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\begin{document}
\title [Metastability Beyond Exponential Asymptotics]
{{Metastability in Glauber Dynamics in the Low-Temperature limit:
Beyond Exponential
Asymptotics}}
\author[A. Bovier]{ Anton Bovier}
\address{Weierstrass-Institut f\"ur
Angewandte Analysis und Stochastik,
Mohrenstrasse 39, 10117 Berlin, Germany}
\email{ bovier@wias-berlin.de}
\author[F. Manzo]{ Francesco
Manzo}
\address{Institut f\"ur Mathematik, TU-Berlin, Strasse des 17. Juni
136, 10623 Berlin, Germany}
\address{ Adress from Aug. 1, 2001: Mathematisches Institut,
Universit\"at Potsdam, PF 601553, Germany}
\email{ manzo@ mat.uniroma2.it }
\thanks{Supported by the DFG through the Graduiertenkolleg ``Stochastische
Prozesse und Probabilistische Analysis''}
\keywords{Metastability, Markov chains,
Glauber dynamics, kinetic Ising model}
\subjclass{60K35, 82C20}
%\vfill
\maketitle
\centerline{\today}
\begin{abstract}
We consider Glauber dynamics of classical spin systems of Ising type
in the limit when the temperature tends to zero in finite volume. We
show that information on the structure of the most profound minima and
the connecting saddle points of the Hamiltonian can be translated into
{\it sharp} estimates on the distribution of the times of {\it
metastable } transitions between such minima as well as the low lying
spectrum of the generator. In contrast with earlier results on
such problems, where only the asymptotics of the exponential rates is
obtained, we compute the precise pre-factors up to multiplicative
errors that tend to 1 as $T\downarrow 0$. As an example we treat the
nearest neighbor Ising model on the 2 and 3 dimensional square
lattice. Our results improve considerably earlier estimates obtained
by Neves-Schonmann [NS] and
Ben Arous-Cerf [BC] and Alonso-Cerf [AC]. Our results employ the methods
introduced by Bovier, Eckhoff, Gayrard, and Klein in [BEGK1,BEGK2].
\end{abstract}
%
\section{ \label{Section 1.} Introduction. } \
Controlling the transitions from metastable states to equilibrium in
the stochastic dynamics of lattice spin systems at low temperatures
has been and still is a subject of considerable interest in
statistical mechanics. The first mathematically rigorous results can
be traced back to the work of Cassandro et al. [CGOV] that initiated
the so-called ``path-wise approach'' to metastability. For a good
review of the earlier literature, see in particular [Va]. All the
mathematical investigations in the subject require some 'small
parameter' that effectively makes the timescales for metastable
phenomena 'large'. The somewhat simplest of these limiting situations
is the case when a system in a finite volume $\L\subset \Z^d$ is
studied for small values of the temperature $T=1/\b$. In systems with
discrete spin space one is then in the situation where the dynamics
can be considered as a small perturbation of a deterministic process,
a situation very similar to what Freidlin and Wentzell [FW] called
'Markov chains with exponentially small transition
probabilities'. Consequently, most of the work concerning this
situation [OS1,OS2,CC,BC,N,NS] can be seen as extensions and
improvements of the {\it large deviation } approach initiated by
Freidlin and Wentzell. This consists essentially in identifying the
most likely path (in the sense of a sequence of transitions) and
proving a large deviation principle on path-space. While this approach
establishes very detailed information on e.g. the typical exit paths
from metastable states, the use of large deviations methods entails a
rather limited precision. Results for e.g. exit times $\t$ are
therefore typically of the following type:
For any $\e>0$,
$$
\P\left(e^{\b (\D-\e)} <\t<\e^{\b(\D+\e)}\right)\uparrow 1,\text{as
$\b\uparrow\infty$}
$$
where $\D$ can be computed explicitly. Similarly, one has results on
eigenvalues
of the generator that are of the form
$$
\lim_{\b\uparrow\infty}\b^{-1}\ln \l_i(\b) =\g_i
$$
with explicit expressions for the $\g_i$ (see e.g. [FW,S]). From many
points of view, the precision of such results is not satisfactory, and
rather than just exponential rates, one would in many situations like
to have precise expressions that also provide the precise
pre-factors. This is particularly important if one wants to understand
the dynamics of systems with a very complex structure of metastable
states, and in particular {\it disordered } systems. (For a rather
dramatic illustration, see e.g. [BBG1,BBG2] where aging phenomena in the
random energy model are studied.) Another drawback of the large
deviation methods employed is that they are rather heavy handed and
require a {\it very detailed knowledge } of the entire energy
landscape, a requirement that frequently cannot be met.
In two recent papers [BEGK1,BEGK2] a somewhat new approach to the
problem of metastability has been initiated aiming at improving the
precision of the results while reducing at the same time the amount of
information necessary to analyze
a given model. To achieve this goal,
the attempt to construct the precise exit paths is largely abandoned,
as are, to a very large extent, large deviation methods.
