E_mail<EMAIL_ADDRESS>
Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sapienza
Università di Roma, via A. Scarpa 16, I–00161, Roma, Italy.

E_mail<EMAIL_ADDRESS>
Dipartimento di Matematica e Informatica “Ulisse Dini”, viale Morgagni 67/a,
50134, Firenze, Italy.

E_mail<EMAIL_ADDRESS>
Institute of Mathematics, University of Utrecht, Budapestlaan 6, 3584 CD
Utrecht, The Netherlands.

# Metastability of synchronous and asynchronous dynamics

Emilio N.M. Cirillo  Vanessa Jacquier  Cristian Spitoni

###### Abstract

Metastability is an ubiquitous phenomenon in nature, which interests several
fields of natural sciences. Since metastability is a genuine non–equilibrium
phenomenon, its description in the framework of thermodynamics and statistical
mechanics has progressed slowly for a long time. Since the publication of the
first seminal paper in which the metastable behavior of the mean field
Curie–Weiss model was approached by means of stochastic techniques, this topic
has been largely studied by the scientific community. Several papers and books
have been published in which many different spin models were studied and
different approaches were developed. In this review we focus on the comparison
between the metastable behavior of synchronous and asynchronous dynamics,
namely, stochastic processes in discrete time in which at each time either all
the spins or one single spin are updated. In particular we discuss how the two
different stochastic implementation of the very same Hamiltonian give rise to
different metastable behaviors.

Keywords: Metastability; Lattice spin systems; Probabilistic cellular
automata; Synchronous dynamics; Asynchronous dynamics.

##  1 Introduction

Metastable states are commonly observed in several diverse fields such as
physics, chemistry, biology, computer science, climatology, and economics. In
order to portray their main features we use, here, the classical example of
super–saturated vapors: under special experimental conditions, a vapor can be
compressed at pressures lower than the value at which liquefaction should
start. The vapor, thus, enters a state different from the thermodynamic
equilibrium phase, but, for small variations of the thermodynamics parameters,
it behaves as if it were in real equilibrium following the laws of
thermodynamics and undergoing small reversible changes. The vapor can remain
in such a state for a very long time, but it can exit such a state, and reach
the real thermodynamic phase (the liquid phase), via internal random
fluctuation or external perturbations [1]. The eventual transition from the
vapor to the liquid phase is irreversible.

Other well known contexts in which metastability is observed are
crystallization of proteins and ferromagnetic materials (branches of the
hysteresis loop where the magnetization is opposite to the external magnetic
field).

The study of metastability has attracted much attention in the last decades
not only for its intrinsic interest, but also because metastable states are an
example of non–equilibrium states of thermodynamical systems. As it is well
known, Statistical Mechanics has developed a complete mathematical formalism,
even on rigorous basis [2], to describe equilibrium states, whereas a complete
consistent theory of systems out of equilibrium is still lacking. Thus,
deriving a rigorous mathematical theory of metastable states can shed same
light on rigorous approaches to non–equilibrium thermodynamics.

Figure 1: Left: the Maxwell equal area rule is used to replace part of the van
der Waals isotherm with an horizontal segment yielding the isotherm describing
the liquid–vapor transition (black curve). The van der Waals isotherm
$(P+a/V^{2})(V-b)=RT$, has been plotted for $a=1.36~{}\ell^{2}$ atm mol-2 and
$b=0.0319~{}\ell$ mol-1, at $T=140^{\textup{o}}$K, with $R=0.082058~{}\ell$
atm mol-1. The equal area rule yields $33.48$ atm as the value of the pressure
at which vapor and liquid coexists. Right: black lines, from the bottom to the
top, are van der Waals isotherms for the same $a$ and $b$ at temperatures
$T=130,140,150,153.94^{\textup{o}}$K. The red and the blue curves are,
respectively, the liquid and the vapor branches of the spinodal curve, which
meet at the critical point $V_{\textup{crit}}=3b=0.0957~{}\ell$ and
$P_{\textup{crit}}=a/(27b^{2})=49.50$ atm on the critical isotherm at
$T_{\textup{crit}}=8a/(27Rb)=153.94^{\textup{o}}$ K. Note that the values of
$a$ and $b$ that we used are the ones valid for the Oxygen, the experimental
value of the critical temperature is $156^{\textup{o}}$K and the vapor
pressure at temperature $140^{\textup{o}}$K is $27.50$ atm [3].

The first attempts to provide a theoretical explanation of metastability are
based on the real gas van der Waals equation [4], which, coupled with the
Maxwell construction [5, 6], can be interpreted as an equation describing the
liquid–vapor transition (see the left panel of figure 1). Below the critical
temperature the van der Waals isotherms have a not monotonic behavior, but,
using the _equal area_ Maxwell rule the kink can be replaced by a segment
joining the high pressure part of the curve representing the liquid and the
low pressure part of the curve representing the vapor.

At the points in the part of the original isotherm curve between the minimum
and the maximum, called the _instability_ branch of the isotherm, the
compressibility would be negative, since $\partial v/\partial p>0$ (see the
green part of the isotherm in the left panel of figure 1). The locus of the
minimum and maximum points of the sub–critical isotherms in the plane $V$–$P$
is called the _spinodal curve_ : the minimum and the maximum points form,
respectively, the liquid and the vapor branches of the curve (red and blue
curves in the right panel of figure 1). All the points below the spinodal
curve cannot represent equilibrium states of the real gas since they follow on
the instability branch of one of the van der Waals isotherm and, thus, are
mechanically unstable since they would have negative compressibility. Indeed,
if a real gas is prepared in one of those states the liquid and the vapor
phase quickly separate through a mechanism called _spinodal_ decomposition. On
the other hand, if the real gas is prepared in states above the spinodal curve
not belonging to the pure phase branches of the Maxwell isotherms, under
particular experimental conditions, it is possible to observe metastable
states, such as super–saturated vapor and super–heated liquid. It is thus
rather natural to interpret those points (respectively, the red and the blue
arcs in the left panel of figure 1) of the van der Walls isotherms as
metastable states.

Although several important studies tried to develop rigorous theories of
metastability in the framework of thermodynamics and Statistical Mechanics of
Gibbsian ensembles [7], it was soon clear that metastability is a genuine
dynamical phenomenon which needs to be described by means of non–equilibrium
Statistical Mechanics ideas. The first kinetical approach to metastability is
the Becker–Doring theory [8], which dates back to 1935. However, almost half a
century was needed to arrive to the first rigorous mathematical discussion of
metastable states in the case of a simple mean–field spin system [9], i.e.,
the Curie–Weiss model. The necessity of a dynamical approach was also pointed
out by many numerical studies; we refer to the review [10] and to references
therein.

In [9] the question of metastable states is posed in the framework of a
stochastic spin system whose evolution is a Markov chain defined as a
reversible Glauber dynamics. An external field is introduced in the
Hamiltonian to select the _stable_ homogeneous phase (corresponding to the
minimum of the Hamiltonian). As initial condition of the dynamics, the
opposite state is considered, and its metastable character is shown by proving
that the time needed by the system to hit for the first time the stable state
is exponentially large with respect to the inverse temperature parameter. The
method proposed in [9] is today known as _pathwise approach_ and has been used
to study metastablity in a large variety of models. The first paper in which
it was applied with success to a model with a physically acceptable short
range interaction is [11] in which the existence of metastable states was
proven for the Ising model.

