# Metastability for Glauber dynamics on
the complete graph with coupling disorder

A. Bovier  Institut für Angewandte Mathematik, Rheinische Friedrich-Wilhelms-
Universität, Endenicher Allee 60, 53115 Bonn, Germany<EMAIL_ADDRESS>,
F. den Hollander  Mathematical Institute, Leiden University, Niels Bohrweg 1,
2333 CA Leiden, The Netherlands<EMAIL_ADDRESS>and  S. Marello
Institut für Angewandte Mathematik, Rheinische Friedrich-Wilhelms-Universität,
Endenicher Allee 60, 53115 Bonn, Germany<EMAIL_ADDRESS>
(Date: August 27, 2024)

###### Abstract.

Consider the complete graph on $n$ vertices. To each vertex assign an Ising
spin that can take the values $-1$ or $+1$. Each spin
$i\in[n]=\\{1,2,\dots,n\\}$ interacts with a magnetic field $h\in[0,\infty)$,
while each pair of spins $i,j\in[n]$ interact with each other at coupling
strength $n^{-1}J(i)J(j)$, where $J=(J(i))_{i\in[n]}$ are i.i.d. non-negative
random variables drawn from a prescribed probability distribution
$\mathcal{P}$. Spins flip according to a Metropolis dynamics at inverse
temperature $\beta\in(0,\infty)$. We show that there are critical thresholds
$\beta_{c}$ and $h_{c}(\beta)$ such that, in the limit as $n\to\infty$, the
system exhibits metastable behaviour if and only if
$\beta\in(\beta_{c},\infty)$ and $h\in[0,h_{c}(\beta))$. Our main result are
sharp asymptotics, up to a multiplicative error $1+o_{n}(1)$, of the average
crossover time from any metastable state to the set of states with lower free
energy. We use standard techniques of the potential-theoretic approach to
metastability. The leading order term in the asymptotics does not depend on
the realisation of $J$, while the correction terms do. The leading order of
the correction term is $\sqrt{n}$ times a centred Gaussian random variable
with a complicated variance depending on $\beta,h,\mathcal{P}$ and on the
metastable state. The critical thresholds $\beta_{c}$ and $h_{c}(\beta)$
depend on $\mathcal{P}$, and so does the number of metastable states. We
derive an explicit formula for $\beta_{c}$ and identify some properties of
$\beta\mapsto h_{c}(\beta)$. Interestingly, the latter is not necessarily
monotone, meaning that the metastable crossover may be re-entrant.

###### Key words and phrases:

Curie-Weiss, Glauber dynamics, disorder, metastability.

###### 2020 Mathematics Subject Classification:

60K35; 60K37; 82B20; 82B44; 82C44

AB and SM were supported through DFG Excellence Cluster GZ 2047/1, Projekt-id
390685813, and DFG SFB 1060, Projektnummer 211504053. FdH was supported
through NWO Gravitation Grant 024.002.003-NETWORKS, and by the Alexander von
Humboldt Foundation during sabbatical leaves in Bonn and Erlangen in the Fall
of 2019 and 2020.

##  1\. Introduction and main results

###  1.1. Background

Interacting particle systems evolving according to a Metropolis dynamics
associated with an energy functional called the Hamiltonian, may be trapped
for a long time near a state that is a local minimum of the free energy, but
not a global minimum. The deepest local minima are called _metastable states_
, the global minimum is called the _stable state_. The transition from a
metastable state to the stable state marks the relaxation of the system to
equilibrium. To describe this relaxation, one needs to identify the set of
critical configurations the system must attain in order to achieve this
transition and to compute the crossover time. These critical configurations
correspond to saddle points in the free energy landscape.

Metastability for interacting particle systems on _lattices_ has been studied
intensively in the past. For a summary, we refer the reader to the monographs
by Olivieri and Vares [12], and Bovier and den Hollander [7]. Successful
attempts towards understanding metastable behaviour in random environments
were made for the random field Curie-Weiss model, first by Bovier and Gayrard
[6] and later by Bianchi, Bovier and Ioffe [1, 2]. Recently, there has been
interest in metastability for interacting particle systems on _random graphs_.
This is challenging, because the crossover times typically depend on the
realisation of the graph. In den Hollander and Jovanovksi [11] and Bovier,
Marello and Pulvirenti [8], Glauber dynamics on dense _Erdős-Rényi_ random
graphs was analysed. Earlier work on metastability for Glauber dynamics on
sparse random graphs can be found in Dommers [9] (random regular graph) and
Dommers, den Hollander, Jovanovski and Nardi [10] (configuration model). The
present paper is a first step towards the study of metastability for Glauber
dynamics on _Chung-Lu_ random graphs.

###  1.2. Glauber dynamics on the complete graph with coupling disorder

Let $\mathcal{K}_{n}$ be the complete graph on $n$ vertices. Each vertex
carries an Ising spin that can take the values $-1$ or $+1$. Let
$\mathcal{S}_{n}=\\{-1,+1\\}^{[n]}$ denote the set of spin configurations on
$\mathcal{K}_{n}$, where $[n]=\\{1,2,\dots,n\\}$. Let $\mathcal{P}$ denote a
probability measure that is supported on a _finite_ subset of $[0,\infty)$.
Let $J=(J(i))_{i\in[n]}$ be i.i.d. random variables with common law
$\mathcal{P}$, and let $H_{n}\colon\mathcal{S}_{n}\to\mathbb{R}$ be defined by

(1.1) $H_{n}(\sigma)\equiv-\frac{1}{n}\sum_{\begin{subarray}{c}i,j\in[n]\\\
i<j\end{subarray}}J(i)J(j)\,\sigma(i)\sigma(j)-h\sum_{i\in[n]}\sigma(i),\qquad\sigma\in\mathcal{S}_{n},$

where $h\in[0,\infty)$ is the magnetic field. We consider _Glauber dynamics_
on $\mathcal{S}_{n}$, defined as the continuous-time Markov process with
transition rates

(1.2)
$r_{n}(\sigma,\sigma^{\prime})=\begin{cases}\mathrm{e}^{-\beta[H_{n}(\sigma^{\prime})-H_{n}(\sigma)]_{+}},&\mbox{if
}\sigma^{\prime}\sim\sigma,\\\
0,&\mbox{otherwise},\end{cases}\qquad\sigma,\sigma^{\prime}\in\mathcal{S}_{n},$

where $\beta\in(0,\infty)$ is the inverse temperature,
$\sigma^{\prime}\sim\sigma$ means that $\sigma^{\prime}$ differs from $\sigma$
by a single spin-flip and $[\cdot]_{+}$ is the positive part. This dynamics is
_reversible_ with respect to the _Gibbs measure_

(1.3) $\mu_{n}(\sigma)\equiv\frac{1}{Z_{n}}\,\mathrm{e}^{-\beta
H_{n}(\sigma)},\qquad\sigma\in\mathcal{S}_{n},$

where the normalising constant $Z_{n}$ is called the partition sum. Note that
the reference measure for (1.3) is the _counting measure_ on
$\mathcal{S}_{n}$. We write

(1.4) $(\sigma_{t})_{t\geq 0},\qquad\sigma_{t}\in\mathcal{S}_{n},$

to denote a path of the Glauber dynamics on $\mathcal{S}_{n}$, and
$\mathbb{P}_{\sigma}$ and $\mathbb{E}_{\sigma}$ to denote probability and
expectation on path space given $\sigma_{0}=\sigma$ (we suppress $J,h,\beta$
and $n$ from the notation).

For fixed $n$, the Hamiltonian in (1.1) achieves a global minimum at
$\sigma\equiv+1$ and a local minimum at $\sigma\equiv-1$. The latter is the
deepest local minimum not equal to the global minimum (at least for $h$ small
enough). However, in the limit as $n\to\infty$, these do _not_ form a
metastable pair of configurations because _entropy_ comes into play.

###  1.3. Metastability on the complete graph with coupling disorder

In this section we state our main results.

####  1.3.1. Empirical magnetisations

The relevant quantity to monitor in order to characterise the metastable
behaviour is the _disorder weighted magnetisation_

(1.5)
$K_{n}(\sigma)=\frac{1}{n}\sum_{i\in[n]}J(i)\sigma(i),\qquad\sigma\in\mathcal{S}_{n}.$

Since $\mathcal{P}$ has finite support, we have

(1.6) $\mathcal{P}=\sum_{\ell\in[k]}\omega_{\ell}\delta_{a_{\ell}}$

for some $k\in\mathbb{N}$, $(a_{\ell})_{\ell\in[k]}\in[0,\infty)^{k}$
distinct, and $(\omega_{\ell})_{\ell\in[k]}\in(0,1)^{k}$ such that
$\sum_{\ell\in[k]}\omega_{\ell}=1$.

The following quantities will be essential for _coarse-graining_. Define the
_level sets_

(1.7) $A_{\ell,n}\equiv\\{i\in[n]\colon\,J(i)=a_{\ell}\\},\qquad\ell\in[k],$

and the _level magnetisations_

(1.8) $m_{\ell,n}(\sigma)\equiv\frac{1}{\left|{A_{\ell,n}}\right|}\sum_{i\in
A_{\ell,n}}\sigma(i),\qquad\ell\in[k],\,\sigma\in\mathcal{S}_{n}.$

Put

(1.9)
$m_{n}(\sigma)=(m_{\ell,n}(\sigma)\big{)}_{\ell\in[k]}\in[-1,1]^{k},\qquad\sigma\in\mathcal{S}_{n},$

and note that
$K_{n}(\sigma)=\frac{1}{n}\sum_{\ell\in[k]}a_{\ell}\,\left|{A_{\ell,n}}\right|\,m_{\ell,n}(\sigma)$
depends on $\sigma$ only through $m_{n}(\sigma)$. Thus, with abuse of
notation, we may define

(1.10)
$K_{n}(m)\equiv\frac{1}{n}\sum_{\ell\in[k]}a_{\ell}\left|{A_{\ell,n}}\right|m_{\ell},\qquad
m=(m_{\ell})_{\ell\in[k]}\in[-1,1]^{k},$

so that $K_{n}(\sigma)=K_{n}(m_{n}(\sigma))$.

####  1.3.2. Thermodynamic limit

With $\mathcal{P}^{n}$-probability tending to $1$ as $n\to\infty$, the random
function $K_{n}$ converges to a deterministic function $K$ given by

(1.11) $K(m)=\sum_{\ell\in[k]}a_{\ell}\,\omega_{\ell}\,m_{\ell},\qquad
m=(m_{\ell})_{\ell\in[k]}\in[-1,1]^{k}.$

Similarly, the random free energy function $F_{n}$ converges to a
deterministic function $F_{\beta,h}$ (see (2.15) and (2.26) below for explicit
formulas). In Section 3, we show that the stationary points of $F_{\beta,h}$
are given by $\mathbf{m}=(\mathbf{m}_{\ell})_{\ell\in[k]}$, where

(1.12)
$\mathbf{m}_{\ell}=\tanh(\beta[a_{\ell}K(\mathbf{m})+h]),\qquad\ell\in[k].$

Note that, via (1.12), the $k$-dimensional vector $\mathbf{m}$ is fully
determined by the real number $K(\mathbf{m})$. Therefore, finding the
stationary points of $F_{\beta,h}$ reduces to finding the solutions of the
equation

(1.13) $K=T_{\beta,h,\mathcal{P}}(K),\qquad
T_{\beta,h,\mathcal{P}}(K)=\sum_{\ell\in[k]}a_{\ell}\,\omega_{\ell}\,\tanh(\beta[a_{\ell}K+h]).$

####  1.3.3. Metastable regime

It turns out that the critical inverse temperature $\beta_{c}$ is given by

(1.14)
$\beta_{c}=\left[\sum_{\ell\in[k]}a_{\ell}^{2}\omega_{\ell}\right]^{-1}.$

Namely, if $\beta\in(0,\beta_{c}]$, then the system is not in the metastable
regime for any $h\in[0,\infty)$, while if $\beta\in(\beta_{c},\infty)$, then,
for $h\in[0,\infty)$ small enough, it is in the metastable regime (i.e.,
(1.13) has more than one solution at which $T_{\beta,h,\mathcal{P}}$ is not
tangent to the diagonal). Given $\beta\in(\beta_{c},\infty)$, the critical
magnetic field $h_{c}(\beta)$ is the minimal value of $h$ for which the system
is not metastable. The _metastable regime_ is thus

(1.15) $\beta\in(\beta_{c},\infty),\qquad h\in[0,h_{c}(\beta)).$

In Section 3, we show that $\beta\mapsto h_{c}(\beta)$ is continuous on
$(\beta_{c},\infty)$, with

(1.16)
$\lim_{\beta\downarrow\beta_{c}}h_{c}(\beta)=0,\qquad\lim_{\beta\to\infty}h_{c}(\beta)=C\in(0,\infty),$

where the explicit value of $C$ is given in (3.12) below. Interestingly,
$\beta\mapsto h_{c}(\beta)$ is not necessarily monotone, i.e., the metastable
crossover may be _re-entrant_.

It turns out that there exists an $\ell\in[k]$ (depending on
$\beta,h,\mathcal{P}$), such that $F_{\beta,h}$ has $2\ell+1$ stationary
points.

####  1.3.4. Metastable crossover

Let $\mathcal{M}_{n}$ be the set of minima of $F_{n}$. Given
$\mathbf{m}\in\mathcal{M}_{n}$, define

(1.17)
$\mathcal{M}_{n}(\mathbf{m})\equiv\\{m\in\mathcal{M}_{n}\backslash\mathbf{m}\colon\,F_{n}(m)\leq
F_{n}(\mathbf{m})\\}.$

Let $\mathcal{G}(A,B)$ be the gate between two disjoint subsets $A$ and $B$ of
$\mathcal{M}_{n}$. We refer to [7, Section 10.1] for a precise definition of
the gate.

Fix $\mathbf{m}_{n}\in\mathcal{M}_{n}$ as the initial magnetisation.
Throughout the paper we assume that the following hypotheses hold for
$\mathbf{m}_{n}$.

######  Hypothesis 1.

  1. (1)

$\mathcal{M}_{n}(\mathbf{m}_{n})$ is non-empty.

  2. (2)

The Hessian of $F_{n}$ has only non-zero eigenvalues at $\mathbf{m}_{n}$ and
at all the points in
$\mathcal{G}(\mathbf{m}_{n},\mathcal{M}_{n}(\mathbf{m}_{n}))$.

  3. (3)

There is a _unique_ point $\mathbf{t}_{n}$ in
$\mathcal{G}(\mathbf{m}_{n},\mathcal{M}_{n}(\mathbf{m}_{n}))$, which will
often be called simply saddle point.

  4. (4)

The saddle point $\mathbf{t}_{n}$ is such that
$r_{\ell}[\left|{A_{\ell,n}}\right|(1-\mathbf{t}_{\ell,n}^{2})]^{-1}$ takes
distinct values for different $\ell\in[k]$, where $r_{\ell}$ is defined in
(4.9) below.

Hypothesis 1(2) and (3) are made to avoid complications. Hypothesis 1(4) is
needed in the proof of Lemma 4.2 below (as in [7, Lemma 14.9]). Neither is
very restrictive: if for some parameter choice they fail, then after an
infinitesimal parameter change they hold. Moreover, if Hypothesis 1(3) fails,
it is sufficient to compute separately the contribution to the crossover time
of the various saddle points in the gate.

Let $\mathcal{S}_{n}[\mathbf{m}_{n}]$ and
$\mathcal{S}_{n}[\mathcal{M}_{n}(\mathbf{m}_{n})]$ denote the sets of
configurations in $\mathcal{S}_{n}$ for which the level magnetisations are
$\mathbf{m}_{n}$ and are contained in $\mathcal{M}_{n}(\mathbf{m}_{n})$,
respectively. Let $\mathbb{A}_{n}(\cdot)$ be the $k\times k$ Hessian matrix
defined in (4.2) below, and $\gamma_{n}$ the unique negative solution of the
equation in (4.20) below. For $A\subset\mathcal{S}_{n}$, write

(1.18) $\tau_{A}=\\{t\geq 0\colon\,\sigma_{t}\in A,\,\sigma_{t-}\notin A\\}$

to denote the first hitting time or return time of $A$.

We next state our main results for the crossover time. Theorem 1.1 provides a
sharp asymptotics for the average crossover time from any metastable state to
the set of states with lower free energy. Theorem 1.2 shows that
asymptotically the crossover time is exponential on the scale of its mean, a
property that is standard for metastable behaviour.

######  Theorem 1.1 (Average crossover time with coupling disorder).

For every $\mathbf{m_{n}}\in\mathcal{M}_{n}$ satisfying Hypothesis 1 and
within the metastable regime (1.15), uniformly in
$\sigma\in\mathcal{S}_{n}[\mathbf{m_{n}}]$, and with
$\mathcal{P}^{n}$-probability tending to $1$,

(1.19)
$\begin{split}\mathbb{E}_{\sigma}\left[\tau_{\mathcal{S}_{n}[\mathcal{M}_{n}(\mathbf{m_{n}})]}\right]&=[1+o_{n}(1)]\,\sqrt{\frac{[-\det(\mathbb{A}_{n}(\mathbf{t}_{n}))]}{\det(\mathbb{A}_{n}(\mathbf{m}_{n}))}}\left(\frac{\pi}{2\beta(-\gamma_{n})}\right)\mathrm{e}^{\beta
n[F_{n}(\mathbf{t}_{n})-F_{n}(\mathbf{m}_{n})]}.\end{split}$

######  Theorem 1.2 (Exponential law with coupling disorder).

For every $\mathbf{m_{n}}\in\mathcal{M}_{n}$ satisfying Hypothesis 1 and
within the metastable regime (1.15), uniformly in
$\sigma\in\mathcal{S}_{n}[\mathbf{m}_{n}]$ and with
$\mathcal{P}^{n}$-probability tending to $1$,

(1.20)
$\mathbb{P}_{\sigma}\left(\tau_{\mathcal{S}_{n}[\mathcal{M}_{n}(\mathbf{m}_{n})]}>t\,\mathbb{E}_{\sigma}\left[\tau_{\mathcal{S}_{n}[\mathcal{M}_{n}(\mathbf{m}_{n})]}\right]\right)=[1+o_{n}(1)]\,\mathrm{e}^{-t},\qquad
t\geq 0.$

As the average crossover time estimated in Theorem 1.1 is a random variable,
we next provide more information on the randomness of the quantity in the
right-hand side of (1.19), which depends on the realisation of the random
variable $J$. The prefactor in (1.19) converges with
$\mathcal{P}^{n}$-probability tending to $1$ to a deterministic limit, which
depends on $\mathcal{P}$ but not on the realisation of $J$. However, the
exponent does not converge to a deterministic limit. In Theorem 1.3 we compute
the exponent up to order $O(1)$. Recall that $F_{n}\to F_{\beta,h}$,
$\mathbf{m}_{n}\to\mathbf{m}$ and $\mathbf{t}_{n}\to\mathbf{t}$ as
$n\to\infty$.

######  Theorem 1.3 (Randomness of the exponent).

For every $\mathbf{m_{n}}\in\mathcal{M}_{n}$ satisfying Hypothesis 1 and
within the metastable regime (1.15), in distribution under the law
$\mathcal{P}^{n}$,

(1.21)
$n[F_{n}(\mathbf{t}_{n})-F_{n}(\mathbf{m}_{n})]=n[F_{\beta,h}(\mathbf{t})-F_{\beta,h}(\mathbf{m})]+Z\sqrt{n}+O(1),$

where $Z$ is a normal random variable with mean zero and variance in
$(0,\infty)$.

The variance of $Z$ turns out to be a complicated function of $\beta$, $h$ and
$\mathcal{P}$. We refer to Section 6.3 for further details. Computing the
exponent up to order $1$ is in principle possible, but the formulas become
rather complicated. Without this precision the prefactor is asymptotically
negligible.

###  1.4. Techniques and outline

In order to prove Theorems 1.1–1.3 we use the potential-theoretic approach to
metastability developed in Bovier, Eckhoff, Gayrard and Klein [3, 4, 5]. More
specifically, we first find a sharp approximation of the Dirichlet form
associated with the coarse-grained dynamics. We use these results, together
with lumpability properties and well-known variation principles, to obtain
sharp capacity estimates that are key quantities in the proof. For a more
detailed overview of the methods, we refer the reader to the monograph by
Bovier and den Hollander [7].

The remainder of the paper is organised as follows. Section 2 provides
quantities and notations that are needed throughout the paper. Section 3
identifies the metastable regime. Section 4 provides a sharp approximation of
the Dirichlet form associated with the Glauber dynamics in the presence of the
disorder. Section 5 provides estimates on capacity and on the metastable
valley measure. Section 6 proves Theorems 1.1–1.3. Appendix A contains a brief
overview on known results for the standard CW model, which corresponds to the
setting without disorder. Appendix B gives numerical evidence for the presence
of multiple metastable states for suitable choices of $\beta$, $h$ and
$\mathcal{P}$. Appendix C contains an example in which $\beta\mapsto
h_{c}(\beta)$ is not increasing, implying the possibility of a re-entrant
metastable crossover. Appendix D provides the limit as $n\to\infty$ of the
prefactor in (1.19).

