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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\NoBlackBoxes
\topmatter
\title
Wulff droplets and the metastable relaxation of kinetic Ising models
\endtitle
\author Roberto H. Schonmann and Senya B. Shlosman
\footnote"*"{The work of R.H.S was partially supported by the
N.S.F. through grants DMS 9100725 and DMS 9400644 and that of S.B.S.
by the N.S.F. through grant DMS 9208029 and by the Russian Fund for
Fundamental Research through grant 930101470 \newline}
\endauthor
\leftheadtext{Roberto H. Schonmann and Senya B. Shlosman}
\rightheadtext{Wulff droplets and metastable relaxation}
\abstract
We consider the kinetic Ising models (Glauber dynamics)
corresponding to the infinite volume Ising model in
dimension 2 with nearest neighbor ferromagnetic interaction
and under a positive external magnetic field $h$. Minimal
conditions on the flip rates are assumed, so that all the common
choices are being considered.
We study the relaxation towards equilibrium when the system
is at an arbitrary subcritical temperature $T$ and the evolution is
started from a distribution which is stochastically lower than
the ($$)phase.
We show that as $\hg$
the relaxation time blows up
as $\exp(\lambda_c(T)/h)$,
with $\lambda_c(T) = w(T)^2/(12 T m^*(T))$.
Here $m^*(T)$ is the spontaneous magnetization
and $w(T)$ is the integrated surface tension of the Wulff body
of unit volume.
Moreover,
for $0 < \lambda < \lambda_c$, the state of the process at time
$\exp(\lambda/h)$ is shown to be close, when $h$ is small,
to the ($$)phase. The difference
between this state and the ($$)phase can be described in terms
of an asymptotic expansion in powers of
the external\corr{field, which can be interpreted as stating
that this state is in a sense a smooth continuation of the Gibbs
distributions with small negative $h$.} field. This expansion can
be interpreted as describing a set of $\Cal C^\infty$
continuations in $h$ of the family of Gibbs distributions
with the negative magnetic fields into the region of positive
fields.
\endabstract
\hyphenation{metastability}
\subjclass Primary 60K35, 82B27
\endsubjclass
\keywords kinetic Ising model, stochastic Ising model,
Glauber dynamics, metastability,
relaxation, nucleation,
droplet growth, Wulff shape, large deviations,
asymptotic expansion, spectral gap estimates,
blockrescaling, percolation techniques, coupling,
attractiveness
\endkeywords
\endtopmatter
%\bigskip
\medskip
\noindent {\bf Contents}
\
\noindent {\bf 1. Introduction}
\roster
\item"1.1" Preliminaries
\item"1.2" Notation and Terminology
\item"1.3" Some Tools and Further Definitions
\item"1.4" Main Results
\item"1.5" Heuristics
\endroster
\
\noindent {\bf 2. Metastable Regime}
\roster
\item"2.1" Preliminaries
\item"2.2" Bottlenecks for the Dynamics
\item"2.3" The Restricted Ensemble
\item"2.4" Asymptotic Expansion
\item"2.5" More General Initial Distributions
\endroster
\
\noindent {\bf 3. Relaxation Regime}
\roster
\item"3.1" Preliminaries
\item"3.2" Inverted Pyramids and Droplet Growth
\item"3.3" Rescaling and Droplet Creation
\item"3.4" DoubleWell Structure of Equilibrium Distributions
\item"3.5" Spectral Gap Estimates
\endroster
\bigskip
\heading{1. Introduction}\endheading
\numsec=1
\numfor=1
\subheading{1.1. Preliminaries}
This paper is a continuation of the paper [\sddr], and contains
substantial strengthening of the results of that paper in the
case of dimension 2.
We refer the reader to [\sddr] for a discussion of the motivation
and background\corr{ on} of the problem.
For introductions to metastability see, e.g., [GD] and [PL].
The precise results in the current
paper can only be stated after enough notation is introduced and
are therefore postponed to Section \thm. We provide next an informal
summary of our results.
Our concern in this paper is with the metastable behavior of the
2 dimensional Ising model, evolving with a reversible spinflip
dynamics, in the proximity of the phasecoexistence line. We
study the system at an arbitrary subcritical temperature $T$ and under
a small positive external magnetic field $h$. The results proven all
refer to limits in which $\hg$. These results fully confirm,
in particular, a conjecture raised by Aizenman and Lebowitz in [AL].
This conjecture was that if started from a typical
configuration of the ($$)phase, for times of order $\exp(\lambda/h)$
with $\lambda$ below a critical value $\lambda_c$ the system would be
in a sort of metastable state, close to the ($$)phase (in spite of
the presence of the positive external field). On the other hand, for
a time of order $\exp(\lambda/h)$ with $\lambda > \lambda_c$ the system
would have already relaxed and so would be close to the (+)phase.
In [\sddr] a weaker version of this conjecture was proven, with
the first scenario occurring for $\lambda < \lambda_1$ and the
second for $\lambda > \lambda_2$, with these two constants $\lambda_1$
and $\lambda_2$ having been explicitly estimated,
but both being nonoptimal.
Moreover the temperature was supposed to be substantially lower
than the critical one and the initial distribution had to be
concentrated on the configuration with all spins down. On the good
side, the results in [\sddr] are valid in arbitrary dimension.
Here we will only consider dimension 2, but strengthen the result in
the following ways:
\
%\roster
%\item"(1)"
\noindent 1)
The constants $\lambda_1$ and $\lambda_2$ are shown to
be identical, the common value being given by
$$
\lambda_c = \lambda_c(T) = \frac{w(T)^2}{12 \, T \, m^*(T)}.
\Eq(lambdac)
$$
Here $m^*(T)$ is the spontaneous magnetization
and $w(T)$ is the integrated surface tension of the Wulff body
of unit volume. Note that all quantities on the righthandside
of \equ(lambdac) pertain to equilibrium statistical mechanics.
%This value for $\lambda_c$ is in agreement with
%a heuristic prediction based on considering the free energy of
%individual droplets of the stable phase in the midst of the
%metastable phase and taking into account droplet growth at a
%fast enough speed. This nonrigorous picture will be discussed in
%section \he.
\
%\item"(2)"
\noindent 2)
\corr{ The temperature can} The (subcritical) temperature $T$ can be
arbitrarily close to the
critical one.
\
%\item"(3)"
\noindent 3)
The initial distribution is only required to be
stochastically lower than the ($$)phase. In particular it
can be a Gibbs distribution under any negative value of the
external field (as it would if the
system were allowed to first relax to equilibrium under a
negative external field and then, suddenly, the field\corr{ were} was
switched
to a small positive value).
\
%\item"(4)"
\noindent 4)
We also show that in a certain technical fashion it makes
sense to say that at times of order $\exp(\lambda/h)$,
with $\lambda < \lambda_c$ rather than being in the
($$)phase the system is better described as being in a
metastable state which is infinitesimally (in $h$)
higher than the ($$)phase. The rigorous result is
presented in the form of an asymptotic expansion in
powers of $h$.
\ccor{ CCC} The metastable states can then be seen as a family of
$ \Cal C^\infty$ continuations into the region of positive external
fields of
the curve of the equilibrium states with negative external fields.
(See also the discussion after the statement of the main result of
this paper in the Section 1.4.)
The result about the $ \Cal C^\infty$ continuations is not in conflict
with the known fact, proven in [Isa],
that at least at low enough temperature, there is no analytic continuation
of the equilibrium states beyond the transition point.
%\endroster
\
Various comments regarding (1) above are in order. To our knowledge,
this is the first rigorous relation established between the
equilibrium Wulff shape and the time evolution of kinetic Ising
models. (The first rigorous relation between the equilibrium
surface tension and the time evolution of kinetic Ising models was,
as far as we know, established in the fundamental paper [Mar],
in the situation in which there is no external field.) It is important
to point out that when we started the investigation which led to the
current paper we could not see any evident reason for \equ(lambdac) to
hold. This doubt was expressed, and to some extent discussed,
in Section 1iii of [\sddr]. Since the doubt\corr{ stemed} stemmed
in part from
the study of the metastable behavior of anisotropic Ising models
in [KO], it is important to stress that regarding the problems
treated in the current paper, our results and methods apply also
to these models. In section \he \ we will present a heuristic
picture which predicts \equ(lambdac) and is
based on considering the freeenergy of
individual droplets of the stable phase in the midst of the
metastable phase and taking into account droplet growth at a
fast enough speed. The aspect of the heuristics which originally
seemed weak to us was the idea that the evolution of these
droplets is governed by their equilibrium freeenergy.
A recent detailed study of computer simulations
of the metastable relaxation of twodimensional
kinetic Ising models in [RTMS], which was done independently
of our work in this paper,
also indicated the validity of \equ(lambdac).
It has become evident over the years that the metastable
behavior of kinetic Ising models is very rich and that precise
mathematical statements can be conjectured and sometimes proven
in various different asymptotic regimes (see, e.g., Section
4 of [\sddr] or a more complete discussion in [Sch2]).
In a recent companion paper, [DS], results were obtained which are
counterparts to some of those presented here, but
in the case in
which the external field is held fixed (positive and small)
and the temperature is scaled to zero.
This paper is divided into three parts. In the\corr{ remaining}
remainder of the first one,
to which this section belongs, we will
be introducing notation and terminology, stating results,
motivating these results heuristically
and presenting some
basic tools. In the second part we prove the results concerning
the metastable regime, i.e., the behavior at times
of order of $\exp(\lambda/h)$ with $0 < \lambda < \lambda_c$. In the
third part we prove the results concerning
the relaxation regime, i.e., the behavior at times of the order
of $\exp(\lambda/h)$ with $\lambda > \lambda_c$.
\bigskip
\noindent
{\it Acknowledgements:}
Over the years during which we have worked on this project, we
have enjoyed the benefit of several stimulating conversations
with various colleagues. We are especially thankful to
A. van Enter,
R. Koteck\'{y},
F. Martinelli,
E. Olivieri
and E. Scoppola.
Part of this work was done while R.H.S. was visiting Rome in the
Fall of 1994, and he thankfully acknowledges the warm hospitality
of the
Physics Department of the University of Rome I and
of the Mathematics Departments of the Universities of Rome II and III.
\bigskip
\subheading{1.2. Notation and terminology}
%\numsec=2
%\numfor=1
In this section we introduce a long sequence of
definitions, notation and techniques.
We tried to make everything as
standard as possible, so that most readers
will browse quickly through this section, finding few
things\corr{ with which they are not familiar.} which they are
not familiar with. Most statements are made without
proof, and we refer readers to the books [Geo] and [Lig], and
other references therein, for explanation. Almost all the notation
introduced below is identical to that in the papers [\sddr] and [SS1].
\medskip
\noindent {\it The lattice} :
%We will consider models on the lattices $\z^d$, where $d$ is the space
%dimensionality.
%Because the dimension $d$ will in general be arbitrary but fixed, we
%will omit it in most of the notation.
The cardinality of a set $\amma \subset \z^2$ will be denoted
by $\amma$.
The expression $\amma \subset \subset \z^2$ will mean that $\amma$
is a finite subset of $\z^2$.
%The family of finite subset of $\z^d$ will be denoted by $\cal{F}$.
For each\corr{ $x \in \z^d$,} $x \in \z^2$, we define the usual norms
$\x\_p = (x_1^p + x_2^p)^{1/p}$, $p>0$ finite, and
$\x\_{\infty} = \max \{x_1 , x_2\}$.
The distance between two sets $\L_1,\L_2 \in \Z^2$ in each one of these
norms will be denoted by
$$
\text{dist}_p(\L_1,\L_2) = \inf\{xy_p : x \in \L_1, y \in \L_2 \}.
$$
In case $\L_1 = \{x\}$, we also write
$\text{dist}_p(\L_1,\L_2) = \text{dist}_p(x,\L_2)$.
The interior and exterior boundaries of a
set\corr{ $\amma \subset \z^d$} $\amma \subset \z^2$ will be denoted,
respectively by
$$
\partial_{\text{int}} \amma =
\{ x \in \amma : \xy\_1 = 1
\text{ for some } y \not \in \amma \},
$$
and
$$
\partial_{\text{ext}} \amma =
\{ x \not \in \amma : \xy\_1 = 1
\text{ for some } y \in \amma \}.
$$
The $p$norm diameter of a set $\L \subset \subset \Z^2$
is defined by
$$
\text{diam}_{p}(\L) = \max\{\text{dist}_p(x,y) : x,y \in \L\}.
$$
Given a set $A \subset \R^2$, we will write $\Lambda(A) =
A \cap \Z^2$. In case $A = [l/2,l/2]^2$ is a\corr{ $l$sided
square} $l\times l$
square centered at the origin, we simplify the notation to
$$
\L(l)=\Lambda(A) = \z^2 \cap [l/2,l/2]^2.
$$
Given $A,B \subset \R^2$, $z \in \R^2$ and $c \in \R$, we define
$A + B =\{x+y : x \in A, y \in B\}$,
$A + z = A + \{z\}$, and $c A = \{cx : x \in A\}$.
The set of bonds, i.e., (unordered) pairs of nearest neighbors
is defined as
$$
\vi = \{ \{x,y\} : x,y \in \z^d \text{ and } \xy\_1 = 1 \}.
$$
Given a set $\amma \subset \subset \Z^2$ we define also
$$
\vi_\amma = \{ \{x,y\} : x,y \in \amma
\text{ and } \xy\_1 = 1 \},
$$
$$
\partial\vi_\amma
= \{ \{x,y\} : x \in \amma, y \not\in \amma
\text{ and } \xy\_1 = 1 \}.
$$
%\vspace{3mm}
\medskip
\noindent {\it Notions from percolation} :
A chain is a sequence of distinct sites
$x_1,\dots,x_n$, with the property that
for $i=1,\dots,n1$,
%$\ \{x_i,x_{i+1}\} \in \vi$. The sites
$\ x_i  x_{i+1}_{1} = 1$. The sites
$x_1$ and $x_n$ are called the endpoints of the chain
$x_1,\dots,x_n$, and $n$ is its length.
A (*)chain, its endpoints and its length
are defined in the same way,
but with $\cdot_{1}$ replaced by $\cdot_{\infty}$.
Informally this means that while chains can only move along
bonds of $\z^2$, (*)chains can also move along diagonals.
A set of sites with the property that each two of them
can be connected by a chain contained in the set
is said to be a connected set.
A chain or (*)chain is said to connect two sets
if it has one endpoint in each set.
A set of sites is said to be simplyconnected
in case it is connected and its complement is also a connected set.
A circuit is a chain such that
$\ x_1  x_{n}_{1} = 1$. Similarly a (*)circuit is a (*)chain
such that $\ x_1  x_{n}_{\infty} = 1$.
%\vspace{3mm}
\medskip
\noindent {\it The configurations and observables} :
At each site in $\z^2$ there is a spin which can take values $1$
and $+1$. The configurations will therefore be elements of the
set $\{1,+1\}^{\z^d} = \Omega$. Given $\sigma \in \Omega$, we
write $\sigma(x)$ for the spin at the site\corr{ $x \in \z^d$.}
$x \in \z^2$.
Two configurations are specially relevant,
the one with all spins $1$ and the one with all spins $+1$.
We will use the simple notation $$ and $+$ to denote them.
The single spin space, $\{1,+1\}$ is endowed with the discrete
topology and $\Omega$ is endowed with the corresponding product
topology.
The following definition will be important when we introduce
finite systems with boundary conditions later on;
given $\amma \subset \subset \Z^2$
and a configuration $\eta \in \Omega$,
we introduce
$$
\Omega_{\amma,\eta} = \{ \sigma \in \Omega : \sigma(x) = \eta(x)
\text{ for all } x \not\in \amma \}.
$$
Realvalued functions with domain in $\Omega$ are called
observables.
For each observable $f$ we use the notation \
$ f_{\infty} = \sup_{\eta \in \Omega} f(\eta)$.
Local observables are those which depend only on
the values of finitely many spins, more precisely, $f : \Omega
\rightarrow \R$ is a local observable if there exists a set
$S \subset \subset \Z^2$ such that $f(\sigma) = f (\eta)$ whenever
$\sigma(x) = \eta(x)$ for all $x \in S$.
The smallest $S$ with this property is called the support of $f$,
denoted $\supp(f)$.
%Clearly, if $f$ is a local observable, then $f_{\infty}
%$< \infty$.
The topology introduced above on $\Omega$, has the nice feature that
it makes the set of local observables be dense in the set of all
continuous observables.
%For the average spin in a set $\amma \subset \subset \Z^2$ we will use
%$$
%X_{\amma}(\sigma) = \frac{1}{\amma} \sum_{x \in \amma} \sigma(x).
%$$
In $\Omega$ the following partial order is\corr{ introduced:
$$
\eta \leq \zeta \ \text{ if } \
\eta(x) \leq \zeta(x) \text{ for all } x \in \z^d.
$$}
introduced:
$$
\eta \leq \zeta \ \text{ if } \
\eta(x) \leq \zeta(x) \text{ for all } x \in \z^2.
$$
A particularly important role will be played in this paper
by the nondecreasing local observables.
%Clearly every
%local observable is of bounded variation, and, as such,
Each local observable can
be written as the difference between two nondecreasing ones.
A ($+$)chain in a configuration $\sigma$
is a chain of sites,
$x_1, \dots, x_n$, as defined above, with the property that
for each $i=1,\dots,n$,
$\ \sigma(x_i) = +1$.
%The ($+$)clusters in a configuration $\sigma$ are the
%connected components of the set of sites where the spin is $+1$
%in the configuration $\sigma$.
%A $$ cluster is called infinite if it
%contains infinitely many sites.
Given $\L_1, \L_2 \subset \Z^2$, we will use the notation
$\{\Lambda_1 \overset+\to\longleftrightarrow \L_2\}$
to denote the set of configurations in which there is a
(+)chain with one endpoint in $\L_1$ and one endpoint
in $\L_2$. In case $\L_1 = \{x\}$ we simplify this notation to
$\{x \overset+\to\longleftrightarrow \L_2\}$, and similarly for
$\L_2$.
Given a configuration $\sigma$,
we say that a site $x$ is (+)connected to a set
$\Lambda \subset \Z^2$ in $\sigma$ if
$\sigma \in \{x \overset+\to\longleftrightarrow \L\}$.
The (+)cluster of a set
$\L \subset \Z^2$ in the configuration $\sigma$
is the set of sites which
are (+)connected to $\L$ in this configuration.
Similar notions can be defined for ($$)connectedness,
(+*)connectedness and ($$*)connectedness.
In particular the notation
$\{\Lambda_1 \overset\to\longleftrightarrow \L_2\}$,
$\{\Lambda_1 \overset+*\to\longleftrightarrow \L_2\}$, and
$\{\Lambda_1 \overset*\to\longleftrightarrow \L_2\}$
should have now selfexplanatory meaning.
