# Disorder Chaos in Short-Range, Diluted, and Lévy Spin Glasses

Wei-Kuo Chen Email<EMAIL_ADDRESS>Partly supported by NSF grants DMS-1752184
and DMS-2246715 and Simons Foundation grant 1027727  Heejune Kim  Email:
<EMAIL_ADDRESS>Partly supported by NSF grants DMS-1752184 and DMS-2246715
Arnab Sen Email<EMAIL_ADDRESS>supported by Simon Foundation MP-
TSM-00002716

###### Abstract

In a recent breakthrough [11], Chatterjee proved site disorder chaos in the
Edwards-Anderson (EA) short-range spin glass model [28] utilizing the Hermite
spectral method. In this paper, we demonstrate the further usefulness of this
Hermite spectral approach by extending the validity of site disorder chaos in
three related spin glass models. The first, called the mixed even $p$-spin
short-range model, is a generalization of the EA model where the underlying
graph is a deterministic bounded degree hypergraph consisting of hyperedges
with even number of vertices. The second model is the diluted mixed $p$-spin
model, which is allowed to have hyperedges with both odd and even number of
vertices. For both models, our results hold under general symmetric disorder
distributions. The main novelty of our argument is played by an elementary
algebraic equation for the Fourier-Hermite series coefficients for the two-
spin correlation functions. It allows us to deduce necessary geometric
conditions to determine the contributing coefficients in the overlap function,
which in spirit is the same as the crucial Lemma 1 in [11]. Finally, we also
establish disorder chaos in the Lévy model with stable index $\alpha\in(1,2)$.

##  1 Introduction and Main Results

The study of chaos in spin glasses began from [7, 31], which is concerned with
the sensitivity of the systems subject to a small change in some external
parameters, such as the temperature, disorder, or external field. One typical
way to capture chaotic phenomenon is to measure the site overlap between two
configurations independently drawn from an unperturbed and a perturbed system,
respectively. The expectation is that with the perturbation, the site overlap
is concentrated near zero if the model is symmetric with respect to the sign
change of the spin configuration and is concentrated near a non-zero value if
the symmetry is broken, for example, due to the presence of the external
field. This is in sharp contrast to the “lack of self-averaging” behavior of
the site overlap in the low temperature phase of many models if the two spin
configurations are independently sampled from the same system (see, e.g.,
Example 1 in [40]). For a modern survey of chaos in spin glasses in the
physics literature, see [43].

Establishing disorder chaos in the Edwards-Anderson (EA) short-range spin
glass model [28] has received great attention recently. In [8], Chatterjee
established a weak form of disorder chaos for the site overlap in the sense
that it is concentrated around its Gibbs average; an alternative proof was
given in Chen-Panchenko [21]. In addition to the site overlap, which aims to
capture chaos through spin interactions on vertices, the bond overlap that
measures chaos through the spin interactions on edges is another quantity of
interest and it was shown in [8] that the expectation of the bond overlap is
strictly positive. In a related direction, Arguin-Newman-Stein [3] made a
connection between bond disorder chaos and incongruent ground states, and
Arguin-Hanson [2] reproved positivity of the expectation of the bond overlap
in [8] in three different perspectives.

In a recent breakthrough, chaos in disorder for the site overlap at zero
temperature in the EA model was settled by Chatterjee [11] through a beautiful
adaptation of the Hermite spectral method. The main ingredient of his proof is
a quantitative version, [11, Theorem 2], of a statement regarding an expected
glassy behavior: “relative orientations of spins with large separation are
sensitive to small changes in the bond strengths”, quoted from Bray-Moore [7].
Following the validity of the site disorder chaos, a number of long-standing
conjectured spin glass behaviors were answered quantitatively.

In the present paper, we aim to investigate chaos in disorder in three spin
glass models. The first, called the mixed $p$-spin short-range model, is a
generalization of the EA model, whose Hamiltonian is defined on a mixture of
hypergraphs of varying lengths $p$ and the spin interactions are placed along
the hyperedges. The second model called the diluted mixed $p$-spin model is a
randomization of the first model, where each $p$-hyperedge is chosen
independently with the same success probability. We assume that the disorder
distribution is symmetric around zero. Therefore, it can always be written as
$\rho(J)$ for an odd function $\rho$ and $J\sim N(0,1)$. For any $t>0,$ we
perturb the disorder in the original systems in continuous and discrete
manners, namely,

$\rho(e^{-t}J+\sqrt{1-e^{-2t}}J^{\prime})$

and

$B\rho(J)+(1-B)\rho(J^{\prime})=\rho(BJ+(1-B)J^{\prime}),$

where $J^{\prime}$ is an independent copy of $J$ and $B$ is Bernoulli with
head probability $e^{-t}$ independent of $J,J^{\prime}.$ Under these two
perturbations, we establish chaos in disorder for the site overlap in the
mixed even $p$-spin short-range model that generalizes the earlier result [11]
in the EA model. Additionally, we show that the same holds in the diluted
mixed $p$-spin model. To the best of our knowledge, disorder chaos in the
diluted models was only known for the diluted even $p$-spin model with large
connectivity, see Chen-Panchenko [23]. Our main result in this direction holds
without this constraint and most importantly, it works for any mixture
including odd $p$.

Our approach to establishing site disorder chaos in the two aforementioned
models is based on the Hermite spectral method; the key ingredient is played
by an elementary algebraic equation stated in Lemma 2.2 for the Fourier-
Hermite series coefficients associated with the two-point spin correlation
function, which allows us to deduce certain geometric conditions, see
Propositions 2.4 and 2.5, to identify the nonzero (contributing) coefficients.
In [11, Lemma 1], a similar condition was derived in the EA model through
flipping the signs of the spins as well as the disorders in the Hamiltonian
depending on the graph structure associated with the given two vertices, which
relies on the fact that the EA model in the absence of external field is
symmetric. In our setting, since the graph structure involves a mixture of
hyperedges of varying lengths, applying the argument in [11, Lemma 1] to
derive the desired geometric conditions appears to be more tedious; such
complication is greatly simplified through our algebraic equations for the
Fourier-Hermite coefficients.

The third model of interest is the Lévy spin glass initially introduced by
Cizeau-Bouchaud [25, 26]. Its Hamiltonian is essentially the same as the
Sherrington-Kirkpatrick (SK) model except that its disorder possesses a power-
law density of order $1<\alpha<2$ so that it only has $\alpha^{\prime}$-moment
for any $\alpha^{\prime}<\alpha.$ Due to this formulation, it is expected that
only the edges with heavy weights will be responsible for the behavior of the
system and as a result, the Lévy model should behave like diluted models in
many aspects, see, [25] for physicists’ predictions. Indeed, it is the case
that one can relate the Lévy model to the diluted $2$-spin model to understand
its free energy and fluctuation, see Chen-Kim-Sen [17]. It is then an
intriguing question to ask whether chaos in disorder holds in the Lévy model
under the same mechanism as the diluted model or not. To this end, we let the
disorder be an arbitrary symmetric heavy-tailed random variable and write the
disorder as $\rho(J)$ for some odd function $\rho$ and again $J\sim N(0,1)$
such that $\rho(J)$ belongs to the domain of attraction of the symmetric
stable distribution of index $\alpha$. Under the continuous perturbation as
before, we show that the Lévy model exhibits disorder chaos. Somewhat
unexpectedly, while one can understand the free energy in the Lévy model via
the diluted model, it is by no means clear if one can use our chaos result for
the diluted model to understand the same objective in the Lévy model as the
Gibbs measures are much more sensitive subject to small perturbation. As we
shall see, our proof makes use of the Hermite spectral method again, but
instead of counting the nonzero Fourier-Hermite coefficients under the same
geometric conditions as before, it is in fact more useful to control the
overlap through the treatment in the SK model, see [10, 18].

###  1.1 Mixed Even $p$-spin short-range Model

Let $\Delta\geq 2$. For $2\leq p\leq\Delta,$ denote by ${[N]\choose p}$ the
collection of all choices of $p$ distinct elements from $[N]$. Let $G=([N],E)$
be a hypergraph with a mixture of hyperedges

$E=\bigcup_{2\leq p\leq\Delta}E_{p},$ (1.1)

for some $E_{p}\subseteq{[N]\choose p}$. An ordinary graph would correspond to
$\Delta=2$, which is the case of the EA model. A ball of radius
$r\in\mathbb{Z}_{+}$ centered at $v\in[N]$ is defined in the usual way as
$B_{r}(v)=\\{v^{\prime}\in[N]:d(v,v^{\prime})\leq r\\}$, where
$d(v,v^{\prime})$ is the hypergraph distance between the vertices $v$ and
$v^{\prime}$ defined in Section 2.1. We assume one of the following growth
assumptions on $G$.

######  Assumption 1.1 (Volume growth assumptions on $G$).

  1. (i)

(Polynomial growth) For some positive constants $A$ and $\theta$ independent
of $N$,

$\max_{i\in[N]}|B_{r}(i)|\leq Ar^{\theta},\ \ r\in\mathbb{N}.$

  2. (ii)

(Exponential growth) For some constants $A$ and $\gamma>1$ independent of $N$,

$\max_{i\in[N]}|B_{r}(i)|\leq A\gamma^{r},\ \ r\in\mathbb{N}.$

For each hyperedge $e\in E$, let $J_{e}$ be an i.i.d. copy from the standard
normal distribution. Let $\rho:\mathbb{R}\to\mathbb{R}$ be an odd function.
With the notation

$\sigma_{e}=\prod_{i\in V(e)}\sigma_{i},\quad\sigma\in\\{\pm 1\\}^{N},$ (1.2)

we consider the Hamiltonian

$H_{J}(\sigma)=\sum_{2\leq
p\leq\Delta}H_{J}^{(p)}(\sigma),\quad\sigma\in\\{\pm 1\\}^{N},$ (1.3)

where $H_{J}^{(p)}(\sigma):=\sum_{e\in E_{p}}\rho(J_{e})\sigma_{e}$ for $2\leq
p\leq\Delta$. Note that we can represent any symmetric disorder as $\rho(J)$.
The Gibbs measure associated to (any) Hamiltonian $H_{J}$ at inverse
temperature $\beta>0$ is given by

$G_{J,\beta}(\sigma)=\frac{e^{\beta
H_{J}(\sigma)}}{Z_{N}},\quad\sigma\in\\{\pm 1\\}^{N},$ (1.4)

where $Z_{N}=Z_{N}(J,\beta)=\sum_{\sigma\in\\{\pm 1\\}^{N}}\exp(\beta
H_{J}(\sigma))$ is the normalizing factor called the partition function.
Denote by $(\sigma^{l})_{l\geq 1}$ i.i.d. samples (replicas) from
$G_{J,\beta}^{\infty}$ and the expectation with respect to these randomness by
$\langle\cdot\rangle_{J,\beta}.$ Note that, when $\beta=\infty$, $G_{J,\beta}$
is a uniform measure supported on the set of ground states, which are spin
configurations achieving the maximum value of $H_{J}$ given the disorder $J$.
In this case, we shall understand $(\sigma^{l})_{l\geq 1}$ as i.i.d. copies
that are sampled uniformly at random from the set of ground states. We often
omit the notation for $J$ and $\beta$ to enhance brevity. Given two spin
configurations $\sigma^{1},\sigma^{2}\in\\{\pm 1\\}^{N}$, we define their
overlap to be

$R(\sigma^{1},\sigma^{2})=\frac{1}{N}\sum_{1\leq i\leq
N}\sigma^{1}_{i}\sigma^{2}_{i}.$

Throughout the article, we consider either one of the following perturbations
of the disorder $\rho(J)$.

######  Definition 1.2 (Two types of perturbations).

  1. (i)

For $t\geq 0$, $J(t)=e^{-t}J+\sqrt{1-e^{-2t}}J^{\prime}$, where $J^{\prime}$
is an indepdent copy of $J$.

  2. (ii)

For $t\geq 0$, $J(t)=BJ+(1-B)J^{\prime}$ where $B\sim\mathrm{Ber}(e^{-t})$,
$J^{\prime}\sim J$, and all the random variables are independent.

We call the perturbation in (i) as a continuous perturbation and the one in
(ii) as a discrete perturbation.

In both types of perturbation, larger $t\geq 0$ corresponds to more
perturbation. The terminology is obvious because, in the first type, we are
continuously deforming the disorder, which contrasts with the discrete
replacement of the disorder with an i.i.d. copy in the second type. The
continuous perturbation is natural when considering Gaussian disorders in view
of the Ornstein-Uhlenbeck (OU) process, see (2.2). The discrete one generally
applies to any disorders, and it is the one often considered in the noise-
sensitivity literature (see, e.g., [35]). We also comment that we may
naturally associate the discrete perturbation with a Markov process called the
independent flip process, a term coined by Chatterjee, see [10, Chapter 7].