The basic idea of the pathwise approach is searching for the optimal path in
the configuration space connecting the metastable state to the stable one and
computing its energy height. The time needed by the system to perform the
transition from the metastable to the stable state, i.e., the _exit time_ , is
a random variable whose mean value can be estimated by an exponential function
of the energy barrier that the system must overcome during the transition. The
method provides also the portion of the configuration space explored by the
system before performing the transition to the stable state and the tube of
trajectories, called the _exit path_ , followed during the exit excursion.
Along the exit path the system necessarily visits those particular
configurations, called the _critical configurations_ , at which the optimal
path attains its maximum. In the original version of the theory this was
achieved via a detailed study of the configuration space and a suitable
definition of _basin of attraction_ of the metastable state based on the
notion of cycles. The pathwise approach was further developed in [12, 13, 14],
see also [15], and several techniques were introduced to simplify the
application of the method. It is worth mentioning that, independently, a
similar cycle theory was derived in [16, 17] and applied to reversible
Metropolis dynamics and to simulated annealing [18, 19].

The general properties of the pathwise approach were further analyzed in [20,
21, 22, 23, 24] to disentangle the study of the transition time from that of
the typical trajectories and to treat irreversible system. This method has
been used to study the metastable behavior of the Ising model with isotropic
and anisotropic interaction, in different dimensions, with different external
magnetic fields evolving according to Glauber dynamics, in [25, 26, 27, 28,
11, 29, 15, 30]. Moreover, it has been used also for a variety of other models
evolving according to Glauber dynamics, such as the Blume Capel model in [31,
21], the Potts model in [32, 33], and hard–core model in [34, 35, 36]. Other
applications of the pathwise approach are present in [37, 38, 39, 40] for the
Metropolis dynamics and in [41, 42, 43] for parallel dynamics.

The focus of this paper is on the comparison between the metastable behavior
of _serial_ or _asynchronous_ dynamics, i.e, stochastic spin systems in which
the configurations are updated a site at a time, and _parallel_ or
_synchronous_ dynamics, i.e., stochastic spin systems in which at each time
step all the spins are simultaneously updated. Although the general ideas are
similar, the parallel case is utterly more puzzling due to the very intricate
structure of the possible paths that can be followed in the configuration
space by the system during its random motion. In Section 2 we consider the
Hamiltonian of the 2D Ising model and we show how it is possible to construct
different dynamics by allowing or forbidding simultaneous spin updating. The
structure of their critical configurations and optimal paths are discussed in
Section 3.

Before closing the Introduction, it is worth mentioning two more approaches to
the rigorous mathematical description of metastability which have been
developed in the last decades. One is known as the _potential–theoretic
approach_ and is based on the seminal papers [44, 45]. We refer to [46] for an
extensive discussion of this method and of its applications to several models.
In the potential–theoretic approach the estimate of the hitting time is
achieved through the use of the Dirichlet form and the spectral properties of
the transition matrix. One of the advantages of this method is that it
provides an estimate of the expected value of the transition time including
the prefactor, by exploiting a detailed knowledge of the critical
configurations, see [47, 46]. This method has been applied in [48, 49, 50, 51,
52, 53] to Metropolis dynamics and in [54] to parallel dynamics.

Finally, we mention the more recent _trace method_ firstly introduced in [55],
that extends in some sense the path–wise approach and the potential theoretic
approach. Differently from the pathwise approach, it does not rely on large
deviations estimates, so that it can be used to study models where the ratio
between the jump rates are not exponential in the scaling parameter (e.g.,
condensing zero–range processes). Moreover, differently from the potential
approach the method does not depend on a reversibility assumption. The main
idea is to consider a reduction of the process by removing rapid fluctuations
from the trajectory. The authors considered indeed the trace of the process on
the metastable states. Then, the metastability behavior is examined through a
martingale problem that controls the convergence of the trace process.
Moreover, based on the martingale characterization of Markov processes, a
sufficient conditions for metastability can be given (see [56] for a general
review on the method and its applications).

##  2 Models

Let $\Lambda$ be a finite square of $\mathbb{Z}^{2}$ with periodic boundary
conditions, namely, a two–dimensional finite torus. With each site
$i\in\Lambda$ it is associated a _spin variable_ $\sigma(i)\in\\{-1,+1\\}$. We
denote by $\Omega=\\{-1,+1\\}^{\Lambda}$ the _configuration_ or _state_ space
and we call _configuration_ or _state_ any element $\sigma\in\Omega$. We say
that two sites are _nearest neighbors_ if and only if their Euclidean distance
is equal to one. Given $i\in\Lambda$, we consider the _shift operator_
$\Theta_{i}:\Omega\to\Omega$ which shifts a configuration so that the site $i$
is mapped to the origin $0$, that is to say
$(\Theta_{i}(\sigma))_{j}=\sigma_{i+j}$ for any $j\in\Lambda$. Given
$\Delta\subset\Lambda$ we denote by $\sigma_{\Delta}$ the restriction to
$\Delta$ of a configuration $\sigma\in\Omega$. Given $\sigma\in\Omega$ we
denote by $\sigma^{s}$, for $s\in\\{-1,+1\\}$, the configuration obtained by
setting to $s$ the value of the spin at the origin, namely, $\sigma^{s}(0)=s$
and $\sigma^{s}(i)=\sigma(i)$ for any $i\in\Lambda\setminus\\{0\\}$.

A _pairwise interaction_ is a collection of real numbers $J_{ij}$, for any
$i,j\in\Lambda$, for $i\neq j$, symmetrical and translationally invariant,
which means that $J_{ij}=J_{ji}$ and $J_{ij}=J_{i+k,j+k}$ for all
$k\in\mathbb{Z}^{2}$. We shall also assume that the interaction is _finite
range_ , i.e., there exists $I\subset\Lambda\setminus\\{0\\}$ (not depending
on the size of $\Lambda$) such that $J_{0i}\neq 0$ for any $i\in I$ and
$J_{0i}=0$ otherwise: it is worth noting that the symmetry of the interaction
implies that $I$ is symmetric with respect to the origin. We thus define the
Hamiltonian

$H(\sigma)=-\frac{1}{2}\sum_{\genfrac{}{}{0.0pt}{}{i,j\in\Lambda:}{i\neq
j}}J_{ij}\sigma(i)\sigma(j)-h\sum_{i\in\Lambda}\sigma(i),$ (2.1)

with $h\in\mathbb{R}$ the _external magnetic field_ , and consider on the
single spin space the probability distribution

$f_{T,\sigma}(s)=\frac{\exp\\{-H(\sigma^{s})/T\\}}{\exp\\{-H(\sigma^{s})/T\\}+\exp\\{-H(\sigma^{-s})/T\\}}$
(2.2)

for any $\sigma\in\Omega$. Note that the probability distribution
$f_{T,\sigma}$ can be rewritten as

$f_{T,\sigma}(s)=\frac{1}{1+\exp\\{[H(\sigma^{s})-H(\sigma^{-s})]/T\\}}=\frac{1}{1+\exp\\{-2s(\sum_{i\in
I}J_{0i}\sigma(i)+h)/T\\}},$ (2.3)

which then implies

$f_{T,\sigma}(s)=\frac{1}{2}\Big{\\{}1+s\tanh\Big{[}\frac{1}{T}\Big{(}\sum_{i\in
I}J_{0i}\sigma(i)+h\Big{)}\Big{]}\Big{\\}}.$ (2.4)

We shall consider stochastic evolutions in $\Omega$ modelled as Markov chains
in which single spins are updated according to the probability distribution
(2.2). The dynamics can be implemented by allowing one spin change at a time
or the simultaneous change of all the spins on the lattice, the dynamics will
be respectively called _asynchronous_ or _synchronous_.

