# A Note on the Replica Symmetric Formula for the SK Model

Christian Brennecke, Horng-Tzer Yau

Department of Mathematics, Harvard University,
One Oxford Street, Cambridge MA 02138, USA

###### Abstract

We provide a simple extension of Bolthausen’s Morita type proof [6] of the
replica symmetric formula for the Sherrington-Kirkpatrick (SK) model and prove
the replica symmetry for all $(\beta,h)$ that satisfy
$\beta^{2}E\operatorname{sech}^{2}(\beta\sqrt{q}Z+h)\leq 1$, where
$q=E\tanh^{2}(\beta\sqrt{q}Z+h)$. Compared to [6], the key of the argument is
to apply the conditional second moment method to a suitably reduced partition
function.

##  1 Introduction

We study systems of $N$ spins $\sigma_{i}$, $i\in\\{1,\dots,N\\}$, with values
in $\\{-1,1\\}$ and with the Hamiltonian $H_{N}:\\{-1,1\\}^{N}\to\mathbb{R}$
defined by

$H_{N}(\sigma)=\frac{\beta}{\sqrt{2}}\sum_{1\leq i,j\leq
N}g_{ij}\sigma_{i}\sigma_{j}+h\sum_{i=1}^{N}\sigma_{i}.$ (1.1)

The interactions $\\{g_{ij}\\}$ are i.i.d. centered Gaussians of variance
$1/N$ for $i\neq j$ and we set $g_{ii}\equiv 0$. $\beta\geq 0$ denotes the
inverse temperature and $h>0$ the external field strength.

Eq. (1.1) corresponds to the Sherrington-Kirkpatrick spin glass model [21] and
we are interested in its free energy $f_{N}$ at high temperature, where

$f_{N}=\frac{1}{N}\log Z_{N},\,\,\text{
}\,\,Z_{N}=\sum_{\sigma\in\\{-1,1\\}^{N}}e^{H_{N}(\sigma)}.$ (1.2)

The mathematical understanding of the SK model has required substantial
efforts until the famous Parisi formula [18, 19] was rigorously established by
Guerra [10] and Talagrand [22]. Later, Panchenko [15, 17] gave another proof
based on the ultrametricity of generic models. For a thorough introduction to
the topic, we refer to [14, 23, 24, 16].

Despite the validity of the Parisi formula, it is an interesting question to
prove the replica symmetry of the SK model at high temperature, as predicted
by de Almeida and Thouless [3]. Replica symmetry is expected for all
$(\beta,h)$ that satisfy

$\beta^{2}E\operatorname{sech}^{4}(\beta\sqrt{q}Z+h)<1,$ (1.3)

where $q$ denotes the unique solution of the self-consistent equation

$q=E\tanh^{2}(\beta\sqrt{q}Z+h).$ (1.4)

In both cases $Z\sim\mathcal{N}(0,1)$ denotes a standard Gaussian and $E$ the
expectation over $Z$.

In the special case that the external field in direction of $\sigma_{i}$ is a
centered Gaussian random variable, $h_{i}=hg_{i}$ for i.i.d.
$g_{i}\sim\mathcal{N}(0,1)$ (independent of the $g_{ij}$), replica symmetry
has been recently shown in [9] for all $(\beta,h)$ that satisfy (1.3) (in
which case $h$ in (1.3) and (1.4) is replaced by $hZ^{\prime}$ for some
$Z^{\prime}\sim\mathcal{N}(0,1)$ independent of $Z$); see also [20] for
previous results in this case. For Hamiltonians $H_{N}$ as in (1.1) (or, more
generally, Hamiltonians with non-centered random external field), however,
replica symmetry is to date only known above the AT line up to a bounded
region in the $(\beta,h)$-phase diagram. This has been analyzed in [13]. Like
[9], this analysis is based on the Parisi variational problem and we refer to
[13] for the details. For previously obtained results based on the Parisi
formula, see also [24, Chapter 13].

In this note, instead of analyzing the high temperature regime in view of the
Parisi variational problem, we give a simple extension of Bolthausen’s
argument [6] and prove the replica symmetric formula for all $(\beta,h)$ that
satisfy

$\beta^{2}E\operatorname{sech}^{2}(\beta\sqrt{q}Z+h)\leq 1.$ (1.5)

Although (1.5) is clearly stronger than the condition (1.3), it already covers
a fairly large region of the high temperature regime, see Fig. 1 for a
schematic. It improves upon the inverse temperature range from [6], where
$\beta$ was assumed to be sufficiently small.

######  Theorem 1.1.

Assume that $(\beta,h)$ satisfies (1.5), then

$\lim_{N\to\infty}\mathbb{E}\frac{1}{N}\log Z_{N}=\log
2+E\log\cosh(\beta\sqrt{q}Z+h)+\frac{\beta^{2}}{4}(1-q)^{2}.$ (1.6)

_Remarks:_

  * 1)

From [11], it is well-known that $\lim_{N\to\infty}\mathbb{E}\frac{1}{N}\log
Z_{N}$ exists and that almost surely $\lim_{N\to\infty}\frac{1}{N}\log
Z_{N}=\lim_{N\to\infty}\mathbb{E}\frac{1}{N}\log Z_{N}$.

  * 2)

It follows from the results of [10] that the right hand side in (1.6) provides
an upper bound to the free energy $\lim_{N\to\infty}\mathbb{E}\frac{1}{N}\log
Z_{N}$, for all inverse temperatures and external fields $(\beta,h)$. To
establish Theorem 1.1, it is therefore sufficient to prove that the right hand
side in (1.6) provides a lower bound to
$\lim_{N\to\infty}\mathbb{E}\frac{1}{N}\log Z_{N}$.

We conclude this introduction with a quick heuristic outline of the main
argument. To this end, consider first the case $h=0$ where the critical
temperature corresponds to $\beta=1$. In this case, it is straight forward
(see e.g. [7, Chapter 1, Section 3]) to see that

$\lim_{N\to\infty}\frac{1}{N}\log\mathbb{E}Z_{N}=\log
2+\frac{\beta^{2}}{4},\hskip
14.22636pt\lim_{N\to\infty}\frac{1}{N}\log\mathbb{E}Z_{N}^{2}=2\log
2+\frac{\beta^{2}}{2}$

for all $\beta<1$. The replica symmetric formula thus follows from the second
moment method, using the Gaussian concentration of the free energy. In fact,
for $h=0$, also the fluctuations of $Z_{N}/\mathbb{E}Z_{N}$ have been known
for a long time [2].

Clearly, it would be desirable to extend this simple argument to the case
$h>0$, but a direct application of the second moment method does not work
here. However, as suggested in [6], one may hope to obtain a model similar to
the case $h=0$ by centering the spins around suitable magnetizations and
viewing $Z_{N}$, up to normalization, as an average over the corresponding
coin-tossing measure. To center the spins correctly, recall that at high
temperature one expects the TAP equations [25] to hold, that is

$m_{i}\approx\tanh\Big{(}h+\beta\sum_{j\neq
i}\bar{g}_{ij}m_{j}-\beta^{2}(1-q)m_{i}\Big{)}\hskip 14.22636pt(\text{for
}i=1,\dots,N),$ (1.7)

where $\bar{g}_{ij}=(g_{ij}+g_{ji})/\sqrt{2}$ and
$m_{i}=Z_{N}^{-1}\sum_{\sigma}\sigma_{i}\,e^{H_{N}(\sigma)}$. The validity of
(1.7) is known for sufficiently small $\beta$ (see [23, 8] and, more recently,
[1]; see also [12] on the TAP equations for generic models, valid at all
temperatures) and expected to be true under (1.3). In [5, 6], Bolthausen has
provided an iterative construction $(\textbf{m}^{(s)})_{s\in\mathbb{N}}$ of
the solution to (1.7) that converges (in a suitable sense) in the full high
temperature regime (1.3). The main result of [6] is a novel proof of (1.6) for
$\beta$ small enough, based on a conditional second moment argument, given the
approximate solutions $(\textbf{m}^{(s)})_{s\in\mathbb{N}}$. It has remained
an open question, however, if the approach can be extended to the region
(1.3).