As mentioned in the introduction, our focus lies in understanding the
sensitivity of our models subject to the perturbation under Definition 1.2. To
make this notion precise, we consider the overlap $R(\sigma,\tau)$ where
$(\sigma,\tau)$ is sampled from the product measure $G_{J}\otimes G_{J(t)}$.
We always sample $(\sigma^{l},\tau^{l})_{l\geq 1}$ from the product measure
$(G_{J}\otimes G_{J(t)})^{\otimes\infty}$. The Gibbs average in this context
will be denoted by $\langle\cdot\rangle_{t}$ to emphasize that $\tau$ is drawn
from the perturbed Gibbs measure $G_{J(t)}$. Note that these are only defined
for the positive temperature case ($\beta<\infty$), and we understand the zero
temperature case ($\beta=\infty$) as $\sigma$ and $\tau$ being independently
sampled uniformly at random from the set of ground states of the Hamiltonian
$H_{J}$ and $H_{J(t)}$, respectively.

Our first main result focuses on the mixed even $p$-spin short-range model,
i.e., all $p$’s are restricted to even values in (1.1).

######  Theorem 1.3.

Consider the mixed even $p$-spin short-range model. Let $t\geq 0$ and
$\beta\in[0,\infty]$. Then, for both types of perturbations and for any
$r\in\mathbb{Z}_{+}$, we have

$\mathbb{E}\langle
R(\sigma,\tau)^{2}\rangle_{t}\leq\frac{1}{N}\max_{i\in[N]}|B_{r}(i)|+e^{-tr}.$
(1.5)

In particular, we can specialize into the following cases.

  1. 1.

Under Assumption 1.1(i), for any $t>0$ and $\beta\in[0,\infty]$,

$\mathbb{E}\langle
R(\sigma,\tau)^{2}\rangle_{t}\leq\frac{1}{N}+\frac{C}{Nt^{\theta}}$

for some constant $C>0$ depending on $A$ and $\theta$.

  2. 2.

Under Assumption 1.1(ii), for any $t>0$ and $\beta\in[0,\infty]$,

$\mathbb{E}\langle
R(\sigma,\tau)^{2}\rangle_{t}\leq\frac{C}{N^{t/(t+\log\gamma)}}$ (1.6)

for some constant $C>0$ depending on $A$ and $\gamma$.

######  Remark 1.4.

It is an easy fact (see, e.g., Lemma 2.1) that $t\mapsto\mathbb{E}\langle
R(\sigma,\tau)^{2}\rangle_{t}$ is non-increasing. Hence, the overlap bounds
given in Theorem 1.3 are more interesting for small $t$, that is, when the
amount to perturbations to disorders is small. In the polynomial growth case,
our result says that $\mathbb{E}\langle R(\sigma,\tau)^{2}\rangle_{t}\to 0$ as
soon as $t\gg N^{-1/\theta}$ (independent of $\beta)$. For $\beta=\infty$,
this matches with the result obtained in [11] for the ground state of the
Edwards-Anderson model on finite boxes of $\mathbb{Z}^{d}$. On the other hand,
for the exponential growth case, our result implies that $\mathbb{E}\langle
R(\sigma,\tau)^{2}\rangle_{t}\to 0$ as soon as $t\gg(\log N)^{-1},$ again
independent of $\beta$.

###  1.2 Diluted Mixed $p$-spin Model

Fix $\Delta\geq 2$. For each $2\leq p\leq\Delta$, each hyperedge from
${[N]\choose p}$ is selected independently with probability

$\frac{\alpha_{p}N}{{N\choose p}}$

and let $E_{p}$ be the collection of chosen hyperedges. Consider the random
hypergraph $([N],E)$ for $E:=\bigcup_{2\leq p\leq\Delta}E_{p}$. The
Hamiltonian is the same as before, namely (1.3), where the disorder is
independent of the randomness of the edge set. While we considered
deterministic graphs in the mixed even $p$-spin short-range model described in
Theorem 1.3, the randomness of the graphs in the diluted mixed $p$-spin model
aids in removing the restriction on the edges being even, as will be apparent
in Proposition 2.5 and Lemma 4.4 below.

Let us fix the notation

$\lambda\colonequals\sum_{2\leq p\leq\Delta}p(p-1)\alpha_{p}.$

######  Theorem 1.5.

Consider the diluted mixed $p$-spin model. Let $t\geq 0$ and
$\beta\in[0,\infty]$. Assume $\lambda>1$. There exists a constant $C>0$
depending only on $(\alpha_{p})_{2\leq p\leq\Delta}$ such that

$\mathbb{E}\langle
R(\sigma,\tau)^{2}\rangle_{t}\leq\frac{C}{N^{t/(t+2\log\lambda)}}.$

######  Remark 1.6.

  1. 1.

Site disorder chaos for a version of the diluted model was studied earlier in
Chen-Panchenko [23], where the result was limited only to the pure even $p$
spin model with very large $\alpha_{p}$ so that it is close to the mean-field
model. Our result here holds without these restrictions.

  2. 2.

The parameter $\lambda$ is the average vertex degree of the underlying random
hypergraph. It is known [44] that the critical threshold is $\lambda=1$. The
case $\lambda>1$ corresponds to the supercritical regime when there is a giant
(linear-sized) component with high probability. On the other hand, if
$\lambda<1$, the hypergraph is subcritical, that is, all connected components
have at most logarithmic size with high probability, which renders the diluted
model less interesting in this regime.

  3. 3.

For the diluted pure $2$-spin model with Gaussian disorder, for any $\beta>0$,
as $N\to\infty$, the unperturbed overlap $R(\sigma^{1},\sigma^{2})$ has a
trivial distribution at zero in the annealed region $\alpha_{2}\leq 1/2$ (see
[36]). On the other hand, if $\alpha_{2}>1/2$, it is expected that there
exists some critical temperature such that below the criticality, the limiting
overlap has a non-trivial distribution. In sharp contrast to this
nontriviality, the result in Theorem 1.5 indicates that the overlap is
trivialized at any temperature as long as the perturbation is in force.

  4. 4.

We can view $\lambda$ as the bound on the (average) growth rate for the
underlying random hypergraph of the diluted model (see Lemma 4.2). However,
let us emphasize that Theorem 1.5 does not assume that hyperedges contain an
even number of vertices. We circumvent the issue of the odd hyperedges by
exploiting the fact that random hypergraphs are locally tree-like. However, in
the implementation of the argument, we lose a multiplicative factor of $2$ in
front of $\log\lambda$ compared to the bound (1.6) obtained in the
deterministic case with no odd hyperedges.

###  1.3 Lévy Model

Let $1<\alpha<2.$ Let $X$ be a symmetric random variable with tail probability

$\mathbb{P}(|X|>x)=\frac{L(x)}{x^{\alpha}},\ \forall x>0,$

where $L:(0,\infty)\to(0,\infty)$ is a slowly varying function at $\infty$. It
is well-known that for any $p<\alpha$,

$\mathbb{E}|X|^{p}<\infty.$ (1.7)

Potter’s bound says that for any $\eta>0$, there exists $y_{0}>0$ such that
for all $y\geq y_{0}$ and $s\geq 1$,

$\displaystyle\frac{1-\eta}{s^{\eta}}<\frac{L(sy)}{L(y)}<\frac{1+\eta}{s^{-\eta}}.$
(1.8)

In particular, (1.8) dictates the behavior of a slowly varying function as
$s\to\infty$. Suppose that $\rho:\mathbb{R}\to\mathbb{R}$ is a non-decreasing
function such that $\rho(J)$ has the same distribution as $X$ for $J\sim
N(0,1).$ Let $(J_{ij})_{1\leq i,j\leq N}$ be i.i.d. copies of $J.$ Define the
Hamiltonian of the Lévy model as

$H_{N}(\sigma)=\frac{1}{a_{N}}\sum_{i,j=1}^{N}\rho(J_{ij})\sigma_{i}\sigma_{j},\quad\sigma\in\\{\pm
1\\}^{N},$

where $a_{N}=\inf\\{x>0:\mathbb{P}(|X|>x)\leq 1/N\\}.$ It is a standard
result, e.g., [6], in the study of heavy-tailed random variables that

$\displaystyle a_{N}=N^{1/\alpha}\ell(N)$ (1.9)

for some slowly varying function $\ell$ at $\infty$.

The Gibbs measure for the Lévy model is defined in the same way as (1.4) with
the above Hamiltonian.

######  Theorem 1.7.

Consider the Lévy model with $1<\alpha<2$ and the continuous perturbation. For
any $0\leq\beta<\infty$ and $0<\varepsilon<2/\alpha-1$, there exist two
constants $c=c(\beta)>0$ and $K=K(\beta,\varepsilon)>0$ such that for any
$t\geq 0$ and $N\geq 1,$

$\mathbb{E}\langle
R(\sigma,\tau)^{2}\rangle_{t}\leq\frac{K}{N^{(\frac{2}{\alpha}-1-\varepsilon)\min(1,ct)}}.$

######  Remark 1.8.

As in the third item of Remark 1.6, the nearly zero behavior of the overlap in
Theorem 1.7 is a sharp transition as opposed to the unperturbed overlap
$R(\sigma^{1},\sigma^{2})$ in the Lévy model, for which it is expected to have
a nontrivial distribution as long as $\beta$ is large enough, see [25, 26].

###  1.4 Related Results and Applications

In addition to the short-range and graphical models, there has been vast
mathematical progress on chaos in mean-field models in recent years. First of
all, chaos in disorder in the SK model as well as its variant, the mixed even
$p$-spin model, was pioneered by Chatterjee [9] (among many other results) in
the absence of external field. Later on, disorder chaos was extended by Chen
[12, 14] in the presence of the external field and was pushed forward to the
zero temperature setting by Chen-Handschy-Lerman [15]. Some efforts (e.g. [14,
20]) were made to go beyond the mixed even $p$-spin model and allow odd
interactions, which culminated in Eldan’s [30] proof in the general mixed
$p$-spin model. On the other hand, for the same class of models, if it was
assumed that the mixture of the $p$-spin Hamiltonians is generic, a form of
chaos in temperature was achieved by Chen-Panchenko [20] and Chen [13] and the
canonical case was settled by Panchenko [41]. Interestingly, in the spherical
setting, while chaos in disorder or external field remains valid, see, Chen et
al. [16], Chen-Sen [24], and Panchenko-Talagrand [42], the validity of chaos
in temperature depends on the mixture of the $p$-spin Hamiltonians; when the
model consisted of a pure $p$-spin Hamiltonian, Subag [45] showed that chaos
in temperature is absent, whereas if one couples the pure $p$-spin model with
a perturbative $q$-spin Hamiltonian for $q\neq p$, Chen-Panchenko [22] and Ben
Arous-Subag-Zeitouni [4] showed that temperature chaos indeed holds. Notably,
the results mentioned here all assumed that the disorders are Gaussian. The
works, Auffinger-Chen [5] and Chen-Lam [19], addressed the universality of
chaos for general settings of the disorders under matching Gaussian moment
assumptions.

While the aforementioned works (and the current paper) measure chaos in
disorder by the chaotic rearrangement of the overlap, it is also possible to
formalize chaos in disorder in terms of other distances between the perturbed
and unperturbed Gibbs measures. For example, El Alaoui-Montanari-Sellke [29]
proposed using the (normalized) 2-Wasserstein distance as a stronger notion
than the overlap-based approach and showed that there is disorder chaos for
the SK model in the low-temperature regime with the Wasserstein distance,
which poses a barrier to stable sampling algorithms. In a subsequent work [1],
the same authors demonstrated the shattering phenomenon in pure $p$-spherical
spin glasses for large enough $p$ and high temperatures, which again implies
disorder chaos in the Wasserstein distance. On the other hand, shattering and
the overlap gap property in the Ising pure $p$-spin case in the high-
temperature regime were investigated by Gamarnik-Jagannath-Kızıldağ [33],
again for large enough $p$ by making a connection to the random energy model.

Finally, we close this subsection by mentioning that the overlap gap and
disorder chaos properties have been shown to be the main features that result
in computational barriers for stable algorithms to achieve near-ground states
in a variety of random optimization problems, including the approximate
message passing, low-degree polynomial, and Lipschitz algorithms (see [32, 34,
37, 38] and references therein).

###  1.5 Open Problems

  1. 1.

For any deterministic sequence of 3-uniform hypergraphs with either a
polynomial or exponential growing number of vertices, does site disorder chaos
hold in the corresponding short-range model?

  2. 2.