The general structure of this approach is as follows. In [BEGK2]
the notion of a {\it set of metastable points } is introduced. The
definition of this set employs only one type of objects, namely
{\it Newtonian capacities} (which may also be interpreted as {\it
escape probabilities}). If such a set of metastable points can be
identified, [BEGK2] provides a general theorem that yields precise
asymptotic formula for the {\it mean exit time} from each
metastable state, shows that this time is asymptotically
exponentially distributed (in a strong sense), and
states that each mean exit time is the inverse of one small
eigenvalue of the generator. While [BEGK2] assumes reversibility
of the dynamics, in [E] is shown that almost the same results can
be obtained in the general case. Thus, the analysis of
metastability is essentially reduced to the computation of
Newtonian capacities. The great advantage of such a result is that
capacities are particularly easy to estimate, due to the fact that
they verify a particularly manageable {\it variational principle}.
This fact is well known and has been exploited in the analysis of
transience versus recurrence properties of Markov chains (see e.g.
[DS]); however, its particular usefulness in the context of
metastability seems to have been noticed only in [BEGK1] where it
was used in the context of reversible discrete diffusion processes
motivated from certain mean field spin systems.
In this short paper we will show that the approach is even more
efficient and simple in the context of the zero temperature limit
of Glauber dynamics of spin systems in finite volume. We will show
that in rather general situations, capacities in this limit can be
computed virtually exactly in terms of properties of the {\it
energy landscape}, and therefore all interesting properties of the
dynamics can be inferred from a (not overly detailed) analysis of
the energy landscape generated by the Hamiltonian considered. As a
particular application that should illustrate the power of our
approach, we apply the general results to the Ising model (in two
and three dimension).
\section{ \label{Section 2.} The general setting and the main theorem. } \
In this section we set up the general context to which our results
will apply. It will be obvious that Glauber dynamics of finite volume
spin systems at low temperatures provide
particular examples. We will consider Markov processes on a finite
state space $\O$ (the {\it configuration space}). To define the
dynamics, we need the following further objects.
\begin{enumerate}
\item A {\it connected graph} $\cG$ on $\O$.
We denote by $E(\cG)$ the set of the edges in
$\cG$.
\item A {\it Hamiltonian} $H : \O \to \R$ also called {\it energy}.
\item The {\it Gibbs measure}
$\Q(x) := \frac{1}{\cZ} \exp(- \b H(x))$, where $\cZ$ is the
normalization factor called partition function, and $\b$ is the {\it
inverse temperature.}
\end{enumerate}
We consider {\it transition probabilities } $P(x,y)$ such that
if $\graffa{x,y}\in E(\cG)$, $P(x,y)>0$, and $P(x,y)=0$ if
$x\neq y$ and $\graffa{x,y}\not\in E(\cG)$. We assume moreover that
the transition probabilities are reversible with respect to the Gibbs
measure, i.e.
\begin{equation}
\label{eq:bilanciodettagliato}
\Q(x) P(x,y) = \Q(y) P(y,x) .
\end{equation}
We will also make the simplifying assumption that any existing
transition in the graph is reasonably strong, i.e. we assume that
there exists a constant $C>0$ such that\footnote{The
constant $C$ will typically be of the order of the inverse of
the maximum coordination number of the graph $\cG$.}
\begin{equation}
\label{eq:haste}
P(x,y) + P(y,x) \geq C \ \ \forall \; \graffa{x,y} \in E(\cG),
\end{equation}
by reversibility, \eqref{eq:haste} is equivalent to
\begin{equation}
\label{eq:haste2}
P(x,y) \geq \frac{C}{1+\exp (-\b(H(y)-H(x)))} \ \
\forall \; \graffa{x,y} \in E(\cG).
\end{equation}
To be able to state our results we need some further notations.
\begin{enumerate}
\item Given a one-dimensional subgraph $\o$, we write $\o: x \to I$
if the subgraph has one end in $x$ and the other end in $I$.
One dimensional subgraphs have a natural parameterization
$\o_0,\dots,\o_K$, where $K:=|\o|-1$,
$\forall k=0,\dots , K-1 \ $ $q(\o_k,\o_{k+1})>0$ and
$\o: \o_0 \to \o_K$.
\item Let $\widetilde H (\{ \o \}) := \max_{z \in \o} H(z)$.
For $x\in \O$ and $I\subset \O$, we introduce the
{\it communication height}, $\widehat H(x,I)$, between $x$ and $I$ as
\begin{equation}
\label{com1} \widehat H (x,I) := \min_{\o : x \to I} \widetilde
H(\{\o\}).
\end{equation}
Moreover we define the set of {\it saddle points} for $x$ and $I$ by
\begin{equation}
\label{eq:selle}
\cS_{x,I} := \graffa{ z \in \O \; ; \; \exists \; \o : x \to I
\text{ with } z \in \o \text{ and } H(z) = \widehat H(x,I) }
\end{equation}
\item Furthermore, we define the set of points
\begin{equation}
\label{com2}
D_x^I := \graffa{z \; ; \; H(\cS_{z,x}) < H(\cS_{z,I})}
\end{equation}
These will be the points that are 'closer' to $x$ than to $I$.
\item For any set $A\subset \O$, we define its {\it outer
boundary} $\partial A $ as the set of all points in $A$ from which an edge
of $\cG$ leads to its complement, $A^c$.
\item For $z \in \partial D_x^I$, let
$\px_z:= \sum_{y' \in D_x^I} P (z,y')$ and
$\py_z:= \sum_{x' \in \O \setminus D_x^I} P (z,x')$.