In this note, while we are not able to resolve this question for all
$(\beta,h)$ satisfying (1.3), we improve the range of $(\beta,h)$ to (1.5) as
follows. [5, 6] show, roughly speaking, that
$\textbf{m}^{(k+1)}\approx\sum_{s=1}^{k}\gamma_{s}\phi^{(s)}$ for certain
orthonormal vectors $\phi^{(s)}\in\mathbb{R}^{N}$ and deterministic numbers
$\gamma_{s}\,(\,\approx\langle\textbf{m}^{(k+1)},\phi^{(s)}\rangle$ with high
probability), where
$\langle\textbf{x},\textbf{y}\rangle=N^{-1}\sum_{i=1}^{N}x_{i}y_{i}$ for
$\textbf{x},\textbf{y}\in\mathbb{R}^{N}$. One also has
$\textbf{g}=\textbf{g}^{(k+1)}+\sum_{s=1}^{k}\rho^{(s)}$ for the interaction
$\textbf{g}=(g_{ij})_{1\leq i,j\leq N}$, where the
$\rho^{(s)}\in\mathbb{R}^{N\times N}$ are measurable w.r.t.
$(\textbf{m}^{(s)})_{s\leq k+1}$ and where the modified interaction
$\textbf{g}^{(k+1)}$ is Gaussian, conditionally on the
$(\textbf{m}^{(s)})_{s\leq k+1}$, with the property that
$\textbf{g}^{(k+1)}\sum_{s=1}^{k}\gamma_{s}\phi^{(s)}=0$. Up to negligible
errors, one obtains with $\widehat{\sigma}=\sigma-\textbf{m}^{(k+1)}$ that

$\begin{split}\frac{1}{N}\log Z_{N}\approx&\,\log
2+E\log\cosh(\beta\sqrt{q}Z+h)\\\
&+\frac{1}{N}\log\sum_{\sigma\in\\{-1,1\\}^{N}}p_{\text{free}}(\sigma)\exp\bigg{[}\frac{N\beta}{\sqrt{2}}\big{\langle}\widehat{\sigma},\textbf{g}^{(k)}\widehat{\sigma}\big{\rangle}+\mathcal{O}\big{(}\max_{s}|\gamma_{s}-\langle\sigma,\phi^{(s)}\rangle|\big{)}\bigg{]},\end{split}$
(1.8)

where $p_{\text{free}}$ denotes the product measure for which
$\sum_{\sigma}p_{\text{free}}(\sigma)\,\sigma=\textbf{m}^{(k+1)}$. A simple
observation is now that we can ignore the error
$\mathcal{O}(\max_{s}|\gamma_{s}-\langle\sigma,\phi^{(s)}\rangle|)$ in (1.8)
by restricting the modified partition function to those $\sigma$ with
$\max_{s}|\gamma_{s}-\langle\sigma,\phi^{(s)}\rangle|\approx 0$. Note that the
probability of the complement of this set is small under $p_{\text{free}}$,
because $\gamma_{s}\approx\langle\textbf{m}^{(k+1)},\phi^{(s)}\rangle$. This
yields a simple lower bound on $\frac{1}{N}\log Z_{N}$ and we can apply the
conditional second moment argument to the restricted partition function. We
show that its first conditional moment equals $\beta^{2}(1-q)^{2}/4$ (up to
negligible errors) in the full high temperature regime (1.3). To dominate its
second moment by the square of the first, on the other hand, we need to impose
the stronger condition (1.5).

Notice that imposing similar orthogonality restrictions on the partition
function has been proved useful before for obtaining lower bounds on the free
energy, like in the TAP analysis of the spherical SK model [4].

Figure 1: Schematic of the $(T,h)$ phase diagram, where $T=1/\beta$ denotes
the temperature. In the blue region, whose boundary corresponds to the AT line
(1.3), the SK model is known to be replica symmetry breaking. The boundary of
the green region corresponds to the condition (1.5). Theorem 1.1 proves the
replica symmetry in the green region.

Although (1.5) covers already a comparably large region of the high
temperature phase as schematically shown in Fig. 1, we do not expect that our
argument can be extended to the full high temperature regime (1.3), for
essentially the same reason as already pointed out by Bolthausen in [6]. In
regimes in which $h\gg 1$ so that we have $m_{i}\approx m\,(\approx 1)$, the
orthogonality relation $\langle\sigma-\textbf{m},\textbf{m}\rangle\approx 0$,
restricts the model to those $\sigma$ with
$|\\{i\in\\{1,\dots,N\\}:\sigma_{i}=1\\}|\approx\frac{N}{2}(1+m)$. As pointed
out in [6], in this toy case the AT condition (1.3) corresponds, at least on a
heuristic level, to the condition $\beta^{2}(1-m^{2})^{2}<1$ and it can be
verified explicitly that the second moment method does not work for
$\beta^{2}\approx(1-m^{2})^{-2}$ and $m\approx 1$.

The paper is structured as follows. In the next two sections we set up the
notation and recall Bolthausen’s iterative construction of the magnetization
[5, 6]. In Section 4, we define the reduced partition function and compute its
first and second moments. In the last section, we apply the conditional second
moment method to prove Theorem 1.1.

Acknowledgements. The work of H.-T. Y. is partially supported by NSF grant
DMS-1855509 and a Simons Investigator award.

##  2 Notation

In this section, we introduce basic notation and conventions. We follow
closely [6].

We usually denote vectors in $\mathbb{R}^{N}$ by boldface or greek letters. If
$\textbf{x}\in\mathbb{R}^{N}$, and $g:\mathbb{R}\to\mathbb{R}$, we define
$g(\textbf{x})$ in the componentwise sense. By
$\langle\cdot,\cdot\rangle:\mathbb{R}^{N}\times\mathbb{R}^{N}\to\mathbb{R}$,
we denote the normalized inner product

$\langle\textbf{x},\textbf{y}\rangle=\frac{1}{N}\sum_{i=1}^{N}x_{i}y_{i}$

and by $\|\cdot\|=\sqrt{\langle\cdot,\cdot\rangle}$ the induced norm. We also
normalize the tensor product
$\textbf{x}\otimes\textbf{y}:\mathbb{R}^{N}\to\mathbb{R}^{N}$ of two vectors
$\textbf{x},\textbf{y}\in\mathbb{R}^{N}$ so that for all
$\textbf{z}\in\mathbb{R}^{N}$

$(\textbf{x}\otimes\textbf{y})(\textbf{z})=\langle\textbf{y},\textbf{z}\rangle\,\textbf{x}.$

Given a matrix $\textbf{A}\in\mathbb{R}^{N\times N}$,
${\textbf{A}}^{T}\in\mathbb{R}^{N\times N}$ denotes its transpose and
$\bar{\textbf{A}}\in\mathbb{R}^{N\times N}$ denotes its symmetrization

$\bar{\textbf{A}}=\frac{\textbf{A}+\textbf{A}^{T}}{\sqrt{2}}.$

We mostly use the letters $Z,Z^{\prime},Z_{1},Z_{2},$ etc. to denote standard
Gaussian random variables independent of the disorder $\\{g_{ij}\\}$ and
independent from one another. When we average over such Gaussians, we denote
the corresponding expectation by $E$, to distinguish it from the expectation
$\mathbb{E}$ with respect to the disorder $\\{g_{ij}\\}$. Unless specified
otherwise, we consider all Gaussians to be centered.

Finally, given two sequences of random variables $(X_{N})_{N\geq
1},(Y_{N})_{N\geq 1}$ that may depend on parameters like $\beta,h,$ etc. we
say that

$X_{N}\simeq Y_{N}$

if and only if there exist positive constants $c,C>0$, which may depend on the
parameters, but which are independent of $N$, such that for every $t>0$ we
have

$\mathbb{P}(|X_{N}-Y_{N}|>t)\leq Ce^{-ct^{2}N}.$

##  3 Bolthausen’s Construction of the Local Magnetizations

In this section, we recall Bolthausen’s iterative construction of the solution
to the TAP equations [5, 6] and list the properties that we will need for the
proof of Theorem 1.1. We follow here the conventions of [6] and we refer to
[6, Sections 2, 4 & 5] for the proofs of the following statements.