While the definition of discrete perturbation is non-ambiguous for general
disorder distribution, there is more than one way to introduce continuous
perturbations to disorders. For example, let us consider the case when the
disorders have a density $e^{-V(x)}$ for some nice (say, continuously
differentiable and convex) function $V$. Let $X$ be drawn from this
distribution and let $X(t)$ be the solution to the stochastic differential
equation,

$X(t)=-V^{\prime}(X(t))dt+\sqrt{2}{dB(t)},\ \ X(0)=X,$

where $(B(t))_{t\geq 0}$ is the standard Brownian motion independent of $X$.
If $p(t,\cdot)$ is the density of $X(t)$, then $p$ satisfies the associated
Fokker-Planck equation

$\frac{\partial p(t,x)}{\partial t}=\frac{\partial}{\partial
x}\Bigl{(}V^{\prime}(x)p(t,x)+\frac{\partial p(t,x)}{\partial x}\Bigr{)},\ \
p(0,x)=e^{-V(x)}.$

Note that $X(0)=X,X(t)\stackrel{{\scriptstyle d}}{{=}}X$ for all
$t\in[0,\infty)$, and $X(\infty)$ is independent of $X$. Therefore, we can
take $X(t)$ as another candidate for continuous perturbation of $X$. In the
case $V(x)=x^{2}/2+2^{-1}\ln(2\pi)$, $(X(t))_{t\geq 0}$ is known as the
Ornstein-Uhlenbeck process and it can be solved as
$X(t)=e^{-t}X+e^{-t}\int_{0}^{t}e^{u}dB(u).$ Consequently, $(X,X(t))$
corresponds to our continuous perturbation $(J,J(t)).$ It would be interesting
to see if chaos in disorder can be studied beyond this case by Fourier
analysis involving orthonormal polynomials in $L^{2}(e^{-V(x)}dx)$.

  3. 3.

For stable index $\alpha>1$, Theorem 1.7 establishes disorder chaos in the
Lévy model under the continuous perturbation. Prove disorder chaos for the
discrete perturbation in the same regime of $\alpha$. Also, the case
$\alpha\in(0,1]$ remains completely open.

##  2 Preliminaries

###  2.1 Hypergraphs

For $\ell\in\mathbb{N}$, a Berge path of length $\ell$ between $v_{1}$ and
$v_{\ell+1}$ is a sequence
$(v_{1},e_{1},v_{2},e_{2},\dots,v_{\ell},e_{\ell},v_{\ell+1})$ of distinct
$\ell+1$ vertices and distinct $\ell$ hyperedges such that
$\\{v_{k},v_{k+1}\\}\subseteq V(e_{k})$ for $k=1,\dots,\ell$. We define the
distance $d:[N]\times[N]\to\mathbb{R}$ between two vertices as the minimum
length among all possible Berge paths connecting those vertices, provided they
are distinct; the distance is defined as $0$ if the vertices are identical. We
assign $\infty$ if there are no paths connecting two distinct vertices. A
(closed) ball of radius $r\in\mathbb{Z}_{+}$ centered at $v\in[N]$ is defined
in the usual way as $B_{r}(v)=\\{v^{\prime}\in[N]:d(v,v^{\prime})\leq r\\}$.
Two vertices are considered connected if there is a Berge path between them,
and this establishes a partition of $[N]$ into connected components. A Berge
cycle of length $\ell$ is a sequence
$(v_{1},e_{1},v_{2},e_{2},\dots,v_{\ell},e_{\ell},v_{\ell+1})$ where
$v_{1},\dots,v_{\ell}$ are distinct vertices, $e_{1},\dots,e_{\ell}$ are
distinct edges, and $\\{v_{k},v_{k+1}\\}\subseteq V(e_{k})$ for
$k=1,\dots,\ell$ with $v_{\ell+1}=v_{1}$. A hypertree is a connected
hypergraph without Berge cycles. For brevity, a path and a cycle will always
mean a Berge path and a Berge cycle, respectively.

For a subset $B\subseteq[N]$, we define its interior and the edge boundary as

$\mathring{B}=\\{e\in E:V(e)\subseteq B\\}\text{ and }\partial B=\\{e\in
E:0<|V(e)\cap B|<|V(e)|\\}.$

Clearly, the edge set has the partition $E=\mathring{B}\cup\partial
B\cup\\{e\in E:|V(e)\cap B|=0\\}$.

Denote $\mathbb{Z}_{+}\colonequals\mathbb{N}\cup\\{0\\}$. For $n=(n_{e})_{e\in
E}\in\mathbb{Z}_{+}^{E}$, we define

$E(n)=\\{e\in E:n_{e}>0\\}\text{ and }V(n)=\\{v\in[N]:v\in V(e)\text{ for some
}e\in E(n)\\}.$

Let $G(n)=(V(n),E(n))$ be the associated sub-hypergraph of $G$. Denote the
connected component of $v\in V(n)$ in $G(n)$ as $C(v,n)$ for any
$n\in\mathbb{Z}_{+}^{E}$.

###  2.2 Hermite Polynomial Expansion

Let $(h_{m})_{m\geq 0}$ be the orthonormal basis of normalized Hermite
polynomials for $L^{2}(\mu)$ where $\mu$ is the standard Gaussian measure on
$\mathbb{R}$ and $h_{0}\equiv 1$. Then, products of the form
$h_{n}(J)\colonequals\prod_{e\in E}h_{n_{e}}(J_{e})$ for
$n\in\mathbb{Z}_{+}^{E}$ is a basis for $L^{2}(J)$. For any $\phi\in
L^{2}(J)$, we have the Hermite polynomial expansion

$\phi(J)=\sum_{n\in\mathbb{Z}_{+}^{E}}\hat{\phi}(n)h_{n}(J),$ (2.1)

where $\hat{\phi}(n)\colonequals\mathbb{E}(\phi(J)h_{n}(J))$. We use the
notation $|n|\colonequals\sum_{e\in E}n_{e}$.

######  Lemma 2.1.

Let $t\geq 0$. For any $\phi\in L^{2}(J)$, we have

$\displaystyle\mathbb{E}\bigl{(}\phi(J)\phi(J(t))\bigr{)}$
$\displaystyle=\sum_{n\in\mathbb{Z}_{+}^{E}}e^{-|n|t}\hat{\phi}(n)^{2},$ (2.2)
$\displaystyle\mathbb{E}\bigl{(}\phi(J)\phi(J(t))\bigr{)}$
$\displaystyle=\sum_{n\in\mathbb{Z}_{+}^{E}}e^{-|E(n)|t}\hat{\phi}(n)^{2}$

for the continuous and discrete perturbations, respectively.

###### Proof.

Our proof relies on the following two equations: for the continuous
perturbation, for any $n\in\mathbb{Z}_{+}^{E},$ we have

$\mathbb{E}(h_{n}(J(t))|J)=\sum_{n\in\mathbb{Z}_{+}^{E}}e^{-|n|t}h_{n}(J),$
(2.3)

whereas for the discrete perturbation,

$\mathbb{E}(h_{n}(J(t))|J)=\sum_{n\in\mathbb{Z}_{+}^{E}}e^{-|E(n)|t}h_{n}(J).$
(2.4)

Equation (2.3) is a standard result of the OU process (see, e.g., [10]).
Equation (2.4) can be obtained by taking expectation in the Bernoulli
randomness.

Now, let $t\geq 0$ and consider the continuous perturbation. From (2.1) and
(2.3), it holds that

$\mathbb{E}(\phi(J(t))|J)=\sum_{n\in\mathbb{Z}_{+}^{E}}\hat{\phi}(n)e^{-|n|t}h_{n}(J).$

Since $\mathbb{E}h_{n}(J)h_{m}(J)=\mathbbm{1}_{n=m}$ for
$n,m\in\mathbb{Z}_{+}^{E}$, we have

$\mathbb{E}(\phi(J)\phi(J(t)))=\mathbb{E}(\phi(J)\mathbb{E}(\phi(J(t))|J))=\sum_{n\in\mathbb{Z}_{+}^{E}}\hat{\phi}(n)^{2}e^{-|n|t}.$

The case of discrete perturbation can similarly be treated using (2.4).

∎

###  2.3 Geometric Conditions for Vanishing Fourier-Hermite Coefficients

This subsection consists of the main ingredient in the present work, in which
we establish the promised elementary algebraic equation for the Fourier-
Hermite series coefficients, see Lemma 2.2 below, and deduce a sufficient
condition for determining the zero coefficients in a systematic way, see
Propositions 2.4 and 2.5 below. In particular, Lemma 2.2 and Proposition 2.4
are in lieu of Lemma 1 and its geometric consequence Corollary 5 in [11],
respectively. While [11]’s trick involves flipping a subset of disorders and
spins, as we will see below, flipping only spins suffices for our purposes.
Nevertheless, we also present Proposition 2.7 as a generalization of the trick
in Lemma 1 of [11].

Recall the notation $\eqref{eq: product notation}.$ For any $a,b\in\\{\pm
1\\}^{N}$, we define the tensors ${\bf a}=(a_{e})_{e\in E}$, ${\bf
b}=(b_{e})_{e\in E}$, and define their Hadamard product ${\bf a}\circ{\bf
b}=(a_{e}b_{e})_{e\in E}$. We denote the Gibbs average of the disorder
$(a_{e}J_{e})_{e\in E}\equalscolon{\bf a}\circ J$ by
$\langle\cdot\rangle_{{\bf a}\circ J}$.

######  Lemma 2.2.

Consider the generalized short-range model with Hamiltonian in (1.3) with
$\beta\in[0,\infty)$. For any $a\in\\{\pm 1\\}^{N}$, $i,j\in[N]$, and
$n\in\mathbb{Z}_{+}^{E}$, it holds that

$\mathbb{E}h_{n}(J)\langle\sigma_{i}\sigma_{j}\rangle=I_{n}(a)\mathbb{E}h_{n}(J)\langle\sigma_{i}\sigma_{j}\rangle,$

where

$I_{n}(a):=a_{i}a_{j}\prod_{e\in E(n)}a_{e}^{n_{e}}.$

######  Remark 2.3.

It is immediate from the lemma that
$\mathbb{E}h_{n}(J)\langle\sigma_{i}\sigma_{j}\rangle=0$ if $I_{n}(a)\neq 1$
for some choice of $a\in\\{\pm 1\\}^{N}$. However, the following example
demonstrates that the converse is not true, even for a connected graph
($p=2$).

Let $G=(V,E)=([4],\\{\\{1,2\\},\\{1,3\\},\\{2,4\\}\\})$. Let
$\phi=\langle\sigma_{1}\sigma_{2}\rangle$. Then one can check that $\phi$ does
not depend on $J_{\\{1,3\\}},J_{\\{2,4\\}}$. Indeed,

$\displaystyle\phi$
$\displaystyle=\frac{\sum_{\sigma}\sigma_{1}\sigma_{2}(1+\tanh(\beta
J_{\\{1,2\\}})\sigma_{1}\sigma_{2})(1+\tanh(\beta
J_{\\{1,3\\}})\sigma_{1}\sigma_{3})(1+\tanh(\beta
J_{\\{2,4\\}})\sigma_{2}\sigma_{4})}{\sum_{\sigma}(1+\tanh(\beta
J_{\\{1,2\\}})\sigma_{1}\sigma_{2})(1+\tanh(\beta
J_{\\{1,3\\}})\sigma_{1}\sigma_{3})(1+\tanh(\beta
J_{\\{2,4\\}})\sigma_{2}\sigma_{4})}$
$\displaystyle=\frac{\sum_{\sigma_{1},\sigma_{2}}\sigma_{1}\sigma_{2}(1+\tanh(\beta
J_{\\{1,2\\}})\sigma_{1}\sigma_{2})}{\sum_{\sigma_{1},\sigma_{2}}(1+\tanh(\beta
J_{\\{1,2\\}})\sigma_{1}\sigma_{2})}=\tanh(\beta J_{\\{1,2\\}}).$

Now let $n\in\mathbb{Z}_{+}^{E}$ with $n_{\\{1,2\\}}=1,$ $n_{\\{1,3\\}}=2,$
and $n_{\\{2,4\\}}=0$. It follows that

$\hat{\phi}(n)=\mathbb{E}[\phi
h_{1}(J_{\\{1,2\\}})h_{2}(J_{\\{1,3\\}})]=\mathbb{E}[\phi
h_{1}(J_{\\{1,2\\}})]\mathbb{E}[h_{2}(J_{\\{1,3\\}})]=0,$

since $\mathbb{E}[h_{2}(J_{\\{1,3\\}})]=0$. On the other hand,

$I_{n}(a)=a_{1}a_{2}(a_{1}a_{2})(a_{1}a_{3})^{2}=1\ \text{ for all }a\in\\{\pm
1\\}^{4}.$

###### Proof of Lemma 2.2.

Note that for any deterministic $a\in\\{-1,1\\}^{N},$ since $\rho$ is an odd
function, we can write

$H_{J}(a\circ\sigma)=\sum_{2\leq p\leq\Delta}\sum_{e\in
E_{p}}\rho(J_{e})a_{e}\sigma_{e}=\sum_{2\leq p\leq\Delta}\sum_{e\in
E_{p}}\rho(a_{e}J_{e})\sigma_{e}=H_{{\bf a}\circ J}(\sigma)$

so that
$\langle\sigma_{i}\sigma_{j}\rangle_{J}=a_{i}a_{j}\langle\sigma_{i}\sigma_{j}\rangle_{{\bf
a}\circ J}.$ From this and the fact that ${\bf a}\circ{\bf a}\circ J=J$,

$\mathbb{E}h_{n}(J)\langle\sigma_{i}\sigma_{j}\rangle_{J}=a_{i}a_{j}\mathbb{E}h_{n}({\bf
a}\circ{\bf a}\circ J)\langle\sigma_{i}\sigma_{j}\rangle_{{\bf a}\circ
J}=a_{i}a_{j}\mathbb{E}h_{n}({\bf a}\circ
J)\langle\sigma_{i}\sigma_{j}\rangle_{J},$

where the second equality used the obvious fact that ${\bf a}\circ
J\stackrel{{\scriptstyle d}}{{=}}J$. Finally, using $h_{n}({\bf a}\circ
J)=\prod_{e\in E}h_{n_{e}}(a_{e}J_{e})=\prod_{e\in
E}a_{e}^{n_{e}}h_{n_{e}}(J_{e})=\prod_{e\in E}a_{e}^{n_{e}}\cdot
h_{n}(J)=\prod_{e\in E(n)}a_{e}^{n_{e}}\cdot h_{n}(J)$, we have

$\mathbb{E}h_{n}(J)\langle\sigma_{i}\sigma_{j}\rangle_{J}=a_{i}a_{j}\prod_{e\in
E(n)}a_{e}^{n_{e}}\cdot\mathbb{E}h_{n}(J)\langle\sigma_{i}\sigma_{j}\rangle_{J}.$

∎

Now, for any $i,j\in[N]$, define
$\phi_{ij}(J)=\langle\sigma_{i}\sigma_{j}\rangle$, which is a bounded
measurable function of the disorder $J$, hence in $L^{2}(J)$.