%\vspace{3mm}
\medskip
\noindent {\it The probability measures} :
We endow $\Omega$ also with the Borel $\sigma$algebra corresponding
to the topology introduced above. In this fashion, each probability
measure $\mu$ in this space can be identified by the corresponding
expected values $\int f d\mu$
of all the local observables $f$. A sequence of
probability measures, $(\mu_n)_{n=1,2,\dots}$,
is said to converge weakly to
the probability measure $\nu$ in case
$$
\lim_{n \rightarrow \infty}\int f d\mu_n = \int f d\nu
\ \text{ for every continuous observable $f$}.
\Eq(conv)
$$
The family of probability measures on $\Omega$ will be partially
ordered by the following relation:
$\mu \leq \nu$ if
$$
\int f d\mu \leq \int f d\nu
\ \text{ for every continuous nondecreasing observable $f$}.
\Eq(sineq)
$$
Because the local observables are dense in the set of continuous
observables, we can restrict ourselves to the local ones in
\equ(conv) and \equ(sineq).
Moreover, because every local observable is the difference
between two nondecreasing ones, we can also restrict ourselves
to those in \equ(conv).
%; we will make heavy use of this observation.
%\vspace{3mm}
\medskip
\noindent {\it The Gibbs measures} :
We will consider always the formal Hamiltonian.
$$
H_h(\sigma) = \frac{1}{2}\sum_{x,y \text{ n.n.}}
\sigma(x) \sigma(y)  \frac{h}{2} \sum_{x} \sigma(x),
\Eq(ham)
$$
where $h \in \R$ is the external field and $\sigma \in \Omega$ is
a generic configuration.
In order
to give precise definitions, we define, for each set $\amma
\subset \subset \Z^2$ and each boundary condition $\eta \in \Omega$,
$$
H_{\amma,\eta,h}(\sigma) = \frac{1}{2}\sum_{\{x,y\} \in \vi_\amma}
\sigma(x) \sigma(y)
\frac{1}{2}\sum
\Sb
\{x,y\} \in \partial \vi_{\amma} \\
y \not\in \amma
\endSb
\sigma(x) \eta(y)
 \frac{h}{2} \sum_{x \in \amma} \sigma(x).
\Eq(ham2)
$$
In what follows the temperature $T$\corr{ will appear} will often
appear explicitly in
the notation, for clarity. Later on in this paper we will
usually be considering a situation in which the temperature
is fixed, while we scale the external field $h$, and then
the temperature will be\corr{ omited} omitted from the notation.
Given $\amma \subset \subset \Z^2$, $\eta \in \Omega$,
$E \subset \Omega$, $T>0$ and $h \in \R$ we write
$$
Z_{\amma,\eta,T,h}(E) = \sum_{\sigma \in \Omega_{\amma, \eta}
\cap E} \exp (\beta H_{\amma,\eta,T,h}(\sigma)),
$$
where $\beta = 1/T$.
We abbreviate $Z_{\amma,\eta,T,h} =Z_{\amma,\eta,T,h}(\Omega)$.
The Gibbs (probability) measure in $\amma$
with boundary condition $\eta$ under external field $h$ and at
temperature $T$ is now defined on $\Omega$ as
$$
\mu_{\amma,\eta,T,h}(\sigma) =
\left\{
%\begin{array}{ll}
\aligned
\frac{\exp(\beta H_{\amma,\eta,h}(\sigma))}{
Z_{\amma,\eta,T,h}}, &
%\sum_{\zeta \in \Omega_{\amma, \eta}}
%\exp(\beta H_{\amma,\eta,h}(\zeta))} &
\text{\quad if $\sigma \in \Omega_{\amma,\eta}$}, \\
0, & \text{ \quad otherwise.}
\endaligned
%\end{array}
\right.
$$
%for $\sigma \in \Omega_{\amma,\eta}$ and 0 otherwise.
%The following property
%is a consequence of the fact that the Hamiltonian only involves
%interactions between nearest neighbors: given
%$\amma \in \cal{F}$,
%if $\eta(x)=\zeta(x)$ for every $x \in
%\partial_{\mbox{ext}}\amma$, then
%\be
%\label{markov0}
%\int f d\mu_{\amma,\eta,h} = \int f d\mu_{\amma,\zeta,h},
%\ee
%for every local observable $f$ whose support
%is contained in $\amma$.
%The next property is known as the DLR equations: given
%$\amma \subset \amma^{\prime} \in \cal{F}$ and a pair
%of configurations $\eta$ and $\eta^{\prime}$ which
%are identical off $\amma^{\prime}$, we have
%\be
%\label{dlr0}
%\mu_{\amma^{\prime},\eta^{\prime},h}(\ \cdot \ 
%\Omega_{\amma,\eta})
%=
%\mu_{\amma,\eta,h}(\cdot).
%\ee
The Gibbs measures satisfy the following monotonicity relations
to which we will refer as the FKGHolley inequalities.
$$
\text{If } \eta \leq \zeta \text{ and } h_1 \leq h_2,
\text{ then, for each $\amma \subset \subset \Z^2$ and $T>0$, } \
\mu_{\amma,\eta,T,h_1} \leq \mu_{\amma,\zeta,T,h_2}.
$$
A Gibbs measure for the infinite system on $\z^2$ is defined
as any probability measure $\mu$ which satisfies the DLR
equations in the sense that for every $\amma \subset \subset \Z^2$
and $\mu$almost all $\eta \in \Omega$
$$
\mu(\ \cdot \ 
\Omega_{\amma,\eta})
=
\mu_{\amma,\eta,T,h}(\ \cdot \ ).
\Eq(dlr1)
$$
Alternatively and equivalently, Gibbs measures can be defined as
elements of the closed convex hull of the set of weak limit points
of sequences of the form \newline
$(\mu_{\amma_i,\eta_i,h})_{i=1,2,\dots}$,
where each $\amma_i$ is finite and $\amma_i \rightarrow \z^d$,
as $i \rightarrow \infty$, in the sense that
$\cup_{i=1}^{\infty} \cap_{j=i}^{\infty} \amma_j = \z^d$.
%Together, (\ref{markov0}) and the DLR equations, (\ref{dlr0})
%and (\ref{dlr1}), imply the Markov property for the Gibbs measures;
%for instance, if $\mu$ is a Gibbs measure for the infinite
%system under external field $h$, then for arbitrary
%$\amma \in \cal{F}$ and $\mu$almost all
%$\eta,\zeta \in \Omega$ such that
%$\eta(x)=\zeta(x)$ for every $x \in \partial_{\mbox{ext}}\amma$,
%$$
%\int f d\mu(\ \cdot \  \Omega_{\amma,\eta}) =
%\int f d\mu(\ \cdot \  \Omega_{\amma,\zeta}),
%$$
%for every local observable $f$ whose support
%is contained in $\amma$.
%The FKGHolley inequalities
%can be used to prove that
For each value of $T$ and $h$, $\mu_{\Lambda(l),,T,h}$
({\it resp.}
$\mu_{\Lambda(l),+,T,h}$) converges weakly, as $l \rightarrow \infty$,
to a probability measure that we will denote by $\mu_{,T,h}$
({\it resp.} $\mu_{+,T,h}$).
If $h \not = 0$ it is known that
$ \mu_{,T,h} = \mu_{+,T,h}$, which will then be denoted simply by
$\mu_{T,h}$; it is also known that this is the only Gibbs measure
for the infinite system in this case.
If $h=0$ the same is true if
the temperature is larger than or equal to a critical value $T_c > 0$,
and is false for $T < T_c$, in which case one says that there is phase
coexistence.
%Moreover, for the values of $T$ and $h$ for which (\ref{uniq})
%holds, any weak limit of any sequence
%of the form ($\mu_{\amma_i,\eta_i,T,h})_{i=1,2,...}$,
%where $\cal{F} \ni \amma^i \rightarrow \z^d$,
%is identical to $\mu_{T,h}$.
%Therefore we conclude that whenever (\ref{uniq}) holds
We use the following abbreviations and names:
$$
\mu_{,T,0} = \mu_{,T} = \text{ the ($$)phase,}
$$
$$
\mu_{+,T,0} = \mu_{+,T} = \text{ the (+)phase.}
$$
Another known fact is that for each fixed $T$
$$
\mu_{T,h} \rightarrow \mu_{+,T} \text{ weakly, as $\hg$},
\Eq(muhtomu+)
$$
and
$$
\mu_{T,h} \rightarrow \mu_{,T}
\text{ weakly, as $h \nearrow 0$}.
\Eq(muhtomu)
$$
%The notation
%$$
%\Cal{U} = \{(h,T) \in \R \times [0,\infty) :
%h \not= 0\ \text{or} \ T > T_c \}
%$$
%will be used to denote the interior of the uniqueness region in the
%$h \times T$ phasediagram, i.e, the set of all the uniqueness points,
%except for the critical point $(0,T_c)$.
For the expected value corresponding to a Gibbs measure
$\mu{...}$, in finite or infinite volume,
we will use the notation
$$
\langle f \rangle_{...} = \int f d \mu{...},
$$
where $...$ stands for arbitrary subscripts.
The corresponding conditional expectation, given the event $E$
will be denoted by $ \langle fE \rangle_{...}$.
The spontaneous magnetization at temperature $T$ is defined as
$$
m^*(T) = \langle \sigma(0) \rangle_{+,T}.
$$
(Here we are using a common and convenient form of abuse of notation:
$\sigma(x)$ is being used to denote the observable which associates
to each configuration the value of the spin at the site $x$ in
that configuration. This notation will also be used in other
places.)\corr{ I do not see any abuse here. Compare with previous
notations.}
It is known that $m^*(T) > 0$ if and only if
$\mu_{,T} \not = \mu_{+,T}$
and also that $ \lim_{T \searrow 0} m^*(T) = 1$.
\medskip
\noindent{\it Surface tension and Wulff shape}:
The direction dependent 0field surface tension is defined in the following
way. First consider on $\R^2 \times \R^2$ the usual inner product
$(x,y) = x_1 y_1 + x_2 y_2$.
Let $\Bbb S ^1 = \{x \in \R^2 : x_2 = 1\}$, and for each vector
$\bold n \in \Bbb S^1$, consider the following configuration, to be
used as a boundary condition
$$
\eta(\bold n) (x) = \cases
+1, & \text{if $(x, \bold n) \geq 0$}, \\
1, & \text{if $(x, \bold n) < 0$}.
\endcases
$$
The surface tension in the direction perpendicular to $\bold n$ is given by
$$
\tau_{T}(\bold n) =
\lim_{l \rightarrow \infty} 
%\frac{1}{\beta d(A_l,B_l)} \log
\frac{1}{\beta y(l)z(l)_{2}} \log
\frac{
Z_{\Lambda(l),\eta(\bold n),T,0}}{
Z_{\Lambda(l),+,T,0}},
$$
where $y(l)$ and $z(l)=y(l)$ are the points where the straight line
$\{x \in \R^2 : (x,\bold n) = 0 \}$
intersects the boundary of the square $\Lambda(l)$.
%and $d(A_l,B_l)$ is the Euclidean distance between these points.
It is known that for each $T < T_c$ the surface tension $\tau_T(\cdot)$ is a
continuous strictly positive and finite function.
We shall use $\Cal D$ to denote the set of all
closed selfavoiding rectifiable
curves $\gamma \subset \Bbb R^2$ that are a boundary
of a bounded region,
$\gamma =\partial V,\,V\subset \Bbb R^2 $.
Let us recall that a curve is called
rectifiable if the supremum of the lengths of polygons, with
edges connecting\corr{ sequentially arbitrary collections of points chosen
on the curve, } arbitrary collections of points chosen
on the curve, in the order inherited from the curve, is finite (and equals
then the\corr{ length of the curve} length, $ \gamma $, of
the curve $ \gamma $),
and that a rectifiable curve has a tangent at almost every point.
It is easy to verify that a curve $\gamma $ that is the
boundary of a convex bounded region belongs to $\Cal D$.
%Let F be a continuous, strictly positive function on the unit
%circle $\Bbb S^1\subset \Bbb R^2$.
We can assign to each curve $\gamma \in\Cal D$ the quantity
$$
\Cal{W}(\gamma )=\Cal{W}_T(\gamma )=\int\limits_\gamma \tau_T(\bold n_s)ds,
%\Eq(tag1)
$$
%where $s$ is a natural parametrization (the length parameter)
%where $s$ is the length parameter of the curve $\gamma$
where $s$ parametrizes the curve $\gamma$ according to Euclidean length
measured along this curve,
and $\bold n_s$ is the
unit outward normal vector to the curve at the point
$s\in\gamma $ (i.e. the vector orthogonal to the
tangent in the considered point and oriented outward the region bounded
by $\gamma$).
The functional $\Cal{W}_T$ will be called
the Wulff functional associated to the zerofield directiondependent
surface tension $\tau_T(\cdot)$. Sometimes we will refer to it also as
the integrated surface tension.
To every vector $\bold n\in \Bbb S^1$ and $\lambda >0$ we assign
the halfplane
$$
L_{T,\bold{n},\lambda }=\left\{x\in\Bbb R^2 :(x,\bold n)\le \lambda
\tau_T(\bold n)\right\}.
%\Eq(tag2)
$$
Let us consider the intersection
$$
W_{T,\lambda} =\bigcap\limits_{\bold n\in \Bbb S^1}L_{T,\bold n,\lambda}.
\Eq(tag3)
$$
These sets clearly satisfy the scaling relation $W_{T,\lambda}
= \lambda W_{T,1}$.
In particular they keep the same shape, as $\lambda$ varies;
this shape is called the Wulff shape.
The Wulff body of volume 1 is defined as $W_T = W_{T,\lambda_0}$,
where $\lambda_0$ is chosen so that its volume is indeed 1.
$W_T$ is clearly convex and thus its boundary $\partial W_T\in\Cal D$.
The following is therefore well defined,
$$
w = w(T) = \Cal{W}_T(\partial W_T).
$$
For each $T0$ and
$0 < c_{\min}(T) \leq c_{\max}(T) < \infty$
such that for all $h \in (h(T),h(T))$ and
$\sigma \in \Omega$
$$
c_{\min}(T) \leq c_h(0,\sigma) \leq c_{\max}(T).
$$
Throughout this paper we will suppose that we have chosen and
kept fixed a set of rates $c_h(x,\sigma)$ which satisfy
the detailed balance conditions, \equ(db0)
and all the hypotheses H(1)  H(4).
This spin flip system will be denoted by
$(\sigma^\eta_{h;t})_{t \geq 0}$, where $\eta$ is the initial
configuration. If this initial configuration is selected at random
according to a probability measure $\nu$, then the resulting
process is denoted by $(\sigma^\nu_{h;t})_{t \geq 0}$.
The probability measure on the space of trajectories of the
process will be denoted by $\p$, and the corresponding
expectation by $\Bbb E$. (Later, when we couple various related
processes, we will also use the symbols $\p$ and $\Bbb E$ to
denote probabilities and expectations in some larger
probability spaces, but no confusion should arise from this.)
The assumption of detailed balance, \equ(db0),
assures that the Gibbs measures are
invariant with respect to the stochastic Ising models. Moreover,
from the assumption of attractiveness, H(3), one obtains the
following convergence\corr{ results
$$
\sigma^_{h;t} \rightarrow \mu_{h,},
$$
and
$$
\sigma^+_{h;t} \rightarrow \mu_{h,+},
$$
weakly,}
results
$$
\sigma^_{h;t} \rightarrow \mu_{,T,h},
$$
and
$$
\sigma^+_{h;t} \rightarrow \mu_{+,T,h},
$$
weakly, as $t \rightarrow \infty$.
We will want to consider, sometimes as a tool,
and sometimes for its
own sake, the counterpart of the stochastic Ising model that we
are considering, on an arbitrary
finite set $\amma \subset \subset \Z^2$, with some
boundary condition $\xi \in \Omega$. This process, which will
be denoted by $(\sigma^\eta_{\amma,\xi,h;t})_{t \geq 0}$,
where $\eta \in \Omega_{\amma,\xi}$ is the initial configuration,
is defined as the spin flip system with rates of flip given by
$$
c_{\amma,\xi,h}(x,\sigma) = \left\{
\aligned
c_{h}(x,\sigma) & \
\text{ \ \ if } \sigma, \sigma^x \in \Omega_{\amma,h,} \\
0 \ \ \ \ \ & \
\text{ \ \ otherwise.}
\endaligned
\right.
$$
When $\sigma, \sigma^x \in \Omega_{\amma,h,}$, \equ(db0) yields,
for all $x \in \z^d$,
$$
\mu_{\amma,\xi,h}(\sigma) c_{\amma,\xi,h}(x,\sigma) =
\mu_{\amma,\xi,h}(\sigma^x) c_{\amma,\xi,h}(x,\sigma^x),
\Eq(db1)
%\label{db1}
$$
which is the usual reversibility condition for finite statespace
Markov processes. (Conversely, if one
requires \equ(db1) to be satisfied
for arbitrary $\amma \in \Cal{F}$ and $\xi \in \Omega$, then one
can deduce that \equ(db0) must hold.)
It is clear from H(4) that
$(\sigma^\eta_{\amma,\xi,h;t})$ is irreducible and hence from
\equ(db1) it follows that, for\corr{ any $\eta$,
$$
\sigma^\eta_{\amma,\xi,h;t} \rightarrow \mu_{\amma,\xi,h},
$$
weakly,}
any $\eta$,
$$
\sigma^\eta_{\amma,\xi,h;t} \rightarrow \mu_{\amma,\xi,T,h},
$$
weakly,
as $t \rightarrow \infty$.
%\vspace{3mm}
\medskip
\noindent {\it Graphical construction} :
In order to prove our claims in this paper, we will use
a standard graphical construction which provides versions of
the whole family of processes at a given temperature $T$, with
arbitrary value of $h \in (h(T),h(T))$,
either on the infinite lattice $\z^2$
or on any of its finite subsets,
with arbitrary boundary conditions and
starting from any initial configuration, all on the same probability
space. This construction is the same one used in [\sddr].
But\corr{ The} the relevance of this construction will be even greater
here than it was in that paper, since in part 3 of our
paper we will use\corr{ this construction} it to define the process on
regions of spacetime which are fairly general, and we will set up a
rescaling procedure based on such objects.
%The type of construction and the proofs that we present next are
%not new, but we could not find in the literature the precise type
%of results that we needed, including the necessary uniformity of
%the estimates in $h \in (h(T),h(T))$, and for this reason we provide
%a selfcontained exposition.
The graphical construction that we use is a specific
version of what is called basic coupling between spin flip processes:
a coupling in which the spins flip together as much as possible,
considering the constraint that they have to flip with certain rates.
%In this paper we will refer to this construction simply as the
%basic coupling.