We let
\begin{equation}
\label{pre-factor}
\pref{x,I}:= \sum_{z \in \cS_{x,I}} \frac{\py_z \px_z}{\py_z + \px_z}.
\end{equation}
\item Let $W_x := \graffa {y \; ; \; H(y) < H(x)} $.
For $x \in \O$, we set
$\G(x) := H(\cS_{x,W_x}) - H(x)$.
If $x$ is not a local minimum of the Hamiltonian, $\G(x)=0$.
If $x$ is a global minimum of the Hamiltonian, we set $\G(x)=\infty$.
\item For the process $X_t$ starting at $x$, we define the
{\it hitting time} to the set $I \subset \O$ as
$\t^x_I:=\inf \{00$, independent of $\b$, such that for
any $x\in\cM$,
\begin{enumerate}
\item [(i)]
\begin{equation}
\label{eq:th31i}
\E \; \t(x)
=
N_x\pref{x}^{-1} e^{\b \G(x)} \err
\end{equation}
\item [(ii)] there exists an eigenvalue
$\l_x$ of $1-P$ such that
\begin{equation}
\label{eq:th31ii}
\l_x= \frac 1{\E \; \t(x)}\err
\end{equation}
\item [(iii)] if $\phi_{x}$ is the right-eigenvector of $P$
corresponding to $\l_x$, normalized so that
$\phi_x(x)=1$, then
\begin{equation}
\label{eq:th31iii}
\phi_x(y)=
\P \tonda{ \t^{y}_{x} <
\t^{y}_{\widetilde \cM_x}} + o(e^{-\d\b}))
\end{equation}
\item [(iv)]
\begin{equation}
\label{eq:th31iv}
\P \tonda{ \t(x) > t \; \E \; \t(x)}
=
e^{- t \err} \err
\end{equation}
\end{enumerate}
\end{thm}
In Section \ref{Section 6.}, we will apply Theorem 2.1 to a well
known situation, the kinetic Ising model,
in the limit of
vanishing temperature.
Let us anticipate our main result about the kinetic Ising model,
referring to Section \ref{Section 6.} for precise definitions and
notation.
\begin{thm}\label{mainising1}
Consider the kinetic
Ising model with Metropolis dynamics in
dimension $d=2$ or $d=3$ in a torus $\L^d(l)$ with diameter $l$.
The magnetic field $00$ is independent of $\b$ (but depends on
arithmetic properties of $h$).
\end{thm}
{\bf Remark.} Note that in our model we flip at most one spin per time step.
In continuous time dynamics the mean transition times would be lowered by a
factor $1/|\L|$.
The above Theorem shows how the results of Theorem \ref{th1} can
be applied (via the analysis of the energy landscape carried out
for the Ising model in [NS] and [AC,BC]) to the so-called
{Freidlin-Wentzell} regime. Notice that the methods of [BEGK2] can
be applied in a very similar way to situations where the volume
grows with $\b$ to compute "exactly" the probability of first
appearance of a critical droplet (a preliminary problem for the
infinite-volume metastability carried out in [DeSc]).
\section{ \label{Section 4.} Basic tools. } \
Theorem \ref{th1} relies on Theorem
1.3 in [BEGK2] that links relative capacities of metastable sets to mean
exit times and to the low lying spectrum of $1-P$. The additional work needed
to
prove Theorem \ref{th1} will be to estimate capacities in terms of the
Hamiltonian $H$, and
to show that the hypotheses of Theorem
1.3 in [BEGK2] are satisfied in our setting.
Let us state Theorem
1.3 in [BEGK2] specialized to our case.
In their context, a set $\cM \in \O$ is called a set of
{\it metastable points} in the sense of [BEGK2] if
\begin{equation}
\label{eq:defmeta}
\frac{
\sup_{{x\not =y \in \cM}} \TT{x}{y} }
{\inf_{z \in \O} \P \tonda{\t^{z}_{\cM} \le \t^{z}_{z}} }
\to 0
\text { as } \b \to \infty.
\end{equation}
The set $\cM$ is {\it generic} in the sense of [BEGK2] if
for any $x,y\in\cM$, and $I\subset \cM$,
$\frac {\P\left(\t^x_I<\t^x_x\right)}{
\P\left(\t^y_I<\t^y_y\right)}$ tends either to zero or to infinity, as
$\b\uparrow\infty$, and if the absolute minimum of the Hamiltonian is
not degenerate.
\begin{thm} {\rm (Theorem 1.3 in [BEGK2])}
\label{th41}
Let $\cM$ be a generic set of metastable states
in the sense of [BEGK2],
and let for $x\in\cM$,
$\cM_x$ and $\t(x)$ be defined as in Theorem \ref{th1}.
Then, for any $x\in\cM$, the following holds:
\begin{enumerate}
\item [(i)]
\begin{equation}
\label{eq:th13i}
\E \; \t(x)
=
\frac{N_x}{\TT{x}{\cM_x}}(1+ o(1))
\end{equation}
\item [(ii)] for any $x \in \cM$, there exists an eigenvalue
$\l_x$ of $1-P$ such that
\begin{equation}
\label{eq:th13ii}
\l_x= \frac{1}{\E \; \t(x)}(1+ o(1)) ,
\end{equation}
moreover, the eigenvalues of $1-P$
not corresponding to any $x \in \cM$
are in the interval
$(c |\O|^{-1} \inf_{z \in \O} \TT{z}{\cM},2]$
for some positive constant $c$.
