# Metastability of the Three-state Potts Model with Asymmetrical External
Field

Jeonghyun Ahn  J. Ahn. Department of Mathematical Sciences, Seoul National
University, Republic of Korea<EMAIL_ADDRESS>
###### Abstract.

In this paper, we explore the metastable behavior of the Glauber dynamics
associated with the three-state Potts model with an asymmetrical external
field at a low-temperature regime. The model exhibits three monochromatic
configurations: a unique stable state and two metastable states with different
stability levels. We investigate metastable transitions, which are transitions
from each metastable state to the ground state, separately and verify that the
two transitions exhibit different behaviors.

For the metastable state with greater stability, we derive large deviation-
type results for metastable transition time, both in terms of probability and
expectation. On the other hand, a particularly intriguing phenomenon emerges
when starting from the other metastable state: the process may fall into the
deep valley of another metastable state with a low probability but remains
trapped there for an exponentially long time. We identify specific conditions
on the external field under which this rare event contributes to the mean
hitting time. To this end, we conduct a detailed analysis of the energy
landscape, revealing a sharp saddle configuration analogous to the Ising
model.

##  1\. Introduction

Metastability is a ubiquitous phenomenon observed when a physical system is
close to a first-order phase transition. It is widely observed in stochastic
systems, including small random perturbations of dynamical systems, condensing
interacting particle systems, and spin systems at low temperatures. For a
detailed explanation of the history of metastability, we refer to the
comprehensive monographs [9, 24]. In the analysis of this phenomenon,
mathematical frameworks to describe the physical system have been established.
Typically, a system is modeled by a mathematical function known as Hamiltonian
$H$, and evolution is followed by specific rules described in (2) known as the
Metropolis-type Glauber dynamics. In essence, the dynamic tends to reduce the
Hamiltonian, while the reverse direction occurs with an exponentially low
probability.

We investigate the metastable behavior of the _Potts model_ , a generalization
of the well-known _Ising model_. Starting from [1, 2], its dynamics have been
extensively studied. The mean-field version of the Potts model has been
discussed in [11, 12, 13, 14]. Additionally, the Potts model with zero
external field has been explored in [5, 17, 18, 21]. Various other
generalizations exist, such as degenerate external fields [6, 7] and general
interaction constants [4].

In this paper, we focus on the three-state Potts model with an asymmetrical
external field, defined on a two-dimensional lattice graph with a periodic
boundary condition.

Recall that the Ising model on a finite graph $\Lambda=(V,E)$ is defined by a
Hamiltonian function

$H^{\mathrm{Ising}}(\sigma)=-J\sum_{(x,y)\in
E}\mathbbm{1}_{\\{\sigma(x)=\sigma(y)\\}}-h\sum_{x\in
V}\mathbbm{1}_{\\{\sigma(x)=1\\}}$

where $\sigma\in\\{1,-1\\}^{V}$ is a spin configuration on $\Lambda$. The
dynamics consist of two parameters, the _interaction constant_ $J>0$ and the
_external field_ $h>0$. The model is said to be _ferromagnetic_ since the
spins favor being aligned.

The Potts model is a generalization of the Ising model to an arbitrary number
of spins, denoted by $q$. There are various ways of generalization, depending
on the form of Hamiltonian $H$. In this paper, we fix the interaction constant
but assign an asymmetrical external field for each spin. Namely, define the
Hamiltonian $H$ by

(1) $H(\sigma)=-J\sum_{(x,y)\in
E}\mathbbm{1}_{\\{\sigma(x)=\sigma(y)\\}}-\sum_{x\in V}h_{\sigma(x)}$

for $\sigma\in\\{1,\ldots,q\\}^{V}$. Each spin $i$ has a different value of
external field denoted by $h_{i}$. This differentiates the energy of $q$
monochromatic states, making exactly one into the most stable state.
Furthermore, we assume $q=3$ to avoid a complex energy landscape.

123•••$\mathcal{G}_{12}$$\mathcal{G}_{23}$$\mathcal{G}_{13}$

$\mathbf{2}$$\mathbf{1}$$\mathbf{3}$

Figure 1.  Schematic picture of the energy landscape of the three-state Potts
model with an asymmetrical external field. An Ising-type reference path
connects each pair of monochromatic states.

Our goal is to investigate the metastable behavior of the three-state Potts
model. Precisely, we show the following results:

  * •

large deviation-type result, using the pathwise approach described in [22],

  * •

the Eyring–Kramers law for transition between monochromatic configurations,
applying potential theory,

  * •

the Markov chain model reduction, introduced in [3].

We start by describing the energy landscape. Denote by $\mathbf{i}$ the
monochromatic configuration with spin $i$. Without loss of generality, let
$h_{3}>h_{2}>h_{1}>0$. In this scenario, we have a unique stable state 3, and
two metastable states 1 and 2, however, with different stability levels. To
see why they differ, it is helpful to examine a simplified model. For a while,
let us only consider configurations with at most two types of spin. Then the
model is a combination of Ising models between two different spins. Hence, the
reference path of the original Ising model gives energy-minimizing paths
between 1, 2, and 3. In fact, we show that these paths are still the energy-
minimizing paths in our Potts model. Hence, the energy barrier of transitions
$\mathbf{1}\to\mathbf{3}$ and $\mathbf{2}\to\mathbf{3}$ is
$\Gamma^{\ast}(J,h_{3}-h_{1})$ and $\Gamma^{\ast}(J,h_{3}-h_{2})$,
respectively. Here, $\Gamma^{\ast}(J,h)$ denotes the energy barrier of the
Ising model with parameters $J$ and $h$ previously discovered in [23]. This
quantity strictly decreases on $h\in(0,J)$, thus 2 has a larger stability
level than 1. Gathering these observations, the overall picture of the energy
landscape is depicted in Figure 1. We also remark that the transition is
_sharp_ in the sense that the transition goes through a single critical
configuration.

Our next focus is the Eyring–Kramers law. Estimation of the mean transition
time from 2 to 3 is straightforward; since 2 has the maximal stability, there
is no chance for the process to be trapped in a valley other than 2 or 3. As
soon as the process escapes the valley of 2 and passes the critical
configuration, the process is likely to fall into 3. This metastable behavior
enables the estimation of the transition time. A technical tool known as
_potential theory_ aids this estimation. In brief, it suffices to estimate a
capacity between 2 and 3, which readily follows from the sharp saddle
configuration. In contrast, the transition from 1 to 3 has a different
behavior since 1 has less stability than 2. In view of the typical path, the
transition takes place away from 2. However, the process may fall into 2 with
low probability, and it takes a long time to escape the deep valley of 2. This
may contribute to the mean hitting time
$\mathbb{E}_{\mathbf{1}}^{\beta}[\tau_{\mathbf{3}}]$ of the transition
$\mathbf{1}\to\mathbf{3}$. We show in Theorem 2.8 that if
$2h_{2}>h_{1}+h_{3}$, then the mean hitting time is dominated by the
transition $\mathbf{1}\to\mathbf{2}\to\mathbf{3}$, having a larger time scale
than a typical transition.

Finally, we show the Markov chain model reduction. Since 1 and 2 have
different depths of valley, say $\mathcal{V}_{\mathbf{1}}$ and
$\mathcal{V}_{\mathbf{2}}$, we may speed up the Markov chain by two different
time scales. When speeded-up by $e^{\beta\mathcal{V}_{\mathbf{1}}}$, only the
transition $\mathbf{1}\to\mathbf{3}$ is observed, and 2 remains isolated. When
speeded-up by $e^{\beta\mathcal{V}_{\mathbf{2}}}$, there is a one way
transition $\mathbf{2}\to\mathbf{3}$, and the state 1 is not observed. Our
model is one example in which multiple time scales appear in a Markov chain
model reduction. See [8, 15, 16, 19] for other examples.

#### Mathematical Obstacles

The main obstacle lies in the analysis of the energy landscape, including the
computation of communication heights and stability levels for various
configurations. In particular, the situation becomes complex when all three
spins are involved in the configuration. To resolve this issue, we employ the
_projection operator_ denoted as $\mathcal{P}_{ij}$ on the configuration
space, which is introduced in [4]. This operator $\mathcal{P}_{ij}$ transforms
$i$ spins into $j$ spins, resulting in configurations with at most two spins.
These reduced configurations can then be analyzed using well-established
knowledge about the Ising model.

An additional obstacle arises when bounding the stability level of
configurations other than 1, 2, and 3. A particularly complex scenario occurs
when a cluster of spin 3 is surrounded by both spins 1 and 2. This appears in
the proof of Theorem 2.3 in Subsection 3.3, specifically in (Case 2) of the
proof. To address this issue, we begin by finding configurations where the
stability level can be bounded through local changes. For instance, in Figure
5, we can either update spins 1 and 2 to spins 3 along the edge or do the
opposite, thereby reducing the overall energy. By eliminating such cases, we
can narrow the scenario to the worst case, which is depicted in Figure 8. To
see how this specific case is managed, we refer to (Case 2-2) in the proof of
Theorem 2.3, which can be found in Subsection 3.3.

We also highlight the use of Proposition 4.10 which states the so-called
_negligibility_ condition of the Markov chain model reduction discussed in
[3]. It is a common belief that we may ignore every shallow valley with a
depth less than that of metastable states. However, to our best knowledge, the
negligibility is typically proven independently for each model. We remark that
Proposition 4.10 gives the proof of negligibility in the Metropolis dynamics
in the most general form. We also remark that this applies to models with
multiple speed-up scales.

#### Relavent Works

We end by introducing other models which have relevance to our setting. In [6,
7], the authors consider the degenerate Potts model, which is the case
$h_{1}\neq h_{2}=h_{3}=\cdots=h_{q}$ in our notation. In this setting, there
are either $q-1$ metastable states and one stable state, or one metastable
state and $q-1$ stable states. The transition from a metastable to a stable
state is studied by classifying critical configurations and the tube of
typical paths. In the model with $q-1$ stable states in [6], a transition
between stable states is also studied, which has an analogy to the zero
external field case.

In [20], the authors investigate the metastability of the two-dimensional
Blume–Capel model. This model has two metastable states and a unique stable
state. A notable behavior of this model is observed on a transition between
two metastable states. With a low probability, the process deviates from a
typical path and falls into a stable state for an exponentially long time.
This rare event dominates the mean hitting time, which is exactly what happens
in our model during a transition $\mathbf{1}\to\mathbf{3}$ under the condition
$2h_{2}>h_{1}+h_{3}$.

Finally, we refer to [4] which deals with a similar setting to this paper. It
works on a three-state Potts model with zero external field and general
interaction constants $J_{ij}$ between arbitrary two spins $i$ and $j$. Some
analogous tools are used in this paper, such as the projection operator
defined in Lemma 3.3.

#### Outline

The paper is organized as follows. In Section 2 we define the three-state
Potts model with an asymmetrical external field, then present our main
results. In Section 3 we briefly recall the notations and preliminaries on the
pathwise approach. We prove the main results related to the pathwise approach.
Finally, in Section 4 we prove the remaining main results using potential
theory.

##  2\. Model Description and Main Results

Let $G=(V,E)$ be a $K\times L$ rectangular lattice graph with a periodic
boundary condition. Precisely, the edge set includes the pair of vertices
lying on the opposite side of the boundary. On each vertex, we assign one of
the three spins 1, 2, and 3. Gathering all possible configurations, we have a
state space denoted by $\mathcal{X}=\\{1,2,3\\}^{V}$. The Hamiltonian function
was previously defined in (1), but we assume $J=1$ for simplicity. Writing
again, the Hamiltonian function $H:\mathcal{X}\to\mathbb{R}$ is defined as

$H(\sigma)=-\sum_{\\{x,y\\}\in
E}\mathbbm{1}_{\\{\sigma(x)=\sigma(y)\\}}-\sum_{x\in V}h_{\sigma(x)}.$

Here, $\sigma(x)$ denotes a spin of configuration $\sigma\in\mathcal{X}$ at
site $x\in V$. Without loss of generality, assume $h_{3}>h_{2}>h_{1}>0$. This
gives asymmetric stability on each spin. Also, we assume $h_{3}<1$ so that the
spin itself has less effect on the energy than the interaction between
adjacent cells.

We impose two assumptions analogous to that of the Ising model, which appears
in [23] as the _standard case_.

Assumption A. $K,L>\frac{3}{h_{3}-h_{2}},\frac{3}{h_{2}-h_{1}}$.

Assumption B. $a(h_{2}-h_{1})+b(h_{3}-h_{2})\notin\mathbb{Z}$ for
$(a,b)\in\mathbb{Z}^{2}\backslash\\{(0,0)\\}$.

Assumption A is required to ensure the existence of a critical droplet.
Assumption B corresponds to the condition $2/h\notin\mathbb{Z}$ of the
original Ising model, where $h$ is the external field of the Ising model.

We say that two states $\sigma,\eta\in\mathcal{X}$ are adjacent if they differ
by exactly one site. The dynamics is defined as a continuous time Markov chain
$\\{X_{\beta}(t)\\}_{t\geq 0}$ on $\mathcal{X}$ with a transition rate

(2)
$c_{\beta}(\sigma,\eta)=\begin{cases}e^{{-\beta[H(\eta)-H(\sigma)]}_{+}}&\text{if
}\sigma\sim\eta\\\ 0&\text{otherwise}\end{cases}$

where $[t]_{+}=\max(t,0)$ and the parameter $\beta>0$ represents the inverse
temperature. The Markov chain has a unique invariant measure called the _Gibbs
measure_ , given as

$\mu_{\beta}(\sigma)=\frac{1}{Z_{\beta}}e^{-\beta H(\sigma)},\quad
Z_{\beta}=\sum_{\sigma\in\mathcal{X}}e^{-\beta H(\sigma)}.$

We see that

$\mu_{\beta}(\sigma)c_{\beta}(\sigma,\eta)=\mu_{\beta}(\eta)c_{\beta}(\eta,\sigma)=\frac{1}{Z_{\beta}}e^{-\beta\max(H(\sigma),H(\eta))},$

so $\sigma_{\beta}(t)$ is a reversible Markov chain. Denote by
$\mathbb{P}^{\beta}_{\sigma}$ and $\mathbb{E}^{\beta}_{\sigma}$ the law and
the expectation, respectively, of the Markov chain $\\{X_{\beta}(t)\\}$
starting from $\sigma\in\mathcal{X}$. We omit the subscript or superscript
$\beta$ if it is clear from the context.

We define some notations that capture essential information from the energy
landscape. First, a _path_ from $\sigma\in\mathcal{X}$ to $\eta\in\mathcal{X}$
is a sequence of states $\\{\omega_{t}\\}_{t=0}^{T}$ such that

$\omega_{0}=\sigma,\,\omega_{T}=\eta,\,c_{\beta}(\sigma_{t},\sigma_{t+1})>0$

for all $0\leq t\leq T-1$. This is simply a possible way to move from $\sigma$
to $\eta$. Denote all possible paths from $\sigma$ to $\eta$ by
$\Omega_{\sigma,\eta}$. For simplicity, we sometimes use the notation
$\gamma:\tau\to\eta$ to indicate $\gamma\in\Omega_{\sigma,\eta}$. Define the
_communication height_ between $\sigma,\eta\in\mathcal{X}$ by

$\Phi(\sigma,\eta)=\min_{(\omega_{t})_{t=0}^{T}\in\Omega_{\sigma,\eta}}\max_{0\leq
t\leq T}H(\omega_{t}).$

Define the set of _optimal path_ between $\sigma,\eta\in\mathcal{X}$ by the
collection of paths attaining the minimal energy, that is,

$\Omega_{\sigma,\eta}^{\operatorname{opt}}=\\{\omega\in\Omega_{\sigma,\eta}:\,\sup_{\xi\in\omega}H(\xi)=\Phi(\sigma,\eta)\\}.$

Accordingly, we define the communication height between disjoint sets
$A,B\subseteq\mathcal{X}$ by

$\Phi(A,B)=\min_{\sigma\in A,\sigma^{\prime}\in
B}\Phi(\sigma,\sigma^{\prime})$

and the set of optimal paths by

$\Omega_{A,B}^{\operatorname{opt}}=\\{\omega\in\Omega_{\sigma,\sigma^{\prime}}^{\operatorname{opt}}:\,\sigma\in
A,\,\sigma^{\prime}\in B,\,\Phi(\sigma,\sigma^{\prime})=\Phi(A,B)\\}.$

Also, define the _energy barrier_ from $\sigma\in\mathcal{X}$ to
$A\subseteq\mathcal{X}$ by

$\Gamma(\sigma,A)=\Phi(\sigma,A)-H(\sigma).$

Next, define the _stability level_ $\mathcal{V}_{\sigma}$ of a state
$\sigma\in\mathcal{X}$ by

$\mathcal{V}_{\sigma}\coloneqq\Gamma(\sigma,I_{\sigma}),$

where

$I_{\sigma}\coloneqq\\{\eta\in\mathcal{X}:\,H(\eta)<H(\sigma)\\}.$

Let $\mathcal{V}_{\sigma}=\infty$ if $I_{\sigma}=\emptyset$.