First of all, we define three sequences $(\alpha_{k})_{k\in\mathbb{N}}$,
$(\gamma_{k})_{k\in\mathbb{N}}$ and $(\Gamma_{k})_{k\in\mathbb{N}}$. Set

$\alpha_{1}=\sqrt{q}\gamma_{1},\hskip
14.22636pt\gamma_{1}=E\tanh(\beta\sqrt{q}Z+h),\hskip
14.22636pt\Gamma_{1}^{2}=\gamma_{1}^{2},$

where here and in the following $q$ denotes the unique solution of (1.4).
Then, we define $\psi:[0,q]\to[0,q]$ by

$\psi(t)=E\tanh(\beta\sqrt{t}Z+\beta\sqrt{q-t}Z^{\prime}+h)\tanh(\beta\sqrt{t}Z+\beta\sqrt{q-t}Z^{\prime\prime}+h)$

and set recursively

$\alpha_{k}=\psi(\alpha_{k-1}),\hskip
14.22636pt\gamma_{k}=\frac{\alpha_{k}-\Gamma^{2}_{k-1}}{\sqrt{q-\Gamma^{2}_{k-1}}},\hskip
14.22636pt\Gamma_{k}^{2}=\sum_{j=1}^{k}\gamma_{k}^{2}.$

The following lemma collects important properties of
$(\alpha_{k})_{k\in\mathbb{N}}$ , $(\gamma_{k})_{k\in\mathbb{N}}$ and
$(\Gamma_{k})_{k\in\mathbb{N}}$.

######  Lemma 3.1.

([5, Lemma 2.2, Corollary 2.3, Lemma 2.4], [6, Lemma 2])

  1. 1)

$\psi$ is strictly increasing and convex in $[0,q]$ with
$0<\psi(0)<\psi(q)=q$. If (1.3) is satisfied, then $q$ is the unique fixed
point of $\psi$ in $[0,q]$.

  2. 2)

The sequence $(\alpha_{k})_{k\in\mathbb{N}}$ is increasing and $\alpha_{k}>0$
for every $k\in\mathbb{N}$. If (1.3) is satisfied, then
$\lim_{k\to\infty}\alpha_{k}=q$ and if (1.3) is satisfied with a strict
inequality, the convergence is exponentially fast.

  3. 3)

For all $k\geq 2$, we have that $0<\Gamma_{k-1}^{2}<\alpha_{k}<q$ and that
$0<\gamma_{k}<\sqrt{q-\Gamma_{k-1}^{2}}$. If (1.3) is true, then
$\lim_{k\to\infty}\Gamma_{k}^{2}=q$ and, as a consequence,
$\lim_{k\to\infty}\gamma_{k}=0$.

Next, we recall Bolthausen’s modified interaction matrix. We define

$\textbf{g}^{(1)}=\textbf{g},\hskip
14.22636pt\phi^{(1)}=\textbf{1}\in\mathbb{R}^{N},\hskip
14.22636pt\textbf{m}^{(1)}=\sqrt{q}\textbf{1}\in\mathbb{R}^{N}.$

Assuming $\textbf{g}^{(s)},\phi^{(s)},\textbf{m}^{(s)}$ are defined for $1\leq
s\leq k$, we set

$\zeta^{(s)}=\bar{\textbf{g}}^{(s)}\phi^{(s)}$

and we define the $\sigma$-algebra $\mathcal{G}_{k}$ through

$\mathcal{G}_{k}=\sigma\big{(}\textbf{g}^{(s)}\phi^{(s)},(\textbf{g}^{(s)})^{T}\phi^{(s)}:1\leq
k\leq s\big{)}.$

Expectations with respect to $\mathcal{G}_{k}$ are denoted by
${\mathbb{E}}_{k}$. Furthermore, we set

$\begin{split}\textbf{h}^{(k+1)}&=h\textbf{1}+\beta\sum_{s=1}^{k-1}\gamma_{s}\zeta^{(s)}+\beta\sqrt{q-\Gamma_{k-1}^{2}}\zeta^{(k)},\hskip
14.22636pt\textbf{m}^{(k+1)}=\tanh(\textbf{h}^{(k+1)}),\\\
\phi^{(k+1)}&=\frac{\textbf{m}^{(k+1)}-\sum_{s=1}^{k}\langle\textbf{m}^{(k+1)},\phi^{(s)}\rangle\phi^{(s)}}{\Big{\|}\textbf{m}^{(k+1)}-\sum_{s=1}^{k}\langle\textbf{m}^{(k+1)},\phi^{(s)}\rangle\phi^{(s)}\Big{\|}}\end{split}$
(3.1)

and we note that $\phi^{(k+1)}$ is $\mathbb{P}-a.s.$ well-defined for all $k$
if $k<N$ [6, Lemma 5]. Finally, the modified interaction matrix
$\textbf{g}^{(k+1)}$ is defined by

$\begin{split}\textbf{g}^{(k+1)}=&\,\textbf{g}^{(k)}-\rho^{(k)},\hskip
14.22636pt\text{where}\\\
\rho^{(k)}=&\,\textbf{g}^{(k)}\phi^{(k)}\otimes\phi^{(k)}+\phi^{(k)}\otimes(\textbf{g}^{(k)})^{T}\phi^{(k)}-\langle\textbf{g}^{(k)}\phi^{(s)},\phi^{(k)}\rangle\,\phi^{(k)}\otimes\phi^{(k)}.\end{split}$

In particular, this means that $\bar{\textbf{g}}^{(k+1)}$ is equal to

$\begin{split}\bar{\textbf{g}}^{(k+1)}=\bar{\textbf{g}}^{(k)}-\bar{\rho}^{(k)},\hskip
14.22636pt\bar{\rho}^{(k)}=\zeta^{(k)}\otimes\phi^{(k)}+\phi^{(k)}\otimes\zeta^{(k)}-\langle\zeta^{(k)},\phi^{(k)}\rangle\,\phi^{(k)}\otimes\phi^{(k)}.\end{split}$

It is clear that $(\phi^{(s)})_{s=1}^{k}$ forms an orthonormal sequence of
vectors in $\mathbb{R}^{N}$ and we denote by $\textbf{P}^{(k)}$ and
$\textbf{Q}^{(k)}$ the corresponding orthogonal projections in
$\mathbb{R}^{N}$, that is

$\textbf{P}^{(k)}=\sum_{s=1}^{k}\phi^{(s)}\otimes\phi^{(s)}=(P^{(k)}_{ij})_{1\leq
i,j\leq N},\hskip
14.22636pt\textbf{Q}^{(k)}=\textbf{1}-\textbf{P}^{(k)}=(Q^{(k)}_{ij})_{1\leq
i,j\leq N}.$

By [6, Lemma 3], $\textbf{m}^{(k)}$ and $\phi^{(k)}$ are
$\mathcal{G}_{k-1}$-measurable for all $k\in\mathbb{N}$ and we also have that

$\textbf{g}^{(k)}\phi^{(s)}=(\textbf{g}^{(k)})^{T}\phi^{(s)}=\bar{\textbf{g}}^{(k)}\phi^{(s)}=0,\,\,\forall\,\,s<k.$

######  Proposition 3.2.

([6, Prop. 4])

  1. 1)

Conditionally on $\mathcal{G}_{k-2}$, $\textbf{g}^{(k)}$ and
$\textbf{g}^{(k-1)}$ are Gaussian with conditional covariance, given
$\mathcal{G}_{k-2}$, equal to

${\mathbb{E}}_{k-2}\,g_{ij}^{(k)}g_{st}^{(k)}=\frac{1}{N}Q^{(k-1)}_{is}Q^{(k-1)}_{jt}.$

  2. 2)

Conditionally on $\mathcal{G}_{k-2}$, $\textbf{g}^{(k)}$ is independent of
$\mathcal{G}_{k-1}$. In particular, conditionally on $\mathcal{G}_{k-1}$,
$\textbf{g}^{(k)}$ is Gaussian with the same covariance as in 1).