##  2\. Preparations

Section 2.1 introduces further notation and writes the Hamiltonian in terms of
the level magnetisations. Section 2.2 introduces the Dirichlet form associated
with the Glauber dynamics and rewrites this in terms of the level
magnetisations. Section 2.3 computes gradients and Hessians of the free energy
as a function of the level magnetisations. Section 2.4 closes with an
approximation of the free energy that will be needed later on.

###  2.1. Hamiltonian

Recall (1.7). Abbreviate

(2.1) $\omega_{\ell,n}=\frac{\left|{A_{\ell,n}}\right|}{n}.$

Since
$(\omega_{\ell,n})_{\ell\in[k]}\to(\omega_{\ell})_{\ell\in[k]}\in(0,\infty)^{k}$
as $n\to\infty$ with $\mathcal{P}^{n}$-probability tending to $1$, we may and
will assume that $A_{\ell,n}\neq\emptyset$ for all $\ell\in[k]$ and all $n$
large enough. Recall (1.8)–(1.9). Note that $m_{\ell,n}(\sigma)$ takes values
in the set

(2.2)
$\Gamma_{\ell,n}=\\{-1,-1+\tfrac{2}{|A_{\ell,n}|},\dots,1-\tfrac{2}{|A_{\ell,n}|},1\\}.$

Hence $m_{n}(\sigma)$ takes values in the set

(2.3) $\Gamma_{n}^{\mathcal{P}}=\bigtimes_{\ell\in[k]}\Gamma_{\ell,n}.$

The configurations corresponding to $M\subseteq\Gamma_{n}^{\mathcal{P}}$ are
denoted by

(2.4) $\mathcal{S}_{n}[M]=\\{\sigma\in\mathcal{S}_{n}\colon\,m_{n}(\sigma)\in
M\\}.$

For singletons $M=\\{m\\}$ we write $\mathcal{S}_{n}[m]$ instead of
$\mathcal{S}_{n}[\\{m\\}]$.

Let

(2.5)
$H_{n}(\sigma)=-\frac{1}{2n}\sum_{i,j\in[n]}J(i)J(j)\,\sigma(i)\sigma(j)-h\sum_{i\in[n]}\sigma(i),\qquad\sigma\in\mathcal{S}_{n},$

which is the Hamiltonian in (1.1), except for the diagonal term
$-\frac{1}{2n}\sum_{i\in[n]}J^{2}(i)$, which is a constant shift. Using the
notation above, we can write the Hamiltonian in (2.5) as

(2.6)
$H_{n}(\sigma)=-n\left[\frac{1}{2}\left(\sum_{\ell\in[k]}a_{\ell}\,\omega_{\ell,n}\,m_{\ell,n}(\sigma)\right)^{2}+h\sum_{\ell\in[k]}\omega_{\ell,n}\,m_{\ell,n}(\sigma)\right]=nE_{n}(m_{n}(\sigma)),$

where we abbreviate

(2.7)
$E_{n}(m)=-\frac{1}{2}\left(\sum_{\ell\in[k]}a_{\ell}\,\omega_{\ell,n}\,m_{\ell}\right)^{2}-h\sum_{\ell\in[k]}\,\omega_{\ell,n}\,m_{\ell},\qquad
m=(m_{\ell})_{\ell\in[k]}\in\Gamma_{n}^{\mathcal{P}}.$

###  2.2. Dirichlet form and mesoscopic dynamics

By (1.2)–(1.3), the _Dirichlet form_ associated with the Glauber dynamics
equals

(2.8)
$\begin{split}\mathcal{E}_{\mathcal{S}_{n}}(h,h)&=\frac{1}{2}\sum_{\sigma,\sigma^{\prime}\in\mathcal{S}_{n}}\mu_{n}(\sigma)r_{n}(\sigma,\sigma^{\prime})\,[h(\sigma)-h(\sigma^{\prime})]^{2}\\\
&=\frac{1}{2Z_{n}}\sum_{\sigma\in\mathcal{S}_{n}}\sum_{\begin{subarray}{c}\sigma^{\prime}\in\mathcal{S}_{n},\\\
\sigma^{\prime}\sim\sigma\end{subarray}}\mathrm{e}^{-\beta\,H_{n}(\sigma)}\mathrm{e}^{-\beta[H_{n}(\sigma^{\prime})-H_{n}(\sigma)]_{+}}[h(\sigma)-h(\sigma^{\prime})]^{2},\end{split}$

where $h$ is a test function on $\mathcal{S}_{n}$ taking values in $[0,1]$.
Because of (2.6), for any $h$ such that $h(\sigma)=\bar{h}(m_{n}(\sigma))$,
with $\bar{h}$ a test function on $\Gamma_{n}^{\mathcal{P}}$, we have

(2.9) $\displaystyle\mathcal{E}_{\mathcal{S}_{n}}(h,h)$
$\displaystyle=\frac{1}{2Z_{n}}\sum_{m\in\Gamma_{n}^{\mathcal{P}}}\sum_{m^{\prime}\in\Gamma_{n}^{\mathcal{P}}}\mathrm{e}^{-\beta\,nE_{n}(m)}\mathrm{e}^{-\beta\,n[E_{n}(m^{\prime})-E_{n}(m)]_{+}}$
$\displaystyle\qquad\qquad\times\big{[}\,\bar{h}(m)-\bar{h}(m^{\prime})\big{]}^{2}\sum_{\begin{subarray}{c}\sigma\in\mathcal{S}_{n},\\\
m_{n}(\sigma)=m\end{subarray}}\,\,\sum_{\begin{subarray}{c}\sigma^{\prime}\in\mathcal{S}_{n},\,\sigma^{\prime}\sim\sigma,\\\
m_{n}(\sigma^{\prime})=m^{\prime}\end{subarray}}1,$

where $m=(m_{\ell})_{\ell\in[k]}$. If $\sigma^{\prime}\sim\sigma$, then
$\sigma^{\prime}=\sigma^{i}$ for some $i\in[n]$, with $\sigma^{i}$ obtained
from $\sigma$ by flipping the spin with label $i$. Let $\ell^{\prime}\in[k]$
be such that $i\in A_{\ell^{\prime},n}$. If $\sigma(i)=\pm 1=-\sigma^{i}(i)$,
then

(2.10)
$m_{\ell,n}(\sigma^{i})=\begin{cases}m_{\ell^{\prime},n}(\sigma)\mp\frac{2}{\left|{A_{\ell^{\prime},n}}\right|},&\ell=\ell^{\prime},\\\
m_{\ell,n}(\sigma),&\ell\neq\ell^{\prime}.\end{cases}$

For $m,m^{\prime}\in\Gamma^{\mathcal{P}}_{n}$, we write $m\sim m^{\prime}$
when there exists an $\ell^{\prime}\in[k]$ such that
$m^{\prime}=m^{\ell^{\prime},+}$ or $m^{\prime}=m^{\ell^{\prime},-}$, where

(2.11)
$m_{\ell}^{\ell^{\prime},\pm}=\begin{cases}m_{\ell^{\prime}}\pm\frac{2}{\left|{A_{\ell^{\prime},n}}\right|},&\ell=\ell^{\prime},\\\
m_{\ell},&\ell\neq\ell^{\prime}.\end{cases}$

Moreover, for $\ell\in[k]$ and $\sigma\in\mathcal{S}_{n}$ with
$m_{n}(\sigma)=m$, the cardinality of the set
$\\{\sigma^{\prime}\in\mathcal{S}_{n}\colon\,\sigma^{\prime}\sim\sigma,\,m_{n}(\sigma^{\prime})=m^{\ell,\pm}\\}$
equals $\frac{1\mp m_{\ell}}{2}|A_{\ell,n}|$, namely, the number of $(\mp
1)$-spins in $\sigma$ with index in $A_{\ell,n}$. Furthermore,

(2.12)
$\left|\\{\sigma\in\mathcal{S}_{n}\colon\,m_{n}(\sigma)=m\\}\right|=\prod_{\ell\in[k]}\binom{|A_{\ell,n}|}{\frac{1+m_{\ell}}{2}|A_{\ell,n}|},\qquad
m\in\Gamma_{n}^{\mathcal{P}},$

as is seen by counting the number of $(-1)$-spins with label in $A_{\ell,n}$
of a configuration with $\ell$-th level magnetisation $m_{\ell}$. Using these
observations, we can rewrite (2.9) as

(2.13)
$\displaystyle\mathcal{E}_{\mathcal{S}_{n}}(h,h)=\frac{1}{2Z_{n}}\sum_{m\in\Gamma_{n}^{\mathcal{P}}}\mathrm{e}^{-\beta\,nE_{n}(m)}\sum_{m^{\prime}\in\Gamma_{n}^{\mathcal{P}}}\mathrm{e}^{-\beta\,n[E_{n}(m^{\prime})-E_{n}(m)]_{+}}\big{[}\,\bar{h}(m)-\bar{h}(m^{\prime})\big{]}^{2}$
$\displaystyle\qquad\times\prod_{\ell\in[k]}\binom{|A_{\ell,n}|}{\frac{1+m_{\ell}}{2}|A_{\ell,n}|}\sum_{\ell\in[k]}|A_{\ell,n}|\left[\frac{1-m_{\ell}}{2}\,\mathds{1}(m^{\prime}=m^{\ell,+})+\frac{1+m_{\ell}}{2}\,\mathds{1}(m^{\prime}=m^{\ell,-})\right].$

Next, abbreviate

(2.14)
$I_{n}(m)=-\frac{1}{n}\log\left[\prod_{\ell\in[k]}\binom{|A_{\ell,n}|}{\frac{1+m_{\ell}}{2}|A_{\ell,n}|}\right],\qquad
m\in\Gamma_{n}^{\mathcal{P}},$

and put

(2.15)
$F_{n}(m)=E_{n}(m)+\frac{1}{\beta}I_{n}(m)=-\frac{1}{2}\left(\sum_{\ell\in[k]}a_{\ell}\,\omega_{\ell,n}\,m_{\ell}\right)^{2}-h\sum_{\ell\in[k]}\,\omega_{\ell,n}\,m_{\ell}+\frac{1}{\beta}I_{n}(m),\quad
m\in\Gamma_{n}^{\mathcal{P}},$

where $E_{n}(m)$ is defined in (2.7). Moreover, define

(2.16) $\displaystyle\bar{r}_{n}(m,m^{\prime})$
$\displaystyle=\mathrm{e}^{-\beta n[E_{n}(m^{\prime})-E_{n}(m)]_{+}}$
$\displaystyle\qquad\times\sum_{\ell\in[k]}|A_{\ell,n}|\left[\frac{1-m_{\ell}}{2}\,\mathds{1}(m^{\prime}=m^{\ell,+})+\frac{1+m_{\ell}}{2}\,\mathds{1}(m^{\prime}=m^{\ell,-})\right].$

With this notation, we can write the _mesoscopic measure_
$\mathcal{Q}_{n}(\cdot)=\mu_{n}\circ m_{n}^{-1}(\cdot)$ on
$\Gamma_{n}^{\mathcal{P}}$, with $\mu_{n}$ defined in (1.3), as

(2.17)
$\mathcal{Q}_{n}(m)=\mu_{n}(\mathcal{S}_{n}[m])=\frac{1}{Z_{n}}\mathrm{e}^{-\beta
nF_{n}(m)},\qquad m\in\Gamma_{n}^{\mathcal{P}},$

and so the Dirichlet form in (2.13) becomes

(2.18)
$\mathcal{E}_{\mathcal{S}_{n}}(h,h)=\frac{1}{2}\sum_{m\in\Gamma_{n}^{\mathcal{P}}}\mathcal{Q}_{n}(m)\,\sum_{m^{\prime}\in\Gamma_{n}^{\mathcal{P}}}\bar{r}_{n}(m,m^{\prime})\,\big{[}\,\bar{h}(m)-\bar{h}(m^{\prime})\big{]}^{2}.$

###  2.3. Gradients and Hessians

Denote the Cramér entropy by

(2.19)
$I_{\mathbf{C}}(x)=\frac{1-x}{2}\log\left(\frac{1-x}{2}\right)+\frac{1+x}{2}\log\left(\frac{1+x}{2}\right).$

Define

(2.20)
$\bar{I}_{n}(m)=\sum_{\ell\in[k]}\omega_{\ell,n}I_{\mathbf{C}}(m_{\ell}).$

Since $\left|{A_{\ell,n}}\right|=[1+o_{n}(1)]\,\omega_{\ell}n$, we can use
Stirling’s formula $N!=[1+o_{N}(1)]\,N^{N}\mathrm{e}^{-N}\sqrt{2\pi N}$ to
obtain

(2.21)
$I_{n}(m)=\bar{I}_{n}(m)+\sum_{\ell\in[k]}\frac{1}{2n}\log\left(\frac{\pi(1-m_{\ell}^{2})|A_{\ell,n}|}{2}\right)+o(n^{-1})=\bar{I}_{n}(m)+O(n^{-1}\log
n),$

where the error term is _uniform_ in $m\in\Gamma_{n}^{\mathcal{P}}$. For
$\ell,\bar{\ell}\in[k]$, we compute

(2.22) $\frac{\partial\bar{I}_{n}(m)}{\partial
m_{\ell}}=\frac{\omega_{\ell,n}}{2}\log\left(\frac{1+m_{\ell}}{1-m_{\ell}}\right)$

and

(2.23)
$\displaystyle\frac{\partial^{2}\bar{I}_{n}(m)}{\partial{m_{\ell}}\partial{m_{\bar{\ell}}}}$
$\displaystyle=0,\qquad\ell\neq\bar{\ell},$
$\displaystyle\frac{\partial^{2}\bar{I}_{n}(m)}{\partial{m_{\ell}}^{2}}$
$\displaystyle=\frac{\omega_{\ell,n}}{1-m_{\ell}^{2}}.$

Recalling (2.7), we compute

(2.24) $\begin{split}\frac{\partial E_{n}(m)}{\partial
m_{\ell}}&=-a_{\ell}\,\omega_{\ell,n}\left(\sum_{\ell^{\prime}\in[k]}a_{\ell^{\prime}}\omega_{\ell^{\prime},n}\,m_{\ell^{\prime}}\right)-\omega_{\ell,n}h.\end{split}$

Define

(2.25)
$\bar{F}_{n}(m)=E_{n}(m)+\frac{1}{\beta}\bar{I}_{n}(m)=-\frac{1}{2}\left(\sum_{\ell^{\prime}\in[k]}a_{\ell^{\prime}}\,\omega_{\ell^{\prime},n}\,m_{\ell^{\prime}}\right)^{2}-h\sum_{\ell^{\prime}\in[k]}\omega_{\ell^{\prime},n}m_{\ell^{\prime}}+\frac{1}{\beta}\bar{I}_{n}(m).$

######  Remark 2.1.

By (2.21), $F_{n}(m)=\bar{F}_{n}(m)+O(n^{-1}\log n)$, where $F_{n}$ is defined
in (2.15). $\spadesuit$

For $m\in[-1,1]^{k}$, define

(2.26)
$\begin{split}F_{\beta,h}(m)&=-\frac{1}{2}\left(\sum_{\ell\in[k]}a_{\ell}\,\omega_{\ell}\,m_{\ell}\right)^{2}-h\sum_{\ell\in[k]}\,\omega_{\ell}\,m_{\ell}+\frac{1}{\beta}\sum_{\ell\in[k]}\omega_{\ell}I_{\mathbf{C}}(m_{\ell}),\end{split}$

which corresponds to $F_{\beta,h}(m)=\lim_{n\to\infty}F_{n}(m)$ for
$m\in\Gamma_{n}^{\mathcal{P}}$. Compute

(2.27) $\frac{\partial\bar{F}_{n}(m)}{\partial
m_{\ell}}=\omega_{\ell,n}\left[\frac{1}{2\beta}\log\left(\frac{1+m_{\ell}}{1-m_{\ell}}\right)-a_{\ell}\left(\sum_{\ell^{\prime}\in[k]}a_{\ell^{\prime}}\,\omega_{\ell^{\prime},n}\,m_{\ell^{\prime}}\right)-h\right]$

and

(2.28)
$\displaystyle\frac{\partial^{2}\bar{F}_{n}(m)}{\partial{m_{\ell}}\,\partial{m_{\ell^{\prime}}}}$
$\displaystyle=-a_{\ell}\,\omega_{\ell,n}\,a_{\ell^{\prime}}\,\omega_{\ell^{\prime},n},\qquad\ell\neq\ell^{\prime},$
$\displaystyle\frac{\partial^{2}\bar{F}_{n}(m)}{\partial{m_{\ell}}^{2}}$
$\displaystyle=\frac{\omega_{\ell,n}}{\beta}\frac{1}{1-m_{\ell}^{2}}-a_{\ell}^{2}\,\omega_{\ell,n}^{2}.$

The same formulas apply for $I_{n},F_{n}$, with an error term $O(n^{-1})$.

###  2.4. Additional computation

We conclude with a computation that will be useful later on. Recalling
(2.11)), we write

(2.29) $\displaystyle n\big{[}\bar{I}_{n}(m^{\ell,\pm})-\bar{I}_{n}(m)\big{]}$
$\displaystyle=n\,\omega_{\ell,n}\Bigg{[}\frac{1+m_{\ell}}{2}\log\left(1\pm\frac{2}{n(1+m_{\ell})}\right)+\frac{1-m_{\ell}}{2}\log\left(1\mp\frac{2}{n(1-m_{\ell})}\right)\pm\frac{1}{n}A_{\ell,n}^{\pm}\Bigg{]}$
$\displaystyle=n\,\omega_{\ell,n}\left[\pm\frac{1}{n}\mp\frac{1}{n}+O(n^{-2})\pm\frac{1}{n}A_{\ell,n}^{\pm}\right]=\Delta_{\ell,n}^{\pm}+O(n^{-1}),$

where

(2.30)
$A_{\ell,n}^{\pm}=\log\left(1+\frac{2m_{\ell}\pm\tfrac{4}{n}}{1-m_{\ell}\mp\tfrac{2}{n}}\right),\qquad\Delta_{\ell,n}^{\pm}=\omega_{\ell,n}A_{\ell,n}^{\pm},$

and we use that $\omega_{\ell,n}\to\omega_{\ell}>0$. The same formula applies
for $I_{n}$ with an error term of order $O(n^{-1})$, and hence

(2.31)
$n\big{[}{I}_{n}(m^{\ell,\pm})-{I}_{n}(m)\big{]}=\Delta_{\ell,n}^{\pm}+O(n^{-1}).$

Note that $\Delta_{\ell,n}^{\pm}=O(1)$ because $\omega_{\ell,n}$ and
$A_{\ell,n}^{\pm}$ are order 1. Therefore, using (2.7), we get

(2.32)
$\begin{split}n\left[E_{n}(m^{\ell,\pm})-E_{n}(m)\right]&=n\left[F_{n}(m^{\ell,\pm})-F_{n}(m)\right]+\frac{1}{\beta}n\left[I_{n}(m^{\ell,\pm})-I_{n}(m)\right]\\\
&=n\left[F_{n}(m^{\ell,\pm})-F_{n}(m)\right]+\frac{1}{\beta}\Delta_{\ell,n}^{\pm}+O(n^{-1}).\end{split}$

##  3\. Metastable regime

Section 3.1 identifies the stationary points of $\bar{F}_{n}$. Section 3.2
identifies the metastable regime. Section 3.3 provides details on the
$1$-dimensional metastable landscape.

###  3.1. Stationary points of $\bar{F}_{n}$ and $F_{\beta,h}$

By (2.27), the critical points $m=(m_{\ell})_{\ell\in[k]}$ of $\bar{F}_{n}$
solve the system of equations (with $\omega_{\ell,n}\neq 0$)

(3.1) $0=\frac{\partial\bar{F}_{n}(m)}{\partial
m_{\ell}}=\omega_{\ell,n}\left[\frac{1}{2\beta}\log\left(\frac{1+m_{\ell}}{1-m_{\ell}}\right)-a_{\ell}\left(\sum_{\ell^{\prime}\in[k]}a_{\ell^{\prime}}\,\omega_{\ell^{\prime},n}\,m_{\ell^{\prime}}\right)-h\right],\quad\ell\in[k].$

Hence

(3.2)
$\frac{1}{2}\log\left(\frac{1+m_{\ell}}{1-m_{\ell}}\right)=\beta\left[a_{\ell}\left(\sum_{\ell^{\prime}\in[k]}a_{\ell^{\prime}}\,\omega_{\ell^{\prime},n}\,m_{\ell^{\prime}}\right)+h\right].$

Since $\mathrm{arctanh\,}x=\tfrac{1}{2}\log\frac{1+x}{1-x}$, $x\in(-1,+1)$,
(3.2) can be rewritten as

(3.3)
$m_{\ell}=\tanh\left(\beta\left[a_{\ell}\left(\sum_{\ell^{\prime}\in[k]}a_{\ell^{\prime}}\,\omega_{\ell^{\prime},n}\,m_{\ell^{\prime}}\right)+h\right]\right),\qquad\ell\in[k].$

Similarly, the critical points $m=(m_{\ell})_{\ell\in[k]}$ of $F_{\beta,h}$
solve the deterministic equation

(3.4)
$m_{\ell}=\tanh\left(\beta\left[a_{\ell}\left(\sum_{\ell^{\prime}\in[k]}a_{\ell^{\prime}}\,\omega_{\ell^{\prime}}\,m_{\ell^{\prime}}\right)+h\right]\right),\qquad\ell\in[k].$

Note that this can also be obtained directly from (3.3) after replacing
$\omega_{\ell,n}$ by its mean value $\omega_{\ell}$.