Finally, we define the _minimal saddle_ between $A,B\subseteq\mathcal{X}$ by

$\mathcal{S}(A,B)\coloneqq\\{\sigma\in\mathcal{X}:\,\sigma\in\mathrm{argmax}_{\omega}H\text{
for some }\omega\in\Omega_{A,B}^{\operatorname{opt}}\\}.$

A set $G\subseteq\mathcal{X}$ is a gate between disjoint subsets
$A,B\subseteq\mathcal{X}$ if $G\subseteq S(A,B)$ and $\omega\cap
G\neq\emptyset$ for all $\omega\in\Omega_{A,B}^{\operatorname{opt}}$.
Moreover, we call a gate between $A$ and $B$ is _minimal_ if it is minimal
with respect to set inclusion amongst all gates between $A$ and $B$. Denote by
$\mathcal{G}(A,B)$ the union of all minimal gates between $A$ and $B$.

While computing the energy barriers in the model, we will use an auxiliary
function $f:(0,1)\to\mathbb{R}$ defined by

$f(h)\coloneqq
4\ell_{c}-h(\ell_{c}(\ell_{c}-1)+1),\quad\ell_{c}=\left\lceil\frac{2}{h}\right\rceil.$

Recall from [23] that $f(h)$ is the energy barrier of the Ising model with
external field $h$, and $\ell_{c}$ appears in the size of the critical
droplet. For $1\leq i<j\leq 3$, denote

(3)
$\Gamma_{ij}^{\star}=f(h_{j}-h_{i}),\quad\ell_{c}^{ij}=\left\lceil\frac{2}{h_{j}-h_{i}}\right\rceil$

and define $\Phi_{ij}^{\star}=\Gamma_{ij}^{\star}+H(\mathbf{i})$ for $1\leq
i\neq j\leq 3$. Note that the quantities with star, such as
$\Gamma_{ij}^{\star}$ and $\Phi_{ij}^{\star}$, are directly computable from
the given parameters $h_{i}$’s, and they may differ from
$\Gamma(\mathbf{i},\mathbf{j})$ and $\Phi(\mathbf{i},\mathbf{j})$,
respectively.

###  2.1. Results on energy landscape

We start by giving an overall picture of the dynamics. Let 1, 2, and 3 denote
the configuration with constant spin 1, 2, and 3, respectively. The dynamics
exhibit three deep valleys, each having 1, 2, and 3 as their bottom. Clearly,
3 is the most stable state. In terms of stability levels, the subsequent most
stable state is 2, followed by 1. Furthermore, there exists an Ising-type
reference path between each state. When these are combined, it results in the
illustration shown in Figure 1.

In this subsection, we justify Figure 1.

######  Theorem 2.1 (Energy barrier).

The communication height between 1, 2, 3 is given by

$\Gamma(\mathbf{1},\mathbf{3})=\Gamma(\mathbf{1},\mathbf{2})=\Gamma_{13}^{\star},\quad\Gamma(\mathbf{2},\mathbf{3})=\Gamma_{23}^{\star}.$

In other words, the communication level of $\mathbf{i}$ and $\mathbf{j}$ is
equal to that of the Ising model, except for between 1 and 2. Since
$\Gamma_{13}^{\star}<\Gamma_{23}^{\star}$, 1 has lower energy barrier than 2
with respect to 3.

######  Theorem 2.2 (Stability level of 1, 2).

The stability level of 1 and 2 is

$\mathcal{V}_{\mathbf{1}}=\Gamma_{13}^{\star},\quad\mathcal{V}_{\mathbf{2}}=\Gamma_{23}^{\star}$

Note that Theorem 2.1 provides an upper bound of the stability level. The
proof is discussed in Subsection 3.2.

The next theorem shows that 2 has the largest stability. Moreover, 1 has the
second-largest stability.

######  Theorem 2.3 (Stability level).

For any state $\sigma\neq\mathbf{1},\mathbf{2},\mathbf{3}$, we have

$\mathcal{V}_{\sigma}<\mathcal{V}_{\mathbf{1}}<\mathcal{V}_{\mathbf{2}}.$

Finally, we determine minimal gates between 1, 2, and 3. Let
$\mathcal{R}^{i}_{j}({\ell},{m})$ be the collection of configurations with
$i$-spin $\ell\times m$ rectangle in a sea of $j$-spin, and
$\mathcal{D}^{i}_{j}({\ell},{m})$ obtained from
$\mathcal{R}^{i}_{j}({\ell},{m})$ by attaching a single $i$-spin on the length
$\ell$ side of the rectangle.

######  Theorem 2.4 (Minimal gates).

Let

$\displaystyle\mathcal{G}_{13}=\mathcal{D}^{3}_{1}({\ell_{c}^{13}+1},{\ell_{c}^{13}}),\quad\mathcal{G}_{23}=\mathcal{D}^{3}_{2}({\ell_{c}^{23}+1},{\ell_{c}^{23}}),$

which is a minimal gate of the Ising model between 1 and 3, 2 and 3,
respectively. Then $\mathcal{G}_{13}$ is the minimal gate for transitions
$\mathbf{1}\to\mathbf{3}$, $\mathbf{1}\to\mathbf{2}$, and
$\mathbf{1}\to\mathbf{2}\cup\mathbf{3}$. Also, $\mathcal{G}_{23}$ is the
minimal gate for transitions $\mathbf{2}\to\mathbf{3}$ and
$\mathbf{2}\to\mathbf{1}\cup\mathbf{3}$.

The proofs of Theorem 2.3 and 2.4 are provided in Subsection 3.3.

###  2.2. Large deviation-type results

In this section, we give an exponential scale estimate on a hitting time of 3.
For a subset $A\subseteq\mathcal{X}$, we denote by $\tau_{A}$ the hitting time
of $A$ and by $\tau_{A}^{+}$ the return time to $A$:

$\displaystyle\tau_{A}$ $\displaystyle\coloneqq\inf\left\\{t\geq 0:X(t)\in
A\right\\},$ $\displaystyle\tau_{A}^{+}$
$\displaystyle\coloneqq\inf\left\\{t>0:\,X(t)\in A,\,X(s)\neq X(0)\text{ for
some }0<s<t\right\\}.$

######  Theorem 2.5 (Large Deviation-type result).

For any $\varepsilon>0$, we have

(4)
$\displaystyle\lim_{\beta\to\infty}\mathbb{P}_{\mathbf{2}}^{\beta}[e^{\beta(\Gamma_{23}^{\star}-\varepsilon)}<\tau_{\mathbf{3}}<e^{\beta(\Gamma_{23}^{\star}+\varepsilon)}]=1$
(5)
$\displaystyle\lim_{\beta\to\infty}\mathbb{P}_{\mathbf{1}}^{\beta}[e^{\beta(\Gamma_{13}^{\star}-\varepsilon)}<\tau_{\mathbf{3}}<e^{\beta(\Gamma_{13}^{\star}+\varepsilon)}]=1$

Moreover, we have

$\lim_{\beta\to\infty}\frac{1}{\beta}\log\mathbb{E}_{\mathbf{2}}^{\beta}[\tau_{\mathbf{3}}]=\Gamma_{23}^{\star}$

and a convergence of distribution under $\mathbb{P}_{\mathbf{2}}^{\beta}$

$\frac{\tau_{\mathbf{3}}}{\mathbb{E}_{\mathbf{2}}^{\beta}[\tau_{\mathbf{3}}]}\xrightarrow{d}\exp(1)$

where $\exp(1)$ denotes the exponential random variable with a mean value 1.

The underlying argument is based on a pathwise approach, which can be found in
[22]. Since 2 is the most metastable state, the energy barrier
$\Gamma_{23}^{\star}$ directly induces an exponential scale of the hitting
time from 2 to 3 as well as the distribution.

In contrast to $\mathbb{P}_{\mathbf{2}}^{\beta}$, the result on
$\mathbb{P}_{\mathbf{1}}^{\beta}$ needs much more detailed knowledge of the
energy landscape. In view of [22, Corollary 3.16], we need to say that the
dynamics avoid a valley deeper than $\mathcal{V}_{\mathbf{1}}$. This can be
done by determining a tube of typical paths from 1 to 3, which is a region
where the dynamics stay with probability close to 1. We saw in Theorem 2.3
that the only valley deeper than $\mathcal{V}_{\mathbf{1}}$ is that of 2, so
it suffices to show that 2 lies outside of the tube. We review various notions
of pathwise approach in Subsection 3.1.

The tube of typical paths also implies the following proposition.

######  Proposition 2.6.

We have that

${\lim_{\beta\to\infty}}\mathbb{P}_{\mathbf{1}}^{\beta}[\tau_{\mathbf{2}}<\tau_{\mathbf{3}}]=0.$

The proof of Theorem 2.5 and Proposition 2.6 is given in Subsection 3.4.

###  2.3. Eyring–Kramers law

We are further interested in the leading coefficient of the exponent term of
the mean hitting time, commonly referred to as the _prefactor_. The potential
theory is applicable to achieve this. Indeed, the estimation of capacity
yields the prefactor, and the capacity can be obtained by examining the saddle
structure. We saw in Theorem 2.4 that the saddle structure between
$\mathbf{1}$ or $\mathbf{2}$ and $\mathbf{3}$ is the same as that of the Ising
model. This will lead us to the computation of the prefactor.

We adopt big-$O$ and little-$o$ notation on $\beta$ where $\beta$ goes to
infinity. Namely, for a function $f:\mathbb{R}^{+}\to\mathbb{R}$ on $\beta$,
we denote $f(\beta)=O_{\beta}(1)$ if there exists constants $\beta_{0}>0$ and
$C>0$ such that

$\left|f(\beta)\right|\leq C$

for all $\beta\geq\beta_{0}$. We denote $f(\beta)=o_{\beta}(1)$ if

${\lim_{\beta\to\infty}}f(\beta)=0.$

######  Theorem 2.7 (Eyring–Kramers law).

As $\beta$ goes to infinity, we have

$\mathbb{E}_{\mathbf{2}}^{\beta}[\tau_{3}]=\kappa_{2}e^{\beta\Gamma_{23}^{\star}}(1+o_{\beta}(1)),\quad\kappa_{2}=\frac{3}{4(2\ell_{c}^{23}-1)}\frac{1}{\left|\mathcal{X}\right|}.$

However, the estimation of
$\mathbb{E}_{\mathbf{1}}^{\beta}[\tau_{\mathbf{3}}]$ is problematic. This is
because the process could fall into 2 with low probability, but it takes a
long time to escape the deep valley of 2. Note that this behavior is not
captured in the Markov chain model reduction discussed in Subsection 2.4.

######  Theorem 2.8.

If $\Gamma_{12}^{\star}<\Gamma_{23}^{\star}$, or equivalently
$2h_{2}>h_{1}+h_{3}$, then

${\lim_{\beta\to\infty}}e^{-\beta\Gamma_{13}^{\star}}\mathbb{E}_{\mathbf{1}}^{\beta}[\tau_{\mathbf{3}}]=\infty.$

Comparing this result to (5) of Theorem 2.5, we indeed see that the rare event
falling to 2 has a significant contribution to the expectation. Theorems 2.7
and 2.8 are proved in Subsections 4.2 and 4.4, respectively.

###  2.4. Markov chain model reduction

In [3], a new approach to describe a metastable behavior was introduced.
Speeding up the process by an appropriate time scale, the Markov chain
converges to a reduced Markov chain. This methodology is called _Markov chain
model reduction_.

In our model, there are two types of speeded-up Markov chains in view of
Theorems 2.5 and 2.7. Let

$\theta_{\beta}^{1}=\kappa_{1}e^{\beta\Gamma_{13}^{\star}},\quad\theta_{\beta}^{2}=\kappa_{2}e^{\beta\Gamma_{23}^{\star}}.$

where

$\kappa_{1}=\frac{3}{4(2\ell_{c}^{13}-1)}\frac{1}{\left|\mathcal{X}\right|},\quad\kappa_{2}=\frac{3}{4(2\ell_{c}^{23}-1)}\frac{1}{\left|\mathcal{X}\right|}.$

Let $\Phi:\mathcal{X}\to\\{\mathbf{1},\mathbf{2},\mathbf{3},[\beta]\\}$ be a
projection operator defined by identity map on
$\\{\mathbf{1},\mathbf{2},\mathbf{3}\\}$, and $[\beta]$ elsewhere. In a sense
of [3, Definition 2.2], the speeded-up chain has the following model
reduction.

######  Theorem 2.9 (Markov chain model reduction).

The speeded-up hidden Markov chain
${\widetilde{X}}_{\beta}(t)=\Phi(X_{\beta}(\theta_{\beta}^{1}t))$ converges to
the Markov chain on $\\{\mathbf{1},\mathbf{2},\mathbf{3}\\}$ with a jump rate

$r(\mathbf{i},\mathbf{j})=\begin{cases}1&(\mathbf{i},\mathbf{j})=(\mathbf{1},\mathbf{3})\\\
0&otherwise.\end{cases}$

Moreover, the time spent outside of $\\{\mathbf{1},\mathbf{2},\mathbf{3}\\}$
is negligible; that is,

${\lim_{\beta\to\infty}}\mathbb{E}_{\xi}^{\beta}\left[\int_{0}^{t}\mathbbm{1}_{\left\\{X(s\theta_{\beta}^{1})\notin\\{\mathbf{1},\mathbf{2},\mathbf{3}\\}\right\\}}ds\right]=0$

for every $\xi\in\mathcal{X}$ and $t>0$.

A similar result holds for speed-up with rate $\theta_{\beta}^{2}$. But the
crucial difference is that the state 1 is not observed.

######  Theorem 2.10 (Markov chain model reduction).

The speeded-up hidden Markov chain
${\widetilde{X}}_{\beta}(t)=\Phi(X_{\beta}(\theta_{\beta}^{2}t))$ converges to
the Markov chain on $\\{\mathbf{2},\mathbf{3}\\}$ with a jump rate

$r(\mathbf{2},\mathbf{3})=1,\quad r(\mathbf{3},\mathbf{2})=0.$

Moreover, the time spent outside of $\\{\mathbf{2},\mathbf{3}\\}$ is
negligible; that is,

${\lim_{\beta\to\infty}}\mathbb{E}_{\xi}^{\beta}\left[\int_{0}^{t}\mathbbm{1}_{\left\\{X(s\theta_{\beta}^{2})\notin\\{\mathbf{2},\mathbf{3}\\}\right\\}}ds\right]=0$

for every $\xi\in\mathcal{X}$ and $t>0$.

Note that the projection $\Phi$ is a projection to
$\\{\mathbf{1},\mathbf{2},\mathbf{3}\\}$ for both propositions. This means
that 1 is not observed when the process is speeded up by $\theta_{\beta}^{2}$.
The proof of Theorems 2.9 and 2.10 is discussed in Subsection 4.3.

######  Remark.

In this paper, we focus on the case $q=3$. The reason we consider the three-
spin case only is because the energy landscape gets more complicated when
$q\geq 4$. To be precise, the difficulty lies in proving Theorem 2.3. In the
proof we present, there are some exceptional cases as depicted in Figure 8-(a)
that we have to handle one by one. There are only a few exceptional cases when
$q=3$, but the possibility increases exponentially as $q$ grows, making the
model unwieldy.

##  3\. Pathwise Approaches and Proofs

The remaining part of the paper is devoted to the proofs of our main results.
In the proofs, two methodologies are mainly used; the pathwise approach and
the potential theory. We divide our main results into two groups based on the
methodology used and prove them separately in the following sections.

In this section, we briefly recall notations and main results of pathwise
approaches. Using this, we prove the results in Subsection 2.1 and 2.2.

###  3.1. Notations and preliminaries

The definitions and results of the pathwise approach are mainly from [22]. For
a nonempty subset $A\subseteq\mathcal{X}$, define the _bottom_ of $A$ as

$\mathcal{F}(A)=\\{\sigma\in A:\,H(\sigma)=\min_{\eta\in A}H(\eta)\\}.$

Also, define its _external boundary_ by

$\partial A=\\{\sigma\notin A:\exists\eta\in A,\,c_{\beta}(\eta,\sigma)>0\\}.$

A nonempty subset $C\subseteq\mathcal{X}$ is called a _cycle_ if it is either
a singleton or it is a connected set such that

$\max_{\eta\in C}H(\eta)<H(\mathcal{F}(\partial C)).$

The depth of a cycle $C$ is defined as

$\Gamma(C)=\left[H(\mathcal{F}(\partial C))-H(\mathcal{F}(C))\right]_{+}.$

The following properties are known about the exit distribution of a cycle. The
proof can be found in [24, Theorem 6.23, Corollary 6.25].