###  2.1 Asynchronous dynamics

The _heat bath_ dynamics is defined as the discrete time Markov chain
$\sigma_{t}\in\Omega$, with $t\in\mathbb{Z}_{+}$, with updating rule defined
as follows: at time $t$ a site $i$ is chosen at random with uniform
probability $1/|\Lambda|$ and the configuration $\sigma_{t}$ is constructed by
letting $\sigma_{t}(j)=\sigma_{t-1}(j)$ for any $j\neq i$ and
$\sigma_{t}(i)=s$ with probability $f_{T,\Theta_{i}\sigma_{t-1}}(s)$. In other
words, the heat bath dynamics is the discrete time Markov chain with
transition probability defined as follows: for $\sigma,\eta\in\Omega$ such
that $\sigma\neq\eta$

$p_{T}(\sigma,\eta)=\left\\{\begin{array}[]{ll}\frac{1}{|\Lambda|}f_{T,\Theta_{i}\sigma}(\eta(i))&\textup{if
}\exists i\in\Lambda:\eta(i)\neq\sigma(i)\textup{ and
}\eta(j)=\sigma(j)\textup{ for }j\neq i\\\
0&\textup{otherwise}\end{array}\right.$ (2.5)

and

$p_{T}(\sigma,\sigma)=1-\sum_{\eta\in\Omega\setminus\\{\sigma\\}}p_{T}(\sigma,\eta).$
(2.6)

As it is well known [57, Section 4.5.1], the heat bath dynamics is
_reversible_ with respect to the _Gibbs measure_ on $\Omega$

$\mu_{T}(\sigma)=\frac{1}{Z_{T}}e^{-H(\sigma)/T}$ (2.7)

where the _partition function_ is

$Z_{T}=\sum_{\sigma\in\Omega}e^{-H(\sigma)/T},$ (2.8)

that is to say, the _detailed balance condition_

$\mu_{T}(\sigma)p_{T}(\sigma,\eta)=\mu_{T}(\eta)p_{T}(\eta,\sigma)$ (2.9)

is satisfied for any $\sigma,\eta\in\Omega$. This, together with the fact that
the Markov chain is irreducible and the state space is finite, implies that
the stationary measure is unique and it is given by the Gibbs measure.

Asynchronous models are typically used in Statistical Mechanics to introduce
stochastic versions of statistical spin systems in order to study how
equilibrium is approached. Often, for efficiency reasons, the Metropolis
algorithm [57, Section 3.1] is preferred to heat bath and to other similar
rules.

A crucial role in the study of the metastable behavior of the dynamics is
played by the so called energy cost. Consider two configurations, $\sigma$ and
$\eta$, differing by only the spin at site $i$; we have that, in the limit
$T\to 0$, $p(\sigma,\eta)\approx 1/|\Lambda|$ if $H(\eta)<H(\sigma)$ and
$p(\sigma,\eta)\approx\exp\\{-[H(\eta)-H(\sigma)]/T\\}/|\Lambda|$ if
$H(\eta)>H(\sigma)$. Then it is reasonable to define the _energy cost_ of the
transition from $\sigma$ to $\eta$ as the quantity $\Delta(\sigma,\eta)=0$ if
$H(\eta)<H(\sigma)$ and $\Delta(\sigma,\eta)=H(\eta)-H(\sigma)$ if
$H(\eta)>H(\sigma)$.

###  2.2 Synchronous dynamics

A model in which all the spins are updated at each time independently and
simultaneously with the probability distribution (2.2) can be defined as the
Markov chain $\sigma_{t}\in\Omega$, with $t\in\mathbb{Z}_{+}$, with transition
matrix

$p_{T}(\sigma,\eta)=\prod_{i\in\Lambda}f_{T,\Theta_{i}\sigma}(\eta(i)).$
(2.10)

This model is an example of _reversible Probabilistic Cellular Automata_
(PCA), see, e.g., [58, 59, 60]. We mention that the class of reversible PCA is
slightly larger, indeed, the constraint that the set $I$ in (2.4) does not
contain the origin can be relaxed; when the origin is considered in $I$, as,
for instance, in [61, 41] its contribution is called _self–interaction_ term.
It does not appear in our derivation of reversible PCA, since we started from
the Hamiltonian (2.1) where it would appear simply as a constant additive
irrelevant contribution [57, Section 4.5.1, equation (4.38)].

As it is proven in [62, 63], reversible Probabilistic Cellular Automata
satisfy the detailed balance condition

$\mu_{T}(\sigma)p_{T}(\sigma,\eta)=\mu_{T}(\eta)p_{T}(\eta,\sigma)$ (2.11)

with respect to the Gibbs measure on $\Omega$

$\mu_{T}(\sigma)=\frac{1}{Z_{T}}e^{-G_{T}(\sigma)/T}$ (2.12)

where

$Z_{T}=\sum_{\sigma\in\Omega}e^{-G_{T}(\sigma)/T}$ (2.13)

is the partition function and

$G_{T}(\sigma)=-h\sum_{i\in\Lambda}\sigma(i)-T\sum_{i\in\Lambda}\log\cosh\Big{[}\frac{1}{T}\Big{(}\sum_{j\in
i+I}J_{ij}\sigma(j)+h\Big{)}\Big{]}$ (2.14)

for any $\sigma\in\Omega$. Thus, even in the synchronous case we have that the
stationary measure is unique, but it is different from the Gibbs measure (2.7)
found for the Statistical Mechanics model considered in the asynchronous case.