###  3.2. Metastable regime

We are interested in identifying the metastable regime, i.e., the set of pairs
$(\beta,h)$ for which $F_{\beta,h}$ has more than one minimum. Put

(3.5) $K=K(m)=\sum_{\ell\in[k]}a_{\ell}\,\omega_{\ell}\,m_{\ell}.$

From the characterisation of the critical points of $F_{\beta,h}$ in (3.4) it
follows that

(3.6) $K=T_{\beta,h,\mathcal{P}}(K),\qquad
T_{\beta,h,\mathcal{P}}(K)=\sum_{\ell\in[k]}a_{\ell}\,\omega_{\ell}\,\tanh(\beta[a_{\ell}K+h]).$

Note that any critical point $m=(m_{\ell})_{\ell\in[k]}\in[-1,1]^{k}$ of
$F_{\beta,h}$ is uniquely determined by $K(m)\in\mathbb{R}$. Consequently, the
problem of solving the $k$-dimensional system in (3.4) can be reduced to
solving the $1$-dimensional equation (3.6). Recalling Hypothesis 1(2), the
system is in the metastable regime if and only if (3.6) has more than one
solution that is not tangent to the diagonal.

Compute

(3.7) $\displaystyle T^{\prime}_{\beta,h,\mathcal{P}}(K)$
$\displaystyle=\beta\sum_{\ell\in[k]}a^{2}_{\ell}\,\omega_{\ell}\,\big{(}1-\tanh^{2}(\beta[a_{\ell}K+h])\big{)},$
$\displaystyle T^{\prime\prime}_{\beta,h,\mathcal{P}}(K)$
$\displaystyle=-2\beta^{2}\sum_{\ell\in[k]}a^{3}_{\ell}\,\omega_{\ell}\,\tanh(\beta[a_{\ell}K+h])\,\big{(}1-\tanh^{2}(\beta[a_{\ell}K+h])\big{)}.$

For $h=0$, the system is metastable when

(3.8) $\beta>\frac{1}{\sum_{\ell\in[k]}a_{\ell}^{2}\,\omega_{\ell}},$

in which case $T_{\beta,h,\mathcal{P}}$ has a unique inflection point at
$K=0$, implying that (3.6) has three solutions $K\in\\{-K^{*},0,+K^{*}\\}$
with $K^{*}>0$. Otherwise (3.6) has only one solution $K=0$.

We proceed with the more interesting case $h>0$.

####  3.2.1. Number of solutions

######  Lemma 3.1 (Number of solutions).

For $h>0$, the number of critical points of $F_{\beta,h}$, i.e., solutions of
(3.6), varies in $\\{1,3,\dots,2\ell+1\\}$, where $\ell\in[k]$ and $2\ell-1$
is the number of inflection points of $F_{\beta,h}$.

###### Proof.

For $h>0$, $\lim_{K\to\infty}T_{\beta,h,\mathcal{P}}^{\prime\prime}(K)<0$ and
$\lim_{K\to-\infty}T_{\beta,h,\mathcal{P}}^{\prime\prime}(K)>0$. This implies
that $T_{\beta,h,\mathcal{P}}$ has at least one inflection point and that the
number of inflection points of $T_{\beta,h,\mathcal{P}}$ cannot be even: it
takes values in $\\{1,3,\dots,2k-1\\}$ depending on $\beta,h$ and
$\mathcal{P}$. Consequently, if $2\ell-1$ ($\ell\in[k]$) is the number of
inflection points, then the cardinality of the solutions of (3.6) takes values
in $\\{1,3,\dots,2\ell+1\\}$ depending on $\beta,h$ and $\mathcal{P}$. ∎

We conjecture that for any finite $k$ there exist $\beta,h$ and $\mathcal{P}$
such that (3.6) has any number of solutions in the set $\\{1,3,\dots,2k+1\\}$.
We found numerical evidence for this fact for $k\in\\{2,3,4\\}$. See Appendix
B.

######  Lemma 3.2 (Unique strictly positive solution).

For every $\beta>0$ and $h>0$, (3.6) has exactly one strictly positive
solution.

###### Proof.

Put $W(K)=T_{\beta,h,\mathcal{P}}(K)-K$. The solutions of $\eqref{eq:Keq}$ are
the roots of $W$. Clearly, $W(0)>0$. Moreover, $\lim_{K\to\infty}W(K)=-\infty$
because
$\lim_{K\to\infty}T_{\beta,h,\mathcal{P}}(K)=\sum_{\ell\in[k]}a_{\ell}\,\omega_{\ell}>0$
is finite. Therefore, by continuity, a root of $W(K)$ exists in $(0,\infty)$.

Let $\tilde{K}$ be the smaller positive root of $W$. We prove that this root
is unique. Indeed, $W(K)^{\prime\prime}<0$ when $K\in[0,\infty)$, meaning that
$K\mapsto W(K)^{\prime}$ is strictly decreasing. By continuity, since $W(K)>0$
for all $K\in[0,\tilde{K})$, we have $W(\tilde{K})^{\prime}\leq 0$ and
$\lim_{K\to\infty}W(K)^{\prime}=-1$. Therefore, $W(K)^{\prime}<0$ for all
$K\in(\tilde{K},\infty)$, and so $W$ is strictly decreasing. Moreover,
$W(K)<W(\tilde{K})=0$ for all $K\in(\tilde{K},\infty)$. Thus, $\tilde{K}$ is
the only positive root of $W$. ∎

####  3.2.2. Metastable regime

######  Lemma 3.3 (Characterisation of the metastable regime).

(3.6) has at least three solutions not tangent to the diagonal if and only if
there exists $\bar{K}<0$ such that $\bar{K}>T_{\beta,h,\mathcal{P}}(\bar{K})$,
i.e.,

(3.9)
$\begin{split}&\bar{K}>\sum_{\ell\in[k]}a_{\ell}\,\omega_{\ell}\,\tanh(\beta[a_{\ell}\bar{K}+h]).\end{split}$

###### Proof.

Using Lemma 3.2, we see that (3.6) has at least three solutions if and only if
it has at least two strictly negative solutions. As above, we define
$W(K)=T_{\beta,h,\mathcal{P}}(K)-K$. The solutions of (3.6) are the roots of
$W$. Now, _assume_ that there exists a $\bar{K}<0$ such that
$\bar{K}>T_{\beta,h,\mathcal{P}}(\bar{K})$. Since $W(\bar{K})<0$ and $W(0)>0$,
$W(K)$ has a root in $(\bar{K},0)$, implying that (3.6) has at least one
solution in $(\bar{K},0)$. Moreover, since
$\lim_{K\to-\infty}T_{\beta,h,\mathcal{P}}(K)=-\sum_{\ell\in[k]}a_{\ell}\,\omega_{\ell}$
is finite, we have $\lim_{K\to-\infty}W(K)=\infty$. Because $W(\bar{K})<0$, it
follows that $W$ has at least one root in $(-\infty,\bar{K})$. With the same
argument it can be shown that the negative roots of $W$ are always even. The
opposite implication is trivial. ∎

######  Remark 3.4.

Applying the intermediate value theorem to the derivative of
$W(K)=T_{\beta,h,\mathcal{P}}(K)-K$, we get that if the condition in Lemma 3.3
is satisfied, then there exists a $\bar{K}<0$ such that
$T_{\beta,h,\mathcal{P}}^{\prime}(\bar{K})=1$ and
$\bar{K}>T_{\beta,h,\mathcal{P}}(\bar{K})$. $\spadesuit$

######  Theorem 3.5 (Metastable regime).

Define, as in (1.14),

(3.10) $\beta_{c}=\frac{1}{\sum_{\ell\in[k]}a^{2}_{\ell}\,\omega_{\ell}}.$

The metastable regime is

(3.11) $\beta\in(\beta_{c},\infty),\qquad h\in\big{[}0,h_{c}(\beta)\big{)},$

with $\beta\mapsto\beta h_{c}(\beta)$ non-decreasing on $[\beta_{c},\infty)$.
Furthermore, if the support of $\mathcal{P}$ is put into increasing order,
i.e., $a_{1}<a_{2}<\dots<a_{k}$, then

(3.12)
$\lim_{\beta\to\infty}h_{c}(\beta)=\min_{\ell\in[k]^{*}}\left(\sum_{\ell^{\prime}=\ell}^{k}a_{\ell}\,a_{\ell^{\prime}}\,\omega_{\ell^{\prime}}-\sum_{\ell^{\prime}=1}^{\ell-1}a_{\ell}\,a_{\ell^{\prime}}\,\omega_{\ell^{\prime}}\right),$

where the minimum is over all $\ell\in[k]$ such that the quantity between
brackets is positive.

###### Proof.

Recalling Lemma 3.3, we look for conditions for the existence of a $K<0$
satisfying (3.9). If such a $K$ exists, then by Remark 3.4 there exists a
$\bar{K}<0$ satisfying (3.9) such that
$T_{\beta,h,\mathcal{P}}^{\prime}(\bar{K})=1$, which reads

(3.13)
$\sum_{\ell\in[k]}a^{2}_{\ell}\,\omega_{\ell}\,\tanh^{2}(\beta[a_{\ell}\bar{K}+h])=\sum_{\ell\in[k]}a^{2}_{\ell}\,\omega_{\ell}-\frac{1}{\beta}.$

Since the right-hand side of (3.13) is positive, it admits solutions only if

(3.14)
$\frac{1}{\beta}<\sum_{\ell\in[k]}a^{2}_{\ell}\,\omega_{\ell}=\frac{1}{\beta_{c}}.$

Therefore, (3.14) is a necessary condition for the metastable regime.

Now assume (3.14). Since $\tanh x\sim x$, $x\to 0$, for
$\left|{K}\right|\ll\beta(\max_{\ell\in[k]}a_{\ell})^{-1}$ and $h\downarrow
0$, we have

(3.15)
$\begin{split}K&=T_{\beta,h,\mathcal{P}}(K)=\sum_{\ell\in[k]}a_{\ell}\,\omega_{\ell}\,\tanh(\beta[a_{\ell}K+h])\sim\sum_{\ell\in[k]}a_{\ell}\,\omega_{\ell}\,\beta[a_{\ell}K+h],\end{split}$

which reads

(3.16)
$K\sim-\left(\sum_{\ell\in[k]}a_{\ell}\,\omega_{\ell}\right)\left(\frac{1}{\beta_{c}}-\frac{1}{\beta}\right)^{-1}h$

and proves the existence of a negative solution. A positive solution is
guaranteed by Lemma 3.2. The existence of a third (strictly negative) solution
of (3.4), for every $\beta>\beta_{c}$ and for $h\downarrow 0$, follows as in
the proof of Lemma 3.3. Therefore, the lower bound on $\beta_{c}$ is sharp.

Since $h\mapsto T_{\beta,h,\mathcal{P}}(K)$ is strictly increasing for every
fixed $\beta>0$ and $K\in\mathbb{R}$, there exists a unique critical curve
$\beta\mapsto h_{c}(\beta)$ such that the system is metastable for $0\leq
h<h_{c}(\beta)$ and not metastable for $h\geq h_{c}(\beta)$. We know that
$h_{c}(\beta)>0$ for $\beta>\beta_{c}$. By passing to the parametrisation
$g=h\beta$, we get that $\beta\mapsto T_{\beta,g,\mathcal{P}}(K)$ is strictly
decreasing for every $g$ and for every $K\in\mathbb{R}$, from which it follows
that $\beta\mapsto g_{c}(\beta)=\beta h_{c}(\beta)$ is non-decreasing.

We next focus on the limit of $h_{c}(\beta)$ as $\beta\to\infty$. By Lemma
3.3, we may focus on the existence of $\bar{K}$ satisfying (3.9). In the limit
as $\beta\to\infty$, $\tanh(\beta[a_{\ell}\bar{K}+h])\to
2\Theta_{-h/a_{\ell}}(\bar{K})-1$, where $\Theta_{x}(\cdot)$ is the Heaviside
function centred in $x$. Thus, for all $\ell\in[k+1]$,

(3.17)
$\lim_{\beta\to\infty}\sum_{\ell^{\prime}\in[k]}a_{\ell^{\prime}}\,\omega_{\ell^{\prime}}\,\tanh(\beta[a_{\ell^{\prime}}K+h])=-\sum_{\ell^{\prime}=\ell}^{k}a_{\ell^{\prime}}\,\omega_{\ell^{\prime}}+\sum_{\ell^{\prime}=1}^{\ell-1}a_{\ell^{\prime}}\,\omega_{\ell^{\prime}},\qquad
K\in\left(-\frac{h}{a_{\ell-1}},-\frac{h}{a_{\ell}}\right),$

and, for all $\ell\in[k]$,

(3.18)
$\lim_{\beta\to\infty}\sum_{\ell^{\prime}\in[k]}a_{\ell^{\prime}}\,\omega_{\ell^{\prime}}\,\tanh(\beta[a_{\ell^{\prime}}K+h])=-\sum_{\ell^{\prime}=\ell+1}^{k}a_{\ell^{\prime}}\,\omega_{\ell^{\prime}}+\sum_{\ell^{\prime}=1}^{\ell-1}a_{\ell^{\prime}}\,\omega_{\ell^{\prime}},\qquad
K=-\frac{h}{a_{\ell}},$

where we set $-\frac{h}{a_{0}}=-\infty$ and $-\frac{h}{a_{k+1}}=\infty$. Thus,
for $\bar{K}\in\left(-\frac{h}{a_{\ell-1}},-\frac{h}{a_{\ell}}\right)$, (3.9)
can be written as

(3.19)
$\bar{K}>-\sum_{\ell^{\prime}=\ell}^{k}a_{\ell^{\prime}}\,\omega_{\ell^{\prime}}+\sum_{\ell^{\prime}=1}^{\ell-1}a_{\ell^{\prime}}\,\omega_{\ell^{\prime}}.$

Therefore, (3.9) has a solution if and only if there exists an $\ell\in[k]$
such that

(3.20)
$-\sum_{\ell^{\prime}=\ell}^{k}a_{\ell^{\prime}}\,\omega_{\ell^{\prime}}+\sum_{\ell^{\prime}=1}^{\ell-1}a_{\ell^{\prime}}\,\omega_{\ell^{\prime}}<-\frac{h}{a_{\ell}},$

in which case a solution $\bar{K}$ of (3.9) exists in

(3.21)
$\left(-\sum_{\ell^{\prime}=\ell}^{k}a_{\ell^{\prime}}\,\omega_{\ell^{\prime}}+\sum_{\ell^{\prime}=1}^{\ell-1}a_{\ell^{\prime}}\,\omega_{\ell^{\prime}},-\frac{h}{a_{\ell}}\right).$

Note that the quantity between brackets in (3.12) is always positive for
$\ell=1$. Thus, the minimum is always finite.

The proof is complete after we show why we may drop the case where
$\bar{K}=-\frac{h}{a_{\ell}}$ for some $\ell\in[k]$. In this case the
condition for $\bar{K}$ to satisfy (3.9) is

(3.22)
$-\sum_{\ell^{\prime}=\ell+1}^{k}a_{\ell^{\prime}}\,\omega_{\ell^{\prime}}+\sum_{\ell^{\prime}=1}^{\ell-1}a_{\ell^{\prime}}\,\omega_{\ell^{\prime}}<-\frac{h}{a_{\ell}},$

which implies (3.20). Thus, if $\bar{K}=\frac{-h}{a_{\ell}}$ satisfies (3.9),
then also some other $K$ in (3.21) satisfies (3.9). Therefore, the condition
in (3.20) is equivalent to having metastability. ∎

######  Lemma 3.6 (Re-entrant crossover).

The function $\beta\mapsto h_{c}(\beta)$ is not necessarily non-decreasing.

###### Proof.

In Appendix C we provide an example of $\beta\mapsto h_{c}(\beta)$ that is not
increasing. ∎

####  3.2.3. Bounds on the inflection points and on the critical curve

######  Lemma 3.7 (Bounds on inflection points).

All solutions of $T_{\beta,h,\mathcal{P}}^{\prime\prime}(K)=0$ are contained
in the interval

(3.23)
$\left[-\frac{h}{\min_{\ell\in[k]}a_{\ell}},-\frac{h}{\max_{\ell\in[k]}a_{\ell}}\right].$

In particular, they are all strictly negative.

###### Proof.

If $K>-\frac{h}{\max_{\ell\in[k]}a_{\ell}}$, then
$\tanh(\beta[a_{\ell}K+h])>0$ for all $\ell\in[k]$, which implies
$T_{\beta,h,\mathcal{P}}^{\prime\prime}(K)<0$. If
$K<-\frac{h}{\min_{\ell\in[k]}a_{\ell}}$, then $\tanh(\beta[a_{\ell}K+h])<0$
for all $\ell\in[k]$, which implies
$T_{\beta,h,\mathcal{P}}^{\prime\prime}(K)>0$. ∎

######  Lemma 3.8 (Upper bound on $h_{c}$).

$\sup_{\beta\in(\beta_{c},\infty)}h_{c}(\beta)<\left(\max_{\ell\in[k]}a_{\ell}\right)\sum_{\ell\in[k]}a_{\ell}\,\omega_{\ell}$.

###### Proof.

Use Lemma 3.3 to characterise the metastable regime and Remark 3.4. We _claim_
that if a solution $\bar{K}$ of (3.9) with
$T_{\beta,h,\mathcal{P}}^{\prime}(\bar{K})=1$ exists, then it must be negative
and strictly less than an inflection point. Using this fact, together with
Lemma 3.7 and the inequality in (3.9), we obtain a necessary upper bound on
$h$:

(3.24)
$\begin{split}\sum_{\ell\in[k]}a_{\ell}\,\omega_{\ell}\,\tanh(\beta[a_{\ell}\bar{K}+h])<-\frac{h}{\max_{\ell\in[k]}a_{\ell}}.\end{split}$

Using that $\tanh(\beta[a_{\ell}\bar{K}+h])>-1$, we conclude the proof.

We are left to prove the claim. By Lemma 3.7, all inflection points are
negative, and $T_{\beta,h,\mathcal{P}}^{\prime\prime}(K)<0$ for $K\geq 0$.
Assume, by contradiction, that $T_{\beta,h,\mathcal{P}}^{\prime\prime}(K)<0$
for all $K\in(\bar{K},\infty)$. Then $T_{\beta,h,\mathcal{P}}^{\prime}$ is
strictly decreasing. Therefore, $T_{\beta,h,\mathcal{P}}^{\prime}(K)<1$ for
all $K\in(\bar{K},\infty)$, which implies
$T_{\beta,h,\mathcal{P}}(K)-T_{\beta,h,\mathcal{P}}(0)<K$. Since
$T_{\beta,h,\mathcal{P}}(0)>0$, there exists a $\tilde{K}\in(\bar{K},0)$ such
that $T_{\beta,h,\mathcal{P}}(\tilde{K})>0>\tilde{K}$. Thus,
$T_{\beta,h,\mathcal{P}}(\tilde{K})-T_{\beta,h,\mathcal{P}}(0)>\tilde{K}$,
which contradicts what we have proved for all $K\in(\bar{K},\infty)$. ∎

###  3.3. Quasi 1-dimensional landscape

Given $K\in\mathbb{R}$, by standard saddle point approximation, the leading
order of

(3.25) $-\frac{1}{\beta
n}\log\mu_{n}\big{(}\\{\sigma\colon\,K_{n}(m_{n}(\sigma))=K\\}\big{)}$

turns out to be the function $G_{n}\colon\mathbb{R}\to\mathbb{R}$ defined by

(3.26) $G_{n}(K)=\inf_{m\colon\,K_{n}(m)=K}\bar{F}_{n}(m).$

Recalling definitions (2.25) and (3.5), using Lagrange multipliers and
integrating the condition $K_{n}(m)=K$, we obtain

(3.27) $G_{n}(K)=-\frac{1}{2}K^{2}-\frac{\log
2}{\beta}-\inf_{t\in\mathbb{R}}\left(Kt+\sum_{\ell\in[k]}\frac{\omega_{\ell,n}}{\beta}\log\cosh\left[\beta(h-ta_{\ell})\right]\right).$

######  Lemma 3.9 (Alternative characterisation for the critical points).

  1. (1)

If $m^{*}$ is a (not maximal) critical point for $F_{n}$, then $K_{n}(m^{*})$
is a critical point for $G_{n}$.

  2. (2)

If $K$ is a critical point for $G_{n}$, then
$m^{*}=(m^{*}_{\ell})_{\ell\in[k]}$ with
$m^{*}_{\ell}=\tanh\left(\beta\left[a_{\ell}K+h\right]\right)$ (recall (3.3))
is a critical point for $F_{n}$.

  3. (3)

$F_{n}(m^{*})=G_{n}(K_{n}(m^{*}))$ for any (not maximal) critical point
$m^{*}$.

###### Proof.

Similar to [3, Lemma 7.4]. ∎

We have already seen that $K_{n}(m)$ fully determines any critical value $m$
of $F_{n}$, and is useful to order them. Lemma 3.9 exhibits the one-
dimensional structure underlying the metastable landscape and provides a tool
to describe the nature of the critical points of $F_{n}$.