  3. 3)

Conditionally on $\mathcal{G}_{k-1}$, the random variables $\zeta^{(k)}$ are
Gaussian with

${\mathbb{E}}_{k-1}\zeta^{(k)}_{i}\zeta^{(k)}_{j}=Q^{(k-1)}_{ij}+\frac{1}{N}\phi^{(k)}_{i}\phi^{(k)}_{j}.$

The main result of [5] is summarized in the following proposition.

######  Proposition 3.3.

([5, Prop. 2.5], [6, Prop. 6]) For every $k\in\mathbb{N}$ and $s<k$, one has

$\langle\textbf{m}^{(k)},\phi^{(s)}\rangle\simeq\gamma_{s},\hskip
14.22636pt\langle\textbf{m}^{(k)},\textbf{m}^{(s)}\rangle\simeq\alpha_{s},\hskip
14.22636pt\langle\textbf{m}^{(k)},\textbf{m}^{(k)}\rangle\simeq q.$

The next lemma collects a few auxiliary results that are helpful in the
sequel.

######  Lemma 3.4.

([6, Lemmas 11, 14, 15(b)])

  1. 1)

For every $k\in\mathbb{N}$,
$\langle\phi^{(k)},\zeta^{(k)}\rangle=\sqrt{2}\,\langle\phi^{(k)},\textbf{g}^{(k)}\phi^{(k)}\rangle$
is unconditionally Gaussian with variance $2/N$.

  2. 2)

For every Lipschitz continuous $f:\mathbb{R}\to\mathbb{R}$ with $|f(x)|\leq
C(1+|x|)$ for some $C>0$, one has for all $k\geq 2$ that

$\lim_{N\to\infty}{\mathbb{E}}\bigg{|}\frac{1}{N}\sum_{i=1}^{N}f(h_{i}^{(k+1)})-Ef(\beta\sqrt{q}Z+h)\bigg{|}=0.$

  3. 3)

For every $k\in\mathbb{N}$ and $t>0$ it holds true that

$\lim_{N\to\infty}\mathbb{P}\big{(}\big{\\{}\|\zeta^{(k)}\|\geq
1+t\big{\\}}\big{)}=0.$

##  4 Conditional Moments of Reduced Partition Function

Using Bolthausen’s magnetizations, we compute in this section the first two
conditional moments of a suitably reduced partition function
$\widetilde{Z}_{N}(S)$, for a subset $S\subset\\{-1,1\\}^{N}$ defined below.
This will suffice to establish Theorem 1.1, as explained in Section 5.

Let $\varepsilon>0$ and $k\in\mathbb{N}$ be fixed. We define the set
$S_{\varepsilon,k}\subset\\{-1,1\\}^{N}$ through

$\begin{split}S_{\varepsilon,k}=\Big{\\{}\sigma\in\\{-1,1\\}^{N}:|\langle\sigma-\textbf{m}^{(k+1)},\phi^{(s)}\rangle|\leq\varepsilon/k,\,\,\,\forall\,\,1\leq
s\leq k\Big{\\}},\end{split}$ (4.1)

with $\phi^{(s)},\gamma_{s}$ from Section 3. We define the reduced partition
function $Z_{N}^{(k+1)}(S_{\varepsilon,k})$ by

$\begin{split}Z^{(k+1)}_{N}(S_{\varepsilon,k})&=\sum_{\sigma\in
S_{\varepsilon,k}}p_{\text{free}}(\sigma)\exp\bigg{(}\frac{N\beta}{\sqrt{2}}\langle\sigma,\textbf{g}^{(k+1)}\sigma\rangle\bigg{)}\\\
&=\sum_{\sigma\in
S_{\varepsilon,k}}p_{\text{free}}(\sigma)\exp\bigg{(}\frac{N\beta}{2}\langle\sigma,\bar{\textbf{g}}^{(k+1)}\sigma\rangle\bigg{)},\end{split}$
(4.2)

where $p_{\text{free}}:\\{-1,1\\}^{N}\to(0,1)$ denotes the coin-tossing
measure

$p_{\text{free}}(\sigma)=\prod_{i=1}^{N}\frac{1}{2}\frac{\exp\big{(}h_{i}^{(k+1)}\sigma_{i}\big{)}}{\cosh\big{(}h_{i}^{(k+1)}\big{)}}.$
(4.3)

The following lemma records that $p_{\text{free}}(S_{\varepsilon,k}^{c})$ is
exponentially small in $N$.

######  Lemma 4.1.

Let $\varepsilon>0$, $k\in\mathbb{N}$, and let $S_{\varepsilon,k}$ and
$p_{\text{free}}$ be defined as in (4.1) and (4.3), respectively. Then, there
exist $c,C>0$, independent of $N$ and $\varepsilon$, such that

$p_{\emph{free}}(S_{\varepsilon,k})\geq 1-Ce^{-cN\varepsilon^{2}}.$ (4.4)

###### Proof.

By a standard union bound, we have that

$\begin{split}p_{\text{free}}(S_{\varepsilon,k}^{c})\leq&\,k\max_{s=1,\dots,k}p_{\text{free}}\Big{(}\Big{\\{}\sigma\in\\{-1,1\\}^{N}:\langle\sigma,\phi^{(s)}\rangle-\langle\textbf{m}^{(k+1)},\phi^{(s)}\rangle>\frac{\varepsilon}{k}\Big{\\}}\Big{)}\\\
&\,+k\max_{s=1,\dots,k}p_{\text{free}}\Big{(}\Big{\\{}\sigma\in\\{-1,1\\}^{N}:\langle\textbf{m}^{(k+1)},\phi^{(s)}\rangle-\langle\sigma,\phi^{(s)}\rangle>\frac{\varepsilon}{k}\Big{\\}}\Big{)},\end{split}$

which implies

$\begin{split}&p_{\text{free}}(S_{\varepsilon,k}^{c})\\\ &\leq
k\max_{s=1,\dots,k}\inf_{\lambda\geq
0}\exp\bigg{[}-N\bigg{(}\frac{\lambda\varepsilon}{k}-\frac{1}{N}\sum_{i=1}^{N}\log\frac{\cosh\big{(}h_{i}^{(k+1)}+\lambda\phi_{i}^{(s)}\big{)}}{\cosh\big{(}h_{i}^{(k+1)}\big{)}}+\lambda\langle\textbf{m}^{(k+1)},\phi^{(s)}\rangle\bigg{)}\bigg{]}\\\
&\hskip 14.22636pt+k\max_{s=1,\dots,k}\inf_{\lambda\geq
0}\exp\bigg{[}-N\bigg{(}\frac{\lambda\varepsilon}{k}-\frac{1}{N}\sum_{i=1}^{N}\log\frac{\cosh\big{(}h_{i}^{(k+1)}-\lambda\phi_{i}^{(s)}\big{)}}{\cosh\big{(}h_{i}^{(k+1)}\big{)}}-\lambda\langle\textbf{m}^{(k+1)},\phi^{(s)}\rangle\bigg{)}\bigg{]}.\\\
\end{split}$

Using the pointwise bound
$\log\cosh(x+y)\leq\log\cosh(x)+y\tanh(x)+\frac{y^{2}}{2}$ for
$x,y\in\mathbb{R}$, $\langle\phi^{(s)},\phi^{(s)}\rangle=1$ and the identity
$\tanh\big{(}\textbf{h}^{(k+1)}\big{)}=\textbf{m}^{(k+1)}$, we obtain

$\begin{split}p_{\text{free}}(S_{\varepsilon,k}^{c})\leq&\,2k\inf_{\lambda\geq
0}\exp\bigg{[}-\frac{N\lambda\varepsilon}{k}+\frac{N\lambda^{2}}{2}\bigg{]}=2ke^{-N\varepsilon^{2}/(2k^{2})}.\end{split}$

This concludes (4.4) for $c=c_{k}=1/(2k^{2}),C=C_{k}=2k$. ∎

We notice that the constants $c,C>0$ in (4.4) are independent of the
realization of the disorder $\\{g_{ij}\\}$. Thus, $a.s.$ in the disorder (so
that $\phi^{(s)}$, $s=1,\dots,k$, and hence $\bar{\textbf{g}}^{(k+1)}$ as well
as $p_{\text{free}}$ are well-defined), $S_{\varepsilon,k}\neq\emptyset$ for
$N$ large enough.

