# Stein’s method and a cubic mean-field model

Peter Eichelsbacher  Faculty of Mathematics, Ruhr University Bochum, Germany.
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study a mean-field spin model with three- and two-body
interactions. In a recent paper [9] by Contucci, Mingione and Osabutey, the
equilibrium measure for large volumes was shown to have three pure states, two
with opposite magnetization and an unpolarized one with zero magnetization,
merging at the critical point. The authors proved a central limit theorem for
the suitably rescaled magnetization. The aim of our paper is presenting a
prove of a central limit theorem for the rescaled magnetization applying the
exchangeable pair approach due to Stein. Moreover we prove (non-uniform)
Berry-Esseen bounds, a concentration inequality, Cramér-type moderate
deviations and a moderate deviations principle for the suitably rescaled
magnetization. Interestingly we analyze Berry-Esseen bounds in case the model-
parameters $(K_{n},J_{n})$ converge to the critical point $(0,1)$ on lines
with different slopes and with a certain speed, and obtain new limiting
distributions and thresholds for the speed of convergence.
Mathematics Subject Classifications (2020): 60E15, 60F05, 60F10, 82B20, 82B26.
Keywords: uniform and non-uniform Berry-Esseen bound, moderate deviations
principle, Cramér-type moderate deviations, concentration inequality, Stein’s
method, exchangeable pairs, Curie-Weiss model, mean-field model, cubic mean-
field Ising model, cubic Curie-Weiss model.

##  1\. Introduction

Stein’s method is a powerful tool to prove distributional approximation. One
of its advantages is that is often automatically provides also a rate of
convergence and may be applied rather effectively also to classes of random
variables that are stochastically dependent. Such classes of random variables
are in natural way provided by spin system in statistical mechanics. The
easiest of such models are mean-field models. Among them the Curie–Weiss model
is well known for exhibiting a number of properties of real substances, such
as spontaneous magnetization or metastability.

The aim of this paper is to apply Stein’s method for exchangeable pairs (see
[26]) for distributional approximations and thereby prove Berry-Esseen bounds
for the suitably rescaled magnetization of a mean-field model with three- and
two-body interaction called Cubic Curie-Weiss model or Cubic mean-field Ising
model. The interest in this model is described in [9]. The interest comes from
condensed matter physics (see e.g. [28]) as well as from complex systems of
socio-technical nature where a three-body interaction entails moving from a
graph-theoretical environment of vertices and edges to a richer hypergraph
setting, see e.g. [1]. The presence of the cubic interaction brings technical
difficulties to the analysis of the model, since now the energy is non-convex.
Techniques like the Hubbard-Stratonovich transform or the Legendre transform
of the logarithm of the moment generating function are prevented by the cubic
power. But a large deviation principle (LDP) for the magnetization per
particle can easily be proved applying Varadhan’s lemma. The rate function of
this principle will play an important role. A similar observation was already
obtained in recent work concerning the fluctuations of the rescaled
magnetization of the $p$-spin Curie Weiss model, see [20] and [21].

In case of the cubic Curie-Weiss model, where a three- and a two-body
interaction with strength $K$ and $J$ we are interested in fluctuations on any
scale different form a law of large numbers or the large deviation principle
of the magnetization per particle. We will ask for moderate deviations,
Cramér-type moderate deviations and Berry-Esseen bounds for normal and
nonnormal approximations. For an overview of results on the Curie-Weiss models
with a pure quadratic interaction and related models, see [14], [16] and [17].
In [9], the authors proved that the central limit theorem holds for a suitably
rescaled magnetization, while its violation with the typical quartic behaviour
appears at the critical point. The last observation was known before.

Very impressive is that in [9] the complete phase diagram in $(K,J)$ of the
equilibrium measure was presented. Outside a curve $\gamma$,
$K\mapsto\gamma(K)=J$, the rate function of the large deviation principle has
a unique minimum. On the curve there are two global minimizers. This
corresponds to the existence of two different thermodynamic equilibrium
phases, a stable paramagnetic state with the absence of spontaneous magnetic
order and the presence of symmetry between the up and down spin, and a
positively polarised state. The jump between these states represents a first-
order phase transition. The quadratic Curie-Weiss model has a second-order
phase transition in the inverse temperature. The new behaviour in the cubic
model reminds of the Curie-Weiss Potts model, where a first order phase
transition was observed, see [18] and [13].

Let us consider $n$ spins $x=(x_{i})_{i\in\mathbb{N}}\in\\{-1,1\\}^{n}$
interacting through an Hamiltonian of the form

$H_{n}(x)=-\frac{K}{3n^{2}}\sum_{i,j,k=1}^{n}x_{i}x_{j}x_{k}-\frac{J}{2n}\sum_{i,j=1}^{n}x_{i}x_{j}-h\sum_{i=1}^{n}x_{i},$
(1.1)

where $K>0$ and $J\in\mathbb{R}$ are interaction parameters and $h$ represents
an external field. We will only consider $h=0$. With

$m(x):=m_{n}(x):=\frac{1}{n}\sum_{i=1}^{n}x_{i}\,\,\text{and}\,\,S_{n}:=S_{n}(x):=\sum_{i=1}^{n}x_{i}$
(1.2)

denoting the magnetization per particle and the total magnetization,
respectively, we obtain the mean-field nature of $H_{n}$ because

$H_{n}(x)=-n\biggl{(}\frac{K}{3}m_{n}^{3}(x)+\frac{J}{2}m_{n}^{2}(x)\biggr{)}=-n\biggl{(}\frac{K}{3}\bigl{(}\frac{S_{n}}{n}\bigr{)}^{3}+\frac{J}{2}\bigl{(}\frac{S_{n}}{n}\bigr{)}^{2}\biggr{)}.$

The Boltzmann-Gibbs probability measure associated to $H_{n}$ is defined by

$\mathbb{P}_{n}((x_{i})):=\mathbb{P}_{n,K,J}((x_{i})):=\mathbb{P}_{n,K,J}\bigl{(}(X_{i})_{i}=(x_{i})_{i}\bigr{)}:=\frac{1}{Z_{n,K,J}}\exp\bigl{(}-H_{n}(x)\bigr{)}\prod_{i=1}^{n}d\varrho(x_{i})$
(1.3)

with $\varrho(-1)=\varrho(1)=\frac{1}{2}$, or
$\varrho=\frac{1}{2}\delta_{-1}+\frac{1}{2}\delta_{1}$. Since the Hamiltonian
is invariant under transformation $K\mapsto-K$ and $x_{i}\mapsto-x_{i}$ for
$i=1,\ldots,n$, the choice $K>0$ is without loss.

We like to point out, that with any distribution $\varrho$ of a single spin,
choosen to be a probability measure on $\mathbb{R}$ endowed with the Borel
$\sigma$-field, the model can be generalized obviously. The $Z_{n,K,J}$ is the
normalizing constant that turns $\mathbb{P}_{n}$ into a probability measure
and is called the partition function.

The case $K=0$ and $\varrho=\frac{1}{2}\delta_{-1}+\frac{1}{2}\delta_{1}$
reduces to the Curie-Weiss model. In (1.3) we set the usual inverse
temperature $\beta$ to 1 without loss since it has been reabsorbed in the
parameters $J$ and $K$. The measures $\mathbb{P}_{n}$ is completely determined
by the value of the magnetization. The total magnetization is therefore called
an order parameter and its behavior will be studied in this paper.

In the classical model ($K=0$ and $\varrho$ is the symmetric Bernoulli
measure), for $0<J<1$, in [25] and [16] the central limit theorem is proved:
$\frac{S_{n}}{\sqrt{n}}=\sqrt{n}m_{n}\to N(0,\sigma^{2}(J))$ in distribution
with respect to the Curie–Weiss finite volume Gibbs states with
$\sigma^{2}(J)=(1-J)^{-1}$. Here $N(0,\sigma^{2}(J))$ denotes a Gaussian
distribution with mean $0$ and variance $\sigma^{2}(J)$. Since for $J=1$ the
variance $\sigma^{2}(J)$ diverges, the central limit theorem fails at the
critical point. In [25] and [16] it is proved that for $J=1$ there exists a
random variable $X$ with probability density proportional to
$\exp(-\frac{1}{12}x^{4})$ such that $\frac{S_{n}}{n^{3/4}}\to X$ as
$n\to\infty$ in distribution with respect to the finite-volume Boltzmann-Gibbs
states. In [9, Theorem 2.1], outside the curve $\gamma$ of values $(K,J)$,
where two minimizers of the rate function coexist, a central limit theorem was
proved for the rescaled magnetization. Since at the critical point of the
model, the cubic part of the Hamiltonian disappears and the inverse
temperature is 1, the quartic behaviour appears, see [25].

Stein introduced in [26] the exchangeable pair approach. Given a random
variable $W$, Stein’s method is based on the construction of another variable
$W^{\prime}$ (some coupling) such that the pair $(W,W^{\prime})$ is
exchangeable, i.e. their joint distribution is symmetric. The approach
essentially uses the elementary fact that if $(W,W^{\prime})$ is an
exchangeable pair, then $\mathbb{E}g(W,W^{\prime})=0$ for all antisymmetric
measurable functions $g(x,y)$ such that the expectation exists. A theorem of
Stein ([26, Theorem 1, Lecture III]) shows that a measure of proximity of $W$
to normality may be provided in terms of the exchangeable pair, requiring
$W-W^{\prime}$ to be sufficiently small. He assumed the linear regression
property

$\mathbb{E}(W-W^{\prime}|W)=\lambda\,W$

for some $0<\lambda<1$. Stein proved that for any uniformly Lipschitz function
$h$, $|\mathbb{E}h(W)-\mathbb{E}h(Z)|\leq\delta\|h^{\prime}\|$ with $Z$
denoting a standard normally distributed random variable and

$\delta=4\mathbb{E}\bigg{|}1-\frac{1}{2\lambda}\mathbb{E}\bigl{(}(W^{\prime}-W)^{2}|W\bigr{)}\bigg{|}+\frac{1}{2\lambda}\mathbb{E}|W-W^{\prime}|^{3}.$
(1.4)

Stein’s approach has been successfully applied in many models, see e.g. [26]
or [27] and references therein. In [22], Rinott and Rotar extended the range
of application by replacing the linear regression property by a weaker
condition, which is

$\mathbb{E}(W-W^{\prime}|W)=\lambda\,W+R$ (1.5)

with $R=R(W)$ being some random reminder term. With Theorem 1.2 in [22], in
the weaker case is holds that for any uniformly Lipschitz function $h$,
$|\mathbb{E}h(W)-\mathbb{E}h(Z)|\leq\delta^{\prime}\|h^{\prime}\|$ with

$\delta^{\prime}=4\mathbb{E}\bigg{|}1-\frac{1}{2\lambda}\mathbb{E}\bigl{(}(W^{\prime}-W)^{2}|W\bigr{)}\bigg{|}+\frac{1}{2\lambda}\mathbb{E}|W-W^{\prime}|^{3}+19\frac{\sqrt{\mathbb{E}R^{2}}}{\lambda}.$
(1.6)

The paper is organized as follows: In Section 2 we cite several recent plug-in
theorems within the theory of the exchangeable pair approach. Due to Berry-
Esseen bounds, we cite results form Eichelsbacher and Löwe [12] and from Shao
and Zhang [23], where in case of an exchangeable pair satisfying (1.5) and
with $|W-W^{\prime}|$ being bounded, one can observe quite simple Berry-Esseen
bounds. We collect recent results of non-uniform Berry-Esseen bounds in the
context of the exchangeable pair approach due to Liu, Li, Wang and Chen, [19].
Moreover we cite a result due to Chatterjee on Stein’s method and
concentration inequalities, [2], and present a theorem how to prove a Cramér-
type moderate deviation result in the context of exchangeable pairs,
originally proved for normal approximation by Chen, Fang and Shao in [5] and
for some nonnormal distributions in [24]. In Section 3 we state some important
results on the cubic mean-field model proved in [9]. In Section 4 we state and
prove (non-uniform) Berry-Esseen bounds for the normal and nonnormal
approximation for the suitably rescaled magnetization in the cubic mean-field
model. Moreover we prove Berry-Esseen bounds in case the model-parameters
$(K_{n},J_{n})$ converge to the critical point $(0,1)$ on lines with different
slopes and with a certain speed, and obtain new limiting distributions and a
threshold for the speed of convergence. In Section 5 we prove a concentration
inequality for the total magnetization, in Section 6 we prove a Cramér-type
moderate deviations for the total magnetization and obtain a moderate
deviations principles.

##  2\. Exchangeable pair approach for distributional approximations and
moderate deviations

Given two real-valued random variables $X$ and $Y$ defined on a common
probability space, we denote the Kolmogorov distance of the distributions of
$X$ and $Y$ by

$d_{\rm{K}}(X,Y):=\sup_{z\in\mathbb{R}}|P(X\leq z)-P(Y\leq z)|.$

###  2.1. Normal approximation

Considering an exchangeable pair $(W,W^{\prime})$ with $|W-W^{\prime}|\leq A$
with $A>0$ the following result in [12, Theorem 2.6] improves [22, Theorem
1.2] with respect to the Berry-Esseen constants:

######  Theorem 2.1.

Let $(W,W^{\prime})$ be an exchangeable pair of real-valued random variables
such that

$\mathbb{E}(W-W^{\prime}|W)=\lambda\,W+R$

for some random variable $R=R(W)$ and with $0<\lambda<1$. Assume that
$|W-W^{\prime}|\leq A$ for $A>0$ and $\mathbb{E}(W^{2})\leq 1$. Let $Z$ be a
random variable with standard normal distribution. Then we have

$\displaystyle d_{\rm{K}}(W,Z)$ $\displaystyle\leq$
$\displaystyle\sqrt{\mathbb{E}\biggl{(}1-\frac{1}{2\lambda}\mathbb{E}[(W^{\prime}-W)^{2}|W]\biggr{)}^{2}}$
(2.1)
$\displaystyle+\biggl{(}\frac{\sqrt{2\pi}}{4}+1.5A\biggr{)}\frac{\sqrt{\mathbb{E}(R^{2})}}{\lambda}+\frac{0.41A^{3}}{\lambda}+1.5A.$

######  Remark 1.

In Theorem 2.1, we assumed $\mathbb{E}(W^{2})\leq 1$. Alternatively, let us
assume that $\mathbb{E}(W^{2})$ is finite. Then the proof of Theorem 2.1
shows, that the third and the fourth summand of the bound change to
$\frac{A^{3}}{\lambda}\bigl{(}\frac{\sqrt{2\pi}}{16}+\frac{\sqrt{\mathbb{E}(W^{2})}}{4}\bigr{)}+1.5A\,\mathbb{E}(|W|)$.