######  Proposition 2.4.

Consider the mixed even $p$-spin short-range model on $G=([N],E)$. Let $i,j$
be distinct vertices in $[N]$ and $n\in\mathbb{Z}_{+}^{E}$. If
$\hat{\phi}_{ij}(n)\neq 0$, then we have $i,j\in V(n)$ and there exists a path
from $i$ to $j$ in $G(n)$. In particular, $|E(n)|\geq d(i,j)$.

###### Proof.

Suppose $i\notin V(n)$. In this case, let $a_{i}=-1$ and $a_{l}=1$ if $l\neq
i$. Then, $i\notin V(n)$ implies $a_{e}=1$ for any $e\in E(n)$, so we can
conclude $\mathbb{E}h_{n}(J)\langle\sigma_{i}\sigma_{j}\rangle=0.$

Now, suppose $i\in V(n).$ Assume that $j\notin C(i,n).$ Let $a_{l}=-1$ if
$l\in C(i,n)$ and $a_{l}=1$ otherwise. We claim that $a_{e}=1$ whenever $e\in
E(n)$. To that end, observe that since every edge has even number of vertices,
we have $a_{e}=1$ whenever $e\in\mathring{C}(i,n)$. For
$e\notin\mathring{C}(i,n)$, $a_{e}=1$ trivially holds. Thus,

$a_{i}a_{j}\prod_{e\in E(n)}a_{e}^{n_{e}}=-1,$

and we can apply Lemma 2.2 to conclude
$\mathbb{E}h_{n}(J)\langle\sigma_{i}\sigma_{j}\rangle=0.$ By changing the
roles of $i$ and $j$, we have shown that $\hat{\phi}_{ij}(n)\neq 0$ implies
$i,j\in V(n)$ are in the same component of $G(n)$. ∎

If we impose that the hypergraph is locally a hypertree, we may allow
hyperedges with odd number of vertices. The following Proposition will be used
in Section 1.2 for the proof of the disorder chaos of the diluted mixed
$p$-spin model,

######  Proposition 2.5.

Consider the generalized short-range model on $G=([N],E)$. Let $i,j$ be
distinct vertices in $[N]$ and $n\in\mathbb{Z}_{+}^{E}$. Assume that the ball
of depth $r\geq 1$ around $i$ in $G$ is a hypertree. If
$\hat{\phi}_{ij}(n)\neq 0$, then $i,j\in V(n)$ and $|E(n)|\geq\min(r,d(i,j)).$

###### Proof.

We note that $I_{n}(a)$ for $\langle\sigma_{i}\sigma_{j}\rangle$ does not
depend on edges with even $n_{e}$. Keeping that in mind, for
$n\in\mathbb{Z}_{+}^{E}$, we let $E_{\mathrm{odd}}(n)\colonequals\\{e\in
E(n):\textrm{$n_{e}$ is odd}\\}$,
$V_{\mathrm{odd}}(n)\colonequals\bigcup_{e\in E_{\mathrm{odd}}(n)}V(e)$ be the
associated vertices, and
$G_{\mathrm{odd}}(n)=(V_{\mathrm{odd}}(n),E_{\mathrm{odd}}(n))$ be the
associated sub-hypergraph. Note that $V_{\mathrm{odd}}(n)\subseteq V(n)$.

If $i\notin V_{\mathrm{odd}}(n)$, then we have $\hat{\phi}_{ij}(n)=0$ by
choosing $a_{i}=-1$ and $a_{l}=1$ for $l\neq i$. The same conclusion holds if
$j\notin V_{\mathrm{odd}}(n)$.

Suppose $i,j\in V_{\mathrm{odd}}(n)$ and let $|E(n)|<\min(r,d(i,j))$. Then,
the connected component $C_{\mathrm{odd}}(i,n)$ of $i$ in the sub-hypergraph
$G_{\mathrm{odd}}(n)$ is a hypertree and $j\notin C_{\mathrm{odd}}(i,n)$.
Choose a hyperleaf $v\in C_{\mathrm{odd}}(i,n)$, i.e., there exists exactly
one edge in $E_{\mathrm{odd}}(n)$ containing $v$. Since all hyperedges contain
at least two vertices, we may assume that $v\neq i$. Define $a_{v}=-1$ and
$a_{l}=1$ if $l\neq v$. Then, $a_{i}a_{j}\prod_{e\in E(n)}a_{e}^{n_{e}}=-1$,
and Lemma 2.2 yields $\hat{\phi}_{ij}(n)=0$.

∎

######  Remark 2.6.

Proposition 2.4 does not always hold if we allow hyperedges with odd number of
vertices. For a counterexample, consider the hypergraph $G=(V,E)$ with vertex
set
$V=\\{0,1^{\prime},2^{\prime},3^{\prime},1^{\prime\prime},2^{\prime\prime},3^{\prime\prime}\\}$
and hyperedge set $E=E^{\prime}\cup
E^{\prime\prime}\cup\\{\\{0,3^{\prime},3^{\prime\prime}\\}\\}$ for

$\displaystyle
E^{\prime}=\\{\\{1^{\prime},2^{\prime},3^{\prime}\\},\\{2^{\prime},3^{\prime}\\}\,\,\mbox{and}\,\,E^{\prime\prime}=\\{\\{1^{\prime\prime},2^{\prime\prime},3^{\prime\prime}\\},\\{2^{\prime\prime},3^{\prime\prime}\\}\\}.$

See Figure 1. Let
$\phi=\langle\sigma_{1^{\prime}}\sigma_{1^{\prime\prime}}\rangle.$

Figure 1: Hypergraph $G$.

Below we write
$\sigma=(\sigma_{0},\sigma^{\prime},\sigma^{\prime\prime})\in\\{-1,1\\}^{V}$
with $\sigma^{\prime}\in\\{-1,1\\}^{\\{1^{\prime},2^{\prime},3^{\prime}\\}}$
and
$\sigma^{\prime\prime}\in\\{-1,1\\}^{\\{1^{\prime\prime},2^{\prime\prime},3^{\prime\prime}\\}}$.
Note that $\phi$ is the ratio of the following two quantities:

$\displaystyle\sum_{\sigma}\sigma_{1^{\prime}}\sigma_{1^{\prime\prime}}\prod_{e\in
E}(1+\tanh(\beta J_{e})\sigma_{e})$
$\displaystyle=\sum_{\sigma^{\prime},\sigma^{\prime\prime}}\sigma_{1^{\prime}}\sigma_{1^{\prime\prime}}\prod_{e\in
E^{\prime}\cup E^{\prime\prime}}(1+\tanh(\beta J_{e})\sigma_{e}),$
$\displaystyle\sum_{\sigma}\prod_{e\in E}(1+\tanh(\beta J_{e})\sigma_{e})$
$\displaystyle=\sum_{\sigma^{\prime},\sigma^{\prime\prime}}\prod_{e\in
E^{\prime}\cup E^{\prime\prime}}(1+\tanh(\beta J_{e})\sigma_{e}),$

where the two equalities hold by summing over the single spin $\sigma_{0}.$
Therefore, $\phi$ decouples as
$\phi=\langle\sigma_{1^{\prime}}\rangle_{E^{\prime}}\langle\sigma_{1^{\prime\prime}}\rangle_{E^{\prime\prime}}$
with

$\displaystyle\langle\sigma_{1^{\prime}}\rangle_{E^{\prime}}$
$\displaystyle=\frac{\sum_{\sigma^{\prime}}\sigma_{1^{\prime}}(1+\tanh(\beta
J_{\\{1^{\prime},2^{\prime},3^{\prime}\\}})\sigma_{1^{\prime}}\sigma_{2^{\prime}}\sigma_{3^{\prime}})(1+\tanh(\beta
J_{\\{2^{\prime},3^{\prime}\\}})\sigma_{2^{\prime}}\sigma_{3^{\prime}})}{\sum_{\sigma^{\prime}}(1+\tanh(\beta
J_{\\{1^{\prime},2^{\prime},3^{\prime}\\}})\sigma_{1^{\prime}}\sigma_{2^{\prime}}\sigma_{3^{\prime}})(1+\tanh(\beta
J_{\\{2^{\prime},3^{\prime}\\}})\sigma_{2^{\prime}}\sigma_{3^{\prime}})}$
(2.5)

and $\langle\sigma_{1^{\prime\prime}}\rangle_{E^{\prime\prime}}$ is defined
similarly. Distributing the product over the sum, we deduce that

$\displaystyle\langle\sigma_{1^{\prime}}\rangle_{E^{\prime}}=\tanh(\beta
J_{\\{1^{\prime},2^{\prime},3^{\prime}\\}})\tanh(\beta
J_{\\{2^{\prime},3^{\prime}\\}}),\quad\langle\sigma_{1^{\prime\prime}}\rangle_{E^{\prime\prime}}=\tanh(\beta
J_{\\{1^{\prime\prime},2^{\prime\prime},3^{\prime\prime}\\}})\tanh(\beta
J_{\\{2^{\prime\prime},3^{\prime\prime}\\}}).$

If we take $n\in\mathbb{Z}_{+}^{E}$ with
$n_{\\{1^{\prime},2^{\prime},3^{\prime}\\}}=n_{\\{2^{\prime},3^{\prime}\\}}=n_{\\{1^{\prime\prime},2^{\prime\prime},3^{\prime\prime}\\}}=n_{\\{2^{\prime\prime},3^{\prime\prime}\\}}=1$
and $n_{\\{0,3^{\prime},3^{\prime\prime}\\}}=0$, then

$\hat{\phi}(n)=\mathbb{E}[\langle\sigma_{1^{\prime}}\rangle_{E^{\prime}}\langle\sigma_{1^{\prime\prime}}\rangle_{E^{\prime\prime}}J_{\\{1^{\prime},2^{\prime},3^{\prime}\\}}J_{\\{2^{\prime},3^{\prime}\\}}J_{\\{1^{\prime\prime},2^{\prime\prime},3^{\prime\prime}\\}}J_{\\{2^{\prime\prime},3^{\prime\prime}\\}}]=(\mathbb{E}[J\tanh(\beta
J)])^{4}>0,$

for any $\beta>0$. However, there is no path between $1^{\prime}$ and
$1^{\prime\prime}$ in $E(n)$, since any such path cannot avoid the hyperedge
$\\{0,3^{\prime},3^{\prime\prime}\\}$. Let us also point out that this example
does not contradict Proposition 2.5 since $1$-neighborhood of vertex
$1^{\prime}$ contains a Berge cycle, namely,
$(2^{\prime},\\{1^{\prime},2^{\prime},3^{\prime}\\},3^{\prime},\\{2^{\prime},3^{\prime}\\},2^{\prime}).$

One can replace the hyperedge $\\{0,3^{\prime},3^{\prime\prime}\\}$ with a
path of arbitrary length of $k$ connecting $3$ and $3^{\prime}$ such as
$\\{3^{\prime},x_{1},y_{1}\\},\\{y_{1},x_{2},y_{2}\\},\\{y_{2},x_{3},y_{3}\\},\ldots,\\{y_{k-1},x_{k},3^{\prime\prime}\\}$
where
$1^{\prime},2^{\prime},3^{\prime},1^{\prime\prime},2^{\prime\prime},3^{\prime\prime}$,
$x_{1},\ldots,x_{k}$, $y_{1},\ldots,y_{k-1}$ are all distinct vertices.
Summing over spins $\sigma_{x_{1}},\ldots,\sigma_{x_{k}}$ as we did in (2.5),
we can still conclude that
$\phi=\langle\sigma_{1^{\prime}}\rangle_{E^{\prime}}\langle\sigma_{1^{\prime\prime}}\rangle_{E^{\prime\prime}}$
and $\hat{\phi}(n)>0$ for the index $n$ where $n_{e}=0$ on this new path and
$n_{e}=1$ on the rest of the four hyperedges. This shows that if we allow odd
hyperedges, then it is possible that $\hat{\phi}(n)$ is non-zero, but $d(i,j)$
is arbitrarily large (but finite) compared to $|E(n)|$. Also, we can have
$d(i,j)=\infty$ and $\hat{\phi}(n)>0$ by simply deleting the hyperedge
$\\{0,3^{\prime},3^{\prime\prime}\\}$ from $E$.