######  Remark 3.10.

The above results extend to the limit $n\to\infty$: replace $F_{n}$ by
$F_{\beta,h}$ and $G_{n}$ by $G_{\beta,\ell}$, obtained after replacing
$\omega_{\ell,n}$ by $\omega_{\ell}$ in (3.27), and $K_{n}(\cdot)$ by
$K(\cdot)$. $\spadesuit$

##  4\. Approximation of the Dirichlet form near the saddle point

In this section we approximate the Dirichlet form associated with the coarse-
grained dynamics near the saddle point. This is a key step to obtain capacity
estimates in the following section. Further details and examples on the
techniques we use here can be found in [7, Chapters 9, 10 and 14].

Section 4.1 introduces some key quantities that are needed to express the
mesoscopic measure. Section 4.2 introduces an approximate mesoscopic measure
that leads to an approximate dynamics. Section 4.3 approximates the harmonic
functions associated with this dynamics. Section 4.4 computes an approximate
Dirichlet form. Section 4.5 uses the latter to approximate the full Dirichlet
form.

###  4.1. Key quantities

Let $\mathbf{m}_{n}=(\mathbf{m}_{\ell,n})_{\ell\in[k]}$ and
$\mathbf{t}_{n}=(\mathbf{t}_{\ell,n})_{\ell\in[k]}$ in
$\Gamma_{n}^{\mathcal{P}}$ be a local minimum of $F_{n}$ and the correspondent
saddle point, respectively, as defined in Section 1.3.4. Note that both
$\mathbf{m}_{n}$ and $\mathbf{t}_{n}$ satisfy (3.3). Consider the
neighbourhood of $\mathbf{t}_{n}$ defined by

(4.1)
$\mathcal{D}_{n}=\left\\{m\in\Gamma_{n}^{\mathcal{P}}\colon\,d(m,\mathbf{t}_{n})\leq
C^{\prime}n^{-1/2}\log^{1/2}n\right\\},$

where $d$ is the Euclidean distance and $C^{\prime}\in(0,\infty)$ is a
constant. Abbreviate the Hessian of $F_{n}$

(4.2) $\mathbb{A}_{n}(m)=(\nabla^{2}F_{n})(m),\qquad
m\in\Gamma_{n}^{\mathcal{P}},$

and put

(4.3) $\mathbb{A}_{n}=\mathbb{A}_{n}(\mathbf{t}_{n}).$

By (2.28),

(4.4) $\displaystyle(\mathbb{A}_{n}(m))_{\ell,\ell^{\prime}}$
$\displaystyle=-a_{\ell}\,\omega_{\ell,n}\,a_{\ell^{\prime}}\,\omega_{\ell^{\prime},n}+O(n^{-1}),\qquad\ell\neq\ell^{\prime},$
$\displaystyle(\mathbb{A}_{n}(m))_{\ell,\ell}$
$\displaystyle=\frac{\omega_{\ell,n}}{\beta}\frac{1}{1-m_{\ell}^{2}}-a_{\ell}^{2}\,\omega_{\ell,n}^{2}+O(n^{-1})=\frac{1}{\beta}\frac{\partial^{2}\bar{I}_{n}(m)}{\partial{m_{\ell}}^{2}}-a_{\ell}^{2}\,\omega_{\ell,n}^{2}+O(n^{-1}).$

Moreover,

(4.5)
$\begin{split}\det\mathbb{A}_{n}(m)&=\sum_{\ell\in[k]}\left(\prod_{\begin{subarray}{c}\ell^{\prime}\in[k],\,\ell^{\prime}\neq\ell\end{subarray}}\frac{1}{\beta}\frac{\omega_{\ell^{\prime},n}}{1-m_{\ell^{\prime}}^{2}}\right)\left(\frac{1}{\beta}\frac{\omega_{\ell,n}}{1-m_{\ell}^{2}}-a_{\ell}^{2}\,\omega_{\ell,n}^{2}\right)+O(n^{-1})\\\
&=\left(\prod_{\begin{subarray}{c}\ell^{\prime}\in[k]\end{subarray}}\frac{1}{\beta}\frac{\omega_{\ell^{\prime},n}}{1-m_{\ell^{\prime}}^{2}}\right)\sum_{\ell\in[k]}\left(1-{\beta}\frac{1-m_{\ell}^{2}}{\omega_{\ell,n}}a_{\ell}^{2}\,\omega_{\ell,n}^{2}\right)+O(n^{-1})\\\
&=\left(1-\sum_{\ell\in[k]}\beta\,a_{\ell}^{2}\,\omega_{\ell,n}[1-m_{\ell}^{2}]\right)\prod_{\begin{subarray}{c}\ell^{\prime}\in[k]\end{subarray}}\frac{1}{\beta}\frac{\omega_{\ell^{\prime},n}}{1-m_{\ell^{\prime}}^{2}}[1+O(n^{-1})].\end{split}$

###  4.2. Approximate dynamics and Dirichlet form

For any two vectors $\mathbf{v},\mathbf{w}\in\mathbb{R}^{k}$, let
$\left\langle{\mathbf{v}},{\mathbf{w}}\right\rangle$ denote their scalar
product. For any $k\times k$ matrix $\mathbf{M}$ and any
$\mathbf{v}\in\mathbb{R}^{k}$, let $\mathbf{M}\cdot\mathbf{v}$ denote their
matrix product, as $\mathbf{v}$ was in $\mathbb{R}^{k}\times 1$.

For $m\in\mathcal{D}_{n}$, define

(4.6) $\tilde{\mathcal{Q}}_{n}(m)=\frac{1}{Z_{n}}\exp\left[-\tfrac{\beta
n}{2}\big{\langle}[m-\mathbf{t}_{n}],\mathbb{A}_{n}\cdot[m-\mathbf{t}_{n}]\big{\rangle}\right]\exp\left[-\beta
nF_{n}(\mathbf{t}_{n})\right],$

and

(4.7) $\displaystyle\tilde{r}_{n}\big{(}m,m^{\prime}\big{)}=$
$\displaystyle\begin{cases}\bar{r}_{n}\big{(}\mathbf{t}_{n},\mathbf{t}_{n}^{\ell,+}\big{)},&m^{\prime}=m^{\ell,+},\\\\[5.69046pt]
\bar{r}_{n}\big{(}\mathbf{t}_{n}^{\ell,-},\mathbf{t}_{n}\big{)}\frac{\tilde{\mathcal{Q}}_{n}(m^{\ell,-})}{\tilde{\mathcal{Q}}_{n}(m)},&m^{\prime}=m^{\ell,-},\\\\[5.69046pt]
0,&\text{else,}\end{cases}$

where $\bar{r}_{n}$ is defined in (2.16). The transition rates $\tilde{r}_{n}$
define a random dynamics on $\mathcal{D}_{n}$ that is reversible with respect
to the mesoscopic measure $\tilde{\mathcal{Q}}_{n}$. The corresponding
_Dirichlet form_ is

(4.8)
$\tilde{\mathcal{E}}_{\mathcal{D}_{n}}(u,u)=\sum_{m\in\mathcal{D}_{n}}\tilde{\mathcal{Q}}_{n}(m)\sum_{\ell\in[k]}\tilde{r}_{n}\big{(}m,m^{\ell,+}\big{)}\left[u(m)-u(m^{\ell,+})\right]^{2},$

where $u$ is a test function on $\mathcal{D}_{n}$. Put

(4.9)
$r_{\ell}=\tilde{r}_{n}\big{(}m,m^{\ell,+}\big{)}=\bar{r}_{n}\big{(}\mathbf{t}_{n},\mathbf{t}_{n}^{\ell,+}\big{)}.$

Using (2.7) and (2.16), we get

(4.10)
$r_{\ell}={\left|{A_{\ell,n}}\right|}\,\frac{1-\mathbf{t}_{\ell,n}}{2}\,\exp\left[-2\beta\left(-h-a_{\ell}\left(\frac{a_{\ell}}{n}+\sum_{\ell^{\prime}\in[k]}a_{\ell^{\prime}}\omega_{\ell^{\prime},n}\mathbf{t}_{\ell^{\prime},n}\right)\right)_{+}\right].$

####  4.2.1. Approximation estimates

Next we estimate how close the pairs $(\bar{r}_{n},\tilde{r}_{n})$ and
$(\mathcal{Q}_{n},\tilde{\mathcal{Q}}_{n})$ are. By Taylor expansion around
$\mathbf{t}_{n}$, we have

(4.11)
$F_{n}(m)-F_{n}(\mathbf{t}_{n})=\tfrac{1}{2}\Big{\langle}[m-\mathbf{t}_{n}],\mathbb{A}_{n}\cdot[m-\mathbf{t}_{n}]\Big{\rangle}+O\big{(}d(m,\mathbf{t}_{n})^{3}\big{)}.$

In particular,

(4.12)
$\begin{split}&F_{n}(\mathbf{t}_{n}^{\ell,\pm})-F_{n}(\mathbf{t}_{n})=\frac{1}{2}\frac{4}{\left|{A_{\ell,n}}\right|^{2}}(\mathbb{A}_{n})_{\ell,{\ell}}+O\big{(}\left|{A_{\ell,n}}\right|^{-3}\big{)}\\\
&=\frac{2}{n^{2}\omega_{\ell,n}^{2}}\left[\frac{\omega_{\ell,n}}{\beta}\frac{1}{1-\mathbf{t}_{\ell,n}^{2}}-a_{\ell}^{2}\,\omega_{\ell,n}^{2}+o\left((n\,\omega_{\ell,n})^{-1}\right)\right]+O\big{(}(n\,\omega_{\ell,n})^{-3}\big{)}\\\
&=\frac{2}{n^{2}}\left(\frac{1}{\beta\,\omega_{\ell,n}(1-\mathbf{t}_{\ell,n}^{2})}-a_{\ell}^{2}\right)+O\big{(}(n\,\omega_{\ell,n})^{-3}\big{)},\end{split}$

where the second equality uses (4.4). Moreover, for $m\in\mathcal{D}_{n}$
($\mathrm{e}_{\ell}$ is the unitary vector in $\mathbb{R}^{k}$ whose $\ell$-th
component is non-zero),

(4.13) $\begin{split}&F_{n}(m^{\ell,\pm})-F_{n}(m)\\\
&=\left\langle\left[\pm\tfrac{2}{\left|{A_{\ell,n}}\right|}\,\mathrm{e}_{\ell}\right],\mathbb{A}_{n}\cdot[m-\mathbf{t}_{n}]\right\rangle+\frac{1}{2}\left\langle\left[\pm\tfrac{2}{\left|{A_{\ell,n}}\right|}\,\mathrm{e}_{\ell}\right],\mathbb{A}_{n}\cdot\left[\pm\tfrac{2}{\left|{A_{\ell,n}}\right|}\,\mathrm{e}_{\ell}\right]\right\rangle+O\big{(}d(m,\mathbf{t}_{n})^{3}\big{)}\\\
&=\pm\frac{2}{\left|{A_{\ell,n}}\right|}\sum_{\ell^{\prime}\in[k]}(\mathbb{A}_{n})_{\ell,{\ell^{\prime}}}(m_{\ell^{\prime}}-\mathbf{t}_{\ell^{\prime},n})+\frac{2}{\left|{A_{\ell,n}}\right|^{2}}(\mathbb{A}_{n})_{\ell,{\ell}}+O\big{(}d(m,\mathbf{t}_{n})^{3}\big{)}\\\
&=\left(\pm\frac{2}{n\omega_{\ell,n}}(m_{\ell}-\mathbf{t}_{\ell,n})+\frac{2}{n^{2}\omega_{\ell,n}^{2}}\right)\left(\frac{\omega_{\ell,n}}{\beta}\frac{1}{1-\mathbf{t}_{\ell,n}^{2}}-a_{\ell}^{2}\,\omega_{\ell,n}^{2}+o(n^{-1})\right)\\\
&\quad\pm\frac{2}{n\omega_{\ell,n}}\sum_{\ell^{\prime}\in[k],\,\ell^{\prime}\neq\ell}(-a_{\ell}\,\omega_{\ell,n}\,a_{{\ell^{\prime}}}\,\omega_{{\ell^{\prime}},n})(m_{\ell^{\prime}}-\mathbf{t}_{\ell^{\prime},n})+O\big{(}n^{-3/2}\log^{3/2}n\big{)}\\\
&=\mp\frac{2}{n}\sum_{\ell^{\prime}\in[k]}a_{\ell}\,a_{{\ell^{\prime}}}\,\omega_{{\ell^{\prime}},n}(m_{\ell^{\prime}}-\mathbf{t}_{\ell^{\prime},n})\pm\frac{2(m_{\ell}-\mathbf{t}_{\ell,n})}{\beta\,n(1-\mathbf{t}_{\ell,n}^{2})}+O\big{(}n^{-3/2}\log^{3/2}n\big{)},\end{split}$

where the third equality uses (4.4). For $m\in\mathcal{D}_{n}$, we have
$d(m,\mathbf{t}_{n})^{3}=O(n^{-3/2}\log^{3/2}n)$. Therefore, combining (2.17),
(4.6) and (4.11), we have

(4.14)
$\left|{\frac{\mathcal{Q}_{n}(m)}{\tilde{\mathcal{Q}}_{n}(m)}-1}\right|\leq
C^{\prime\prime}n^{-1/2}\log^{3/2}n,\qquad m\in\mathcal{D}_{n},$

for some $C^{\prime\prime}\in(0,\infty)$ constant. Using (2.16) and (2.32), we
can write

(4.15) $\displaystyle\bar{r}_{n}(m,m^{\ell,\pm})$
$\displaystyle=\exp{\left[-\beta\left[n\left[F_{n}(m^{\ell,\pm})-F_{n}(m)\right]+\frac{1}{\beta}\Delta_{\ell,n}^{\pm}+O\left(n^{-1}\right)\right]_{+}\right]}\frac{1\mp
m_{\ell}}{2},$

where $\Delta_{\ell,n}^{\pm}$ is defined in (2.31).

Using (4.7), (4.12), (4.13) and (4.15), we find that, for all
$m\in\mathcal{D}_{n}$,

(4.16)
$\displaystyle\left|{\frac{\bar{r}_{n}\left(m,m^{\ell,+}\right)}{\tilde{r}_{n}\left(m,m^{\ell,+}\right)}-1}\right|=\left|{\frac{\bar{r}_{n}\left(m,m^{\ell,+}\right)}{\bar{r}_{n}\big{(}\mathbf{t}_{n},\mathbf{t}_{n}^{\ell,+}\big{)}}-1}\right|$
$\displaystyle=\left|{\frac{(1-m_{\ell})\,\exp{\left\\{-\left[I_{1}+O(n^{-1/2}\log^{3/2}n)+\Delta_{\ell,n}^{\pm}+o_{n}(1)\right]_{+}\right\\}}}{(1-\mathbf{t}_{\ell,n})\,\exp{\left\\{-\left[I_{2}+O(n^{-2}\,\omega_{\ell,n}^{-3})+\Delta_{\ell,n}^{\pm}+o_{n}(1)\right]_{+}\right\\}}}-1}\right|$
$\displaystyle=\left|{\frac{(1-m_{\ell})\,\exp{\left\\{-\left[I_{1}+\Delta_{\ell,n}^{\pm}+o_{n}(1)\right]_{+}\right\\}}}{(1-\mathbf{t}_{\ell,n})\,\exp{\left\\{-\left[\Delta_{\ell,n}^{\pm}+o_{n}(1)\right]_{+}\right\\}}}-1}\right|\leq
C^{\prime\prime\prime}n^{-1/2}\log^{1/2}n,$

where $C^{\prime\prime\prime}\in(0,\infty)$ is a constant and we abbreviate

(4.17) $\displaystyle I_{1}$
$\displaystyle=-2\beta\sum_{\ell^{\prime}\in[k]}a_{\ell}\,a_{{\ell^{\prime}}}\,\omega_{{\ell^{\prime}},n}(m_{\ell^{\prime}}-\mathbf{t}_{\ell^{\prime},n})+\frac{2(m_{\ell}-\mathbf{t}_{\ell,n})}{1-\mathbf{t}_{\ell,n}^{2}},$
$\displaystyle I_{2}$
$\displaystyle=\frac{2}{n}\left(\frac{1}{\omega_{\ell,n}(1-\mathbf{t}_{\ell,n}^{2})}-\beta\,a_{\ell}^{2}\right).$

Equations (4.14) and (4.16) are relevant for the following approximation.

###  4.3. Approximate harmonic function

Let $\mathbb{B}_{n}$ be the $k\times k$ matrix defined by

(4.18)
$(\mathbb{B}_{n})_{\ell\ell^{\prime}}=\frac{\sqrt{r_{\ell}r_{\ell^{\prime}}}}{n\,\omega_{\ell,n}\omega_{\ell^{\prime},n}}\,(\mathbb{A}_{n})_{\ell\ell^{\prime}},$

where $\mathbb{A}_{n}$ is defined in (4.3). Note that

(4.19)
$\det\mathbb{B}_{n}=(\det\mathbb{A}_{n})\,\prod_{\ell\in[k]}\frac{r_{\ell}}{n\,\omega_{\ell,n}^{2}}.$

Let $\gamma_{n}^{(\ell)}$, $\ell\in[k]$, be the eigenvalues of
$\mathbb{B}_{n}$, ordered in increasing order. Let
$\gamma_{n}=\gamma_{n}^{(1)}$ denote the unique negative eigenvalue of
$\mathbb{B}_{n}$, and $\hat{v}$ the corresponding unitary eigenvector. Define
$v=(v_{\ell})_{\ell\in[k]}$ by
$v_{\ell}=\hat{v}_{\ell}\frac{\omega_{\ell,n}\sqrt{n}}{\sqrt{r_{\ell}}}$.

######  Remark 4.1.

As in [7, Remark 10.4], it follows by Hypothesis 1 that $\mathbb{A}_{n}$ has
all strictly positive eigenvalues but one strictly negative. It can be seen
that the same property holds for the eigenvalues of $\mathbb{B}_{n}$.
$\spadesuit$

######  Lemma 4.2 (Eigenvalue).

The eigenvalue $\gamma_{n}$ is the unique solution of the equation

(4.20)
$\frac{1}{n}\sum_{\ell\in[k]}\frac{a_{\ell}^{2}}{\frac{1}{n\beta\,\omega_{\ell,n}(1-\mathbf{t}_{\ell,n}^{2})}-\frac{\gamma_{n}}{r_{\ell}}}=1+O(n^{-1}).$

###### Proof.

We follow the line of proof of [7, Lemma 14.9], using the last point in
Hypothesis 1. In our case, [7, Eq. (14.7.12)] reads

(4.21)
$-\frac{1}{n}a_{\ell}\sqrt{r_{\ell}}\sum_{\ell^{\prime}\in[k]}a_{\ell^{\prime}}\sqrt{r_{\ell^{\prime}}}u_{\ell^{\prime}}+\left(r_{\ell}\frac{1}{n\beta\omega_{\ell,n}(1-\mathbf{t}_{\ell,n})^{2}}-\gamma_{n}\right)u_{\ell}+O(n^{-1})=0,\qquad\ell\in[k].$

∎

######  Remark 4.3.

As in [7, Lemma 14.9], since the left-hand side of (4.20) is increasing in
$\gamma_{n}$ for $\gamma_{n}\geq 0$, a negative solution of (4.20) exists if
and only if

(4.22)
$\beta\sum_{\ell\in[k]}a_{\ell}^{2}\,\omega_{\ell,n}(1-\mathbf{t}_{\ell,n}^{2})>1.$

Using (4.5), (4.22) holds if and only if $\det\mathbb{A}_{n}<0$. By Remark 4.1
the latter holds true.  $\spadesuit$

Define $f\colon\,\mathbb{R}\to[0,1]$ as

(4.23) $f(x)=\sqrt{\frac{(-\gamma_{n})\beta
n}{2\pi}}\int_{-\infty}^{x}\mathrm{e}^{-\tfrac{1}{2}(-\gamma_{n})\beta
nu^{2}}\mathrm{d}u$

and $g\colon\,\mathbb{R}^{k}\to[0,1]$ as

(4.24) $g(m)=f(\left\langle{v},{m-\mathbf{t}_{n}}\right\rangle).$

Recall the definition of $\mathcal{M}_{n}(\mathbf{m}_{n})$ given in (1.17).