An important step in the development of Stein’s method was made in [23]. If
there is given an exchangeable pair satisfying (1.5) for some constant
$\lambda\in(0,1)$ and $\Delta:=W-W^{\prime}$. Then

$d_{\rm{K}}(W,Z)\leq\mathbb{E}\bigg{|}1-\frac{1}{2\lambda}\mathbb{E}(\Delta^{2}|W)\bigg{|}+\frac{\mathbb{E}|R|}{\lambda}+\frac{1}{\lambda}\mathbb{E}\big{|}\mathbb{E}(\Delta\,\Delta^{*}|W)\big{|},$

where $\Delta^{*}:=\Delta^{*}(W,W^{\prime})$ is any random variable satisfying
$\Delta^{*}(W,W^{\prime})=\Delta^{*}(W^{\prime},W)$ and
$\Delta^{*}\geq|\Delta|$.

The following consequence of the last result is stated in [23, Corollary 2.1],
which will be the result which will be applied in this paper:

######  Theorem 2.2.

If there is given an exchangeable pair satisfying (1.5) for some constant
$\lambda\in(0,1)$ and $\Delta:=W-W^{\prime}$. If moreover $|W-W^{\prime}|\leq
A$ with $A>0$ and $\mathbb{E}|W|\leq 2$, then

$d_{\rm{K}}(W,Z)\leq\mathbb{E}\bigg{|}1-\frac{1}{2\lambda}\mathbb{E}(\Delta^{2}|W)\bigg{|}+\frac{\mathbb{E}|R|}{\lambda}+3A.$
(2.2)

Notice that the term $\frac{A^{3}}{\lambda}$ in (2.1) does not appear in the
preceding theorem at the cost of assuming $\mathbb{E}|W|\leq 2$, which is
easily satisfied.

In recent years Stein’s method was also successfully applied to prove non-
uniform Berry-Esseen bounds. In case of sums of independent or locally
dependent random variables, Chen and Shao proved such non-uniform bounds with
the help of concentration inequalities in [7] and [8]. Recently Liu, Li, Wang
and Chen obtained in [19] a plug-in theorem for non-uniform Berry-Esseen
bounds in the case of the exchangeable pair approach. If there is given an
exchangeable pair $(W,W^{\prime})$ satisfying (1.5) for some constant
$\lambda\in(0,1)$ and $\Delta:=W-W^{\prime}$ and $\mathbb{E}W^{2}<\infty$.
Then for any $z\in\mathbb{R}$, one is able to obtain

$|\mathbb{P}(W\leq z)-\mathbb{P}(Z\leq
z)|\leq\frac{c}{1+|z|}\biggl{(}\mathbb{E}\bigg{|}1-\frac{1}{2\lambda}\mathbb{E}(\Delta^{2}|W)\bigg{|}+\frac{\mathbb{E}|R|}{\lambda}+\frac{1}{\lambda}\mathbb{E}\big{|}\mathbb{E}(\Delta\,\Delta^{*}|W)\big{|}\biggr{)},$
(2.3)

where $\Delta^{*}:=\Delta^{*}(W,W^{\prime})$ is any random variable satisfying
$\Delta^{*}(W,W^{\prime})=\Delta^{*}(W^{\prime},W)$ and
$\Delta^{*}\geq|\Delta|$. In the bounded case we will apply the following
result, which is an obvious combination of [19, Theorem 2.1] and Theorem 2.2.

######  Theorem 2.3.

If there is given an exchangeable pair satisfying (1.5) for some constant
$\lambda\in(0,1)$ and $\Delta:=W-W^{\prime}$. If moreover $|W-W^{\prime}|\leq
A$ and $\mathbb{E}W^{2}\leq 2$, then

$|\mathbb{P}(W\leq z)-\mathbb{P}(Z\leq
z)|\leq\frac{c}{1+|z|}\biggl{(}\mathbb{E}\bigg{|}1-\frac{1}{2\lambda}\mathbb{E}(\Delta^{2}|W)\bigg{|}+\frac{\mathbb{E}|R|}{\lambda}+3A\biggr{)},$
(2.4)

where $c$ is a constant which depends only on $\mathbb{E}W^{2}$.

######  Remark 2.

In case the random variable $R$ in (1.5) does not appear, the result in [19]
can simply be improved. Whenever $W$ satisfies the assumption
$\mathbb{E}\bigl{(}W^{2p}\bigr{)}<\infty$ for $p\geq 1$ being an integer, then
(2.3) can be replaced by

$|\mathbb{P}(W\leq z)-\mathbb{P}(Z\leq
z)|\leq\frac{c}{1+|z|^{p}}\biggl{(}\mathbb{E}\bigg{|}1-\frac{1}{2\lambda}\mathbb{E}(\Delta^{2}|W)\bigg{|}+\frac{1}{\lambda}\mathbb{E}\big{|}\mathbb{E}(\Delta\,\Delta^{*}|W)\big{|}\biggr{)}.$

The reason is quite simple. Considering the proof of Theorem 2.1 in [19], with
the help of Markov’s inequality with respect to the $2p$-th moment, is is
possible to improve (3.7) and (3.12) in [19] to

$\sqrt{\mathbb{E}|f_{z}^{\prime}(W)|^{2}}\leq\frac{c_{1}}{1+|z|^{p}}\,\,\text{and}\,\,\sqrt{\mathbb{E}|Wf_{z}(W)|^{2}}\leq\frac{c_{2}}{1+|z|^{p}},$

with constants $c_{1}$ and $c_{2}$ only depending on $\mathbb{E}(W^{2p})$.
Here, $f_{z}$ with fixed $z\in\mathbb{R}$ denotes the solution of the Stein
equation

$f_{z}^{\prime}(x)-xf_{z}(x)=1_{\\{x\leq z\\}}-P(Z\leq z),\,\,x\in\mathbb{R}.$

In case the reminder $R$ appears, one has to bound
$\sqrt{\mathbb{E}|f_{z}(W)|^{2}}$, but its bound $\frac{1}{|z|}$ cannot be
improved, see [24], Lemma 4.1 and it’s proof.

###  2.2. Nonnormal approximation

Interesting enough, in [23] Berry-Esseen bounds for nonnormal approximation
are considered. With respect to the exchangeable pair approach, this is a
further development of results in [4] and [12]. Let $W$ be random variable and
let $(W,W^{\prime})$ be an exchangeable pair satisfying

$\mathbb{E}(W-W^{\prime}|W)=\lambda g(W)+R,$ (2.5)

where $g(x):=g_{k}(x)=a_{1}x^{2k-1}+a_{2}x^{2k+1}$ for an integer $k\geq 1$,
$a_{1},a_{2}\in\mathbb{R}$, $\lambda\in(0,1)$ and $R$ is a random variable.
Consider $G_{k}(y):=\int_{-\infty}^{y}g_{k}(t)dt$ and
$p_{k}(y):=c_{k}\exp(-G_{k}(y))$ and $c_{k}$ is the constant so that
$\int_{\mathbb{R}}p_{k}(y)dy=1$. Let $Y_{k}$ be a random variable with the
probability density $p_{k}$, and let $\Delta:=W-W^{\prime}$. Interesting
enough [23, Theorem 2.2] reads as

$d_{\rm{K}}(W,Y_{k})\leq\mathbb{E}\bigg{|}1-\frac{1}{2\lambda}\mathbb{E}(\Delta^{2}|W)\bigg{|}+\frac{\mathbb{E}|R|}{c_{k}\lambda}+\frac{1}{\lambda}\mathbb{E}\big{|}\mathbb{E}(\Delta\,\Delta^{*}|W)\big{|},$

where again $\Delta^{*}:=\Delta^{*}(W,W^{\prime})$ is any random variable
satisfying $\Delta^{*}(W,W^{\prime})=\Delta^{*}(W^{\prime},W)$ and
$\Delta^{*}\geq|\Delta|$.

We will apply the following consequence of the last bound:

######  Theorem 2.4.

If there is given an exchangeable pair satisfying (2.5) for some constant
$\lambda\in(0,1)$, $g_{k}(x)=a_{1}x^{2k-1}+a_{2}x^{2k+1}$ for some integer
$k\geq 1$, $a_{1},a_{2}\in\mathbb{R}$ and $\Delta:=W-W^{\prime}$. If moreover
$|W-W^{\prime}|\leq A$ and $\mathbb{E}|W|\leq 2$, then

$d_{\rm{K}}(W,Y_{k})\leq\mathbb{E}\bigg{|}1-\frac{1}{2\lambda}\mathbb{E}(\Delta^{2}|W)\bigg{|}+\frac{\mathbb{E}|R|}{c_{k}\lambda}+3A.$
(2.6)

Also a non-uniform Berry-Esseen bound is available, see [19]. A combination of
Theorem 2.1 in [19] with Theorem 2.2 and Corollary 2.1 in [23] leads to:

######  Theorem 2.5.

If there is given an exchangeable pair satisfying (2.5) for some constant
$\lambda\in(0,1)$, $g_{k}(x)=a_{1}x^{2k-1}+a_{2}x^{2k+1}$ for some integer
$k\geq 1$, $a_{1},a_{2}\in\mathbb{R}$ and $\Delta:=W-W^{\prime}$. If moreover
$|W-W^{\prime}|\leq A$ and $\mathbb{E}W^{2}\leq 2$, then

$|\mathbb{P}(W\leq z)-\mathbb{P}(Y_{k}\leq
z)|\leq\frac{c}{1+|g(z)|}\biggl{(}\mathbb{E}\bigg{|}1-\frac{1}{2\lambda}\mathbb{E}(\Delta^{2}|W)\bigg{|}+\frac{\mathbb{E}|R|}{c_{k}\lambda}+3A\biggr{)}.$
(2.7)

######  Remark 3.

As pointed out in Remark 2, in case the random variable $R$ in (2.5) does not
appear, the result in [19] can simply be improved. Whenever $W$ satisfies the
assumption $\mathbb{E}\bigl{(}W^{2p}\bigr{)}<\infty$, with $p\geq 1$ an
integer, then (2.7) can be replaced by

$|\mathbb{P}(W\leq z)-\mathbb{P}(Y_{k}\leq
z)|\leq\frac{c}{1+|g(z)|^{p}}\biggl{(}\mathbb{E}\bigg{|}1-\frac{1}{2\lambda}\mathbb{E}(\Delta^{2}|W)\bigg{|}+\frac{1}{\lambda}\mathbb{E}\big{|}\mathbb{E}(\Delta\,\Delta^{*}|W)\big{|}\biggr{)}.$

Considering the proof of Theorem 2.1 in [19], with the help of Markov’s
inequality with respect to the $2p$-th moment, is is possible to improve (3.7)
and (3.12) in [19] to

$\sqrt{\mathbb{E}|f_{z}^{\prime}(W)|^{2}}\leq\frac{c_{1}}{1+|g(z)|^{p}}\,\,\text{and}\,\,\sqrt{\mathbb{E}|g(W)f_{z}(W)|^{2}}\leq\frac{c_{2}}{1+|g(z)|^{p}},$

with constants $c_{1}$ and $c_{2}$ only depending on $\mathbb{E}(W^{2p})$.

###  2.3. Concentration inequalities and Cramér-type moderate deviations via
exchangeable pairs

With the exchangeable pair approach it is also possible to prove a
concentration inequality for the magnetization in the cubic mean-field model.
Therefore we will apply the following result due to Chatterjee, [2, Theorem
1.5].

######  Theorem 2.6.

Let $(X,X^{\prime})$ be an exchangeable pair of real-valued random variables,
let $f:\mathbb{R}\to\mathbb{R}$ and
$F:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ are square-integrable functions
such that $F$ is antisymmetric and $\mathbb{E}(F(X,X^{\prime})|X)=f(X)$ almost
surely. Let

$\Delta(X):=\frac{1}{2}\mathbb{E}\bigl{(}|(f(X)-f(X^{\prime}))F(X,X^{\prime})|\big{|}X\bigr{)}.$

Assume that $\mathbb{E}(e^{\theta f(X)}|F(X,X^{\prime})|)<\infty$ for all
$\theta$. If there exists nonnegative constants $B$ and $C$ such that
$\Delta(X)\leq B\,f(X)+C$, then for any $t>0$,

$\mathbb{P}\bigl{(}f(X)\geq
t\bigr{)}\leq\exp\biggl{(}-\frac{t^{2}}{2C+2Bt}\biggr{)}\,\,\text{and}\,\,\mathbb{P}\bigl{(}f(X)\leq-t\bigr{)}\leq\exp\biggl{(}-\frac{t^{2}}{2C}\biggr{)}.$

Stein’s method was also applied to obtain a general Cramér-type moderate
deviation results, starting with [5] in the context of normal approximation
and [24] in the context of nonnormal approximation. Moderate deviations date
back to Cramér in 1938, proving that for independent and identically
distributed random variables $X_{1},\ldots,X_{n}$ with $\mathbb{E}(X_{1})=0$
and variance 1 such that $\mathbb{E}\bigl{(}e^{t_{0}|X_{1}|}\bigr{)}\leq
c<\infty$ for some $t_{0}>0$, it follows that

$\frac{\mathbb{P}(W_{n}>x)}{\mathbb{P}(Z>x)}=1+{\mathcal{O}}(1)(1+x^{3})/\sqrt{n}$

for $0\leq x\leq a_{0}n^{1/6}$, where $W_{n}:=(X_{1}+\cdots+X_{n})/\sqrt{n}$,
$Z$ is standardnormal distributed, $a_{0}>0$ depends on $c$ and $t_{0}$ and
${\mathcal{O}}(1)$ is bounded by a constant depending on $c$ and $t_{0}$. The
range $0\leq x\leq a_{0}n^{1/6}$ and the order of the error term
${\mathcal{O}}(1)(1+x^{3})/\sqrt{n}$ are optimal.