We present a proposition that is of independent interest, but is not used
elsewhere in the paper.

######  Proposition 2.7.

Let $G=([N],E)$ be a graph and $i\neq j\in[N]$. If $S\subseteq E$ does not
contain a path in $S$ that connects $i$ and $j$, then

$\mathbb{E}[\langle\sigma_{i}\sigma_{j}\rangle|J_{r},r\in S]=0.$

###### Proof.

Let $\phi=\langle\sigma_{i}\sigma_{j}\rangle$. Then

$\mathbb{E}[\phi|J_{r},r\in S]=\sum_{n\in\mathbb{Z}_{+}^{E},n|_{E\setminus
S}=0}\hat{\phi}(n)h_{n}(J).$

Now for any $n|_{E\setminus S}=0$, the edge set $E(n)=\\{e\in
E:n_{e}>0\\}\subseteq S$ and hence, it does not contain a path joining $i$ and
$j$. Therefore, $\hat{\phi}(n)=0$. The proof follows. ∎

##  3 Proof of Theorem 1.3

###### Proof of Theorem 1.3.

It suffices to prove the theorem for finite $\beta\geq 0$. The case
$\beta=\infty$ would then immediately follows from sending $\beta\to\infty$.
From Lemma 2.1, Proposition 2.4, and Parseval’s identity
$\sum_{n\in\mathbb{Z}_{+}^{E}}\hat{\phi}_{ij}(n)^{2}=\mathbb{E}\langle\sigma_{i}\sigma_{j}\rangle^{2}$,
and the fact that $|n|\geq|E(n)|$ for $n\in\mathbb{Z}_{+}^{E}$, we have for
both types of perturbations

$\mathbb{E}\langle\sigma_{i}\sigma_{j}\tau_{i}\tau_{j}\rangle_{t}\leq
e^{-td(i,j)}\mathbb{E}\langle\sigma_{i}\sigma_{j}\rangle^{2}$

for any $i,j\in[N]$. Therefore, for any $r\in\mathbb{Z}_{+}$,

$\displaystyle\mathbb{E}\langle R(\sigma,\tau)^{2}\rangle_{t}$
$\displaystyle=\frac{1}{N^{2}}\sum_{1\leq i,j\leq
N}\mathbb{E}\langle\sigma_{i}\sigma_{j}\tau_{i}\tau_{j}\rangle_{t}\leq\frac{1}{N^{2}}\sum_{1\leq
i,j\leq N}e^{-td(i,j)}\mathbb{E}\langle\sigma_{i}\sigma_{j}\rangle^{2}$
$\displaystyle\leq\frac{1}{N}\max_{i\in[N]}\sum_{j:d(i,j)\leq
r}e^{-td(i,j)}\mathbb{E}\langle\sigma_{i}\sigma_{j}\rangle^{2}+\frac{1}{N^{2}}\sum_{(i,j):d(i,j)>r}e^{-td(i,j)}\mathbb{E}\langle\sigma_{i}\sigma_{j}\rangle^{2}$
(3.1)
$\displaystyle\leq\frac{1}{N}\max_{i\in[N]}|B_{r}(i)|+e^{-tr}\mathbb{E}\langle
R(\sigma^{1},\sigma^{2})^{2}\rangle,$ (3.2)

which proves (1.5) since $|R(\sigma^{1},\sigma^{2})|\leq 1$.

Under (i) of Assumption 1.1, setting $r=N+1$ so that the second term vanishes
in (3.1), we have

$\displaystyle\mathbb{E}\langle R(\sigma,\tau)^{2}\rangle_{t}$
$\displaystyle\leq\frac{1}{N}\max_{i\in[N]}\sum_{1\leq j\leq
N}e^{-td(i,j)}\leq\frac{1}{N}+\frac{1}{N}\max_{i\in[N]}\sum_{j\in[N]\setminus\\{i\\}}e^{-td(i,j)},$
(3.3)

which we bound below by grouping vertices according to the values of $\lfloor
td(i,j)\rfloor,$

$\displaystyle\frac{1}{N}+\frac{1}{N}\max_{i\in[N]}\sum_{k=0}^{\infty}\sum_{\begin{subarray}{c}j:\lfloor
td(i,j)\rfloor=k\\\ j\neq i\end{subarray}}e^{-k}$
$\displaystyle\leq\frac{1}{N}+\frac{1}{N}\sum_{k=0}^{\infty}e^{-k}\max_{i\in[N]}|\\{j\neq
i:kt^{-1}\leq d(i,j)<(k+1)t^{-1}\\}|$

Note that $|\\{j\neq i:kt^{-1}\leq d(i,j)<(k+1)t^{-1}\\}|\leq
A(k+1)^{\theta}t^{-\theta},$ which also holds true if $(k+1)<t$, since in this
case the set becomes empty.

Thus, (3.3) is bounded by

$\frac{1}{N}+\sum_{k=0}^{\infty}e^{-k}A(k+1)^{\theta}t^{-\theta}\leq\frac{1}{N}+\frac{C}{Nt^{\theta}},$

for some constant $C>0$ depending only on $A$ and $\theta$.

Under (ii) of Assumption 1.1, then (3.2) is bounded above by

$\displaystyle\mathbb{E}\langle R(\sigma,\tau)^{2}\rangle_{t}$
$\displaystyle\leq\frac{1}{N}\max_{i\in[N]}|B_{r}(i)|+e^{-tr}\leq\frac{1}{N}A\gamma^{r}+e^{-tr}.$

Setting $r=\lceil(t+\log\gamma)^{-1}\log N\rceil$, we obtain the bound

$\mathbb{E}\langle
R(\sigma,\tau)^{2}\rangle_{t}\leq\frac{C}{N^{t/(t+\log\gamma)}}$

for some $C>0$ depending only on $A$ and $\gamma$.

∎

######  Remark 3.1.

In the exponential growth case, we can retain $\mathbb{E}\langle
R(\sigma^{1},\sigma^{2})^{2}\rangle$ in the bound (3.2) to obtain

$\mathbb{E}\langle
R(\sigma,\tau)^{2}\rangle_{t}\leq\frac{1}{N}A\gamma^{r}+e^{-tr}\mathbb{E}\langle
R(\sigma^{1},\sigma^{2})^{2}\rangle.$

By taking $r=\lceil(t+\log\gamma)^{-1}\log(N\mathbb{E}\langle
R(\sigma^{1},\sigma^{2})^{2}\rangle)\rceil$, we arrive at the bound

$\mathbb{E}\langle R(\sigma,\tau)^{2}\rangle_{t}\leq C(N\mathbb{E}\langle
R(\sigma^{1},\sigma^{2})^{2}\rangle)^{-t/(t+\log\gamma)}\mathbb{E}\langle
R(\sigma^{1},\sigma^{2})^{2}\rangle$

for some constant $C$ depending on $A$ and $\gamma$. Of course, this bound
offers no improvement over (1.6) if $\mathbb{E}\langle
R(\sigma^{1},\sigma^{2})^{2}\rangle\geq c$ for some positive constant $c$.
However, in the case when $\mathbb{E}\langle
R(\sigma^{1},\sigma^{2})^{2}\rangle\ll 1$, instead of the trivial bound
$\mathbb{E}\langle R(\sigma,\tau)^{2}\rangle_{t}\leq\mathbb{E}\langle
R(\sigma^{1},\sigma^{2})^{2}\rangle$, we can conclude from the above bound
that

$\mathbb{E}\langle R(\sigma,\tau)^{2}\rangle_{t}\ll\mathbb{E}\langle
R(\sigma^{1},\sigma^{2})^{2}\rangle,$

if $t\gg 1/\log(N\mathbb{E}\langle R(\sigma^{1},\sigma^{2})^{2}\rangle).$

##  4 Proof of Theorem 1.5

We prove Theorem 1.5 using a spectral method similar to the proof of Theorem
1.3. The main difference lies in the utilization of Proposition 2.5 instead of
Proposition 2.4. Due to the diluted nature of our hypergraph, we demonstrate
in Lemma 4.4 that the hypergraph looks locally like a hypertree up to depth
$(2\log\lambda)^{-1}(1-\varepsilon)\log N$ with probability at least
$1-N^{-\varepsilon}$ for any $0<\varepsilon<1$. We achieved this via the use
of the exploration process defined in Section 4.1, which allows us to control
the growth rate and the probability of producing a cycle at each depth of a
ball. The use of the exploration process to study the component size and their
properties of random hypergraphs is standard, see for example, [27, 39, 44].
However, we could not locate Lemma 4.4 in the literature in its exact form,
and we chose to include its proof for completeness. Combined with Proposition
2.5, we show that the low-frequency coefficients in the Hermite polynomial
expansion are zero with high probability. The remainder of the proof follows
as in Theorem 1.3.

###  4.1 Exploration Process

We first describe the exploration process on the random hypergraph which
starts from a single vertex and conducts a breadth-first search on its
connected component. Fix a vertex $i\in[N]$. We define the exploration process
$(R_{t},I_{t},S_{t},E_{t})_{t\in\mathbb{Z}_{+}}\in[N]^{3}\times E$ inductively
as follows. Put $R_{0}=\emptyset$, $I_{0}=\\{i\\}$,
$S_{0}=[N]\setminus\\{i\\}$, and $E_{0}=\emptyset$. Given
$\mathcal{F}_{t}\colonequals\sigma(R_{s},I_{s},S_{s},E_{s})_{0\leq s\leq t}$,
we define

$\displaystyle R_{t+1}$ $\displaystyle=R_{t}\cup I_{t},$ $\displaystyle
E_{t+1}$ $\displaystyle=\\{e\in E:V(e)\cap I_{t}\neq\emptyset,\ V(e)\cap
R_{t}=\emptyset\\},$ $\displaystyle I_{t+1}$ $\displaystyle=\bigcup_{e\in
E_{t+1}}V(e)\cap S_{t},$ $\displaystyle S_{t+1}$ $\displaystyle=S_{t}\setminus
I_{t+1}.$

Heuristically, at step $t$, $R_{t}$ represents the removed (or recovered)
vertices, $I_{t}$ represents the newly infected (or explored) vertices,
$S_{t}$ represents the susceptible (or unexplored) vertices. $E_{t}$
represents the set of hyperedges that we ‘reveal’ during the step $t$ of the
algorithm. It consists of hyperedges with the property that they contain at
least one vertex from $I_{t-1}$ and all of their vertices belong to
$I_{t-1}\cup S_{t-1}$.

Below we state some simple properties of the exploration process without a
proof. Here $\dot{\cup}$ denotes the disjoint union.

######  Proposition 4.1 (Properties of exploration processes).

  1. (i)

$R_{t}=\dot{\bigcup}_{s<t}I_{s}$ for any $t\in\mathbb{Z}_{+}.$

  2. (ii)

$R_{t}\dot{\cup}I_{t}\dot{\cup}S_{t}=[N]$ for any $t\in\mathbb{Z}_{+}$.

  3. (iii)

$V(e)\subseteq I_{t}\cup I_{t+1}$ for any $t\in\mathbb{Z}_{+}$ and $e\in
E_{t+1}$.

  4. (iv)

$\bigcup_{t\in\mathbb{Z}_{+}}R_{t}=V(C)$ and
$\dot{\bigcup}_{t\in\mathbb{Z}_{+}}E_{t}=E(C)$ where $C$ is the connected
component of $i$.

  5. (v)

$B_{t}=R_{t+1}=\dot{\bigcup}_{s\leq t}I_{s}$ for any $t\in\mathbb{Z}_{+}$,
where $B_{t}=B_{t}(i)$ is the ball of radius $t$ centered at $i$.

Recall $\lambda=\sum_{2\leq p\leq\Delta}p(p-1)\alpha_{p}$. Let us also define
$\lambda^{\prime}=\sum_{2\leq p\leq\Delta}p(p-1)(p-2)\alpha_{p}$.