\item [(iii)] if $\phi_{x}$ is the right-eigenvector of $P$
corresponding to $\l_x$, normalized so that
$\phi_x(x)=1$, then
\begin{equation}
\label{eq:th13iii}
\phi_x(y)=
\P \tonda{\t^{y}_{x} < \t^{y}_{\cM_x}} + o(1)
\end{equation}
\item [(iv)] for any $x \in \cM$, for any $t>0$,
\begin{equation}
\label{eq:th13iv}
\P \tonda{\t(x) > t \; \E\; \t(x)}
=
e^{- t (1+ o(1))} (1+ o(1)).
\end{equation}
\end{enumerate}
Here $o(1)$ stands for a small error that depends only on the small parameters
introduced via \eqref{eq:defmeta} and the non-degeneracy condition following
it.
\end{thm}
We leave it to the reader to verify that this theorem is indeed a special
case of the more general result stated in [BEGK2].
Theorem \ref{th1} will follow from Theorem \ref{th41}
since in the finite-volume and $\b \to \infty$ regime, we
compute $\TT{x}{\cM_x}$ and show that local minima of the
Hamiltonian are metastable states giving at the same time the
value of the nucleation rate in the limit $\b \to \infty$.
The key estimate is the following Lemma.
\begin{lemma}
\label{lemma1}
$\forall x,y \in \wt\cM$
such that $\cS_{x,y}$ is a set of isolated single points,
\begin{equation}
\label{eq:lemma1}
\TT{x}{y}
=
\pref{x,y} e^{-\b (H(\cS_{x,y}) - H(x))} (1+o(\expd)).
\end{equation}
\end{lemma}
We will explain in the appendix how our method can be extended to
situations where the saddles are degenerate. In this case the
pre-factor $\pref{x,y}$ does not have the nice form in
\eqref{pre-factor} but
can still be computed explicitly in terms of small
"local variational problems".
\begin{lemma} \label{cor1} Let $x$ be a minimum for the
Hamiltonian. Then, $x$ is a metastable state (in the sense of
[BEGK2]) in the set $\cM:=\graffa{y \; ; \; \G (y) \geq \G(x) }$.
\end{lemma}
Clearly, Theorem \ref{th1} immediately follows from Theorem \ref{th41},
Lemma \ref{lemma1} and Lemma \ref{cor1}.
\section{ \label{Section 5.} Proof of Lemmata
\ref{lemma1} and \ref{cor1}. } \
In order to prove Lemmata \ref{lemma1} and \ref{cor1},
we make use of many ideas
contained in [BEGK1].
The following Lemma corresponds to Theorem 6.1 in [Li].
\begin{lemma}
\label{lemma2}
(Dirichlet representation).\acapo
Let $\cH^x_y := \graffa{h: \O \to [0,1] \; ; \; h(x)=0, h(y)=1}$ and
\begin{equation}
\label{eq:dirichlet}
\Phi(h):= \frac{1}{\cZ} \sum_{x',x'' \in \O}
e^{-\b H(x')} P(x',x'') [h(x')-h(x'')]^2.
\end{equation}
Then,
\begin{equation}
\label{eq:lemma2}
\frac{ e^{-\b H(x)}}{\cZ } \TT{x}{y}
=
\frac{1}{2}\inf_{h \in \cH^x_y} \Phi(h)
\end{equation}
\end{lemma}
\begin{proof}
See [Li], Chapter II.6.
\end{proof}
Note that the left-hand side of \eqref{eq:lemma2}
has the potential-theoretic
interpretation of the {\it Newtonian capacity} of the point $y$ relative to
$x$ (i.e. the electric charge induced on the grounded site $x$
when the potential is set to 1 on the site $y$).
The Dirichlet form is just the
electric energy, and the minimizer $h^*$ is the {\it equilibrium
potential}, with the probabilistic interpretation
$h^*(z)=\P\left(\t^z_y<\t^z_x\right)$.
The strength of this variational representation comes from the monotonicity
of the Dirichlet form in the variables $P(x',x'')$,
expressed in the next Lemma, known as Rayleigh's short-cut rule
(see Lemma 2.2 in [BEGK1]):
\begin{lemma}
\label {lemma3}
Let $\D$ be a subgraph of $\cG$
and let $\widetilde \P_\D $ denote the law of
the Markov chain with transition rates, for $u \not = v$,
defined by
$\widetilde P_\D (u,v) := P(u,v) \Bbb I \graffa{ \graffa{u,v} \in E(\D)}$.
If $x$ and $y$ are vertices in $\D$, then
\begin{equation}
\label{eq:short}
\TT{x}{y} \ge \TTT{x}{y}{\D}
\end{equation}
\end {lemma}
\begin{proof}
The proof follows directly from Lemma \ref{lemma2}
and can be found in [BEGK1].
\end{proof}
The following Lemma corresponds to Lemma 2.3 in [BEGK1] and is a well known
fact (see e.g. [DS]).
\begin{lemma}
\label {lemma4} {\rm (The one dimensional case)}.