The next lemma determines the first conditional moment of the reduced
partition function $Z^{(k+1)}_{N}(S_{\varepsilon,k})$ and is valid in the full
high temperature regime (1.3).

######  Lemma 4.2.

Let $\varepsilon>0$, $k\in\mathbb{N}$ and let $S_{\varepsilon,k}$,
$Z^{(k+1)}_{N}(S_{\varepsilon,k})$ and $p_{\text{free}}$ be as in (4.1), (4.2)
and (4.3), respectively. Assume that $(\beta,h)$ satisfy the AT condition
(1.3). Then

$\lim_{\varepsilon\to
0}\lim_{k\to\infty}\limsup_{N\to\infty}\mathbb{E}\bigg{|}\frac{1}{N}\log{\mathbb{E}}_{k}\,Z_{N}^{(k+1)}(S_{\varepsilon,k})-\frac{\beta^{2}}{4}(1-q)^{2}\bigg{|}=0$
(4.5)

###### Proof.

By Proposition 3.2, we have that

$\begin{split}{\mathbb{E}}_{k}Z^{(k+1)}_{N}(S_{\varepsilon,k})&=\sum_{\sigma\in
S_{\varepsilon,k}}p_{\text{free}}(\sigma)\exp\bigg{(}\frac{\beta^{2}N^{2}}{4}{\mathbb{E}}_{k}\langle\sigma,\textbf{g}^{(k+1)}\sigma\rangle^{2}\bigg{)}\\\
&=\sum_{\sigma\in
S_{\varepsilon,k}}p_{\text{free}}(\sigma)\exp\bigg{[}\frac{\beta^{2}N}{4}\bigg{(}1-\sum_{s=1}^{k}\langle\sigma,\phi^{(s)}\rangle^{2}\bigg{)}^{2}\bigg{]}.\end{split}$

Centering around $\textbf{m}^{(k+1)}$ yields

$\begin{split}1-\sum_{s=1}^{k}\langle\sigma,\phi^{(s)}\rangle^{2}&=1-\sum_{s=1}^{k}\langle\textbf{m}^{(k+1)},\phi^{(s)}\rangle^{2}-2\sum_{s=1}^{k}\langle\sigma-\textbf{m}^{(k+1)},\phi^{(s)}\rangle\langle\textbf{m}^{(k+1)},\phi^{(s)}\rangle\\\
&\hskip
14.22636pt-\sum_{s=1}^{k}\langle\sigma-\textbf{m}^{(k+1)},\phi^{(s)}\rangle^{2}\end{split}$

so that

$\begin{split}&\sup_{\sigma\in
S_{\varepsilon,k}}\bigg{|}\bigg{(}1-\sum_{s=1}^{k}\langle\sigma,\phi^{(s)}\rangle^{2}\bigg{)}-\bigg{(}1-\sum_{s=1}^{k}\langle\textbf{m}^{(k+1)},\phi^{(s)}\rangle^{2}\bigg{)}\bigg{|}\\\
&\hskip 142.26378pt\leq C\sup_{\sigma\in
S_{\varepsilon,k}}\sum_{s=1}^{k}|\langle\sigma-\textbf{m}^{(k+1)},\phi^{(s)}\rangle|\leq
C\varepsilon\end{split}$

for some $C>0$, independent of $N$ and $k$. This implies with Lemma 4.1 that

$\begin{split}&\bigg{|}\frac{1}{N}\log{\mathbb{E}}_{k}Z^{(k+1)}_{N}(S_{\varepsilon,k})-\frac{\beta^{2}}{4}\bigg{(}1-\sum_{s=1}^{k}\langle\textbf{m}^{(k+1)},\phi^{(s)}\rangle^{2}\bigg{)}^{2}\bigg{|}\leq
C\beta^{2}\varepsilon+\frac{C}{N}|\log(1-Ce^{-cN\varepsilon^{2}})|.\end{split}$

Moreover, by Proposition 3.3, we have that
$\lim_{N\to\infty}\sum_{s=1}^{k}\langle\textbf{m}^{(k+1)},\phi^{(s)}\rangle^{2}=\Gamma_{k}^{2}$
in $L^{p}(d\mathbb{P})$, for any $p\in[1;\infty)$, so that

$\limsup_{N\to\infty}\mathbb{E}\bigg{|}\frac{1}{N}\log{\mathbb{E}}_{k}Z^{(k+1)}_{N}(S_{\varepsilon,k})-\frac{\beta^{2}}{4}\big{(}1-\Gamma_{k}^{2}\big{)}^{2}\bigg{|}\leq
C\beta^{2}\varepsilon.$

Finally, since $\lim_{k\to\infty}\Gamma_{k}^{2}=q$ under the AT condition
(1.3), by Lemma 3.1, we let $N\to\infty$, then $k\to\infty$ and then
$\varepsilon\to 0$ which implies

$\lim_{\varepsilon\to
0}\lim_{k\to\infty}\limsup_{N\to\infty}\mathbb{E}\bigg{|}\frac{1}{N}\log{\mathbb{E}}_{k}\,Z_{N}^{(k+1)}(S_{\varepsilon,k})-\frac{\beta^{2}}{4}(1-q)^{2}\bigg{|}=0.$

∎

The following lemma computes the second conditional moment of
$Z_{N}^{(k+1)}(S_{\varepsilon,k})$ under the stronger high temperature
condition (1.5).

######  Lemma 4.3.

Let $\varepsilon>0$, $k\in\mathbb{N}$ and let $S_{\varepsilon,k}$,
$Z^{(k+1)}_{N}(S_{\varepsilon,k})$ and $p_{\text{free}}$ be as in (4.1), (4.2)
and (4.3), respectively. Assume that $(\beta,h)$ satisfy the condition (1.5).
Then

$\lim_{\varepsilon\to
0}\lim_{k\to\infty}\limsup_{N\to\infty}\mathbb{E}\bigg{|}\frac{1}{N}\log{\mathbb{E}}_{k}\,\Big{[}\big{(}Z_{N}^{(k+1)}(S_{\varepsilon,k})\big{)}^{2}\Big{]}-\frac{\beta^{2}}{2}(1-q)^{2}\bigg{|}=0.$
(4.6)

###### Proof.

Proceeding as in the previous proposition, we compute

$\begin{split}&{\mathbb{E}}_{k}\,\Big{[}\big{(}Z_{N}^{(k+1)}(S_{\varepsilon,k})\big{)}^{2}\Big{]}\\\
&=\sum_{\sigma,\tau\in
S_{\varepsilon,k}}p_{\text{free}}(\sigma)p_{\text{free}}(\tau)\exp\bigg{(}\frac{\beta^{2}N^{2}}{4}{\mathbb{E}}_{k}\Big{(}\langle\sigma,\textbf{g}^{(k+1)}\sigma\rangle+\langle\tau,\textbf{g}^{(k+1)}\tau\rangle\Big{)}^{2}\bigg{)}\\\
&=\sum_{\sigma,\tau\in
S_{\varepsilon,k}}p_{\text{free}}(\sigma)p_{\text{free}}(\tau)\exp\bigg{[}\frac{\beta^{2}N}{4}\bigg{(}1-\sum_{s=1}^{k}\langle\sigma,\phi^{(s)}\rangle^{2}\bigg{)}^{2}+\frac{\beta^{2}N}{4}\bigg{(}1-\sum_{s=1}^{k}\langle\tau,\phi^{(s)}\rangle^{2}\bigg{)}^{2}\bigg{]}\\\
&\hskip
108.12054pt\times\exp\bigg{[}\frac{\beta^{2}N}{2}\bigg{(}\langle\sigma,\tau\rangle-\sum_{s=1}^{k}\langle\sigma,\phi^{(s)}\rangle\langle\phi^{(s)}\tau\rangle\bigg{)}^{2}\bigg{]}\\\
&=\sum_{\sigma,\tau\in
S_{\varepsilon,k}}p_{\text{free}}(\sigma)p_{\text{free}}(\tau)\exp\bigg{[}\frac{\beta^{2}N}{2}\langle\sigma,\textbf{Q}^{(k)}\tau\rangle^{2}\bigg{]}\\\
&\hskip
42.67912pt\times\exp\bigg{[}\frac{\beta^{2}N}{2}\bigg{(}1-\sum_{s=1}^{k}\langle\textbf{m}^{(k+1)},\phi^{(s)}\rangle^{2}\bigg{)}^{2}+N\mathcal{O}(\beta^{2}\varepsilon)\bigg{]}.\end{split}$