In the context of the exchangeable pair approach, Theorem 3.1 from [5] reads
as following:

######  Theorem 2.7.

Assume that $(W,W^{\prime})$ is an exchangeable pair of real-valued random
variables such that

$\mathbb{E}(W-W^{\prime}|W)=\lambda W+R$

for some random variable $R=R(W)$ and with $0<\lambda<1$. Assume that
$|W-W^{\prime}|\leq A$ for some $A>0$ and let
$D:=\frac{(W-W^{\prime})^{2}}{2\lambda}$. Suppose that there exist constants
$\delta_{1},\delta_{2}$ and $\theta\geq 1$ such that

$|E(D|W)-1|\leq\delta_{1}(1+|W|),$ (2.8)
$\displaystyle\big{|}E\bigl{(}\frac{R}{\lambda}|W\bigr{)}\big{|}$
$\displaystyle\leq$ $\displaystyle\delta_{2}(1+|W|)\,\,\text{or}$
$\displaystyle\big{|}E\bigl{(}\frac{R}{\lambda}|W\bigr{)}\big{|}$
$\displaystyle\leq$
$\displaystyle\delta_{2}(1+W^{2})\,\,\text{and}\,\,\delta_{2}|W|\leq\alpha<1$
(2.9)

and

$|E(D|W)|\leq\theta.$ (2.10)

Then

$\frac{\mathbb{P}(W>x)}{\mathbb{P}(Z>x)}=1+{\mathcal{O}}_{\alpha}(1)\theta^{3}(1+x^{3})(A+\delta_{1}+\delta_{2})$
(2.11)

for $0\leq
x\leq\theta^{-1}\min(A^{-1/3},\delta_{1}^{-1/3},\delta_{2}^{-1/3})$, where
${\mathcal{O}}_{\alpha}(1)$ denotes a quantity whose absolute value is bounded
by a universal constant which depends on $\alpha$ only under the second
alternative of (2.7).

In [24], Theorem 2.1, a Cramér-type moderate deviation result was established
for nonnormal approximation, especially for exchangeable pairs. In [29,
Theorem 2.1], a Cramér-type moderate deviation result was established for
normal approximation for unbounded exchangeable pairs. We could apply this
Theorem in our situation, but prefer to apply Theorem 2.7.

##  3\. Important facts on the cubic Ising mean-field model

We appeal to the theory of large deviations to define the set of canonical
equilibrium macrostates. In order to state a large deviations principle (LDP)
(for a definition see [10], Section 1.2) for the spin per site for the Cubic
mean-filed Ising model we remind that Cramér’s theorem [10, Theorem 2.2.3]
states that with respect to the product measure of
$\frac{1}{2}\delta_{-1}+\frac{1}{2}\delta_{1}$ the sequence
$\bigl{(}\frac{S_{n}}{n}\bigr{)}_{n}=(m_{n})_{n}$ satisfies the LDP on
$[-1,1]$ with speed $n$ and rate function

$I(m)=\frac{1-m}{2}\log\biggl{(}\frac{1-m}{2}\biggr{)}+\frac{1+m}{2}\log\biggl{(}\frac{1+m}{2}\biggr{)},$
(3.1)

sometimes called the (binary) relative entropy. Having this LDP, the LDP for
$\bigl{(}\frac{S_{n}}{n}\bigr{)}_{n}=(m_{n})_{n}$ with respect to
$\mathbb{P}_{n}=\mathbb{P}_{n,K,J}$ is a consequence of [15, Theorem 2.4]:

######  Theorem 3.1 (Large deviation principle for the total magnetization).

For all $(K,J)\in{\mathbb{R}}^{2}$ the following conclusion holds: with
respect to $\mathbb{P}_{n,K,J}$,
$\bigl{(}\frac{S_{n}}{n}\bigr{)}_{n}=(m_{n})_{n}$ satisfies the LDP on
$[-1,1]$ with speed $n$ and rate function

$I_{K,J}(m)=I(m)-\frac{K}{3}m^{3}-\frac{J}{2}m^{2}-\inf_{y\in\mathbb{R}}\biggl{\\{}I(y)-\frac{K}{3}y^{3}-\frac{J}{2}y^{2}\biggr{\\}},$

with $I$ taken from (3.1).

As a consequence only the points $m\in[-1,1]$ satisfying $I_{K,J}(m)=0$ do not
have an exponentially small probability of being observed. These points form
the set of the so-called equilibrium macrostates, which is accordingly defined
as

${\mathcal{M}}_{K,J}=\\{m\in[-1,1]:I_{K,J}(m)=0\\}.$

This set was already analyzed in [9]. The critical points $m$ with

$I(m)-\frac{K}{3}m^{3}-\frac{J}{2}m^{2}=\inf_{y\in\mathbb{R}}\biggl{\\{}I(y)-\frac{K}{3}y^{3}-\frac{J}{2}y^{2}\biggr{\\}}$

satisfy the equation $I^{\prime}(m)-Km^{2}-Jm=0$, which is equivalent to

$\tanh(Km^{2}+Jm)=m.$ (3.2)

Proposition 2.2 in [9] states that for any $K>0$, there exists a function
$\gamma$ and $J=\gamma(K)$ such that ${\mathcal{M}}_{K,J}$ consists of a
unique phase, which is a unique maximum point $m^{*}$ of
$y\mapsto\frac{K}{3}y^{3}+\frac{J}{2}y^{2}-I(y)$ for
$(K,J)\in(\mathbb{R}_{+}\times\mathbb{R})\setminus\\{(K,\gamma(K)),K>0\\}$.
Moreover on the curve $\gamma$ there are two distinct phases. Interesting
enough the function $\gamma$ exists as a consequence of the intermediate value
theorem, but the expression is not given explicitly. Moreover the limit, as
$K\to 0$ of $\gamma(K)$ identifies a critical point $(K_{c},J_{c})=(0,1)$
where the magnetization takes the value $m_{c}=0$. Stated in [9, Corollary
3.1], the set ${\mathcal{M}}_{K,J}$ consists of a unique pure phase
$m^{*}(K,J)$ given by:

$m^{*}:=m^{*}(K,J)=\left\\{\begin{array}[]{ll}m_{0}=0,&\text{if
}J<\gamma(K),\\\ m_{1}(K,J)&\text{if }\gamma(K)<J<J_{c}=1,\\\
m_{2}(K,J),&\text{if }J\geq J_{c}=1.\end{array}\right.$

It is know from [9, Proposition 3.1] that $m_{1}(K,J)$ and $m_{2}(K,J)$ are
strictly positive for $K>0$. Of big interest for the model is the
understanding of critical exponents. By the law of large numbers the average
value of the magnetization is $m^{*}$. In Theorem 2.1 in [9] the law of large
numbers was proved. Outside of the critical curve, the strong law of large
numbers follows as a direct consequence of the large deviations principle
Theorem 3.1. Interesting enough is the critical behaviour of $m^{*}$ when
$(K,J)\to(K_{c},J_{c})=(0,1)$, proved in Proposition 2.2 in [9]. The result
will be used to observe a threshold for the speed of convergence, see Theorem
4.3. Given the unique value $m^{*}(K,J)$ for all $(K,J)$ outside the curve
$\gamma$, consider the affine lines

$J(K)=1+\alpha K,\,\,K>0$

and $m^{*}(K):=m^{*}(K,J(K))$. Then for $K\to 0^{+}$, the following holds

$m^{*}(K)\sim\left\\{\begin{array}[]{ll}\sqrt{3\alpha}\sqrt{K},&\textrm{for}\,\,\alpha>0\\\
3K,&\textrm{for}\,\,\alpha=0\\\
0,&\textrm{for}\,\,\alpha<0.\end{array}\right.$

For $\alpha<0$ the statement is $m^{*}(K)=0$, if $K$ is small enough.

##  4\. (non-uniform) Berry-Esseen bounds for limit theorems for the cubic
mean-field model

Now we consider

$\phi(x):=I(x)-\frac{K}{3}x^{3}-\frac{J}{2}x^{2}.$ (4.1)

Hence $\phi^{\prime}(x)=\frac{1}{2}\log((1+x)/(1-x))-Kx^{2}-Jx$ and
$\phi^{\prime\prime}(x)=\frac{1}{1-x^{2}}-2Km-J$. Notice that for any pure
phase $m^{*}$ we have
$\phi^{\prime\prime}(m^{*})=\frac{1-(1-(m^{*})^{2})(2Km^{*}+J)}{1-(m^{*})^{2}}$,
and we define

$\sigma^{2}:=\frac{n}{\phi^{\prime\prime}(m^{*})}=\frac{n(1-(m^{*})^{2})}{1-(1-(m^{*})^{2})(2Km^{*}+J)}.$
(4.2)

and

$W:=W_{n}:=\frac{S_{n}-nm^{*}}{\sigma}=\frac{S_{n}-nm^{*}}{\sqrt{n/\phi^{\prime\prime}(m^{*})}}.$
(4.3)

In Theorem 2.1 (part 1.) of [9] the authors proved a law of large numbers and
a central limit theorem: For
$(K,J)\in(\mathbb{R}_{+}\times\mathbb{R})\setminus(\gamma\cup(K_{c},J_{c}))$,
the sequence $(S_{n}/n)_{n}$ converges in distribution to the Dirac measure
$\delta_{m^{*}}$ and the sequence $(W_{n})_{n}$ converges in distribution to a
standard normal distributed random variable.

First of all we will prove the following Berry-Esseen bound for the rescaled
magnetization of the cubic mean-field model. The central limit theorem was
proved in [9, Theorem 2.1 (1)].

######  Theorem 4.1 ((non-uniform) Berry-Esseen bounds for the magnetization).

Let $Z$ be a standardnormal distributed random variable. Consider the
classical cubic mean-field model given by the probability measure (1.3) and
consider the magnetization per particle $m_{n}$ (1.2) and the normalized
random variable $W_{n}$ in (4.3). Assume that
$(K,J)\in(\mathbb{R}_{+}\times\mathbb{R})\setminus(\gamma\cup(K_{c},J_{c}))$,
then there exists a constant $c=c(K,J)$ such that

$d_{K}(W_{n},Z)\leq\frac{c}{\sqrt{n}},$

and there exists a constant $c=c(K,J,\mathbb{E}(W_{n}^{2}))$ such that for any
$z\in\mathbb{R}$

$|\mathbb{P}_{n}(W_{n}\leq z)-\mathbb{P}(Z\leq
z)|\leq\frac{c}{(1+|z|)\sqrt{n}}.$

######  Remark 4.

Choosing the critical values $(K_{c}=0,J_{c}=1)$ the model is no longer a
cubic model, but the 2-spin Curie Weiss model at the critical temperature
$\beta_{c}=J_{c}=1$. It is well known that at this critical point,
$n^{1/4}m_{n}$ converges in distribution to a random variable $Y$, where it’s
distribution has the density $C\exp\bigl{(}-\frac{x^{4}}{12}\bigr{)}$ with
$C=\frac{\sqrt[4]{3}\,\Gamma(\frac{1}{4})}{\sqrt{2}}$, see [16]. This is the
density $p_{2}$ in Section 2.2 choosing $g_{3}(x)=x^{3}$. In [12] and [4], the
Berry-Essen bound was proved to be $\frac{C}{\sqrt{n}}$. Hence the statement
in Part 3. of Theorem 2.1 in [9] follows immediately and a Berry-Esseen rate
is known as well. The non-uniform bound was proved in [19], Theorem 4.2:

$\big{|}\mathbb{P}\bigl{(}n^{1/4}m_{n}\leq z\bigr{)}-\mathbb{P}(Y\leq
z)\big{|}\leq\frac{c}{(1+\frac{1}{3}|z|^{3})\,\sqrt{n}}.$

In Theorem 4.2 in [19] the speed was considered to be $n^{-1/4}$. But in [4]
as well as in [12] the speed $n^{-1/2}$ was already considered in the
symmetric 2-spin Curie-Weiss model. Moreover, in [23], the Berry-Esseen bound
at the critical value of $J$ was considered also for non-symmetric single-spin
measures and this is of order $n^{-1/2k}$ in case of $g(x)=x^{2k-1}$. But in
the case of any symmetric single-spin distribution, one is able to obtain a
Berry-Esseen bound of order $n^{-1/k}$, see also Remark 3.1 in [24].

For every pair $(K,J)$ on the curve $\gamma$ there are two global maximizers
of $\phi$. In this case we will prove Berry-Esseen bounds in addition to the
central limit theorem in [9, Theorem 2.1 (2)]:

######  Theorem 4.2 (local Berry-Esseen bounds).

If $(K,J)$ is any point on the curve $\gamma$, then for both distinct phases
$m_{0}$ and $m_{1}$, we have

$\sup_{x\in\mathbb{R}}\bigg{|}\mathbb{P}_{n}\bigl{(}W_{n}\leq x\big{|}m_{n}\in
A_{i}\bigr{)}-\mathbb{P}(Z\leq x)\bigg{|}\leq\frac{c}{\sqrt{n}}$

for a constant $c=c(K,\gamma(K))$ and $A_{i}$ being a neighborhood of $m_{i}$
whose closure does not contain the other phase, for $i=0,1$.