Let $W_{0}$ be a strip in $\Gamma_{n}^{\mathcal{P}}$ of width
$Cn^{-1/2}\log^{1/2}n$ such that $\mathbf{t}_{n}\in W_{0}$,
$\mathcal{M}_{n}(\mathbf{m}_{n})\cap W_{0}$ is empty and $W_{0}^{c}$ consists
in two non-neighbouring parts: $W_{1}$ containing $\mathbf{m}_{n}$ and $W_{2}$
containing $\mathcal{M}_{n}(\mathbf{m}_{n})$. Moreover, we require that, for
some fixed constant $c>1$,
$W_{0}\cap\mathcal{D}_{n}^{c}\subseteq\\{m\in\Gamma_{n}^{\mathcal{P}}\colon
F_{n}(m)>F_{n}(\mathbf{t}_{n})+cn^{-1}\log n\\}$. Define

(4.25) $\tilde{g}(m)=\begin{cases}0,&m\in W_{1},\\\ 1,&m\in W_{2},\\\
g(x),&m\in W_{0}\cap\mathcal{D}_{n},\\\ 0,&m\in
W_{0}\cap\mathcal{D}_{n}^{c}.\end{cases}$

By choosing $W_{0}$ and $\mathcal{D}_{n}$ suitably we have, for $m\sim
m^{\prime}$ (i.e., $\bar{r}_{n}(m,m^{\prime})>0$) and $c\in(0,\infty)$ large
enough (coming from the definition of $W_{0}$),

(4.26)
$\displaystyle\mathcal{Q}_{n}(m)\leq\mathcal{Q}_{n}(\mathbf{t}_{n})n^{-c\beta},$
$\displaystyle m\in W_{0}\cap\mathcal{D}_{n}^{c},$ (4.27)
$\displaystyle(\tilde{g}(m)-\tilde{g}(m^{\prime}))^{2}\,\bar{r}_{n}(m,m^{\prime})\mathcal{Q}_{n}(m)\leq\mathcal{Q}_{n}(\mathbf{t}_{n})n^{-c\beta},$
$\displaystyle m\in W_{0}\cap\mathcal{D}_{n},m^{\prime}\in W_{0}^{c}.$

###  4.4. Computation of the approximate Dirichlet form

In this section we follow [7, Sections 10.2.2–10.2.3] to approximate
$\tilde{\mathcal{E}}_{\mathcal{D}_{n}}(g,g)$ defined in (4.8). As in [7, Eq.
(10.2.24)], for $m\in\mathcal{D}_{n}$ and $\ell\in[k]$ such that
$m^{\ell,+}\in D_{n}$, compute

(4.28) $\begin{split}&g(m^{\ell,+})-g(m)\\\
&\quad=\frac{2}{\left|{A_{\ell,n}}\right|}v_{\ell}f^{\prime}(\left\langle{v},{m-\mathbf{t}_{n}}\right\rangle)+\frac{2}{\left|{A_{\ell,n}}\right|^{2}}v_{\ell}^{2}f^{\prime\prime}(\left\langle{v},{m-\mathbf{t}_{n}}\right\rangle)+\frac{4}{3\left|{A_{\ell,n}}\right|^{3}}v_{\ell}^{3}f^{\prime\prime\prime}(\left\langle{v},{\tilde{m}-\mathbf{t}_{n}}\right\rangle)\\\
&\quad=v_{\ell}\sqrt{\frac{2(-\gamma_{n})\beta}{\pi
n\,\omega_{\ell,n}^{2}}}\exp\left(-\frac{\beta
n}{2}(-\gamma_{n})\left\langle{v},{m-\mathbf{t}_{n}}\right\rangle^{2}\right)\\\
&\quad\qquad\times\left(1-\frac{1}{\omega_{\ell,n}}v_{\ell}(-\gamma_{n})\beta\left\langle{v},{m-\mathbf{t}_{n}}\right\rangle+O(\omega_{\ell,n}^{-2}\,n^{-1}\log
n)\right).\end{split}$

Recalling (4.8)–(4.9), we have

(4.29)
$\begin{split}&\tilde{\mathcal{E}}_{\mathcal{D}_{n}}(g,g)=\sum_{m\in\mathcal{D}_{n}}\tilde{\mathcal{Q}}_{n}(m)\sum_{\ell\in[k]}r_{\ell}\left[g(m^{\ell,+})-g(m)\right]^{2}\\\
&=\frac{1}{Z_{n}}\sum_{m\in\mathcal{D}_{n}}\exp\left[-\tfrac{\beta
n}{2}\big{\langle}[m-\mathbf{t}_{n}],\mathbb{A}_{n}\cdot[m-\mathbf{t}_{n}]\big{\rangle}\right]\mathrm{e}^{-\beta
nF_{n}(\mathbf{t}_{n})}\\\
&\quad\times\sum_{\ell\in[k]}r_{\ell}v_{\ell}^{2}\,\frac{2(-\gamma_{n})\beta}{\pi
n\,\omega_{\ell,n}^{2}}\exp\left(-\beta
n(-\gamma_{n})\left\langle{v},{m-\mathbf{t}_{n}}\right\rangle^{2}\right)\\\
&\qquad\times\left(1-v_{\ell}(-\gamma_{n})\beta\left\langle{v},{m-\mathbf{t}_{n}}\right\rangle+O(\omega_{\ell,n}^{-2}\,n^{-1}\log
n)\right)^{2}\\\
&=\frac{1}{Z_{n}}\frac{2(-\gamma_{n})\beta}{\pi}\sum_{m\in\mathcal{D}_{n}}\exp\left[-\tfrac{\beta
n}{2}\big{\langle}[m-\mathbf{t}_{n}],\mathbb{A}_{n}\cdot[m-\mathbf{t}_{n}]\big{\rangle}\right]\mathrm{e}^{-\beta
nF_{n}(\mathbf{t}_{n})}\\\ &\quad\times\exp\left(-\beta
n(-\gamma_{n})\left\langle{v},{m-\mathbf{t}_{n}}\right\rangle^{2}\right)\left[1+O\left(\omega_{\ell,n}^{-1}\,n^{-1/2}\log^{1/2}n\,\right)\right]\\\
&=\frac{1}{Z_{n}}\frac{2(-\gamma_{n})\beta}{\pi}\left[1+O\left(\omega_{\ell,n}^{-1}\,\,n^{-1/2}\log^{1/2}n\,\right)\right]\mathrm{e}^{-\beta
nF_{n}(\mathbf{t}_{n})}\left(\prod_{\ell\in[k]}\frac{\left|{A_{\ell,n}}\right|}{2}\right)\\\
&\quad\times\int_{\mathcal{D}_{n}}\mathrm{d}m\exp\left[-\tfrac{\beta
n}{2}\big{\langle}[m-\mathbf{t}_{n}],\mathbb{A}_{n}\cdot[m-\mathbf{t}_{n}]\big{\rangle}\right]\exp\left(-\beta
n(-\gamma_{n})\left\langle{v},{m-\mathbf{t}_{n}}\right\rangle^{2}\right)\\\
&=\frac{1}{Z_{n}}\mathrm{e}^{-\beta
nF_{n}(\mathbf{t}_{n})}\frac{(-\gamma_{n})n}{\sqrt{[-\det\mathbb{A}_{n}]}}\left(\frac{\pi
n}{2\beta}\right)^{\frac{k}{2}-1}\left(\prod_{\ell\in[k]}\frac{1}{\omega_{\ell,n}}\right)\left[1+O\left(\omega_{\ell,n}^{-1}\,n^{-1/2}\log^{1/2}n\,\right)\right],\end{split}$

where we use [7, Eq. (10.2.33)] with $\varepsilon=\frac{1}{\beta n}$ and
$d=k$. Here $\tfrac{1}{2}\left|{A_{\ell,n}}\right|$ is the inverse of the step
in the $\ell$–direction, while in [7, Eq. (10.2.33)] the step is
$\varepsilon$.

######  Remark 4.4.

Note that

(4.30)
$\tilde{\mathcal{E}}_{\mathcal{D}_{n}}(g,g)=\tilde{\mathcal{E}}_{\mathcal{D}_{n}}(\tilde{g},\tilde{g})\,[1+o(1)]$

because $\tilde{g}(m)=g(m)\,[1+o(1)]$ for all $m\in
W_{0}^{c}\cap\mathcal{D}_{n}$. The latter can be proved by approximating the
Gaussian integral by $0$ or $1$ when
$\left\langle{v},{m-\mathbf{t}_{n}}\right\rangle$ is proportional to
$-n^{-1/2}\log^{1/2}n$ or $n^{-1/2}\log^{1/2}n$, respectively. $\spadesuit$

###  4.5. Final Dirichlet form approximation

We are now ready to compare ${\mathcal{E}}_{\mathcal{S}_{n}}$ with
$\tilde{\mathcal{E}}_{\mathcal{D}_{n}}$. Let $h\colon\mathcal{S}_{n}\to[0,1]$
be such that $h(\sigma)=\tilde{g}(m_{n}(\sigma))$, $\sigma\in\mathcal{S}_{n}$.
We split the sum in (2.18) into four subsets of
$\Gamma_{n}^{\mathcal{P}}\times\Gamma_{n}^{\mathcal{P}}$: $m\in
W_{0}\cap\mathcal{D}_{n}^{c}$, $m^{\prime}\in\Gamma_{n}^{\mathcal{P}}$; $m\in
W_{0}\cap\mathcal{D}_{n}$, $m^{\prime}\in W_{1}$; $m\in
W_{0}\cap\mathcal{D}_{n}$, $m^{\prime}\in W_{2}$; $m\in
W_{0}\cap\mathcal{D}_{n}$, $m^{\prime}\in W_{0}\cap\mathcal{D}_{n}$. Then,
using (4.25)–(4.27), we obtain

(4.31)
$\begin{split}{\mathcal{E}}_{\mathcal{S}_{n}}(h,h)&=O(n^{-c\beta})+\frac{1}{2}\sum_{m\in
W_{0}\cap\mathcal{D}_{n}}\sum_{m^{\prime}\in
W_{0}\cap\mathcal{D}_{n}}\mathcal{Q}_{n}(m)\,\bar{r}_{n}\big{(}m,m^{\prime}\big{)}\,\big{[}\,\tilde{g}(m)-\tilde{g}(m^{\prime})\big{]}^{2}.\end{split}$

Using (4.14) and (4.16), we obtain

(4.32)
$\begin{split}{\mathcal{E}}_{\mathcal{S}_{n}}(h,h)&=O(n^{-c\beta})+\frac{1}{2}\sum_{m\in
W_{0}\cap\mathcal{D}_{n}}\left[1+O\big{(}n^{-1/2}\log^{3/2}n\big{)}\right]\tilde{\mathcal{Q}}_{n}(m)\\\
&\quad\times\sum_{m^{\prime}\in
W_{0}\cap\mathcal{D}_{n}}\,\left(1+O\big{(}n^{-1/2}\log^{1/2}n\big{)}\right)\tilde{r}_{n}\big{(}m,m^{\prime}\big{)}\,\big{[}\,\tilde{g}(m)-\tilde{g}(m^{\prime})\big{]}^{2}\\\
&=\left[1+O\big{(}n^{-1/2}\log^{1/2}n\big{)}\right]\frac{1}{2}\sum_{m,m^{\prime}\in
W_{0}\cap\mathcal{D}_{n}}\tilde{\mathcal{Q}}_{n}(m)\tilde{r}_{n}\big{(}m,m^{\prime}\big{)}\,\big{[}\,\tilde{g}(m)-\tilde{g}(m^{\prime})\big{]}^{2}\\\
&=\left[1+O\big{(}n^{-1/2}\log^{1/2}n\big{)}\right]\frac{1}{2}\sum_{m,m^{\prime}\in\mathcal{D}_{n}}\tilde{\mathcal{Q}}_{n}(m)\tilde{r}_{n}\big{(}m,m^{\prime}\big{)}\,\big{[}\,\tilde{g}(m)-\tilde{g}(m^{\prime})\big{]}^{2}\\\
&=\tilde{\mathcal{E}}_{\mathcal{D}_{n}}(\tilde{g},\tilde{g})\left[1+O\big{(}n^{-1/2}\log^{1/2}n\big{)}\right]\\\
&=[1+o_{n}(1)]\,\frac{1}{Z_{n}}\exp\left[-\beta
nF_{n}(\mathbf{t}_{n})\right]\frac{(-\gamma_{n})n}{\sqrt{[-\det\mathbb{A}_{n}]}}\left(\frac{\pi
n}{2\beta}\right)^{\frac{k}{2}-1}\left(\prod_{\ell\in[k]}\frac{1}{\omega_{\ell,n}}\right),\end{split}$

where the third equality follows from (4.25)–(4.27) together with (4.14), and
the last equality follows from (4.29)–(4.30).

##  5\. Capacity and valley estimates

Section 5.1 provides sharp asymptotic upper bounds and lower bounds on the
capacity of the metastable pair between which the crossover is being
considered. These estimates use the results of the Section 4 together with the
Dirichlet Principle and the Berman-Konsowa Principle, which are variational
representations of capacity. Section 5.2 provides a sharp asymptotic estimate
for the mesoscopic measure of the valleys of the minima of $F_{n}$, which
leads to a sharp asymptotic estimate for $F_{n}$ inside this valley.

###  5.1. Capacity estimates

Given a Markov process $(x_{t})_{t\geq 0}$ with state space $S$, a key
quantity in the potential-theoretic approach to metastability is the
_capacity_ $\hbox{\rm cap}(A,B)$ of two disjoint subsets $A,B$ of $S$. This is
defined by (see [7, Eq. (7.1.39)])

(5.1) $\hbox{\rm cap}(A,B)=\sum_{x\in
A}\mu(x)\mathbb{P}_{x}(\tau_{B}<\tau_{A}),$

where $\mu$ is the invariant measure and $\mathbb{P}_{x}$ is the probability
distribution of the Markov process starting in $x$.

Recall that $\mathcal{M}_{n}$ is the set of local minima of $F_{n}$.

######  Proposition 5.1 (Asymptotics of the capacity).

Let $\mathbf{m}_{n}=(\mathbf{m}_{\ell,n})_{\ell\in[k]}\in\mathcal{M}_{n}$ and
$M_{n}\subset\mathcal{M}_{n}\backslash\mathbf{m}_{n}$, such that the gate
$\mathcal{G}(\mathbf{m}_{n},M_{n})$ consists of a unique point
$\mathbf{t}_{n}=(\mathbf{t}_{\ell,n})_{\ell\in[k]}$. Suppose that
$\beta\in(\beta_{c},\infty)$ and $h\in[0,h_{c}(\beta))$. Then, as
$n\to\infty$,

(5.2) $\displaystyle\hbox{\rm
cap}(\mathcal{S}_{n}[\mathbf{m}_{n}],\mathcal{S}_{n}[M_{n}])$
$\displaystyle\quad=[1+o_{n}(1)]\,\frac{1}{Z_{n}}\mathrm{e}^{-\beta
nF_{n}(\mathbf{t}_{n})}\frac{(-\gamma_{n})\,n}{\sqrt{[-\det(\mathbb{A}_{n}(\mathbf{t}_{n}))]}}\left(\frac{\pi
n}{2\beta}\right)^{\frac{k}{2}-1}\left(\prod_{\ell\in[k]}\frac{1}{\omega_{\ell,n}}\right).$

######  Remark 5.2.

Proposition 5.1 holds for any subset
$M_{n}\subseteq\mathcal{M}_{n}\backslash\mathbf{m}_{n}$, separated from
$\mathbf{m}_{n}$ by $\mathbf{t}_{n}$, independently on the values of $F_{n}$
on $M_{n}$. $\spadesuit$

####  5.1.1. Upper bound: Dirichlet Principle

An important characterisation of the capacity between two disjoint sets is
given by the _Dirichlet principle_. For our quantity of interest this states
that

(5.3) $\hbox{\rm
cap}(\mathcal{S}_{n}[\mathbf{m}_{n}],\mathcal{S}_{n}[M_{n}])=\inf_{u\in\tilde{\mathcal{H}}}{\mathcal{E}}_{\mathcal{S}_{n}}(u,u),$

where $\tilde{\mathcal{H}}$ is the set of functions from $\mathcal{S}_{n}$ to
$[0,1]$ that are equal to $1$ on $\mathcal{S}_{n}[\mathbf{m}_{n}]$ and $0$ on
$\mathcal{S}_{n}[M_{n}]$.

Given that, by assumption,
$\mathcal{G}(\mathbf{m}_{n},M_{n})=\\{\mathbf{t}_{n}\\}$, we use the Dirichlet
principle in (5.3) to obtain an upper bound on the capacity. We take as test
function $h\in\tilde{\mathcal{H}}$ defined in Section 4.5) and, using (4.32),
we obtain

(5.4) $\begin{split}&\hbox{\rm
cap}(\mathcal{S}_{n}[\mathbf{m}_{n}],\mathcal{S}_{n}[M_{n}])\leq{\mathcal{E}}_{\mathcal{S}_{n}}(h,h)\\\
&=[1+o_{n}(1)]\,\frac{1}{Z_{n}}\mathrm{e}^{-\beta
nF_{n}(\mathbf{t}_{n})}\frac{(-\gamma_{n})n}{\sqrt{[-\det(\mathbb{A}_{n}(\mathbf{t}_{n}))]}}\left(\frac{\pi
n}{2\beta}\right)^{\frac{k}{2}-1}\left(\prod_{\ell\in[k]}\frac{1}{\omega_{\ell,n}}\right).\end{split}$

####  5.1.2. Lower bound: Berman-Konsowa Principle

We first note that the process $(\sigma_{t})_{t\geq 0}$ is lumpable. Indeed,
the process $(m_{n}(\sigma_{t}))_{t\geq 0}$ is Markovian because the
Hamiltonian $H_{n}(\sigma)$ depends on $m_{n}(\sigma)$ only (see (2.6)).
Therefore, for $\mathbf{A}=\mathcal{S}_{n}[{A}]$ and
$\mathbf{B}=\mathcal{S}_{n}[{B}]$ with $A$ and $B$ disjoint subsets of
$\Gamma_{n}^{\mathcal{P}}$,

(5.5) $\hbox{\rm cap}(\mathbf{A},\mathbf{B})=\hbox{\rm cap}_{\Gamma}(A,B),$

where $\hbox{\rm cap}_{\Gamma}$ denotes the capacity for the process
$(m_{n}(\sigma_{t}))_{t\geq 0}$, i.e., the projection of the process
$(\sigma_{t})_{t\geq 0}$ on the magnetisation space
$\Gamma_{n}^{\mathcal{P}}$. We write $\mathbb{P}^{\Gamma}$ to denote the law
of $(m_{n}(\sigma_{t}))_{t\geq 0}$ induced by the law $\mathbb{P}$ of
$(\sigma_{t})_{t\geq 0}$. By the lumpability, we can focus on the dynamics on
$\Gamma_{n}^{\mathcal{P}}$.

Following the line of argument in [7, Section 10.3] (with
$\varepsilon=\frac{2}{n}$ and $d=k$), we obtain the lower bound

(5.6) $\begin{split}&\hbox{\rm
cap}(\mathcal{S}_{n}[\mathbf{m}_{n}],\mathcal{S}_{n}[M_{n}])=\hbox{\rm
cap}_{\Gamma}(\mathbf{m}_{n},M_{n})\geq\tilde{\mathcal{E}}_{\mathcal{D}_{n}}(\tilde{g},\tilde{g})\left[1+O(\,n^{-1/2}\log^{1/2}n)\right]\\\
&=\frac{1}{Z_{n}}\mathrm{e}^{-\beta
nF_{n}(\mathbf{t}_{n})}\frac{(-\gamma_{n})n}{\sqrt{[-\det(\mathbb{A}_{n}(\mathbf{t}_{n}))]}}\left(\frac{\pi
n}{2\beta}\right)^{\frac{k}{2}-1}\left(\prod_{\ell\in[k]}\frac{1}{\omega_{\ell,n}}\right)\left[1+o_{n}(1)\right],\end{split}$

where we use (4.29) and (4.30).

We sketch the proof. The main idea is to use the Berman-Konsowa Principle for
a suitable defective flow. More precisely, given disjoint subsets $A,B$ of the
state space, for any _defective loop-free unit flow_ $f_{A,B}$ from $A$ to $B$
with defect function $\delta$ (as defined in [7, Definition 9.2]), we can
estimate (see [7, Lemma 9.4], and notation therein)

(5.7) $\hbox{\rm cap}(A,B)\geq\prod_{i=1}^{M}\left(1+\left[\max_{y\in
A_{i}}\frac{\delta(y)}{\mathcal{F}(y)}\right]_{+}\right)^{-1}\sum_{\gamma}\mathbb{P}^{f_{A,B}}(\gamma)\left[\left(\sum_{(x,y)\in\gamma}\frac{f_{A,B}((x,y))}{\mu(x)p(x,y)}\right)^{-1}\right],$

where $[\cdot]_{+}$ denotes the positive part and $\gamma$ is a self-avoiding
path from $A$ to $B$. It turns out that, with a suitable choice of the flow
$f$, the product in the right-hand side of (5.7) is bounded from below by
$1+O(\,n^{-1/2}\log^{1/2}n)$, and the sum over $\gamma$ from below by
$\tilde{\mathcal{E}}_{\mathcal{D}_{n}}(\tilde{g},\tilde{g})[1+o_{n}(1)]$. This
proves (5.6).

We give a sketch of the test flow definition in our setting. Here
$A=\\{\mathbf{m}_{n}\\}$ and $B=M_{n}$. Let $v^{*}$ be the eigenvector
corresponding to the unique negative eigenvalue of the Hessian of $F_{n}$ at
the saddle point $\mathbf{t}_{n}$ (unique gate point in
$\mathcal{G}(\\{\mathbf{m}_{n}\\},M_{n})$). Let $G_{n}$ be the cylinder in
$\mathbb{R}^{k}$ intersected with $\Gamma_{n}^{\mathcal{P}}$, centred at
$\mathbf{t}_{n}$, with axis $v^{*}$, radius $\rho=C\,n^{-1/2}\log^{1/2}n$ and
length $\rho^{\prime}=C^{\prime}\,n^{-1/2}\log^{1/2}n$. We will denote by
$\partial_{B}G_{n}$ the base facing $B$ and by $\partial_{A}G_{n}$ the central
part of radius $C^{\prime\prime}\,n^{-1/2}\log^{1/2}n$ of the base facing $A$,
with $C^{\prime\prime}<C$. Choose the constants so that $G_{n}$ is contained
in $\mathcal{D}_{n}$ defined in (4.1).