The construction is carried out by
first associating to each site $x \in \z^2$ two independent
Poisson processes, each one with
rate $c_{\max}(T)$. We will denote the successive arrival times
(after time $0$) of
these Poisson processes $(\tau^+_{x,n})_{n=1,2,\dots}$ and
$(\tau^_{x,n})_{n=1,2,\dots}$.
Assume that the Poisson processes
associated to different sites are also mutually independent.
We say that at each point in spacetime of the form
$(x, \tau^+_{x,n})$ there is an upward mark and that
at each point of the form $(x, \tau^_{x,n})$ there is
a downward mark.
%It is clear that almost surely the random times
%$\tau^*_{x,n}$, \ $x \in \z^2$, \ $n=1,2,\dots$, \ $*=+,$,
%are all distinct.
Next we associate to each arrival time
$\tau^*_{x,n}$, where $*$ stands for $+$ or $$, a random
variable $U^*_{x,n}$ with uniform distribution between 0 and 1.
All these random variables are supposed to be independent
among themselves and independent from the previously introduced
Poisson processes. This finishes the construction of the
probability space.
The corresponding probability and expectation will be denoted,
respectively, by $\p$ and $\E$.
We have to say now how the various processes are constructed
on this probability space. For finite $\amma$ and arbitrary
$\xi$, the process $(\sigma_{\amma,\xi,h;t}^\eta)$
is constructed as follows.
We know that almost surely the random times
$\tau^*_{x,n}$, $x \in \amma$, $n=1,2,\dots$, $*=+,$,
are all distinct, and
we update the state of the process at each time when
there is a mark at some $x \in \amma$
according to the following rules. If the mark
that we are considering is at the point
$(x, \tau^*_{x,n})$,
%occurs at the site $x$ at the time $t$,
and the configuration immediately before time
$\tau^*_{x,n}$ was $\sigma$, then
%\vspace{2mm}
\noindent i) The spins not at $x$ do not change.
%\vspace{2mm}
\noindent ii) If $\sigma(x) = 1$
%immediately before time $t$ the spin at $x$ was $$,
({\it resp.} $\sigma(x) = +1$),
then the spin at $x$ can only flip if the mark is of upward type
({\it resp.} downward type).
%\vspace{2mm}
\noindent iii) If the mark is upward and $\sigma(x)=1$,
or if the mark is downward and $\sigma(x)=+1$, then
we flip the spin at $x$ if and only if $c_{\amma,\xi,h}(x,\sigma)
> U^*_{x,n} c_{\max}$.
%\vspace{2mm}
\noindent One can readily see that the process
constructed in this fashion has the correct rates of flip.
In principle, one would like to construct the processes on
the infinite lattice $\z^2$ in a
similar fashion, with $c_h(x,\sigma)$ replacing
$c_{\amma,\xi,h}(x,\sigma)$ in (iii). Some extra care has to be
taken, because during any nondegenerate interval of time
infinitely many marks occur. This is not a real
problem, because of the assumption
that the range of the interaction, $R$, is finite. Starting from a
configuration $\eta$ at time $0$, we have to say how the spin at
a generic site $x$ at a time $t$ is obtained.
Using percolation arguments one can argue that
on a set of probability 1 in the probability
space where the marks were
defined, for any fixed $x$ and $t$,
if we take any boundary condition $\xi$, then the sequence
$(\sigma^\eta_{\Lambda(l),\xi,h;t}(x))_{l=1,2,\dots}$ will
converge as $l \rightarrow \infty$
(i.e., will become constant for large $l$), to a limit which does not
depend on $\xi$. This limit can then be taken to be the value
of $\sigma^\eta_{h;t}(x)$, and it is clear that the\corr{ version of
the process $(\sigma^\eta_{h;t})$
constructed in this fashion has the correct
flip rates.} process thus constructed has the correct flip rates
and is therefore a version of
the process $(\sigma^\eta_{h;t})$.\ccor{ CCC} The expected value of a
function of that process will be denoted by
$\langle \cdot \rangle ^\eta_{h;t}$.
A standard proof of the claim above about insensitivity
to receding boundary conditions can be found in [\sddr].
%we introduce
%the events $E(x,t,l)$ that there exists a sequence of points in
%spacetime
%$(x_0,0),(x_1,t_1),\dots,(x_n,t_n)$ with the properties that
%$0 0$ associates $\langle f \rangle_{h}$, is infinitely
differentiable at $h=0$.
Moreover, for $j=1,2,...$, the following identity holds:
$$
\left.
%\frac{1}{j!}
\frac{d^j\langle f \rangle_{+,h}}{d h^j}
\right_{h = 0_{+}}
=
\(\frac{\beta}{2}\)^j
\sum_{x_1,...,x_j \in \Z^2}
\langle f;\s(x_1);...;\s(x_j) \rangle_{+}.
\Eq(jthderiv)
$$
In this expression, which will be justified below, the
quantities that appear inside of the summation are
called generalized Ursell functions, and are defined next.
We start by defining the generalized Ursell functions for a
Gibbs measure
$\mu_{\L,\eta, h}( \ \cdot \ )$, where $\L$ is a
finite set. For this purpose we consider a generalization of the
Hamiltonian, in which at each site $x$ the external applied field
may be different, and takes the value $h_x$. If $\underline{h}$
is the function that to each $x \in \Z^2$ associates $h_x$, then
we will denote by $\mu_{\L,\eta,\underline{h}}$ the corresponding Gibbs
distribution in $\L$ with boundary condition $\eta$.
With a local observable $f$ given we define
$$
\langle f;\s(x_1);...;\s(x_j) \rangle_{\L,\eta,h}
=
\left(\frac{2}{\beta}\right)^j
\left.
\frac{\partial^j \ \langle f \rangle_{\L,\eta,\underline{h}} }
{\partial h_{x_1} \dots
\partial h_{x_j}} \right_{\underline{h} \equiv h},
$$
where $\underline{h} \equiv h$ means that
the function $\underline{h}$ is identically $h$.
It is easy to see, by induction on $j$, that
$\langle f;\s(x_1);...;\s(x_j) \rangle_{\L,\eta,h}$
is a linear combination of products of
$\mu_{\L,\eta,h}$expected values
of local observables, all of them with support contained in
$\Supp(f) \cup \{x_1,...,x_j\} $.
Convergence of the generalized Ursell functions as $\L \rightarrow
\Z^2$ follows from this, for arbitrary $T$, $h$ and $\eta$, as long
as $\mu_{\L,\eta,h}$ converges weakly to some distribution. In case
$T < T_c$, $h \geq 0$ and $\eta = +$, we can do better and
obtain from \equ(expmu+) the bound
$$
\align
 \langle f;\s(x_1); & ...;\s(x_j) \rangle_{\L,+,h}

\langle f;\s(x_1);...;\s(x_j) \rangle_{+,h} 
\\ & \leq \
C_{j}(f) \exp\Big(C(T) \ \dist(
\Supp(f) \cup \{x_1,...,x_j\},\L^c)\Big).
\teq(expursell)
\endalign
$$
Observe that in particular this estimate is uniform in $h\geq0$.
The proof of \equ(jthderiv) can be found in Section 2 of [ML]. In
that paper the result was proven at low enough temperature, but
this was so simply because the estimate \equ(correl),
which\corr{ derives
from} is a consequence of \equ(CCS) was not available then.
Replacing Theorem 1 in
[ML] with \equ(correl), the proof in that paper applies up to
$T_c$. The basic estimate used in [ML], and to which we will
return in section \CCD, states that the exponential decay of correlations
\equ(correl) implies a similar exponential decay for the generalized
Ursell functions, when the diameter of the set
$\Supp(f) \cup \{x_1,...,x_j\}$ becomes large:
$$
\langle f;\s(x_1);...;\s(x_j) \rangle_{\L,+,h}
\leq
C_j(T,f) \exp \( C(T) \frac{ \diam(\Supp(f) \cup \{x_1,...,x_j \}) }{j}\).
\Eq(ML)
$$
Observe that in particular this estimate is uniform in $h\geq0$ and
in $\L$.
The proof that \equ(ML) follows from \equ(correl)
in the special case in which $f$ is of the form $f(\s) =
\s(y_1)\cdots\s(y_m)$ for some set of sites $\{y_1,...,y_m\}$ can be
found in the Appendix B of [ML]. The general case
follows immediately from this one, since any local function $f$
can be written as a linear combination of functions of this special
form, with $\{y_1,...,y_m\}$ running over all the subsets of $\Supp(f)$.
As explained in the proof of Theorem 4 in [ML], \equ(jthderiv)
follows in a standard fashion from \equ(ML).
In connection to \equ(jthderiv) it is worth mentioning that for $h
> 0$ the function $\langle f \rangle_{h}$ is analytic,
as follows from the LeeYang theorem. Moreover,
in this case the Gibbs distributions are completely analytic in
an appropriate sense (see [SS1] and references
therein).\corr{? } The estimate \equ(ML) can be replaced for $h
> 0$ by a stronger one:
$$
\langle f;\s(x_1);...;\s(x_j) \rangle_{\L,+,h}
\leq
C'_j(T,h,f) \exp \( C(T,h) \text{ dist} _{ \text{tree} }
(\Supp(f),x_1,...,x_j ) \),
$$
where $\text{ dist} _{ \text{tree} }
(\Lambda _1,..., \Lambda _j )$ is the length of the
shortest tree in $\Bbb R^2$, connecting all the sets
$\Lambda _1,..., \Lambda _j \subset \Bbb Z^2$. However,
the constants $ C'_j(T,h,f)$ explode as $h\to 0$.
It is also important to recall that,
on the other hand, it has been proven in
[Isa] that at low enough temperatures there is an essential
singularity nevertheless at $h=0$; this is expected to be so
up to $T_c$, but no proof of that claim is available, as far as
we know.
The identity \equ(jthderiv) and
the various related statements that we made above
have, of course, analogues for
$h \leq 0$. Those are the ones that will be relevant for us
when we study metastability under small $h>0$, since the
``metastable state'' should then be a ``continuation'' of the
equilibrium states with $h \leq 0$.
\bigskip
%\input thm
\subheading{1.4. Main results}
Recall that we are considering a kinetic Ising model for the formal
Hamiltonian \equ(ham) in dimension 2, which is supposed to satisfy
conditions (H1), (H2), (H3) and (H4) of Section 1.2. Recall also that
for $T < T_c$ we define
$\lambda_c = \lambda_c(T) = w(T)^2 / (12 \, T \, m^*(T))$.
The following theorem is our main result.
\proclaim{Theorem 1}
Suppose $T < T_c$. For every
probability distribution $\nu \leq \mu_{}$
the following happens.
\
\noindent i) If $0<\lambda < \lambda_c$,
then for each $n \in \{1,2,...\}$
and for each local observable $f$,
$$
\E\(f\(\sigma_{h;\exp(\lambda/h)}^{\nu}\)\)
=
\sum_{j=0}^{n1} b_j(f) h^j + O(h^n)
$$
for $h > 0$, where
$$
b_j(f) = \left. \frac{1}{j!}
\frac{d^j\langle f \rangle_{,h}}{d h^j}
\right_{h = 0_{}}
= \frac{1}{j!}
\(\frac{\beta}{2}\)^j
\sum_{x_1,...,x_j \in \Z^2}
\langle f;\s(x_1);...;\s(x_j) \rangle_{},
$$
and $O(h^n)$ is a function of $f$ and $h$ which
satisfies $\limsup_{h \searrow 0} O(h^n) / h^n < \infty$.
\
\noindent ii) If $\lambda > \lambda_c$, then
for any finite positive $C$ there is a finite positive
$C_1$ such that for every local observable $f$,
$$ \left \E\(f\(\sigma_{h;\exp(\lambda/h)}^{\nu}\)\)
 \langle f \rangle_{h} \right
\ \leq \
C_1 \ f_{\infty} \ \exp \(\frac{C}{h}\),
$$
for all $h > 0$.
\endproclaim
From this theorem, the simple Proposition 1
in [\sddr], and the fact that
$\langle f \rangle_h \rightarrow \langle f \rangle_+$
as $\hg$ (see \equ(muhtomu+), or the paragraph which
precedes \equ(jthderiv)),
the following corollary is obtained.
\proclaim{Corollary 1}
Suppose $T < T_c$. For every
probability distribution $\nu \leq \mu_{}$
the following happens.
If we let $\hg$ and $t \rightarrow \infty$ together,
then for every local observable $f$
\noindent i)
$ \ \E\(f\(\sigma_{h;t}^{\nu}\)\)
\rightarrow \langle f \rangle_{} \ \text{ if } \
\limsup h \log t < \lambda_c(T)$.
\noindent ii)
$ \ \E\(f\(\sigma_{h;t}^{\nu}\)\)
\rightarrow \langle f \rangle_{+} \ \text{ if } \
\liminf h \log t > \lambda_c(T)$.
\endproclaim
In other words, we are stating that the law of the random configuration
$\sigma^{\nu}_{h;t}$
converges weakly to $\mu_{}$ in case (i) and to $\mu_+$ in case (ii).
This corollary is already an important strengthening of Theorem 1 in
[\sddr] in the 2 dimensional case. The following aspects of that
theorem are improved here: 1) There is a single constant $\lambda_c$
separating
the regimes (i) and (ii). 2) The temperature is now only required to
be below
$T_c$. 3) The initial distribution is much more general than in [\sddr],
where it was supposed to be concentrated on the configuration with all
spins down. Note that, by the FKGHolley inequalities, for each $h < 0$
the distribution $\mu_h$ satisfies the condition above on the initial
distribution $\nu$.
To illustrate and clarify the way in which Theorem 1(i) improves even
further the statement in Corollary 1(i), let us take
the local observable given by $f(\sigma)
= \sigma(0)$ and $n=2$. We have then, when $0<\lambda < \lambda_c$
$$
\E\(\sigma_{h;\exp(\lambda/h)}^{\nu}(0)\)
=
m^* + \chi h +
O(h^2),
$$
when $h > 0$. Here
$$
\chi =
b_1(f) = \left.
\frac{d\langle \sigma(0) \rangle_{,h}}{d h}
\right_{h = 0_{}}
=
\(\frac{\beta}{2}\)
\sum_{x \in \Z^2}
\langle \sigma(0);\s(x) \rangle_{},
$$
is the susceptibility at $h = 0_{}$. This means that when $h>0$ is
small the function $m^* + \chi h$ is a better approximation to
$\E\(\sigma_{h;\exp(\lambda/h)}^{\nu}(0)\)$ than the constant
function identical to $m^*=\langle f \rangle_{}$. This function
$m^* + \chi h$ is the smooth linear continuation into the region
$h \geq 0$ of the function
which to $h < 0$ associates the equilibrium expectation
$\langle f \rangle_{h}$.
Similar interpretations can be given for larger values of $n$ and
arbitrary $f$.\ccor{ CCC}
Another way to express part of the content
of the Theorem 1 is to observe that it claims
that for any $\lambda$, $0<\lambda < \lambda_c$ and any
probability distribution $\nu \leq \mu_{}$
the branch of states
$\langle \cdot \rangle _{h;\exp(\lambda/h)}^{\nu}$ for $h>0$
is a $ \Cal C^\infty$ continuation of the
family $\langle \cdot \rangle_{h}$ for $h<0$.
That interpretation suggests that the phenomenon of
metastability should be understood dynamically, in
which case the physically meaningful smooth continuations
through the critical point $h=0$ become possible.
In the Physics literature (see, e.g., [BM]), one sometimes
relates the metastable relaxation of a system to the presence
of a ``plateau'' in the graph corresponding to the time
evolution of a quantity of the type of
$\E\(f\(\sigma_{h;t}^{\nu}\)\)$.
Of course, strictly speaking there is no ``plateau'', and
generically the slope of such a function is never 0. Still,
from the experimental point of view a rough ``plateau''
can be seen and described as follows.
In a relatively short time
$\E\(f\(\sigma_{h;t}^{\nu}\)\)$
seems to converge to a value close to
$\langle f \rangle_{}$;
after this, one sees an
apparent flatness in the relaxation curve over a period of
time which may be quite long compared with the time needed
to first approach this value. But eventually the relaxation
curve starts to deviate from this almost constant value and move
towards the true asymptotic limit, close to
$\langle f \rangle_{+}$.
The experimentally almost flat
portion of the relaxation curve is referred to as a ``plateau''.
Theorem 1 can be seen
to some extent as giving some precise meaning to such a
``plateau'', and we discuss now two ways in which this can be
done. First note that if $0 < \lambda' < \lambda'' < \lambda_c$,
then from Part (i) of the Theorem we have
$$
\E\(f\(\sigma_{h;\exp(\lambda'/h)}^{\nu}\)\)

\E\(f\(\sigma_{h;\exp(\lambda''/h)}^{\nu}\)\)
\rightarrow 0,
$$
faster than any power of $h$.
Observe that we are considering times which are of different
order of magnitudes, when $h$ is small, and still we are
observing a nearly constant
$\E\(f\(\sigma_{h;t}^{\nu}\)\)$.
For a second way in which Theorem 1 can be seen as expressing
the presence of a ``plateau'', we can think of plotting
$\E\(f\(\sigma_{h;t}^{\nu}\)\)$ versus $\log(t)$, rather than
versus $t$. This is somewhat the natural graph to consider,
if one is interested in the order of magnitude of the relaxation
time. If the $\log(t)$axis is drawn in the proper scale,
amounting to replacing it with $h \log(t)$, then, when $h$ is
small, Theorem 1 tells us that the graph should be close to
that of a step function which jumps at the point $\lambda_c$,
from the value
$\langle f \rangle_{}$
to the value
$\langle f \rangle_{+}$.
Readers who are familiar with [\sddr] and [Sch2] can expect that also
Theorem 4 and Corollary 1 in [\sddr], which refer to finite systems
with ($$) boundary conditions and sizes which are scaled as $\hg$,
have stronger versions along the lines of the current paper. This is
indeed the case, but for brevity we will omit the statements of these
theorems, which can easily be obtained and can be proved with the
techniques introduced in this paper.
\bigskip
%\input he
\subheading{\he \ Heuristics}
One of the appealing features of the results proven in this paper
is that some of them can be correctly predicted based
on a very simple and naivelooking heuristics. It is probably a
challenge to a historian of science to trace back the origin and
evolution of this nonrigorous approach to the problem, to give
the proper credit to the people involved
and to elucidate how the interplay among empirical
observation of metastable systems, theoretical analysis and computer
simulation led to the reasoning described in a simple fashion below.
Here we will make no attempt to clarify the history of the subject.
The interested reader will find a great deal of references to the
earlier literature on the subject in the paper [RTMS].
It is worth stressing that certain parts of the heuristics
were rediscovered more than once in different forms and contexts,
so that giving proper credit is a very difficult task.