######  Lemma 4.2.

For $t\in\mathbb{Z}_{+}$, we have

$\displaystyle\mathbb{E}|I_{t}|$ $\displaystyle\leq\lambda^{t},$
$\displaystyle\mathbb{E}|I_{t}|^{2}$
$\displaystyle\leq\sum_{k=t}^{2t}\lambda^{k}+\lambda^{\prime}\sum_{k=t-1}^{2(t-1)}\lambda^{k}.$

###### Proof.

Let $t\in\mathbb{Z}_{+}$. Observe that $v\in I_{t+1}$ only if there is a
$p$-hyperedge in $E_{t+1}$ connecting $v$ to some vertex in $I_{t}$ for some
$2\leq p\leq\Delta$, and similarly, for $v\neq v^{\prime}\in S_{t}$,
$\\{v,v^{\prime}\\}\subseteq I_{t+1}$ only if either there is a $p$-hyperedge
in $E_{t+1}$ connecting $\\{v,v^{\prime}\\}$ to some vertex in $I_{t}$ for
some $2\leq p\leq\Delta$ or there is a pair $(e_{1},e_{2})$ of a
$p_{1}$-hyperedge $e_{1}\in E_{t+1}$ and a $p_{2}$-hyperedge $e_{2}\in
E_{t+1}$ for some $2\leq p_{1},p_{2}\leq\Delta$ such that $v\in
V(e_{1})\setminus V(e_{2})$ and $v^{\prime}\in V(e_{2})\setminus V(e_{1})$.

Figure 2: (Left panel): $v,v^{\prime}$ belonging to the same hyperedge. (Right
panel): $v,v^{\prime}$ belonging to different hyperedges.

Hence,

$\displaystyle\mathbb{P}(v\in I_{t+1}|\mathcal{F}_{t})$ $\displaystyle\leq
q_{t},$ $\displaystyle\mathbb{P}(v,v^{\prime}\in I_{t+1}|\mathcal{F}_{t})$
$\displaystyle\leq q^{\prime}_{t},$ (4.1)

where

$\displaystyle q_{t}$ $\displaystyle\colonequals 1-\prod_{2\leq
p\leq\Delta}\Bigl{(}1-\frac{\alpha_{p}N}{{N\choose p}}\Bigr{)}^{{N-2\choose
p-2}|I_{t}|}$ $\displaystyle q^{\prime}_{t}$ $\displaystyle\colonequals
1-\prod_{3\leq p\leq\Delta}\Bigl{(}1-\frac{\alpha_{p}N}{{N\choose
p}}\Bigr{)}^{{N-3\choose p-3}|I_{t}|}+\Bigl{(}1-\prod_{2\leq
p\leq\Delta}\Bigl{(}1-\frac{\alpha_{p}N}{{N\choose p}}\Bigr{)}^{{N-3\choose
p-2}|I_{t}|}\Bigr{)}^{2}.$

Using the elementary bound $1-\prod_{2\leq
p\leq\Delta}(1-x_{p})^{n_{p}}\leq\sum_{2\leq p\leq\Delta}n_{p}x_{p}$ for
$n_{p}\in\mathbb{Z}_{+}$ and $x_{p}\in(0,1),$ which follows from the convexity
of the function $t\mapsto\prod_{2\leq p\leq\Delta}(1-tx_{p})^{n_{p}}$ on the
interval $(0,1)$,

$\displaystyle q_{t}$ $\displaystyle\leq\sum_{2\leq
p\leq\Delta}\frac{\alpha_{p}N}{{N\choose p}}{N-2\choose
p-2}|I_{t}|=\frac{\lambda|I_{t}|}{N-1},$ $\displaystyle q^{\prime}_{t}$
$\displaystyle\leq\sum_{3\leq p\leq\Delta}\frac{\alpha_{p}N}{{N\choose
p}}{N-3\choose p-3}|I_{t}|+\Bigl{(}\sum_{2\leq
p\leq\Delta}\frac{\alpha_{p}N}{{N\choose p}}{N-3\choose
p-2}|I_{t}|\Bigr{)}^{2}\leq\frac{\lambda^{\prime}|I_{t}|}{(N-1)(N-2)}+\frac{\lambda^{2}|I_{t}|^{2}}{(N-1)^{2}}.$

From the previous display and the trivial bound $|S_{t}|\leq N-1$, we have

$\displaystyle q_{t}|S_{t}|$ $\displaystyle\leq\lambda|I_{t}|,$ $\displaystyle
q_{t}^{\prime}|S_{t}|(|S_{t}|-1)$
$\displaystyle\leq\lambda^{\prime}|I_{t}|+\lambda^{2}|I_{t}|^{2}.$

Now, since $|I_{t+1}|=\sum_{v\in S_{t}}\mathbbm{1}_{\\{v\in I_{t+1}\\}}$,
(4.1) implies the recursions

$\displaystyle\mathbb{E}|I_{t+1}|$ $\displaystyle\leq
q_{t}\mathbb{E}|S_{t}|\leq\lambda\mathbb{E}|I_{t}|,$ (4.2)
$\displaystyle\mathbb{E}|I_{t+1}|^{2}$ $\displaystyle\leq
q_{t}\mathbb{E}|S_{t}|+q^{\prime}_{t}\mathbb{E}|S_{t}|(|S_{t}|-1)\leq(\lambda+\lambda^{\prime})\mathbb{E}|I_{t}|+\lambda^{2}\mathbb{E}|I_{t}|^{2}.$
(4.3)

Equation (4.2) and the initial condition $|I_{0}|=1$ readily implies
$\mathbb{E}|I_{t}|\leq\lambda^{t}$. Plugging this into (4.3), we have

$\mathbb{E}|I_{t+1}|^{2}\leq(\lambda+\lambda^{\prime})\lambda^{t}+\lambda^{2}\mathbb{E}|I_{t}|^{2},$

and an induction easily yields

$\mathbb{E}|I_{t}|^{2}\leq(\lambda+\lambda^{\prime})\sum_{k=0}^{t-1}\lambda^{t-1+k}+\lambda^{2t}.$

∎

######  Lemma 4.3.

Assume that $\lambda>1$. Then there exists a constant $C$ depending only on
$(\alpha_{p})_{2\leq p\leq\Delta}$ such that for any $t\in\mathbb{Z}_{+}$,

$\mathbb{E}|B_{t}(i)|\leq C\lambda^{t}.$

###### Proof.

By Proposition 4.1(v), $|B_{t}(i)|=\sum_{s=0}^{t}|I_{s}|$. The lemma now
follows from the estimate of $\mathbb{E}|I_{s}|$ in Lemma 4.2. ∎

######  Lemma 4.4.

Assume $\lambda>1$. Let $0<\varepsilon<1$ and
$\delta=(2\log\lambda)^{-1}(1-\varepsilon)$. We have

$\mathbb{P}(\text{$B_{\lfloor\delta\log N\rfloor}(i)$ contains a Berge
cycle})\leq CN^{-\varepsilon}$

for some constant $C>0$ depending only on $(\alpha_{p})_{2\leq p\leq\Delta}$.

###### Proof.

We run the exploration process
$(R_{t},I_{t},S_{t},E_{t})_{t\in\mathbb{Z}_{+}}$ starting at the vertex $i$.
Suppose there is no cycle until step $t-1$. Because of (iii) of Proposition
4.1, there are only two ways to create a potential cycle at step $t\geq 1$:

Case 1. A hyperedge $e\in E_{t}$ contains two distinct vertices $v\neq
v^{\prime}$ in $I_{t-1}$.

Case 2. Two distinct hyperedges $e_{1}\neq e_{2}$ in $E_{t}$ have a common
vertex $v$ in $I_{t}$.

Figure 3: Left panel depicts Case 1 and right panel depicts Case 2.

Let $A_{t}$ and $D_{t}$ denote the number of hyperedges corresponding to Case
1 and Case 2 respectively. Given $\mathcal{F}_{t-1}$, $A_{t}$ is
stochastically dominated by $\sum_{2\leq p\leq\Delta}Y_{p}$, where
$(Y_{p})_{2\leq p\leq\Delta}$ are independent binomial random variables with
distribution

$Y_{p}\sim\text{Bin}\Biggl{(}|I_{t-1}|^{2}{N-2\choose
p-2},\frac{\alpha_{p}N}{{N\choose p}}\Biggr{)},$

so by linearity of expectation, we have

$\mathbb{E}(A_{t}|\mathcal{F}_{t-1})\leq\sum_{p=2}^{\Delta}\mathbb{E}(Y_{p}|\mathcal{F}_{t-1})=\sum_{p=2}^{\Delta}|I_{t-1}|^{2}{N-2\choose
p-2}\frac{\alpha_{p}N}{{N\choose p}}=\frac{\lambda|I_{t-1}|^{2}}{N-1}.$ (4.4)

Let us write

$D_{t}=\sum_{2\leq p_{1},p_{2}\leq\Delta}\sum_{|V(e_{l})|=p_{\ell},\
\ell=1,2}\mathbbm{1}_{\\{(e_{1},e_{2})\text{ satisfies Case 2}\\}},$

and note that the indicators in the previous display may be dependent if
$p_{1}=p_{2}$. Nonetheless, the number of all such pairs in Case 2 is at most

$\sum_{p_{1},p_{2}=2}^{\Delta}N|I_{t-1}|^{2}{N-2\choose p_{1}-2}{N-2\choose
p_{2}-2},$

so we have

$\mathbb{E}(D_{t}|\mathcal{F}_{t-1})\leq\sum_{p_{1},p_{2}=2}^{\Delta}N|I_{t-1}|^{2}{N-2\choose
p_{1}-2}{N-2\choose p_{2}-2}\frac{\alpha_{p_{1}}N}{{N\choose
p_{1}}}\frac{\alpha_{p_{2}}N}{{N\choose
p_{2}}}=\frac{\lambda^{2}N|I_{t-1}|^{2}}{(N-1)^{2}}.$ (4.5)

By Markov’s inequality, (4.4), and (4.5), we have

$\mathbb{P}(A_{t}\geq 1\text{ or }D_{t}\geq
1|\mathcal{F}_{t-1})\leq\mathbb{E}(A_{t}|\mathcal{F}_{t-1})+\mathbb{E}(D_{t}|\mathcal{F}_{t-1})\leq\frac{\lambda|I_{t-1}|^{2}}{N-1}+\frac{\lambda^{2}N|I_{t-1}|^{2}}{(N-1)^{2}}.$

Taking another expectation and summing over $1\leq t\leq 1+\delta\log N$, we
have that for sufficiently large $N$ and $C>0$ depending only on $\lambda$,

$\mathbb{P}(\text{$B_{\lfloor\delta\log N\rfloor}(i)$ contains a
cycle})\leq\sum_{1\leq t\leq 1+\delta\log N}\mathbb{P}(A_{t}\geq 1\text{ or
}D_{t}\geq 1)\leq\frac{C}{N}\sum_{0\leq t\leq\delta\log
N}\mathbb{E}|I_{t}|^{2}.$ (4.6)

On the other hand, from Lemma 4.2, there is another constant $C$ depending
only on $\lambda$ and $\lambda^{\prime}$ such that

$\sum_{0\leq t\leq\delta\log N}\mathbb{E}|I_{t}|^{2}\leq
CN^{2\delta\log\lambda}.$

Together with (4.6), this finishes the proof. ∎

Similar to the proof of the disorder chaos in the mixed even $p$-spin short-
range model (Section 3), for any $i,j\in[N]$, we view
$\langle\sigma_{i}\sigma_{j}\rangle$ as a measurable function $\phi_{ij}(J,E)$
of the disorder $J$ and the edge set $E$, which is bounded in modulus by $1$.
Let $\hat{\phi}_{ij}(n,E)\colonequals\mathbb{E}_{J}(\phi_{ij}(J,E)h_{n}(J))$
for any $i,j\in[N]$.

Now, we combine the results obtained so far. Below $\mathbb{E}$ represents the
expectation with respect to two layers of randomness: the disorder and the
random graph.

######  Lemma 4.5.

Let $\varepsilon>0$ and $\delta=(2\log\lambda)^{-1}(1-\varepsilon)$. Then, for
both types of perturbations and any $i,j\in[N]$, we have

$\mathbb{E}\langle\sigma_{i}\sigma_{j}\tau_{i}\tau_{j}\rangle_{t}\leq
CN^{-\varepsilon}+\mathbb{E}e^{-t\min(d(i,j),\delta\log N)},$

where $C>0$ is the constant in Lemma 4.4.