Next we analyze Berry-Esseen bounds in case the model-parameters
$(K_{n},J_{n})$ converge to the critical point $(K_{c},J_{c})$ with a certain
speed and obtain a threshold for the speed of convergence. Consider
$(K_{n},J(K_{n}))$ being points on the affine line

$J(K_{n})=1+\alpha K_{n}$ (4.4)

for $(K_{n})_{n}$ with $K_{n}>0$, $n\in\mathbb{N}$, and fix the slope
$\alpha<0$. It is known from Proposition 2.3 in [9] that for $K_{n}$ small
enough that $J(K_{n})<\gamma(K_{n})$ (with $\gamma$ is the curve such that for
$(K_{n},\gamma(K_{n}))$ there are two phases) and that there exists a
$n_{0}\in\mathbb{N}$ such that $m(K_{n},J(K_{n}))=0$ for all $n\geq n_{0}$. We
will see, that with speed $K_{n}=1/\sqrt{n}$, although we have convergence to
the critical point $(0,1)$, the sequence $\frac{S_{n}}{n^{3/4}}$ converges in
distribution to a mixture of a normal distribution and the quartic
distribution at the critical point. If we speed up, the limiting distribution
will be the quartic distribution, if we slow down, we obtain the Gaussian
approximation for $\frac{S_{n}}{n^{3/4}}$ and for all
$\frac{S_{n}}{n^{\varepsilon}}$ with $\frac{1}{2}<\varepsilon<\frac{3}{4}$,
although we are reaching the critical point.

In Remark 9 we discuss the cases where the affine line has slope $\alpha=0$
($J(K_{n})=1$ for every $n\in\mathbb{N}$) and a positive slope $\alpha>0$.
Interesting enough, the results for $\alpha=0$ differ.

######  Theorem 4.3 (Mixed distributions near the critical point).

Consider $(K_{n},J(K_{n}))$ with (4.4) and fix $\alpha<0$. Choose $n_{0}$ such
that $m=m(K_{n},J(K_{n}))=0$ for all $n\geq n_{0}$.

(1). Let $K_{n}=\frac{1}{\sqrt{n}}$ and

$W:=W_{n}:=\frac{S_{n}}{n^{3/4}}.$

Then we have

$d_{K}(W_{n},Y_{\alpha,J(K_{n})})\leq\frac{c}{n^{1/4}}$

for $n\geq n_{0}$, $c=c(\alpha)$ a constant, and where $Y_{\alpha,J(K_{n})}$
is a random variable with probability density

$c_{\alpha,J(K_{n})}\exp\biggl{(}-(-\alpha)\frac{x^{2}}{2}-J(K_{n})^{3}\frac{x^{4}}{12}\biggr{)},$

with $c_{\alpha,J(K_{n})}$ being the normalization factor to obtain a
probability density. Moreover we have for any $z\in\mathbb{R}$

$\big{|}\mathbb{P}_{n}(W_{n}\leq z)-\mathbb{P}(Y_{\alpha,J(K_{n})}\leq
z)\big{|}\leq\frac{c}{(1+|g(z)|)n^{1/4}}$

with $g(z)=-\alpha z+\frac{1}{3}J(K_{n})^{3}z^{3}$.

(2). If $K_{n}<<\frac{1}{\sqrt{n}}$, $W_{n}$ converges in distribution to a
random variable with density
$c_{J(K_{n})}\exp\biggl{(}-J(K_{n})^{3}\frac{x^{4}}{12}\biggr{)}$, with
$c_{J(K_{n})}$ being the normalization factor to obtain a probability density.
If $K_{n}={\mathcal{O}}\bigl{(}\frac{1}{n^{3/4}}\bigr{)}$, the following
Berry-Esseen bound is true for $n\geq n_{0}$ and a constant $c$:

$d_{K}(W_{n},Y_{0,J(K_{n})})\leq\frac{c}{n^{1/4}}.$

Moreover we have for any $z\in\mathbb{R}$

$\big{|}\mathbb{P}_{n}(W_{n}\leq z)-\mathbb{P}(Y_{0,J(K_{n})}\leq
z)\big{|}\leq\frac{c}{(1+|g(z)|)n^{1/4}}$

with $g(z)=\frac{1}{3}J(K_{n})^{3}z^{3}$.

(3). If $K_{n}>>\frac{1}{\sqrt{n}}$, and let $W_{n}=\frac{S_{n}}{\sigma}$ with
$\sigma=\frac{\sqrt{n}}{\sqrt{1-J(K_{n})}}$, then $W_{n}$ converges in
distribution to a standardnormal distributed random variable $Z$. If we choose
$K_{n}=n^{-2\delta}$ with $1/6<\delta<1/4$ , the following Berry-Esseen bound
is true for $n\geq n_{0}$ and a constant $c$:

$d_{K}(W_{n},Z)\leq\frac{c}{n^{1/2-\delta}}.$

Moreover we have for any $z\in\mathbb{R}$

$\big{|}\mathbb{P}_{n}(W_{n}\leq z)-\mathbb{P}(Z\leq
z)\big{|}\leq\frac{c}{(1+|z|)n^{1/2-\delta}}.$

If we choose $K_{n}=n^{-2\delta}$ with $0<\delta\leq 1/6$, the following
Berry-Esseen bound is true for $n\geq n_{0}$ and a constant $c$:

$d_{K}(W_{n},Z)\leq\frac{c}{n^{1-4\delta}}.$

Moreover we have for any $z\in\mathbb{R}$

$\big{|}\mathbb{P}_{n}(W_{n}\leq z)-\mathbb{P}(Z\leq
z)\big{|}\leq\frac{c}{(1+|z|)n^{1-4\delta}}.$

(4). If we choose $K_{n}=n^{-2\delta}$ with $0<\delta<1/4$ (hence
$K_{n}>>\frac{1}{\sqrt{n}}$, see (3).), and let $W_{n}=\frac{S_{n}}{n^{3/4}}$,
then $W_{n}$ converges in distribution to a standardnormal distributed random
variable $Z$. For any $0<\delta<\frac{1}{8}$ we obtain

$d_{K}(W_{n},Z)\leq\frac{c}{n^{1/4}},$

where $c$ is a positive constant. The non-uniform Berry-Esseen bound with
prefactor $1/(1+|z|)$, $z\in\mathbb{R}$, holds as well.

######  Remark 5.

In the classical Curie-Weiss model $K=0$, the Berry-Esseen bound in part (1)
is known to be of order $1/\sqrt{n}$. We will see, that for $K\not=0$, the
bound of order $n^{-1/4}$ cannot be improved within the method we are
applying, see Remark 8.

The fundamental construction of the paper is to obtain the exchangeable pair
$(W_{n},W_{n}^{\prime})$. For simplicity we will denote $W:=W_{n}$ and
$W^{\prime}:=W_{n}$. Suppose that $X=(X_{i})_{i}$ is drawn from the Cubic
mean-field measure $\mathbb{P}_{n}$, then let $I$ be a uniformly distributed
random variable on the set $\\{1,\ldots,n\\}$, independent of $X$. Given
$I=i$, $X^{\prime}$ is constructed by taking one step in the heat-bath Glauber
dynamics: coordinate $X_{i}$ is replaced by $X_{i}^{\prime}$ drawn from the
conditional distribution of $X_{i}$ given $(X_{j})_{j\not=i}$. We denote bei
$W^{\prime}$ the version of $W$ evaluated at $X^{\prime}$, this is

$W^{\prime}=W+\frac{X_{I}^{\prime}-X_{I}}{\sigma}.$

For simplicity we denote $m:=m^{*}=m^{*}(K,J)$ for any unique pure phase.
Hence we obtain that

$\mathbb{E}(W-W^{\prime}|W)=\frac{1}{\sigma}\mathbb{E}(X_{I}-X_{I}^{\prime}|W)=\biggl{(}\frac{W}{n}+\frac{m}{\sigma}\biggr{)}-\frac{1}{\sigma}\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}(X_{i}^{\prime}|W).$

######  Lemma 4.4.

We have

$\mathbb{E}(X_{i}^{\prime}|W)=\tanh\bigl{(}Jm_{i}(X)+Km_{i}^{2}(X)+\frac{K}{3n^{2}}\bigr{)},$

where

$m_{i}(x):=\frac{1}{n}\sum_{j\not=i}x_{j}:=\frac{1}{n}\sum_{j=1,j\not=i}^{n}x_{j}.$

###### Proof.

We compute the conditional density $g(x_{1}|(X_{i})_{i\geq 2})$ of
$X_{1}=x_{1}$ given $(X_{i})_{i\geq 2}$ under the measure $\mathbb{P}_{n}$:

$g(x_{1}|(X_{i})_{i\geq
2})=\frac{\exp\biggl{(}\frac{J}{2n}\bigl{(}x_{1}^{2}+2\sum_{j=2}^{n}x_{1}X_{j}\bigr{)}+\frac{K}{3n^{2}}\bigl{(}x_{1}^{3}+3\sum_{j=2}^{n}x_{1}^{2}X_{j}+3\sum_{j,k=2}^{n}x_{1}X_{j}X_{k}\bigr{)}\biggr{)}}{\exp\bigl{(}\frac{J}{2n}+\frac{Km_{1}(X)}{n}\bigr{)}\cosh(Jm_{1}(X)+Km_{1}^{2}(X)+\frac{K}{3n^{2}})}$

Hence we can compute $\mathbb{E}(X_{1}^{\prime}|W)$ as

$\displaystyle\mathbb{E}(X_{1}^{\prime}|W)$ $\displaystyle=$
$\displaystyle\frac{\frac{1}{2}\exp\bigl{(}Jm_{1}(X)+Km_{1}^{2}(X)+\frac{K}{3n^{2}}\bigr{)}-\frac{1}{2}\exp\bigl{(}-Jm_{1}(X)-Km_{1}^{2}(X)-\frac{K}{3n^{2}}\bigr{)}}{\cosh(Jm_{1}(X)+Km_{1}^{2}(X)+\frac{K}{3n^{2}})}$
$\displaystyle=$
$\displaystyle\frac{\sinh(Jm_{1}(X)+Km_{1}^{2}(X)+\frac{K}{3n^{2}})}{\cosh(Jm_{1}(X)+Km_{1}^{2}(X)+\frac{K}{3n^{2}})}.$

The computation of any $\mathbb{E}(X_{i}^{\prime}|W)$ is the same. ∎

In what follows we will apply the Taylor-expansion of the function
$x\mapsto\tanh(J(m+x)+K(m+x)^{2})$ up to the fourth order.

######  Lemma 4.5.

Let $f(x):=\tanh\bigl{(}J(m+x)+K(m+x)^{2}\bigr{)}$ for
$(K,J)\in\mathbb{R}_{+}\times\mathbb{R}_{+}$ and $m\in\mathbb{R}$. The Taylor-
expansion of $f$ around $0$ up to the order four reads as

$f(x)=\sum_{k=0}^{3}c_{k}(J,K,m)\frac{x^{k}}{k!}+f^{(4)}(\eta)\frac{x^{4}}{4!}$

with $\eta\in(0,x)$ and

$c_{0}(J,K,m)=f(0)=\tanh(Jm+Km^{2}),$
$c_{1}(J,K,m)=f^{\prime}(0)=(J+2Km)\,\tanh^{\prime}(Jm+Km^{2}),$
$c_{2}(J,K,m)=f^{\prime\prime}(0)=2K\,\tanh^{\prime}(Jm+Km^{2})+(J+2Km)^{2}\,\tanh^{\prime\prime}(Jm+Km^{2}),$

and

$c_{3}(J,K,m)=f^{\prime\prime\prime}(0)=6K\,(J+2Km)\,\tanh^{\prime\prime}(Jm+Km^{2})+(J+2Km)^{3}\,\tanh^{\prime\prime\prime}(Jm+Km^{2}).$

###### Proof of Theorem 4.1.

On the way of finding an exchangeable pair identity (1.5), we obtain

$\displaystyle\mathbb{E}(W-W^{\prime}|W)$ $\displaystyle=$
$\displaystyle\biggl{(}\frac{W}{n}+\frac{m}{\sigma}\biggr{)}-\frac{1}{\sigma}\frac{1}{n}\sum_{i=1}^{n}\tanh\bigl{(}Jm_{i}(X)+Km_{i}^{2}(X)+\frac{K}{3n^{2}}\bigr{)}$
$\displaystyle=$
$\displaystyle\biggl{(}\frac{W}{n}+\frac{m}{\sigma}\biggr{)}-\frac{1}{\sigma}\tanh\bigl{(}Jm(X)+Km^{2}(X)\bigr{)}+r_{1}(X)$

with total magnetization $m(X)$ from (1.2) and

$r_{1}(X)=\frac{1}{\sigma}\frac{1}{n}\sum_{i=1}^{n}\biggl{(}\tanh\bigl{(}Jm(X)+Km^{2}(X)\bigr{)}-\tanh\bigl{(}Jm_{i}(X)+Km_{i}^{2}(X)+\frac{K}{3n^{2}}\bigr{)}\biggr{)}.$
(4.5)

Using $m(X)=\frac{\sigma W}{n}+m$ we will consider

$\frac{1}{\sigma}\tanh\biggl{(}J\bigl{(}m+\frac{\sigma
W}{n}\bigr{)}+K\bigl{(}\frac{\sigma W}{n}+m\bigr{)}^{2}\biggr{)}.$

With the Taylor expansion of $x\mapsto\tanh\bigl{(}J(m+x)+K(m+x)^{2}\bigr{)}$
around $0$ of order 2, see Lemma 4.5, we have