It is interesting to note that the Gibbs measure (2.12) is such that
$G_{T}=-T\log(Z_{T}\mu_{T})$ depends on the temperature, which is not the case
in standard Statistical Mechanics models. On the other hand, the global minima
of the function

$H(\sigma)=\lim_{T\to
0}G_{T}(\sigma)=-h\sum_{i\in\Lambda}\sigma(i)-\sum_{i\in\Lambda}\Big{|}\sum_{j\in
i+I}J_{ij}\sigma(j)+h\Big{|}$ (2.15)

configurations are the configurations in which the dynamics will be trapped at
low temperature. By abusing the notation, the function $H$ will be called
_Hamiltonian_ or _energy_ and its global minima will be called _ground
states_. Moreover, a crucial role in our discussion will be played by the
_cost function_

$\Delta(\sigma,\eta)=-\lim_{T\to
0}T\log\pi_{T}(\sigma,\eta)=\sum_{\genfrac{}{}{0.0pt}{}{i\in\Lambda:}{\eta(i)[\sum_{j\in
i+I}J_{ij}\sigma(j)+h]<0}}2\Big{|}\sum_{j\in i+I}J_{ij}\sigma(j)+h\Big{|},$
(2.16)

since, as it as been proven in [41, Section 2.6], for the transition matrix it
is possible to prove the so–called Friedlin–Wentzel condition

$e^{-\Delta(\sigma,\eta)/T-\gamma(T)/T}\leq\pi_{T}(\sigma,\eta)\leq
e^{-\Delta(\sigma,\eta)/T+\gamma(T)/T}$ (2.17)

with $\gamma(T)\to 0$ as $T\to 0$. Equation (2.17), together with the equality

$H(\sigma)+\Delta(\sigma,\eta)=H(\eta)+\Delta(\eta,\sigma),$ (2.18)

which follows immediately from (2.11), (2.12), (2.15), (2.16), allows us to
interpret $\Delta(\sigma,\eta)$ as the energy cost that the chain has to pay
in the jump from $\sigma$ to $\eta$. We note that it is possible that both
$\Delta(\sigma,\eta)$ and $\Delta(\eta,\sigma)$ are positive. This is not the
case in the asynchronous case, in which only one of the two can be positive.

###  2.3 The Ising case

In this paper we shall focus our discussion to the Ising case, namely, we
assume that the interaction is $1$ for nearest neighbors spins and $0$
otherwise, which mean that the set $I$ is equal to set of the four neighboring
sites of the origin. With this choice the Hamiltonian (2.1) of the
asynchronous model is the standard 2D Ising Hamiltonian

$H(\sigma)=-\frac{1}{2}\sum_{\genfrac{}{}{0.0pt}{}{i,j\in\Lambda:}{|i-j|=1}}\sigma(i)\sigma(j)-h\sum_{i\in\Lambda}\sigma(i).$
(2.19)

On the other hand, the energy of the synchronous version of the dynamics reads

$H(\sigma)=-h\sum_{i\in\Lambda}\sigma(i)-\sum_{i\in\Lambda}\Big{|}\sum_{\genfrac{}{}{0.0pt}{}{j\in\Lambda:}{|j-i|=1}}\sigma(j)+h\Big{|}.$
(2.20)

In Statistical Mechanics models, Hamiltonians are usually written in terms of
coupling constants. Following [64], see also [65, 66], for the Hamiltonian
(2.20) we get

$\begin{array}[]{ll}{\displaystyle{H(\sigma)=}}&{\displaystyle{\\!\\!\\!-J_{.}\sum_{x\in\Lambda}\sigma(x)-J_{{}_{\langle\langle\rangle\rangle}}\sum_{\langle\langle
xy\rangle\rangle}\sigma(x)\sigma(y)-J_{{}_{\langle\langle\langle\rangle\rangle\rangle}}\sum_{\langle\langle\langle
xy\rangle\rangle\rangle}\sigma(x)\sigma(y)}}\\\
&{\displaystyle{\\!\\!\\!-J_{{}_{\triangle}}\sum_{\triangle_{xyz}}\sigma(x)\sigma(y)\sigma(z)-J_{{}_{\diamondsuit}}\sum_{\diamondsuit_{xywz}}\sigma(x)\sigma(y)\sigma(w)\sigma(z)}}\\\
\end{array}$ (2.21)

where the coupling constants

$J_{.}=\frac{5}{2}h,\;J_{{}_{\langle\langle\rangle\rangle}}=1-\frac{1}{4}h,\;J_{{}_{\langle\langle\langle\rangle\rangle\rangle}}=\frac{1}{2}-\frac{1}{8}h,\;J_{{}_{\triangle}}=-\frac{1}{8}hJ_{{}_{\diamondsuit}}=-\frac{1}{2}+\frac{3}{8}h,\;$
(2.22)

refer, respectively, to single site, next to the nearest neighbor pairs, third
neighbor pairs, triangle, and diamond clusters (see, figure 2). Note that the
diamond cluster coupling is negative for small magnetic field, this yields an
anti–ferromagnetic behavior of the interaction that will give rise to very
peculiar phenomena that will be discussed in the following sections. The
triangle cluster coupling is negative, as well, but it becomes negligible for
small $h$.

Figure 2: Schematical representation of coupling constants (2.22).

We stress that implementing the dynamics in asynchronous or synchronous ways
yield completely different stochastic models. We have already noticed that the
Hamiltonians are different, the simple 2D Ising Hamiltonian has to be compared
to the complicated Hamiltonian (2.21) found in the synchronous case. Even the
cost functions are pretty different: in the asynchronous case it reduces to
the positive part of the energy difference between configurations differing
for the value of one spin, whereas in the synchronous case it is defined for
any pair of configurations and it is given by the baroque formula (2.16). But
we want to remark that in the latter case the energy cost is defined for any
pair of configurations because in a single step the system can jump from any
configuration to any other. This is the essential difficulty of studying
metastability for synchronous dynamics, the fact that the structure of the
trajectories followed by the system in the configuration space is utterly
complicated due to the fact that any transition is possible in a single time
step.

##  3 Metastable state and critical configurations

As already mentioned in Section 1 the first rigorous and full description of
metastable behavior from the mathematical point of view dates back to the
paper [9] where the theory of the pathwise approach was firstly introduced.
Other widely applied methods are nowadays known as the potential theoretic
approach and the trace method, respectively proposed in [44, 45] and [55]. For
a comprehensive discussion of the pathwise and the potential theoretic
approach we refer the reader to the books [15, 46].

Here, we shall review some metastability results adopting the pathwise point
of view, in particular we shall follow the strategy refined in [20] for the
synchronous dynamics with Metropolis updating algorithm and extended in [22]
to a much larger class of Markov chains, including not reversible dynamics.
This theory applies both to the heat bath Glauber dynamics and to the
reversible PCA. We warn the reader that we use the same notation ($H$ for
Hamiltonian, $\Delta$ for the energy cost, etc.) for the two cases, but,
depending on the model one has in mind, the correct related quantities should
be used.

$\omega_{1}$$\omega_{2}$$\omega_{3}$$\omega_{4}$$\omega_{5}$$\Phi_{\omega}-H(\omega_{1})$
Figure 3: Schematical representation of the height of a path.