Arguing as in the previous lemma, we therefore see that it is enough to show
that

${\mathbb{E}}\bigg{|}\frac{1}{N}\log\sum_{\sigma,\tau\in
S_{\varepsilon,k}}p_{\text{free}}(\sigma)p_{\text{free}}(\tau)\exp\bigg{[}\frac{\beta^{2}N}{2}\langle\sigma,\textbf{Q}^{(k)}\tau\rangle^{2}\bigg{]}\bigg{|}$

vanishes when $N\to\infty$. To this end, recall that by definition of
$S_{\varepsilon,k}$, we have that

$\sup_{\tau\in
S_{\varepsilon,k}}\big{|}\|\textbf{Q}^{(k)}\tau\|^{2}-\big{(}1-\langle\textbf{m}^{(k+1)},\textbf{P}^{(k)}\textbf{m}^{(k+1)}\rangle\big{)}\big{|}\leq
C\varepsilon$ (4.7)

for some $C>0$ independent of $N$ and $k$. For fixed $\tau\in
S_{\varepsilon,k}$, we therefore find with
$\log\cosh(x+y)\leq\log\cosh(x)+y\tanh(x)+y^{2}/2$ for $x,y\in\mathbb{R}$ and
Markov’s inequality

$\begin{split}p_{\text{free}}(S_{\varepsilon,k})&\leq\sum_{\sigma\in
S_{\varepsilon,k}}p_{\text{free}}(\sigma)\exp\bigg{[}\frac{\beta^{2}N}{2}\langle\sigma,\textbf{Q}^{(k)}\tau\rangle^{2}\bigg{]}\\\
&\leq
p_{\text{free}}(S_{\varepsilon,k})+\int_{0}^{1}dt\,N\beta^{2}t\,e^{\frac{N}{2}\beta^{2}t^{2}}\,p_{\text{free}}\big{(}\big{\\{}\sigma\in\\{-1,1\\}^{N}:|\langle\sigma,\textbf{Q}^{(k)}\tau\rangle|>t\big{\\}}\big{)}\\\
&\leq
p_{\text{free}}(S_{\varepsilon,k})+2\int_{0}^{1}dt\,N\beta^{2}t\,e^{\frac{Nt^{2}}{2(1-q)}\big{(}\beta^{2}(1-q)-2+\frac{\|\textbf{Q}^{(k)}\tau\|^{2}}{(1-q)}\big{)}}e^{\frac{Nt}{1-q}\|\textbf{Q}^{(k)}\textbf{m}^{(k+1)}\|}.\end{split}$

In particular, by (4.7) and because

$\lim_{N\to\infty}\langle\textbf{m}^{(k+1)},\textbf{P}^{(k)}\textbf{m}^{(k+1)}\rangle=\Gamma_{k}^{2},\hskip
14.22636pt\lim_{N\to\infty}\|\textbf{Q}^{(k)}\textbf{m}^{(k+1)}\|^{2}=q-\Gamma_{k}^{2},$

in $L^{p}(d\mathbb{P})$ for $p\in[1;\infty)$, we obtain under the condition
(1.5), i.e. $\beta^{2}(1-q)\leq 1$, that

$\lim_{\varepsilon\to
0}\lim_{k\to\infty}\limsup_{N\to\infty}{\mathbb{E}}\bigg{|}\frac{1}{N}\log\sum_{\sigma,\tau\in
S_{\varepsilon,k}}p_{\text{free}}(\sigma)p_{\text{free}}(\tau)\exp\bigg{[}\frac{\beta^{2}N}{2}\langle\sigma,\textbf{Q}^{(k)}\tau\rangle^{2}\bigg{]}\bigg{|}=0,$

which implies (4.6). ∎

##  5 Proof of Theorem 1.1

In this section we prove Theorem 1.1, based on Lemmas 4.2 and 4.3. Before we
start, let us first re-center the Hamiltonian $H_{N}$ appropriately, as
outlined in the introduction.

Using the notation of Section 3, we have that

$\begin{split}\frac{H_{N}(\sigma)}{N}&=\frac{\beta}{2}\langle\sigma,\bar{\textbf{g}}\sigma\rangle+\langle
h,\sigma\rangle=\frac{\beta}{2}\langle\sigma,\bar{\textbf{g}}^{(k+1)}\sigma\rangle+\frac{\beta}{2}\sum_{s=1}^{k}\langle\sigma,\bar{\rho}^{(s)}\sigma\rangle+\langle
h,\sigma\rangle.\end{split}$

In contrast to [6], instead of centering the spins $\sigma$ around
$\textbf{m}^{(k+1)}$, we center the spins in
$\langle\sigma,\bar{\rho}^{(s)}\sigma\rangle$ around $\gamma_{s}\phi^{(s)}$ in
order to produce the right cavity field $\textbf{h}^{(k+1)}$. Notice that the
remaining term $\langle\sigma,\bar{\textbf{g}}^{(k+1)}\sigma\rangle$ contains
automatically centered spins around $\sum_{s=1}^{k}\gamma_{s}\phi^{(s)}$
(which approximately equals $\textbf{m}^{(k+1)}$), as
$\bar{\textbf{g}}^{(k+1)}\phi^{(s)}=0$ for $s<k+1$. We thus write

$\begin{split}\langle\sigma,\bar{\rho}^{(s)}\sigma\rangle&=2\gamma_{s}\langle\sigma,\bar{\rho}^{(s)}\phi^{(s)}\rangle+\langle\sigma-\gamma_{s}\phi^{(s)},\bar{\rho}^{(s)}(\sigma-\gamma_{s}\phi^{(s)})\rangle-\gamma_{s}^{2}\langle\phi^{(s)},\bar{\rho}^{(s)}\phi^{(s)}\rangle\\\
&=2\gamma_{s}\langle\sigma,\zeta^{(s)}\rangle+\langle\widehat{\sigma}^{(s)},\bar{\rho}^{(s)}\widehat{\sigma}^{(s)}\rangle-\gamma_{s}^{2}\langle\phi^{(s)},\zeta^{(s)}\rangle,\end{split}$

which follows from $\bar{\rho}^{(s)}\phi^{(s)}=\zeta^{(s)}$ and where we set
$\widehat{\sigma}^{(s)}=\sigma-\gamma_{s}\phi^{(s)}$. Hence

$\begin{split}\frac{H_{N}(\sigma)}{N}&=\frac{\beta}{2}\langle\sigma,\bar{\textbf{g}}^{(k+1)}\sigma\rangle+\langle\textbf{h}^{(k+1)},\sigma\rangle\\\
&\hskip
14.22636pt+\frac{\beta}{2}\sum_{s=1}^{k}\langle\widehat{\sigma}^{(s)},\bar{\rho}^{(s)}\widehat{\sigma}^{(s)}\rangle-\frac{\beta}{2}\sum_{s=1}^{k}\gamma_{s}^{2}\langle\phi^{(s)},\zeta^{(s)}\rangle+\beta\Big{(}\gamma_{k}-\sqrt{q-\Gamma_{k-1}^{2}}\Big{)}\langle\sigma,\zeta^{(k)}\rangle.\end{split}$

Since an exact evaluation of the free energy seems rather involved, let us
notice here that for configurations $\sigma\in S_{\varepsilon,k}$ as defined
in (4.1) instead, we have approximately
$H_{N}(\sigma)/N\approx\frac{\beta}{2}\langle\sigma,\bar{\textbf{g}}^{(k+1)}\sigma\rangle+\langle\textbf{h}^{(k+1)},\sigma\rangle$.
Indeed, we find that