$\tanh\biggl{(}Jm+\frac{J\sigma W}{n}+K\bigl{(}\frac{\sigma
W}{n}+m\bigr{)}^{2}\biggr{)}=c_{0}(J,K,m)+c_{1}(J,K,m)\frac{\sigma
W}{n}+r_{2}(\eta)\frac{\sigma^{2}W^{2}}{2n^{2}}$

with

$\displaystyle r_{2}(\eta)=f^{\prime\prime}(\eta)$ $\displaystyle=$
$\displaystyle 2K\tanh^{\prime}\bigl{(}J(m+\eta)+K(m+\eta)^{2}\bigr{)}$
$\displaystyle\hskip 28.45274pt+$
$\displaystyle(J+2K(m+\eta))^{2}\tanh^{\prime\prime}\bigl{(}J(m+\eta)+K(m+\eta)^{2}\bigr{)}$

for $\eta\in(0,\sigma W/n)$. For any equilibrium macrostate $m$ we know that
$c_{0}(J,K,m)=\tanh(Jm+Km^{2})=m$, see (3.2). Moreover with the definition of
$I$ and $\phi$ ((3.1), (4.1)) we obtain that
$\tanh^{-1}(x)=Jx+Kx^{2}+\phi^{\prime}(x)$, which is
$x=\tanh(Jx+Kx^{2}+\phi^{\prime}(x))$ and differentiating leads to

$1=\tanh^{\prime}(Jx+Kx^{2}+\phi^{\prime}(x))(J+2Kx+\phi^{\prime\prime}(x)).$
(4.7)

For $x=m$, using $\phi^{\prime}(m)=0$ for any equilibrium macrostate, we have

$\tanh^{\prime}(Jm+Km^{2})=\bigl{(}J+2Km+\phi^{\prime\prime}(m)\bigr{)}^{-1}=1-m^{2}.$

Now it follows that

$c_{1}(J,K,m)=(J+2Km)\,\tanh^{\prime}(Jm+Km^{2})=(J+2Km)\,(1-m^{2})=1-(1-m^{2})\phi^{\prime\prime}(m).$
(4.8)

Differentiating (4.7) once more and choosing $x=m$ for any equilibrium
macrostate leads to $\tanh^{\prime\prime}(Jm+Km^{2})=-2m(1-m^{2})$ and it
follows that

$\displaystyle c_{2}(J,K,m)$ $\displaystyle=$ $\displaystyle
2K(1-m^{2})-2m(1-m^{2})(J+2Km)^{2}$ (4.9) $\displaystyle=$ $\displaystyle
2m(1-m^{2})\phi^{\prime\prime}(m)-(1-m^{2})\phi^{\prime\prime\prime}(m).$

Summarzing we obtain

$-\frac{1}{\sigma}\tanh\biggl{(}Jm+\frac{J\sigma W}{n}+K\bigl{(}\frac{\sigma
W}{n}+m\bigr{)}^{2}\biggr{)}=-\frac{m}{\sigma}-\frac{W}{n}(J+2Km)\,(1-m^{2})-r_{2}(\eta)\frac{\sigma
W^{2}}{2n^{2}}.$

Now we have checked the linear regression condition of our exchangeable pair:

######  Lemma 4.6.

In the cubic mean-field model we obtain:

$\mathbb{E}(W-W^{\prime}|W)=\lambda W+R$ (4.10)

with

$\lambda=\frac{1}{n}\bigl{(}1-(J+2Km)\,(1-m^{2})\bigr{)}=\frac{1}{n}\frac{\phi^{\prime\prime}(m)}{\phi^{\prime\prime}(m)+(J+2Km)}=\frac{1}{n}(1-m^{2})\phi^{\prime\prime}(m)$
(4.11)

and

$R=r_{1}(X)-r_{2}(\eta)\frac{\sigma W^{2}}{2n^{2}}$ (4.12)

with $r_{1}$ defined in (4.5) and $r_{2}$ defined in (4) and $\eta\in(0,\sigma
W/n)$.

######  Remark 6.

We observe that $\lambda$ in (4.11) satisfies $\lambda\in(0,1)$ using the fact
that $\phi^{\prime\prime}(m)>0$ for any pure phase $m$, see [9, Proposition
3.2 (b)]. Hence, we expect for any pure phase a central limit theorem for the
rescaled magnetization $W_{n}$.

######  Remark 7.

If $K_{c}=0$ and $J_{c}=1$ one can see that with $m=0$ the factor
$\bigl{(}1-(1-m^{2})(J+2Km)\bigr{)}=0$, hence the variance of $W$ does not
exist. This case corresponds to the 2-spin Curie-Weiss model at critical
temperature $J_{c}=1$. It is possible to expand the $\tanh$ function to the
third order to be able to observe that $n^{1/4}m_{n}(X)$ converges in
distribution and to obtain Berry-Esseen bounds, see [4] and [12].

Now we are ready to apply Theorem 2.2 and Theorem 2.3. Therefore we will
calculate $\frac{1}{2\lambda}\mathbb{E}((W-W^{\prime})^{2}|W)$. Applying Lemma
4.4 we have

$\displaystyle\mathbb{E}((W-W^{\prime})^{2}|W)$ $\displaystyle=$
$\displaystyle\frac{1}{\sigma^{2}}\mathbb{E}(X_{I}^{2}-2X_{I}X_{I}^{\prime}+(X_{I}^{\prime})^{2}|W)$
$\displaystyle=$
$\displaystyle\frac{2}{\sigma^{2}}-\frac{2}{\sigma^{2}}\frac{1}{n}\sum_{i=1}^{n}X_{i}\mathbb{E}(X_{i}^{\prime}|W)$
$\displaystyle=$
$\displaystyle\frac{2}{\sigma^{2}}-\frac{2}{\sigma^{2}}\frac{1}{n}\sum_{i=1}^{n}X_{i}\tanh\biggl{(}Jm_{i}(X)+Km_{i}^{2}(X)+\frac{K}{3n^{2}}\biggr{)}$
$\displaystyle=$
$\displaystyle\frac{2}{\sigma^{2}}-\frac{2}{\sigma^{2}}\frac{1}{n}\sum_{i=1}^{n}X_{i}\tanh\biggl{(}J\bigl{(}m+\frac{\sigma
W}{n}\bigr{)}+K\bigl{(}m+\frac{\sigma W}{n}\bigr{)}^{2}\biggr{)}+r_{3}(X)$
$\displaystyle=$
$\displaystyle\frac{2}{\sigma^{2}}-\frac{2}{\sigma^{2}}\biggl{(}m+\frac{\sigma
W}{n}\biggr{)}\tanh\biggl{(}J\bigl{(}m+\frac{\sigma
W}{n}\bigr{)}+K\bigl{(}m+\frac{\sigma W}{n}\bigr{)}^{2}\biggr{)}+r_{3}(X),$

where

$r_{3}(X):=\frac{2}{\sigma^{2}}\frac{1}{n}\sum_{i=1}^{n}X_{i}\biggl{(}\tanh\bigl{(}Jm_{i}(X)+Km_{i}^{2}(X)+\frac{K}{3n^{2}}\bigr{)}-\tanh\bigl{(}Jm(X)+Km^{2}(X)\bigr{)}\biggr{)}.$
(4.14)

With the Taylor expansion of $x\mapsto\tanh\bigl{(}J(m+x)+K(m+x)^{2}\bigr{)}$
around $0$ of order 1, see Lemma 4.5, we have

$\displaystyle\tanh\biggl{(}J\bigl{(}m+\frac{\sigma
W}{n}\bigr{)}+K\bigl{(}m+\frac{\sigma W}{n}\bigr{)}^{2}\biggr{)}$
$\displaystyle=$ $\displaystyle\tanh(Jm+Km^{2})+f^{\prime}(\eta)\,\frac{\sigma
W}{n}$ $\displaystyle=$ $\displaystyle
m+(J+2K(m+\eta))\tanh^{\prime}(J(m+\eta)+K(m+\eta)^{2})\frac{\sigma W}{n}$

with $\eta\in(0,(\sigma W)/n)$. Using that $\tanh^{\prime}(x)$ is bounded by
1, we obtain that

$\displaystyle\tanh\biggl{(}J\bigl{(}m+\frac{\sigma
W}{n}\bigr{)}+K\bigl{(}m+\frac{\sigma
W}{n}\bigr{)}^{2}\biggr{)}=\biggl{(}m+\frac{\sigma W}{n}r_{4}(\eta)\biggr{)}$

with

$r_{4}(\eta):=(J+2K(m+\eta))\tanh^{\prime}(J(m+\eta)+K(m+\eta)^{2}).$

Summarizing we see that

$\displaystyle\mathbb{E}((W-W^{\prime})^{2}|W)$ $\displaystyle=$
$\displaystyle\frac{2}{\sigma^{2}}-\frac{2}{\sigma^{2}}m^{2}-\frac{2m\,W}{n\sigma}(1+r_{4}(\eta))-\frac{2W^{2}}{n^{2}}r_{4}(\eta)+r_{3}(X)$
$\displaystyle=$
$\displaystyle\frac{2}{\sigma^{2}}(1-m^{2})+\frac{m\,W}{n\sigma}{\mathcal{O}}(1)+\frac{W^{2}}{n^{2}}{\mathcal{O}}(1)+r_{3}(X).$

With $\frac{2}{\sigma^{2}}(1-m^{2})=2\lambda$ we obtain

$\bigg{|}\frac{1}{2\lambda}\mathbb{E}((W-W^{\prime})^{2}|W)-1\bigg{|}=\bigg{|}{\mathcal{O}}(1)\frac{m\,W}{2\lambda
n\sigma}+{\mathcal{O}}(1)\frac{W^{2}}{2\lambda
n^{2}}+\frac{r_{3}(X)}{2\lambda}\bigg{|}.$ (4.15)

Note that $\lambda n\sigma$ is of order $\sqrt{n}$, hence with
$\mathbb{E}W^{2}<\infty$ we see that $\mathbb{E}\big{|}\frac{m\,W}{2\lambda
n\sigma}\big{|}={\mathcal{O}}(\frac{1}{\sqrt{n}})$ and
$\mathbb{E}\big{|}\frac{W^{2}}{2\lambda
n^{2}}\big{|}={\mathcal{O}}(\frac{1}{n})$.

######  Lemma 4.7.

We have

$\mathbb{E}\bigg{|}\frac{r_{3}(X)}{2\lambda}\biggr{|}={\mathcal{O}}(1/n).$

###### Proof.

Using $|X_{i}|\leq 1$ and by the 1-Lipschitz property of $\tanh$,

$\displaystyle\bigg{|}\tanh\bigl{(}Jm_{i}(X)+Km_{i}^{2}(X)+\frac{K}{3n^{2}}\bigr{)}-\tanh\bigl{(}Jm(X)+Km^{2}(X)\bigr{)}\bigg{|}\leq$
$\displaystyle\hskip
28.45274pt\big{|}J(m(X)-m_{i}(X))\big{|}+\big{|}K(m^{2}(X)-m_{i}^{2}(X))\big{|}+\frac{|K|}{3n^{2}}\leq\frac{J}{n}+{\mathcal{O}}(1/n)={\mathcal{O}}(1/n),$

we observe that
$|r_{3}(X)|={\mathcal{O}}\bigl{(}\frac{1}{\sigma^{2}n}\bigr{)}={\mathcal{O}}\bigl{(}\frac{1}{n^{2}}\bigr{)}$.
Since $\lambda$ is of order $n$, the Lemma is proved. ∎

Summarizing we obtain that

$\mathbb{E}\bigg{|}\frac{1}{2\lambda}\mathbb{E}((W-W^{\prime})^{2}|W)-1\bigg{|}={\mathcal{O}}\bigl{(}\frac{1}{\sqrt{n}}\bigr{)}.$

######  Lemma 4.8.

The reminder term $R$ in (4.12) satisfies

$\frac{\mathbb{E}|R|}{\lambda}={\mathcal{O}}\bigl{(}\frac{1}{\sqrt{n}}\bigr{)}.$

###### Proof.

We have seen that $R=r_{1}(X)-r_{2}(\eta)\,\frac{\sigma W^{2}}{2n^{2}}$ with
$r_{1}(X)$ defined in (4.5) and $r_{2}(\eta)$ in (4). The function $r_{2}$ is
bounded, since $\tanh^{\prime}(x)$ is bounded by 1 and
$\tanh^{\prime\prime}(x)$ is bounded by $4/3^{3/2}$. This can be seen by
$\tanh^{\prime\prime}(x)=-2\tanh(x)(1-\tanh^{2}(x))$, which has exactly two
extrema $\pm x^{*}$ on the real line, where $x^{*}$ solves the equation
$\tanh^{2}(x)=\frac{1}{3}$. Next we bound the expectation of the modulus of
$r_{1}(X)$. But with the same argument as in Lemma 4.7 we obtain
$\mathbb{E}|r_{1}(X)|={\mathcal{O}}\bigl{(}\frac{1}{n^{3/2}}\bigr{)}$. Since
the order of $\lambda$ is $n$, the lemma is proved. ∎

We observe that
$|W-W^{\prime}|\leq\frac{2}{\sigma}={\mathcal{O}}(1/\sqrt{n})$. Applying
Theorems 2.2 and 2.3, the proof of Theorem 4.1 is given under the assumption
that $\mathbb{E}|W|\leq 2$. Without this assumption, the proof for the uniform
Berry-Esseen bound is done, applying Theorem 2.1. ∎

###### Proof of Theorem 4.3.

Proof of part (1):

To be able to see the quartic part of the density, we have to apply a fourth
order Taylor expansion for $f(x)=\tanh(Jx+Kx^{2})$, see Lemma 4.5. In analogy
to the start of the proof of Theorem 4.1, we obtain for every equilibrium
macrostate $m$ that for $W=\frac{1}{n^{3/4}}(S_{n}-nm)$

$\mathbb{E}(W-W^{\prime}|W)=\biggl{(}\frac{W}{n}+\frac{m}{n^{3/4}}\biggr{)}-\frac{1}{n^{3/4}}\tanh\bigl{(}Jm(X)+Km^{2}(X)\bigr{)}+\widetilde{r_{1}}(X)$

with $\widetilde{r_{1}}(X)=\frac{\sigma}{n^{3/4}}r_{1}(X)$ with $r_{1}(X)$ as
in (4.5). After a careful look we obtain