Let $\o$ be a one-dimensional subgraph of $\cG$,
$K:=|\o|-1$
and let
$\{\o_n\}_n:\{0,\dots,K\} \to \O$
be such that
$\forall n \le K$,
$q(\o_n,\o_{n-1}) > 0$
\begin{equation}
\label{eq:dim1}
\TTT{\o_0}{\o_{K}}{\o}
=
\quadra{ \sum_{n=0}^{K-1}
\frac{e^{-\b(H(\o_0)-H(\o_n))}}
{P(\o_{n},\o_{n+1})}
}^{-1}
\end{equation}
\end{lemma}
Remark: Lemmata \ref{lemma3}, \ref{lemma4} and \eqref{eq:haste}
immediately give the following bound: $\forall x',I$ s.t.
$\cS_{x',I}$ is made of simple points,
\begin{eqnarray}
\TT{x'}{I}
& \ge &
\quadra{
\sum_{n=0}^{K-1}
\frac{
e^{-\b(H(x')-H(\o_n))} }
{P \tonda{\o_{n},\o_{n+1} } }
}^{-1}
\ge
\nonumber\\
\label{eq:apriori1}
& \ge &
C \quadra{
\sum_{n=0}^{K-1}
\tonda{e^{-\b(H(x')-H(\o_n))}
+ e^{- \b (H(x')- H(\o_{n+1}))} }
}^{-1}
\nonumber\\
\label{eq:apriori}
& \ge &
\frac{C}{2} e^{- \b (H(\cS_{x',I})- H(x'))} \tonda{1-\expd}
\end{eqnarray}
for any choice of the subgraph $\o : x' \to I$
having its maximum energy in $\cS_{x',I}$.
The constant $C$ is the same as in \eqref{eq:haste2}.
\begin{proof} [{Proof of Lemma \ref{lemma1}.}]
Let $\G :=H(\cS_{x,y}) - H(x)$.
We consider the surface $\sselle:= \partial D_x^y$.
Notice that:
\begin{enumerate}
\item $\cS:= \cS_{x,y} \subset \sselle $
\item $\exists \d > 0$ such that
$\forall z \in \sselle \setminus \cS$,
$H(z) \ge H(\cS) + \d$.
\item $\sselle$ is the outer boundary of a connected set
that contains $x$.
\end{enumerate}
{\bf Remark:} In what follows, any other surface with properties 1, 2, and 3
would give the bounds we need for the proof.
The quantity $\cC'_{x,y}$ defined with respect to the new surface
differs from $\pref{x,y}$ by a factor $1+o(\expd)$.
We set
$D_x := D_x^y$,
$D_y := \O \setminus (\sselle \cup D_x)$,
$\sselle^- := \partial \sselle \cap D_x$ and
$\sselle^+ := \partial \sselle \cap D_y$.
\acapo
(1) The upper bound.
We use Lemma \ref{lemma2} with
$h(x'):=0$ if $x' \in D_x$ and
$h(y'):=1$ if $y' \in D_y$;
we choose $h(z)$ for $z \in \sselle$
in an optimal way. In the rest of the space we choose
$h(x)=1$.
We have
\begin{equation*}
\TT{x}{y}
\le
\frac{\cZ e^{\b H(x) }}{2 } \Phi(h)=
\end{equation*}
\begin{equation}
\label{eq:low1}
=
\sum_{z \in \sselle}
e^{-\b (H(z)-H(x))}
P (z,x)
\tonda{
\px_z h^2(z) + \py_z (1-h(z))^2} +o(e^{-\b\d}),
\end{equation}
where we used reversibility. The small error comes from the mismatch
on the boundary of $D_x$ that lies higher than the saddle hight.
The quadratic form
$\px h^2 + \py (1-h)^2$
has a minimum for
$h=\frac{\py }{\py + \px}$.
Hence, we can saturate the inequality
\eqref{eq:low1} and get
\begin{equation}
\label{eq:low}
\text{(l.h.s. of \eqref{eq:low1}) }
\le
\pref{x,y} e^{-\b \G} \tonda{1+\expd}.
\end{equation}
\acapo
(2) The lower bound.
We consider the subgraph $\D$ obtained by
cutting all the connections to the vertices in
$\sselle \setminus \cS$.
We use Lemma \ref{lemma3} to bound the original process by the
restricted process.
We use \eqref{eq:apriori} to estimate the probability
to reach $x' \in \sselle^-$
and the probability to go from $y'\in \sselle^+$ to $y$: \acapo
By the strong Markov property at time $\t^x_\cS$, we have
\begin{eqnarray}
\label{up1}
\TTT{x}{y}{\D}
& =&
\sum_{z \in \cS}
\widetilde \P_{\D} \tonda{\t^x_{z} \le \t^x_{\cS \cup x}}
\widetilde \P_{\D} \tonda{\t^{z}_y < \t^{z}_{x}}
\nonumber\\\label{upa}
& =&
e^{-\b \G}
\sum_{z \in \cS}
\widetilde \P_{\D} \tonda{\t^z_{x} < \t^z_{\cS}}
\widetilde \P_{\D} \tonda{\t^{z}_y < \t^{z}_{x}},
\end{eqnarray}
where we used reversibility.
Now,
\begin{equation}
\label{eq:parte1}
\widetilde \P_{\D} \tonda{\t^z_{x} < \t^z_{\cS}}
=
\sum_{x' \in \sselle^-}
P \tonda{z,x'}
\widetilde \P_{\D} \tonda{\t^{x'}_x < \t^{x'}_{\cS}}.