$\begin{split}\langle\widehat{\sigma}^{(s)},\bar{\rho}^{(s)}\widehat{\sigma}^{(s)}\rangle=2\langle\sigma-\gamma_{s}\phi^{(s)},\zeta^{(s)}\rangle(\langle\phi^{(s)},\sigma\rangle-\gamma_{s})-\langle\phi^{(s)},\zeta^{(s)}\rangle(\langle\phi^{(s)},\sigma\rangle-\gamma_{s})^{2}\end{split}$

with $\gamma_{s}\simeq\langle\textbf{m}^{(k+1)},\phi^{(s)}\rangle$. Similarly,
recall that $\langle\phi^{(s)},\zeta^{(s)}\rangle\sim\mathcal{N}(0,2/N)$ for
each $s$ and

$\Big{|}\Big{(}\gamma_{k}-\sqrt{q-\Gamma_{k}^{2}}\Big{)}\sup_{\sigma\in\\{-1,1\\}^{N}}\langle\sigma,\zeta^{(k)}\rangle\Big{|}\leq\Big{(}\gamma_{k}+\sqrt{q-\Gamma_{k}^{2}}\Big{)}\|\zeta^{(k)}\|$

with $\lim_{k\to\infty}|q-\Gamma_{k}^{2}|=\lim_{k\to\infty}\gamma_{k}=0$ under
(1.3), by Lemmas 3.1 and 3.4.

Thus, we obtain the simple lower bound

$\begin{split}\frac{1}{N}\log
Z_{N}&=\frac{1}{N}\log\sum_{\sigma\in\\{-1,1\\}^{N}}e^{H_{N}(\sigma)}\\\
&=\log
2+\frac{1}{N}\sum_{i=1}^{N}\log\cosh(h^{(k+1)}_{i})+\frac{1}{N}\log\sum_{\sigma\in\\{-1,1\\}^{N}}p_{\text{free}}(\sigma)e^{H_{N}(\sigma)-N\langle\textbf{h}^{(k+1)},\sigma\rangle}\\\
&\geq\log
2+\frac{1}{N}\sum_{i=1}^{N}\log\cosh(h^{(k+1)}_{i})+\frac{1}{N}\log\sum_{\sigma\in
S_{\varepsilon,k}}p_{\text{free}}(\sigma)e^{H_{N}(\sigma)-N\langle\textbf{h}^{(k+1)},\sigma\rangle}\\\
\end{split}$

so that

$\begin{split}\frac{1}{N}\log Z_{N}&\geq\log
2+\frac{1}{N}\sum_{i=1}^{N}\log\cosh(h^{(k+1)}_{i})+\frac{1}{N}\log
Z_{N}^{(k+1)}(S_{\varepsilon,k})\\\ &\hskip
11.38092pt-C\frac{\beta\varepsilon}{k}\sum_{s=1}^{k}\|\zeta^{(s)}\|-C\beta\sum_{s=1}^{k}\|\zeta^{(s)}\|\big{|}\langle\phi^{(s)},\textbf{m}^{(k+1)}\rangle-\gamma_{s}\big{|}\\\
&\hskip
11.38092pt-\frac{\beta}{2}\sum_{s=1}^{k}\gamma_{s}^{2}\langle\phi^{(s)},\zeta^{(s)}\rangle-\beta\gamma_{k}\|\zeta^{(k)}\|-\beta\sqrt{q-\Gamma_{k}^{2}}\|\zeta^{(k)}\|.\end{split}$
(5.1)

We have now all necessary preparations for the proof of Theorem 1.1.

###### Proof of Theorem 1.1.

Up to minor modifications, we follow [6, Section 3] and we also abbreviate
$RS(\beta,h)=\log 2+E\log\cosh(\beta\sqrt{q}Z+h)+\beta^{2}(1-q)^{2}/4.$

By the Paley-Zygmund inequality, we have that

$\mathbb{P}_{k}\Big{(}Z_{N}^{(k+1)}(S_{\varepsilon,k})\geq{\mathbb{E}}_{k}Z_{N}^{(k+1)}(S_{\varepsilon,k})/2\Big{)}\geq\frac{\big{(}{\mathbb{E}}_{k}(Z_{N}^{(k+1)}(S_{\varepsilon,k})\big{)}^{2}}{4{\mathbb{E}}_{k}\big{[}(Z_{N}^{(k+1)}(S_{\varepsilon,k}))^{2}\big{]}}$

Given $\delta_{1}>0$, Lemmas 4.2 and 4.3 imply

$\mathbb{P}\bigg{(}\frac{2}{N}\log{\mathbb{E}}_{k}\,Z_{N}^{(k+1)}(S_{\varepsilon,k})\geq\frac{1}{N}\log
4\,{\mathbb{E}}_{k}\big{(}Z_{N}^{2}(S_{\varepsilon,k})\big{)}-\delta_{1}\bigg{)}\geq\frac{1}{2}$

if we choose $\varepsilon>0$ sufficiently small, $k$ sufficiently large and
$N\geq N_{1}(\varepsilon,k)\in\mathbb{N}$ sufficiently large. This also
implies that

$\begin{split}\mathbb{P}\bigg{(}\mathbb{P}_{k}\bigg{(}\frac{1}{N}\log
Z_{N}^{(k+1)}(S_{\varepsilon,k})\geq\frac{1}{N}\log{\mathbb{E}}_{k}Z_{N}^{(k+1)}(S_{\varepsilon,k})-\frac{\log
2}{N}\bigg{)}\geq e^{-\delta_{1}N}\bigg{)}\geq\frac{1}{2}.\end{split}$

On the other hand, applying Lemma 3.4 _3)_ , we note that

$\lim_{N\to\infty}\mathbb{P}\bigg{(}\frac{1}{k}\sum_{s=1}^{k}\|\zeta^{(s)}\|>2\bigg{)}\leq\lim_{N\to\infty}\sum_{s=1}^{k}\mathbb{P}\Big{(}\|\zeta^{(s)}\|>2\Big{)}=0$

and that

$\lim_{N\to\infty}\mathbb{P}\Big{(}\|\zeta^{(k+1)}\|>2\Big{)}=0.$

Moreover, Lemma 3.4 and the fact that
$\langle\phi^{(s)},\textbf{m}^{(k+1)}\rangle\simeq\gamma_{s}$ by Proposition
3.3 imply

$\lim_{N\to\infty}\bigg{(}C\beta\sum_{s=1}^{k}\|\zeta^{(s)}\|\big{|}\langle\phi^{(s)},\textbf{m}^{(k+1)}\rangle-\gamma_{s}\big{|}+\frac{\beta}{2}\sum_{s=1}^{k}\gamma_{s}^{2}\langle\phi^{(s)},\zeta^{(s)}\rangle\bigg{)}=0$

as well as

$\lim_{N\to\infty}\frac{1}{N}\sum_{i=1}^{N}\log\cosh\big{(}h^{(k+1)}_{i}\big{)}=E\log\cosh(\beta\sqrt{q}Z+h)$

in probability. Now, the lower bound (5.1) implies that

$\begin{split}&\mathbb{P}\bigg{(}\mathbb{P}_{k}\bigg{(}\frac{1}{N}\log
Z_{N}\geq\log 2+\frac{1}{N}\sum_{i=1}^{N}\log\cosh(\beta\sqrt{q}Z+h)\\\
&\hskip
85.35826pt+\frac{1}{N}\log{\mathbb{E}}_{k}Z_{N}^{(k+1)}(S_{\varepsilon,k})-\frac{\log
2}{N}-\mathcal{E}_{\varepsilon,k}\bigg{)}\geq
e^{-\delta_{1}N}\bigg{)}\geq\frac{1}{2},\end{split}$

where we defined the error $\mathcal{E}_{\varepsilon,k}$ by

$\begin{split}\mathcal{E}_{\varepsilon,k}&=\frac{C\beta\varepsilon}{k}\sum_{s=1}^{k}\|\zeta^{(s)}\|+C\beta\sum_{s=1}^{k}\|\zeta^{(s)}\|\big{|}\langle\phi^{(s)},\textbf{m}^{(k+1)}\rangle-\gamma_{s}\big{|}\\\
&\hskip
11.38092pt+\frac{\beta}{2}\sum_{s=1}^{k}\gamma_{s}^{2}\langle\phi^{(s)},\zeta^{(s)}\rangle+\beta\gamma_{k}\|\zeta^{(k)}\|+\beta\sqrt{q-\Gamma_{k}^{2}}\|\zeta^{(k)}\|.\end{split}$