######  Proposition 3.1.

The following statements hold.

  1. (a)

There exists $k>0$ such that for all $\beta$ sufficiently large,

$\sup_{\eta\in C}\mathbb{P}_{\eta}^{\beta}\left[X_{\beta}\left(\tau_{\partial
C}\right)\notin\mathcal{F}(\partial C)\right]\leq e^{-k\beta}.$

  2. (b)

Let $\sigma\in\partial C$. For any $\varepsilon>0$, for all $\beta$
sufficiently large,

$\inf_{\eta\in C}\mathbb{P}_{\eta}^{\beta}\left[X_{\beta}(\tau_{\partial
C})=\sigma\right]\geq e^{-\beta(H(\sigma)-H(\mathcal{F}(C))+\varepsilon)}.$

For a nonempty subset $A\subseteq\mathcal{X}$ and
$\eta\in\mathcal{X}\backslash A$, define the initial cycle $C_{A}(\eta)$ by

$C_{A}(\eta)=\\{\eta\\}\cup\\{\sigma\in\mathcal{X}:\Phi(\eta,\sigma)<\Phi(\eta,A)\\}.$

In fact, the initial cycle is the maximal cycle containing $x$ and disjoint to
$A$. For fixed $A\subseteq\mathcal{X}$, we may observe that initial cycles are
either disjoint or are the same. This gives a partition of
$\mathcal{X}\setminus A$ into maximal cycles. We denote it as

$\mathcal{M}(\mathcal{X}\backslash
A)=\\{C_{A}(\eta)\\}_{\eta\in\mathcal{X}\backslash A}.$

Now we define a typical path from $\eta$ and $A$. This is done by determining
which transitions between maximal cycles are the most likely ones. It will
appear that the typical path provides a bound for hitting time from $\eta$ to
$A$.

A _cycle-path_ is a finite sequence of cycles $C_{1},\ldots,C_{m}$ such that

$C_{i}\cap C_{i+1}=\emptyset,\quad\partial C_{i}\cap C_{i+1}\neq\emptyset$

for $i=1,\ldots,m-1$. Note that given a path $\gamma=(\gamma_{i})_{i=1}^{m}$
from $x$ to $A$, there is a corresponding cycle-path
$(C_{A}(\gamma_{i}))_{i=1}^{m}$.

We say that a cycle-path $(C_{1},\ldots,C_{m})$ is _connected via typical
jumps to $A$_, or simply _vtj-connected_ to $A$ if

$\mathcal{F}(\partial C_{i})\cap
C_{i+1}\neq\emptyset,\quad\mathcal{F}(\partial C_{m})\cap A\neq\emptyset.$

A path is called _typical_ if its corresponding cycle-path is vtj-connected. A
union of all vtj-connected cycle-path from $\eta$ to $A$ is called the
_typical tube_ $T_{A}(\eta)$. It is called typical in the sense of the next
lemma, which is found in [22, Lemma 3.13].

######  Lemma 3.2.

Let $\eta\in\mathcal{X}\setminus A$. There exists $k>0$ such that for $\beta$
sufficiently large,

$\mathbb{P}_{\eta}^{\beta}\left[\tau_{\partial
T_{A}(\eta)}\leq\tau_{A}\right]\leq e^{-k\beta}.$

###  3.2. Energy landscape

We begin by introducing the _projection operator_ , originally introduced in
[4], which will be useful when examining the energy landscape.

######  Lemma 3.3 (Projection).

Let $\sigma\in\mathcal{X}$ be a configuration. Let $\mathcal{P}_{ij}\sigma$ be
the configuration obtained by substituting $i$ spins in $\sigma$ to $j$ spins.
Then the following inequality holds.

$\displaystyle H(\mathcal{P}_{12}\sigma)\leq H(\sigma),$ $\displaystyle
H(\mathcal{P}_{13}\sigma)\leq H(\sigma),$ $\displaystyle
H(\mathcal{P}_{23}\sigma)\leq H(\sigma).$

Moreover, each equality holds if and only if $\mathcal{P}_{ij}\sigma=\sigma$
for corresponding $i,j$.

###### Proof.

Projection reduces the number of adjacent cells with different spins. Also,
the energy of each cell decreases along the given projection, so we have the
desired inequalities. ∎

Now we prove Theorem 2.1.

###### Proof of Theorem 2.1.

Consider a path $\gamma$ from 1 to 3. The projected path
$\mathcal{P}_{23}\gamma$ gives a path in the Ising model with spins of type 1
and 3, of which the communication height between 1 and 3 is
$H(1)+\Gamma_{13}^{\star}$ as described in Appendix A.1. Together with Lemma
3.3, we have

$\sup_{\sigma\in\gamma}H(\sigma)\geq\sup_{\sigma\in\gamma}H(\mathcal{P}_{23}(\sigma))\geq
H(\mathbf{1})+\Gamma_{13}^{\star}.$

Hence we obtain $\Gamma(\mathbf{1},\mathbf{3})\geq\Gamma_{13}^{\star}$.
Projection by $\mathcal{P}_{13}$ similarly gives
$\Gamma(\mathbf{2},\mathbf{3})\geq\Gamma_{23}^{\star}$. On the other hand,
consider the _reference path_ of the Ising model described in Appendix A.1.
Assumption A ensures that such path exists in our model. This gives an exact
equality for the previous inequality. Thus, we obtain

$\Gamma(\mathbf{1},\mathbf{3})\leq\Gamma_{13}^{\star},\quad\Gamma(\mathbf{2},\mathbf{3})\leq\Gamma_{23}^{\star}.$

Finally, it remains to compute $\Gamma(\mathbf{1},\mathbf{2})$. Projecting a
path from 1 to 2 by $\mathcal{P}_{13}$ gives a path $\mathcal{P}_{13}\gamma$
from 1 to 3 in the Ising model with spins of type 1 and 3. By the same
argument as above, we have
$\Gamma(\mathbf{1},\mathbf{2})\geq\Gamma_{13}^{\star}$. Also, consider a path,
denoted by $\gamma_{0}$, concatenating the reference path of the Ising model
between 1 and 3 with the reference path of the Ising model between 3 and 2.
The communication height of this path is the maximum of the communication
height of two reference paths, which is

$\max(H(\mathbf{1})+\Gamma_{13}^{\star},H(\mathbf{2})+\Gamma_{23}^{\star})=H(\mathbf{2})+\Gamma_{23}^{\star}$

following by the computation in the Lemma A.2. Therefore, the equality
$\Gamma(\mathbf{1},\mathbf{2})=\Gamma_{13}^{\star}$ holds. ∎

The same argument can be applied to compute $\mathcal{V}_{1}$ and
$\mathcal{V}_{2}$.

######  Proposition 3.4.

The stability level of 1 and 2 is given by

$\mathcal{V}_{\mathbf{1}}=\Gamma^{\star}_{13},\quad\mathcal{V}_{\mathbf{2}}=\Gamma^{\star}_{23}.$

###### Proof of Theorem 2.2.

Let $\gamma$ be a path from 1 to $\sigma$ with $H(\sigma)<H(\mathbf{1})$. Then
the projection by $\mathcal{P}_{23}$ gives a path $\mathcal{P}_{23}\gamma$
from 1 to $\mathcal{P}_{23}(\sigma)$ in the Ising model with spins 1 and 3.
Also, note that $H(\mathcal{P}_{23}(\sigma))\leq H(\sigma)<H(\mathbf{1})$ by
Lemma 3.3. Hence, the energy of the projected path is bounded by the stability
level of 1 in the Ising model, which is $\Gamma_{13}^{\star}$. Together, we
have

$\sup_{\sigma\in\gamma}H(\sigma)-H(\mathbf{1})\leq\sup_{\sigma\in\gamma}H(\mathcal{P}_{23}(\sigma))-H(\mathbf{1})\leq\Gamma_{13}^{\star}.$

Taking a minimum of over all possible $\gamma$, we have
$\mathcal{V}_{\mathbf{1}}\geq\Gamma_{13}^{\star}$. On the other hand, the
reference path of the Ising model between 1 and 3 gives the equality of the
above inequality. Therefore, $\mathcal{V}_{\mathbf{1}}=\Gamma_{13}^{\star}$.
Similarly, we may apply $\mathcal{P}_{13}$ to obtain
$\mathcal{V}_{\mathbf{2}}=\Gamma_{23}^{\star}$. ∎

###  3.3. Proof of stability level

In this subsection, we give a proof of Theorem 2.3. We clarify that the entire
subsection is designed to include the proof of Theorem 2.3, without declaring
the beginning and the end of the proof. We use this structure due to a number
of lemmas and cases involved in the proof.

From the previous proposition and Appendix A.2,
$\mathcal{V}_{\mathbf{1}}<\mathcal{V}_{\mathbf{2}}$ immediately follows. Thus,
it remains to prove $\mathcal{V}_{\sigma}<\mathcal{V}_{\mathbf{1}}$ for
$\sigma\in\mathcal{X}\backslash\\{\mathbf{1},\mathbf{2},\mathbf{3}\\}$. A
simple computation in the Appendix A.2 shows that
$\mathcal{V}_{\mathbf{1}}=\Gamma_{13}^{\star}>2$. Thus, it suffices to check
the case where $\mathcal{V}_{\sigma}>2$.

$u_{1}$$v$$u_{2}$3 Figure 2.  Figure for Lemma 3.5. Flipping the spin of $v$
to 3 decreases the energy

######  Lemma 3.5.

Suppose that $\mathcal{V}_{\sigma}>0$. Then each 3-cluster, a connected
component of 3-spins, of $\sigma$ forms a rectangle. Moreover, each rectangle
has a distance of at least 2 from the others.

###### Proof.

If a cell has at least two neighborhoods with 3-spins, flipping its spin to 3
decreases the energy. Consider a 3-cluster $D$ of $\sigma$. Suppose $D$ has
internal angle $\frac{3}{2}\pi$ as in Figure 2 where $u_{1},u_{2}\in C$,
$v\notin C$. Then flipping the spin of $v$ to 3 decreases the energy, so
$\mathcal{V}_{\sigma}=0$. Therefore $D$ must have internal angles $\pi$ or
$\frac{1}{2}\pi$, which is a rectangle.

Similarly, if two rectangles of 3-spins are spaced 2 units apart, we can find
a cell with a spin other than 3 that is adjacent to both rectangles. Changing
its spin into 3 decreases the energy, so $\mathcal{V}_{\sigma}=0$, which is a
contradiction. ∎

Now we divide into two cases.

(Case 1: $\sigma$ has no 3-cluster) Assuming $\mathcal{V}_{\sigma}>0$, we may
imitate the proof of Lemma 3.5 to show that 2-clusters form a rectangle.
Suppose $\sigma$ has both 1 and 2 spins, let $D^{\prime}$ be a 2-cluster of
size $l\times m$.

If $l<\frac{2}{h_{2}-h_{1}}$, we may consecutively flip 2-spins into 1-spins
along the length $l$ side of $D^{\prime}$. This gives a 2-cluster of size
$l\times(m-1)$. Each flip increases the energy by $h_{2}-h_{1}$ except for the
last flip which decreases the energy by $2-(h_{2}-h_{1})$, and the energy
difference in total is $l(h_{2}-h_{1})-2<0$. So we have
$\mathcal{V}_{\sigma}<(l-1)(h_{2}-h_{1})<2$.

If $l>\frac{2}{h_{2}-h_{1}}$, we flip 1-spins into 2-spins along the length
$l$ side of $D^{\prime}$ to obtain a 2-cluster of size $l\times(m+1)$. The
first flip increases the energy by $2-(h_{2}-h_{1})$, then it decreases by
$h_{2}-h_{1}$. The final energy difference is $2-l(h_{2}-h_{1})<0$, so we have
$\mathcal{V}_{\sigma}<2$.

(Case 2: $\sigma$ has 3-cluster) Let $D$ be a 3-cluster. By Lemma 3.5, we may
assume $D$ is a rectangle of size $l\times m$.

$u_{1}$$v$$u_{2}$3333$\cdots$$v_{1}$$v_{i}$$v_{l}$ Figure 3.  Energy
decreasing path for a configuration with two-sided 3-cluster. Flipping the
cells $v_{1},\ldots,v_{l}$ in order decreases the energy

First, we consider the case where $D$ is two-sided as in Figure 3. Let
$v_{1},\ldots,v_{l}$ be the cells adjacent to one side of $D$. Define the path
$\omega=(\omega_{0},\ldots,\omega_{l})$ by flipping the spins of $v_{i}$
consecutively to 3. As a result, the width of $D$ increases by one. Denote the
spin of $v_{i}$ in $\sigma$ as $\sigma(v_{i})$. Then a simple computation
shows that

$H(\omega_{i})-H(\omega_{0})\leq 2,\quad
H(\omega_{l})-H(\omega_{0})=\sum_{i=1}^{l}(h_{\sigma(v_{i})}-h_{3})\leq 0.$

If $H(\omega_{l})=H(\omega_{0})$, we may repeat the same action which
increases the width of $D$. This eventually decreases the energy. Hence
$\mathcal{V}_{\sigma}\leq 2$.

Now we may assume that $D$ has four sides. We focus on the neighborhood cells
of $D$ including the corners, which are $2(l+m+2)$ cells in total. By Lemma
3.5, these cells have spin 1 or 2. Among $2(l+m+2)$ edges lying between these
cells, we focus on the edge where the spin changes. We call them _colored_
edges. Formally, we say that the edge is _colored_ if two cells containing the
edge have different spins.

$u_{1}$$v$$u_{2}$3333$\cdots$$v_{1}$$v_{i}$$v_{l}$32213221322$v_{1}$$v_{n}$(a)(b)(c)
Figure 4. Illustration regarding the proof of Lemma 3.6.

We claim that Figure 4-(a) has a stability level at most 1. Denote by $D$ the
rectangular 3-cluster in the configuration.

######  Lemma 3.6.

If two or more colored edge exist on one side of $D$, then
$\mathcal{V}_{\sigma}\leq 1$.

###### Proof.

Let $v_{1},\ldots,v_{n}$ be the maximal consecutive cells on one side of $D$
with the same spin, as in Figure 4. By assumption, neither $v_{1}$ nor $v_{n}$
is a corner of $D$. For simplicity, we assume that the spins of $v_{i}$’s are
1. It will turn out that the proof is independent of the spin type.

Define the path $\omega=(\omega_{0},\ldots,\omega_{n})$ by flipping the spins
of $v_{i}$ consecutively to 3. To compute $H(\omega_{i})-H(\omega_{0})$, it
suffices to compare $v_{1},\ldots,v_{i}$. Counting the number of the colored
edges of $v_{1},\ldots,v_{i}$, we see that $\omega_{0}$ contains at least
$i+1$ colored edges, and $\omega_{i}$ contains at most $i+2$ colored edges for
$0<i<n$. The red edges in Figure 4-(a), (b) show the possible colored edges.
Thus, we have

$H(\omega_{i})-H(\omega_{0})\leq 1-i(h_{3}-h_{1})\leq 1.$

If $i=n$, $\omega_{0}$ contains at least $n+2$ colored edges among the edges
of $v_{1},\ldots,v_{n}$, and $\omega_{n}$ contains at most $n+2$ colored
edges, see the red edges in Figure 4-(a), (c). Thus, we have

$H(\omega_{n})-H(\omega_{0})\leq-n(h_{3}-h_{1})<0.$

Therefore, we have $\mathcal{V}_{\sigma}\leq 1$. ∎

$u_{1}$$v$$u_{2}$3333$\cdots$$v_{1}$$v_{i}$$v_{l}$32213221322$v_{1}$$v_{n}$(a)(b)(c)312$v_{1}$$v_{a}$$w_{1}$$w_{b}$$f_{1}$$f_{2}$$e$312(a)(b)312312(c)(d)
Figure 5.  Figure for Lemma 3.7 where $a,b\neq 0$. The red edges indicate the
possible boundaries between different spins. The top two figures show the
process of flipping the spins of $v_{1},\ldots,v_{a}$ and $w_{1},\ldots w_{b}$
successively. The bottom two figures are obtained through a similar process.
Either one of these processes decreases the total energy of the configuration.
$u_{1}$$v$$u_{2}$3333$\cdots$$v_{1}$$v_{i}$$v_{l}$32213221322$v_{1}$$v_{n}$(a)(b)(c)312$v_{1}$$v_{a}$$w_{1}$$w_{b}$$f_{1}$$f_{2}$$e$312(a)(b)312312(c)(d)32$f_{2}$3211323112(a)(b)(c)(d)
Figure 6.  Figure for Lemma 3.7 where $a=0$. The right two figures show the
resulting configurations by flipping the spins along the rows. If $f_{2}$ is
not colored, either of these processes decreases the total energy of the
configuration.