\end{equation}
We bound the last factor using a standard renewal argument (see
e.g. [BEGK1] Corollary 1.6) that yields if $z' \in D_x$ the last
term is exponentially close to $1$:
\begin{eqnarray}
\label{eq:b1}
\Tzyx{x'}{\cS}{x}{\D}
& =&
\frac{ \Tzyx{x'}{\cS}{x \cup x'}{\D} }
{ \Tzyx{x'}{x \cup \cS}{x'}{\D} }
\le
\frac{ e^{- \b \G}
\sum_{z \in \cS}
\Tzyx{z}{x'}{x \cup \cS}{\D} }
{ \Tzyx{x'}{x}{x'}{\D} } \nonumber\\
\label{eq:b2}
&\le&
\frac{ | \cS | e^{- \b \G} }
{ C e^{-\b (\G - \d')} \tonda{1 -e^{-\b \d'}}}
\le \expd,
\end{eqnarray}
where we used \eqref{eq:apriori}. By putting together
\eqref{eq:parte1} and \eqref{eq:b2} we get
\begin{equation}
\label{p1}
\widetilde \P_{\D} \tonda{\t^z_{x} < \t^z_{\cS}}
\ge
\px_z \tonda{1-\expd}
\end{equation}
We use the same procedure to bound the last term in
\eqref{upa}:
\begin{equation}
\label{eq:parte2}
\Tzyx{z}{y}{x}{\D}
\ge
\sum_{y' \in \sselle^+}
P \tonda{z,y'}
\Tzyx{y'}{y}{x}{\D}.
\end{equation}
Again, the same arguments leading to \eqref{eq:b2} show that
the last term in this sum is exponentially close to 1:
\begin{equation}
\label{eq:c1}
\Tzyx{y'}{x}{y}{\D}
\le
\frac{ | \cS | e^{- \b (H(z) - H(y')) } }
{ e^{- \b (H(\cS_{y,y'}) - H(y')) }
\tonda{1 - e^{-\b\d'} } }
\le
\expd
\end{equation}
We put together \eqref{eq:parte2} and \eqref{eq:c1}
and get
\begin{equation}
\label{p2}
\widetilde \P_{\D} \tonda{\t^{z}_y < \t^{z}_{x}}
\ge
\frac{\py_z}{\py_z+\px_z} (1-\expd)
\end {equation}
Going back to \eqref{up1},
we get from \eqref{p1}, \eqref{p2}
\begin{equation}
\label{up2}
\widetilde \P_{\D} \tonda{\t^x_y < \t^x_x}
\ge
\pref{x,y} e^{- \b \G} \tonda{1- \expd}
\end {equation}
\end {proof}
\begin{proof}[Proof of Lemma \ref{cor1}]
For any $z \not \in \cM$
we know that by definition of $\cM$, we have that
$\G(z)<\min_{x\in\cM}\G(x)\equiv \G$. In view of Lemma \ref{lemma2} and
the lower
bound (4.5) we only need to show that this implies that
$\widehat H(z,\cM)-H(z)<\G$.
Now let $u\not\in\cM$ be the point that realizes the minimum of
the energy among the states such that
$\widehat{H}(z,u) < \widehat{H}(z,\cM)$.
For such a point, by definition, $\widehat
H(u,\cM)-H(u)=\G(u)<\G$. But clearly,
$\widehat{H}(z,\cM)-H(z) \le \widehat{H}(u,W_u)-H(u) < \G(x)$, and we are
done.
\end{proof}
\section{ \label{Section 6.} The Ising case. } \
In this section we want to illustrate the strength of Theorem
\ref{th1} in a well known context, namely the stochastic Ising
model on the $d$-dimensional lattice. In this case the state space
is $\O=\{-1,+1\}^\L$, where $\L=\L(L)$ is a torus in
$\Z^d$ with side-length $L$.
For $\s \in \O$, the Hamiltonian is
then given by
\begin{equation}
\label{isiham}
H(\s)\equiv H_\L(\s)
=-\frac{1}{2} \sum_{* \in\L}
\s_i\s_j-h\sum_{i\in\L}\s_i,
\end{equation}
where the first sum concerns all the pairs of nearest neighbor
sites in $\L$.
Let $\s^i$ be the configuration that differs from $\s$ only in the
value of the spin of site $i$ and $[a]_+$ denote the positive part
of the real number $a$.
We will consider for definiteness only the
case of the Metropolis dynamics, i.e. the transition probabilities are
chosen
\begin{equation}
\label{metro1} P(\s,\s')=\frac {e^{-\b[H(\s')-H(\s)]_+}}{|\L|},
\,\hbox{ if }\, \s'=\s^i, i\in\L
\end{equation}
\begin{equation}
\label{metro2}
P(\s,\s)= 1-\sum_{i\in\L} P(\s,\s^i)
\end{equation}
and all others are zero.
We will use the estimate given in
Theorem \ref{th1} to analyze this dynamics in a finite volume
$\L$, under a positive magnetic field, in the limit when
$\b\uparrow\infty$.
Let $\minus$ and $\plus$ be the configurations full of minuses
or full of pluses, respectively.
We will show in Lemma \ref{pozzi} that $\graffa{\minus,\plus}$
is a set metastable states.