###  3.1 General results

A _path of length_ $n\geq 1$ is a sequence
$\\{\omega_{1},\dots\omega_{n}\\}\in\Omega^{n}$ such that
$p_{T}(\omega_{i},\omega_{i+1})>0$ for any $i=1,\dots,n-1$. We denote by
$\mathcal{F}$ the set of all loop free paths and, given
$A,A^{\prime}\subset\Omega$ not empty, we let $\mathcal{F}(A,A^{\prime})$ be
the set of all loop free paths with first configuration in $A$ and last
configuration in $A^{\prime}$. Given a path $\omega$ of length $n$, we define
the _height_ of the path as

$\Phi_{\omega}=\max_{i=1,\dots,n-1}[H(\omega_{i})+\Delta(\omega_{i},\omega_{i+1})].$
(3.23)

Given $A,A^{\prime}\subset\Omega$, the _communication height_
$\Phi(A,A^{\prime})$ between $A$ and $A^{\prime}$ is defined as

$\Phi(A,A^{\prime})=\min_{\omega\in\mathcal{F}(A,A^{\prime})}\Phi_{\omega}$
(3.24)

We define the _stability level_ $V_{\sigma}$ of any configuration
$\sigma\in\Omega$ as the minimal height, with respect to $H(\sigma)$, that
must be overcome by paths connecting $\sigma$ to the set of configurations at
energy smaller than $H(\sigma)$, namely,

$V_{\sigma}=\Phi(\\{\sigma\\},\\{\eta\in\Omega:\,H(\eta)<H(\sigma)\\})-H(\sigma).$
(3.25)

We are now ready to give the key notion of metastable state. We let
$\Omega^{\textup{s}}$ be the set of the absolute minima of the Hamiltonian,
namely, the set of ground states, and define the _maximal stability level_

$\Gamma=\max_{\sigma\in\Omega\setminus\Omega^{\textup{s}}}V_{\sigma}>0$ (3.26)

and the set of _metastable_ states

$\Omega^{\textup{m}}=\\{\sigma\in\Omega\setminus\Omega^{\textrm{s}}:\,V_{\sigma}=\Gamma\\}.$
(3.27)

The set $\Omega^{\textup{m}}$ deserves the name of set of metastable states
since it is possible to prove the following theorem [22, Theorem 2.1]: for any
$\sigma\in\Omega^{\textup{m}}$, for any $\varepsilon>0$ we have that

$\lim_{T\to
0}\mathbb{P}_{\sigma}(e^{(\Gamma-\varepsilon)/T}<\tau_{\Omega^{\textup{s}}}<e^{(\Gamma+\varepsilon)/T})=1,$
(3.28)

where $\mathbb{P}_{\sigma}$ is the probability for the chain $\sigma_{t}$
started at $\sigma$ and the random variable $\tau_{\Omega^{\textup{s}}}$ is
the _first hitting time_ to $\Omega^{\textup{s}}$ for the dynamics started at
$\sigma$, i.e., $\tau_{\Omega^{\textup{s}}}=\inf\\{t\geq
0:\,\sigma_{t}\in\Omega^{\textup{s}}\\}$. In words, equation (3.28) states
that the time needed by the system to exit the metastable state and touch the
ground state is, controlled in probability, of order $\exp\\{\Gamma/T\\}$.
Thus, (3.28) gives a mathematically rigorous meaning to the statement “the
exit time from the metastable state is of order $\exp\\{\Gamma/T\\}$”.

The pathwise approach provides also an estimate for the mean value
$\mathbb{E}_{\sigma}[\tau_{\Omega^{\textup{s}}}]$, indeed, it is possible to
prove that

$\lim_{T\to 0}T\log\mathbb{E}_{\sigma}[\tau_{\Omega^{\textup{s}}}]=\Gamma,$
(3.29)

see [22, Theorem 2.2].

The exit time is, for sure, the main property of metastable states that one
want to compute. Another relevant property concerns the path followed by the
system to exit the metastable state. Think to a super–saturated vapor: how
does it perform the transition to the liquid stable phase? Will it happen
through the coalescence of small droplets of liquid phase appeared throughout
the whole volume occupied by the system? Or will it happen through a sudden
formation of a sufficiently large droplet? These questions can be answered in
the framework of the pathwise approach, indeed it is possible to identify
configurations, called _critical_ , that must be necessarily visited during
the excursion from the metastable to the stable state. These special
configurations will give a clear indication of the mechanism of the transition
from the metastable to the stable state.

To make this formal, given $A,A^{\prime}\subset\Omega$ not empty, we define
the set of _optimal paths_ connecting $A$ to $A^{\prime}$, and denote it by
$\mathcal{F}_{\textup{o}}(A,A^{\prime})$, as the set of loop free paths
$\omega\in\mathcal{F}(A,A^{\prime})$ such that
$\Phi_{\omega}=\Phi(A,A^{\prime})$. That is to say, an optimal path connecting
$A$ to $A^{\prime}$ is a path starting in $A$, ending in $A^{\prime}$, and
having maximal height equal to the communication height between $A$ and
$A^{\prime}$.

Morally, the critical configurations are those configurations where the
optimal paths attain the maximal height. Unfortunately, as remarked above and
depicted in figure 3, the maximal height of a path in general does not
correspond to the energy of one of the configurations forming the path. This
is true in the case of Glauber dynamics [20], but it is not necessarily true
in a more general set–up including the PCA case. Thus, a more sophisticated
notion of critical configuration is needed [22]: given $\eta\in\Omega$ and
$A^{\prime}\subset\Omega$ the set of _saddles_
$\mathcal{S}(\\{\eta\\},A^{\prime})$ between $\eta$ and $A^{\prime}$ is the
set of configurations $\xi$ such that there exists an optimal path
$\omega\in\mathcal{F}_{\textup{o}}(\\{\eta\\},A^{\prime})$ and a configuration
$\zeta$ such that $\xi$ follows $\zeta$ in the path $\omega$ and
$H(\zeta)+\Delta(\zeta,\xi)=\Phi(\\{\eta\\},A^{\prime})$.

Among all the possible saddles, a relevant role is played by those that must
be necessarily visited by optimal paths: given $\eta\in\Omega$ and
$A^{\prime}\subset\Omega$ a subset
$W\subset\mathcal{S}(\\{\eta\\},A^{\prime})$ is a _gate_ for $\eta$ and
$A^{\prime}$ if and only if every optimal path in
$\mathcal{F}_{\textup{o}}(\\{\eta\\},A^{\prime})$ intersects $W$. Moreover, a
gate $W$ is _minimal_ if and only if for any $W^{\prime}\subset W$ and
$W^{\prime}\neq W$ there exists an optimal path which does not intersect
$W^{\prime}$. The fact that gates must be necessarily visited during the
transition from the metastable to the stable state is proven in [22, Theorem
2.4] stating that given $\sigma\in\Omega^{\textup{m}}$ and $W$ a minimal gate
for $\sigma$ and $\Omega^{\textup{s}}$, we have that there exists $c>0$ such
that

$\mathbb{P}_{\sigma}[\tau_{W}>\tau_{\Omega^{\textup{s}}}]\leq e^{-c/T},$
(3.30)

where $\tau_{W}$ is the first hitting time to $W$ for the dynamics started at
$\sigma$.

We mention that in the framework of the pathwise approach it is also possible
to characterize the behavior of the system before it performs the transition
to the metastable state. Indeed, by using the theory of cycles, it is possible
to define the basin of attraction of the metastable state and to prove that in
a time smaller than the exit time the system is confined to move within such a
basin of attraction.

Finally, we remark that the estimate provided by the pathwise approach for the
exit time is rather rough, in the sense that it is given at the level of
logarithmic equivalence. A more refined estimate can be provided in the
framework of the potential theoretic approach and using the trace method. The
typical result that can be proven is the following: for $T\to 0$,

$\frac{\mathbb{E}_{T}[\tau_{\Omega^{\textup{s}}}]}{\exp\\{\Gamma/T\\}}=C(\Lambda)[1+o(1)]$
(3.31)

where $C(\Lambda)$ is a constant, depending on the volume $\Lambda$, which can
be computed in terms of the number of saddles between the starting metastable
state $\sigma$ and the set of ground states $\Omega^{\textup{s}}$.