The next lemma shows that a colored edge on one side forces the existence of a
colored edge on the other side.

######  Lemma 3.7.

Suppose that $\mathcal{V}_{\sigma}>1$. Along one side of the 3-cluster $D$,
suppose there is a unique colored edge $e$ splitting the side of $D$ with
length $a$ and $b$. Consider the cells around $D$, which share an edge or a
vertex with $D$. Starting from the edge $e$, denote these cells in clockwise
order by $w_{1},\ldots,w_{b+2}$. Also, starting from $e$, denote the cells by
$v_{1},\ldots,v_{a+2}$ in counterclockwise order. Let $f_{1}$ be the edge
between $v_{a+1}$ and $v_{a+2}$, and $f_{2}$ be the edge between $w_{b+1}$ and
$w_{b+2}$. Refer to Figure 5-(a) for the precise illustrations.

Suppose $a,b>0$. If $e$ is colored, then $f_{1}$ or $f_{2}$ is colored.
Moreover, if $a=0$, then $f_{2}$ is colored.

###### Proof.

Suppose $e$ is colored but $f_{1},f_{2}$ are not colored. By Lemma 3.6 and the
assumption $\mathcal{V}_{\sigma}>2$, the cells $v_{1},\ldots,v_{a+2}$ have
same spins and $w_{1},\ldots,w_{b+2}$ have same spins. Without loss of
generality, let $v_{i}$’s have spin 1 and $w_{j}$’s have spin 2.

Let $\sigma_{1}$ be a configuration obtained by flipping the spins of
$v_{1},\ldots,v_{a}$ and $w_{1},\ldots,w_{b}$ to 3, as in Figure 5-(b). Let
$\sigma_{2}$ be a configuration obtained by flipping the spins of cells below
$v_{1},\ldots,v_{a}$ to 1, and the cells below $w_{1},\ldots,w_{b}$ to 2 as in
Figure 5-(d). We compare the energy of these two states with $\sigma$.

Figure 5-(a),(b) compare $\sigma_{1}$ and $\sigma$. The figure shows the
following observations.

  * •

Among the edges of the flipped cells, $\sigma$ has at least $l+1$ colored
edges.

  * •

Among the edges of the flipped cells, $\sigma_{1}$ has exactly $l+2$ colored
edges. This is because the distances between 3-clusters are at least 2, as
stated in Lemma 3.5.

As before, possible colored edges are colored red in Figure 5. Hence we have

$H(\sigma_{1})-H(\sigma)\leq-(h_{3}-h_{1})a-(h_{3}-h_{2})b+1.$

Figure 5-(c),(d) compare $\sigma_{2}$ and $\sigma$. The figure shows the
following observations.

  * •

Among the edges of the flipped cells, $\sigma$ has exactly $l+2$ colored
edges.

  * •

Among the edges of the flipped cells, $\sigma_{2}$ has at most $l+1$ colored
edges, due to the case where the height of $D$ is 1.

Hence we have

$H(\sigma_{2})-H(\sigma)\leq(h_{3}-h_{1})a+(h_{3}-h_{2})b-1.$

Recall from Assumption B that $(h_{3}-h_{1})a+(h_{3}-h_{2})b-1$ cannot be
zero. Hence, we have either $H(\sigma_{1})<H(\sigma)$ or
$H(\sigma_{2})<H(\sigma)$.

Finally, we construct a path from $\sigma$ to $\sigma_{1}$ and $\sigma_{2}$ by
flipping the spins of consecutive cells. By bounding the energy of the path by
$H(\sigma)+1$, one can show that $\mathcal{V}_{\sigma}\leq 1$ as in the proof
of Lemma 3.6. First, suppose $H(\sigma_{1})<H(\sigma)$. Consider a path from
$\sigma$ to $\sigma_{1}$ by flipping $v_{1},\ldots,v_{a}$, then
$w_{1},\ldots,w_{b}$ in order. The first flip increases the energy by
$1-(h_{3}-h_{1})$, then the following steps decrease the energy. So the energy
of the path is bounded by $H(\sigma)+1$ and we have $V_{\sigma}\leq 1$. On the
other hand, suppose $H(\sigma_{2})<H(\sigma)$. Construct a path from $\sigma$
to $\sigma_{2}$ by sequentially flipping cells from the outermost to the
innermost. Begin by flipping the 2-spins. At each step, when a 1-spin or
2-spin is transformed into a 3-spin, the energy increases by at most
$h_{3}-h_{1}$ or $h_{3}-h_{2}$, respectively. Meanwhile, the last flip
decreases the energy by $1-(h_{3}-h_{2})$. Thus, the energy of the path is
bounded by $H(\sigma_{2})+1$, which is less than $H(\sigma)+1$.

Therefore, we have a path from $\sigma$ to a state with lower energy, where
the energy of the path is bounded by $H(\sigma)+1$. This contradicts the
assumption $\mathcal{V}_{\sigma}>1$, so $f_{1}$ or $f_{2}$ must be colored.

We may handle the case $a=0$ similarly. Suppose that $f_{2}$ is not colored.
Note that $f_{1}$ may be colored here. With the same notation as above, Figure
6 shows that

$H(\sigma_{1})-H(\sigma)\leq(h_{3}-h_{2})l-1,\quad
H(\sigma_{2})-H(\sigma)\leq-(h_{3}-h_{2})l+1.$

Either $\sigma_{1}$ and $\sigma_{2}$ have energy less than $H(\sigma)$, and
both have a path from $\sigma$ obtained by flipping consecutive cells as
before. This path has energy less than $H(\sigma)+1$, so $V_{\sigma}<1$, which
is a contradiction. Therefore $f_{2}$ must be colored. ∎

The previous lemma provides a restriction on the possible position of the
colored edge along the boundary of $D$. Using the notations from Lemma 3.7 and
Figure 5-(a), we claim that there is no colored edge $e$ with $a$ and $b$ both
greater than 0.

Suppose to the contrary that such $e$ exists, which we will denote as $e_{1}$.
Now, we reuse the notations of Figure 5-(a) where $e_{i+1}$ takes on the role
of $e$ and $a=0$. Define the edge corresponding to $f_{2}$ as $e_{i+1}$ for
$i=1,2,3,4$, as shown in Figure 7. By the previous lemma, $e_{2}$ is colored.
Then inductively, $e_{3}$, $e_{4}$, $e_{5}$ must be colored. Since $e_{1}$ and
$e_{5}$ is colored, this contradicts $V_{\sigma}>2$ by Lemma 3.6.

Hence we may assume that there is no colored edge at the middle of the side of
$D$. The remaining case is when the 3-cluster is surrounded by a single type
of spin, or when the neighborhood spins are located as Figure 8-(a).

$u_{1}$$v$$u_{2}$3333$\cdots$$v_{1}$$v_{i}$$v_{l}$32213221322$v_{1}$$v_{n}$(a)(b)(c)312$v_{1}$$v_{a}$$w_{1}$$w_{b}$$f_{1}$$f_{2}$$e$312(a)(b)312312(c)(d)32$f_{2}$3211323112(a)(b)(c)(d)312$e_{2}$$e_{1}$$e_{3}$$e_{4}$$e_{5}$
Figure 7.  The figure shows that two spins cannot lie on the same side. If
$e_{1}$ is colored, so are $e_{2},\ldots,e_{5}$ inductively by Lemma 3.7.

(Case 2-1: 3-cluster is surrounded by a single type of spin) Let the
surrounding spin type be $i$. We flip 3-spin at the down left corner to
$i$-spin. Then we flip $i$-spin at the opposite side to 3-spin. Repeat this
along the left side. As a result, the 3-cluster is shifted to the right by
one. Along this operation, the energy is bounded above by $H(\sigma)+2$.
Moreover, the final configuration has the same number of spin as before, and
the number of colored edges remains the same or decreases. Repeating this
operation, the energy eventually decreases if the 3-cluster meets another
cluster. So we have $V_{\sigma}<2$ in this case. If the 3-cluster is the only
cluster in the configuration, we may apply the same argument as in Case 1.
Therefore $V_{\sigma}<2$.

$u_{1}$$v$$u_{2}$3333$\cdots$$v_{1}$$v_{i}$$v_{l}$32213221322$v_{1}$$v_{n}$(a)(b)(c)312$v_{1}$$v_{a}$$w_{1}$$w_{b}$$f_{1}$$f_{2}$$e$312(a)(b)312312(c)(d)32$f_{2}$3211323112(a)(b)(c)(d)312$e_{2}$$e_{1}$$e_{3}$$e_{4}$$e_{5}$3112231122211(a)(b)(c)
Figure 8.  Figure for Case 2-2 in the proof of Theorem 2.3. Starting from (a),
2-spins are flipped to 3-spins from left to right column, downwards in each
column. The resulting configuration (c) has lower energy than (a).

(Case 2-2: 3-cluster is surrounded by two type of spins) Lemma 3.7 asserts
that Figure 8-(a) is the only possible configuration. Consider the following
two operations.

  1. (1)

Successively flip the 1-spins along the side of length $a$ to 3-spins. This
changes the energy no more than 2. The final configuration $\sigma_{a}$ has at
most $a+2$ colored edge around the flipped cells. The initial state has at
least $a+1$ colored edge, so we have

$H(\sigma_{a})-H(\sigma)\leq-(h_{3}-h_{1})a+1.$

If this bound is below zero, we have $V_{\sigma}\leq 2$.

  2. (2)

Do the same operation on the side of length $b$. Then we have either
$V_{\sigma}\leq 2$ or $b\leq\frac{1}{h_{3}-h_{2}}$.

Hence we may assume $a\leq\frac{1}{h_{3}-h_{1}}$ and
$b\leq\frac{1}{h_{3}-h_{2}}$. Now, consider the path shown in Figure 8. Flip
the 3-spins of the rectangle into 2-spins from the left column to the right,
each column from the bottom to the top. Denote this path by
$\\{\gamma_{n}\\}_{n=0,\ldots,ab}$. Let $0\leq q\leq a-1$ and $0<r\leq b$ be
integers such that $n=bq+r$. Then we have

$H(\gamma_{n})-H(\sigma)=\begin{cases}(h_{3}-h_{2})(bq+r)+\mathbbm{1}_{\\{r\neq
0\\}}&q<a-1\\\ (h_{3}-h_{2})(bq+r)-(2r-1)&q=a-1,\quad r<b\\\
(h_{3}-h_{2})ab-2b&q=a-1,\quad r=b.\end{cases}$

We have

$H(\gamma_{ab})-H(\sigma)=\left((h_{3}-h_{2})a-2\right)b<0,\text{ and}$
$\max_{n=0,\ldots,ab}H(\gamma_{n})-H(\sigma)=(h_{3}-h_{2})(a-1)b<(h_{3}-h_{2})\cdot\frac{1}{(h_{3}-h_{2})}\cdot\frac{1}{(h_{3}-h_{1})}<V_{1}.$

Therefore, $V_{\sigma}<V_{1}$. This completes the proof of Theorem 2.3.

Finally, we determine the minimal gates between 1, 2, and 3.

###### Proof of Theorem 2.4.

Recall the proof of Theorem 2.1. For a path $\gamma$ from 1 to 3, we observed
that $\mathcal{P}_{23}\gamma$ gives a path in the Ising model with spins of
type 1 and 3, and thus,

$\sup_{\sigma\in\gamma}H(\sigma)\geq\sup_{\sigma\in\gamma}H(\mathcal{P}_{23}(\sigma))\geq
H(\mathbf{1})+\Gamma_{13}^{\star}.$

For the optimal path $\gamma\in\Omega_{\mathbf{1},\mathbf{3}}$, each equality
of the above inequalities must hold. Thus, $\mathcal{P}_{23}\gamma$ must be an
optimal path of the Ising model. So
$\mathcal{P}_{23}(\eta)\in\mathcal{G}_{13}$ for some $\eta\in\gamma$. Then, we
have

$\Phi(\mathbf{1},\mathbf{3})=\sup_{\sigma\in\gamma}H(\mathcal{P}_{23}\sigma)=H(\mathcal{P}_{23}\eta)\leq
H(\eta)\leq\sup_{\sigma\in\gamma}H(\sigma)=\Phi(\mathbf{1},\mathbf{3}),$

so $H(\eta)=H(\mathcal{P}_{13}(\eta))$ are both equal to
$\Phi(\mathbf{1},\mathbf{3})$. By Lemma 3.3, this is the case only if
$\eta=\mathcal{P}_{13}(\eta)$. Therefore $\gamma$ meets $\mathcal{G}_{13}$.

Now we consider the reference path of the Ising model between 1 and 3. Note
that each configuration in
$\mathcal{G}_{13}=\mathcal{D}^{3}_{1}({\ell_{c}^{13}+1},{\ell_{c}^{13}})$ is
contained in some reference path, denoted by $\gamma_{13}$. Since
$\mathcal{P}_{23}(\gamma_{13})=\gamma_{13}$, $\gamma_{13}$ is an optimal path
in $\Omega_{\mathbf{1},\mathbf{3}}$ in our model. Relying on the knowledge on
the reference path provided in the Appendix A.1, $\gamma_{13}$ has a unique
saddle configuration. Namely, $|\gamma_{13}\cap\mathcal{G}_{13}|=1$ and

$\sup_{\sigma\in\gamma_{13},\sigma\notin\mathcal{G}_{13}}H(\sigma)<\Phi(\mathbf{1},\mathbf{3}).$

This tells that each configuration in $\mathcal{G}_{13}$ is _essential_ ,
which means that it must be contained in any gate between 1 and 3. Hence,
$\mathcal{G}_{13}$ is a minimal gate between 1 and 3.

Similarly, we may apply $\mathcal{P}_{13}$ and $\mathcal{P}_{23}$ to show the
remaining statements. ∎

###  3.4. Large deviation-type results

We prove Theorem 2.5 and Proposition 2.6 using a pathwise approach described
in Section 3.1. To achieve this, we identify the tube of typical paths and
show that 2 does not appear in such paths. It is worth noting that in other
literature, such precise study is not needed since transition time, a usual
interest, can be estimated by a rough knowledge such as minimal gates.
However, we need that 2 is not contained in all typical paths, which requires
a more precise analysis.

We start by classifying the cycles along the typical path.

######  Lemma 3.8.

Recall the definition of $\ell_{c}^{13}$ from (3). Let $i,j$ be at least
$\ell_{c}^{13}=\lceil 2/(h_{3}-h_{1})\rceil$ and $\eta_{0}$ be a single
configuration $\mathcal{R}^{3}_{1}({i},{j})$. Let $C$ be a set of
configurations characterized by the following properties, depicted in Figure
9.

  1. (1)

The configuration does not contain 2-spins.

  2. (2)

There is a unique 3-cluster, and the contour of the 3-cluster is _simple_ , in
the sense that every intersection between parallel or vertical lines within
the interior of the 3-cluster is connected.

  3. (3)

The unique 3-cluster of the configuration is contained in the rectangular
3-cluster of $\eta_{0}$.

  4. (4)

The number of 3-spins is at least $ij-\ell_{c}^{13}+2$.

Then $C$ is a maximal cycle disjoint from 3 and containing $\eta_{0}$. In
particular, we have the following properties on $C$.

  1. (a)

The bottom of $C$ is $\eta_{0}$. The depth of $C$ is $2-(h_{3}-h_{1})$.

  2. (b)

The bottom of $\partial C$ lies in
$\mathcal{D}^{3}_{1}({i},{j})\cup\mathcal{D}^{3}_{1}({j},{i})$.

$u_{1}$$v$$u_{2}$3333$\cdots$$v_{1}$$v_{i}$$v_{l}$32213221322$v_{1}$$v_{n}$(a)(b)(c)312$v_{1}$$v_{a}$$w_{1}$$w_{b}$$f_{1}$$f_{2}$$e$312(a)(b)312312(c)(d)32$f_{2}$3211323112(a)(b)(c)(d)312$e_{2}$$e_{1}$$e_{3}$$e_{4}$$e_{5}$3112231122211(a)(b)(c)$\mathbf{3}$$\mathbf{1}$
Figure 9.  A 3-cluster and the surrounding rectangle. Lemma 3.8-(2) claims
that for the configurations in the cycle $C$, the surrounding rectangle
remains fixed.

###### Proof.

First, we show that $C$ is a cycle. It suffices to show the following claims.