Apart from this characterization, we will only use the description
of the energy landscape given
in [NS], [AC,BC] and [N] in dimension 2, 3 or larger, respectively.
We will show that the methods of Theorem \ref{th1} allow to improve
the known estimates without requiring further analysis of the
energy landscape.
In dimension 2 and 3, the improvement amounts to the computation
of the exact (including the pre-factor $\pref{\minus,\plus}$) asymptotic
value of the expected transition time $\t^-_+$
needed to reach $\plus$
starting from $\minus$
(that by Theorem \ref{th41} is the inverse
of the spectral gap of $P$).
In higher dimension, where our knowledge of the energy landscape is
not so detailed, we cannot compute the pre-factor but we show that
it is a constant independent of $\b$, while previous results
only gave sub-exponential bounds.
We remark that, unlike the exponential factor $\exp (- \b \G(\minus))$
(that only depends on the
graph structure), the pre-factor
$\pref{\minus,\plus}$ is related to the particular
Glauber dynamics we
choose.
We consider $00$,
for any $\s\neq \minus$, $\G(\s)<\G(\minus)$,
i.e. for any $\s \not \in
\graffa{\minus,\plus}$, there exists a configuration $\s'$ such
that
\begin{enumerate}
\item $H(\s') < H(\s)-\d$
\item $\widehat H(\s,\s') - H(\s) <
\widehat H(\minus,\plus) - H(\minus)-\d$.
\end{enumerate}
For $\eta \in \O$, let $|\eta|$ and $\wp(\eta)$ be the number of
pluses and the number of pairs of nearest neighbors with different
spin (namely, the perimeter, or cardinality of the contour),
respectively. It is a well known fact that the Hamiltonian of the
Ising model can be written as
\begin{equation}
\label{eq:h2}
H(\eta)=\wp(\eta) - h |\eta| +H(\minus)
\end{equation}
Let
$m:=\min \graffa{k \ge 1 ; \exists \eta \text{ with } |\eta|=k
\text{ and } H(\eta) \le H(\minus)}$.
Let
$\o:\minus \to \plus$
be a monotone one-dimensional subgraph such that $\o_k \in \cB^d(k)$
that reaches its maximal energy in
$\cS (\minus, \plus)$.
Clearly, $H(\o_m) 3$,
there exists a constant $c_d$ such that
\begin{equation}\label{d23i}
\E \t_{+}^{-} = c_d e^{\b \G_d} \err
\end{equation}
\end{thm}
If Conjecture \ref{conj1} holds and $sk/h$ is not an integer for all
$k=1,\dots,d-1$,
then the pre-factor $c_d$ is equal to
\begin{equation}
\label{eq:cd}
\tonda{d! \frac{2^d}{3} \tonda{1-\frac{2}{\ell_2}}
\prod_{k=1}^{d-1}
\tonda{\ell_{k+1} - \ell_k +1}^k }^{-1}
\end{equation}
\begin{lemma}
\label{numselle}
The number of candidate saddles in dimension $d$
contained in a $d$-dimensional cube
of side-length $l \ge \ell_d$ is
\begin{equation}
\label{num}
\cN_d(l) = 2^{d-1} d! \tonda{l - \ell_{d} +1}^d
\prod_{k=1}^{d-1} \tonda{\ell_{k+1} - \ell_{k} +1}^k
\end{equation}
All the candidate saddles have $\px = l^{-d}$, while $\py $ can
take the value $l^{-d}$ or $2l^{-d}$.
The fraction of candidate
saddles with $\py =l^{-d}$ is $\frac{2}{\ell_2}$, independently of $d$
and $l$.
\end{lemma}
\begin{proof}
Let $\cN_d:=\cN_d(\ell_d)$.
The key observation is that the pluses of a candidate saddle
are contained in exactly one cube of side-length $\ell_d$.
A $d-dimensional$ cube of side-length $l \ge \ell_d$ contains
$(l-\ell_d+1)^d$ of such cubes.
Hence, $\cN_d(l)=(l-\ell_d+1)^d \cN_d$.
Given a cube of side-length $l$,
there are $2 d$ possible choices for the incomplete face and
$\cN_{d-1}(\ell_d)$ ways to arrange the ($d-1$)-dimensional
candidate droplet on this face.
Hence,
$
\cN_d=2 d \cN_{d-1}(\ell_d) =
2 d (\ell_d-\ell_{d-1}+1)^{d-1} \cN_{d-1}
$.
Since $\cN_1=1$, a simple calculation gives \eqref{num}.
The computation of the number $m_d(l)$ of candidate saddles with
$\py =l^{-d}$ is very similar: In dimension two, $m_2 (\ell_2) =
8$ i.e. the number of configurations made of a quasi-square plus a
protuberance at one end of one of the longest sides. All other
candidate saddles have $\py =2l^{-d}$, since there are two neighbors of
the protuberance that can be occupied. All candidate saddles
have $\px=l^{-d}$, since we can void the occupied site and reach a
quasi-cube in $D_-$. In general, for $d>1$, the only sites with
$d$ plus-neighbors are in an incomplete face of the \ddim critical
cube. Hence, $m_d(\ell_d)$ is equal the number of
($d-1$)-dimensional critical squares on the faces of the \ddim
critical cube times $m_{d-1}(\ell_d)$ namely, $ m_d(\ell_d) =2 d
(\ell_d-\ell_{d-1}+1)^{d-1} m_{d-1}(\ell_{d-1}) $. On the other
hand, $m_d(l) =2 d (l-\ell_d+1)^d m_d(\ell_d)$. Thus, the ratio
$m_d(l) / \cN_d(l)$ does not depend on $d$ or on $l$ and is equal
to $2/ \ell_2$.