The reader may want to compare the heuristics present below with the
one presented in [Sch1] and [Sch2]. The main difference is that here
we are being more ambitious by basing the heuristics on the computation
of the freeenergy of a droplet with an arbitrary shape, rather than
on the energy of a square droplet. At the time that those two former
papers were written, it seemed to its author that there was no
compelling reason to believe that the equilibrium freeenergy of droplets
would predict correctly the value of $\lambda_c$, i.e, the behavior
of the relaxation time to the level of the correct rate of exponential growth
with $1/h$.
The first ingredient of the heuristics is the idea of looking at an
individual droplet of the stable phase (roughly the (+)phase,
since $h$ is small) in a background given by the metastable phase
(roughly the ($$)phase).\corr{ In principle let the shape of the droplet
be arbitrary and indicate it by S in the notation below.} Let S be
the shape of that droplet, which a priori can be arbitrary.
Say that $l^2$ is the volume (i.e., the number of sites) of the
droplet, and let us find an expression for the freeenergy of such
a droplet.
This freeenergy may be seen as coming from two main
contributions. There should be a bulk term, proportional to $l^2$.
This term should be
obtained by multiplying $l^2$ by the difference in freeenergy per site
between the (+)phase and the ($$)phase in the presence of a small
magnetic field $h>0$. This difference in the freeenergy per site
of the two
phases should come only from the term in the Hamiltonian which couples
the spins to the external field and should therefore be given by $2m^* h/2
= m^* h$. The other relevant contribution to the freeenergy of the
droplet should come from its surface, where there is an interface between
the (+)phase and the ($$)phase. This contribution\corr{ should be}
is proportional
to the length of the interface, which is of the order
of $l$.\corr{ This} It
should be multiplied by a constant
$w_{\text{S}}$
which depends on the shape of the droplet. This constant
$w_{\text{S}}$\corr{ should represent} represents the excess freeenergy
per unit of
length integrated over the surface of the droplet when its scale is
changed so that its volume becomes 1. Therefore, since the external
field $h$ is small, we can take for $w_{\text{S}}$ the value
$\Cal{W}(\gamma)$, where $\gamma$ is the boundary of the droplet,
rescaled in this fashion. In particular, $w_{\text{S}}$ is minimized
when the droplet has the Wulff shape, and in this case
$w_{\text{S}} = w$. Adding the pieces, we obtain for the freeenergy
of the droplet the expression
$$
\Phi_{\text{S}} (l) = m^* h l^2 + w_{\text{S}} l.
$$
The two terms in this expression
become of the same order of magnitude, in case $l$ is of the order
of $1/h$. Therefore, for later convenience we write $l = b/h$, with
a new variable $b \geq 0$. This yields
$$
\Phi_{\text{S}} (b/h) = \frac{\phi_{\text{S}} (b)}{h},
$$
where
$$
\phi_{\text{S}} (b) = m^* b^2 + w_{\text{S}} b.
$$
This very simple function takes the value 0 at $b=0$, grows with
$b$ on the interval $[0,B_c^{\text{S}},]$, where
$B_c^{\text{S}} = B_c^{\text{S}}(T) = \frac{w_{\text{S}}}{2m^*}$,
reaching its absolute maximum
$\phi_{\text{S}}(B_c^{\text{S}}) = \frac{(w_{\text{S}})^2}{4 m^*}
= A^{\text{S}}(T) = A^{\text{S}}$
at the end of this interval and decreases with $b$ on the
semiinfinite interval $[B_c^{\text{S}}, \infty)$. It crosses the value
0 at the point $B_0^{\text{S}} (T) = B_0^{\text{S}}
= \frac{w_{\text{S}}}{m^*} = 2B_c$.
Metastability is then ``understood'' from the fact that systems in contact
with a heat bath move towards lowering their freeenergy, so that the
presence of a freeenergy barrier which needs to be overcome in order to
create a large droplet of the stable phase with
any shape keeps the system close to the metastable phase.
Subcritical droplets are constantly being created by thermal fluctuations,
in the metastable phase, but they tend to shrink, as dictated by the
freeenergy landscape. On the other hand, once a supercritical droplets
is created due to a larger fluctuation, it will grow and drive the system
to the stable phase, possibly colliding and coalescing in its growth
with other supercritical droplets created elsewhere.
As a function of $h$, the linear size of a critical droplet,
$B_c^{\text{S}} / h $, blows up as $h \searrow 0$. One can then, in
a somewhat circular, but heuristicallymeaningful way, say that the
macroscopic freeenergy of droplets is indeed a relevant object of
consideration. One can also hope then that sharp theorems could be
conjectured and possibly proven regarding the asymptotic behavior of
quantities of interest in the limit $h \searrow 0$.
Regarding
the shape of the droplet, the height of this barrier is minimized by
minimizing the value of the constant $w_{\text{S}}$, i.e., by considering
Wulffshaped droplets. This singles out the Wulff shape as the most relevant
one in the heuristics above. We will simplify the notation
by\corr{ omiting} omitting
the\corr{ mention to the shape} subscript S when\corr{ it is} talking
about the Wulff shape.\corr{!! } In particular,
$$ B_c= \frac{w}{2m^*},\quad
A = \frac{w^2}{4 m^*} . \Eq(Bc) $$
Based on the expression above for the freeenergy barrier, one predicts
the rate of creation of supercritical droplets with center at a given
place\corr{ as} to be $\exp \(\frac{\beta A}{h} \)$.
In what follows now we write $d$ instead of 2, to make the role of the
dimension clear in the geometric argument which comes next.
We are concerned with an infinite
system, and we are observing it through a local function $f$, which
depends, say, on the spins in a finite set
$\Supp(f)$.
For us the system
will have relaxed to equilibrium when $\Supp(f)$ is covered by a big
droplet of the plusphase, which appeared spontaneously somewhere
and then grew, as discussed above. We want to estimate how long we
have to wait for the probability of such an event to be large. If
we suppose that the radius of supercritical droplets grows
with a speed $v$, then we can see that the region in spacetime
where a droplet which covers $\Supp(f)$ at time $t$ could have appeared
is, roughly speaking, a cone with vertex in
$\Supp(f)$ and which has as
base the set of points which have timecoordinate 0 and are at
most at distance $tv$ from $\Supp(f)$.
The volume of such a cone is of the order of
$ (vt)^d t$.
%Now, from the discussion in the first part of the heuristics, one
%can infer that ``the rate with which supercritical droplets
%appear by thermal fluctuations'' at a given location
%should be of the order of $\exp(\beta E_{\max})$.
The order of magnitude of the relaxation time,
$t_{\text{rel}}$, before which the
region $\Supp(f)$ is unlikely to have been covered by a large
droplet and after which
the region $\Supp(f)$ is likely to have been covered by such
an object can now be obtained by solving the equation
$$
(vt_{\text{rel}})^d \, t_{\text{rel}} \,
\exp\(\frac{\beta A}{h}\) = 1.
\Eq(trelv)
$$
This gives us
$$
t_{\text{rel}} = v^{d/(d+1)} \
\exp\(\frac{\beta A}{(d+1)\ h}\).
\Eq(trelv)
$$
In order to use this relation to predict the way in which the
relaxation time scales with $h$, one needs to figure out
the way in which $v$ scales with $h$. If we suppose, for
instance, that $v$ does not scale with $h$, or at least that if
it goes to 0, as $\hg$, it does it so slowly that
$$
\lim_{\hg}\ h^{d1} \log v = 0,
\Eq(v)
$$
then we can predict that
$$
t_{\text{rel}} =
\exp\(\frac{\beta A}{(d+1) \ h }\) =
\exp\(\frac{\lambda_c}{h}\),
%\label{heu}
$$
where
$$
\lambda_c = \frac{\beta A}{d+1} =
\frac{\beta w^2}{(d+1)\ 4 m^*} =
\frac{\beta w^2}{12 \ m^*},
$$
in agreement with our result \equ(lambdac).
In Section 1iii of [Sch1] and more explicitly in
Section 4 of [Sch2] an argument was
given in support of the conjecture that $v \sim C h$ as
$\hg$, a much stronger conjecture than \equ(v). In the
paper [RTMS] (see display (9) there)
a different nonrigorous argument is described, in which
the same conclusion is derived from
an ``AllenCahn approximation''.
In part 3 of this paper we will introduce a rescaling procedure
and obtain results which can be seen as rigorous counterparts
to \equ(v). It is interesting to compare this feature of the
regime of fixed $T$ and $\hg$ with the case of fixed $h>0$ and
$T \searrow 0$, studied in [DS]. In that case the analogue of
\equ(v) is false, and consequently
the\corr{ term in $v$} $v$term in \equ(trelv) is of greater relevance than
it is here.
%@@ Papers with dispute on result for T close to T_c
%\bye
\bigskip
%\input CC
%@@ COMMENTS ON LEMMA \CCC.6 (?).
\numsec=\CC
\numfor=1
%\leftheadtext{CC.metastable regime}
%\rightheadtext{CC.metastable regime}
\heading{\CC. Metastable regime}\endheading
\subheading{\CCA. Preliminaries}
In this part of the paper we will prove part (i) of Theorem 1.
%For this purpose we suppose that $0 < \lambda < \lambda_c$.
The first step will be to prove, in Sections \CCB \ and \CCC,
the following proposition.
\proclaim{Proposition \CCA.1}
Suppose that $T < T_c$ and $0 < \lambda < \lambda_c$. Then for
each constant $a \in (0,1/4)$, there is a positive finite constant
$C$ such that for each local observable $f$ there is a positive
finite constant $C(f)$ such that for $h > 0$
$$
\left \E\(f\(\sigma^_{h;\exp(\lambda/h)}\)\)
 \langle f \rangle_{\Lambda(1/h^a),,h} \right
\leq
%C_1 f_{\infty} \exp(C_2/h^a),
C(f) \exp(C/h^a).
$$
%for small enough $h>0$.
\endproclaim
We will not try to optimize the constants $C$ and $C(f)$
in this proposition.
But we observe that
from our proof, if the inequality displayed above is only required to hold
for $h \leq h_0$ for some $h_0 > 0$ depending on $f$, then
we can take $ C(f) = C' f_{\infty} \Sf$, where
$C'$ does not depend on $f$.
%and $\Sf$ is the smallest lattice rectangle which contains $\Supp(f)$.
Proposition \CCA.1 transforms our dynamical problem into an equilibrium
one, in case the initial distribution is concentrated on the
configuration with all spins down. In Section \CCD \
we will study the
behavior of $\langle f \rangle_{\Lambda(1/h^a),,h}$ for small $h>0$
and show that it gives rise to the asymptotic expansion claimed in
part (i) of Theorem 1.\corr{ It is worth noting, however that it is not
hard to see} Let us note here that if our goal were only to prove
Corollary 1(i),
with the initial distribution having all spins down, then
Proposition \CCA.1 would have reduced our task to a very simple one.
First, from the heuristic viewpoint,
with $a>0$ small,
the box $\Lambda(1/h^a)$ is too small for any supercritical droplet to
fit inside, so that one should expect to see the ($$)phase inside it.
A rigorous argument to the effect that for $00$, acting on each spin, to be able
to win over the effect of the negative spins at the boundary.
The extension of
Theorem 1(i) to arbitrary initial
distributions $\nu \leq \mu_{}$ will be obtained in Section \CCE.
Interestingly enough a basic tool there will be a result obtained
in [Mar], and extended to arbitrary subcritical temperatures in
[CGMS], concerning the 2 dimensional Ising model evolving in the
absence of an external field.
\bigskip
%% END OF PART 1
%% BEGINNING OF PART 2
\subheading{\CCB. Bottlenecks for the dynamics}
We start now the proof of Proposition \CCA.1.
Several times in the proof of this proposition we will use arguments which
are only true for small enough $h > 0$, but the constant $C(f)$ can be
adjusted so that we do not have to require $h$ to be small in the statement
of the proposition.
In order to prove Proposition \CCA.1 there is no loss in generality in
supposing that $f$ is increasing and that it has
$f_{\infty} \leq 1$. For the remainder of the proof of this
proposition we will make these assumptions.
To simplify the notation we set\corr{ $t_h = \exp(\lambda/h)$.}
$$t_h = \exp(\lambda/h). \Eq(thh) $$
We turn first to the proof of the easy half of Proposition \CCA.1.
We will show that for small $h>0$
$$
\E(f(\sigma^_{h;t_h}))  \langle f \rangle_{\Lambda(1/h^a),,h}
\geq
 \exp\Big(\exp\Big(\frac{\lambda}{2h}\Big)\Big).
\Eq(Prop.CC.1.easy)
$$
For this, observe first
that from the basiccoupling inequalities we have
$$
\E(f(\sigma^_{h;t_h}))
\geq
\E(f(\sigma^_{\Lambda(1/h^a),,h;t_h})).
\Eq(Prop.CC.1.easy.1)
$$
Let the process
$(\sigma^_{\Lambda(1/h^a),,h;t})$
and the stationary process
$(\sigma^{ \mu_{\Lambda(1/h^a),,h} }_{\Lambda(1/h^a),,h;t})$
evolve on the probability space defined by the graphical
construction, so that in particular once these processes
hit each other they remain together forever. Note that,
for some positive $\epsilon$,
\corr{ during each unit time interval these two processes have a
probability at least $\epsilon ^{1/h^{2a}}$ of hitting each other,}
the probability that these two processes will hit each other during
any unit time interval is
at least $\epsilon ^{1/h^{2a}}$,
regardless of their states at the begining of this time interval.
\corr{! } Also, $f(\sigma^_{\Lambda(1/h^a),,h;t})\le
f(\sigma^{ \mu_{\Lambda(1/h^a),,h} }_{\Lambda(1/h^a),,h;t})$
with probability one.
From these remarks it is clear that\corr{ next formula is now a
trifle stronger}
$$
0\le\langle f \rangle_{\Lambda(1/h^a),,h}
 \E(f(\sigma^_{\Lambda(1/h^a),,h;t_h}))
\leq
\big( 1  \epsilon^{1/h^{2a}} \big)^{t_h1}
\leq
\exp\Big(\exp\Big(\frac{\lambda}{2h}\Big)\Big),
\Eq(Prop.CC.1.easy.2)
$$
for small $h>0$. The inequality \equ(Prop.CC.1.easy) follows from
\equ(Prop.CC.1.easy.1) and \equ(Prop.CC.1.easy.2).
The main task in this and the next
sections is to prove the other half of Proposition
\CCA.1, i.e., the inequality
$$
\E(f(\sigma^_{h;t_h}))  \langle f \rangle_{\Lambda(1/h^a),,h}
\leq
C(f) \exp(C/h^a).
\Eq(Prop.CC.1.hard)
$$
We approach this problem borrowing some ideas from
[Sch1].
As in that paper, set\corr{ !}
$$\Lambda_h = \Lambda(\exp(\lambda_{\text{c}}/h)) \Eq(Lh) $$
and observe that
Lemma 1.2.1 (which is the same as Lemma 1 in [Sch1])
gives us the following stronger version of Lemma 2
in\corr{ [Sch1].} [Sch1]:\corr{ the constant is changed!}
$$
 \E(f(\sigma^_{h;t_h})) 
\E(f(\sigma^_{\Lambda_h,,h;t_h})) 
\leq C_1 (f) \exp(C_2\exp(\lambda/h)),
\Eq(finite)
$$
where\corr{ $C_1$} $C_1(f)$ and $C_2$ are positive
and\corr{ finite constants.} finite.
Next we will introduce a restricted set of configurations, in
a way similar to [Sch1], and inspired there by [CCO] and
by the heuristic idea of critical droplets. To make this idea precise
one uses the standard notion of contours,
on the dual lattice $\Z^2 + (1/2,1/2)$,
which separate spins $1$ from $+1$.
In the definition of these contours, we adopt here the splitting rules
used in, e.g.,
[DKS] (see Section 3.1 there), which allow one to take the contours
as selfavoiding\corr{ closed
curves, when} curves, which are closed, when the boundary conditions
are, e.g., $()$, as is our case.
We will denote by $\Gamma$ the length of the\corr{ contour $\Gamma$}
contour $\Gamma$, by $\text{Int } \Gamma$ the set of sites it surrounds,
and by $\V(\Gamma)$ the number of spins that it surrounds,
which we call the volume of $\Gamma$.
\corr{ More notations}
As usual, a contour is called an external contour if it is not enclosed
by any other contour. If $ \Gamma $ is such a contour of the configuration
$\sigma $, and the boundary conditions are ($$), then at certain
sites $x$, attached to $\Gamma $,
the values $\sigma (x)$ are uniquely defined by the presence of
$\Gamma $. The set of such sites will be denoted by $\partial \Gamma $.
We have
$$\partial \Gamma= \partial_{} \Gamma \cup_{} \partial_{+}
\Gamma, \Eq(pm) $$
where $\sigma _{\partial_{\pm}} \Gamma =\pm 1.$
\corr{ end addition} We will use the notation
$$\Omega_{} = \bigcup_{l=1}^{\infty} \Omega_{\L(l),}.$$
Our restricted set of configurations is defined as\corr{ !}
$$
\Cal{R} = \Big\{\sigma \in \Omega_{} :
\text{each contour $\Gamma$ in $\sigma$ has $\V(\Gamma)
\leq \Big( \frac{B_{\text{c}}}{h} \Big)^2$} \Big\},
\Eq(defR)
$$
where $B_c$ is defined in \equ(Bc).
We want to argue that up to a time as large as $t_h$
the system evolving in the box
$ \Lambda_h $
with $()$ boundary conditions and starting with
all spins $1$, will be unlikely to
\corr{ have escaped} escape from $\Cal{R}$,\corr{ !} in which
case the system indeed would look very much like the ($$) phase.
In order to do this we introduce a
modified dynamics evolving in
$\Omega_{\Lambda_h,}$,
in which large droplets cannot, by definition, be formed
and then we couple the unrestricted dynamics to this modified one,
in a natural way.
The modified dynamics is simply defined as the Markov
process on
$\Omega_{\Lambda_h,}$
which evolves as the
original stochastic Ising model in $\Lambda_h$, with $()$ boundary
conditions, but for which all jumps
out of $\Cal{R}$ are suppressed. In other words, the rates,
$\tilde{c}_{\Lambda_h,,h}(x,\sigma)$,
of the new process are identical to
$c_{\Lambda_h,,h}(x,\sigma)$
in case
$\sigma,\sigma^x \in \Omega_{\L_h,}\cap\Cal{R}$
and are 0 otherwise. We will denote this
modified process, which is restricted to the state space
$\Cal{R}$,
by\corr{ $(\tilde\sigma^{\eta}_{\Lambda,,h;t})
_{t \geq 0}$, } $(\tilde\sigma^{\eta}_{\Lambda_h,,h;t})_{t \geq 0}$,where
$\eta \in \Cal{R}$ is the initial
configuration. It is well known, and very easy to
prove, that such a modified process is also reversible and that
since it is, in our case, irreducible, its unique invariant
probability measure is
$\tilde\mu_{\Lambda_h,,h}$
given by
$$
\tilde\mu_{\Lambda_h,,h}(\ \cdot \ )
= \mu_{\Lambda_h,,h} (\ \cdot \  \Cal{R}).
$$
This distribution is sometimes called a ``restricted ensemble'',
and, informally speaking, represents the ``metastable state''.