$\displaystyle\mathbb{E}(W-W^{\prime}|W)$ $\displaystyle=$
$\displaystyle\frac{W}{n}\bigl{(}1-(1-m^{2})(J+2Km)\bigr{)}-\frac{W^{3}}{n^{3/2}}\frac{c_{3}(J,K,m)}{6}$
(4.16) $\displaystyle-$
$\displaystyle\frac{W^{2}}{n^{5/4}}\frac{c_{2}(J,K,m)}{2}-f^{(4)}(\eta)\frac{W^{4}}{n^{7/4}}+\widetilde{r_{1}}(X).$

Here we have used that $m(X)=W/n^{1/4}+m$. Since we are considering the case
$\alpha<0$, we have $m=m(K_{n},J(K_{n}))=0$ for $n$ sufficiently large. We
will apply the general identity (4.16) in a discussion in Remark 9. For $m=0$
we have $c_{2}(J,K,0)=2K$ (see (4.9)) and $c_{3}(J,K,0)=-2J^{3}$, such that
for every $(K_{n},J(K_{n}))$

$\mathbb{E}(W-W^{\prime}|W)=\frac{W}{n}\bigl{(}1-J(K_{n}))\bigr{)}+\frac{W^{3}}{n^{3/2}}\frac{J(K_{n})^{3}}{3}-\frac{K_{n}\,W^{2}}{n^{5/4}}\
-f^{(4)}(\eta)\frac{W^{4}}{n^{7/4}}+\widetilde{r_{1}}(X).$ (4.17)

Remark that $f^{(4)}(\eta)$ is depending on $(K_{n},J(K_{n}))$ as well. With
$K_{n}=1/\sqrt{n}$ we have $(1-J(K_{n}))=-\alpha/\sqrt{n}$, thus the first
term is $\frac{-\alpha W}{n^{3/2}}$. With $g(x):=-\alpha
x+\frac{1}{3}J(K_{n})^{3}x^{3}$, the regression identity is

$\mathbb{E}(W-W^{\prime}|W)=\lambda g(W)+R$

with $\lambda=1/n^{3/2}$ and

$R:=-\frac{K_{n}\,W^{2}}{n^{5/4}}\
-f^{(4)}(\eta)\frac{W^{4}}{n^{7/4}}+\widetilde{r_{1}}(X).$ (4.18)

Now we will apply Theorem 2.4. With the definition of $W$ we observe that

$\mathbb{E}((W-W^{\prime})^{2}|W)=\frac{2}{n^{3/2}}-\frac{2}{n^{3/2}}\biggl{(}m+\frac{W}{n^{1/4}}\biggr{)}\tanh\biggl{(}J\bigl{(}m+\frac{W}{n^{1/4}}\bigr{)}+K\bigl{(}m+\frac{W}{n^{1/4}}\bigr{)}^{2}\biggr{)}+\widetilde{r_{3}}(X)$

with $\widetilde{r_{3}}(X)=\frac{\sigma^{2}}{n^{3/2}}r_{3}(X)$, where
$r_{3}(X)$ is defined in (4.14).

Analogously to the proof of Theorem 4.1 we obtain by first order Taylor
expansion around 0, that

$\mathbb{E}((W-W^{\prime})^{2}|W)=\frac{2}{n^{3/2}}-\frac{2}{n^{3/2}}m^{2}+\frac{m\,W}{n^{7/4}}{\mathcal{O}}(1)+\frac{W^{2}}{n^{2}}{\mathcal{O}}(1)+\widetilde{r_{3}}(X).$
(4.19)

Now with $m=0$ and $\frac{2}{n^{3/2}}=2\lambda$, we get

$\bigg{|}\frac{\mathbb{E}((W-W^{\prime})^{2}|W)}{2\lambda}-1\bigg{|}=\bigg{|}\frac{W^{2}}{2\lambda
n^{2}}{\mathcal{O}}(1)+\frac{\widetilde{r_{3}}(X)}{2\lambda}\bigg{|}.$

Since $\mathbb{E}|\frac{W^{2}}{2\lambda
n^{2}}|={\mathcal{O}}\bigl{(}1/\sqrt{n}\bigr{)}$ and with the help of Lemma
4.7
$\mathbb{E}\big{|}\widetilde{r_{3}}(X)\big{|}={\mathcal{O}}\bigl{(}1/n^{5/2}\bigr{)}$,
we have

$\mathbb{E}\bigg{|}\frac{\mathbb{E}((W-W^{\prime})^{2}|W)}{2\lambda}-1\bigg{|}={\mathcal{O}}\bigl{(}1/\sqrt{n}\bigr{)}.$

The second term in the bound of Theorem 2.4 is
$\frac{\mathbb{E}|R|}{c_{k}\lambda}$. Here $R$ is given by (4.18). With
$K_{n}=1/\sqrt{n}$ and $\lambda=n^{-3/2}$ we get
$\mathbb{E}\big{|}\frac{K_{n}W^{2}}{n^{5/4}\lambda}\big{|}={\mathcal{O}}(1/n^{1/4})$.
Next we have
$\frac{\mathbb{E}|\widetilde{r_{1}}(X)|}{2\lambda}={\mathcal{O}}(1/n^{1/4})$.
Finally we observe that $f^{(4)}(\eta)$ is bounded (using the boundedness of
$\eta$), and therefore the last term can be bounded by
$\frac{1}{\lambda}\mathbb{E}\big{|}f^{(4)}(\eta)W^{4}/n^{7/4}\big{|}={\mathcal{O}}(1/n^{1/4})$.
Finally $|W-W^{\prime}|\leq\frac{2}{n^{3/4}}=:A$. Thus with Theorem 2.4 and
Theorem 2.5, part (1) of the Theorem is proved.

Proof of part (2):

Now we assume that $K_{n}$ converges faster to zero than $1/\sqrt{n}$. Now the
first summand on the right hand side of the equality (4.17) is

$\frac{W}{n}\bigl{(}1-J(K_{n}))=-\frac{\alpha K_{n}W}{n}$

and $\mathbb{E}\bigg{|}\frac{-\alpha K_{n}W}{\lambda
n}\bigg{|}={\mathcal{O}}(K_{n}\sqrt{n})=o(1)$. Hence this term will be a new
member of the reminder term and this observation leads to the new regression
identity

$\mathbb{E}(W-W^{\prime}|W)=\lambda\widetilde{g}(W)+R-\frac{\alpha K_{n}W}{n}$

with $\widetilde{g}(x)=\frac{1}{3}J(K_{n})^{3}x^{3}$ and $R$ defined in
(4.18). Now we obtain convergence in distribution for any $K_{n}<<1/\sqrt{n}$
applying Theorem 2.4. Moreover we obtain the Berry-Esseen bound of order
$n^{-1/4}$ if we choose $K_{n}=n^{-3/4}$. If we let $K_{n}$ converge to zero
even faster, this does not influence the order of the bound of $R$. Hence the
rate of convergence $n^{-1/4}$ seem to be optimal within the method we have
applied.

Proof of part (3):

Now we change to the random variable $W:=\frac{S_{n}}{\sigma}$ with
$\sigma^{2}=\frac{n}{1-J(K_{n})}$. We went back to the proof of Theorem 4.1.
We apply Taylor’s expansion of third order with the help of Lemma 4.5. Now let
$\lambda=\frac{1-J(K_{n})}{n}=-\frac{\alpha K_{n}}{n}$ with $\alpha<0$ and
$m=0$, we obtain the following representation of $\mathbb{E}(W-W^{\prime}|W)$:

$\mathbb{E}(W-W^{\prime}|W)=\frac{W}{n}(1-J(K_{n}))-\frac{K_{n}\sigma
W^{2}}{n^{2}}-\frac{1}{6}\frac{\sigma^{2}W^{3}}{n^{3}}r_{5}(\eta)+r_{1}(X),$

where $r_{5}(\eta)$ is given by

$\displaystyle r_{5}(\eta)$ $\displaystyle=$ $\displaystyle
6K_{n}(J(K_{n})+2K_{n}\eta)\tanh^{\prime\prime}(J(K_{n})\eta+K_{n}\eta^{2})$
$\displaystyle\hskip 28.45274pt+$
$\displaystyle(J(K_{n})+2K_{n}\eta)^{3}\tanh^{\prime\prime\prime}(J(K_{n})\eta+K_{n}\eta^{2})$

with $\eta\in(0,\sigma W/n)$. We have
$|W-W^{\prime}|\leq\frac{2}{\sigma}=\frac{2\sqrt{1-J(K_{n})}}{\sqrt{n}}=\frac{2\sqrt{-\alpha}\sqrt{K_{n}}}{\sqrt{n}}=:A$,
which is converging to zero by our assumption on $K_{n}$. Since $m=0$, it
follows from (4.15) that

$\bigg{|}\frac{1}{2\lambda}\mathbb{E}((W-W^{\prime})^{2}|W)-1\bigg{|}=\bigg{|}{\mathcal{O}}(1)\frac{W^{2}}{2\lambda
n^{2}}+\biggl{(}\frac{r_{3}(X)}{2\lambda}\biggr{)}\bigg{|}.$

We have

$\mathbb{E}\bigg{|}\frac{r_{3}(X)}{\lambda}\bigg{|}={\mathcal{O}}((\sigma^{2}n\lambda)^{-1})={\mathcal{O}}(n^{-1})\quad\text{and}\quad\mathbb{E}\bigg{|}\frac{W^{2}}{\lambda
n^{2}}\bigg{|}={\mathcal{O}}\bigl{(}\frac{1}{nK_{n}}\bigr{)},$

and the last bound is converging to zero by our assumption on $K_{n}$.
Finally, since $r_{5}(\eta)$ is bounded by a constant, we consider the
following three bounds:

$\mathbb{E}\bigg{|}\frac{r_{1}(X)}{\lambda}\bigg{|}={\mathcal{O}}((\sigma
n\lambda)^{-1})={\mathcal{O}}\bigl{(}\frac{1}{\sqrt{nK_{n}}}\bigr{)},$
$\mathbb{E}\bigg{|}\frac{K_{n}\sigma W^{2}}{\lambda
n^{2}}\bigg{|}={\mathcal{O}}\bigl{(}\frac{1}{\sqrt{nK_{n}}}\bigr{)}\quad\text{and}\quad\mathbb{E}\bigg{|}\frac{\sigma^{2}W^{3}}{\lambda
n^{3}}\bigg{|}={\mathcal{O}}\bigl{(}\frac{1}{nK_{n}^{2}}\bigr{)}.$

With $K_{n}>>1/\sqrt{n}$ all terms converge to zero, hence convergence to the
standardnormal distribution is proved.

Since we assume that $K_{n}$ converges to zero slower than $n^{-1/2}$, we
choose without loss of generality $K_{n}=n^{-2\delta}$ for $0<\delta<1/4$. It
follows that $\lambda=\frac{1}{n}(1-J(K_{n}))=-\frac{\alpha
K_{n}}{n}=-\alpha\frac{1}{n^{1+2\delta}}$. With $\sigma^{2}=\lambda^{-1}$ we
obtain the following bounds:

$\frac{|W-W^{\prime}|}{\sigma}=\frac{|X_{I}-X_{I}^{\prime}|}{\sigma}\leq(2\sqrt{-\alpha})\frac{1}{n^{1/2+\delta}}=:A_{n},$

and

${\mathcal{O}}\bigl{(}\frac{1}{nK_{n}}\bigr{)}={\mathcal{O}}\bigl{(}\frac{1}{n^{1-2\delta}}\bigr{)},\,\,{\mathcal{O}}\bigl{(}\frac{1}{\sqrt{nK_{n}}}\bigr{)}={\mathcal{O}}\bigl{(}\frac{1}{n^{1/2-\delta}}\bigr{)},\,\,{\mathcal{O}}\bigl{(}\frac{1}{nK_{n}^{2}}\bigr{)}={\mathcal{O}}\bigl{(}\frac{1}{n^{1-4\delta}}\bigr{)}.$

Summarizing the rate of convergence is $\frac{1}{n^{1-4\delta}}$ or
$\frac{1}{n^{1/2-\delta}}$ for $0<\delta<\frac{1}{4}$. Hence the rate of
convergence is $\frac{1}{n^{1/2-\delta}}$ for $0<\delta\leq\frac{1}{6}$ and
the rate of convergence is $\frac{1}{n^{1-4\delta}}$ for
$\frac{1}{6}<\delta<\frac{1}{4}$. In the first case it is slower than
$1/\sqrt{n}$, in the second case it is slower than $1/n^{1/3}$.