######  Claim.

For $\sigma\in C$, we have $H(\eta_{0})\leq
H(\sigma)<H(\eta_{0})+2-(h_{3}-h_{1})$.

###### Proof of Claim.

Observe that the perimeter of the 3-cluster of $\sigma$ is $2(i+j)$, which is
same as that of $\eta_{0}$. So counting the number of 1-spins yields

$\displaystyle 0\leq H(\sigma)-H(\eta_{0})=(h_{3}-h_{1})\left(ij-\sum_{x\in
V}\mathbbm{1}_{\\{\sigma(x)=1\\}}\right)\leq(h_{3}-h_{1})(\ell_{c}-2)<2-(h_{3}-h_{1})$

where $V$ denotes the set of cells of the configuration. Therefore the claim
holds. ∎

######  Claim.

For $\sigma\in\partial C$, we have $H(\sigma)\geq
H(\eta_{0})+2-(h_{3}-h_{1})$.

###### Proof of Claim.

Let $\sigma^{\prime}$ be a configuration in C adjacent to $\sigma$. We show by
considering the possible shapes of $\sigma$. Suppose $\sigma$ is obtained from
$\sigma$ by changing a $i$-spin to a 2-spin. Then the number of the pairs of
adjacent distinct spins increases by at least 2, so

$H(\sigma^{\prime})-H(\sigma)\geq 2+(h_{i}-h_{2})>2-(h_{3}-h_{1}).$

Next, suppose $\sigma$ is obtained from $\sigma$ by changing a 3-spin to a
1-spin. Note that the perimeter of the 3-cluster cannot decrease. If this is
the case, we need to remove at least $\min(i,j)\geq\ell_{c}^{13}$ 3-spins
compared to $\eta_{0}$. This contradicts the property (4) of the set $C$. If
the perimeter of the 3-cluster increases, it increases by at least 2. So the
energy increases by at least $2+(h_{3}-h_{1})$, so we are done. If the
perimeter is preserved, we see that $\sigma^{\prime}$ must be a shape of
Figure 9, or explicitly, satisfy (1), (2), and (3) of the above properties.
Hence $\sigma^{\prime}$ consists of exactly $ij-\ell_{c}^{13}+1$ 3-spins, and
we have

$H(\sigma^{\prime})-H(\eta_{0})=(h_{3}-h_{1})(\ell_{c}^{13}+1)>2-(h_{3}-h_{1}).$

Finally, suppose $\sigma^{\prime}$ is obtained from $\sigma$ by replacing a
1-spin by 3-spin. Then the parameter of the 3-cluster is either preserved or
increases. If the perimeter increases, the energy difference is bounded by

(6) $H(\sigma^{\prime})-H(\eta_{0})\geq H(\sigma^{\prime})-H(\sigma)\geq
2-(h_{3}-h_{1}),$

so the claim holds. If the perimeter is preserved, we again see that
$\sigma^{\prime}$ must be as in Figure 9, satisfying properties (1), (2), and
(3). Moreover, the number of 3-spin decreases, so (4) is satisfied. So
$\sigma^{\prime}\in C$, which contradicts $\sigma^{\prime}\in\partial C$. ∎

From the above two claims, we have that $C$ is a cycle. Moreover, the claims
directly imply (a) of the statement. To see (b), it suffices to see when the
equality holds while showing $H(\sigma)\leq H(\eta_{0})+2-(h_{3}-h_{1})$ for
$\sigma\in\partial C$ in the claim. Following the proof, (6) is the only
possible case. This occurs when $\sigma^{\prime}$ is obtained from $\sigma$ by
changing a 1-spin attached to a boundary of 3-cluster into 3-spin, and
moreover $H(\sigma)=H(\eta_{0})$. Since $\eta_{0}$ is the unique bottom of
$C$, we have $\sigma=\eta_{0}$. Therefore, $\sigma^{\prime}$ lies in
$\mathcal{D}^{3}_{1}({i},{j})\cup\mathcal{D}^{3}_{1}({j},{i})$.

Finally, we show that $C$ is a maximal cycle containing $\eta_{0}$ and
disjoint from 3. Showing that $\Gamma(\eta_{0},\mathbf{3})=2-(h_{3}-h_{1})$,
which is a depth of $C$, gives the result. Indeed, a cycle containing $C$ has
a larger depth, so it must contain 3. The cycle $C$ itself gives a lower bound
$\Gamma(\eta_{0},\mathbf{3})\geq 2-(h_{3}-h_{1})$, so it suffices to construct
an optimal path $\gamma$ of height $H(\eta_{0})+2-(h_{3}-h_{1})$. The
construction is similar to the reference path of the Ising model. Precisely,
starting from $\eta_{0}$, add a single 1-spin to the edge of the 3-cluster.
Then, flip the rest of the 1-spins on the edge to 3-spins, which strictly
decreases the energy. This operation gives a 3-cluster of size $(i+1)\times
j$, and decreases the energy by $2-j(h_{3}-h_{1})<0$ in total. Repeating the
same operation, we may construct a path with the maximal energy less than
$\Phi(\eta_{0},\mathbf{3})$. Hence, $C$ is the maximal cycle. ∎

We remark that in the proof, we did not use the relation $h_{1}<h_{2}$. Thus a
similar statement holds between spins of type 1 and 2, where $\ell_{c}^{13}$
is replaced by $\ell_{c}^{12}$ and the depth of the cycle is replaced by
$2-(h_{3}-h_{2})$.

Now we can fully describe typical paths from 1 to 3. This will lead us to the
proof of Proposition 2.6.

###### Proof of Proposition 2.6.

Consider a state in $\mathcal{D}^{3}_{1}({i},{j})$ where $i$ and $j$ are both
greater than $\ell_{c}^{13}$. In this state, the only way to decrease the
energy is by filling the 3-spins on the length $i$ side of the 3-cluster or
returning to $\mathcal{R}^{3}_{1}({i},{j})$. Therefore, we move from
$\mathcal{R}^{3}_{1}({i},{j})$ to either $\mathcal{R}^{3}_{1}({i+1},{j})$ or
$\mathcal{R}^{3}_{1}({i},{j+1})$ with a probability of $1-o_{\beta}(1)$. This
follows from Proposition 3.1 and the previous lemma.

Now we determine a typical path from 1 to 3. The process exits the cycle of 1
through
$\mathcal{G}_{13}=\mathcal{D}^{3}_{1}({\ell_{c}^{13}+1},{\ell_{c}^{13}})$ with
a probability of $1-o_{\beta}(1)$. Then, it follows a downhill path towards
$\mathcal{R}^{3}_{1}({\ell_{c}^{13}+1},{\ell_{c}^{13}})$. This sequence of
steps is repeated until we reach 3. This provides a comprehensive description
of a typical path. Let $P$ be a union of all typical paths from 1 to 3. Note
that $P$ does not contain any states with 2-spins. By Lemma 3.2, we obtain

${\lim_{\beta\to\infty}}\mathbb{P}_{\mathbf{1}}^{\beta}[\tau_{\mathbf{2}}<\tau_{\mathbf{3}}]\leq{\lim_{\beta\to\infty}}\mathbb{P}_{\mathbf{1}}^{\beta}[\tau_{\mathcal{X}\backslash
P}<\tau_{\mathbf{3}}]=0.$

∎

The main difficulty of the proof was that the tube of typical trajectories
from 1 to 3 avoids 2. With this fact, one can also derive a large deviation-
type result on a transition from 1 to 3.

###### Proof of Theorem 2.5.

A result on $\mathbf{2}\to\mathbf{3}$ transition readily follows since 2 is
the state with maximal stability. Indeed, [22, Theorems 3.17 and 3.19,
Propositions 3.18 and 3.20] directly gives

$\lim_{\beta\to\infty}\frac{1}{\beta}\log\mathbb{E}_{\mathbf{2}}^{\beta}[\tau_{\mathbf{3}}]=\Gamma_{23}^{\star}$

and

$\frac{\tau_{\mathbf{3}}}{\mathbb{E}_{\mathbf{2}}^{\beta}[\tau_{\mathbf{3}}]}\xrightarrow{d}\exp(1).$

To show the remaining result, we may apply [22, Corollary 3.16]. It states
that if a typical tube $T_{A}(\sigma)$ avoids a cycle deeper than
$C_{A}(\sigma)$, then we may obtain an estimate

${\lim_{\beta\to\infty}}\mathbb{P}_{\sigma}^{\beta}\left[e^{\beta(\Theta(\sigma,A)-\varepsilon)}<\tau_{A}<e^{\beta(\Theta(\sigma,A)+\varepsilon)}\right]=1$

for all $\varepsilon>0$ where $\Theta(\sigma,A)$ is the depth of the cycle
$C_{A}(\sigma)$. The transition $\mathbf{2}\to\mathbf{3}$ clearly satisfies
this condition since $\mathbf{2}$ has the maximal stability. In the case of
transition $\mathbf{1}\to\mathbf{3}$, we saw in the previous proof that the
tube of typical paths avoids 2. Since 1 is the state with the second greatest
stability level, as proved in Theorem 2.3, transition
$\mathbf{1}\to\mathbf{3}$ also satisfies the required condition. Hence, we
have the desired results. ∎

##  4\. Potential Theory and Proofs

###  4.1. Notations and preliminaries

The potential theory provides a way to compute the prefactor of the mean
hitting time. We recall several concepts from potential theory. Given a
function $f:\mathcal{X}\to\mathbb{R}$, the _Dirichlet form_
$\mathcal{D}_{\beta}(f)$ is defined as

$\mathcal{D}_{\beta}(f)=\frac{1}{2}\sum_{\sigma,\eta\in\mathcal{X}}\mu_{\beta}(\sigma)c_{\beta}(\sigma,\eta)\left[f(\sigma)-f(\eta)\right]^{2}.$

For disjoint subsets $A,B\subseteq\mathcal{X}$, define the _capacity_ between
$A$ and $B$ as

$\operatorname{cap}_{\beta}(A,B)=\inf_{h:\mathcal{X}\to\mathbb{R}}\mathcal{D}_{\beta}(h)$

where the infimum runs over a function $h$ satisfying $h|_{A}=1$ and
$h|_{B}=0$. The infimum attains a unique minimum at the _potential function_
$h_{A,B}$, which is defined as

$h_{A,B}(\sigma)=\mathbb{P}_{\sigma}^{\beta}[\tau_{A}<\tau_{B}]$

for $\sigma\notin A\cup B$.

Recall from [3, Proposition 6.10] the following proposition.

######  Proposition 4.1.

Let $A$ and $B$ be two disjoint subsets of $\mathcal{X}$. For a
$\mu$-integrable function $g:E\to\mathbb{R}$, we have

$\mathbb{E}_{\nu_{AB}}^{\beta}\left[\int_{0}^{\tau_{B}}g(X_{t})dt\right]=\frac{\langle
g,h_{A,B}\rangle_{\mu}}{\operatorname{cap}_{\beta}(A,B)}.$

Here, $\nu_{AB}$ is the equibilium measure on $A$ defined as

$\nu_{AB}(\sigma)=\frac{\mu_{\beta}(\sigma)\mathbb{P}_{\sigma}^{\beta}[\tau_{B}^{+}<\tau_{A}^{+}]}{\operatorname{cap}_{\beta}(A,B)}.$

By putting $A=\\{\eta\\}$, $B=\\{\eta^{\prime}\\}$, and $g$ as a constant
function, we obtain a formula on the mean hitting time.

######  Proposition 4.2.

For any $\eta,\eta^{\prime}\in\mathcal{X}$, we have

$\mathbb{E}_{\eta}^{\beta}[\tau_{\eta^{\prime}}]=\frac{1}{\operatorname{cap}_{\beta}(\eta,\eta^{\prime})}\sum_{\sigma\in\mathcal{X}}\mu_{\beta}(\sigma)\mathbb{P}_{\sigma}^{\beta}[\tau_{\eta}<\tau_{\eta^{\prime}}].$

The next two lemmas give an estimation of the terms appearing in the previous
proposition. We refer to [9, Lemma 8.4] for the first lemma, so-called the
_renewal estimate_.

######  Lemma 4.3.

For two disjoint sets $A,B\subseteq\mathcal{X}$, we have that

$\mathbb{P}_{\sigma}^{\beta}[\tau_{A}<\tau_{B}]\leq\frac{\operatorname{cap}_{\beta}(\sigma,A)}{\operatorname{cap}_{\beta}(\sigma,B)}.$

We refer to [9, Lemma 16.11] for the following lemma.

######  Lemma 4.4.

For two disjoint sets $A,B\subseteq\mathcal{X}$, there exists $0<C_{1}\leq
C_{2}<\infty$ such that

$C_{1}<e^{\beta\Phi(A,B)}Z_{\beta}\operatorname{cap}_{\beta}(A,B)<C_{2}$

for all $\beta\in(0,\infty)$.

###  4.2. Capacity estimation

In this section, we estimate the capacities between 1, 2, and 3. This is based
on the work of [10], which computes the prefactor of the Ising model with an
external field. However, due to the convenience of notations, we refer to the
proof given in Section 16.3 of [9]. Although the statements found in [9] are
between unique metastable and stable states, we may generalize them into
arbitrary states by imposing some nice conditions. With the conditions below,
the proof of the statements is merely the same.

Let $\xi$ and $\xi^{\prime}$ be arbitrary distinct states in $\mathcal{X}$,
which appear to be metastable and stable states in the original works, and
define $\Gamma^{\star}=\Phi(\xi,\xi^{\prime})-H(\xi)$. Let $\mathcal{G}$ be a
subset of $\mathcal{X}$ with the following property.

  1. (1)

For all $\sigma\in\mathcal{G}$, $H(\sigma)=\Gamma^{\star}$.

  2. (2)

For all $\gamma\in\Omega_{\xi,\xi^{\prime}}$,
$\gamma\cap\mathcal{G}\neq\emptyset$.

  3. (3)

For all $\sigma\in\mathcal{G}$, there exists a path
$\\{\gamma_{i}\\}_{i=1}^{n}\in\Omega_{\xi,\xi^{\prime}}$ passing $\sigma$ such
that $H(\gamma_{i})<\Gamma^{\star}$ for all $\gamma_{i}\neq\sigma$.

We may directly see from the definition that $\mathcal{G}$ is the unique
minimal gate from $\xi$ to $\xi^{\prime}$. Define the following subsets of
$\mathcal{X}$.

  * •

$S^{\star}$ is obtained from $\mathcal{X}$ by removing all vertices $\sigma$
with $H(\sigma)>\Gamma^{\star}$ and edges adjacent to these vertices, then
taking the connected component of $m$ and $s$.

  * •

$S^{\star\star}$ is obtained from $S^{\star}$ by removing $\mathcal{G}$.

Consider the connected components of $S^{\star\star}$. Note that by property
(2) of $\mathcal{G}$, $\xi$ and $\xi^{\prime}$ lie in different components.
Let $S_{\xi}$ and $S_{\xi^{\prime}}$ be the connected components containing
$\xi$ and $\xi^{\prime}$, respectively. Denote other components by $S_{i}$,
$i=1,\ldots,m$. Then we have

######  Lemma 4.5.

([9, Lemma 16.16]) Recall the notation
$h_{\xi,\xi^{\prime}}(\sigma)=\mathbb{P}_{\sigma}^{\beta}[\tau_{\xi}<\tau_{\xi^{\prime}}]$.
There exist constants $C<\infty$ and $\delta>0$ such that for any two states
$\sigma,\sigma^{\prime}$ in a same component $S_{i}$,

$\left|h_{\xi,\xi^{\prime}}(\sigma)-h_{\xi,\xi^{\prime}}(\sigma^{\prime})\right|\leq
Ce^{-\beta\delta}$

for all $\beta\in(0,\infty)$.

######  Lemma 4.6.

([9, Lemma 16.17]) The following estimate on the capacity holds as
$\beta\to\infty$.

$Z_{\beta}\operatorname{cap}_{\beta}(\xi,\xi^{\prime})=[1+o_{\beta}(1)]\Theta
e^{-\beta\Phi(\xi,\xi^{\prime})}$

where

$\Theta=\min_{h}\frac{1}{2}\sum_{\sigma,\sigma^{\prime}\in
S^{\star}}\mathbbm{1}_{\\{\sigma\sim\sigma^{\prime}\\}}[h(\sigma)-h(\sigma^{\prime})]^{2}$

subject to a function $h:S^{\star}\to[0,1]$ satisfying $h|_{S_{\xi}}=1$,
$h|_{S_{\xi^{\prime}}}=0$, and $h|_{S_{i}}$ constant for each $i=1,\ldots,m$.