%Now we couple
%the process $(\sigma_{\Lambda_h,,h;t}^)$
%to the stationary process
%$(\tilde\sigma^{\tilde\mu_{\Lambda_h,,h}}_{\Lambda_h,,h;t})$
%in the following way.
%First we couple
%$(\sigma_{\Lambda_h,,h;t}^)$
%to
%$(\sigma^{\tilde\mu^c_{\Lambda_h,,h}}_{\Lambda_h,,h;t})$
%using the basic coupling, so that for all $t \geq 0$
%we have
%$$
%\sigma_{\Lambda_h,,h;t}^
%\leq
%\sigma^{\tilde\mu_{\Lambda_h,,h}}_{\Lambda_h,,h;t}.
%$$
%Now we enlarge the space on which this construction was made to accommodate
%$(\tilde\sigma^{\tilde\mu_{\Lambda_h,,h}}_{\Lambda_h,,h;t})$
%which evolves together with
%$(\sigma^{\tilde\mu_{\Lambda_h,,h}}_{\Lambda_h,,h;t})$
%up to the moment when this one escapes from
%$\Cal{R}$; at this moment
%$(\tilde\sigma^{\tilde\mu_{\Lambda_h,,h}}_{\Lambda_h,,h;t})$
%does not move (we say that it has a suppressed jump),
%and afterwards it evolves independently of the other two processes,
%$(\sigma_{\Lambda_h,,h;t}^)$
%and $(\sigma^{\tilde\mu_{\Lambda_h,,h}}_{\Lambda_h,,h;t})$.
For each initial\corr{ configuration $\eta \in
\Omega$,} configuration $\eta \in \Cal R$, the process
$(\tilde\sigma^{\eta}_{\Lambda_h,,h;t})$
can be constructed on the same probability space
corresponding to the graphical construction introduced
in Section 1.2. For this purpose it is enough to suppress all
jumps out of $\Cal{R}$. In other words, Poisson marks which
should cause such a jump are just ignored.
The important fact about this construction is that if we introduce
$$
\tau = \inf \{t \geq 0 :
\text{the process
$(\tilde\sigma^ {\tilde\mu_{\Lambda_h,,h}}_{\Lambda_h,,h;t})$
has a suppressed jump at time $t$} \},
$$
then
$$
\sigma_{\Lambda_h,,h;t}^
\leq
\tilde\sigma^{\tilde\mu_{\Lambda_h,,h}}_{\Lambda_h,,h;t}
\ \text{ for all } \ t < \tau.
\Eq(tilde.control)
$$
\corr{ noindent, also brackets}
(Readers who are familiar with the argumentation in [Sch1], have
noted that while most of what we introduced above in connection
to the restricted ensemble is similar to its counterparts in that
paper, the notions are not strictly parallel to those there. The
reason is that, while we are still pursuing the idea that the
boundary of $\Cal{R}$ is a bottleneck, the arguments used in
[Sch1] to prove Lemma 5(i) there would give us an estimate that,
while good enough to obtain Corollary 1 (i),
would not be good enough
to obtain an estimate as sharp as Proposition \CCA.1, which is needed
for the proof of Theorem 1(i).)\corr{ indent}
We introduce now a family of sets
whose union is the inner boundary of $\Cal{R}$. For each site
$x \in \Z^2$ define
$$
\Fx = \Big\{\sigma \in \Cal{R} :
\sigma^x \not\in \Cal{R} \Big\}.
$$
Set also
$$
\pphi = \sup_{x \in \Lambda_h} \tilde\mu_{\Lambda_h,,h}(\Fx).
$$
\corr{ indent} We will now consider a discrete time Markov chain
embedded into the
stationary process
$(\tilde\sigma^{\tilde\mu_{\Lambda_h,,h}}_{\Lambda_h,,h;t})$.\corr{ !}
It is formed by the configurations of our process between the
successive jumps.
For this purpose order all the Poisson marks in the graphical
construction which occur on\corr{ $\Lambda_h$.} $\Lambda_h$, according
to the time they occur. Let $N(t)$ be the number
of such marks from time 0 up to time $t$. Let $M_{x,k}$ be the event
that the $k$th such mark occurs at the site $x$, and $F_{x,k}$
be the event that immediately before this $k$th mark the process
$(\tilde\sigma^{\tilde\mu_{\Lambda_h,,h}}_{\Lambda_h,,h;t})$ is
in $\Fx$. Note that $M_{x,k}$ and $F_{x,k}$ are independent
events and that $\P(M_{x,k}) = 1/\Lambda_h$, while by stationarity
$\P(F_{x,k})=\tilde\mu_{\Lambda_h,,h}(\Fx) \leq \pphi$
for all $x$ and $k$.
Set also $K = \lfloor 2 \, \Lambda_h \, c_{\text{max}}(T)
\, t_h \rfloor$
to obtain the estimate
$$\align
\P(\tau \leq t_h)
& \leq
\P(N(t_h) > K) + \sum_{x \in \Lambda_h, k=1,...,K}
\P(M_{x,k} \cap F_{x,k}) \\
& \leq
C_1 \exp(C_2 \exp(\lambda/h)) + C_3 \Lambda_h t_h \pphi,
\teq(tau)
\endalign$$
where in the second inequality a standard large deviation\corr{ estimates
for Poisson random variables were} estimate
for Poisson random variables was used.
From \equ(finite), \equ(tilde.control) and \equ(tau) it follows
that\corr{ new coefficient}
$$
\E(f(\sigma^_{h;t_h})) \leq
\int f d\tilde\mu_{\Lambda_h,,h} + C_3 \Lambda_h t_h \pphi +
C_4 (f) \exp(C_5\exp(\lambda/h)).
\Eq(equilibrium)
$$
All the quantities in the righthandside of \equ(equilibrium) pertain
to equilibrium statistical mechanics, so that we have reduced our
dynamical problem of proving \equ(Prop.CC.1.hard) to the equilibrium
problems of proving the following two claims concerning
the\corr{ measure $\tilde\mu_{\Lambda_h,,h}$.
For} measure $\tilde\mu_{\Lambda_h,,h}$:
\noindent i) for all $\epsilon > 0$ and $h>0$\corr{ some additions}
small enough
$$
\pphi \leq C_{1} \exp \left( \beta (1\epsilon)\frac{A}{h} \right),
\Eq(claim1)
$$
where $A$ is defined by \equ(Bc);
\noindent ii) for $h>0$
$$
\int f d\tilde\mu_{\Lambda_h,,h}
\leq
\langle f \rangle_{\Lambda(1/h^a),,h}
+ C(f) \exp(C_{2}/h^a).
\Eq(claim2)
$$
\corr{ end changes}
It is interesting to note that the term
$C_3 \Lambda_h t_h \pphi$
in \equ(equilibrium) has a direct connection with the heuristics in
Section 1.4. The quantity $\pphi$ plays the role of the rate of
nucleation, while $\Lambda_h t_h$ is the spacetime volume of
a cylinder which plays a role similar to the spacetime cone in
the heuristics. The absence here of the velocity factor which appears
in \equ(trelv) is a consequence of our using an upper bound of
order 1 for the velocity of propagation of effects, through \equ(finite).
Once \equ(claim1) is proven, it\corr{ follows} follows from the
definitions \equ(lambdac), \equ(Bc), \equ(thh) and \equ(Lh)
that $\Lambda_h t_h \pphi$
vanishes exponentially fast in $1/h$ as $\hg$.
\bigskip
\subheading{\CCC. The restricted ensemble}
We start our study of
the measure $\tilde\mu_{\Lambda_h,,h}$ by observing that
it is sufficient to study measures of the type
$\tilde\mu_{\Lambda,,h}$ on %simplyconnected
subsets $\Lambda$ of $\Z^2$
which are much smaller than $\Lambda_h$.
The definition of $\tilde\mu_{\Lambda,,h}$ is analogous to that of
$\tilde\mu_{\Lambda_h,,h}$, with $\L$ replacing $\L_h$.
Suppose that for each sufficiently small
value of $h>0$ we have an event $E_h$ which
only depends on the values of the spins inside the box
$x + \Lambda(1/h^3)$. Consider the larger box
$x + \Lambda(2/h^3)$ and condition on what the
set\corr{ of contours} of exterior contours
which surround at least one site in its
\corr{ complement,$(x + \Lambda(2/h^3))^c$,
is.} complement, $(x + \Lambda(2/h^3))^c$,
is.\corr{ I added more explanations.} Let that set consist of
contours $ \{ \Gamma _ j \} $. Denote by $ \Lambda (\{ \Gamma _ j \})$
the connected component of the complement, $(x + \Lambda(2/h^3))
\setminus \bigcup_{ j }
(\text{Int } \Gamma _ j \cup_{} \partial ( \Gamma _j ))$, which
contains the set $x + \Lambda(1/h^3)$.
Then one obtains, when $h$ is small, that
$$
\tilde\mu_{\Lambda_h,,h}(E_h)
= \sum_{i} \alpha_i \cdot
\tilde\mu_{\Lambda^{i},,h}(E_h),
\Eq(mixture)
$$
where the $\Lambda ^i$s denote different $ \Lambda (\{ \Gamma _ j \})$s,
the index $i$ runs over a finite set,
$\alpha_i > 0$ and $\sum_i \alpha_i = 1$.\corr{ endaddition} The choices
of the scales above and the need that $h$ be small for
$\equ(mixture)$ to hold, are clearly related to the fact that
we are conditioning the Gibbs measure on the absence of any contour
with volume larger than $(B_{\text{c}}/h )^2$. Therefore, with
the choices above, we are sure that for small $h$ the support
of $E_h$ will be disjoint from the set of sites surrounded by
any contour which also surrounds any site in
$(x + \Lambda(2/h^3))^c$.
The sets $\L^{i}$ which appear in \equ(mixture) have an additional
property, which will be important\corr{later on.} later: they are
simplyconnected.
\corr{ For lack of a better name,
we will say that a finite set $\Lambda \subset \Z^2$ is simple in case
for each configuration in $\Omega_{\L,}$ no contour surrounds any site
outside of $\L$. Simplyconnected sets are simple, but they do not
exhaust this class of sets.
%due to the fact that we are using splitting
%rules in the definition of our contours.
Because contours cannot cross
each other, it is easy to see that all our sets $\L^{i}$ are simple.}
%Because they are obtained
%from the set $\L_h \cap x + \L(2/h^3)$ via the removal of sites which
%have their value
For each value of $h>0$ and each
$x \in \Lambda_h$ the event $\Fx$ satisfies the condition
above on $E_h$, so that to derive \equ(claim1) it is enough
to obtain a corresponding upper bound:
$$
\sup_{x \in \Lambda_h}
\sup_{\Lambda \in \Cal{L}_{x,h}}
\tilde\mu_{\Lambda,,h}(\Fx) \leq
C_{1} \exp \left( \beta (1\epsilon)\frac{A}{h} \right),
\Eq(claim1')
$$
for small $h > 0$. Here $\Cal{L}_{x,h}$ is the family of simplyconnected
sets $\L$ which satisfy
$(x + \Lambda(1/h^3)) \cap \L_h
\subset \Lambda \subset x + \Lambda(2/h^3)$.
Similarly, the derivation of \equ(claim2) is reduced to that of the
following,
$$
\sup_{\Lambda \in \Cal{L}_{0,h}}
\int f d\tilde\mu_{\Lambda,,h}
\leq
\langle f \rangle_{\Lambda(1/h^a),,h}
+ C(f) \exp(C_{2}/h^a),
\Eq(claim2')
$$
for $h > 0$.
In order to derive \equ(claim1') and \equ(claim2') we will use the
notion of the skeleton of a contour, as introduced in [DKS].
In what follows $b$ is a fixed but
arbitrary number in $(a,1/4)$ and $\anu$ is a fixed but also arbitrary
number in $(0,b/2) \cap (0,a)$.
A contour $\Gamma$ will be said to be
$h$vertebrate if $\V(\Gamma) > (1/h)^{2b}$, otherwise $\Gamma$ will be said to
be $h$invertebrate.
(Usually one says that $\Gamma$ is large in the former
case and small in the latter one, but this terminology would be confusing
in the present paper, since ``large'' and ``small'' contours may also
be used in connection with being supercritical or subcritical, with the
threshold volume being the quantity $(B_{\text{c}}/h )^2$, which is much
larger than $(1/h)^{2b}$.)
Often we will omit the mention to $h$
when referring to a vertebrate contour.
Given now a vertebrate contour $\Gamma$
one can associate to it, in an algorithmic way,
a sequence of sites, $(x_1,...,x_J)$ of the dual lattice $\Z^2 + (1/2,1/2)$.
We think of the sites $x_1,...,x_J$ as the ordered vertices
of a closed polygonal
curve, with possible selfintersections (see fig. 5.3 on p.166 in [DKS]);
we will denote this curve by $\gamma$ in what follows and call it the
skeleton of $\Gamma$.
For the construction of $\gamma$, given $\Gamma$, the reader is referred
to Chapter 5 of [DKS]; here we will limit ourselves to reviewing some of
the basic properties that we can guarantee the skeleton to have:
\roster
\item"(S.1)" $x_i \in \Gamma$ for each $i$\corr{!! II}, moreover,
the points $x_i$ are consecutive on $\Gamma$ (for one of the
orientations of it).
\item"(S.2)" The
length of each edge of $\gamma$ is bounded between $C_{1} (1/h)^{\anu}$
and $C_{2} (1/h)^\anu$, where $0 < C_1 < C_2 < \infty$ are fixed
appropriate constants.
\item"(S.3)" The Hausdorff
distance between $\Gamma$ and $\gamma$ satisfies
$$
\rho_H(\Gamma, \gamma) \leq (1/h)^\anu.
\Eq(hausdorff)
$$
\endroster
The length, $\gamma$ of a skeleton $\gamma$ is defined as the sum of the
Euclidean lengths of its edges.
To each skeleton $\gamma$ we associate its Wulff functional,
$\Cal{W}(\gamma)$,
defined by summing over the edges of $\gamma$ the
product of the Euclidean length of
each edge by the surface tension in the direction defined by the edge,
i.e., $\tau_T(\bold n)$, with $\bold n$ perpendicular to the edge.
Observe that from the fact that the surface tension
$\tau_T(\bold n)$ is bounded away from 0 and $\infty$ uniformly in $\bold n$,
$$
C_{3} \Cal{W}(\gamma) \leq \gamma \leq C_{4} \Cal{W}(\gamma).
\Eq(bounds)
$$
From (S.2) and \equ(bounds) it follows that the number $J(\gamma)$ of
vertices in $\gamma$ satisfies
$$
C_{5} \Cal{W}(\gamma)h^{\anu} \leq J(\gamma)
\leq C_{6} \Cal{W}(\gamma)h^{\anu}.
\Eq(bounds')
$$
\corr{ As usual, a contour is called an external contour if it is not enclosed
by any other contour. Moved to p. 22.}
To each configuration $\sigma \in \Omega_{}$ we can associate
the collection $G = \{\Gamma_1,...\Gamma_n\}$ of its
external vertebrate contours.
To this collection we can associate the collection $S=\{\gamma_1,...,
\gamma_n\}$ of their skeletons. The Wulff functional associated to the
configuration $\sigma$ is then defined as
$$
\Cal{W}(S) = \sum_{i=1}^{n} \Cal{W}(\gamma_i),
$$
with the convention that $ \Cal{W}(\emptyset) = 0$.
Next we want to consider the volume surrounded by the external vertebrate
contours $\Gamma_1,...,\Gamma_n$
and say that it has to be close to the volume surrounded
by the collection of skeletons $\gamma_1,..., \gamma_n$. A difficulty lies in
the fact that while the volume surrounded by the contours is easily defined as
$$
\V(G) = \sum_{i=1}^{n} \V(\Gamma_i),
$$
the fact that the skeletons can selfintersect and also intersect with each
other makes the notion of the volume that they surround more delicate.
Fortunately the notion of ``phase volume'', as defined
in Section 2.10 of [DKS], solves this difficulty. This definition is as
follows (a look at fig.2.5 on p.37 of [DKS] will probably
lead the reader to guess correctly the definition).
The set $ \R^2 \backslash \cup\gamma_i$ splits up into a collection of
connected components $Q_{\alpha}$ with exactly one unbounded component
among them. A component $Q_{\alpha}$ will be called a minuscomponent
if any path that connects its interior points with points of the
unbounded component and intersects the curves from $S$ in a finite
number of points, intersects them in an odd number of points (counted
with multiplicities). The phase volume of $S$, denoted by $\pV(S)$, is
defined as the joint volume of all the minuscomponents.
Motivated by \equ(hausdorff), we want to show that $\V(G)$ and $\pV(S)$
have to be also relatively close to each other.
If we remove from $\R^2$ all the points
which are at a distance not larger than $(1/h)^\anu$ from $\cup \Gamma_i$,
then the remaining set also splits up into connected components
with exactly one unbounded component among them. It is easy to see that all
the bounded components are subsets of minuscomponents in the splitting
produced by $S$, while the unbounded component is a subset of the unbounded
component in the splitting produced by $S$. It is also clear that the
bounded components in this splitting are inside contours of $G$, while the
unbounded component in this splitting is completely outside the contours
of $G$. Hence
$$
\V(G)  \pV(S)  \leq C_{7} \left(\sum_{i=1}^n \Gamma_i\right)
(1/h)^{2 \anu}.
\Eq(volume)
$$
Similarly, by removing from $\R^2$ all the points
which are at a distance not larger than $(1/h)^\anu$ from $\cup \gamma_i$,
one can also derive
$$
\V(G)  \pV(S)  \leq C_{8} \left(\sum_{i=1}^n \gamma_i\right)
(1/h)^{2 \anu}
\leq
C_{9} \Cal{W}(S) (1/h)^{2 \anu},
\Eq(volume')
$$
where in the second inequality use of \equ(bounds) was made.
For convenience we introduce also another measure of the ``volume''
of a collection of skeletons, which is motivated by the procedure described
in the explanation of why \equ(volume) and \equ(volume') hold. We
define $\T(S)$ as the number of sites inside the minus
components $Q_ \alpha $,
which are at a distance larger than $(1/h)^\anu$ from $\cup \gamma_i$.
Clearly we have
$$
\T(S) \leq \min\{ \V(G), \pV(S) \},
\Eq(volume'')
$$
and
$$
\T(S) \geq \max \{ \V(G), \pV(S) \}  C_{10} \Cal{W}(S) (1/h)^{2\anu}
\Eq(volume''')
$$
\corr{ Given a finite set of $h$vertebrate
contours $G$} Given a finite set $G$ of $h$vertebrate
contours we denote by $\SG$ the set of configurations
which belong to $\Omega_{}$ and which have as their collection of
external $h$vertebrate
contours the set $G$. We say that $G$ is a compatible set of
external $h$vertebrate
contours in case $\SG$ is not empty.