Proof of part (4): With $W_{n}=\frac{S_{n}}{n^{3/4}}$ we jump back to (4.17).
With $K_{n}=n^{-2\delta}$, we obtain the regression identity

$\mathbb{E}(W-W^{\prime}|W)=\lambda W+\widetilde{R}$

with $\lambda=-\alpha\frac{1}{n^{1+2\delta}}$ and
$\widetilde{R}:=R-\frac{W^{3}}{n^{3/2}}\frac{J(K_{n})^{3}}{3}$ with $R$
defined in (4.18). We have that $|W-W^{\prime}|\leq{2}{n^{3/4}}$, and
$\mathbb{E}\big{|}\frac{R}{\lambda}\big{|}={\mathcal{O}}\bigl{(}n^{2\delta-1/2}\bigr{)}$.
This is because we have

$\mathbb{E}\bigg{|}\frac{K_{n}W^{2}}{\lambda
n^{5/4}}\bigg{|}={\mathcal{O}}\bigl{(}n^{-1/4}\bigr{)},\quad\mathbb{E}\bigg{|}\frac{W^{3}}{\lambda
n^{3/2}}\bigg{|}={\mathcal{O}}\bigl{(}n^{-1/2+2\delta}\bigr{)},$

and

$\mathbb{E}\bigg{|}\frac{f^{(4)}(\eta)W^{4}}{\lambda
n^{7/4}}\bigg{|}={\mathcal{O}}\bigl{(}n^{-3/4+2\delta}\bigr{)},\quad\mathbb{E}\bigg{|}\frac{\widetilde{r_{1}}(X)}{\lambda}\bigg{|}={\mathcal{O}}\bigl{(}n^{-3/4+2\delta}\bigr{)}.$

With

$\mathbb{E}\bigg{|}\frac{W^{2}}{2\lambda
n^{2}}\bigg{|}={\mathcal{O}}\bigl{(}n^{-1+2\delta}\bigr{)},\quad\mathbb{E}\bigg{|}\frac{\widetilde{r_{3}}(X)}{2\lambda}\bigg{|}={\mathcal{O}}\bigl{(}n^{-3/2+2\delta}\bigr{)},$

we obtain
$\mathbb{E}\big{|}\frac{\mathbb{E}((W-W^{\prime})^{2}|W)}{2\lambda}-1\big{|}={\mathcal{O}}\bigl{(}n^{-1+2\delta}\bigr{)}$.
Hence the convergence in distribution is proved and with $0<\delta<1/8$ we
obtain the (non-uniform) Berry-Esseen bound. ∎

######  Remark 8.

In the proof of part (1), we have seen that every term of the remainder $R$ in
(4.18) is of order $n^{-1/4}$. The other terms are of order $1/\sqrt{n}$ or
even smaller. We saw that
$\frac{\mathbb{E}|\widetilde{r_{1}}(X)|}{2\lambda}={\mathcal{O}}(1/n^{1/4})$,
using the Lipschitz property of $\tanh(x)$. Applying for any
$x,y\in\mathbb{R}$

$|\tanh(x)-\tanh(y)-(x-y)(\cosh y)^{-2}|\leq\frac{2(x-y)^{2}}{3^{3/2}}$

(see [4, page 480]), one can do some calculations to obtain that
$\frac{\mathbb{E}|\widetilde{r_{1}}(X)|}{2\lambda}={\mathcal{O}}(1/n^{1/2})$.
Moreover, the fourth derivative $f^{(4)}(0)$ has a prefactor $K_{n}$, hence if
we apply Taylor expansion in (4.16) up to the fifth order, we can obtain that
$\frac{1}{\lambda}\mathbb{E}\big{|}f^{(4)}(0)W^{4}/n^{7/4}\big{|}={\mathcal{O}}(1/n^{1/2})$
and that
$\frac{1}{\lambda}\mathbb{E}\big{|}f^{(5)}(\eta)W^{5}/n^{2}\big{|}={\mathcal{O}}(1/n^{1/2})$.
Nevertheless, the fact that $c_{2}(J(K_{n}),K_{n},0)=2K_{n}\not=0$ determines
the Berry-Esseen bound of order $n^{-1/4}$. The exchangeable pair approach is
limited at that point.

######  Remark 9.

Let us briefly discuss the statements of Theorem 4.3 in the case $\alpha\geq
0$. This slope of $J(K_{n})=1+\alpha K_{n}$ implies $J(K_{n})\geq J_{c}=1$ and
therefore $m^{*}(K_{n})=m_{2}(J(K_{n}),K_{n})$ is the only positive (!)
solution of $\tanh(Km^{2}+Jm)=m$.

If $\alpha=0$, Proposition 2.3 in [9] gives $m^{*}(K_{n})\sim 3K_{n}$ by
Taylor expansion. With the notion $m(n):=m^{*}(K_{n})\sim 3K_{n}$, we consider
the expansion in (4.16). We observe that $1-(1-m(n)^{2})(1+2K_{n}m(n))\sim
3K_{n}^{2}$, $c_{3}(1,K_{n},m(n))=-2+o(1)$ and $c_{2}(1,K_{n},m(n))\sim
2K_{n}$. Hence we approximately have

$\mathbb{E}(W-W^{\prime}|W)\sim\frac{W}{n}3K_{n}^{2}+\frac{W^{3}}{3n^{3/2}}-\frac{W^{2}}{n^{5/4}}K_{n}-f^{(4)}(\eta)\frac{W^{4}}{n^{7/4}}+\widetilde{r_{1}}(X).$

Now the choice $K_{n}=1/n^{1/4}$ leads to the same prefactor $1/n^{3/2}$ for
the first 3 terms, but we cannot handle the function $g(x)=3x+x^{3}/3-x^{2}$
within the approach of exchangeable pairs. $g$ does not satisfy the
assumptions in [12], [4] or [23]. The choice $K_{n}=1/n^{1/2}$ implies the
quartic limiting behaviour, the Gaussian part does not survive. Every sequence
$K_{n}$ which converges to zero slower than $n^{-1/2}$ can be presented as
$K_{n}=n^{-1/2+\varepsilon}$ with $0<\varepsilon<1/2$. Now the prefactor of
$W$ is of order $n^{-2+2\varepsilon}$ and the choice
$\lambda=n^{-2+2\varepsilon}$ with $1/4<\varepsilon<1/2$ leads to a normal
approximation of $W$ since $2-2\varepsilon<3/2$ and
$2-2\varepsilon<7/4-\varepsilon$. Here the quartic limit does not survive.
Moreover, like in part (3). of Theorem 4.3, normal approximation for
$W_{n}=S_{n}/\sigma$ follows for $K_{n}=n^{-1/2+\varepsilon}$.

If $\alpha>0$, Proposition 2.3 in [9] gives $m^{*}(K_{n})\sim\sqrt{3\alpha
K_{n}}$. Again we consider the expansion in (4.16). Now we have
$1-(1-m(n)^{2})(1+2K_{n}m(n))\sim 2\alpha K_{n}$,
$c_{3}(1,K_{n},m(n))=-2+o(1)$ and $c_{2}(1,K_{n},m(n))\sim 2K_{n}$. Hence
approximately we get

$\mathbb{E}(W-W^{\prime}|W)\sim\frac{W}{n}2\alpha
K_{n}+\frac{W^{3}}{3n^{3/2}}-\frac{W^{2}}{n^{5/4}}K_{n}-f^{(4)}(\eta)\frac{W^{4}}{n^{7/4}}+\widetilde{r_{1}}(X).$

Now the choice $K_{n}=1/\sqrt{n}$ lead to the same results as in part (1) of
Theorem 4.3, in exactly the same manner. Moreover the statements of part (2)
\- (4) of Theorem 4.3 should be the same as well. The details need some
technical precisions. We omit the details.

For the proof of Theorem 4.2, it will be helpful to use an alternative
representation of $\mathbb{E}(W-W^{\prime}|W)$ which will be developed in the
proof of the Cramér-type moderate deviations, see Theorem 6.1 in Section 6.
The proof of Theorem 4.2 will therefore be positioned in Section 6.

##  5\. Concentration inequality for the magnetization in the cubic mean-field
model

In the cubic mean-field model we consider
$F(X,X^{\prime}):=\sum_{i=1}^{n}(X_{i}-X_{i}^{\prime})=X_{I}-X_{I}^{\prime}$,
where $X$ are the spins under $\mathbb{P}_{n}$ and $X^{\prime}$ is the
constructed sequence due to the heat-bath Glauber dynamics. We know that

$f(X)=\mathbb{E}(F(X,X^{\prime})|X)=m(X)-\frac{1}{n}\sum_{i=1}^{n}\tanh\bigl{(}Jm_{i}(X)+Km_{i}^{2}(X)+\frac{K}{3n^{2}}\bigr{)}.$

We know that $|F(X,X^{\prime})|\leq 2$ because $X$ and $X^{\prime}$ differ at
only one coordinate. Moreover, since $\tanh(x)$ is 1-Lipschitz, we obtain

$|f(X)-f(X^{\prime})|\leq|m(X)-m(X^{\prime})|+J|m_{i}(X)-m_{i}(X^{\prime})|+K|m_{i}^{2}(X)-m_{i}^{2}(X^{\prime})|.$

We observe that $|m(X)-m(X^{\prime})|\leq\frac{2}{n}$. The difference
$|m_{i}^{2}(X)-m_{i}^{2}(X^{\prime})|$ is only nonzero if in
$\frac{1}{n^{2}}\sum_{i,j=1}^{n}(X_{i}X_{j}-X_{i}^{\prime}X_{j}^{\prime})$ one
index is $I$ and the other not. Hence the difference is equal to
$\frac{1}{n}(X_{I}-X_{I}^{\prime})m(X_{I})$ and appears twice, so it can be
bounded by $\frac{4}{n}$. Summarizing we obtain

$|f(X)-f(X^{\prime})|\leq(J+1)\frac{2}{n}+K\frac{4}{n}=\frac{2(J+1+2K)}{n}.$

It follows that
$\Delta(X)\leq|E(|f(X)-f(X^{\prime})||X)\leq\frac{2(J+1+2K)}{n}=:C$. Theorem
2.6 implies

$\mathbb{P}_{n}\biggl{(}|f(X)|\geq\frac{t}{\sqrt{n}}\biggr{)}\leq
2\exp\biggl{(}-\frac{t^{2}}{4(1+J+2K)}\biggr{)}.$

Once more using the 1-Lipschitz property of $\tanh(x)$ we obtain that

$\bigg{|}\frac{1}{n}\sum_{i=1}^{n}\tanh\bigl{(}Jm_{i}(X)+Km_{i}^{2}(X)+\frac{K}{3n^{2}}\bigr{)}-\tanh(Jm(X)+Km^{2}(X))\bigg{|}\leq\frac{J+4K}{n}.$

We have proved:

######  Theorem 5.1 (Concentration inequality for the total magnetization).

The magnetization per particle $m(X)$ in the cubic mean-field model satisfies
the following concentration inequality for all $J>0$, $K\geq 0$, $n\geq 1$ and
$t\geq 0$

$\mathbb{P}_{n}\biggl{(}|m(X)-\tanh(Jm(X)+Km^{2}(X))|\geq\frac{J+4K}{n}+\frac{t}{\sqrt{n}}\biggr{)}\leq
2\exp\biggl{(}-\frac{t^{2}}{4(1+J+2K)}\biggr{)}.$

######  Remark 10.

The critical case $(K,J)=(0,1)$ is the 2-spin Curie-Weiss model in the
critical temperature $J=J_{c}=1$. Here in [3] it was proved that

$P\bigl{(}n^{1/4}|m(X)|\geq t\bigr{)}\leq 2\exp\bigl{(}-c\,t^{4}\bigr{)}$

for any $n\geq 1$ and $t\geq 0$. $c>0$ is an absolute constant.

##  6\. Cramér-type moderate deviations for the cubic mean-field model

We will prove the following deviations result:

######  Theorem 6.1 (Cramér-type moderate deviations for the total
magnetization).

Let $Z$ be a standardnormal distributed random variable. Consider the
classical cubic mean-field model given by the probability measure (1.3) and
consider the magnetization per particle $m_{n}$ and the normalized random
variable $W_{n}$ in (4.3). Assume that
$(K,J)\in(\mathbb{R}_{+}\times\mathbb{R})\setminus(\gamma\cup(K_{c},J_{c}))$,
then there exists a constant $c>0$ such that

$\frac{\mathbb{P}_{n}(W_{n}>x)}{\mathbb{P}(Z>x)}=1+{\mathcal{O}}(1)(1+x^{3})\bigl{(}\frac{1}{\sqrt{n}}\bigr{)}$

for all $x\in[0,cn^{1/6}]$. If $(K,J)$ is any point on the curve $\gamma$,
then for both distinct phases $m_{0}$ and $m_{1}$, then there exists constants
$c_{i}>0$, $i=0,1$, such that

$\frac{\mathbb{P}_{n}\bigl{(}W_{n}>x\big{|}m_{n}\in
A_{i}\bigr{)}}{\mathbb{P}(Z>x)}=1+{\mathcal{O}}(1)(1+x^{3})\bigl{(}\frac{1}{\sqrt{n}}\bigr{)}$
(6.1)

for all $x\in[0,c_{i}n^{1/6}]$, and $A_{i}$ being a neighborhood of $m_{i}$
whose closure does not contain the other phase, for $i=0,1$.

######  Remark 11.

For the classical Curie-Weiss model ($K=0$), in [29] the convergence range was
proved to be $x\in[0,cn^{1/2}]$ instead of $x\in[0,cn^{1/6}]$.

######  Remark 12.

At the critical point $(K,J)=(0,1)$, we are considering the classical Curie-
Weiss model at the critical temperature $J_{c}=1$. Corollary 3.1 in [24]
states that

$\frac{\mathbb{P}_{n,0,1}(S_{n}/n^{3/4}>x)}{\mathbb{P}(Y>x)}=1+{\mathcal{O}}(1)(1+x^{6})\bigl{(}\frac{1}{\sqrt{n}}\bigr{)}$

for $x\in[0,cn^{1/12}]$, where $Y$ has the density
$C\exp\bigl{(}-\frac{x^{4}}{12}\bigr{)}$.

###### Proof of Theorem 6.1.

We will adapt the truncation argument presented in [5]. Let
$\widetilde{W_{n}}$ have the conditional distribution of $W_{n}$ given
$|W_{n}|\leq d\sqrt{n}$, where $d$ is to be determined. If we can prove that

$\frac{\mathbb{P}_{n}(\widetilde{W_{n}}>x)}{\mathbb{P}(Z>x)}=1+{\mathcal{O}}(1)(1+x^{3})\bigl{(}\frac{1}{\sqrt{n}}\bigr{)}$

for all $x\in[0,cn^{1/6}]$, we observe the result from the fact that

$\mathbb{P}_{n}(|W_{n}|>K\sqrt{n})\leq e^{-nC(K)}$

for any positive $K$ and a positive constant $C(K)$. The fact is an immediate
consequence of the LDP in Theorem 3.1, since we have a unique phase $m$ which
is the only value such that the rate function $I_{K,J}(m)=0$. Consider the
inequality
$\mathbb{P}_{n}(W_{n}>x)\leq\mathbb{P}_{n}(\widetilde{W_{n}}>x)+\mathbb{P}_{n}(\delta_{2}|W_{n}|>1/2)$
with $\delta_{2}={\mathcal{O}}(1/\sqrt{n})$.