We define a defective flow $f_{A,B}$ from $A$ to $B$ consisting of three
parts: $f_{A}$, a unitary flow from $A$ to $\partial_{A}G_{n}$; $f$, a
defective loop-free unit flow from $\partial_{A}G_{n}$ to $\partial_{B}G_{n}$
inside $G_{n}$; $f_{B}$, a unitary flow from $\partial_{B}G_{n}$ to $B$. This
choice implies that the sum over $\gamma$ in (5.7) is relevant only on the
paths entering $G_{n}$ in $\partial_{A}G_{n}$, exiting $G_{n}$ in
$\partial_{B}G_{n}$, and afterwards reaching $B$ without going back to
$G_{n}$. For this purpose we choose $f_{A}$ and $f_{B}$ such that
$f_{A}((x,y))$ and $f_{B}((x,y))$ are proportional to $\mathcal{Q}_{n}(x)$.
For $m\in G_{n}$ such that $m^{\ell,+}\in G_{n}$, define

(5.8)
$f((m,m^{\ell,+}))=\frac{\tilde{\mathcal{Q}}_{n}(m)r_{\ell}\left[g(m^{\ell,+})-g(m)\right]_{+}}{N(g)},$

where $g$ is defined in (4.24), $\tilde{\mathcal{Q}}_{n}$ in (4.6), $r_{\ell}$
in (4.9) and

(5.9)
$N(g)=\sum_{m\in\partial_{A}G_{n}}\sum_{\begin{subarray}{c}\ell\in[k]:\\\
m^{\ell,+}\in
G_{n}\end{subarray}}\tilde{\mathcal{Q}}_{n}(m)r_{\ell}\left[g(m^{\ell,+})-g(m)\right]_{+}.$

The contribution to the sum in brackets in (5.7) turns out to be negligible
outside $G_{n}$. Therefore, no further conditions on the flows $f_{A}$ and
$f_{B}$ are necessary, provided the total flow out of $A$ is $1$ and the total
flow $f_{A,B}$ is defective and loop-free.

###  5.2. Measure of the valley

In order to prove Theorem 1.1, we need the following estimate on the measure
of the valley of the minima of $F_{n}$. For
$\mathbf{m}_{n}\in\mathcal{M}_{n}$, let
$A(\mathbf{m}_{n})\subset\Gamma_{n}^{\mathcal{P}}$ be the valley of
$\mathbf{m}_{n}$ as defined in [7, Eq. (8.2.10)].

######  Lemma 5.3 (Gibbs weight of the valley).

Given $\mathbf{m}_{n}\in\mathcal{M}_{n}$,

(5.10)
$\mathcal{Q}_{n}(A(\mathbf{m}_{n}))=\frac{1}{Z_{n}}\frac{\exp\left(-\beta
nF_{n}(\mathbf{m}_{n})\right)}{\sqrt{\det(\mathbb{A}_{n}(\mathbf{m}_{n}))}}\left(\frac{n\pi}{2\beta}\right)^{\frac{k}{2}}\left(\prod_{\ell\in[k]}\frac{1}{\omega_{\ell,n}}\right)\left[1+O(\,n^{-1/2}\log^{3/2}n)\right],$

where $\mathcal{Q}_{n}$ is the mesoscopic measure defined in (2.17), and
$\mathbb{A}_{n}(\mathbf{m}_{n})$ is the $k\times k$ Hessian matrix defined in
(4.2).

###### Proof.

The proof follows that of [7, Lemma 10.12 and (10.2.33)]. The relevant
contribution to $\mathcal{Q}_{n}(A(\mathbf{m}_{n}))$ is given by the measure
of a ball $B_{\rho}$ of radius $\rho=C\,n^{-1/2}\log^{1/2}n$ centred in
$\mathbf{m}_{n}$, with $C$ constant, contained in $A(\mathbf{m}_{n})$. Indeed,
if $y\in A(\mathbf{m}_{n})$ and $d(\mathbf{m}_{n},y)>\rho$, then by Taylor
expansion of $F_{n}$ around $\mathbf{m}_{n}$ we have

(5.11) $\begin{split}\mathcal{Q}_{n}(y)&=\frac{1}{Z_{n}}\exp[-\beta
nF_{n}(y)]=\frac{1}{Z_{n}}\exp\left[-\beta
n[F_{n}(\mathbf{m}_{n})+c\,d(\mathbf{m}_{n},y)^{2}]\right]\\\
&\leq\frac{1}{Z_{n}}\exp\left[-\beta
n[F_{n}(\mathbf{m}_{n})+c\rho^{2}]\right]=\frac{n^{-\beta
cC^{2}}}{Z_{n}}\exp\left[-\beta nF_{n}(\mathbf{m}_{n})\right],\end{split}$

where $c$ is a constant. The condition $y\in A(\mathbf{m}_{n})$ is needed to
ensure that $F_{n}(y)>F_{n}(\mathbf{m}_{n})$, implying that $c$ is positive.
Therefore, we obtain the rough estimate

(5.12) $\mathcal{Q}_{n}(A(\mathbf{m}_{n})\backslash B_{\rho})\leq
n^{k}\frac{n^{-\beta cC^{2}}}{Z_{n}}\exp\left[-\beta
nF_{n}(\mathbf{m}_{n})\right],$

where we use that $|\Gamma_{n}^{\mathcal{P}}|\leq n^{k}$. The bound in (5.12)
is sufficient to show that $\mathcal{Q}_{n}(A(\mathbf{m}_{n})\backslash
B_{\rho})$ is negligible in $\mathcal{Q}_{n}(A(\mathbf{m}_{n}))$.

Compute

(5.13) $\begin{split}&Z_{n}\mathcal{Q}_{n}(A(\mathbf{m}_{n})\cap
B_{\rho})=Z_{n}\mathcal{Q}_{n}(B_{\rho})=Z_{n}\sum_{y\in
B_{\rho}}\mathcal{Q}_{n}(y)=\sum_{y\in B_{\rho}}\mathrm{e}^{-\beta
nF_{n}(y)}\\\ &=\mathrm{e}^{-\beta nF_{n}(\mathbf{m}_{n})}\sum_{y\in
B_{\rho}}\exp\left[-\frac{\beta
n}{2}\left\langle{y-\mathbf{m}_{n}},{(\mathbb{A}_{n}(\mathbf{m}_{n}))\cdot(y-\mathbf{m}_{n})}\right\rangle+O(n\rho^{3})\right]\\\
&=\mathrm{e}^{-\beta nF_{n}(\mathbf{m}_{n})}[1+O(n\rho^{3})]\sum_{y\in
B_{\rho}}\exp\left[-\frac{\beta
n}{2}\left\langle{y-\mathbf{m}_{n}},{(\mathbb{A}_{n}(\mathbf{m}_{n}))\cdot(y-\mathbf{m}_{n})}\right\rangle\right]\\\
&=\mathrm{e}^{-\beta
nF_{n}(\mathbf{m}_{n})}\left(\prod_{\ell\in[k]}\frac{\left|{A_{\ell,n}}\right|}{2}\right)[1+O(n\rho^{3})]\\\
&\times\int_{B_{\rho}}\mathrm{d}y\exp\left[-\frac{\beta
n}{2}\left\langle{y-\mathbf{m}_{n}},{(\mathbb{A}_{n}(\mathbf{m}_{n}))\cdot(y-\mathbf{m}_{n})}\right\rangle\right]\\\
&=\mathrm{e}^{-\beta
nF_{n}(\mathbf{m}_{n})}\left(\frac{n}{2}\right)^{k}\left(\prod_{\ell\in[k]}\frac{1}{\omega_{\ell,n}}\right)[1+O(n\rho^{3})]\left(\frac{2\pi}{n\beta}\right)^{\frac{k}{2}}\sqrt{\frac{1}{\det(\mathbb{A}_{n}(\mathbf{m}_{n}))}}\\\
&=\frac{\mathrm{e}^{-\beta
nF_{n}(\mathbf{m}_{n})}}{\sqrt{\det(\mathbb{A}_{n}(\mathbf{m}_{n}))}}\left(\frac{n\pi}{2\beta}\right)^{\frac{k}{2}}\left(\prod_{\ell\in[k]}\frac{1}{\omega_{\ell,n}}\right)[1+O(n\rho^{3})],\end{split}$

where we use the Taylor expansion

(5.14)
$F_{n}(y)=F_{n}(\mathbf{m}_{n})+\frac{1}{2}\left\langle{y-\mathbf{m}_{n}},{(\nabla^{2}F_{n})(\mathbf{m}_{n})\cdot(y-\mathbf{m}_{n})}\right\rangle+O(\rho^{3}),\qquad
y\in B_{\rho},$

and the approximation of the sum by an integral is correct up to an error
$1+O(\rho)$. In the last lines we approximated the Gaussian integral on
intervals $[-\rho,\rho]$ by the Gaussian integral on $\mathbb{R}$, with an
error $1+O(n^{-c})$. We conclude by looking at (5.12) and (5.13), and noting
that for $C$ large enough $\mathcal{Q}_{n}(A(\mathbf{m}_{n})\backslash
B_{\rho})$ is negligible compared to $\mathcal{Q}_{n}(A(\mathbf{m}_{n})\cap
B_{\rho})$. ∎

##  6\. Proof of the theorems

In this section we prove Theorems 1.1–1.3. Section 6.1 uses the asymptotics
for the capacity of the metastable pair from Section 5.1 and the asymptotics
for the mesoscopic measure from Section 5.2 to prove Theorem 1.1. Section 6.2
proves Theorem 1.2. Section 6.3 proves Theorem 1.3.

###  6.1. Average crossover time

Let us return to the notation of Theorem 1.1, where
$\mathbf{m}_{n}\in\mathcal{M}_{n}$ and
$\mathcal{M}_{n}(\mathbf{m}_{n})=\\{m\in\mathcal{M}_{n}\backslash\mathbf{m}_{n}\colon\,F_{n}(m)\leq
F_{n}(\mathbf{m}_{n})\\}$. To prove Theorem 1.1 we use the relation

(6.1)
$\mathbb{E}^{\Gamma}_{\mathbf{m}_{n}}(\tau_{\mathcal{M}_{n}(\mathbf{m}_{n})})=[1+o_{n}(1)]\,\frac{\mu(A(\mathbf{m}_{n}))}{\hbox{\rm
cap}_{\Gamma}(\mathbf{m}_{n},\mathcal{M}_{n}(\mathbf{m}_{n}))}.$

Because $F_{n}(m)\leq F_{n}(\mathbf{m}_{n})$ for all
$m\in\mathcal{M}_{n}(\mathbf{m}_{n})$, (6.1) follows from [7, Theorem 8.15]
after proving that $\mathcal{M}_{n}$ is a set of metastable points is the
sense of [7, Definition 8.2]. The latter follows along the lines of the proof
of [7, Theorem 10.6], where similar values of capacities and invariant
measures occur.

Using (6.1) in combination with Proposition 5.1 and Lemma 5.3, we obtain that,
for all $\sigma\in\mathcal{S}_{n}[\mathbf{m}_{n}]$,

(6.2)
$\begin{split}\mathbb{E}_{\sigma}(\tau_{\mathcal{S}_{n}[\mathcal{M}_{n}(\mathbf{m}_{n})]})&=[1+o_{n}(1)]\,\mathbb{E}^{\Gamma}_{\mathbf{m}_{n}}(\tau_{\mathcal{M}_{n}(\mathbf{m}_{n})})=\frac{\mathcal{Q}_{n}(A(\mathbf{m}_{n}))}{\hbox{\rm
cap}_{\Gamma}(\mathbf{m}_{n},\mathcal{M}_{n}(\mathbf{m}_{n}))}\\\
&=[1+o_{n}(1)]\,\frac{\mathcal{Q}_{n}(A(\mathbf{m}_{n}))}{\hbox{\rm
cap}(\mathcal{S}_{n}[\mathbf{m}_{n}],\mathcal{S}_{n}[\mathcal{M}_{n}(\mathbf{m}_{n})])}\\\
&=[1+o_{n}(1)]\,\frac{\frac{1}{Z_{n}}\frac{\exp\left(-\beta
nF_{n}(\mathbf{m}_{n})\right)}{\sqrt{\det(\mathbb{A}_{n}(\mathbf{m}_{n}))}}\left(\frac{n\pi}{2\beta}\right)^{\frac{k}{2}}\left(\prod_{\ell\in[k]}\frac{1}{\omega_{\ell,n}}\right)}{\frac{1}{Z_{n}}\exp\left[-\beta
nF_{n}(\mathbf{t}_{n})\right]\frac{(-\gamma_{n})n}{\sqrt{[-\det(\mathbb{A}_{n}(\mathbf{t}_{n}))]}}\left(\frac{\pi
n}{2\beta}\right)^{\frac{k}{2}-1}\left(\prod_{\ell\in[k]}\frac{1}{\omega_{\ell,n}}\right)}\\\
&=[1+o_{n}(1)]\,\sqrt{\frac{[-\det(\mathbb{A}_{n}(\mathbf{t}_{n}))]}{\det(\mathbb{A}_{n}(\mathbf{m}_{n}))}}\left(\frac{\pi}{2\beta(-\gamma_{n})}\right)\exp\left[\beta
n(F_{n}(\mathbf{t}_{n})-F_{n}(\mathbf{m}_{n}))\right],\end{split}$

where we use that the dynamics depends on the starting configuration
$\sigma\in\mathcal{S}_{n}[\mathbf{m}_{n}]$ only, through its level
magnetisations $m_{n}(\sigma)=\mathbf{m}_{n}$ (see (2.6)), and also use the
lumpability.

###  6.2. Exponential law

In this section we prove Theorem 1.2. Since the dynamics depends on the
starting configuration $\sigma\in\mathcal{S}_{n}[\mathbf{m}_{n}]$ through its
level magnetisation $m_{n}(\sigma)=\mathbf{m}_{n}$ only (see (2.6)), we have

(6.3)
$\begin{split}&\lim_{n\to\infty}\mathbb{P}_{\sigma}\left(\tau_{\mathcal{S}_{n}[\mathcal{M}_{n}(\mathbf{m}_{n})]}>t\,\mathbb{E}_{\sigma}\left[\tau_{\mathcal{S}_{n}[\mathcal{M}_{n}(\mathbf{m}_{n})]}\right]\right)=\lim_{n\to\infty}\mathbb{P}^{\Gamma}_{\mathbf{m}_{n}}\left(\bar{\tau}_{\mathcal{M}_{n}(\mathbf{m}_{n})}>t\,\mathbb{E}^{\Gamma}_{\mathbf{m}_{n}}\left[\bar{\tau}_{\mathcal{M}_{n}(\mathbf{m}_{n})}\right]\right),\end{split}$

where $\bar{\tau}$ is the hitting time of the process projected on
$\Gamma_{n}^{\mathcal{P}}$. Given the non-degeneracy hypothesis (Hypothesis 1
in Section 1.3.4) and the one-dimensional landscape analysis (in Section 3.3),
we can apply [7, Theorem 8.45] to the right-hand side of (6.3) and conclude
the proof.

###  6.3. Randomness of the exponent

In this section we prove Theorem 1.3. In particular, we compute
$F_{n}(\mathbf{t}_{n})-F_{n}(\mathbf{m}_{n})-[F_{\beta,h}(\mathbf{t})-F_{\beta,h}(\mathbf{m})]$
to leading order.

Recalling definitions (2.26) and (3.5), we have

(6.4)
$F_{\beta,h}(m)=-\frac{1}{2}K(m)^{2}-h\sum_{\ell\in[k]}\,\omega_{\ell}\,m_{\ell}+\frac{1}{\beta}\sum_{\ell\in[k]}\omega_{\ell}I_{\mathbf{C}}(m_{\ell}).$

Let
$\mathbf{m}=(\mathbf{m}_{\ell})_{\ell\in[k]},\mathbf{t}=(\mathbf{t}_{\ell})_{\ell\in[k]}\in[-1,1]^{k}$
be the critical points of $F_{\beta,h}$ closest to
$\mathbf{m}_{n},\mathbf{t}_{n}$ (i.e., the critical points of $F_{n}$ defined
above), respectively. Note that $\mathbf{m}$ and $\mathbf{t}$ satisfy (3.4),
while $\mathbf{m}_{n}$ and $\mathbf{t}_{n}$ satisfy (3.3). Using (2.21), we
get

(6.5)
$\begin{split}&F_{n}(\mathbf{t}_{n})-F_{\beta,h}(\mathbf{t}_{n})=-\frac{1}{2}[K_{n}(\mathbf{t}_{n})^{2}-K(\mathbf{t}_{n})^{2}]-h\sum_{\ell\in[k]}[\omega_{\ell,n}-\omega_{\ell}]\,\mathbf{t}_{\ell,n}\\\
&\quad+\frac{1}{\beta}\left[\sum_{\ell\in[k]}[\omega_{\ell,n}-\omega_{\ell}]I_{\mathbf{C}}(\mathbf{t}_{\ell,n})+\sum_{\ell\in[k]}\frac{1}{2n}\log\left(\frac{\pi(1-\mathbf{t}_{\ell,n}^{2})}{2}\right)\omega_{\ell,n}-\frac{k}{2n}+o\left(n^{-1}\right)\right]\end{split}$

and

(6.6)
$\begin{split}&F_{\beta,h}(\mathbf{t}_{n})-F_{\beta,h}(\mathbf{t})=-\frac{1}{2}[K(\mathbf{t}_{n})^{2}-K(\mathbf{t})^{2}]+\frac{1}{\beta}\sum_{\ell\in[k]}\omega_{\ell}[I_{\mathbf{C}}(\mathbf{t}_{\ell,n})-I_{\mathbf{C}}(\mathbf{t}_{\ell})].\end{split}$

By (3.2), we have

(6.7)
$\displaystyle\frac{1}{2}\log\left(\frac{1+\mathbf{t}_{\ell,n}}{1-\mathbf{t}_{\ell,n}}\right)$
$\displaystyle=\beta\left[a_{\ell}K_{n}(\mathbf{t}_{n})+h\right],$
$\displaystyle\frac{1}{2}\log\left(\frac{1+\mathbf{t}_{\ell}}{1-\mathbf{t}_{\ell}}\right)$
$\displaystyle=\beta\left[a_{\ell}K(\mathbf{t})+h\right].$

Thus,

(6.8)
$\begin{split}I_{\mathbf{C}}(\mathbf{t}_{\ell,n})-I_{\mathbf{C}}(\mathbf{t}_{\ell})&=(\mathbf{t}_{\ell,n}-\mathbf{t}_{\ell})I_{\mathbf{C}}^{\prime}(\mathbf{t}_{\ell})+O((\mathbf{t}_{\ell,n}-\mathbf{t}_{\ell})^{2})\\\
&=(\mathbf{t}_{\ell,n}-\mathbf{t}_{\ell})\frac{1}{2}\log\left(\frac{1+\mathbf{t}_{\ell}}{1-\mathbf{t}_{\ell}}\right)+O((\mathbf{t}_{\ell,n}-\mathbf{t}_{\ell})^{2})\\\
&=(\mathbf{t}_{\ell,n}-\mathbf{t}_{\ell})\beta\left[a_{\ell}K(\mathbf{t})+h\right]+O((\mathbf{t}_{\ell,n}-\mathbf{t}_{\ell})^{2}).\end{split}$

Moreover,

(6.9)
$\begin{split}&K(\mathbf{t}_{n})^{2}-K(\mathbf{t})^{2}=\sum_{\ell,\ell^{\prime}\in[k]}a_{\ell}\,a_{\ell^{\prime}}\,\omega_{\ell}\,\omega_{\ell^{\prime}}[\mathbf{t}_{\ell,n}\mathbf{t}_{\ell^{\prime},n}-\mathbf{t}_{\ell}\mathbf{t}_{\ell^{\prime}}]\\\
&=\sum_{\ell,\ell^{\prime}\in[k]}a_{\ell}\,a_{\ell^{\prime}}\,\omega_{\ell}\,\omega_{\ell^{\prime}}\left(\mathbf{t}_{\ell}[\mathbf{t}_{\ell^{\prime},n}-\mathbf{t}_{\ell^{\prime}}]+\mathbf{t}_{\ell^{\prime}}[\mathbf{t}_{\ell,n}-\mathbf{t}_{\ell}]+[\mathbf{t}_{\ell,n}-\mathbf{t}_{\ell}][\mathbf{t}_{\ell^{\prime},n}-\mathbf{t}_{\ell^{\prime}}]\right)\end{split}$

and

(6.10)
$\begin{split}&K_{n}(\mathbf{t}_{n})^{2}-K(\mathbf{t}_{n})^{2}=\sum_{\ell,\ell^{\prime}\in[k]}a_{\ell}\,a_{\ell^{\prime}}\,[\omega_{\ell,n}\,\omega_{\ell^{\prime},n}-\omega_{\ell}\,\omega_{\ell^{\prime}}]\mathbf{t}_{\ell,n}\mathbf{t}_{\ell^{\prime},n}\\\
&=\sum_{\ell,\ell^{\prime}\in[k]}a_{\ell}\,a_{\ell^{\prime}}\,\mathbf{t}_{\ell,n}\mathbf{t}_{\ell^{\prime},n}\left(\omega_{\ell}[\omega_{\ell^{\prime},n}-\omega_{\ell^{\prime}}]+\omega_{\ell^{\prime}}[\omega_{\ell,n}-\omega_{\ell}]+[\omega_{\ell,n}-\omega_{\ell}][\omega_{\ell^{\prime},n}-\omega_{\ell^{\prime}}]\right).\end{split}$