%By $\Cal{G}$
%we denote the class of all $G$ which are compatible finite
%sets of external vertebrate contours
Similarly, given a finite set of skeletons $S$,
we define $\CSh$ as the class of all $G$
which are compatible sets of external $h$vertebrate
contours and which have as
their set of skeletons the set $S$. We define also
$$
\SS = \bigcup_{G \in \CSh} \SG.
$$
$\SS$ is the set of configurations
which belong to $\Omega_{}$ and which have as their collection of
skeletons corresponding to their external $h$vertebrate contours the set $S$.
We say that $S$ is a compatible set of skeletons
in\corr{ case $\SG$ is} case $\SS$ is not empty.
Again similarly, given an interval of real numbers, $I$,
we define $\SW$ as the set of configurations which
belong to $\Omega_{}$ and for which the collection $S$ of
skeletons corresponding to their external $h$vertebrate contours satisfies
$\W(S) \in I$.
One would like to say that the volume of a collection of skeletons is
the sum of the volumes of the individual skeletons. One proper version
of this is the following relation, which follows from the argumentation
used to show \equ(volume) and \equ(volume'). In this relation, and also
later on, we use the
simplified notation $\T(\gamma_i)$ in place of the more cumbersome
$\T(\{\gamma_i\})$.
$$
\text{If $S=\{\gamma_1,..., \gamma_n\}$ is a compatible set of
skeletons, then
$\T(S) = \sum_{i=1}^n \T(\gamma_i)$}.
\Eq(sumvolumes)
$$
A fundamental fact about the Wulff shape and the associated quantity
$w$ is the variational characterization \equ(var).
By scaling lengths we obtain easily from this and \equ(volume'')
that for any skeleton
$\gamma$
$$
w \sqrt {\T(\gamma)} \leq \W(\gamma).
\Eq(var')
$$
This inequality will now be used to derive another one, which is of
central relevance in this paper. This is the content of the next lemma.
\proclaim{Lemma \CCC.1}
%If the collection of skeletons $S$ is such that $\SS \cap \Cal{R}_{\L}
%\neq \emptyset$, then
For each configuration in $\Cal{R}$,
the associated set of skeletons $S$,
corresponding to its set of vertebrate external contours satisfies
$$
\W(S) \geq 2 m^* h \T(S).
$$
\endproclaim
\demo{Proof}
Say that the collection of external vertebrate
contours for the configuration with which we are concerned is
$G = \{\Gamma_1,...,\Gamma_n \}$
and that $\Gamma_i$ has skeleton $\gamma_i$, for $i=1,...,n$. From
\equ(volume'') and the definition \equ(defR) of $\Cal{R}$ we have
for each $i$
$$
\sqrt {\T(\gamma_i)} \leq \frac{B_{\text{c}}}{h}.
\Eq(var'')
$$
Multiplying the inequalities \equ(var') (with $\gamma =\gamma_i$)
and \equ(var'') by each other,
and using the fact that $B_{\text{c}}=w/(2 m^*)$ we obtain
$$
\W(\gamma_i) \geq 2 m^* h \T(\gamma_i).
$$
The thesis now follows by adding over $i$, using \equ(sumvolumes).
\cqd
\enddemo
In order to show \equ(claim1'), we need to know that the skeletons
of configurations in $\Fx$ are associated with sufficiently large
values of $\W(\cdot)$. This is the content of the next lemma.
\proclaim{Lemma \CCC.2}
Given $\epsilon >0$ there is $h_0 > 0$ such that for all $0 <
h \leq h_0$
and all $x \in \Z^2$ the following holds.
For each configuration in $\Fx$,
the associated set of skeletons $S$,
corresponding to its set of vertebrate external contours satisfies
$$
\W(S) \geq \frac{2 A (1\epsilon)}{h}.
$$
\endproclaim
\demo{Proof}
Let $\sigma \in \Fx$
be the configuration with which we are concerned.
By definition, $\sigma^x \in \Cal{R}^c$,
so the configuration $\sigma^x$
has an external contour $\Gamma$ with
$\V(\Gamma) > (B_{\text{c}}/h )^2$. On the other hand,
$\sigma \in \Cal{R}$, so the contour $\Gamma$ of the
configuration $\sigma^x$ is attached to the site $x$. Let
$sq(x)$ be the $3\times 3$ square of the dual lattice, centered
at $x$. Consider the set $ \widetilde G(x, \sigma )$ of all
dual bonds that are either in $sq(x)$ (there are 24 of them) or
belong to a contour of $ \sigma $, which contains some bonds
of $sq(x)$. Let $ G(x, \sigma) \subset \widetilde G(x,
\sigma)$ be these bonds, which are ``visible from infinity'', i.e.,
are not screened away from it by other bonds in $\widetilde
G(x, \sigma)$. Evidently, the set of bonds $ G(x, \sigma)$
serves as a set of bonds of exterior contours of some new
configuration $\eta=\eta(x, \sigma)$, which in fact has no
interior contours. Let these contours be $G_1, G_2,..., G_k$.
Clearly, $k\le 6$, since each contour $G_i$ passes through at
least two boundary sites of $sq(x)$. Note that $ \widetilde
G(x, \sigma ) =\widetilde G(x, \sigma^x )$, so the same is true
also for $ G(x, \sigma^x)$, $\eta(x, \sigma^x)$ and $G_1, G_2,...,
G_k$.
The idea of the proof is to observe that $ \text{Int }\Gamma
\subset \cup_{i} \text{Int } G_i$, so in particular $ \sum_i
\V(G_i) \ge \V(\Gamma)$. That implies that the contours
$G_i$ should be quite long. On the other hand, the set of dual bonds
$ G(x, \sigma) \setminus sq(x)$ belongs to exterior contours
of $ \sigma $, so they should be long as well.
The implementation of the above argument is as follows. Let $
{\widetilde S}$ be the set of skeletons of the subset of all
vertebrate contours among $G_1, G_2,..., G_k$. Denote the
corresponding subset of indices by $ \text{ver } \subset
\{1,...,k\}$. We have then that $ {\widetilde S}=\{\gamma_i,i\in
\text{ver
}\}$. Note that the volume of any invertebrate contour is $o(
{1\over h}) $, so we have:
$$\sum_{ i\in \text{ver }}\V(G_i)\ge
\Big(\frac{B_{\text{c}}}{h}\Big)^2 o\({1\over h}\).
$$
Hence the estimate \equ(volume''') implies that also
$$
\sum_{ i\in \text{ver }}\T( \gamma _i)
\geq \Big(\frac{B_{\text{c}}}{h}\Big)^2 o\({1\over h}\).
$$
Using now \equ(var') we have
$$\W( {\widetilde S}) \ge w\sum_{ i\in \text{ver }} \sqrt
{\T( \gamma _i)}
\ge w\sqrt {\Big(\frac{B_{\text{c}}}{h}\Big)^2 o\({1\over h}\)}=
{2A\over h}  o\({1\over h}\), \Eq(skel)
$$
where we are using an evident property of the square root and
also the definitions \equ(Bc).
The only remaining problem now is the relation between the
skeletons $S$ and $ {\widetilde S}$. One would like to argue
that in some sense $ {\widetilde S} \subset S$, which would
imply our claim. However, the last inclusion is almost always
violated. The way out of this unlucky circumstance is the
following. Note, that in fact the skeleton of a contour is not
uniquely defined; it should just be a closed polygon satisfying
the properties of the type (S.1)(S.3) above. Once this is the
case, any such polygon satisfies any statement above made
about any skeleton. We are going to use this nonuniqueness in
order to prepare a special family of the skeletons $ \widetilde
{\widetilde S}$ of the family $\{G_i, i\in \text{ver }\}$. This
family is going to be constructed in such a way as to use as big
pieces of the family $S$ as possible. Namely, note that every
maximal connected arc $k_j$ of $ G(x, \sigma) \setminus
sq(x)$ is also an arc of some external contour of the
configuration $ \sigma $, and so it inherits a portion $ \kappa
_j$ of the skeleton $S$. (Note that there are at most 6 such
arcs.) Namely, define $ \kappa _j$ to be the maximal
subpolygon of $S$ with both endpoints in $k_j$. (It might
happen that some $ \kappa _j$ are empty; for example, it will
be the case when the external contour in question would be
invertebrate.) It is immediate to see that the family of the
(open) polygons $ \cup_{} \kappa _j$ can be made into a
skeleton family of the contour family $\{G_i, i\in \text{ver
}\}$ by adding at most six extra edges to it, which addition
might require the prior removal of some ending edges from $
\cup_{} \kappa _j$ (also in the amount of at most six). This is the
skeleton family $ \widetilde {\widetilde S}$ sought. Now
\equ(skel), being valid for every possible skeleton family of the
contour family $\{G_i, i\in \text{ver }\}$, implies that
$$\W( \widetilde {\widetilde S}) \ge
{2A\over h}  o\({1\over h}\).
$$
On the other hand,
$$\W(S) \ge\W( \widetilde {\widetilde S}) o\({1\over h}\),
$$
since every edge of $\widetilde {\widetilde S} $ except a finite
number belongs to $S$. (Note that in fact the family $S$
might be much longer than $\widetilde {\widetilde S} $, though
it is immaterial for our argument).
\cqd
\enddemo
The next lemma shows that vertebrate contours have a minimum cost.
\proclaim{Lemma \CCC.3} There is a constant $h_0>0$ such that
for $0 0$; moreover $\sum_i \alpha_i = 1$.
The event that the box $\L^{i}$ is free of $h$vertebrate contours
is a decreasing
event, while $\CPxa$ is an increasing event, so by the FKGHolley
inequalities we obtain
$$
\mu^{\emptyset,h}_{\Lambda^{i},,h'}\left(\CPxa\right)
\leq
\mu_{\Lambda^{i},,h'}\left(\CPxa\right).
$$
Using now the fact that for each $i$, $\L^{i} \leq 9/h^{4b}$, that
$b<1/4$ and that $00$ } is $h_0= h_0(g)>0$
such that for all $0 < h' \leq h \leq h_0$ and
all finite $\L \subset \Z^2$
$$
\langle g\rangle^{\emptyset,h}_{\L,,h'}
 \langle g\rangle_{\L \cap (\Sg + \L(1/h^a)),,h'}
\leq C_1 g_{\infty} \Sg \exp ( C_2 / h^a),
$$
\endproclaim
\demo{Proof}
Without loss in generality we suppose that $g$ is increasing and
has $g_{\infty} \leq 1$.
Set
$$
E_1 = \bigcup_{x \in \Supp (g)} \CPxa
$$
Once more we use an argument similar to the one used to prove
\equ(mixture), this time
%with the box
%$\Supp(g) + \L(1/h^{a})$ replacing the box $x + \Lambda(2/h^3)$.
%In other words,
we condition on what the (+,*)cluster of the set
%set of contours which surround at least one site in
$(\Supp (g) + \L(1/h^{a}))^c$ is.
%Since at the boundary of any contour of a configuration
%in $\Omega_{}$ there is a (+,*)chain,
In this fashion we obtain
the following\corr{ equality.} equality:\corr{
$$
\langle g\rangle^{\emptyset,h}_{\L,,h'} =
\left( \sum_{i} \alpha_i \cdot
\langle g \rangle^{\emptyset,h}_{\Lambda^{i},,h'} \right)
\mu^{\emptyset,h}_{\Lambda,,h'}((E_1)^c) \
+ \
\langle g  (E_1)^c \rangle^{\emptyset,h}_{\Lambda^{i},,h'} \
\mu^{\emptyset,h}_{\Lambda,,h'}(E_1),
$$ 4 changes in that formula}
$$
\langle g\rangle^{\emptyset,h}_{\L,,h'} =
\left( \sum_{i} \alpha_i \cdot
\langle g \rangle^{\emptyset,h}_{\Lambda^{i},,h'} \right)
\mu^{\emptyset,h}_{\Lambda,,h'}((E_1)^c) \
+ \
\langle g  E_1 \rangle^{\emptyset,h}_{\Lambda,,h'} \
\mu^{\emptyset,h}_{\Lambda,,h'}(E_1),
$$
where the index $i$ runs over a finite set, for each of its
values $\Lambda^i$ is a
subset of $\L \cap (\Sg+\Lambda(1/h^{a}))$
which contains $\L \cap \Sg$ and
$\alpha_i > 0$; moreover $\sum_i \alpha_i = 1$.
Note that for small enough $h_0$ (depending on $\Sg$) the fact
that $a < b$ implies that no $h$vertebrate contour can fit inside
$\L \cap (\Sg+\Lambda(1/h^{a}))$, so that for each value of $i$
we have
$ \langle \ \cdot \ \rangle^{\emptyset,h}_{\Lambda^{i},,h'} =
\langle \ \cdot \ \rangle_{\Lambda^{i},,h'} $. The equality displayed
above and Lemma \CCC.4 \ now lead to
$$
\align
\left\langle g\rangle^{\emptyset,h}_{\L,,h'}
 \sum_{i} \alpha_i \cdot
\langle g \rangle_{\Lambda^{i},,h}\right
& \leq 2 \mu^{\emptyset,h}_{\Lambda,,h'}(E_1) \cr
%& \leq 2 \mu^{\emptyset,h}_{\Lambda,,h}(E_1)
& \leq C_1 \Sg \exp ( C_2 / h^a),
\teq(mixture'')
\endalign
$$
for a proper choice of $h_0$.
Similarly,
consider now the event
$$
E_2 = \bigcup_{x \in \Supp (g) + \L(1/h^a)} \CPxa,
$$
and condition on what the
(+,*)cluster of
%set of contours which surround at least one site in
$(\Supp (g) + \L(2/h^{a}))^c$
is. The same reasoning above, gives us,
for small enough $h_0$,
$$
\align
\left\langle g\rangle^{\emptyset,h}_{\L,,h'}
 \sum_{i} \alpha_j \cdot
\langle g \rangle_{\Lambda^{j},,h}\right
& \leq 2 \mu^{\emptyset,h}_{\Lambda,,h'}(E_2) \cr
%& \leq 2 \mu^{\emptyset,h}_{\Lambda,,h}(E_2)
&\leq C_1 \Sg \exp ( C_2 / h^a),
\teq(mixture''')
\endalign
$$
where the index $j$ runs over a finite set, for each of its
values $\Lambda^j$ is a
subset of $\L \cap (\Sg+\Lambda(2/h^{a}))$
which contains
$\L \cap (\Sg+\Lambda(1/h^{a}))$ and
$\alpha_j > 0$; moreover $\sum_j \alpha_j = 1$.
Thanks to the fact that $g$ is being supposed to be increasing, we
have for all $i$ and all $j$ as above
$$
\langle g \rangle_{\Lambda^{i},,h}
\leq
\langle g \rangle_{\L \cap (\Sg + \L(1/h^a)),,h}
\leq
\langle g \rangle_{\Lambda^{j},,h}.
\Eq(sandwich)
$$
The lemma follows from \equ(mixture''), \equ(mixture''') and
\equ(sandwich).
\cqd
\enddemo
%\input bas
The next lemma is a fundamental\corr{ step} step, in which an aspect of the
heuristics about
droplets and their surface and bulk freeenergies is made into
a rigorous results.
\proclaim{Lemma \CCC.6}
For any $p > 0$, given
$\epsilon > 0$ there is a finite positive constant $h_0$
such that for any $0 < h \leq h_0$,
any collection of skeletons $S$, and any simplyconnected set
$\Lambda \subset \Z^2$ which satisfies
$\Lambda \leq 1/h^p$,
$$\align
\tilde\mu_{\Lambda,,h}(\SS) & \leq
\exp \left( \beta \Big(
(1\epsilon) \Cal{W}(S) 
(1+\epsilon) h m^* \T(S)
\Big) \right) \\
& \leq
\exp \left( \beta
(13 \epsilon) \frac{\Cal{W}(S)}{2}
\right) .
\endalign$$
\endproclaim
\corr{ Here I made your $C=1$. I suggest to implement it. Also:}
\demo{Note} In this and some further statements the restriction
$\Lambda \leq 1/h^p$ is not really necessary. Still, we are using it
since it simplifies some arguments, and is enough for our purposes.
\enddemo
\demo{Proof}
The second inequality is immediate from Lemma \CCC.1, so that we only
have to prove the first one. It is enough to consider the case of nonempty
skeleton $S$.
We start with
$$ \align
\tilde\mu_{\Lambda,,h} & (\SS)
\leq
\sum_{G \in \CSh} \frac{Z_{\L,,h}(\SG)}
{Z_{\L,,h}(\Sempty)} \cr
&
=
\sum_{G \in \CSh} \frac{Z_{\L,,0}(\SG)}
{Z_{\L,,0}(\Sempty)}
\exp \left( \frac{\beta}{2} \int_0^h \sum_{x \in \L} \left[
\langle \sigma(x) \rangle_{\L,,h'}^{G,h}

\langle \sigma(x) \rangle_{\L,,h'}^{\emptyset,h}
\right] dh' \right) , \cr
&
\teq(remh)
\endalign
$$
where in the first step we used the fact that $\Cal{R} \supset
\Sempty$ when $h$ is small, while in the second step
we used the fact that for an arbitrary $E \subset \Omega_{\L,}$,
$$
\frac{d}{dh} \log Z_{\L,,h}(E) = \frac{\beta}{2} \sum_{x \in \L}
\langle \sigma(x)  E \rangle_{\L,,h}.
$$
Next we will show that given $\epsilon > 0$ it
is possible to take $h_0>0$ small enough so that
if $G \in \CSh$,
$0 < h' \leq h \leq h_0$,
and $\L$ is simplyconnected and satisfies $\Lambda \leq 1/h^p$,
then
$$
\sum_{x \in \L}
\left[ \langle \sigma(x) \rangle_{\L,,h'}^{G,h}

\langle \sigma(x) \rangle_{\L,,h'}^{\emptyset,h} \right]
\leq
2 (1 + \epsilon) m^* \T(S)
+ C \W (S)/ h^{2a} +1.
\Eq(diffs)
$$
(The reader should not be confused by the fact that $a$ seemingly does not
enter the l.h.s. In fact, it enters, because the restriction $G \in \CSh$
depends on the related parameter $r$.)
\corr{ !!}
For this purpose let
$\partial_{} G$ ({\it resp.} $\partial_{+} G$)
be the set of sites where each configuration
in $\Omega_{}$ with the set of external
contours equal to $G$ is doomed to be $1$ ({\it resp.} +1).
Let $\Lambda^G_{\text{ext}}$ and $\Lambda^G_{\text{int}}$ be the
components of $\L \backslash
(\partial_{} G \cup \partial_{+} G )$
which are, respectively, external and internal
to the contours in $G$.