The proof of two lemmas is exactly the same as in [9]. Lemma 4.5 follows from
Lemma 4.4 and Lemma 4.3 and that $\Phi(\sigma,\sigma^{\prime})$ is less than
$\Phi(\xi,\xi^{\prime})$. This implies that the potential function is nearly
constant in each $S_{i}$. Now consider the equation

$\operatorname{cap}_{\beta}(\xi,\xi^{\prime})=\frac{1}{2Z_{\beta}}\sum_{\sigma,\sigma^{\prime}\in\mathcal{X}}e^{-\beta\Phi(\sigma,\sigma^{\prime})}[h(\sigma)-h(\sigma^{\prime})]^{2},$

Lemma 4.5 implies that the potential function can be estimated by a constant
in each component $S_{i}$. With this in mind, Lemma 4.6 follows by direct
computation of Dirichlet form.

Finally, we obtain the capacity estimates between 1, 2, and 3 from the
previous lemma.

######  Proposition 4.7.

We have

$\displaystyle\operatorname{cap}_{\beta}(\mathbf{1},\mathbf{3})=\kappa_{1}^{-1}e^{-\beta\Gamma^{\star}_{13}}(1+o_{\beta}(1))$
$\displaystyle\operatorname{cap}_{\beta}(\mathbf{2},\mathbf{3})=\kappa_{2}^{-1}e^{-\beta\Gamma^{\star}_{23}}(1+o_{\beta}(1))$
$\displaystyle\operatorname{cap}_{\beta}(\mathbf{1},\mathbf{2}\cup\mathbf{3})=\kappa_{1}^{-1}e^{-\beta\Gamma^{\star}_{13}}(1+o_{\beta}(1))$
$\displaystyle\operatorname{cap}_{\beta}(\mathbf{2},\mathbf{1}\cup\mathbf{3})=\kappa_{2}^{-1}e^{-\beta\Gamma^{\star}_{23}}(1+o_{\beta}(1))$

where

$\displaystyle\kappa_{1}=\frac{3}{4(2\ell_{c}^{13}-1)}\frac{1}{\left|\mathcal{X}\right|},\quad\kappa_{2}=\frac{3}{4(2\ell_{c}^{23}-1)}\frac{1}{\left|\mathcal{X}\right|}.$

$u_{1}$$v$$u_{2}$3333$\cdots$$v_{1}$$v_{i}$$v_{l}$32213221322$v_{1}$$v_{n}$(a)(b)(c)312$v_{1}$$v_{a}$$w_{1}$$w_{b}$$f_{1}$$f_{2}$$e$312(a)(b)312312(c)(d)32$f_{2}$3211323112(a)(b)(c)(d)312$e_{2}$$e_{1}$$e_{3}$$e_{4}$$e_{5}$3112231122211(a)(b)(c)$\mathbf{3}$$\mathbf{1}$3$a$$b$$c$13$a$$b$1(a)(b)
Figure 10.  Two types of minimal gates between 1 and 3. Figure (a) has three
adjacent states with lower energy. Figure (b) has two adjacent states with
lower energy.

###### Proof.

For the first and third statements, assign $\mathcal{G}=\mathcal{G}_{13}$.
Otherwise, assign $\mathcal{G}=\mathcal{G}_{23}$. Then $\mathcal{G}$ satisfies
the above three conditions. This is followed by Theorem 2.4.

To apply the previous lemma, we need to determine the connected components of
$S^{\star}\backslash\mathcal{G}$. Each component must be adjacent to
$\mathcal{G}$, so it suffices to examine adjacent states of $\mathcal{G}$ with
energy no more than $\Gamma^{\star}$. There are two types of elements in
$\mathcal{G}$, see Figure 10.

  1. _Type 1_ :

Consider Figure 10-(a). If we add a different spin, the energy increases by at
least $2-(h_{3}-h_{1})$. Hence there are exactly three adjacent configurations
lowering the energy: flipping the cells $a$, $b$, and $c$ into 3-spin. One
lies in $S_{m}$, and two lie in $S_{s}$.

  2. _Type 2_ :

Consider Figure 10-(b). Two adjacent configurations lower the energy, marked
as $a,b$. One lies in $S_{m}$, and one lies in $S_{s}$.

Therefore $S_{m}$ and $S_{s}$ are the only connected components of
$S^{\star}\backslash\mathcal{G}$. Now we apply the variation principle to
compute the capacity. We have

$\Theta=\min_{h:\mathcal{G}\to\mathbb{R}}\frac{1}{2}\Big{[}\sum_{\begin{subarray}{c}\xi\in\mathcal{G}\\\
\mathcal{\xi}:\text{ type
1}\end{subarray}}(1-h(\xi))^{2}+2h(\xi)^{2}+\sum_{\begin{subarray}{c}\xi^{\prime}\in\mathcal{G}\\\
\mathcal{\xi}^{\prime}:\text{ type
2}\end{subarray}}(1-h(\xi^{\prime}))^{2}+h(\xi^{\prime})^{2}\Big{]}.$

There are $2|\mathcal{X}|$ rectangles of size $(\ell_{c}-1)\times\ell_{c}$ or
$(\ell_{c}-1)\times\ell_{c}$, and there are 4 sites of type 1, $2\ell_{c}-4$
sites of type 2. By Cauchy-Schwartz inequality, assigning $h\equiv\frac{1}{3}$
on type 1 states and $h\equiv\frac{1}{2}$ on type 2 states gives the minimum

$\Theta=\left[\frac{1}{2}\cdot 4+\frac{2}{3}(2\ell_{c}-4)\right]\cdot
2\left|\mathcal{X}\right|=\left|\mathcal{X}\right|\frac{4}{3}(2\ell_{c}-1).$

Since $Z_{\beta}=e^{-\beta H(\mathbf{3})}(1+o_{\beta}(1))$, we have the
desired results. ∎

Now, Theorem 2.7 follows immediately.

###### Proof of Theorem 2.7.

By Proposition 4.2

$\mathbb{E}_{2}^{\beta}[\tau_{3}]=\frac{1}{\operatorname{cap}_{\beta}(\mathbf{2},\mathbf{3})}\sum_{\sigma\in\mathcal{X}}\mu_{\beta}(\sigma)\mathbb{P}_{\sigma}^{\beta}[\tau_{2}<\tau_{3}].$

To bound the right-hand side, we show that the summation is dominated by
$\mu_{\beta}(\mathbf{2})$. For each $\sigma\neq\mathbf{2},\mathbf{3}$, we have
two cases:

(Case 1: $H(\sigma)>H(\mathbf{2})$) Then

${\lim_{\beta\to\infty}}\frac{1}{\mu_{\beta}(\mathbf{2})}\mu_{\beta}(\sigma)h_{2,3}(\sigma)\leq
e^{-\beta(H(\sigma)-H(\mathbf{2}))}=0,$

so $\mu_{\beta}(\sigma)h_{2,3}(\sigma)$ is dominated by
$\mu_{\beta}(\mathbf{2})$.

(Case 2: $H(\sigma)\leq H(\mathbf{2})$) This implies that $\sigma$ lies
outside of the domain of attraction

$\\{\eta:\Phi(\eta,\mathbf{2})<\Phi(\mathbf{2},\mathbf{3})\\},$

so $\Phi(\sigma,\mathbf{2})\geq\Phi(\mathbf{2},\mathbf{3})$. Also,
$\mathcal{V}_{\sigma}<\mathcal{V}_{23}=\Gamma^{\star}_{23}$. So we can move to
a state with less energy, without visiting states with energy more than
$H(\sigma)+\mathcal{V}_{\sigma}$. Since 2 has maximal stability, we may repeat
this until we reach 3. Thus
$\Phi(\sigma,\mathbf{3})<H(\sigma)+\Gamma^{\star}_{23}$ and we have

$\displaystyle\mu_{\beta}(\sigma)h_{2,3}(\sigma)$
$\displaystyle\leq\mu_{\beta}(\mathbf{2})e^{\beta(H(\mathbf{2})-H(\sigma))}\frac{\operatorname{cap}_{\beta}(\sigma,\mathbf{2})}{\operatorname{cap}_{\beta}(\sigma,\mathbf{3})}$
$\displaystyle\leq\mu_{\beta}(\mathbf{2})e^{\beta(H(\mathbf{2})-H(\sigma))}e^{-\beta(\Phi(\sigma,\mathbf{2})-\Phi(\sigma,\mathbf{3}))}(1+O_{\beta}(1))$

by Lemma 4.4. The power of the exponent is

$\beta\left(H(\mathbf{2})-\Phi(\sigma,\mathbf{2})-H(\sigma)+\Phi(\sigma,\mathbf{3})\right)<\beta(H(\mathbf{2})-\Phi(\mathbf{2},\mathbf{3})+\Gamma_{23}^{\star})=0.$

Hence we again have

${\lim_{\beta\to\infty}}\frac{1}{\mu_{\beta}(\mathbf{2})}\mu_{\beta}(\sigma)h_{2,3}(\sigma)=0.$

Gathering two cases gives an estimation

$\displaystyle\mathbb{E}_{\mathbf{2}}^{\beta}[\tau_{\mathbf{3}}]=\frac{1}{\operatorname{cap}_{\beta}(\mathbf{2},\mathbf{3})}\mu_{\beta}(\mathbf{2})(1+o_{\beta}(1))=\kappa_{2}e^{\beta\Gamma^{\star}_{23}}(1+o_{\beta}(1)).$

∎

###  4.3. Markov chain model reduction

In this section, we prove the model reduction stated in Section 2.4. In [3,
Theorem 2.4], it states the sufficient condition to show model reduction:
convergence of rate of the trace process, and negligibility of the time spent
in non-metastable states. We show two conditions are satisfied in our model
with respect to two different speed-up scales. Recall that two speed-up scales
$\theta_{\beta}^{1}$ and $\theta_{\beta}^{2}$ are given by

$\theta_{\beta}^{1}=\kappa_{1}e^{\beta\Gamma_{13}^{\star}},\quad\theta_{\beta}^{2}=\kappa_{2}e^{\beta\Gamma_{23}^{\star}}.$

Recall from [3, Subsection 2.1] the definition of _trace process_. In brief, a
trace process on $S\subseteq\mathcal{X}$ is the original process but the clock
stopped outside of $S$. This defines a continuous time Markov chain on $S$,
and its jump rate can be computed in terms of capacities.

######  Proposition 4.8 (Theorem 2.9: Convergence of the rate).

Consider the trace process on $\\{\mathbf{1},\mathbf{2},\mathbf{3}\\}$. Let
$r_{\beta}(\mathbf{i},\mathbf{j})$ be the transition rate from $\mathbf{i}$ to
$\mathbf{j}$ of the trace process. Then the scaled jump rate
$\theta_{\beta}^{1}r_{\beta}(\mathbf{i},\mathbf{j})$ converges to

${\lim_{\beta\to\infty}}\theta_{\beta}^{1}r_{\beta}(\mathbf{i},\mathbf{j})=\begin{cases}1&(\mathbf{i},\mathbf{j})=(\mathbf{1},\mathbf{3})\\\
0&otherwise.\end{cases}$

###### Proof.

From [3, Lemma 6.8], we have

$\displaystyle\mu_{\beta}(\mathbf{1})r_{\beta}(\mathbf{1},\mathbf{2})=\frac{1}{2}[\operatorname{cap}_{\beta}(\mathbf{1},\mathbf{2}\cup\mathbf{3})+\operatorname{cap}_{\beta}(\mathbf{2},\mathbf{1}\cup\mathbf{3})-\operatorname{cap}_{\beta}(\mathbf{3},\mathbf{1}\cup\mathbf{2})]$
$\displaystyle\mu_{\beta}(\mathbf{1})r_{\beta}(\mathbf{1},\mathbf{3})=\frac{1}{2}[\operatorname{cap}_{\beta}(\mathbf{1},\mathbf{2}\cup\mathbf{3})+\operatorname{cap}_{\beta}(\mathbf{3},\mathbf{1}\cup\mathbf{2})-\operatorname{cap}_{\beta}(\mathbf{2},\mathbf{1}\cup\mathbf{3})]$

and

$\frac{r_{\beta}(\mathbf{1},\mathbf{2})}{r_{\beta}(\mathbf{1},\mathbf{3})}=\frac{\mathbb{P}_{\mathbf{1}}^{\beta}[\tau_{\mathbf{2}}<\tau_{\mathbf{3}}]}{\mathbb{P}_{\mathbf{1}}^{\beta}[\tau_{\mathbf{3}}<\tau_{\mathbf{2}}]}.$

This gives

$\displaystyle e^{\beta\Gamma^{\star}_{13}}r_{\beta}(\mathbf{1},\mathbf{3})$
$\displaystyle=e^{\beta\Gamma^{\star}_{13}}\mathbb{P}_{\mathbf{1}}^{\beta}[\tau_{\mathbf{3}}<\tau_{\mathbf{2}}](r_{\beta}(\mathbf{1},\mathbf{2})+r_{\beta}(\mathbf{1},\mathbf{3}))$
$\displaystyle=e^{\beta\Gamma^{\star}_{13}}\mathbb{P}_{\mathbf{1}}^{\beta}[\tau_{\mathbf{3}}<\tau_{\mathbf{2}}]\frac{\operatorname{cap}_{\beta}(\mathbf{1},\mathbf{2}\cup\mathbf{3})}{\mu_{\beta}(\mathbf{1})}$

Together with Propositions 4.2 and 2.6, this converges to

$\kappa_{1}\mathbb{P}_{\mathbf{1}}^{\beta}[\tau_{3}<\tau_{2}](1+o_{\beta}(1))\to\kappa_{1}$

as $\beta\to\infty$. Similarly, as $\beta\to\infty$,

$\displaystyle e^{\beta\Gamma^{\star}_{13}}r_{\beta}(\mathbf{1},\mathbf{2})$
$\displaystyle=e^{\beta\Gamma^{\star}_{13}}\frac{1}{\mu(\mathbf{1})}\mathbb{P}_{\mathbf{1}}^{\beta}[\tau_{\mathbf{2}}<\tau_{\mathbf{3}}]\operatorname{cap}_{\beta}(\mathbf{1},\mathbf{2}\cup\mathbf{3})$
$\displaystyle=\kappa_{1}\mathbb{P}_{\mathbf{1}}^{\beta}[\tau_{\mathbf{3}}<\tau_{\mathbf{2}}](1+o_{\beta}(1))\to
0$

On the other hand, [3, Lemma 6.7] gives that as $\beta\to\infty$,

$\displaystyle e^{\beta\Gamma^{\star}_{13}}r_{\beta}(\mathbf{2},\mathbf{3})$
$\displaystyle=e^{\beta\Gamma^{\star}_{13}}\frac{\operatorname{cap}_{\beta}(\mathbf{2},\mathbf{3})}{\mu(\mathbf{2})}=\kappa_{2}e^{\beta(\Gamma^{\star}_{13}-\Gamma^{\star}_{23})}(1+o_{\beta}(1))\to
0,\text{ and }$ $\displaystyle
e^{\beta\Gamma^{\star}_{13}}r_{\beta}(\mathbf{2},\mathbf{1})$
$\displaystyle=e^{\beta\Gamma^{\star}_{13}}\frac{\operatorname{cap}_{\beta}(\mathbf{1},\mathbf{2})}{\mu(\mathbf{2})}\leq
Ce^{\beta(\Gamma^{\star}_{13}+\Phi(\mathbf{1},\mathbf{2})-H(\mathbf{2}))}\to
0.$

∎

######  Proposition 4.9 (Theorem 2.10: Convergence of the rate).

Consider a trace process on $\\{\mathbf{2},\mathbf{3}\\}$. Let
$r_{\beta}(\mathbf{i},\mathbf{j})$ be the transition rate from $\mathbf{i}$ to
$\mathbf{j}$ of the trace process. Then the scaled jump rate
$\theta_{\beta}^{2}r_{\beta}(\mathbf{i},\mathbf{j})$ converges to

${\lim_{\beta\to\infty}}\theta_{\beta}^{2}r_{\beta}(\mathbf{2},\mathbf{3})=1,\quad{\lim_{\beta\to\infty}}\theta_{\beta}^{1}r_{\beta}(\mathbf{3},\mathbf{2})=0.$

###### Proof.

It follows from [3, Lemma 6.7] that as $\beta\to\infty$, we have

$\displaystyle
e^{\beta\Gamma^{\star}_{23}}r_{\beta}(\mathbf{2},\mathbf{3})=e^{\beta\Gamma^{\star}_{23}}\frac{\operatorname{cap}_{\beta}(\mathbf{2},\mathbf{3})}{\mu_{\beta}(\mathbf{2})}\to\kappa_{2}\text{
and
}e^{\beta\Gamma^{\star}_{23}}r_{\beta}(\mathbf{3},\mathbf{2})=e^{\beta\Gamma^{\star}_{23}}\frac{\operatorname{cap}_{\beta}(\mathbf{3},\mathbf{2})}{\mu_{\beta}(\mathbf{3})}\to
0.$

∎

To finish the proof of Theorems 2.9 and 2.10, it remains to show the
negligibility. We propose a general form of the statement.