\end{proof}
\begin{corollary}
\label{prefattore}
The number of candidate saddles in dimension
$d$ contained in a $d$-dimensional torus of side-length $l \ge
\ell_d$ is
\begin{equation}
\label{num2}
\widetilde \cN_d(l) = 2^{d-1} d! \; l^d \;
\prod_{k=1}^{d-1} \tonda{\ell_{k+1} - \ell_{k} +1}^k
\end{equation}
\begin{equation}
\label{eq:stima}
= d! 2^{\frac{d^2+d-2}{2}} h^{-\frac{d^2-d}{2}}
\tonda{1+o(h)}
\end{equation}
All the candidate saddles have $\px = l^{-d}$, while $\py $ can
take the value $l^{-d}$ or $2l^{-d}$. The fraction of candidate
saddles with $\py =l^{-d}$ is $\frac{2}{\ell_2}$, independently of
$d$ and $l$.
\end{corollary}
\begin{proof}
The result is a straightforward consequence of
lemma \ref{numselle} and of the fact that the number of
$d$-dimensional cubes of side-length $\ell_d$
that can be put into the torus is $l^d$.
The estimate in \eqref{eq:stima} comes from the approximation
$\ell_d - \ell_{d-1} + 1 = \frac{2}{h}\tonda{1+o(h)}$
and hence \eqref{eq:stima}.
\end{proof}
\begin{proof}[Proof of Theorems \ref{mainising1} and \ref{mainising}:]
The results of Theorem \ref{mainising1} are straightforward
consequences of Theorem \ref{th1}, Lemma \ref{2e3}, Lemma \ref{pozzi},
and Corollary \ref{prefattore}.
In higher dimension,
the corresponding result comes
from Theorem \ref{th1}
and Lemma \ref{pozzi}.
If conjecture \ref{conj1} holds, Corollary \ref{prefattore}
gives the estimate in \ref{eq:cd}.
\end{proof}
\bigskip
In conclusion, let us notice that the form of the
quantities $\TT{x}{y}$ in the case of the
Metropolis dynamics may offer an interpretation in terms
of ``free energy of the set of saddle points''.
Indeed, every point in $\sselle$ gives a contribution
to the pre-factor $\pref{x,y}$
that does not depend on $\b$ and can be bounded by a
constant $c$.
With the arguments in the proof of Lemma \ref{lemma1},
we get
\begin{equation}
\label{eq:entropia}
\TT{x}{y}
\sim
c \cN e^{-\b \G} =
c \exp \tonda{ -\b (\G - T \log \cN)},
\end{equation}
where $\cN$ is the number of saddles
between $x$ and $y$ and $T=\b^{-1}$.
The logarithm of $\cN$ can be interpreted as an entropy.
This interpretation could be related to
the results by Schonmann and Shlosman (see [ScSh])
on the connections between
Wulff droplets and the metastable relaxation of kinetic Ising model.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\label{Appendix} Appendix.}
In this appendix we briefly explain how our general approach can be
generalized to situations than the saddles are more complicated
when the isolated single points assumed in Section
2. The point we want to make is that in such a case it is still
possible to localize the problem to the understanding of the
neighborhood of the saddle points and to thus reduce the analysis of
the capacities to a `local' variational problem. Let us consider a
situation when in the computation of a transition from $x$ to $y$ we
encounter a set of saddles $\cS_{x,y}$ that can be decomposed into
a collection of disconnected subsets $\cS^{(k)}$, $k=1,\dots,L$.
By definition, it must be true that each of the sets $\cS^{(k)}$ is
connected to two subsets $\cR^{(k)}$ and $\cN^{(k)} $ of $D^y_x$ and
$D^x_y$, respectively. Let us define
\begin{equation}
\label{app1}
C(k):=\sum_{i\in \cN^{(k)}}e^{-\b(H(x)-H(i))}
\wt\P\left(\t^i_{\cR^{(k)}}<\t^{i}_{\cN^{(k)}}\right)
\end{equation}
where $\wt \P$ is the law of the chain where all the edges exiting
from the sets $S^{(k)}$ not leading to $\cN^{(k)}$ or $\cR^{(k)}$ are
cut. Note that it is not difficult to see that
\begin{equation}
\label{app2}
C^{(k)} = \inf_{h\in \cH^{\cR}_{\cN}}\wt\Phi(h)
\end{equation}
Repeating the steps of the proof of Lemma 4.2, one obtains then that
\begin{lemma}
\label{applem}
In the situation described above we have that
\begin{equation}
\label{app3}
\P\left(\t^x_y<\t^x_x\right)= \sum_{k=1}^L C^{(k)}\left(1+o(e^{-\b
\d})\right)
\end{equation}
\end{lemma}
{\bf Acknowledgements:} We thank Frank den Hollander, Enzo Olivieri,
Elisabetta Scoppola, and Raphael Cerf for useful discussions.
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