The results that we have discussed in in this section are general, but the
model dependent inputs that will be presented in Sections 3.2 and 3.3 are
necessary if one wants to describe the metastable behavior of a specific
system. This is precisely the idea on which the paper [20] is based and which
was further developed in [22]: disentangling the proof of the general
properties of metastable states from that of the model dependent inputs. These
model dependent inputs, necessary to achieve the full characterization of the
metastable behavior of a particular model, are the set of metastable states
and the optimal paths connecting such states to the set of stables one. Once
we get this, all the properties follow from the general theory.

###  3.2 Metastable behavior of the 2D Ising model

We consider the asynchronous dynamics (heat bath) introduced in Section 2.1
for the standard 2D Ising Hamiltonian considered in Section 2.3 with $0<h\ll
1$ such that $2/h$ is not integer. This assumption reduces the number of
degenerate critical configurations; we refer to [20] for a thorough discussion
of the case in which $2/h$ is integer and to [11], page 213, where some
comments on this singular point are reported. Although the metastable behavior
for such a model was first studied in [11] in the case of the Metropolis
single site updating rule, we describe here the main results using the
language developed in the previous sections.

d${\scriptstyle\blacksquare}$${\scriptstyle
8-2h}$${\scriptstyle\blacksquare}$${\scriptstyle\blacksquare}$${\scriptstyle
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4-2h}$${\scriptstyle 2h}$ Figure 4: Schematical representation of the optimal
path between d and u for the Ising model with indication of energy
differences. Black squares represent pluses.

We denote by d and u the two homogeneous configurations with spins
respectively equal to $-1$ and $+1$. It is not surprising that
$\Omega^{\textrm{s}}=\\{\textbf{u}\\}$ and
$\Omega^{\textrm{m}}=\\{\textbf{d}\\}$, provided the Hamiltonian is the
standard 2D Ising Hamiltonian defined in (2.19). The optimal path connecting d
to u is depicted in figure 4: four minuses are flipped to plus one after the
other to form a two by two square, the sides of the droplet grow one after the
other in order to obtain alternately a rectangle of pluses with side length
difference equal to one and a square of pluses till u is reached. A side is
added to a rectangle (or a square) by flipping to plus a minus spin adjacent
to the rectangle (adding one protuberance) and then flipping, one after the
other, minus spins with two adjacent pluses until a rectangular or square
shape is recovered.

When a plus protuberance is added to the side of the plus rectangle (or
square) of length $\ell$ the energy increases by $4-2h$; on the other hand,
when this very side is filled by pluses the energy decreases by $2h(\ell-1)$.
The energy difference related to the process of adding a side is equal to
$4-2h-2h(\ell-1)$, which is positive provided $\ell<\ell_{\textup{c}}$, where
$\ell_{\textup{c}}=\lfloor 2/h\rfloor+1$, where, for any real number $a$,
$\lfloor a\rfloor$ is its integer part, i.e., the largest integer smaller than
$a$. Thus the optimal path achieves its maximal energy at the configuration p,
see figure 4, made of a $\ell_{\textup{c}}(\ell_{\textup{c}}-1)$ rectangle of
pluses with a unit protuberance on one of its longest sides and the
communication height between d and u is

$\Gamma=H(\textbf{p})-H(\textbf{d})=8\ell_{\textup{c}}-2h[\ell_{\textup{c}}(\ell_{\textup{c}}-1)+1]\sim\frac{8}{h},$
(3.32)

where $H$ is the Ising Hamiltonian (2.19) and the estimate is valid for $h\to
0$.

With the model dependent ingredients summarized in this section, the results
reported in Section 3.1 provide a full description of the metastable behavior
of the Ising model: the state d is metastable and, if the system is prepared
in d, at small temperature the typical time necessary to reach the stable
state u is of order $\exp\\{8/(hT)\\}$. Moreover, during the transition from
the metastable to the stable state, with high probability, the system visits
the configuration p. This last remark is very important, since it means that
the transition to the stable state is performed through the sudden nucleation
of a large critical droplet and not via the coalescence of many small droplets
distributed throughout the whole volume $\Lambda$.

###  3.3 Metastable behavior of the reversible nearest–neighbor PCA

We consider the synchronous dynamics (PCA) introduced in Section 2.2 for the
standard 2D Ising Hamiltonian considered in Section 2.3 with $0<h\ll 1$ such
that $2/h$ is not integer. We also assume that the side length of the lattice
$\Lambda$ is an even number. The metastable behavior for such a model was
first studied in [67]; we describe here the main results using the language
developed above.

As in the previous section, we denote by d and u the two homogeneous
configurations with spins respectively equal to $-1$ and $+1$. Moreover, we
consider the two chessboard configurations such that all the spins associated
with sites on the even sub–lattice are equal and opposite to the spins
associated with sites on the odd sub–lattice. These two identified
configurations are denoted by c. The two chessboards are identified, since,
with high probability at low temperature, the chain started at one of them is
trapped in a continuous flip–flop among the two which is performed with no
energy cost. The presence of this flip–flopping configurations is a signature
of the parallel (synchronous) character of the dynamics. It is proven in [67]
that $\Omega^{\textrm{s}}=\\{\textbf{u}\\}$ and
$\Omega^{\textrm{m}}=\\{\textbf{d},\textbf{c}\\}$, with respect to the
Hamiltonian (2.20). The fact that chessboard configurations play a crucial
role in the metastable behavior of the model is also due to the
antiferromagnetic term $J_{\diamond}$ which is present in the Hamiltonian
(2.21).

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Figure 5: Schematical representation of the optimal path between c and u for
the PCA model with indication of the energy cost computed using (2.16) with
$I$ the set of nearest neighbors and $J_{ij}=1$ for $i$ and $j$ nearest
neighbors. Black squares represent pluses, white ones minuses.

The optimal path connecting c to u is constructed, see also [43, Section 5.2,
case A3–A4], by letting the spins to perform a flip–flop at each time except
for some pluses that are kept fixed in such a way to eventually invade the
whole lattice (see figure 5): at the first step two arbitrary
next–to–the–nearest pluses (Euclidean distance $\sqrt{2}$) are kept fixed, so
that a two by two plus square is formed in the chessboard sea. At the second
step the pluses in the square and a plus adjacent to the square is kept fixed
in order to form a two by three rectangle. At the third step the pluses in the
rectangle are kept fixed together with the plus at the center of one
chessboard side adjacent to the longest side of the rectangle. In this way a
three by three plus rectangle is formed. Before taking the fourth step the
chessboard sea is let flip–flop, if necessary, so that a plus appears at the
site in the center of one of the chessboard sides adjacent to the plus square,
thus such a spin is kept fixed and a three times four rectangle is formed. In
the fifth step one of the pluses adjacent to the longest side of the plus
rectangle is kept fixed and a double or triple plus protuberance is formed on
such a side. In the following steps this protuberance is kept fixed together
with the plus in the three by four rectangle till the four by four plus
rectangle is formed. This path is then followed alike alternating plus squares
to plus rectangles whose side lengths differ by one.

