of Theorem 1.1. $\Box$