Let also\corr{ $\LGic$ and $\LGoc$ } $\LGoc$ and $\LGic$ be, respectively,
the subsets of\corr{ $\LGi$
and $\LGo$} $\LGo$
and $\LGi$ obtained by removing from these sets all sites which are
at a distance not larger than $2/h^a$ from any point in any contour
of $G$.
First we consider the sites $x$ which are neither in $\LGic$ nor in
$\LGoc$. For these we have, using \equ(hausdorff) %(S.3),
\equ(bounds) and the fact
that $\anu < a$, that when $h_0$ is small,
$$
\sum_{x \in \L \backslash (\LGic \cup \LGoc)}
\left[ \langle \sigma(x) \rangle_{\L,,h'}^{G,h}

\langle \sigma(x) \rangle_{\L,,h'}^{\emptyset,h} \right]
\leq
2 \L \backslash (\LGic \cup \LGoc)
\leq
C \W (S)/ h^{2a}.
\Eq(diffs1)
$$
Regarding now the sites $x \in \LGoc$, we observe that for these sites
$$
\langle \sigma(x) \rangle_{\L,,h'}^{G,h}
=
\langle \sigma(x) \rangle_{\LGo,,h'}^{\emptyset,h}.
$$
But for each such $x$ we have $\LGo \cap (x + \L(1/h^a)) =
\L \cap (x + \L(1/h^a))$, so that a double application of
Lemma \CCC.5 gives us, for small $h_0$, that
$$
 \langle \sigma(x) \rangle_{\LGo,,h'}^{\emptyset,h} 
\langle \sigma(x) \rangle_{\L,,h'}^{\emptyset,h}
\leq C_1 \exp(C_2 / h^a).
$$
Combining the last two displays, and using the fact
that $  \LGoc \leq \L \leq 1/h^p$, we obtain for small enough $h_0$,
$$
\sum_{x \in \LGoc}
\left [\langle \sigma(x) \rangle_{\L,,h'}^{G,h}

\langle \sigma(x) \rangle_{\L,,h'}^{\emptyset,h} \right]
\leq
1.
\Eq(diffs2)
$$
Finally, regarding now the sites $x \in \LGic$, we observe that for these
sites
$$
\langle \sigma(x) \rangle_{\L,,h'}^{G,h}
=
\langle \sigma(x) \rangle_{\LGi,+,h'}.
$$
But since each such $x$ is separated from the boundary of the set $\LGi$
by a minimal distance $1/h^a$, when $h$ is small, we obtain from
the FKGHolley inequalities, \equ(expmu+) and \equ(muhtomu+)
$$
\langle \sigma(x) \rangle_{\LGi,+,h'}
\leq
\langle \sigma(x) \rangle_{\LGi,+,h}
\leq
m(h) + C_1 \exp(C_2/ h^a)
\leq
m^* (1 + \epsilon),
%\leq
%\leq m^* + C_1 \exp(C_2/ h^a).
$$
provided $h_0$ is chosen small enough.
On the other hand, since $\L$ is simplyconnected,
for each $x \in \LGic$ we have $\L \cap (x + \L(1/h^a)) =
x + \L(1/h^a)$, so that Lemma \CCC.5 gives us, for small $h_0$,
$$
\langle \sigma(x) \rangle^{\emptyset,h}_{\L,,h'}
 \langle \sigma(x) \rangle_{x + \L(1/h^a),,h'}
\leq C_1 \exp ( C_2 / h^a).
$$
By another application of the FKGHolley inequalities
and \equ(expmu+)
(with +1 and $1$ switched) we have
$$
\langle \sigma(x) \rangle_{x + \L(1/h^a),,h'}
\geq
\langle \sigma(x) \rangle_{x + \L(1/h^a),,0}
\geq m^*  C_1 \exp(C_2/ h^a)
\geq m^* (1 + \epsilon),
$$
provided $h_0$ is chosen small enough.
Combining the last four displays, and using the fact
that $  \LGic \leq \T(S)$,
for small enough $h_0$
we obtain
$$
\sum_{x \in \LGic}
\left [\langle \sigma(x) \rangle_{\L,,h'}^{G,h}

\langle \sigma(x) \rangle_{\L,,h'}^{\emptyset,h} \right]
\leq
2 (1 + \epsilon) m^* \T(S).
\Eq(diffs3)
$$
By adding \equ(diffs1), \equ(diffs2) and \equ(diffs3), we obtain \equ(diffs).
Keeping in mind that our concern is with the r.h.s. of \equ(remh),
we observe now that
$$
\sum_{G \in \CSh} \frac{ Z_{\L,,0}(\SG)}
{Z_{\L,,0}(\Sempty)}
=\frac{\mu_{\L,,0}(\SS)}{\mu_{\L,,0}(\Sempty)}.
\Eq(h=0)
$$
The numerator of this fraction is controlled in the fundamental
Lemma 10.1 in [Pfi], where it is shown to satisfy
$$
\mu_{\L,,0}(\SS) \leq \exp( \beta \W(S) + C J(S)),
$$
where $J(S) = \sum J(\gamma_i)$, in case $S = \{\gamma_1,...,\gamma_n\}$,
with $J(\gamma)$ being the number of vertices of the skeleton $\gamma$.
In reality, in the statement of Lemma 10.1 in
[Pfi] the assumption that the temperature is low enough is made. But,
as pointed out in [Iof2], this assumption is actually not needed for
Pfister's elegant proof, based on a clever use of duality and Griffiths'
inequalities, to work.
Using \equ(bounds') we can now write that given
$\epsilon >0$ there is $h_0>0$ such that for $0 < h \leq h_0$
$$
\mu_{\L,,0}(\SS) \leq \exp(\beta (1\epsilon/2) \W(S)).
\Eq(h=0a)
$$
Turning now to the denominator in the r.h.s. of \equ(h=0), observe that
at the boundary of any external vertebrate contour of a configuration
in $\Omega_{}$ there is a
(+,*)chain with $l^{\infty}$diameter at least $1/h^{b}$.
If $\sigma \in (\Sempty)^c$, then in $\sigma$ there is such a
(+,*)chain, and \equ(CCS)
(with +1 and $1$ switched)
can be used in combination with $\L \leq 1/h^p$ to conclude that
$$
\mu_{\L,,0}(\Sempty)
\geq \frac{1}{2},
\Eq(h=0b)
$$
for small enough $h$.
The first inequality claimed in the statement of the lemma follows from
\equ(remh), \equ(diffs), \equ(h=0), \equ(h=0a) and
\equ(h=0b),\corr{ since
$C h^{1a} < \epsilon/2$} since
$C h^{12a} < \epsilon/2$ if $h$ is small.
\cqd
\enddemo
%\input ent
The next lemma takes care of some remaining entropy.
\proclaim{Lemma \CCC.7}
For any $p>0$, given $\epsilon > 0$
there exists $h_0 > 0$ such that for all
$u>0$ and $D>0$,
there is a finite positive constant $C$
such that for any $0 \leq h \leq h_0$,
and any simplyconnected set
$\Lambda \subset \Z^2$ which satisfies
$\Lambda \leq 1/h^p$,
$$
\tilde\mu_{\Lambda,,h}\left(\SWDu\right) \leq
C \exp \left( \beta (1\epsilon) \frac{D}{2h^u} \right) .
$$
\endproclaim
\demo{Proof}
There is no loss in generality in supposing that $ 0 < \epsilon < 1$
and that $u > \anu$; the second of these claims being justified by
Lemma \CCC.3 and the fact that $\anu < b$.
We start by estimating
$\tilde\mu_{\Lambda,,h}\left(\SWDuk\right)$, for $k=1,2,...$.
To bound these quantities from above, using Lemma \CCC.6, all that we
need is an upper bound on the number of choices of skeletons $S$ which
correspond to configurations in $\Omega_{\L,}$ and have $\W(S)
\in [\frac{D}{h^u}k, \frac{D}{h^u}(k+1))$.
Using \equ(bounds'), the number $J(S)$ of
vertices that $S$ can have is bounded above by $N_k =
C_1 D (k+1) / h^{u\anu} \leq
C_2 D k / h^{u\anu}$.
Let $V$ be the set of distinct
points which are possible vertices of $S$.
The cardinality of $V$ is bounded above by $4 \L \leq 4/h^p$.
We consider now ordered $N_k$tuples of points\corr{ in $S$} in $V$ and associate
to each point in such an $N_k$tuple one of the words ``continue",
``close", or ``quit".\corr{ In this way we obtain collections of polygonal
lines constructed as follows.} To every such object we associate a collection
of closed polygons in the following way. We start from the first point in the
$N_k$tuple and while we\corr{ only see} keep seeing the word ``continue", we
do the following. We join the first point
to the second point, this one to the third point, and so on. When
we first reach a point were we read ``close", we connect it to the
first point of the $N_k$tuple, closing a first polygonal line,
and then we jump to the point which in the $N_k$tuple follows the
point where we just read ``close". While we do not see the word
``quit", we proceed in this fashion, understanding that the word
``close" means that we close the polygonal line which we are
currently constructing, and that then we jump to the next point in the
$N_k$tuple and start the next polygonal line from there. The word
``quit" is selfexplanatory: we stop the\corr{ procedure,} procedure by
closing the last polygon and disregard
the remaining points of the $N_k$tuple. The procedure that we
described generates all the collections of skeletons with which we
are concerned and plenty of additional garbage. At any rate, counting
the number of options here gives us an upper bound on the quantity
we are interested in. This upper bound on the number of choices of $S$
is bounded above by\corr{ !!!}
$$
V^{N_k} 3^{N_k} \leq
\left( \frac{12}{h^p} \right)^{\frac{C_2 D}{h^{u  \anu}} k }
\leq
\exp\left( \beta \frac{\epsilon D}{4 h^u} k \right),
$$
for small enough $h$.
Combining this estimate with Lemma \CCC.6, gives us that for small $h_0$
$$
\multline
\tilde\mu_{\Lambda,,h}\left(\SWDu\right) =
\sum_{k=1}^{\infty}
\tilde\mu_{\Lambda,,h}\left(\SWDuk\right)
\\ \leq
\sum_{k=1}^{\infty}
C \exp \left( \beta \left(1\frac{\epsilon}{2}\right)
\frac{D}{2 h^u} k \right)
\exp\left( \beta \frac{\epsilon D}{4 h^u} k \right)
\leq
C' \exp \left( \beta (1\epsilon) \frac{D}{2h^u} \right),
\endmultline
$$
provided $\epsilon < 1$.
\cqd
\enddemo
%\input t
We are now close to completing the proof of Proposition\corr{ \CCC.1.} 2.1.1. The
inequality \equ(claim1') is a direct consequence of
Lemmas \CCC.2 and \CCC.7.
For use in Section \ip, we observe that the same argument
gives for each $p>0$ and $\epsilon>0$ the existence of\corr{ of}
some finite $C_1$ so that
$$
\sup
\Sb
\Lambda \text{ simplyconnected}
\\
\L  \leq (1/h)^p
\endSb
\,
\sup_{x \in \Lambda}
\,
\tilde\mu_{\Lambda,,h}(\Fx) \leq
C_{1} \exp \left( \beta (1\epsilon)\frac{A}{h} \right),
\Eq(claim1'')
$$
for small $h$.
Turning to \equ(claim2'), Lemma \CCC.5, with $h = h'$,
gives us that when $h$ is small, for all $\L \in \Cal{L}_{0,h}$
\corr{ $$
\langle f \rangle^{\emptyset}_{\L,,h}
 \langle f \rangle_{\L(1/h^a)),,h}
\leq C(f) \exp ( C_1 / h^a).
$$}
$$
\langle f \rangle^{\emptyset,h}_{\L,,h}
 \langle f \rangle_{\L(1/h^a)),,h}
\leq C(f) \exp ( C_1 / h^a).
\Eq(for2'a)
$$
We will be done once we replace the conditional Gibbs\corr{ distribution
$\mu^{\emptyset}_{\L,,h}$,} distribution
$\mu^{\emptyset,h}_{\L,,h}$, implicit in this expression,
with the conditional Gibbs distribution
$\tilde\mu_{\L,,h}$. To this end we combine Lemmas \CCC.3 and \CCC.7 to write
$$
\tilde\mu_{\L,,h} \left( (\Sempty)^c \right) \leq C_2 \exp (C_3/h^b ).
$$
Combining this inequality with the equalities
$$
\int f d\tilde\mu_{\L,,h} =
\int_{\Sempty} f d\tilde\mu_{\L,,h} +
\int_{\left(\Sempty\right)^c} f d\tilde\mu_{\L,,h},
$$
and\corr{
$$
\langle f \rangle^{\emptyset}_{\L,,h} =
\frac{ \int_{\Sempty} f d\tilde\mu_{\L,,h} }{ \tilde\mu_{\L,,h}(\Sempty)},
$$}
$$
\langle f \rangle^{\emptyset,h}_{\L,,h} =
\frac{ \int_{\Sempty} f d\tilde\mu_{\L,,h} }{ \tilde\mu_{\L,,h}(\Sempty)},
$$
gives us\corr{ three items
$$
\left\int f d\tilde\mu_{\L,,h}

\langle f \rangle^{\emptyset}_{\L,,h}\right
\leq C_4 \exp ( C_3 / h^b)
\leq C_4 \exp ( C_3 / h^a).
$$}
$$
\left\int f d\tilde\mu_{\L,,h}

\langle f \rangle^{\emptyset,h}_{\L,,h}\right
\leq C_4(f) \exp ( C_3 / h^b)
\leq C_4(f) \exp ( C_3 / h^a).
\Eq(for2'b)
$$
Together \equ(for2'a) and \equ(for2'b) give us \equ(claim2'), and Proposition
\corr{ \CCC.1} 2.1.1 is proved.
%\bye
\bigskip
%\input CCD
\subheading{\CCD. Asymptotic expansion}
\proclaim{Proposition \CCD.1}
Suppose the $T < T_c$. Then for
each constant $a \in (0,1/2)$
and local observable $f$, for $n=1,2,3,...$, the
following expansion holds when $h>0$:
$$
\langle f \rangle_{\Lambda(1/h^a),,h}
=
\sum_{j=0}^{n1} b_j(f) h^j + O(h^n),
$$
where
$$
b_j(f) = \left. \frac{1}{j!}
\frac{d^j\langle f \rangle_{,h}}{d h^j}
\right_{h = 0_{}}
= \frac{1}{j!}
\(\frac{\beta}{2}\)^j
\sum_{x_1,...,x_j \in \Z^2}
\langle f;\s(x_1);...;\s(x_j) \rangle_{},
$$
and $O(h^n)$ is a function of $f$ and $h$ which
satisfies $\limsup_{h \searrow 0} O(h^n) / h^n < \infty$.
\endproclaim
The existence of the derivatives\corr{ which define} $b_j(f)$ and
their relations with the summations over generalized Ursell
functions are contained in \equ(jthderiv), modulo the
interchange of the role of +s and $$s. Below we will also
be using various other relations from Section \per \ modulo this
symmetry; we will do it without further warning.
From \equ(corrperc) and \equ(CCS), in combination
with an idea already explained in Section \CCA\corr{ ,} in
connection with the derivation of \equ(equila<1/2)\corr{ ,}
and also exploited
at the end of the proof
of Lemma \CCC.4 (and which amounts to an estimate on a
RadonNikodym derivative),
we have for each $a \in (0,1/2)$, $T < T_c$,
$0 \leq h' \leq h \leq 1$ and local observables $f$ and $g$,
\corr{ ' added}
$$
\align
\langle f ; g \rangle_{\L(1/h^a),,h'}
& \leq
%C \ f_{\infty} \ g_{\infty} \
C(f,g) \
\mu_{\L(1/h^a),,h'}
\left( \Supp (f) \overset+\to\longleftrightarrow \Supp (g) \right) \\
& =
C(f,g) \
\frac{Z_{\L(1/h^a),,h'}
\left( \Supp (f) \overset+\to\longleftrightarrow \Supp (g) \right)
}{Z_{\L(1/h^a),,h'}} \\
& \leq
C(f,g) \
\frac{Z_{\L(1/h^a),,0}
\left( \Supp (f) \overset+\to\longleftrightarrow \Supp (g) \right)
}{Z_{\L(1/h^a),,0}} \exp(\beta\L(1/h^a)h') \\
& \leq
C(f,g) \ e^{\beta} \
\mu_{\L(1/h^a),,0}
\left( \Supp (f) \overset+\to\longleftrightarrow \Supp (g) \right) \\
& \leq
C(f,g,T) \exp\Big(C(T) \ \dist (\Supp (f) , \Supp (g))\Big).
\teq(correlmeta)
\endalign
$$
As with \equ(ML) in Section \per, the argument in Appendix B of
[ML] shows that
from the exponential decay of correlations in
\equ(correlmeta) a similar exponential decay follows for the generalized
Ursell functions. With $a$, $T$, $h$, $h'$ and $f$ as above,
$$
\langle f;\s(x_1);...;\s(x_j) \rangle_{\L(1/h^a),,h'}
\leq
C_j(T,f) \exp \(C(T) \frac{ \diam(\Supp(f) \cup \{x_1,...,x_j \}) }{j}\).
\Eq(MLmeta)
$$
If we keep $0 0$
$$
\E(f(\sigma^{\nu}_{h;t}))
\geq
\E(f(\sigma^{}_{h;t})).
$$
The claim therefore will be proven once we show that for all
$\lambda$ and $\lambda'$ which satisfy $0 < \lambda < \lambda'$,
there are finite positive constants $C_1$ and $C_2$ such that
for $h>0$
$$
\E(f(\sigma^{\nu}_{h;\exp(\lambda/h)}))
\leq
\E(f(\sigma^{}_{h;\exp(\lambda'/h)}))
+ C_1 \exp (C_2/h).
\Eq(goalnu)
$$
To prove this inequality we first note that for arbitrary
$s,t \geq 0$,
$$
\E(f(\sigma^{\nu}_{h;t}))
\leq
\E(f(\sigma^{\mu_{}}_{h;t}))
=
\int
\P(\sigma^{\mu_{}}_{0;s} \in d\zeta)
\E(f(\sigma^{\zeta}_{h;t})).
$$
From Lemma 1.2.1 there exists finite positive $C_3$, $C_4$
and $C_5$ so that we have
$$
\multline
\int
\P(\sigma^{\mu_{}}_{0;s} \in d\zeta)
\E(f(\sigma^{\zeta}_{h;t}))
\leq
\int
\P(\sigma^{}_{0;s} \in d\zeta)
\E(f(\sigma^{\zeta}_{h;t})) \\
+
\P\(\sigma^{}_{0;s}(x) \neq \sigma^{\mu_{}}_{0;s}(x)
\text{ for some $x \in \L(C_3t)$} \)
\ + \
C_4 \exp (C_5 t).