Similar equalities hold after we replace $\mathbf{t}$ by $\mathbf{m}$ and
$\mathbf{t}_{n}$ by $\mathbf{m}_{n}$. Using the previous computations, we
obtain

(6.11)
$\begin{split}&F_{n}(\mathbf{t}_{n})-F_{n}(\mathbf{m}_{n})-[F_{\beta,h}(\mathbf{t})-F_{\beta,h}(\mathbf{m})]\\\
&=F_{n}(\mathbf{t}_{n})-F_{\beta,h}(\mathbf{t}_{n})+F_{\beta,h}(\mathbf{t}_{n})-F_{\beta,h}(\mathbf{t})-[F_{n}(\mathbf{m}_{n})-F_{\beta,h}(\mathbf{m}_{n})+F_{\beta,h}(\mathbf{m}_{n})-F_{\beta,h}(\mathbf{m})]\\\
&=-\frac{1}{2}\sum_{\ell,\ell^{\prime}\in[k]}a_{\ell}\,a_{\ell^{\prime}}\\\
&\times\left[\mathbf{t}_{\ell,n}\mathbf{t}_{\ell^{\prime},n}-\mathbf{m}_{\ell,n}\mathbf{m}_{\ell^{\prime},n}\right]\left(\omega_{\ell}[\omega_{\ell^{\prime},n}-\omega_{\ell^{\prime}}]+\omega_{\ell^{\prime}}[\omega_{\ell,n}-\omega_{\ell}]+[\omega_{\ell,n}-\omega_{\ell}][\omega_{\ell^{\prime},n}-\omega_{\ell^{\prime}}]\right)\\\
&\quad-\frac{1}{2}\sum_{\ell,\ell^{\prime}\in[k]}a_{\ell}\,a_{\ell^{\prime}}\,\omega_{\ell}\,\omega_{\ell^{\prime}}\left[\mathbf{t}_{\ell,n}\mathbf{t}_{\ell^{\prime},n}-\mathbf{t}_{\ell}\mathbf{t}_{\ell^{\prime}}+\mathbf{m}_{\ell}\mathbf{m}_{\ell^{\prime}}-\mathbf{m}_{\ell,n}\mathbf{m}_{\ell^{\prime},n}\right]\\\
&\quad-h\sum_{\ell\in[k]}[\omega_{\ell,n}-\omega_{\ell}]\left[\mathbf{t}_{\ell,n}-\mathbf{m}_{\ell,n}\right]\\\
&\quad+\frac{1}{\beta}\sum_{\ell\in[k]}[\omega_{\ell,n}-\omega_{\ell}]\left[I_{\mathbf{C}}(\mathbf{t}_{\ell,n})-I_{\mathbf{C}}(\mathbf{m}_{\ell,n})\right]+\frac{1}{\beta}\sum_{\ell\in[k]}\frac{1}{2n}\log\left(\frac{1-\mathbf{t}_{\ell,n}^{2}}{1-\mathbf{m}_{\ell,n}^{2}}\right)\\\
&\quad+\frac{1}{\beta}\sum_{\ell\in[k]}\omega_{\ell}\left[I_{\mathbf{C}}(\mathbf{t}_{\ell,n})-I_{\mathbf{C}}(\mathbf{t}_{\ell})+I_{\mathbf{C}}(\mathbf{m}_{\ell})-I_{\mathbf{C}}(\mathbf{m}_{\ell,n})\right]+o\left(n^{-1}\right).\end{split}$

Using(6.8), we find

(6.12)
$\begin{split}&[F_{n}(\mathbf{t}_{n})-F_{n}(\mathbf{m}_{n})]-[F_{\beta,h}(\mathbf{t})-F_{\beta,h}(\mathbf{m})]\\\
&=-\frac{1}{2}\sum_{\ell,\ell^{\prime}\in[k]}a_{\ell}\,a_{\ell^{\prime}}\left[\mathbf{t}_{\ell,n}\mathbf{t}_{\ell^{\prime},n}-\mathbf{m}_{\ell,n}\mathbf{m}_{\ell^{\prime},n}\right]\\\
&\quad\times\left(\omega_{\ell}[\omega_{\ell^{\prime},n}-\omega_{\ell^{\prime}}]+\omega_{\ell^{\prime}}[\omega_{\ell,n}-\omega_{\ell}]+[\omega_{\ell,n}-\omega_{\ell}][\omega_{\ell^{\prime},n}-\omega_{\ell^{\prime}}]\right)\\\
&\quad-\frac{1}{2}\sum_{\ell,\ell^{\prime}\in[k]}a_{\ell}\,a_{\ell^{\prime}}\,\omega_{\ell}\,\omega_{\ell^{\prime}}\left[\mathbf{t}_{\ell,n}\mathbf{t}_{\ell^{\prime},n}-\mathbf{t}_{\ell}\mathbf{t}_{\ell^{\prime}}+\mathbf{m}_{\ell}\mathbf{m}_{\ell^{\prime}}-\mathbf{m}_{\ell,n}\mathbf{m}_{\ell^{\prime},n}\right]\\\
&\quad-h\sum_{\ell\in[k]}[\omega_{\ell,n}-\omega_{\ell}]\left[\mathbf{t}_{\ell,n}-\mathbf{m}_{\ell,n}\right]\\\
&\quad+\frac{1}{\beta}\sum_{\ell\in[k]}[\omega_{\ell,n}-\omega_{\ell}]\left[I_{\mathbf{C}}(\mathbf{t}_{\ell,n})-I_{\mathbf{C}}(\mathbf{m}_{\ell,n})\right]+\frac{1}{\beta}\sum_{\ell\in[k]}\frac{1}{2n}\log\left(\frac{1-\mathbf{t}_{\ell,n}^{2}}{1-\mathbf{m}_{\ell,n}^{2}}\right)\\\
&\quad+\frac{1}{\beta}\sum_{\ell\in[k]}\omega_{\ell}\Big{[}(\mathbf{t}_{\ell,n}-\mathbf{t}_{\ell})\beta\left[a_{\ell}K(\mathbf{t})+h\right]+O((\mathbf{t}_{\ell,n}-\mathbf{t}_{\ell})^{2})\\\
&\qquad-(\mathbf{m}_{\ell,n}-\mathbf{m}_{\ell})\beta\left[a_{\ell}K(\mathbf{m})+h\Big{]}+O((\mathbf{m}_{\ell,n}-\mathbf{m}_{\ell})^{2})\right]\\\
&\quad+o\left(n^{-1}\right).\end{split}$

Since

(6.13)
$\mathbf{t}_{\ell,n}\mathbf{t}_{\ell^{\prime},n}-\mathbf{t}_{\ell}\mathbf{t}_{\ell^{\prime}}=\left(\mathbf{t}_{\ell}[\mathbf{t}_{\ell^{\prime},n}-\mathbf{t}_{\ell^{\prime}}]+\mathbf{t}_{\ell^{\prime}}[\mathbf{t}_{\ell,n}-\mathbf{t}_{\ell}]+[\mathbf{t}_{\ell,n}-\mathbf{t}_{\ell}][\mathbf{t}_{\ell^{\prime},n}-\mathbf{t}_{\ell^{\prime}}]\right),$

we focus on estimating $\mathbf{t}_{\ell,n}-\mathbf{t}_{\ell}$.

From Taylor expansion, we get

(6.14)
$\begin{split}&\mathbf{t}_{\ell,n}-\mathbf{t}_{\ell}=\tanh\left(\beta\left[a_{\ell}\sum_{\ell^{\prime}\in[k]}\beta\,a_{\ell^{\prime}}\,\omega_{\ell^{\prime},n}\,\mathbf{t}_{\ell^{\prime},n}+h\right]\right)-\tanh\left(\beta\left[a_{\ell}\sum_{\ell^{\prime}\in[k]}a_{\ell^{\prime}}\,\omega_{\ell^{\prime}}\,\mathbf{t}_{\ell^{\prime}}+h\right]\right)\\\
&=\beta\,a_{\ell}\sum_{\ell^{\prime}\in[k]}a_{\ell^{\prime}}[\omega_{\ell^{\prime},n}\,\mathbf{t}_{\ell^{\prime},n}-\omega_{\ell^{\prime}}\,\mathbf{t}_{\ell^{\prime}}]\left[1-\tanh\left(\beta\left[a_{\ell}\sum_{\ell^{\prime}\in[k]}a_{\ell^{\prime}}\,\omega_{\ell^{\prime}}\,\mathbf{t}_{\ell^{\prime}}+h\right]\right)^{2}\right]\\\
&\quad-\beta^{2}\,a_{\ell}^{2}\left(\sum_{\ell^{\prime}\in[k]}a_{\ell^{\prime}}[\omega_{\ell^{\prime},n}\,\mathbf{t}_{\ell^{\prime},n}-\omega_{\ell^{\prime}}\,\mathbf{t}_{\ell^{\prime}}]\right)^{2}\tanh\left(\beta\left[a_{\ell}\sum_{\ell^{\prime}\in[k]}a_{\ell^{\prime}}\,\omega_{\ell^{\prime}}\,\mathbf{t}_{\ell^{\prime}}+h\right]\right)\\\
&\quad\times\left[1-\tanh\left(\beta\left[a_{\ell}\sum_{\ell^{\prime}\in[k]}a_{\ell^{\prime}}\,\omega_{\ell^{\prime}}\,\mathbf{t}_{\ell^{\prime}}+h\right]\right)^{2}\right]\\\
&\quad+O\left(a_{\ell}^{3}\left(\sum_{\ell^{\prime}\in[k]}a_{\ell^{\prime}}[\omega_{\ell^{\prime},n}\,\mathbf{t}_{\ell^{\prime},n}-\omega_{\ell^{\prime}}\,\mathbf{t}_{\ell^{\prime}}]\right)^{3}\right).\end{split}$

Since

(6.15)
$\begin{split}\omega_{\ell^{\prime},n}\,\mathbf{t}_{\ell^{\prime},n}-\omega_{\ell^{\prime}}\,\mathbf{t}_{\ell^{\prime}}&=(\omega_{\ell^{\prime},n}\,-\omega_{\ell^{\prime}})\,\mathbf{t}_{\ell^{\prime}}+\omega_{\ell^{\prime},n}(\mathbf{t}_{\ell^{\prime},n}-\mathbf{t}_{\ell^{\prime}}),\end{split}$

we have

(6.16)
$\begin{split}\mathbf{t}_{\ell,n}-\mathbf{t}_{\ell}&=\beta\,a_{\ell}\left[1-\mathbf{t}_{\ell}^{2}\right]\sum_{\ell^{\prime}\in[k]}a_{\ell^{\prime}}[(\omega_{\ell^{\prime},n}\,-\omega_{\ell^{\prime}})\,\mathbf{t}_{\ell^{\prime}}+\omega_{\ell^{\prime},n}(\mathbf{t}_{\ell^{\prime},n}-\mathbf{t}_{\ell^{\prime}})]\\\
&\quad-\beta^{2}\,a_{\ell}^{2}\,\mathbf{t}_{\ell}\left[1-\mathbf{t}_{\ell}^{2}\right]\left(\sum_{\ell^{\prime}\in[k]}a_{\ell^{\prime}}[(\omega_{\ell^{\prime},n}\,-\omega_{\ell^{\prime}})\,\mathbf{t}_{\ell^{\prime}}+\omega_{\ell^{\prime},n}(\mathbf{t}_{\ell^{\prime},n}-\mathbf{t}_{\ell^{\prime}})]\right)^{2}\\\
&\quad+O\left(a_{\ell}^{3}\left(\sum_{\ell^{\prime}\in[k]}a_{\ell^{\prime}}[(\omega_{\ell^{\prime},n}\,-\omega_{\ell^{\prime}})\,\mathbf{t}_{\ell^{\prime}}+\omega_{\ell^{\prime},n}(\mathbf{t}_{\ell^{\prime},n}-\mathbf{t}_{\ell^{\prime}})]\right)^{3}\right).\end{split}$

Suppose that
$\mathbf{t}_{\ell,n}-\mathbf{t}_{\ell}\sim\frac{Y_{\ell}^{\mathbf{t}}}{\sqrt{n}}$.
By the Central Limit Theorem,
$\omega_{\ell,n}-\omega_{\ell}\sim\frac{Z_{\ell}}{\sqrt{n}}$, where $Z_{\ell}$
is the normal random variable $N(0,\omega_{\ell}(1-\omega_{\ell}))$. Hence

(6.17)
$\begin{split}\frac{Y_{\ell}^{\mathbf{t}}}{\sqrt{n}}&=\beta\,a_{\ell}\left[1-\mathbf{t}_{\ell}^{2}\right]\sum_{\ell^{\prime}\in[k]}a_{\ell^{\prime}}\left[\frac{Z_{\ell^{\prime}}}{\sqrt{n}}\mathbf{t}_{\ell^{\prime}}+\left(\frac{Z_{\ell^{\prime}}}{\sqrt{n}}+\omega_{\ell^{\prime}}\right)\frac{Y_{\ell^{\prime}}^{\mathbf{t}}}{\sqrt{n}}\right]\\\
&\quad-\beta^{2}\,a_{\ell}^{2}\,\mathbf{t}_{\ell}\left[1-\mathbf{t}_{\ell}^{2}\right]\left(\sum_{\ell^{\prime}\in[k]}a_{\ell^{\prime}}\left[\frac{Z_{\ell^{\prime}}}{\sqrt{n}}\,\mathbf{t}_{\ell^{\prime}}+\left(\frac{Z_{\ell^{\prime}}}{\sqrt{n}}+\omega_{\ell^{\prime}}\right)\frac{Y_{\ell^{\prime}}^{\mathbf{t}}}{\sqrt{n}}\right]\right)^{2}\\\
&\quad+O\left(a_{\ell}^{3}\left(\sum_{\ell^{\prime}\in[k]}a_{\ell^{\prime}}\left[\frac{Z_{\ell^{\prime}}}{\sqrt{n}}\,\mathbf{t}_{\ell^{\prime}}+\left(\frac{Z_{\ell^{\prime}}}{\sqrt{n}}+\omega_{\ell^{\prime}}\right)\frac{Y_{\ell^{\prime}}^{\mathbf{t}}}{\sqrt{n}}\right]\right)^{3}\right)\\\
&=\frac{1}{\sqrt{n}}\beta\,a_{\ell}\left[1-\mathbf{t}_{\ell}^{2}\right]\sum_{\ell^{\prime}\in[k]}a_{\ell^{\prime}}\left(\mathbf{t}_{\ell^{\prime}}Z_{\ell^{\prime}}+\omega_{\ell^{\prime}}Y_{\ell^{\prime}}^{\mathbf{t}}\right)\\\
&\quad+\frac{1}{n}\beta\,a_{\ell}\left[1-\mathbf{t}_{\ell}^{2}\right]\sum_{\ell^{\prime}\in[k]}a_{\ell^{\prime}}Z_{\ell^{\prime}}\left(Y_{\ell^{\prime}}^{\mathbf{t}}-\beta\,a_{\ell}\mathbf{t}_{\ell}\mathbf{t}_{\ell^{\prime}}\sum_{\ell^{\prime\prime}\in[k]}\,a_{\ell^{\prime\prime}}\omega_{\ell^{\prime\prime}}Y_{\ell^{\prime\prime}}^{\mathbf{t}}\right)+o(n^{-1})\end{split}$

and so

(6.18)
$\begin{split}Y_{\ell}^{\mathbf{t}}&=\beta\,a_{\ell}\left[1-\mathbf{t}_{\ell}^{2}\right]\frac{\sum_{\ell^{\prime}\in[k]}a_{\ell^{\prime}}\mathbf{t}_{\ell^{\prime}}Z_{\ell^{\prime}}}{1-\beta\sum_{\ell^{\prime}\in[k]}a_{\ell^{\prime}}^{2}\omega_{\ell^{\prime}}\left[1-\mathbf{t}_{\ell^{\prime}}^{2}\right]}+O(n^{-\frac{1}{2}}),\end{split}$

where the denominator does not vanish because of Remark 4.3. Thus, up to a
factor $O(n^{-\frac{1}{2}})$, $Y_{\ell}^{\mathbf{t}}$ is a normal random
variable with mean $0$ and variance

(6.19)
$\beta^{2}\,a_{\ell}^{2}\left[1-\mathbf{t}_{\ell}^{2}\right]^{2}\frac{\sum_{\ell^{\prime}\in[k]}a_{\ell^{\prime}}^{2}\mathbf{t}_{\ell^{\prime}}^{2}\omega_{\ell^{\prime}}(1-\omega_{\ell^{\prime}})}{\left(1-\beta\sum_{\ell^{\prime}\in[k]}a_{\ell^{\prime}}^{2}\omega_{\ell^{\prime}}\left[1-\mathbf{t}_{\ell^{\prime}}^{2}\right]\right)^{2}}.$

Similar results hold after we replace $\mathbf{t}$ by $\mathbf{m}$.

Going back to (6.12), using (6.13) and (6.18), and inserting
$\mathbf{t}_{\ell,n}-\mathbf{t}_{\ell}\sim\frac{Y_{\ell}^{\mathbf{t}}}{\sqrt{n}}$
and
$\mathbf{m}_{\ell,n}-\mathbf{m}_{\ell}\sim\frac{Y_{\ell}^{\mathbf{m}}}{\sqrt{n}}$
and $\omega_{\ell,n}-\omega_{\ell}\sim\frac{Z_{\ell}}{\sqrt{n}}$, we obtain

(6.20)
$\begin{split}&[F_{n}(\mathbf{t}_{n})-F_{n}(\mathbf{m}_{n})]-[F_{\beta,h}(\mathbf{t})-F_{\beta,h}(\mathbf{m})]\\\
&\sim-\frac{1}{2}\sum_{\ell,\ell^{\prime}\in[k]}a_{\ell}\,a_{\ell^{\prime}}\left[\left(\mathbf{t}_{\ell}+\frac{Y_{\ell}^{\mathbf{t}}}{\sqrt{n}}\right)\left(\mathbf{t}_{\ell^{\prime}}+\frac{Y_{\ell^{\prime}}^{\mathbf{t}}}{\sqrt{n}}\right)-\left(\mathbf{m}_{\ell}+\frac{Y_{\ell}^{\mathbf{m}}}{\sqrt{n}}\right)\left(\mathbf{m}_{\ell^{\prime}}+\frac{Y_{\ell^{\prime}}^{\mathbf{m}}}{\sqrt{n}}\right)\right]\\\
&\qquad\times\left(\omega_{\ell}\frac{Z_{\ell^{\prime}}}{\sqrt{n}}+\omega_{\ell^{\prime}}\frac{Z_{\ell}}{\sqrt{n}}+\frac{Z_{\ell}Z_{\ell^{\prime}}}{n}\right)\\\
&\quad-\frac{1}{2}\sum_{\ell,\ell^{\prime}\in[k]}a_{\ell}\,a_{\ell^{\prime}}\,\omega_{\ell}\,\omega_{\ell^{\prime}}\left(\mathbf{t}_{\ell}\frac{Y_{\ell^{\prime}}^{\mathbf{t}}}{\sqrt{n}}+\mathbf{t}_{\ell^{\prime}}\frac{Y_{\ell}^{\mathbf{t}}}{\sqrt{n}}+\frac{Y_{\ell}^{\mathbf{t}}Y_{\ell^{\prime}}^{\mathbf{t}}}{n}-\mathbf{m}_{\ell}\frac{Y_{\ell^{\prime}}^{\mathbf{m}}}{\sqrt{n}}-\mathbf{m}_{\ell^{\prime}}\frac{Y_{\ell}^{\mathbf{m}}}{\sqrt{n}}-\frac{Y_{\ell}^{\mathbf{m}}Y_{\ell^{\prime}}^{\mathbf{m}}}{n}\right)\\\
&\quad-h\sum_{\ell\in[k]}\frac{Z_{\ell}}{\sqrt{n}}\left(\mathbf{t}_{\ell}+\frac{Y_{\ell}^{\mathbf{t}}}{\sqrt{n}}-\mathbf{m}_{\ell}-\frac{Y_{\ell}^{\mathbf{m}}}{\sqrt{n}}\right)\\\
&\quad+\frac{1}{\beta}\sum_{\ell\in[k]}\frac{Z_{\ell}}{\sqrt{n}}\left[I_{\mathbf{C}}\left(\mathbf{t}_{\ell}+\frac{Y_{\ell}^{\mathbf{t}}}{\sqrt{n}}\right)-I_{\mathbf{C}}\left(\mathbf{m}_{\ell}+\frac{Y_{\ell}^{\mathbf{m}}}{\sqrt{n}}\right)\right]+\frac{1}{\beta}\sum_{\ell\in[k]}\frac{1}{2n}\log\left(\frac{1-\left(\mathbf{t}_{\ell}+\frac{Y_{\ell}^{\mathbf{t}}}{\sqrt{n}}\right)^{2}}{1-\left(\mathbf{m}_{\ell}+\frac{Y_{\ell}^{\mathbf{m}}}{\sqrt{n}}\right)^{2}}\right)\\\
&\quad+\frac{1}{\beta}\sum_{\ell\in[k]}\omega_{\ell}\left[\frac{Y_{\ell}^{\mathbf{t}}}{\sqrt{n}}\beta\left[a_{\ell}K(\mathbf{t})+h\right]+O\left(\frac{(Y_{\ell}^{\mathbf{t}})^{2}}{n}\right)-\frac{Y_{\ell}^{\mathbf{m}}}{\sqrt{n}}\left[a_{\ell}K(\mathbf{m})+h\right]+O\left(\frac{(Y_{\ell}^{\mathbf{m}})^{2}}{n}\right)\right]\\\
&\qquad+o\left(n^{-1}\right).\\\ \end{split}$