We still denote $\widetilde{W_{n}}$ by $W$. Now we consider an interval
$[a,b]$ such that mean value $m(X)$ of the spins under $\mathbb{P}_{n}$ takes
it possible values in $[a,b]$, the only phase $m$ is in $[a,b]$ and where
$[a,b]$ is choosen such that $|W_{n}|\leq d\sqrt{n}$. Now we consider a
slightly different point of view when calculating
$\mathbb{E}(W-W^{\prime}|W)$, where again
$W-W^{\prime}=\frac{X_{I}-X_{I}^{\prime}}{\sigma}$ with $\sigma$ defined in
(4.2). We have

$\displaystyle\mathbb{E}(W-W^{\prime}|W)$ $\displaystyle=$
$\displaystyle\frac{1}{\sigma}\mathbb{E}(X_{I}-X_{I}^{\prime}|W)=\frac{2}{\sigma}\mathbb{E}\bigl{(}1_{\\{X_{I}=1,X_{I}^{\prime}=-1)\\}}-1_{\\{X_{I}=-1,X_{I}^{\prime}=1)\\}}|W\bigr{)}$
$\displaystyle=$
$\displaystyle\frac{2}{\sigma}\frac{1}{n}\sum_{i=1}^{n}\biggl{(}1_{\\{X_{i}=1\\}}\mathbb{P}_{n}(X_{i}^{\prime}=-1|W)-1_{\\{X_{i}=-1\\}}\mathbb{P}_{n}(X_{i}^{\prime}=1|W)\biggr{)}.$

The number of cases $1_{\\{X_{i}=1\\}}$ is equal to
$\frac{S_{n}}{2}+\frac{n}{2}$, and the number of cases $1_{\\{X_{i}=-1\\}}$ is
$-\frac{S_{n}}{2}+\frac{n}{2}$. Remark that
$\frac{S_{n}}{2}+\frac{n}{2}=\frac{1}{2}(\sigma W+nm+n)$ and
$-\frac{S_{n}}{2}+\frac{n}{2}=\frac{1}{2}(n-\sigma W-nm)$. For $X_{i}=1$ we
have

$\sum_{i=1}^{n}\mathbb{P}_{n}(X_{i}^{\prime}=-1|W)=A(m(X)):=\frac{e^{-Jm(X)-Km(X)^{2}}}{2\cosh(Jm(X)+Km(X)^{2})}$

and

$\sum_{i=1}^{n}\mathbb{P}_{n}(X_{i}^{\prime}=1|W)=B(m(X)):=\frac{e^{Jm(X)+Km(X)^{2}}}{2\cosh(Jm(X)+Km(X)^{2})}.$

If $X_{i}=1$ and $X_{i}^{\prime}=-1$ we have to ensure that $S_{n}-2\geq an$,
define $A_{a}:=\\{S_{n}-2\geq an\\}$. If $X_{i}=-1$ and $X_{i}^{\prime}=1$ we
have to ensure that $S_{n}+2\leq bn$, define $A_{b}:=\\{S_{n}+2\leq bn\\}$.
Summarizing we obtain

$\displaystyle\mathbb{E}(W-W^{\prime}|W)$ $\displaystyle=$
$\displaystyle\frac{2}{\sigma n}\biggl{(}\frac{A(m(X))}{2}(\sigma
W+nm+n)1_{A_{a}}-\frac{B(m(X))}{2}(n-\sigma W-nm)1_{A_{b}}\biggr{)}$
$\displaystyle=$
$\displaystyle\bigl{(}A(m(X))+B(m(X))\bigr{)}\biggl{(}\frac{W}{n}+\frac{m}{\sigma}\biggr{)}+\frac{1}{\sigma}\bigl{(}A(m(X))-B(m(X))\bigr{)}+R_{1}(X)$
$\displaystyle=$
$\displaystyle\biggl{(}\frac{W}{n}+\frac{m}{\sigma}\biggr{)}-\frac{1}{\sigma}\tanh\bigl{(}Jm(X)+Km(X)^{2}\bigr{)}+R_{1}(X)$

with

$R_{1}(X)=-A(m(X))\frac{\sigma
W+nm+n}{n\sigma}1_{A_{a}^{c}}+B(m(X))\frac{n-\sigma
W-nm}{n\sigma}1_{A_{b}^{c}}.$

Now we proceed as in the proof of Theorem 4.1 and obtain by Taylor expansion

$\mathbb{E}(W-W^{\prime}|W)=\lambda W+R$ (6.2)

with

$R=R_{1}(X)-r_{2}(\eta)\frac{\sigma W^{2}}{2n^{2}},$

where $\eta\in(0,\sigma W/n)$ and $r_{2}(\eta)$ is given as in (4). Next we
consider

$\mathbb{E}((W-W^{\prime})^{2}|W)=\frac{4}{\sigma^{2}}\mathbb{E}\bigl{(}1_{\\{X_{I}=1,X_{I}^{\prime}=-1)\\}}-1_{\\{X_{I}=-1,X_{I}^{\prime}=1)\\}}|W\bigr{)}.$

Similarly to (4), we have

$\mathbb{E}((W-W^{\prime})^{2}|W)=\frac{2}{\sigma^{2}}(1-m^{2})+\frac{mW}{n\sigma}{\mathcal{O}}(1)+\frac{W^{2}}{n^{2}}{\mathcal{O}}(1)+{\mathcal{O}}\biggl{(}\frac{1_{A_{a}^{c}\cup
A_{b}^{c}}}{\sigma^{2}}\biggr{)}$ (6.3)

and with $\frac{2}{\sigma^{2}}(1-m^{2})=2\lambda$ we obtain the identity

$\bigg{|}\frac{1}{2\lambda}\mathbb{E}((W-W^{\prime})^{2}|W)-1\bigg{|}=\bigg{|}{\mathcal{O}}(1)\frac{m\,W}{2\lambda
n\sigma}+{\mathcal{O}}(1)\frac{W^{2}}{2\lambda
n^{2}}+{\mathcal{O}}\biggl{(}\frac{1_{A_{a}^{c}\cup
A_{b}^{c}}}{2\lambda\sigma^{2}}\biggr{)}\bigg{|}.$

Since
$\big{|}\frac{W}{n}\big{|}=\big{|}\frac{m(X)-m}{\sigma}\big{|}={\mathcal{O}}(1/\sigma)$
we have that ${\mathcal{O}}(1)\big{|}\frac{W^{2}}{2\lambda
n^{2}}\big{|}={\mathcal{O}}(1)\big{|}\frac{W}{2\lambda n\sigma}\big{|}$. We
know already that $\mathbb{E}\big{|}\frac{W}{2\lambda
n\sigma}\big{|}={\mathcal{O}}(1/\sqrt{n})$. Moreover if $|W_{n}|\leq
d\sqrt{n}$, then $|m(X)-m|\leq\frac{d}{\phi^{\prime\prime}(m)}$, see (4.2).
Since $m\in(a,b)$ we can find a constant $\delta>0$ such that $A_{a}^{c}\cup
A_{b}^{c}\subset\\{|m(X)-m|>\delta\\}$. Now we choose $d$ small enough such
that $\\{|m(X)-m|>\delta\\}$ is empty. Summarizing we obtain

$\bigg{|}\frac{1}{2\lambda}\mathbb{E}((W-W^{\prime})^{2}|W)-1\bigg{|}={\mathcal{O}}(1/\sqrt{n})(1+|W|),$

hence condition (2.8) of Theorem 2.7 is satisfied with
$\delta_{1}={\mathcal{O}}(1/\sqrt{n})$. With $|W|={\mathcal{O}}(\sqrt{n})$,
condition(2.10) of Theorem 2.7 is satisfied with $\theta={\mathcal{O}}(1)$.
Finally we have

$\big{|}\mathbb{E}\bigl{(}\frac{R}{\lambda}|W\bigr{)}\big{|}\leq{\mathcal{O}}(1/\sqrt{n})(1+W^{2})$

and with $\delta_{2}={\mathcal{O}}(1/\sqrt{n})$ we can choose the $d$ in
condition $|W_{n}|\leq d\sqrt{n}$ such that $\delta_{2}|W|\leq\frac{1}{2}$.
Hence the second alternative of (2.7) in Theorem 2.7 is satisfied with
$\alpha=\frac{1}{2}$. From Theorem 2.7, we have proved the result.

For $(K,J)$ being any point on the curve $\gamma$, we consider
$\mathbb{P}_{n}(W_{n}|m_{n}\in A_{i})$, where $A_{i}$ is a neighborhood of
$m_{i}$ whose closure does not contain the other phase. But know, intersect
$A_{i}$ with the event $\\{|W_{n}|\leq d\sqrt{n}\\}$, where $d$ is to be
determined. Again an interval $[a,b]$ is choosen such that the mean value of
the spins under this conditional distribution of $W_{n}$ takes it possible
values in $[a,b]$, meaning that only one of the two phases is in the interval
and the interval is choosen such that $|W_{n}|\leq d\sqrt{n}$. Now all the
steps of the proof for values $(K,\gamma(K))$ are exactly the same. This
completes the proof of (6.1). ∎

###### Proof of Theorem 4.2.

To obtain the conditional probability $\mathbb{P}_{n}(W_{n}\leq x|m_{n}\in
A_{i})$ we proceed as in the Proof of Theorem 2.7. We choose an interval
$[a,b]$ such that the mean value of the spins under this conditional
distribution of $W_{n}$ takes it possible values in $[a,b]$, meaning that only
one of the two phases is in the interval. Now we consider the representation
(6.2) of $\mathbb{E}(W-W^{\prime}|W)$ and (6.3) of
$\mathbb{E}((W-W^{\prime})^{2}|W)$ and proceed as in the proof of Theorem 4.1.
This completes the proof of the Theorem. ∎

The Cramér type moderate deviation results in Theorem 2.7 imply so called
moderate deviations principles. Note that for $K=0$, moderate deviation
principles for the magnetization in the Curie-Weiss model were proved in [11].

######  Theorem 6.2 (Moderate deviation principle for the total
magnetization).

Consider any sequence of real numbers $(a_{n})_{n}$ with $1<<a_{n}<<n^{1/6}$.
Consider the classical cubic mean-field model given by the probability measure
(1.3) and consider the magnetization per particle $m_{n}$ and the normalized
random variable $W_{n}$ in (4.3). Assume that
$(K,J)\in(\mathbb{R}_{+}\times\mathbb{R})\setminus(\gamma\cup(K_{c},J_{c}))$.
Then $(\frac{1}{a_{n}}W_{n})_{n}$ satisfies the moderate deviations principle,
which is a large deviations principle with speed $a_{n}^{2}$ and rate function
$\frac{x^{2}}{2}$ in the sense of the definition [10, Section 1.2]. In
particular we have for any $x\in\mathbb{R}$

$\lim_{n\to\infty}\frac{1}{a_{n}^{2}}\mathbb{P}_{n}\bigl{(}W_{n}/a_{n}>x)=-\frac{x^{2}}{2}.$

If $(K,J)$ is any point on the curve $\gamma$, then for both distinct phases
$m_{0}$ and $m_{1}$ let $A_{i}$ being a neighborhood of $m_{i}$ whose closure
does not contain the other phase, for $i=0,1$. Then we obtain a conditional
moderate deviations principle, including

$\lim_{n\to\infty}\frac{1}{a_{n}^{2}}\mathbb{P}_{n}\bigl{(}W_{n}/a_{n}>x|m_{n}\in
A_{i})=-\frac{x^{2}}{2},i=0,1,$

for any $x\in\mathbb{R}$.

###### Proof of Theorem 6.2.

We start with $x\geq 0$ and obtain

$\displaystyle\big{|}\log\mathbb{P}_{n}(W_{n}>a_{n}x)-\log\mathbb{P}(Z>a_{n}x)\big{|}$
$\displaystyle=$
$\displaystyle\bigg{|}\log\biggl{(}1+{\mathcal{O}}(1)(1+(a_{n}x)^{3})\frac{1}{\sqrt{n}}\biggr{)}\bigg{|}$
$\displaystyle\leq$
$\displaystyle{\mathcal{O}}(1)(1+(a_{n}x)^{3})\frac{1}{\sqrt{n}}.$

For $x\geq 0$ it is known that

$\frac{1}{1+\sqrt{2\pi}x}\leq e^{x^{2}/2}\mathbb{P}(Z>x)\leq\frac{1}{2}.$

The monotonicity of the logarithm implies

$\displaystyle\bigg{|}\log\mathbb{P}_{n}(W_{n}>a_{n}x)+\frac{(a_{n}x)^{2}}{2}\bigg{|}$
$\displaystyle\leq$
$\displaystyle\bigg{|}\log\bigl{(}e^{\frac{(a_{n}x)^{2}}{2}}P(Z>a_{n}x)\bigr{)}\bigg{|}+{\mathcal{O}}(1)(1+(a_{n}x)^{3})\frac{1}{\sqrt{n}}$
$\displaystyle\leq$
$\displaystyle\bigg{|}\log\biggl{(}\frac{1}{2+\sqrt{2\pi}a_{n}x}\biggr{)}\bigg{|}+{\mathcal{O}}(1)(1+(a_{n}x)^{3})\frac{1}{\sqrt{n}}$
$\displaystyle\leq$
$\displaystyle\log(2+\sqrt{2\pi}a_{n}x)+{\mathcal{O}}(1)(1+(a_{n}x)^{3})\frac{1}{\sqrt{n}},$

and it follows by our assumption on $(a_{n})_{n}$ that

$\lim_{n\to\infty}\frac{1}{a_{n}^{2}}\log\mathbb{P}_{n}(W_{n}>a_{n}x)=-\frac{x^{2}}{2}.$

It is easy to verify, that the Cramér-type moderate deviations implies

$\frac{\mathbb{P}_{n}(W_{n}<-x)}{\mathbb{P}(Z<-x)}=1+{\mathcal{O}}(1)(1+x^{3})\bigl{(}\frac{1}{\sqrt{n}}\bigr{)}.$

Our bounds can be carried forward to a full moderate deviation principle
similarly to the proof of Theorems 4.1.11 and 1.2.18 in [10]. ∎

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