######  Proposition 4.10.

Suppose $\mathcal{M}\subseteq\mathcal{X}$ consists of states with stability
levels strictly greater than states not in $\mathcal{M}$, that is,

$\mathcal{V}_{\sigma}<\mathcal{V}_{i}\quad\text{ for all
}i\in\mathcal{M},\,\sigma\notin\mathcal{M}.$

Let $\mathcal{V}=\min_{i\in\mathcal{M}}\mathcal{V}_{i}$ and
$\theta_{\beta}=e^{\beta\mathcal{V}}$. Denote
$\Delta=\mathcal{X}\backslash\mathcal{M}$. Then we have

${\lim_{\beta\to\infty}}\frac{1}{\theta_{\beta}}\mathbb{E}_{\xi}^{\beta}\left[\int_{0}^{t\theta_{\beta}}\mathbbm{1}_{\\{X_{\beta}(s)\in\Delta\\}}ds\right]=0$

for every $\xi\in\mathcal{X}$ and $t>0$. In other words, the chain speeded up
by $\theta_{\beta}$ spends a negligible amount of time outside of
$\mathcal{M}$.

###### Proof.

We first prove the statement for $\xi$ in $\mathcal{M}$, then extend the
result to $\xi$ that are not in $\mathcal{M}$. This will be based on the
observation that the state not in $\mathcal{M}$ falls to $\mathcal{M}$ in a
negligible time.

_Step 1._ The statement holds for $\xi\in\mathcal{M}$.

We prove the statement inductively, in an energy-increasing order. First, let
$\xi\in\mathcal{M}$ be a state with minimal energy. Let
$\mathbb{P}_{\mu_{\beta}}$ denote the law of the Metropolis dynamics starting
from the initial distribution $\mu_{\beta}$. Then, for any time $s>0$, we have

$\mathbb{P}_{\xi}^{\beta}[X_{\beta}(s)\in\Delta]\leq\frac{1}{\mu_{\beta}(\xi)}\mathbb{P}_{\mu_{\beta}}[X_{\beta}(s)\in\Delta]=\frac{\mu_{\beta}(\Delta)}{\mu_{\beta}(\xi)}.$

By Fubini’s theorem, we have

$\displaystyle\frac{1}{\theta_{\beta}}\mathbb{E}_{\xi}^{\beta}\left[\int_{0}^{t\theta_{\beta}}\mathbbm{1}_{\\{X_{\beta}(s)\in\Delta\\}}ds\right]=\frac{1}{\theta_{\beta}}\int_{0}^{t\theta_{\beta}}\mathbb{P}_{\xi}^{\beta}\left[X_{\beta}(s)\in\Delta\right]\leq
t\cdot\frac{\mu_{\beta}(\Delta)}{\mu_{\beta}(\xi)}=o_{\beta}(1).$

Hence, the statement holds for the state with minimal energy.

Next, we take $\xi\in\mathcal{M}$, and define
$\mathcal{M}^{\prime}=\\{\sigma\in\mathcal{M}:\,H(\sigma)<H(\xi)\\}$. By the
induction hypothesis, the statement holds for states in
$\mathcal{M}^{\prime}$. Since $\mathcal{V}_{\xi}\geq\mathcal{V}$ by
assumption, we divide the case into two.

(Case 1: $\mathcal{V}_{\xi}>\mathcal{V}$) Let $C$ be the cycle of depth
$\mathcal{V}_{\xi}$ containing $\xi$ as a bottom. By [22, Theorem 3.2],
$\mathbb{P}_{\xi}[\tau_{\partial C}<t\theta_{\beta}]=o_{\beta}(1)$. So we may
assume that the dynamics take place in $C$ on time $[0,t\theta_{\beta}]$. By
the same argument as the previous case, Fubini’s theorem gives

$\displaystyle\frac{1}{\theta_{\beta}}\mathbb{E}_{\xi}^{\beta}\left[\int_{0}^{t\theta_{\beta}}\mathbbm{1}_{\\{X_{\beta}(s)\in\Delta\\}}ds;\tau_{\partial
C}>t\theta_{\beta}\right]$
$\displaystyle=\frac{1}{\theta_{\beta}}\int_{0}^{t\theta_{\beta}}\mathbb{P}_{\xi}^{\beta}\left[X_{\beta}(s)\in\Delta,\,\tau_{\partial
C}>t\theta_{\beta}\right]$ $\displaystyle\leq
t\cdot\frac{\mu_{\beta}(\Delta\cap C)}{\mu_{\beta}(\xi)}=o_{\beta}(1)$

(Case 2: $\mathcal{V}_{\xi}=\mathcal{V}$) Putting $g=\mathbbm{1}_{\Delta}$ in
Proposition 4.1, we have

$\frac{1}{\theta_{\beta}}\mathbb{E}_{\xi}^{\beta}\left[\int_{0}^{\tau_{\mathcal{M}^{\prime}}}\mathbbm{1}_{\\{X_{\beta}(s)\in\Delta\\}}ds\right]=\frac{1}{\theta_{\beta}\operatorname{cap}_{\beta}(\xi,\mathcal{M}^{\prime})}\sum_{\sigma\in\Delta}\mathbb{P}_{\sigma}^{\beta}[\tau_{\xi}<\tau_{\mathcal{M}^{\prime}}]\mu_{\beta}(\sigma).$

Define $\mathcal{V}^{\prime}=\max_{\sigma\notin M}\mathcal{V}_{\sigma}$, which
is less than $\mathcal{V}$ by assumption. It is clear that
$H(\xi)+\mathcal{V}_{\xi}\leq\Phi(\xi,\mathcal{M}^{\prime})$. Moreover,
retaining the energy lower than $H(\xi)+\mathcal{V}_{\xi}$, we may move from
$\xi$ to a state with lower energy. Repeat this until the process hits
$\mathcal{M}^{\prime}$. Every state visited must have a stability level at
most $\mathcal{V}^{\prime}<\mathcal{V}_{\xi}$, so we have that
$\Phi(\xi,\mathcal{M}^{\prime})\leq H(\xi)+\mathcal{V}_{\xi}$. Hence,
$\Phi(\xi,\mathcal{M}^{\prime})=H(\xi)+\mathcal{V}_{\xi}$ and equivalently,
$\Gamma(\xi,\mathcal{M}^{\prime})=\mathcal{V}_{\xi}$. Together with Lemma 4.4,
we have

$\operatorname{cap}_{\beta}(\xi,\mathcal{M}^{\prime})\geq
Ce^{-\beta\Phi(\xi,\mathcal{M}^{\prime})}Z_{\beta}^{-1}=Ce^{-\beta(\mathcal{V}_{\xi}+H(\xi))}Z_{\beta}^{-1}=Ce^{-\beta\mathcal{V}_{\xi}}\mu_{\beta}(\xi)$

for some $C>0$. This gives

$\frac{1}{\theta_{\beta}}\mathbb{E}_{\xi}^{\beta}\left[\int_{0}^{\tau_{\mathcal{M}^{\prime}}}\mathbbm{1}_{\\{X_{\beta}(s)\in\Delta\\}}ds\right]\leq
C^{-1}\sum_{\sigma\in\Delta}\mathbb{P}_{\sigma}^{\beta}[\tau_{\xi}<\tau_{\mathcal{M}^{\prime}}]\frac{\mu_{\beta}(\sigma)}{\mu_{\beta}(\xi)}.$

We divide the sum over $\sigma\in\Delta$ into two parts and show that each
term vanishes as $\beta\to 0$.

  1. (1)

$H(\sigma)>H(\xi)$: Bounding the probability by 1, the term exponentially
vanishes.

  2. (2)

$H(\sigma)\leq H(\xi)$: Starting from $\sigma$, we may reach a state with
lower energy without exceeding $H(\sigma)+\mathcal{V}^{\prime}$. Repeating
this until we reach $\mathcal{M}^{\prime}$, we have that
$\Gamma(\sigma,\mathcal{M}^{\prime})\leq\mathcal{V}^{\prime}$. Also, observe
that
$\Gamma(\xi,\mathcal{M}^{\prime})\leq\max(\Gamma(\xi,\sigma),\Gamma(\sigma,\mathcal{M}^{\prime}))$
and

$\Gamma(\sigma,\mathcal{M}^{\prime})\leq\mathcal{V}^{\prime}<\mathcal{V}_{\xi}=\Gamma(\xi,\mathcal{M}^{\prime}).$

Hence $\Gamma(\xi,\mathcal{M}^{\prime})\leq\Gamma(\xi,\sigma)$ and
$\Gamma(\sigma,\mathcal{M}^{\prime})<\Gamma(\xi,\sigma)$. By Lemma 4.3 on
$\mathbb{P}_{\sigma}^{\beta}[\tau_{\xi}<\tau_{\mathcal{M}^{\prime}}]$,

$\displaystyle\mathbb{P}_{\sigma}[\tau_{\xi}<\tau_{\mathcal{M}}]\frac{\mu_{\beta}(\sigma)}{\mu_{\beta}(\xi)}\leq\frac{\operatorname{cap}_{\beta}(\sigma,\xi)e^{\beta
H(\xi)}}{\operatorname{cap}_{\beta}(\sigma,\mathcal{M}^{\prime})e^{\beta
H(\sigma)}},$

and by Lemma 4.4 on both capacities, the above term is bounded by

$Ce^{-\beta(\Phi(\xi,\sigma)-H(\xi))+\beta(\Phi(\sigma,\mathcal{M}^{\prime})-H(\sigma))}=Ce^{-\beta(\Gamma(\xi,\sigma)-\Gamma(\sigma,\mathcal{M}^{\prime}))}=o_{\beta}(1).$

for some constant $C>0$.

Gathering two bounds, we obtain

$\frac{1}{\theta_{\beta}}\mathbb{E}_{\xi}^{\beta}\left[\int_{0}^{\tau_{\mathcal{M}^{\prime}}}\mathbbm{1}_{\\{X_{\beta}(s)\in\Delta\\}}ds\right]=o_{\beta}(1).$

Finally, we bound the integral over $[0,t\theta_{\beta}]$. In the case of
$\tau_{\mathcal{M}^{\prime}}>t\theta_{\beta}$, the previous result gives

$\frac{1}{\theta_{\beta}}\mathbb{E}_{\xi}^{\beta}\left[\int_{0}^{t\theta_{\beta}}\mathbbm{1}_{\\{X_{\beta}(s)\in\Delta\\}}ds;\,\tau_{\mathcal{M}^{\prime}}>t\theta_{\beta}\right]\leq\frac{1}{\theta_{\beta}}\mathbb{E}_{\xi}^{\beta}\left[\int_{0}^{\tau_{\mathcal{M}^{\prime}}}\mathbbm{1}_{\\{X_{\beta}(s)\in\Delta\\}}ds\right]=o_{\beta}(1).$

In the case of $\tau_{\mathcal{M}^{\prime}}>t\theta_{\beta}$, we split the
integral over $[0,\tau_{\mathcal{M}^{\prime}}]$ and
$[\tau_{\mathcal{M}^{\prime}},\theta_{\beta}]$. The former integral is again
$o_{\beta}(1)$ by the previous result. The latter integral can be written in

$\displaystyle\frac{1}{\theta_{\beta}}\mathbb{E}_{\xi}^{\beta}\left[\int_{\tau_{\mathcal{M}^{\prime}}}^{t\theta_{\beta}}\mathbbm{1}_{\\{X_{\beta}(s)\in\Delta\\}}ds;\,\tau_{\mathcal{M}^{\prime}}<\theta_{\beta}\right]\leq\sum_{\sigma\in\mathcal{M}^{\prime}}\frac{1}{\theta_{\beta}}\mathbb{P}_{\xi}^{\beta}[\tau_{\mathcal{M}^{\prime}}=\tau_{\sigma}]\mathbb{E}_{\sigma}^{\beta}\left[\int_{0}^{t\theta_{\beta}}\mathbbm{1}_{\\{X_{\beta}(s)\in\Delta\\}}ds\right].$

by strong Markov property on $\tau_{\mathcal{M}^{\prime}}$. Applying the
induction hypothesis to each $\sigma\in\mathcal{M}^{\prime}$, the summation is
bounded by $o_{\beta}(1)$. Summing up the results, we obtain

$\frac{1}{t\theta_{\beta}}\mathbb{E}_{\xi}^{\beta}\left[\int_{0}^{\tau_{\mathcal{M}^{\prime}}}\mathbbm{1}_{\\{X_{\beta}(s)\in\Delta\\}}ds\right]=o_{\beta}(1),$

completing the proof of the statement for $\xi\in\mathcal{M}$.

_Step 2_. The statement holds for $\xi\notin\mathcal{M}$.

To see this, consider a maximal cycle consisting of
$\mathcal{X}\backslash\mathcal{M}$. Since the stability level is bounded by
$\mathcal{V}^{\prime}$, the depths of cycles are bounded by
$\mathcal{V}^{\prime}$. By [22, Proposition 3.7], for sufficiently small
$\varepsilon>0$

$\mathbb{P}_{\xi}^{\beta}\left[\tau_{\mathcal{M}}>t\theta_{\beta}\right]<\mathbb{P}_{\xi}^{\beta}\left[\tau_{\mathcal{M}}>e^{\beta(\mathcal{V}^{\prime}+\varepsilon)}\right]=o_{\beta}(1).$

We split the integral into two parts according to $\tau_{\mathcal{M}}$ and
$t\theta_{\beta}$. The first part becomes

$\frac{1}{\theta_{\beta}}\mathbb{E}_{\xi}^{\beta}\left[\int_{0}^{t\theta_{\beta}}\mathbbm{1}_{\\{X_{\beta}(s)\in\Delta\\}}ds;\,\tau_{\mathcal{M}}>t\theta_{\beta}\right]\leq
t\mathbb{P}_{\xi}^{\beta}[\tau_{\mathcal{M}}>t\theta_{\beta}]=o_{\beta}(1).$

For the second part, we use the strong Markov property on
$\tau_{\mathcal{M}}$. Then

$\frac{1}{\theta_{\beta}}\mathbb{E}_{\xi}^{\beta}\left[\int_{0}^{t\theta_{\beta}}\mathbbm{1}_{\\{X_{\beta}(s)\in\Delta\\}}ds;\,\tau_{\mathcal{M}}<t\theta_{\beta}\right]\leq\frac{1}{\theta_{\beta}}\sum_{m\in\mathcal{M}}\mathbb{P}_{\xi}^{\beta}[\tau_{\mathcal{M}}=\tau_{m}]\mathbb{E}_{m}^{\beta}\left[\int_{0}^{t\theta_{\beta}}\mathbbm{1}_{\\{X_{\beta}(s)\in\Delta\\}}ds\right].$

The desired result follows from the case where $\xi\in\mathcal{M}$. ∎

###  4.4. Computation of transition time

Recall from Proposition 4.2 the equality

(7)
$\mathbb{E}_{1}^{\beta}[\tau_{3}]=\frac{1}{\operatorname{cap}_{\beta}(\mathbf{1},\mathbf{3})}\sum_{\sigma\in\mathcal{X}}\mu_{\beta}(\sigma)\mathbb{P}_{\sigma}^{\beta}[\tau_{1}<\tau_{3}].$

First, we show that states other than 1, 2, and 3 do not contribute to the
summation. The general form of this statement is given in [20, Lemma 3.3],
which includes our case as shown in the next lemma.