When one plus adjacent from the exterior to a square or a rectangle of pluses
is kept during the flip–flop, the energy cost is equal to $4-2h$. This is the
total cost payed to add a plus slice, since the following steps have no cost.
On the way back, eroding a plus slice of length $\ell$ has a cost $2h(\ell-1)$
since at each step the external pluses, having only two neighboring pluses,
are flipped paying the cost 2h The last step is cost free, because the last
plus has just one neighboring plus. By repeating the computation performed in
the asynchronous case, one finds again the critical length
$\ell_{\textup{c}}=\lfloor 2/h\rfloor+1$ for the plus droplets in the
chessboard sea. Thus the optimal path achieves its maximal height in the jump
from the $\ell_{\textup{c}}\times(\ell_{\textup{c}}-1)$ plus rectangle to the
configuration in which a double or triple plus protuberance is added to the
slab adjacent to the longest side of the same rectangle. Finally the
communication height between c and u is

$\Gamma=H(\textbf{q}_{2})-H(\textbf{c})+2h(\ell_{\textup{c}}-1)=-2h\ell_{\textup{c}}^{2}+(8+2h)\ell_{\textup{c}}+4h\sim\frac{8}{h},$
(3.33)

where $\textbf{q}_{2}$ is the $\ell_{\textup{c}}\times\ell_{\textup{c}}$ plus
droplet in the sea of chessboard, $H$ is the Hamiltonian (2.20) and the
estimate is valid for $h\to 0$.

We do not describe in detail the optimal path between d and u, we just mention
that it is made of two parts: the first part from d to c realizes the growth
of the chessboard square droplet $\textbf{q}_{1}$ of size $\lfloor
2/h\rfloor+1$ in the sea of minuses and its eventual growth to c, see [43,
Section 5.2, case A1–A2], the second part is precisely the optimal path from c
to u, see also [43, Section 5.2, case A3–A4]. Its maximal height is equal to
the value $\Gamma$ computed in (3.33) and this explains why this model have
two metastable states.

With the model dependent ingredients summarized in this section, the results
reported in Section 3.1 provide a full description of the metastable behavior
of the Ising model: the states d and c are metastable and, if the system is
prepared in d, at small temperature the typical time necessary to reach the
stable state u is of order $\exp\\{8/(hT)\\}$. Moreover, during the transition
from the metastable to the stable state, with high probability, the system
visits the configurations $\textbf{q}_{1}$, c, and $\textbf{q}_{2}$.

This situation has been called in the literature as “series of metastable
states”. We refer the interested reader to the papers [51] for some general
results and their applications to the case of the Blume–Capel model with zero
chemical potential, [68] for the discussion of this phenomenon in the context
of Probabilistic Cellular Automata, and to [43] for the extension of these
results to an arbitrary series of metastable states. The case of the
Blume–Capel model is also discussed in [69].

###  3.4 Numerical simulations

We illustrate by means of numerical simulations the nucleation phenomenon
described on a rigorous basis above. We refer to [70] for a detailed numerical
study of the exit time for the 2D Ising model.

We first consider the asynchronous model and we simulate the stochastic Ising
model on the $512\times 512$ lattice with magnetic field $h=0.2$. In figure 6
we show the nucleation of the plus phase starting from the minus metastable
state at inverse temperature $1/T=0.78$. The sequence of configurations shows
that the nucleation is performed via the formation of a single droplet of
pluses in the sea of minuses. Due to the fact that in the simulation we could
not consider a too small value of $T$, the droplet is not perfectly
rectangular and the sea of minuses is full of very small sub–critical droplets
of pluses.

Figure 6: Nucleation of the plus phase in the asynchronous Ising model with
$h=0.2$ and $1/T=0.78$. From the left to the right the configurations after
$800$, $1000$, $1400$, $2000$, and $3200$ Monte Carlo full lattice sweeps of
the lattice are reported. Black and white spots correspond to plus and minus
spins, respectively.

In figure 7 we illustrate the more complex nucleation phenomenon for the
asynchronous nearest neighbor PCA model on the $512\times 512$ lattice with
magnetic field $h=0.3$ and $1/T=0.90$. In order to show the nucleation of the
chessboard configuration, we represent the whole configuration on the
$256\times 256$ lattice and we associate with each site of the new lattice the
sum of the spins in a $2\times 2$ tile of the original lattice. The new block
variables, taking values in $\\{-4,-2,0,+2,+4\\}$, are plotted using a
grayscale paddle ranging from white ($-4$) to black ($+4$). The sequence of
configurations shows that the nucleation of the chessboard (gray)
configuration is performed via the formation of a single droplet in the sea of
minuses. Then, the plus phase is nucleated in the sea of chessboard via the
formation of a droplet of pluses. Due to the relative high value of the
temperature that we used in the simulations the formation of two plus droplets
is observed.

Figure 7: Nucleation of the plus phase in the synchronous nearest neighbor PCA
model with $h=0.3$ and $1/T=0.90$. From the left to the right the
configurations after $3000$, $3600$, $6000$, $7600$, and $9600$ Monte Carlo
full lattice sweeps of the lattice are reported. Configurations on $2\times 2$
tiles are reported using grayscale from $-4$ (white) to $+4$ (black).

In order to give a vivid idea of the time scales involved in the nucleation
process, we plot in figure 8 the magnetization, namely, the sum of all the
spins on the lattice, versus the number of full Monte Carlo sweeps for the
synchronous and asynchronous dynamics started at the minus configuration. A
full Monte Carlo sweep corresponds to one step of the dynamics in the
synchronous case and to a sequence of steps equal to the number of sites of
the lattice in the asynchronous case.

It is not reasonable to compare the time scales observed in the figure to the
result (3.29) which is an exact prediction for the exit time in the limit
$T\to 0$. Indeed, in simulating metastability effects it is necessary to use
not very small values of the temperature, otherwise the dynamics would stick
to the metastable state and nothing would be observed. We refer to [37, Figure
3] for the numerical estimate of the exit time for the Ising model with free
boundary conditions on a small lattice.

Nevertheless, it is very interesting to remark how neatly the simulations
show, in both cases, that the exit time from the metastable state increases
dramatically when $T$ becomes smaller and smaller (note that in the figure the
horizontal scale is logarithmic). Moreover, the data in the pictures also show
that the exit from the metastable state is an abrupt phenomenon: the system is
trapped in the metastable state for a long time, but when it performs the
transition, this process is completed in a very small time.