###### Proof.

For both types of perturbations and any $i,j\in[N]$, Lemma 2.1 implies

$\displaystyle\mathbb{E}_{J}\langle\sigma_{i}\sigma_{j}\tau_{i}\tau_{j}\rangle_{t}\leq$
$\displaystyle\sum_{n:|E(n)|<\min(d(i,j),\delta\log
N)}e^{-|E(n)|t}\hat{\phi}_{ij}(n,E)^{2}$
$\displaystyle\qquad+\sum_{n:|E(n)|\geq\min(d(i,j),\delta\log
N)}e^{-|E(n)|t}\hat{\phi}_{ij}(n,E)^{2}.$ (4.7)

Note that by Parseval’s identity
$\sum_{n\in\mathbb{Z}_{+}^{E}}\hat{\phi}_{ij}(n,E)^{2}=\mathbb{E}_{J}\phi_{ij}(J,E)^{2}\leq
1$. Thus, an application of Proposition 2.5 with $r=\delta\log n$ yields

$\sum_{n:|E(n)|<\min(d(i,j),\delta\log
N)}e^{-|E(n)|t}\hat{\phi}_{ij}(n,E)^{2}\leq\mathbbm{1}_{\\{\text{$B_{\lfloor\delta\log
N\rfloor}(i)$ contains a berge cycle}\\}},$

since on the complement of the event on the RHS above, $B_{\delta\log n}(i)$
is a hypertree and hence, $\hat{\phi}_{ij}(n,E)=0$ whenever
$|E(n)|<\min(d(i,j),\delta\log N)$. Thanks to Lemma 4.4, the first term in
(4.7), after taking an expectation with respect to the randomness of the
hypergraph, is bounded above by $CN^{-\varepsilon}$. The second term in (4.7)
can be trivially bounded above by $\mathbb{E}\exp(-t\min(d(i,j),\delta\log
N))$ from another use of Parseval’s identity. ∎

###### Proof of Theorem 1.5.

By symmetry of the diluted mixed $p$-spin model and Lemma 4.5, we have

$\displaystyle\mathbb{E}\langle R(\sigma,\tau)^{2}\rangle_{t}$
$\displaystyle=\frac{1}{N^{2}}\sum_{1\leq i,j\leq
N}\mathbb{E}\langle\sigma_{i}\sigma_{j}\tau_{i}\tau_{j}\rangle_{t}=\frac{1}{N}\sum_{1\leq
j\leq N}\mathbb{E}\langle\sigma_{1}\sigma_{j}\tau_{1}\tau_{j}\rangle_{t}$
$\displaystyle\leq CN^{-\varepsilon}+\frac{1}{N}\sum_{1\leq j\leq
N}\mathbb{E}e^{-t\min(d(1,j),\delta\log N)}$ $\displaystyle\leq
CN^{-\varepsilon}+\frac{1}{N}\mathbb{E}|B_{\lfloor\delta\log
N\rfloor}(1)|+\frac{1}{N}\mathbb{E}\sum_{j:d(1,j)>\delta\log N}e^{-t\delta\log
N}$ $\displaystyle\leq
CN^{-\varepsilon}+\frac{1}{N}\mathbb{E}|B_{\lfloor\delta\log
N\rfloor}(1)|+N^{-\delta t}.$ (4.8)

By Lemma 4.3, the second term of (4.8) is bounded by
$\mathbb{E}|B_{\lfloor\delta\log N\rfloor}(1)|\leq CN^{\delta\log\lambda}$.
Plugging into (4.8), we obtain

$\mathbb{E}\langle R(\sigma,\tau)^{2}\rangle_{t}\leq
CN^{-\varepsilon}+CN^{-1+\delta\log\lambda}+N^{-\delta t}.$

Finally, the proof is complete by recalling the definition
$\delta=(2\log\lambda)^{-1}(1-\varepsilon)$ to write the previous display as

$\mathbb{E}\langle R(\sigma,\tau)^{2}\rangle_{t}\leq
2CN^{-\varepsilon}+N^{-t(1-\varepsilon)/(2\log\lambda)}$

for some constant $C>0$, and then optimizing over $\varepsilon$. ∎

##  5 Proof of Theorem 1.7

Our proof of Theorem 1.7 is inspired by Chatterjee’s semigroup method for the
SK model in [10] that consisted of two major steps. The first is similar to
Lemma 5.1$(ii)$ established below that allows one to control
$\mathbb{E}\langle R(\sigma,\tau)^{2}\rangle_{t}$ for small $t$ via its
behavior for large $t$. The second step aims to control the large $t$ case,
which relies on controlling of the derivatives of $\mathbb{E}\langle
R(\sigma,\tau)^{2}\rangle_{t}$ for any orders via the Gaussian integration by
parts. However, this step is not relevant to our case as the heavy-tailed
distribution does not have a finite second moment. To tackle this difficulty,
we adapt the convexity argument from Chen-Lam [19] instead and the control is
made possible through a critical use of the Gaussian hypercontractivity.

We begin our proof by the following lemma that is essentially taken from
Chatterjee [10]. Let $k\geq 1$ be a fixed integer. Let $J$ be a
$k$-dimensional standard normal random vector and ${J}^{1}$ and ${J}^{2}$
i.i.d. copies of $J.$ For the sake of symmetry, we set
$J^{1}(t)=e^{-t/2}{J}+\sqrt{1-e^{-t}}{J}^{1}$ and
$J^{2}(t)=e^{-t/2}{J}+\sqrt{1-e^{-t}}{J}^{2}$ for $t\geq 0$. Notice that the
Gaussian vector $(J^{1}(t),J^{2}(t))$ has the same distribution as $(J,J(t))$
in Definition 1.2.

######  Lemma 5.1.

Let $m\geq 1$ and $L_{1},\ldots,L_{m}:\mathbb{R}^{k}\to\mathbb{R}$ be
measurable functions. Assume that

$\max_{1\leq r\leq m}\mathbb{E}|L_{r}({J})|^{2}<\infty.$

Let $\phi(t):=\sum_{r=1}^{m}\mathbb{E}L_{r}(J^{1}(t))L_{r}(J^{2}(t))$. For any
$0\leq s\leq t$, we have

  1. $(i)$

$0\leq\phi(t)\leq\phi(s),$

  2. $(ii)$

$\phi(s)\leq\phi(t)^{\frac{s}{t}}\phi(0)^{1-\frac{s}{t}}.$

###### Proof.

It is immediate to see that Lemma 2.1 implies $(i)$. Moreover, $(ii)$ follows
from Hölder’s inequality:

$\displaystyle\phi(s)$
$\displaystyle=\sum_{r=1}^{m}\sum_{n\in\mathbb{Z}_{+}^{k}}e^{-s|n|}\widehat{L}_{r}(n)^{2}$
$\displaystyle\leq\Bigl{(}\sum_{r=1}^{m}\sum_{n\in\mathbb{Z}_{+}^{k}}e^{-s|n|\frac{t}{s}}\widehat{L}_{r}(n)^{2}\Bigr{)}^{\frac{s}{t}}\Bigl{(}\sum_{r=1}^{m}\sum_{n\in\mathbb{Z}_{+}^{k}}1\cdot\widehat{L}_{r}(n)^{2}\Bigr{)}^{1-\frac{s}{t}}=\phi(t)^{\frac{s}{t}}\phi(0)^{1-\frac{s}{t}}.$

∎

For any $t\in[0,1]$ and $\lambda>0,$ consider the coupled free energy

$\displaystyle F_{N}(t,\lambda)$
$\displaystyle:=\frac{1}{N}\mathbb{E}\ln\sum_{\sigma,\tau}\exp\bigl{(}\beta
H_{N,t}^{1}(\sigma)+\beta H_{N,t}^{2}(\tau)+\lambda
NR(\sigma,\tau)^{2}\bigr{)}.$

Let $\langle\cdot\rangle_{t,\lambda}$ be the Gibbs expectation associated to
this free energy. We use the following consequence of the Gaussian
hypercontractive inequality (see, e.g., page 45 in [10]): for any function
$f:\mathbb{R}^{k}\to\mathbb{R}_{+}$ such that $f\in L^{1+e^{-t}}(J)$,

$\mathbb{E}f(J^{1}(t))f(J^{2}(t))\leq(\mathbb{E}f(J)^{1+e^{-t}})^{2/(1+e^{-t})}.$

######  Lemma 5.2.

For any $\lambda>0,$ $t>-\ln(\alpha-1)$, and $1<b<\alpha/(1+e^{-t})$, we have
that

$\displaystyle F_{N}(t,\lambda)-F_{N}(\infty,\lambda)$
$\displaystyle\leq\frac{2N\beta^{2}}{a_{N}^{2}}\frac{1}{1-e^{-t}}\bigl{(}\mathbb{E}|J|^{b/(b-1)}\bigr{)}^{2(b-1)/b}\bigl{(}\mathbb{E}|\rho|^{b(1+e^{-t})}(J)\bigr{)}^{2/((1+e^{-t})b)}.$

###### Proof.

Case 1: Smooth $\rho$. For any $R>0,$ let $\zeta_{R}:\mathbb{R}\to[0,1]$ be a
smooth function satisfying that $\zeta_{R}\equiv 1$ on $[-R,R]$ and
$\zeta_{R}\equiv 0$ on $[-(R+1),R+1]^{c}$. We further assume that
$\sup_{R>0}\|\zeta_{R}^{\prime}\|_{\infty}<\infty.$ Let
$\rho_{R}=\rho\zeta_{R}$. Let $F_{N}^{R}(t,\lambda)$ be the same as
$F_{N}(t,\lambda)$ with the replacement of $\rho$ by $\rho_{R}$ and let
$G_{N,t,\lambda}^{R}$ be the Gibbs measure associated to
$F_{N}^{R}(t,\lambda).$ Denote by
$(\sigma^{1},\sigma^{2}),(\sigma^{1},\tau^{1}),(\sigma^{2},\tau^{2})$ the
i.i.d. samples of $G_{N,t,\lambda}^{R}$ and by
$\langle\cdot\rangle_{t,\lambda}^{R}$ the corresponding Gibbs expectation.
Note that $\rho_{R}$ is twice differentiable and its derivatives are uniformly
bounded. A direct computation gives that

$\displaystyle\partial_{t}F_{N}^{R}(t,\lambda)$
$\displaystyle=\frac{\beta}{2Na_{N}}\sum_{i,j=1}^{N}\mathbb{E}\Bigl{\langle}\bigl{(}-e^{-t/2}J_{ij}+e^{-t}(1-e^{-t})^{-1/2}J_{ij}^{1}\bigr{)}\rho_{R}^{\prime}(J_{ij}^{1}(t))\sigma_{i}\sigma_{j}$
$\displaystyle\qquad\qquad\qquad\qquad+\bigl{(}-e^{-t/2}J_{ij}+e^{-t}(1-e^{-t})^{-1/2}J_{ij}^{2}\bigr{)}\rho_{R}^{\prime}(J_{ij}^{2}(t))\tau_{i}\tau_{j}\Big{\rangle}_{t,\lambda}^{R}.$

Here, using the Gaussian integration by parts yields

$\displaystyle\mathbb{E}J_{ij}\bigl{\langle}\rho_{R}^{\prime}(J_{ij}^{1}(t))\sigma_{i}\sigma_{j}\bigr{\rangle}_{t,\lambda}^{R}$
$\displaystyle=\frac{e^{-t/2}\beta}{a_{N}}\mathbb{E}\rho_{R}^{\prime}(J_{ij}^{1}(t))\rho_{R}^{\prime}(J_{ij}^{1}(t))\bigl{\langle}\sigma_{i}^{1}\sigma_{j}^{1}(\sigma_{i}^{1}\sigma_{j}^{1}-\sigma_{i}^{2}\sigma_{j}^{2})\bigr{\rangle}_{t,\lambda}^{R}$
$\displaystyle+\frac{e^{-t/2}\beta}{a_{N}}\mathbb{E}\rho_{R}^{\prime}(J_{ij}^{1}(t))\rho_{R}^{\prime}(J_{ij}^{2}(t))\bigl{\langle}\sigma_{i}^{1}\sigma_{j}^{1}(\tau_{i}^{1}\tau_{j}^{1}-\tau_{i}^{2}\tau_{j}^{2})\bigr{\rangle}_{t,\lambda}^{R}$
$\displaystyle+e^{-t/2}\mathbb{E}\bigl{\langle}\rho_{R}^{\prime\prime}(J_{ij}^{1}(t))\sigma_{i}\sigma_{j}\bigr{\rangle}_{t,\lambda}^{R}.$
(5.1)

and

$\displaystyle(1-e^{-t})^{-1/2}\mathbb{E}J_{ij}^{1}\bigl{\langle}\rho_{R}^{\prime}(J_{ij}^{1}(t))\sigma_{i}\sigma_{j}\bigr{\rangle}_{t,\lambda}^{R}$
$\displaystyle=\mathbb{E}\bigl{\langle}\rho_{R}^{\prime\prime}(J_{ij}^{1}(t))\sigma_{i}\sigma_{j}\bigr{\rangle}_{t,\lambda}^{R}$
$\displaystyle+\frac{\beta}{a_{N}}\mathbb{E}\rho_{R}^{\prime}(J_{ij}^{1}(t))\rho_{R}^{\prime}(J_{ij}^{1}(t))\bigl{\langle}\sigma_{i}^{1}\sigma_{j}^{1}(\sigma_{i}^{1}\sigma_{j}^{1}-\sigma_{i}^{2}\sigma_{j}^{2})\bigr{\rangle}_{t,\lambda}^{R}.$
(5.2)