######  Lemma 4.11.

Suppose $\mathcal{M}\subseteq\mathcal{X}$ consists of states with stability
levels strictly greater than states not in $\mathcal{M}$, which is

$\mathcal{V}_{\sigma}<\mathcal{V}_{\eta}\quad\text{ for all
}\eta\in\mathcal{M},\,\sigma\notin\mathcal{M}.$

Then, for any $B\subset\mathcal{M}$ and $\xi\in\mathcal{M}\backslash B$,

$\mathbb{E}_{\xi}^{\beta}[\tau_{B}]=(1+o_{\beta}(1))\frac{1}{\operatorname{cap}_{\beta}(\xi,B)}\sum_{\sigma\in\mathcal{M}}\mu(\sigma)\mathbb{P}_{\sigma}^{\beta}[\tau_{\xi}<\tau_{B}].$

###### Proof.

For a positive functions $f(\beta)$ and $g(\beta)$ on $\beta$, we use the
notation $f(\beta)\ll g(\beta)$ if
${\lim_{\beta\to\infty}}f(\beta)/g(\beta)=0$. We define $f(\beta)\gg g(\beta)$
in a similar fashion. By [20, Lemma 3.3], it suffices to check that for
$\sigma\notin\mathcal{M}$, $\eta\in\mathcal{M}$ such that $H(\sigma)\leq
H(\eta)$,

$\frac{\operatorname{cap}_{\beta}(\sigma,\eta)}{\mu(\eta)}\ll\frac{\operatorname{cap}_{\beta}(\sigma,\mathcal{M})}{\mu(\sigma)},$

or equivalently $\Gamma(\eta,\sigma)>\Gamma(\sigma,\mathcal{M})$. If
$H(\eta)<H(\sigma)$, then $\Gamma(\eta,\sigma)\geq\mathcal{V}_{\eta}$ by
definition, and
$\Gamma(\sigma,\mathcal{M})\leq\max_{i\in\mathcal{X}\backslash\mathcal{M}}\mathcal{V}_{i}$
by starting from $\sigma$ and repeatedly moving to a state with lower energy
until hitting $\mathcal{M}$. Hence, the desired condition holds. If
$H(\eta)=H(\sigma)$, then the energy barrier $\Gamma(\eta,\sigma)$ must be
greater than $\mathcal{V}_{\eta}$, or unless
$\mathcal{V}_{\sigma}=\mathcal{V}_{\eta}$ and this contradicts
$\sigma\notin\mathcal{M}$ and $\eta\in\mathcal{M}$. Hence
$\Gamma(\eta,\sigma)>\mathcal{V}_{\eta}\geq\max_{\sigma^{\prime}\in\mathcal{X}\backslash\mathcal{M}}\mathcal{V}_{\sigma^{\prime}}\geq\mathcal{V}_{\sigma}$
as before. Therefore, the desired condition holds for either case. ∎

Finally, we prove Theorem 2.8. Applying the previous lemma to (7), we have

$\mathbb{E}_{\mathbf{1}}^{\beta}[\tau_{\mathbf{3}}]=(1+o_{\beta}(1))\frac{1}{\operatorname{cap}_{\beta}(\mathbf{1},\mathbf{3})}\left(\mu_{\beta}(\mathbf{1})+\mu_{\beta}(\mathbf{2})\mathbb{P}_{\mathbf{2}}^{\beta}[\tau_{\mathbf{1}}<\tau_{\mathbf{3}}]\right).$

Putting the estimate of $\operatorname{cap}_{\beta}(\mathbf{1},\mathbf{3})$
from Proposition 4.7,

$e^{-\beta\Gamma_{13}^{\star}}\mathbb{E}_{\mathbf{1}}^{\beta}[\tau_{\mathbf{3}}]=(\kappa_{1}+o_{\beta}(1))\left(1+e^{\beta(H(\mathbf{1})-H(\mathbf{2}))}\mathbb{P}_{\mathbf{2}}^{\beta}[\tau_{\mathbf{1}}<\tau_{\mathbf{3}}]\right).$

Hence, it suffices to obtain a sufficient lower bound of
$\mathbb{P}_{\mathbf{2}}^{\beta}[\tau_{\mathbf{1}}<\tau_{\mathbf{3}}]$. This
can be done by constructing a cycle-path from 2 to 1 disjoint to 3, then
computing the probability of following the path. Readers can find a similar
construction of paths in [20, Lemma 6.1], for example.

###### Proof of Theorem 2.8.

To obtain a lower bound of
$\mathbb{P}_{\mathbf{2}}^{\beta}[\tau_{\mathbf{1}}<\tau_{\mathbf{3}}]$, we
construct a cycle-path from 2 to 1 disjoint to 3. This is done by reversing a
reference path of the Ising model of spin 1 and 2. Precisely, we divide the
path into the following four parts.

  * •

Let $\gamma_{0}$ be a path from 2 to $\mathcal{R}^{2}_{1}({K-1},{L-1})$
obtained by adding 1-spin along one row, then adding 1-spin along one column.
We see that the energy strictly increases.

  * •

For $1\leq i\leq K-L-1$, let $\gamma_{i}$ be a path from
$\mathcal{R}^{2}_{1}({K-i},{L-1})$ to $\mathcal{R}^{2}_{1}({K-i-1},{L-1})$
obtained by consecutively adding 1-spin along a vertical side of 2-cluster.

  * •

For $K-L\leq i\leq K+L-1$, let $\gamma_{i}$ be a path adding a line of 1-spin
to transform a 2-cluster of size $\ell\times\ell$ into size
$\ell\times(\ell-1)$, or 2-cluster of size $(\ell+1)\times\ell$ into size
$\ell\times\ell$, alternatively. We end at
$\mathcal{R}^{2}_{1}({\ell_{c}^{12}},{\ell_{c}^{12}})$. Finally, let
$\gamma_{K+L-1+2\ell_{c}^{12}+1}$ be a path connecting
$\mathcal{R}^{2}_{1}({\ell_{c}^{12}},{\ell_{c}^{12}})$ and
$\mathcal{D}^{2}_{1}({\ell_{c}^{12}-1},{\ell_{c}^{12}})$, which is the saddle
configuration of the Ising model between 1 and 2.

  * •

Let $\beta_{0}$ be a length 2 path from
$\mathcal{D}^{2}_{1}({\ell_{c}^{12}-1},{\ell_{c}^{12}})$ to
$\mathcal{R}^{2}_{1}({\ell_{c}^{12}-1},{\ell_{c}^{12}})$. For $1\leq i\leq
2\ell_{c}^{12}$, let $\beta_{i}$ be a path defined the same as above,
connecting $\mathcal{R}^{2}_{1}({\ell_{c}^{12}},{\ell_{c}^{12}})$ and
$\mathcal{R}^{2}_{1}({0},{0})$, which is 1.

Let $\gamma$ be a path obtained by concatenating
$\gamma_{0},\ldots,\gamma_{K+L-1+2\ell_{c}^{12}+1}$, and $\beta$ be a path
concatenating $\beta_{0},\ldots,\beta_{2\ell_{c}^{12}}$. Since the energy
fluctuates along the path $\gamma\cup\beta$, computing the transition rate
along the path does not yield a desirable lower bound. Nevertheless, we may
remedy this by considering a cycle-path and then computing the probability of
the process following the cycle-path. Namely, for each state in
$\gamma\cup\beta$, consider a maximal cycle disjoint from 3. This gives a
cycle-path from $C(\mathbf{2})$ to $C(\mathbf{1})$. Here, $C(\sigma)$ denotes
the maximal cycle disjoint from 3 and containing $\sigma$. We claim that the
energy increases along the cycle-path of $\gamma$, and that energy decreases
along the cycle-path of $\beta$.

First, we prove the claim on each path $\gamma_{k}$. Denote the path as
$\gamma_{k}=(\eta_{0},\ldots,\eta_{n})$. If $k=0$, the energy of the state
strictly increases, so we are done. Thus, we may assume $k>0$. Then
$\eta_{0}\in\mathcal{R}^{2}_{1}({n},{m+1})$ and
$\eta_{n}\in\mathcal{R}^{2}_{1}({n},{m})$ for $n,m\geq\ell_{c}^{12}$.

From $\eta_{0}$ to $\eta_{n-1}$, the energy increase by $h_{2}-h_{1}$. To
compare $\eta_{n-1}$ and $\eta_{n}$, recall the variant of Lemma 3.8 on 1 and
2 that $\eta_{n-1}$ is a state with minimal energy in $\partial C(\eta_{n})$.
Hence, the energy monotone increases along the cycle-path obtained by
$(\eta_{0},\ldots,\eta_{n})$. Therefore, we have the claim for $\gamma$.

We may prove the claim similarly on each path $\beta_{k}$. In this case, each
$\beta_{k}$ is contained in a single cycle. These cycles are adjacent and the
energy decreases along $\beta$.

Finally, we compute the probability of the process following the cycle-path
constructed above. Let $(C_{1},\ldots,C_{a})$ be a uphill cycle-path from
$C_{1}=C(\mathbf{2})$ to
$C_{a}=C(\mathcal{D}^{2}_{1}({{\ell_{c}^{12}-1}},{\ell_{c}^{12}}))$, and
$(D_{1},\ldots,D_{b})$ be a downhill cycle-path from $D_{1}=C_{a}$ to
$D_{b}=C(\mathbf{1})$. Note that this cycle-path is disjoint from 3. We have

$\displaystyle\mathbb{P}_{\mathbf{2}}[\tau_{\mathbf{1}}<\tau_{\mathbf{3}}]\geq\mathbb{P}_{2}^{\beta}[\tau_{\partial
C_{0}}=\tau_{C_{1}}]\prod_{i=1}^{a-1}\sup_{\sigma\in
C_{i}}\mathbb{P}_{\sigma}^{\beta}[\tau_{\partial
C_{i}}=\tau_{C_{i+1}}]\prod_{j=1}^{b-1}\sup_{\sigma\in
D_{i}}\mathbb{P}_{\sigma}^{\beta}[\tau_{\partial D_{i}}=\tau_{D_{i+1}}].$

From Proposition 3.1, for any $\varepsilon>0$, for all $\beta$ sufficiently
large,

$\sup_{\sigma C_{i}}\mathbb{P}_{\sigma}^{\beta}[\tau_{\partial
C_{i}}=\tau_{C_{i+1}}]\geq e^{-\beta[H(\mathcal{F}(\partial
C_{i+1}))-H(\mathcal{F}(\partial C_{i}))+\varepsilon]_{+}}.$

Therefore, we have

$\displaystyle\mathbb{P}_{\mathbf{2}}^{\beta}[\tau_{\mathbf{1}}<\tau_{\mathbf{3}}]$
$\displaystyle\geq
e^{-\beta[H(\mathcal{D}^{2}_{1}({\ell_{c}^{13}+1},{\ell_{c}^{13}}))-H(C(\mathbf{2}))+\varepsilon]}$
$\displaystyle=e^{-\beta[(H(\mathbf{1})+\Gamma^{\star}_{12})-(H(\mathbf{2})+\Gamma^{\star}_{23})+\varepsilon]}.$

Assuming $\Gamma^{\star}_{12}>\Gamma^{\star}_{23}$, we have

$\mu(2)\mathbb{P}_{2}^{\beta}[\tau_{1}<\tau_{3}]\gg\mu(1),$

so the left-hand-side probability majorizes the summation (7). Hence,

${\lim_{\beta\to\infty}}e^{-\beta\Gamma^{\star}_{13}}\mathbb{E}_{1}^{\beta}[\tau_{3}]=\infty.$

∎

##  Appendix A Appendix

###  A.1. Reference path of the Ising model

Recall the Ising model on a lattice graph $\Lambda=(V,E)$ defined by a
Hamiltonian function

$H(\sigma)=-J\sum_{(x,y)\in
E}\mathbbm{1}_{\\{\sigma(x)=\sigma(y)\\}}-h\sum_{x\in
V}\mathbbm{1}_{\\{\sigma(x)=1\\}}$

where $\sigma\in\\{-1,1\\}^{V}$ is spin configuration on $\Lambda$ and
$J,h>0$. For simplicity, we put $J=1$. The model has a unique metastable state
$\mathbf{-1}$, the configuration with constant spin $-1$, and a unique stable
state $\mathbf{+1}$, the configuration with constant spin $1$.

We describe the well-known _reference path_ , which is the energy minimizing
path from $\mathbf{-1}$ to $\mathbf{+1}$. While it was originally introduced
in [23], it has become a well-established concept and is explained in various
literature. As an example, one can find a detailed explanation in [9, Section
17].

  * •

Starting from $\mathbf{-1}$, flip a single $-1$ spin to make a 1-cluster, a
connected component of 1-spin, of size $1\times 1$.

  * •

Having an 1-cluster of size $i\times i$ or $i\times(i+1)$ in a sea of
$-1$-spins, flip a $-1$-spin adjacent to an edge of length $i$. This process
increases the energy by $2-h$. Accordingly, flip the rest of the $-1$-spins
into $1$-spins along the edge. We end up having an 1-cluster of size
$i\times(i+1)$ or $(i+1)\times(i+1)$, respectively. Note that each step
decreases the energy by $-h$.

  * •

Repeat the previous step until the 1-cluster grows up to the whole lattice.

Along the reference path, the Hamiltonian function has a unique maximum when
adding a spin to a 1-cluster of size $(\ell_{c}-1)\times\ell_{c}$. This
configuration is called the _critical droplet_. Moreover, the direct
computation gives a energy barrier from $\mathbf{-1}$ to $\mathbf{+1}$,
written by

$f(h)\coloneqq
4\ell_{c}-h(\ell_{c}(\ell_{c}-1)+1),\quad\ell_{c}=\left\lceil\frac{2}{h}\right\rceil.$

###  A.2. Auxiliary function

In this section, we provide a comparison between several quantities appearing
throughout the paper. Let $f(h)$ be a function on $h$ defined by the above
equation. Recall from (3) the quantities

$\Gamma_{ij}^{\star}=f(h_{j}-h_{i}),\quad\Phi_{ij}^{\star}=f(h_{j}-h_{i})+H(\mathbf{i})$

which were used to represent the energy barrier and communication height of
our model.

######  Lemma A.1.

A function $f(h)$ is continuous and strictly decreasing on $h\in(0,\infty)$.
Moreover, $5<f(h)<\frac{8}{h}$ for $h\in(0,1)$.

###### Proof.

Let $h\in[\frac{2}{m+1},\frac{2}{m})$ for $m\in\mathbb{Z}^{+}$. Then
$\ell_{c}=m+1$ and we have

$f(h)=4(m+1)-h(m(m+1)+1),$

which is a continuous decreasing function on $[\frac{2}{m+1},\frac{2}{m})$.
Also, note that

$\lim_{h\to\frac{2}{m}+}f(h)=4m-\frac{2}{m}\left(m(m-1)+1\right)=2m+2-\frac{2}{m}$

and

$\lim_{h\to\frac{2}{m}-}f(h)=4(m+1)-\frac{2}{m}\left(m(m+1)+1\right)=2m+2-\frac{2}{m}.$

Therefore, $f(h)$ is continuous and strictly decreasing on $(0,\infty)$.

Finally, we have that

$f(h)>f(1)=5,\quad f(h)=4m-h(m(m-1)+1)<4m<\frac{8}{h}$

for $h\in(0,1)$. This proves the lemma. ∎

The lemma directly implies that $\Gamma_{13}^{\star}<\Gamma_{23}^{\star}$ and
$\Gamma_{13}^{\star}<\Gamma_{12}^{\star}$. Moreover,

$\Gamma_{13}^{\star}=f(h_{3}-h_{1})>5$

by the assumption $0<h_{1}<h_{2}<h_{3}<1$ in Section 2.

Finally, we compare the Hamiltonian between states of our interest: three
monochromatic configurations 1, 2, and 3, and saddle configurations between
each two of these.

######  Lemma A.2.

The following inequality holds.

$\Phi_{12}^{\star}>\Phi_{13}^{\star}>H(\mathbf{1})>\Phi_{23}^{\star}>H(\mathbf{2})>H(\mathbf{3}).$

We also remark that the lemma shows that the six states of Figure 1 are
depicted in the correct order of energy level.

###### Proof.

Recall that $K\times L$ is the size of the lattice graph the model lives. We
have assumed in Assumption A that
$K,L>\frac{3}{h_{3}-h_{2}},\,\frac{3}{h_{2}-h_{1}}$.

The first three quantities can be compared using the previous lemma. Indeed,
$h_{2}-h_{1}<h_{3}-h_{1}<1$ implies

$\Gamma_{12}^{\star}=f(h_{2}-h_{1})>\Gamma_{13}^{\star}=f(h_{3}-h_{1})>0.$

Adding $H(\mathbf{1})$ to each side gives the desired inequality.

Next, we show $H(\mathbf{1})>\Phi_{23}^{\star}$. This is equivalent to showing

$(h_{2}-h_{1})KL>\Gamma_{23}^{\star}=f(h_{3}-h_{2}).$

By Assumption A, we have

$(h_{2}-h_{1})KL>(h_{2}-h_{1})\cdot\frac{3}{h_{2}-h_{1}}\cdot\frac{3}{h_{3}-h_{2}}>f(h_{3}-h_{2})$

where the last inequality follows from the previous lemma. Hence, we have
$H(\mathbf{1})>\Phi_{23}^{\star}$.

Finally, $\Phi_{23}^{\star}>H(\mathbf{2})$ follows from
$\Gamma_{23}^{\star}>0$, and clearly $H(\mathbf{2})>H(\mathbf{3})$. This
completes the proof. ∎

#### Acknowledgement

This research is supported by the National Research Foundation of Korea (NRF)
grant funded by the Korea government (MSIT) (No. 2023R1A2C100517311). The
author thanks Insuk Seo and Seonwoo Kim for introducing the problem and
offering enlightening insights.

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