Thus,

(6.21)
$\begin{split}&[F_{n}(\mathbf{t}_{n})-F_{n}(\mathbf{m}_{n})]-[F_{\beta,h}(\mathbf{t})-F_{\beta,h}(\mathbf{m})]\\\
&=-\frac{1}{2}\sum_{\ell,\ell^{\prime}\in[k]}a_{\ell}\,a_{\ell^{\prime}}\left[\mathbf{t}_{\ell}\mathbf{t}_{\ell^{\prime}}-\mathbf{m}_{\ell}\mathbf{m}_{\ell^{\prime}}\right]\left(\omega_{\ell}\frac{Z_{\ell^{\prime}}}{\sqrt{n}}+\omega_{\ell^{\prime}}\frac{Z_{\ell}}{\sqrt{n}}\right)\\\
&\quad-\frac{1}{2}\sum_{\ell,\ell^{\prime}\in[k]}a_{\ell}\,a_{\ell^{\prime}}\,\omega_{\ell}\,\omega_{\ell^{\prime}}\left(\mathbf{t}_{\ell}\frac{Y_{\ell^{\prime}}^{\mathbf{t}}}{\sqrt{n}}+\mathbf{t}_{\ell^{\prime}}\frac{Y_{\ell}^{\mathbf{t}}}{\sqrt{n}}-\mathbf{m}_{\ell}\frac{Y_{\ell^{\prime}}^{\mathbf{m}}}{\sqrt{n}}-\mathbf{m}_{\ell^{\prime}}\frac{Y_{\ell}^{\mathbf{m}}}{\sqrt{n}}\right)\\\
&\quad-h\sum_{\ell\in[k]}\left[\mathbf{t}_{\ell}-\mathbf{m}_{\ell}\right]\frac{Z_{\ell}}{\sqrt{n}}+\frac{1}{\beta}\sum_{\ell\in[k]}\frac{Z_{\ell}}{\sqrt{n}}\left[I_{\mathbf{C}}\left(\mathbf{t}_{\ell}+\frac{Y_{\ell}^{\mathbf{t}}}{\sqrt{n}}\right)-I_{\mathbf{C}}\left(\mathbf{m}_{\ell}+\frac{Y_{\ell}^{\mathbf{m}}}{\sqrt{n}}\right)\right]\\\
&\quad+\sum_{\ell\in[k]}\omega_{\ell}\left[\frac{Y_{\ell}^{\mathbf{t}}}{\sqrt{n}}\left[a_{\ell}K(\mathbf{t})+h\right]-\frac{Y_{\ell}^{\mathbf{m}}}{\sqrt{n}}\left[a_{\ell}K(\mathbf{m})+h\right]\right]+O\left(n^{-1}\right).\end{split}$

Since the random variables $Y_{\ell}^{\mathbf{t}}$, $Y_{\ell}^{\mathbf{m}}$,
$Z_{\ell}$ are centred normal, this concludes the proof of Theorem 1.3.

From (6.21) it is possible to compute explicitly the variance of $Z$ defined
in Theorem 1.3, because the variances of all the random variables involved are
known (at least to leading order).

##  Appendix A Metastability on the complete graph without disorder

We give a brief overview of well-known results for the standard Curie-Weiss
model. We refer to [7, Chapter 13] for more details.

The Glauber dynamics is defined as in Section 1.2, but with $J\equiv 1$. For
convenience we write the Curie-Weiss Hamiltonian as

(A.1)
$H_{n}(\sigma)=-\frac{1}{2n}\sum_{i,j\in[n]}\sigma(i)\sigma(j)-h\sum_{i\in[n]}\sigma(i),\qquad\sigma\in\mathcal{S}_{n},$

which is as (2.5) when $J\equiv 1$. What makes this case easier than the one
with disorder is that the interaction is _mean-field_. Indeed, we may write

(A.2)
$H_{n}(\sigma)=n\big{[}-\tfrac{1}{2}m_{n}(\sigma)^{2}-hm_{n}(\sigma)\big{]},$

with

(A.3) $m_{n}(\sigma)=\frac{1}{n}\sum_{i\in[n]}\sigma(i)\in[-1,1]$

the magnetisation. In this case the magnetisation process $(m_{n}(t))_{t\geq
0}$, defined by

(A.4) $m_{n}(t)=m_{n}(\sigma_{t}),$

is Markovian. More specifically, it is a nearest-neighbour random walk on the
grid

(A.5)
$\Gamma_{n}=\left\\{-1,-1+\tfrac{2}{n},\ldots,+1-\tfrac{2}{n},+1\right\\}.$

In the limit as $n\to\infty$, (A.4) converges to a Brownian motion on
$[-1,+1]$ in the potential $F_{\beta,h}$ given by

(A.6) $F_{\beta,h}(m)=-\frac{1}{2}m^{2}-hm+\frac{1}{\beta}I(m),$

with

(A.7)
$I(m)=\frac{1-m}{2}\log\left(\frac{1-m}{2}\right)+\frac{1+m}{2}\log\left(\frac{1+m}{2}\right)$

the relative entropy of the Bernoulli measure on $\\{-1,+1\\}$ with parameter
$m$ with respect to the counting measure on $\\{-1,+1\\}$. $F_{\beta,h}(m)$ is
the _free energy_ at magnetisation $m$, consisting of an _energy term_
$-\frac{1}{2}m^{2}-hm$ and an _entropy term_ $\frac{1}{\beta}I(m)$. See [7,
Chapter 13] for more details.

Since

(A.8)
$F^{\prime}_{\beta,h}(m)=-m-h+\frac{1}{2\beta}\log\left(\frac{1+m}{1-m}\right),\qquad
F^{\prime\prime}_{\beta,h}(m)=-1-\frac{1}{\beta}\frac{m}{1-m^{2}},$

the stationary points of $F_{\beta,h}$ are the solutions to the equation

(A.9) $m=T_{\beta,h}(m),\qquad T_{\beta,h}(m)=\tanh[\beta(m+h)].$

Since

(A.10) $T^{\prime}_{\beta,h}(m)=\beta\big{[}1-T^{2}_{\beta,h}(m)\big{]},$

$T_{\beta,h}$ is strictly increasing and has a unique inflection point at
$m=-h$. Consequently, (A.9) has either one or three solutions. The latter
occurs if and only if

(A.11) $\beta\in(\bar{\beta}_{c},\infty)\qquad\text{and }\qquad
h\in(0,h_{c}(\beta)),$

where $\bar{\beta}_{c}=1$ is the _critical inverse temperature_ and
$\bar{h}_{c}(\beta)$ is the _critical magnetic field_ , i.e., the unique value
of $h$ for which $T_{\beta,h}$ touches the diagonal at a unique value of the
magnetisation, say $-m(\beta)$. Clearly, $1=\beta(1-m^{2}(\beta))$, i.e.,

(A.12) $m(\beta)=\sqrt{1-\beta^{-1}},$

and so $\bar{h}_{c}(\beta)$ solves the equation
$T_{\beta,\bar{h}_{c}(\beta)}(-m(\beta))=-m(\beta)$. Hence (see Fig. 1)

(A.13)
$\bar{h}_{c}(\beta)=m(\beta)-\frac{1}{2\beta}\log\left(\frac{1+m(\beta)}{1-m(\beta)}\right),\qquad\beta\geq
1.$

$1$$1$$\bar{h}_{c}(\beta)$$\beta$ Figure 1. Plot of
$\beta\mapsto\bar{h}_{c}(\beta)$.

The range of parameters in (A.11) represents the _metastable regime_ in which
$F_{\beta,h}$ has a _double-well_ shape and, in the limit as $n\to\infty$, the
Gibbs measure $\mu_{n}$ in (1.3) has two phases given by the two minima of
$F_{\beta,h}$: the _metastable phase_ with magnetisation $\mathbf{m}<0$ and
the _stable phase_ with magnetisation $\mathbf{s}>0$. The unique _saddle
point_ in the gate $\mathcal{G}(\mathbf{m},\mathbf{s})$ has magnetisation
$\mathbf{t}<0$ (see Fig. 2).

$m$$F_{\beta,h}(m)$$\mathbf{m}$$\mathbf{s}$$\mathbf{t}$$-h$$1$$-1$ Figure 2.
Plot of $m\mapsto F_{\beta,h}(m)$ for $\beta,h$ in the metastable regime.

Theorems A.1–A.2 can be found in Bovier and den Hollander [7, Chapter 13].
Here the notation is the same as the one in Section1. Let
$\mathcal{S}_{n}[\mathbf{m}]$, $\mathcal{S}_{n}[\mathbf{s}]$ denote the sets
of configurations in $\mathcal{S}_{n}$ for which the magnetisation is closest
to $\mathbf{m}$, $\mathbf{s}$, respectively.

######  Theorem A.1 (Average crossover time).

Subject to (A.11), uniformly in $\sigma\in\mathcal{S}_{n}[\mathbf{m}]$,

(A.14)
$\mathbb{E}_{\sigma}\left[\tau_{\mathcal{S}_{n}[\mathbf{s}]}\right]=[1+o_{n}(1)]\,\frac{\pi}{1-\mathbf{t}}\sqrt{\frac{1-\mathbf{t}^{2}}{1-\mathbf{m}^{2}}}\frac{1}{\beta\sqrt{F_{\beta,h}^{\prime\prime}(\mathbf{m})[-F_{\beta,h}^{\prime\prime}(\mathbf{t})]}}\,\mathrm{e}^{\beta
n[F_{\beta,h}(\mathbf{t})-F_{\beta,h}(\mathbf{m})]}.$

######  Theorem A.2 (Exponential law).

Subject to (A.11), uniformly in $\sigma\in\mathcal{S}_{n}[\mathbf{m}]$,

(A.15)
$\mathbb{P}_{\sigma}\left(\tau_{\mathcal{S}_{n}[\mathbf{s}]}>t\,\mathbb{E}_{\sigma}\left[\tau_{\mathcal{S}_{n}[\mathbf{s}]}\right]\right)=[1+o_{n}(1)]\,\mathrm{e}^{-t},\qquad
t\geq 0.$

Fig. 2 illustrates the setting: the average crossover time from
$\mathcal{S}_{n}[\mathbf{m}]$ to $\mathcal{S}_{n}[\mathbf{s}]$ depends on the
energy barrier $F_{\beta,h}(\mathbf{t})-F_{\beta,h}(\mathbf{m})$ and on the
curvature of $F_{\beta,h}$ at $\mathbf{m}$ and $\mathbf{t}$. The crossover
time is exponential on the scale of its average.

##  Appendix B Examples with multiple metastable states

We provide examples of distributions and parameter choices (in the metastable
regime) for which the model with disorder has multiple critical points. More
specifically, we provide numerical evidence that, for $k\in\\{2,3,4\\}$, (3.6)
can have any number of solutions in the set $\\{3,5\ldots,2k+1\\}$. The cases
with strictly more than 3 solutions present multiple minimal critical points,
i.e. multiple metastable states.

###  B.1. Case k=2

(a)

(b)

Figure 3. $T_{\beta,h,\mathcal{P}}$, $k=2$.

  * •

Figure 3a: 3 critical points, parameters $a_{1}=77$, $a_{2}=45$,
$\omega_{1}=0.688$, $h=1740$, $\beta=113\,\beta_{c}$.

  * •

Figure 3b: 5 critical points, parameters $a_{1}=774$, $a_{2}=36.84$,
$\omega_{1}=0.59$, $h=1740$, $\beta=131\,\beta_{c}$.

###  B.2. Case k=3

(a)

(b)

(c)

Figure 4. $T_{\beta,h,\mathcal{P}}$, $k=3$.

  * •

Figure 4a: 3 critical points, parameters $a_{1}=77$, $a_{2}=45$, $a_{3}=33.5$,
$\omega_{1}=0.688$, $\omega_{2}=0.15$, $h=1740$, $\beta=113\,\beta_{c}$.

  * •

Figure 4b: 5 critical points, parameters $a_{1}=77$, $a_{2}=45$, $a_{3}=27$,
$\omega_{1}=0.59$, $\omega_{2}=0.15$, $h=1740$, $\beta=113\,\beta_{c}$.

  * •

Figure 4c: 7 critical points, parameters $a_{1}=77$, $a_{2}=45$, $a_{3}=33.5$,
$\omega_{1}=0.59$, $\omega_{2}=0.15$, $h=1740$, $\beta=113\,\beta_{c}$.

###  B.3. Case k=4

(a)

(b)

(c)

(d)

Figure 5. $T_{\beta,h,\mathcal{P}}$, $k=4$.

  * •

Figure 5a: 3 critical points, parameters $a_{1}=12$, $a_{2}=16$,
$a_{3}=139.5$, $a_{4}=24.5$, $\omega_{1}=0.474$, $\omega_{2}=0.22$,
$\omega_{3}=0.111$, $h=178$, $\beta=3.8\,\beta_{c}$.

  * •

Figure 5b: 5 critical points, parameters $a_{1}=14$, $a_{2}=27$, $a_{3}=57$,
$a_{4}=24.5$, $\omega_{1}=0.366$, $\omega_{2}=0.1$, $\omega_{3}=0.13$,
$h=262$, $\beta=38.4\,\beta_{c}$.

  * •

Figure 5c: 7 critical points, parameters $a_{1}=2.32$, $a_{2}=4.92$,
$a_{3}=5$, $a_{4}=11.32$, $\omega_{1}=0.6$, $\omega_{2}=0.096$,
$\omega_{3}=0.033$, $h=7.6$, $\beta=95.2\,\beta_{c}$.

  * •

Figure 5d: 9 critical points, parameters $a_{1}=12$, $a_{2}=16$, $a_{3}=50.5$,
$a_{4}=24.5$, $\omega_{1}=0.474$, $\omega_{2}=0.22$, $\omega_{3}=0.111$,
$h=178$, $\beta=63.2\,\beta_{c}$.

##  Appendix C Example of $h_{c}(\beta)$ not increasing

We provide here an example of choice of $\mathcal{P}$ for which the critical
threshold $\beta\mapsto h_{c}(\beta)$ is not monotone increasing. This implies
the possibility of a re-entrant metastable crossover.

For $k=4$, pick $a_{1}=12$, $a_{2}=16$, $a_{3}=50.5$, $a_{4}=24.5$ and
$\omega_{1}=0.474$, $\omega_{2}=0.22$, $\omega_{3}=0.111$. Take $h=100$, and
plot the function $K\mapsto T_{\beta,h,\mathcal{P}}(K)$ varying $\beta$. For
$\beta_{1}=4\,\beta_{c}=0.00762336$ the system is metastable:
$T_{\beta,h,\mathcal{P}}$ intersects the diagonal three times (see Figure 6a),
which implies that $h<h_{c}(\beta_{1})$. For
$\beta_{2}=21\,\beta_{c}=0,04002264>\beta_{1}$ the system is not metastable:
$T_{\beta,h,\mathcal{P}}$ intersects the diagonal only once (see Figure 6b),
which implies that $h>h_{c}(\beta_{2})$. This shows that $h_{c}(\beta)$ is not
necessarily an increasing function of $\beta$.

(a)

(b)

Figure 6. $T_{\beta,h,\mathcal{P}}$, fixed $h$ and $\mathcal{P}$, varying
$\beta$.

##  Appendix D Limit of the prefactor

Below Theorem 1.2 we stated that the prefactor in (1.19) converges. For
completeness, in this Appendix we compute its limit, although, as we mentioned
after Theorem 1.3, it is negligible because of the order of approximation of
the exponent.

We focus first on $\gamma_{n}$. Recall notation in (1.10), (1.11) and (2.1).
Then (4.20) can be written as

(D.1)
$\begin{split}&1+O(n^{-1})=\sum_{\ell\in[k]}\frac{a_{\ell}^{2}\omega_{\ell,n}(1-\mathbf{t}_{\ell,n})\exp\left[-2\beta\left(-a_{\ell}\left(\frac{a_{\ell}}{n}+K_{n}(\mathbf{t}_{n})\right)-h\right)_{+}\right]}{\frac{\exp\left[-2\beta\left(-a_{\ell}\left(\frac{a_{\ell}}{n}+K_{n}(\mathbf{t}_{n})\right)-h\right)_{+}\right]}{\beta(1+\mathbf{t}_{\ell,n})}-2\gamma_{n}}\\\
&=\sum_{\ell\in[k]}\frac{a_{\ell}^{2}\omega_{\ell,n}\left(1-\tanh\left(\beta\left[a_{\ell}\,K_{n}(\mathbf{t}_{n})+h\right]\right)\right)\exp\left[-2\beta\left(-a_{\ell}\left(\frac{a_{\ell}}{n}+K_{n}(\mathbf{t}_{n})\right)-h\right)_{+}\right]}{\frac{\exp\left[-2\beta\left(-a_{\ell}\left(\frac{a_{\ell}}{n}+K_{n}(\mathbf{t}_{n})\right)-h\right)_{+}\right]}{\beta\left(1+\tanh\left(\beta\left[a_{\ell}K_{n}(\mathbf{t}_{n})+h\right]\right)\right)}-2\gamma_{n}}.\end{split}$

In the first equality we use (3.3) for $\mathbf{t}_{n}$, i.e., the
approximation of the stationary points of $F_{n}$ by the stationary points of
$\bar{F}_{n}$. This makes $\mathbf{t}_{\ell,n}$ independent of $\ell$, so that
we can use the law of large numbers in the limit as $n\to\infty$. Thus, we
obtain that $\gamma_{n}$ converges to $\gamma$, the solution of the equation

(D.2) $\mathfrak{E}\left(\frac{J(1)^{2}(1+\tanh
U)\,\mathrm{e}^{-2U_{+}}}{\frac{1}{\beta(1-\tanh
U)}\,\mathrm{e}^{-2U_{+}}-2\gamma}\right)=1,$

where $\mathfrak{E}$ denotes expectation with respect to $\mathcal{P}$, the
law of the components of $J$, and $U=-\beta[J(1)\,K(\mathbf{t})+h]$, with
$\mathbf{t}$ solving (3.4). Note that (D.2) is similar to [7, Eq. (14.4.14)].

We are left to find the limit of the determinants ratio. By (4.5),

(D.3)
$\begin{split}\det\mathbb{A}_{n}(m)&=\left(1-\sum_{\ell\in[k]}\beta\,a_{\ell}^{2}\,\omega_{\ell,n}[1-(m_{\ell})^{2}]\right)\prod_{\ell^{\prime}\in[k]}\frac{1}{\beta}\frac{\omega_{\ell^{\prime},n}}{1-(m_{\ell^{\prime}})^{2}}\left[1+O(n^{-1})\right].\end{split}$

Using (3.3) for $m\in\\{\mathbf{t}_{n},\mathbf{m}_{n}\\}$, we have

(D.4)
$\begin{split}&\sum_{\ell\in[k]}\beta\,a_{\ell}^{2}\,\omega_{\ell,n}[1-(m_{\ell,n})^{2}]\\\
&=\sum_{\ell\in[k]}\beta\,a_{\ell}^{2}\,\omega_{\ell,n}\left[1-\tanh^{2}\left(\beta\left[a_{\ell}\sum_{\ell^{\prime}\in[k]}a_{\ell^{\prime}}\,\omega_{\ell^{\prime},n}\,m_{\ell^{\prime},n}+h\right]\right)\right].\end{split}$

Using the law of large numbers as above and with the same notation, we find

(D.5)
$\begin{split}&\lim_{n\to\infty}\frac{[-\det(\mathbb{A}_{n}(\mathbf{t}_{n}))]}{\det(\mathbb{A}_{n}(\mathbf{m}_{n}))}=\frac{-1+\mathfrak{E}\left(\beta\,J(1)^{2}\,\left[1-\tanh^{2}\left[U(\mathbf{t})\right]\right]\right)}{1-\mathfrak{E}\left(\beta\,J(1)^{2}\,\left[1-\tanh^{2}\left[U(\mathbf{m})\right]\right]\right)}\prod_{\ell^{\prime}\in[k]}\frac{1-(\mathbf{m}_{\ell^{\prime}})^{2}}{1-(\mathbf{t}_{\ell^{\prime}})^{2}},\end{split}$

where $U(\mathbf{x})=-\beta(J(1)K(\mathbf{x})+h)$.

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