It is also very interesting to observe that, in the PCA case, the dynamics
started at the minus configuration, before reaching the plus state, spends a
huge time in a configuration with zero magnetization, namely, the chessboard
metastable state. This effect is lost if the temperature is so large (purple
curve) that fluctuations dominate with respect to the behavior driven by the
energy landscape. The fact that the intermediate zero magnetization state is
precisely the flip–flopping chessboard configuration is demonstrated by the
right panel where we plotted the staggered magnetization, i.e., the absolute
value of the difference between the sum of the spins on the even sub–lattice
and the sum of the spin on the odd one, versus the number of full sweeps

Figure 8: Magnetization and staggered magnetization versus number of full
Monte Carlo lattice sweeps. Left: magnetization of the asynchronous Ising
model on the $512\times 512$ lattice with $h=0.2$ and
$1/T=0.70,0.79,0.80,0.85,0.90$ (respectively from the left to the right, i.e.,
purple, green, blue, orange, yellow). Center: magnetizations of the
synchronous nearest neighbor reversible PCA on the $512\times 512$ lattice
with $h=0.3$ and $1/T=0.85,0.90,0.95,1.00$ (respectively from the left to the
right, i.e., purple, green, blue, orange). Right: staggered magnetization for
the same model as in the center panel.

##  4 Concluding remarks

We have shown that starting from the same spin system Hamiltonian (2.1)
depending on how the spin changes are implemented, asynchronously or
synchronously, two different stochastic models are found. The two models
differ not only for their stationary behavior, i.e., the stationary Gibbs
measure, but also for their dynamical behavior. In particular we have
discussed these dynamical aspects in the framework of the metastability theory
and, in this connection, we have shown that, depending on how the dynamics is
implemented, different metastable states can be found. Moreover, the critical
configurations and the exit time also turn out to be different.

In this review we have compared the asynchronous heat bath dynamics for the
Ising model with its natural parallel implementation. However, one could have
considered the parallel implementation of an asynchronous Metropolis dynamics,
as done in [71]. Here indeed a 2D Ising model with Hamiltonian (2.19) is
considered and the spins on the even (odd) sub–lattice are simultaneously
updated by using the Metropolis weights [38]. Given a random set of sites $I$
belonging to the even (odd) sub–lattice, and sampled under $\nu$, the
transition matrix is given by

$p_{T}(\sigma,\eta)=\left\\{\begin{array}[]{ll}\\!\\!\frac{\nu(I)}{|\Lambda|}e^{-[H(\eta)-H(\sigma)]_{+}/T}&\\!\\!\textup{if
}\eta(j)=-\sigma(j)\textup{ for }j\in I,\eta(j)=\sigma(j)\textup{ for
}j\in\Lambda\setminus I,\\\
\\!\\!0&\\!\\!\textup{otherwise}\end{array}\right.,$ (4.34)

where, for any $a\in\mathbb{R}$, the positive part $[a]_{+}$ of $a$ is equal
to $a$ if $a>0$ and to $0$ otherwise. The Markov chain defined by (4.34) is
reversible with respect to the Gibbs measure defined in (2.7). Despite the
fact that new jumps have been allowed by the dynamics, in [71] has been shown
that the metastable character of the one–site Metropolis dynamics is
preserved. The key observation is that the flip of the spins in $I$ can be
substituted by a suitable sequence of sequential single spin flip events.
Since the sites in $I$ belong to the same sub–lattice, the author could prove
that a configuration is a local minimum for the multi–spin dynamics if and
only if it is a local minimum in the single spin flip case. A natural
generalization of the model might be to consider a general set $I$, not
necessarily belonging to a sub–lattice. In this case the set $I$ might be
decomposed in a connected (in the sense of nearest neighbors) part and in a
non–connected part. The non–connected part might behave as the case in [71],
while the connected cluster poses extra challenges and might change
drastically the metastability scenarios.

As we have mentioned in Section 2.2, in this review we have not considered the
case of PCA with non–null _self–interaction_ , that is to say the case in
which $J_{00}\neq 0$ in equation (2.3) (i.e., the set $I$ contains the
origin), since its dynamics cannot be written in terms of heat bath rates.
However, for $J_{00}\neq 0$ interesting and different metastability scenarios
arise: the case $J_{00}=1$ has been studied in [41] showing a similar
metastable behavior to the Ising model with the asynchronous dynamics. For
$0<J_{00}<1$ the paper [42] showed that, quite surprisingly, results similar
to those found in [31] for the Blume–Capel model are obtained.

As we have stressed in our manuscript, the synchronous dynamics converges to a
measure that is different from the Ising Gibbs measure. Therefore, an
interesting question is whether these two measure are close with respect to a
suitable metric. A result in this direction is obtained in [72]. The role of
the _self–interaction_ has been indeed further investigated, in order to
derive an efficient method for approximating the sampling for the Ising
stationary measure. In [72, equation (8)] a PCA dynamics is introduced via the
_lifted Hamiltonian_

$H(\sigma,\eta)=-\sum_{\genfrac{}{}{0.0pt}{}{i,j\in\Lambda}{i\neq
j}}\bar{J}_{ij}\sigma(i)\eta(j)+\sum_{i\in\Lambda}q(1-\sigma(i)\eta(i))$
(4.35)

with $\bar{J}_{ij}$ a positive symmetric matrix and $q>0$ a positive
parameter. The PCA dynamics is then the Markov chain defined by the transition
probabilities

$p_{T,q}(\sigma,\eta)=\frac{e^{-H(\sigma,\eta)/T}}{\sum_{\zeta\in\Omega}e^{-H(\sigma,\zeta)/T}}\;.$
(4.36)

The parameter $q$ can be interpreted as an inertial term: for large values of
$q$ the dynamics is very slow, flipping few spins at each time. The transition
probabilities (4.36) can be written as a product of single site updating
probabilities, indeed,

$p_{T,q}(\sigma,\eta)=\prod_{i\in\Lambda}\bar{f}_{T,\Theta_{i}\sigma}(\eta(i))$
(4.37)

with

$\bar{f}_{T,\Theta_{i}\sigma}(\eta(i))=\frac{\exp\Big{\\{}\frac{1}{T}\eta(i)\left[\sum_{j\in\Lambda\setminus\\{i\\}}\bar{J}_{ij}\sigma(j)+q\sigma(i)\right]\Big{\\}}}{2\cosh\Big{\\{}\frac{1}{T}\left[\sum_{j\in\Lambda\setminus\\{i\\}}\bar{J}_{ij}\sigma(j)+q\sigma(i)\right]\Big{\\}}}\;.$
(4.38)

In [72] they posed for this model the question of Gibbsianess [73, 74], it has
been proved that the total variation distance between the invariant measure of
the PCA defined in (4.36) and the Ising Gibbs measure goes to zero as the
volume and the control parameter $q$ goes to infinity. Notice that these
approximation results do not apply directly to the model considered in the
present review, which, indeed, is recovered for $q=h=0$ and
$\bar{J}_{ij}=J_{ij}$. Afterwards, [75], [76] extended the results of [72]
also to the case of _weakly irreversible_ PCA dynamics, and [77] beyond the
high temperature Dobrushin regime.

Finally, we mention the so called _shaken_ dynamics, which have been
introduced in [78] via a suitable lifted Hamiltonian having only down–left
interactions and depending on a parameter $q$ that tunes the geometry of the
system, that allows to interpolate between different lattices. For $q$ large
the geometry is indeed the square lattice, for finite $q$ the system lives on
the hexagonal lattice, while for small $q$ the system becomes the product of
independent one dimensional lsing systems. Also for this dynamics the
stationary measure tends to the Ising Gibbs measure in the thermodynamic
limit.

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