Given $\delta_{2}>0$, we may choose $\varepsilon>0$ sufficiently small,
$k\in\mathbb{N}$ sufficiently large and $N\geq
N_{2}(k,\varepsilon)\in\mathbb{N}$ sufficiently large such that

$\mathbb{P}\bigg{(}|\mathcal{E}_{\varepsilon,k}|\leq\frac{\delta_{2}}{4},\,\bigg{|}\frac{1}{N}\sum_{i=1}^{N}\log\cosh(h^{(k+1)}_{i})-E\log\cosh(\beta\sqrt{q}Z+h)\bigg{|}\leq\frac{\delta_{2}}{4}\bigg{)}\geq\frac{7}{8}$

and, by Lemma 4.2, also such that

$\mathbb{P}\bigg{(}\frac{1}{N}\log{\mathbb{E}}_{k}Z_{N}^{(k+1)}(S_{\varepsilon,k})\geq\frac{\beta^{2}}{4}(1-q)^{2}-\frac{\delta_{2}}{4}\bigg{)}\geq\frac{7}{8}.$

Combining the above observations, we find that

$\mathbb{P}\bigg{(}\mathbb{P}_{k}\bigg{(}\frac{1}{N}\log Z_{N}\geq
RS(\beta,h)-\delta_{2}\bigg{)}\geq e^{-\delta_{1}N}\bigg{)}\geq\frac{1}{4}$

for all $N\geq\max(N_{1}(\varepsilon,k),N_{2}(\varepsilon,k),4\log
2/\delta_{2})$. This implies

$\mathbb{P}\bigg{(}\frac{1}{N}\log Z_{N}\geq
RS(\beta,h)-\delta_{2}\bigg{)}\geq\frac{1}{4}e^{-\delta_{1}N}.$

By Gaussian concentration of the free energy, i.e.

$\mathbb{P}\bigg{(}\bigg{|}\frac{1}{N}\log Z_{N}-\frac{1}{N}{\mathbb{E}}\log
Z_{N}\bigg{|}\leq\delta_{3}\bigg{)}\geq 1-e^{-N\delta_{3}^{2}/\beta^{2}},$

we may choose $\delta_{1}<\delta_{3}^{2}/(2\beta^{2})$ to conclude for large
enough $N$ that

${\mathbb{E}}\frac{1}{N}\log Z_{N}\geq RS(\beta,h)-\delta_{2}-\delta_{3}.$

Since $\delta_{2},\delta_{3}>0$ were arbitrary, this shows that the right hand
side in (1.6) is a lower bound to $\lim_{N\to\infty}\mathbb{E}\frac{1}{N}\log
Z_{N}$ and by _Remark_ 2), this proves Theorem 1.1. ∎

## References

  * [1] A. Adhikari, C. Brennecke, P. von Soosten, H.-T. Yau. Dynamical Approach to the TAP Equations for the Sherrington-Kirkpatrick Model. _J. Stat. Phys._ 183, 35 (2021). https://doi.org/10.1007/s10955-021-02773-7
  * [2] M. Aizenman, J. L. Lebowitz, D. Ruelle. Some rigorous results on the Sherrington-Kirkpatrick spin glass model. _Comm. Math. Phys._ 11 (1987), pp. 3-20.
  * [3] J. R. L. de Almeida, D. J. Thouless. Stability of the Sherrington–Kirkpatrick solution of a spin glass model. _J. Phys. A: Math. Gen._ 11 (1978), pp. 983-990.
  * [4] D. Belius, N. Kistler. The TAP-Plefka Variational Principle for the Spherical SK Model. _Comm. Math. Phys._ 367 (2019), pp. 991-1017.
  * [5] E. Bolthausen. An Iterative Construction of Solutions of the TAP Equations for the Sherrington–Kirkpatrick Model. _Comm. Math. Phys._ 325 (2014), pp. 333-366.
  * [6] E. Bolthausen. A Morita Type Proof of the replica-symmetric Formula for SK. In: _Statistical Mechanics of Classical and Disordered Systems_ (2018), Springer Proceedings in Mathematics & Statistics, pp. 63-93. arXiv:1809.07972.
  * [7] E. Bolthausen, A. Bovier. Spin Glasses. Lecture Notes in Mathematics, 1900 (2007), _Springer Verlag Berlin–Heidelberg_.
  * [8] S. Chatterjee. Spin glasses and Stein’s method. _Probab. Theory Relat. Fields_ 148 (2010), pp. 567-600.
  * [9] W.-K. Chen. On the Almeida-Thouless transition line in the SK model with centered Gaussian external field. Preprint: arXiv:2103.04802.
  * [10] F. Guerra. Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model. _Comm. Math. Phys._ 233 (2003), pp. 1-12.
  * [11] F. Guerra, F.L. Toninelli. The Thermodynamic Limit in Mean Field Spin Glass Models. _Commun. Math. Phys._ 230 (2002), pp. 71-79.
  * [12] A. Auffinger, A. Jagannath. Thouless-Anderson-Palmer equations for generic p-spin glasses. _Ann. Probab._ 47 (2019), no. 4, pp. 2230–2256.
  * [13] A. Jagannath, I. Tobasco. Some properties of the phase diagram for mixed p-spin glasses. _Probab. Theory Relat. Fields_ 167 (2017), pp. 615-672.
  * [14] M. Mézard, G. Parisi, M. A. Virasoro. Spin Glass Theory and Beyond. World Scientific Lecture Notes in Physics Vol.9 (1987), _World Scientific, Singapore New Jersey Hong Kong_.
  * [15] D. Panchenko. The Parisi ultrametricity conjecture. _Ann. of Math._ 177 (2013), Issue 1, pp. 383-393.
  * [16] D. Panchenko. The Sherrington-Kirkpatrick Model. Springer Monographs in Mathematics (2013), _Springer Verlag, New York_.
  * [17] D. Panchenko. The Parisi formula for mixed $p$-spin models. _Ann. of Probab._ 42 (2014), no. 3, pp. 946-958.
  * [18] G. Parisi. Infinite number of order parameters for spin-glasses. _Phys. Rev. Lett._ 43 (1979), pp. 1754-1756.
  * [19] G. Parisi. A sequence of approximate solutions to the S-K model for spin glasses. _J. Phys. A_ 13 (1980), pp. L-115.
  * [20] M. Shcherbina. On the Replica-Symmetric Solution for the Sherrington-Kirkpatrick Model. _Helv. Phys. Acta_ 70 (1997), pp. 838-853.
  * [21] D. Sherrington, S. Kirkpatrick. Solvable model of a spin glass. _Phys. Rev. Lett._ 35 (1975), pp. 1792-1796.
  * [22] M. Talagrand. The Parisi formula. _Ann. of Math._ 163 (2006), no. 1, pp. 221-263.
  * [23] M. Talagrand. Mean Field Models for Spin Glasses. Volume I: Basic Examples. A Series of Modern Surveys in Mathematics, Vol. 54 (2011), _Springer Verlag Berlin–Heidelberg_.
  * [24] M. Talagrand. Mean Field Models for Spin Glasses. Volume II: Advanced Replica-Symmetry and Low Temperature. A Series of Modern Surveys in Mathematics, Vol. 54 (2011), _Springer Verlag Berlin–Heidelberg_.
  * [25] D. J. Thouless, P. W. Anderson, R. G. Palmer. Solution of ’solvable model in spin glasses’. _Philos. Magazin_ 35 (1977), pp. 593-601.