Adding these together leads to

$\displaystyle\mathbb{E}\Bigl{\langle}\bigl{(}-e^{-t/2}J_{ij}+e^{-t}(1-e^{-t})^{-1/2}J_{ij}^{1}\bigr{)}\rho^{\prime}(J_{ij}^{1}(t))\sigma_{i}\sigma_{j}\Bigr{\rangle}_{t,\lambda}^{R}$
$\displaystyle=\frac{-e^{-t}\beta}{a_{N}}\mathbb{E}\rho_{R}^{\prime}(J_{ij}^{1}(t))\rho_{R}^{\prime}(J_{ij}^{2}(t))\bigl{\langle}\sigma_{i}^{1}\sigma_{j}^{1}(\tau_{i}^{1}\tau_{j}^{1}-\tau_{i}^{2}\tau_{j}^{2})\bigr{\rangle}_{t,\lambda}^{R}.$

By symmetry, we then have

$\displaystyle\partial_{t}F_{N}^{R}(t,\lambda)$
$\displaystyle=\frac{-e^{-t}\beta^{2}}{Na_{N}^{2}}\sum_{i,j=1}^{N}\mathbb{E}\rho_{R}^{\prime}(J_{ij}^{1}(t))\rho_{R}^{\prime}(J_{ij}^{2}(t))\bigl{\langle}\sigma_{i}^{1}\sigma_{j}^{1}(\tau_{i}^{1}\tau_{j}^{1}-\tau_{i}^{2}\tau_{j}^{2})\bigr{\rangle}_{t,\lambda}^{R},$
(5.3)

which implies that for $b>1$ such that $b(1+e^{-t})<\alpha$ and its conjugate
exponent $a=b/(b-1)$,

$\displaystyle F_{N}^{R}(t,\lambda)-F_{N}^{R}(\infty,\lambda)$
$\displaystyle=-\int_{t}^{\infty}\partial_{s}F_{N}^{R}(s,\lambda)ds\leq\int_{t}^{\infty}\frac{2Ne^{-s}\beta^{2}}{a_{N}^{2}}\mathbb{E}\rho_{R}^{\prime}(J^{1}(s))\rho_{R}^{\prime}(J^{2}(s))ds$
$\displaystyle\leq\frac{2N\beta^{2}}{a_{N}^{2}}\mathbb{E}\rho_{R}^{\prime}(J^{1}(t))\rho_{R}^{\prime}(J^{2}(t))=\frac{2N\beta^{2}}{a_{N}^{2}}\frac{1}{1-e^{-t}}\mathbb{E}J^{1}J^{2}\rho_{R}(J^{1}(t))\rho_{R}(J^{2}(t))$
$\displaystyle\leq\frac{2N\beta^{2}}{a_{N}^{2}}\frac{1}{1-e^{-t}}(\mathbb{E}|J|^{a})^{2/a}(\mathbb{E}|\rho_{R}|^{b}(J^{1}(t))|\rho_{R}|^{b}(J^{2}(t)))^{1/b}$
$\displaystyle\leq\frac{2N\beta^{2}}{a_{N}^{2}}\frac{1}{1-e^{-t}}(\mathbb{E}|J|^{a})^{2/a}(\mathbb{E}|\rho_{R}|^{b(1+e^{-t})}(J))^{2/((1+e^{-t})b)},$
(5.4)

where the inequalities used Lemma 5.1$(i),$ Gaussian integration by parts,
Hölder’s inequality, and Gaussian hypercontractive inequality.

Finally, we argue that our assertion follows by taking $R\to\infty.$ To see
this, note that

$\displaystyle\bigl{|}F_{N}(t,\lambda)-F_{N}^{R}(t,\lambda)\bigr{|}$
$\displaystyle=\frac{1}{N}\Bigl{|}\mathbb{E}\ln\Bigl{\langle}\exp\frac{\beta}{a_{N}}\sum_{i,j=1}^{N}\Bigl{(}\bigl{(}\rho(J^{1}_{ij}(t))-\rho_{R}(J^{1}_{ij}(t))\bigr{)}\sigma_{i}\sigma_{j}+\bigl{(}\rho(J^{2}_{ij}(t))-\rho_{R}(J^{2}_{ij}(t))\bigr{)}\tau_{i}\tau_{j}\Bigr{)}\Bigr{\rangle}_{t,\lambda}^{R}\Bigr{|}$
$\displaystyle\leq\frac{\beta}{Na_{N}}\sum_{i,j=1}^{N}\mathbb{E}\bigl{(}\bigl{|}\rho(J^{1}_{ij}(t))-\rho_{R}(J^{1}_{ij}(t))\bigr{|}+\bigl{|}\rho(J^{2}_{ij}(t))-\rho_{R}(J^{2}_{ij}(t))\bigr{|}\bigr{)}$
$\displaystyle=2\beta
Na_{N}^{-1}\mathbb{E}\bigl{|}\rho(J)-\rho_{R}(J)\bigr{|},$ (5.5)

where the inequality used the Jensen inequality. Since
$\mathbb{E}|\rho(J)|<\infty$ implies

$\displaystyle\mathbb{E}\bigl{|}\rho(J)-\rho_{R}(J)\bigr{|}$
$\displaystyle\leq\mathbb{E}\bigl{[}|\rho(J)|;|J|\geq R\bigr{]}\to
0,\,\,\mbox{as $R\to\infty$,}$

we have $\lim_{R\to\infty}F_{N}^{R}(t,\lambda)=F_{N}(t,\lambda)$ for any
$t\geq 0.$ Additionally, from (1.7), we have
$\mathbb{E}|\rho|^{b(1+e^{-t})}(J)<\infty$ and

$\lim_{R\to\infty}\mathbb{E}\bigl{(}|\rho|^{b(1+e^{-t})}(J)-|\rho_{R}|^{b(1+e^{-t})}(J)\bigr{)}=0,$

hence the right-hand side of (5.4) converges as well.

Case 2: General $\rho$. Define $\rho_{(y)}(x)=\mathbb{E}\rho(x+yJ^{\prime})$
for $x,y\in\mathbb{R}$ where $J^{\prime}$ is standard normal. As $\rho_{(y)}$
is smooth, it belongs to the first case of the proof and enjoys the inequality
of the lemma. Since $\mathbb{E}|\rho(J)|^{p}<\infty$, from a standard
approximation argument, $\rho_{(y)}$ approximates $\rho(J)$ in $L^{p}$ for any
$1\leq p<\alpha$, i.e., $\mathbb{E}|\rho(J)-\rho_{(y)}(J)|^{p}\to 0$ as
$y\downarrow 0$ for any $1\leq p<\alpha$. Define $F^{(y)}_{N}(t,\lambda)$ to
be $F_{N}(t,\lambda)$ with $\rho$ replaced by $\rho_{(y)}$. The calculations
in (5) show that as $y\downarrow 0$,

$|F^{(y)}_{N}(t,\lambda)-F_{N}(t,\lambda)|\leq 2\beta
Na_{N}^{-1}\mathbb{E}|\rho(J)-\rho_{(y)}(J)|\to 0.$

This completes our proof.

∎

######  Remark 5.3.

It might look like that (5.3) already holds for $F_{N}$ following the same
argument for $F_{N}^{R}$ and one does not need the approximate argument in the
proof above. However, in view of (5.1) and (5.2), it does not seem to be easy
to see that

$\mathbb{E}\rho^{\prime}(J^{1}_{ij}(t))\rho^{\prime}(J^{1}_{ij}(t))\bigl{\langle}\sigma_{i}^{1}\sigma_{j}^{1}(\sigma_{i}^{1}\sigma_{j}^{1}-\sigma_{i}^{2}\sigma_{j}^{2})\bigr{\rangle}_{t,\lambda}$

is finite since it might well be the case
$\mathbb{E}\rho^{\prime}(J)^{2}=\infty$.

###### Proof of Theorem 1.7.

Let $t_{0}>-\log(\alpha-1)$ and $0<\lambda_{0}<1/2$ be fixed. Note that (1.7)
ensures that $\mathbb{E}|\rho(J)|^{1+e^{-t_{0}}}<\infty.$ From the convexity
of $F(t_{0},\cdot)$, $F_{N}(t_{0},0)=F_{N}(0,0),$ and Lemma 5.2, there exists
a constant $K>0$ independent of $N$ such that

$\displaystyle\lambda_{0}\mathbb{E}\langle R(\sigma,\tau)^{2}\rangle_{t_{0}}$
$\displaystyle\leq F_{N}(t_{0},\lambda_{0})-F_{N}(t_{0},0)$ $\displaystyle\leq
F_{N}(\infty,\lambda_{0})-F_{N}(\infty,0)+\frac{KN}{a_{N}^{2}}$
$\displaystyle=\frac{1}{N}\mathbb{E}\ln\bigl{\langle}\exp\lambda_{0}NR(\sigma,\tau)^{2}\bigr{\rangle}_{\infty}+\frac{KN}{a_{N}^{2}}$
$\displaystyle\leq\frac{1}{N}\ln\mathbb{E}\bigl{\langle}\exp\lambda_{0}NR(\sigma,\tau)^{2}\bigr{\rangle}_{\infty}+\frac{KN}{a_{N}^{2}}.$

Observe that since $(\rho(J_{ij}^{1}))_{i,j\in[N]}$ and
$(\rho(J_{ij}^{2}))_{i,j\in[N]}$ are i.i.d. and symmetric, we see that with
respect to the measure $\mathbb{E}\langle\cdot\rangle_{\infty}$,
$NR(\sigma,\tau)$ is equal to $B_{1}+\cdots+B_{N}$ in distribution for i.i.d.
Rademacher$(1/2)$ random variables $B_{1},\ldots,B_{N}.$ Consequently,

$\displaystyle\mathbb{E}\bigl{\langle}\exp\lambda_{0}NR(\sigma,\tau)^{2}\bigr{\rangle}_{\infty}$
$\displaystyle=\mathbb{E}\exp\frac{\lambda_{0}}{N}(B_{1}+\cdots+B_{N})^{2}\leq\frac{1}{\sqrt{1-2\lambda_{0}}}.$

Observe that

$\langle
R(\sigma,\tau)^{2}\rangle_{t}=\frac{1}{N^{2}}\sum_{i,j=1}^{N}\phi_{ij}(J^{1}(t))\phi_{ij}(J^{2}(t)),$

where for each $(i,j)\in[N]^{2}$, we define the bounded function
$\phi_{ij}:\mathbb{R}^{N^{2}}\to\mathbb{R}$ by

$\phi_{ij}(x)=\Bigl{(}\sum_{\sigma}\exp\Bigl{(}\frac{\beta}{a_{N}}\sum_{k,l=1}^{N}\rho(x)\sigma_{k}\sigma_{l}\Bigr{)}\Bigr{)}^{-1}\sum_{\sigma}\sigma_{i}\sigma_{j}\exp\Bigl{(}\frac{\beta}{a_{N}}\sum_{k,l=1}^{N}\rho(x)\sigma_{k}\sigma_{l}\Bigr{)},\quad
x\in\mathbb{R}^{N^{2}}.$

Hence, there exists a constant $K^{\prime}$ such that for any $t\geq t_{0}$
and $N\geq 1,$

$\displaystyle\mathbb{E}\langle
R(\sigma,\tau)^{2}\rangle_{t}\leq\mathbb{E}\langle
R(\sigma,\tau)^{2}\rangle_{t_{0}}$
$\displaystyle\leq\frac{1}{\lambda_{0}}\Bigl{(}\frac{1}{N}\ln\frac{1}{\sqrt{1-2\lambda_{0}}}+\frac{KN}{a_{N}^{2}}\Bigr{)}\leq\frac{K^{\prime}N}{a_{N}^{2}},$

where the first inequality used Lemma 5.1$(i)$. Note that $\mathbb{E}\langle
R(\sigma,\tau)^{2}\rangle_{0}\leq 1$. From Lemma 5.1$(ii),$ for any $0\leq
t\leq t_{0}$ and $N\geq 1,$

$\displaystyle\mathbb{E}\langle R(\sigma,\tau)^{2}\rangle_{t}$
$\displaystyle\leq(\mathbb{E}\langle
R(\sigma,\tau)^{2}\rangle_{t_{0}})^{t/t_{0}}\leq
K^{\prime\prime}\Bigl{(}\frac{N}{a_{N}^{2}}\Bigr{)}^{t/t_{0}},$

where $K^{\prime\prime}\geq K^{\prime}$ is a constant independent of
$t\in[0,t_{0}]$ and $N\geq 1.$ Putting these together, we conclude that

$\displaystyle\mathbb{E}\langle R(\sigma,\tau)^{2}\rangle_{t}$
$\displaystyle\leq
K^{\prime\prime}\Bigl{(}\frac{N}{a_{N}^{2}}\Bigr{)}^{\min(1,t/t_{0})}.$

The proof is now complete in view of the estimate for $a_{N}$ obtained from
(1.8) and (1.9). ∎

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