\endmultline
$$
From the basiccoupling inequalities and the Markov property,
$$
\int
\P(\sigma^{}_{0;s} \in d\zeta)
\E(f(\sigma^{\zeta}_{h;t}))
\leq
\int
\P(\sigma^{}_{h;s} \in d\zeta)
\E(f(\sigma^{\zeta}_{h;t}))
=
\E(f(\sigma^{}_{h;s+t})).
$$
Combining the last three displays, and taking $t=s=\exp(\lambda/h)$
we obtain\corr{ (0) deleted in the second line}
$$
\align
\E(f(\sigma^{\nu}_{h;\exp(\lambda/h)}))
& \leq
\E(f(\sigma^{}_{h;2\exp(\lambda/h)})) \\
& \quad + \L(C_3\exp(\lambda/h)) \
\P\(\sigma^{}_{0;\exp(\lambda/h)}(0) \neq
\sigma^{\mu_{}}_{0;\exp(\lambda/h)}(0)\) \\
& \quad + C_4 \exp (C_5 \exp (\lambda/h)) \\
& \leq
\E(f(\sigma^{}_{h;\exp(\lambda'/h)})) \\
& \quad + C_6 \exp(2\lambda/h) \
\( \E\(\sigma^{+}_{0;\exp(\lambda/h)}(0)\)
 m^* \) \\
& \quad + C_4 \exp (C_5 \exp (\lambda/h)),
\endalign
$$
when $h$ is small.
In the second inequality above we used the basiccoupling inequalities\corr{
twice and} twice (which imply in particular the monotonicity of
$\E(f(\sigma^{}_{h;t}))$ in $t$), and also
the spinreversal symmetry in case $h=0$.
To complete the proof of \equ(goalnu) we have to show that
$\E(\sigma^{+}_{0;u}(0)) \rightarrow m^*$ fast enough as
$u \rightarrow \infty$. The following lemma, which states
that this happens faster than any power of $1/u$, is clearly
sufficient for our purpose.
\proclaim {Lemma \CCE.1}
Suppose $T < T_c$. Then for each $p > 0$ there is a
positive finite constant $C$ such that
$$
0 \leq \E\(\sigma^{+}_{0;u}(0)\)  m^*
\leq C u^{p}.
$$
\endproclaim
\demo{Proof}
The lower bound is a standard application of a basiccoupling inequality.
To prove the upper bound, we will
first also use basiccoupling inequalities, in order to compare the
infinite system with finite ones with (+) boundary conditions. For
these finite systems we will then use a result in [Mar], which
was extended up to $T_c$ in [CGMS].
%Our argument is similar to that used
%in the proof of Theorem 3.2 in [Mar].
The result of [Mar] and [CGMS] that we will use refers to
the spectral gap of the generator a kinetic Ising model
in a finite box.
For each finite $\L \subset \Z^2$, $\eta \in
\Omega$ and $h$, the process\corr{ here and in the following I replaced a
dot (which is not visible enough) by a star}
$(\s^{{\bold \cdot}}_{\L,\eta,h;t})_{t \geq 0}$
is a finitestatespace reversible irreducible Markov process
and its generator has its (discrete) spectrum contained in
the interval $(\infty, 0]$, with $0$ being in the spectrum.
The spectral gap, denoted by $\gap(\L,\eta,h)$,
is then simply the\corr{ largest nonnull} absolute value of the largest
nonzero number in the spectrum.
It is shown in [Mar] (Theorem 3.1) that for low enough $T$,
given $\epsilon \in (0,1/2)$, there exists a finite positive
$C$ so that
$$
\gap(\L(l),+,0) \geq \exp(C l^{1/2 + \epsilon}).
%\Eq(MarCCE)
$$
The common belief is that even up to $T_c$ the lower
bound\corr{ on $\gap(\L(L),+,0)$} on $\gap(\L(l),+,0)$ in this inequality is far from
optimal. No rigorous result in this direction is available,
and for temperatures close to $T_c$ only a weaker bound has
been proven. That bound, which is implicitly derived in
[CGMS] (see the introduction of that paper), states that
for any $\epsilon > 0$\corr{
$$
\gap(\L(L),+,0) \geq \exp(\epsilon l),
$$}
$$
\gap(\L(l),+,0) \geq \exp(\epsilon l),
\Eq(CGMS)
$$
for large $l$.
Fortunately for us, this estimate is suffices for our purpose
here.
For any $l > 0$, we
can write, using basiccoupling inequalities,
a standard estimate for the relaxation to equilibrium of
expected values of observables in terms of the spectral gap
(see, e.g., inequality (59) in [Sch1]), and \equ(expmu+),
$$
\align
\E\(\sigma^{+}_{0;u}\)  m^*
& \leq
\( \E\(\sigma^{+}_{\L(l),+,0;u}(0)\) 
\langle \sigma(0) \rangle_{\L(l),+,0} \)
+
\(\langle \sigma(0) \rangle_{\L(l),+,0}  m^* \) \\
& \leq
\frac{e^{\gap(\L(l),+,0) \, u}}{\mu_{\L(l),+,0}(+)}
+
C_1 \exp(C_2 l).
\teq(CCE1)
\endalign
$$
We use now \equ(CGMS) with
$\epsilon = C_2/(2p)$, and we choose
$ l =(\log u) /(2\epsilon) $.
%which is the same as saying $ u = \exp(2\epsilon l)$.
Since
\newline
\corr{ 2 items !! $\mu_{\L(l),+,0}(+) \leq \exp(C_3 l^2)$,}
$\mu_{\L(l),+,0}(+) \ge \exp(C_3 l^2)$, the first term in the
right hand side of \equ(CCE1) is bounded above by
$\exp(\sqrt{u}/2)$, when $u$ is large.
This finishes the proof, since the
second term in the
right hand side of \equ(CCE1) is of the claimed form and
this upper bound on the first term goes to 0 even faster.
\cqd
\enddemo
%\bye
\bigskip
%\input PP
\numsec=\PP
\numfor=1
%\leftheadtext{PP.First part}
%\rightheadtext{PP.First part}
%\NoBlackBoxes
\heading{\PP. Relaxation regime}\endheading
\subheading{\PPp. Preliminaries}
In this part of the paper we will prove part (ii) of Theorem 1.
For this we suppose that $\lambda > \lambda_{\text{c}}$ is
fixed and that $\nu \leq \mu_$. Once more, there is no
loss in generality in
supposing that $f$ is increasing and that it has
$f_{\infty} \leq 1$. For the remainder of the proof we will
make these assumptions.
Half of our goal is trivial, since the basiccoupling
inequalities give for each $t \geq 0$
$$
\E(f(\s^{\nu}_{h;t})) \leq
\E(f(\s^{\mu_h}_{h;t}))
= \langle f \rangle_h.
$$
Our goal is therefore reduced to proving that for all $C > 0$
there exists a finite $C_1$ such that
$$
\E(f(\s^{\nu}_{h;\exp(\lambda/h)})) \geq
\langle f \rangle_h  C_1 \exp(C/h).
\Eq(goalPP)
$$
At this point there is also no loss in supposing that $\nu$ is
concentrated on the configuration with all spins down, and therefore
also that $\lambda_{\text{c}} < \lambda < 2 \lambda_{\text{c}}$.
In our argumentation this second assumption will simplify things.
In Section \ip \ we will introduce certain spacetime structures,
which we call inverted pyramids, and which will be used to
obtain statements concerning droplet growth. To be able to use these
results in order to prove \equ(goalPP), we will need to use
such inverted pyramids as building blocks of a rescaling
procedure, and will also need to obtain a mathematical
counterpart to the notion of droplet creation at the correct
rate; these two topics will be covered in Section \re. In
both sections, \ip \ and \re, we will need to use some lemmas
which can be seen as rigorous counterparts to the notion that
the function $\phi(b) = wb m^*b^2$ gives the freeenergy of
optimallyshaped
droplets, and that equilibrium distributions can be studied
based on this heuristics. To avoid distracting the reader with
the technicalities behind these lemmas, they are only presented
later in the paper, in Section \dw. Some of the results in that
section were already contained in the paper [SS1], but here we
will need substantial strengthenings of them; moreover the techniques
used here will be different from those in [SS1] and so provide an
alternative to parts of that paper. Finally some estimates on the
spectral gap of the generator of the dynamics of some kinetic Ising
models on some finite sets will also be needed in Sections \ip \ and
\re. Again, those will be postponed to the final section, \ga, in
this part of the paper, in order to avoid distracting the reader's
attention from the main ideas in Sections \ip \ and \re.
The remainder of the
paper is written having in mind a reader who will be following
it in the order in which the sections are presented. With this in
mind we tried to motivate and explain heuristically in Sections \ip \
and \re \ the results which are used there but will only be proven
later, in Sections \dw \ and \ga. Readers who prefer following the
lemmas and propositions in a strictly logical order, and who do not
worry about the motivation behind each lemma can read the sections
in the following order: \dw, \ga, \ip, \re. Considering this
possibility and also the length of the paper,
we repeat some definitions from Sections \ip \ and
\re \ in Sections \dw \ and \ga.
%% END OF PART 2
%% BEGINNING OF PART 3
\subheading{\ip. Inverted pyramids and droplet growth}
In this section we will introduce two propositions which are
counterparts to
the statement: ``If we start with a large enough Wulffshaped droplet
of the (+)phase in the midst of the ($$)phase,
then it is likely to grow with a linear speed larger than any
negative exponential of $1/h$''. In the first of these
propositions ``large enough'' will
be substantially larger than just ``supercritical'' (it will mean that
the droplet has a negative freeenergy); in the second of these
propositions this aspect will be improved,
at the cost of extra technicalities and extra work in the proof.
We will need to generalize the notion of a process
$(\s^{{\bold \cdot}}_{\L,\eta,h;t})_{t \geq 0}$ evolving in a box $\L$, with
boundary condition $\eta$. We will have to consider the time
evolution to occur in boxes which may change with time. Since
we will construct all of our generalized processes using the
graphical construction, their definition is very elementary.
The spacetime regions with which we will be concerned will
all be of the following type. Let $t_02$ then for
all $b_1$, $b_2$ and $h$ as above there is a sequence,
$$
\SEQ(b_1,b_2;h) = (\L_0, ... , \L_N),
$$
of boxes with the required properties, and with
$$
N = N(b_1,b_2;h) \leq C \left( \frac{ b_2}{h} \right)^2
\Eq(N)
$$
for some finite constant $C$.
(Of course,
more than one such sequence typically exists,
and we suppose that some rule to choose one from among them
is being used). Given also $\delta$, the
notation
$$\IP = \IP(b_1,b_2;h;\delta)$$ will be used
for the inverted pyramid which we described
above and which also has $t_0 = 0$.
The technical counterpart to the idea of the growth of a droplet
will be contained in an event that we define next. With $b_1$, $b_2$,
$h$ and $\delta$ fixed
(and\corr{ omited} omitted from the notation in several places, for simplicity),
and also an initial configuration $\eta
\in \Omega_{\L(\frac{b_1}{h}W),}$ given, we let $G_{\eta}$ be
the following event:
$$
G_{\eta} = \left\{
\s^{0,\eta}_{\IP,,h;t_{N+1}}
=
\s^{t_N,+}_{\IP,,h;t_{N+1}}
\right\}.
$$
Observe that
it is clear from the basiccoupling inequalities that, regardless of
what $\eta$ is,
$\s^{0,\eta}_{\IP,,h;t_{N+1}}
\leq
\s^{t_N,+}_{\IP,,h;t_{N+1}}$, so that $G_{\eta}$ only really requires
that the complementary inequality holds.
Informally, $G_{+}$ can be seen
as the event of a droplet of +1 spins of
linear size proportional to $b_1 / h$ at time 0
growing to become a droplet of linear size proportional to $b_2 / h$
at time $t_N = N \exp(\delta/h) \leq C \left( { b_2}/{h} \right)^2
\exp(\delta/h)$.
Recall from Section \he \
that heuristically the freeenergy of a Wulffshaped droplet
$(b/h)W$ of the (+)phase is given by $\phi(b)/h$, where
$ \phi(b) = w b  m^* b^2$,
and that $B_c$ is the value of $b$ which maximizes this function,
while $B_0 = 2B_c$ is the value of $b$ above which this function
becomes negative. In Section \he \ this was used to predict
heristically the behavior of the relaxation time as $\hg$. Similarly,
the freeenergy of such droplets can also be used to predict the
typical aspect of a Gibbs distribution
$\mu_{\L\(\frac{b}{h}W\),,h}(\ \cdot \ )$, when $h$ is small, and
this will be of great relevance in this paper.
When $b < B_0$, one should expect
this Gibbs distribution to resemble the ($$)phase,
since droplets of the (+)phase would all have positive freeenergies.
On the other hand, when $b > B_0$, one should expect
this Gibbs distribution to resemble the ($+$)phase, separated from
the ($$) spins at the boundary by a large contour,
since a single droplet of the (+)phase of the size of the system itself
would have the lowest possible freeenergy. Rigorous results of this
type were obtained in [SS1]. Unfortunately, for our purposes in the
current paper we will need technically stronger results than those in
[SS1]. As mentioned in Section \PPp, these technical results will only
be presented and proven in Section \dw. If the reader accepts the
picture which we just presented and justified heuristically as
reasonable, then, he or she should have no difficulty believing in the
specific statements which appear in this and in the next section
and are proven only in Section \dw.
Next we state the first of the two main claims\corr{ in} of this section.
\proclaim{Proposition \ip.1}
Given $B_0 < b_1 < b_2$ and $\delta > 0$ there are positive finite
constants $C_1$ and $C_2$ such that for $h>0$
$$
\int d\mu_{\L(\frac{b_1}{h}W),,h} (\eta) \P(G_{\eta})
\geq 1  C_1 \exp(C_2 / h).
\Eq(goal0)
$$
In particular
$$
\P(G_{+})
\geq 1  C_1 \exp(C_2 / h).
\Eq(goal0')
$$
Moreover the choice of $C_2$ does not depend on $\de$ and $b_2$
and it can be taken arbitrarily large, provided $b_1$ is large enough.
\endproclaim
Observe that from the heuristic picture described before\corr{ of the
statement of Proposition \ip.1,} the last proposition,
we can see that in \equ(goal0)
we are starting from a droplet of the (+)phase, and the statement
is that it is likely to grow at a speed which is controlled in a
useful way.
In comparison, for $B_c < b_1 < B_0$, \equ(goal0) should be false,
since then in the Gibbs distribution
$\mu_{\L(\frac{b_1}{h}W),,h} (\ \cdot \ )$
no droplet of the (+)phase should be present. On the other hand,
we should expect \equ(goal0') to be true also in this case, since
there we are starting from a droplet not just of the (+)phase,
but actually a solid droplet of (+) spins, with a supercritical
size. This claim is contained in the next proposition. In this
proposition we will use the following object.
Recall the definition \equ(defR) of $\Cal{R}$ and set
$$
\ha\mu_{\L,,h} (\ \cdot \ ) =
\mu_{\L,,h} (\ \cdot\  \Cal{R}^c ).
$$
This is the Gibbs measure conditioned on the presence of a supercritical
droplet. The following should be expected to\corr{ happen}
happen, based on the
heuristics.
When $\L = \L(\frac{b}{h}W)$ and $b > B_0$ the conditioning has
no major effect;
but if $B_c < b < B_0$, then the conditioning produces a droplet of
the (+)phase, of roughly the size of the whole system,
separated from
the ($$) spins at the boundary by a large contour. This is so
since a single droplet of the (+)phase of the size of the system itself
would have the lowest possible freeenergy compatible with the
conditioning.
It is important to note that in the next proposition,
in addition to having to modify \equ(goal0)
by introducing the conditioning on $\Cal{R}^c$, also the
way in which $\delta$ can be chosen is different. The reason for this
difference will be clarified in the proof of the proposition.
\proclaim{Proposition \ip.2}
Given $B_c < b_1 < b_2$ there are positive finite
constants $\de_0$, $C_1$ and $C_2$ such that if $0 < \de < \de_0$,
for $h>0$
$$
\int d\ha\mu_{\L(\frac{b_1}{h}W),,h} (\eta) \P(G_{\eta})
\geq 1  C_1 \exp(C_2 / h).
\Eq(goal0ha)
$$
In particular
$$
\P(G_{+})
\geq 1  C_1 \exp(C_2 / h).
\Eq(goal0'ha)
$$
\endproclaim
In the remainder of this section we will prove Propositions
\ip.1 and\corr{ \ip2.} \ip.2.
Besides leaving some technical lemmas which
concern equilibrium distributions to Section \dw,
also some technical results which concern the
kinetic Ising models run in certain finite boxes, including
those of the type of $\L\(\frac{b}{h}W\)$, will have their
proofs postponed to Section \ga. These results, when used in the
current section, will be heuristically motivated, though.
\medskip
%\input pip
%\subheading{\PP.5. Proofs of propositions on inverted pyramids}
\demo{Proof of Proposition \ip.1
(modulo results in Sections \dw \ and \ga ) }
The second claim, \equ(goal0'),
follows from the first one, \equ(goal0), and the basiccoupling
inequalities. Our task is to prove \equ(goal0)
and the claims about the value of the constant $C_2$. There is
no loss in supposing that $h$ is small, and we will assume that
$0 < h \leq 1$.
For $i=1,...,N$, and arbitrary $\zeta \in \Omega_{\L_{i1},}$ set
$$
G^i_{\zeta} = \left\{
\s^{t_{i1},\zeta}_{\IP,,h;t_{i+1}}
=
\s^{t_{i},+}_{\IP,,h;t_{i+1}}
\right\}.
$$
Note that
$$
G_{\eta}
\ \supset \
G^1_{\eta} \cap \left( \cap_{i=2}^N G^i_{+} \right).
\Eq(chain)
$$
Our goal now is to prove that for some positive finite $C_1$ and $C_2$,
as in the statement of the\corr{ lemma,} proposition,
for $i=1,....,N$,
$$
\int d\mu_{\L_{i1},,h} (\zeta) \P((G^i_{\zeta})^c)
\leq C_1 \exp(C_2 / h).
\Eq(goal1)
$$
In particular this implies, as in the first paragraph in this proof, that
$$
\P((G^i_{+})^c)
\leq C_1 \exp(C_2 / h).
\Eq(goal1')
$$\corr{
Together, \equ(goal1) (used for $i=1$), \equ(goal1')
(used for $i = 2,...,N$), \equ(chain) and \equ(N) imply
\equ(goal0).} By putting together \equ(goal1) (used for $i=1$), \equ(goal1')
(used for $i = 2,...,N$), \equ(chain) and \equ(N) we obtain
\equ(goal0).
The sets $\L_{i1}$ and $\L_{i}$ differ only in that the latter has
one extra site, say $x$, that the former does not have.
For an arbitrary $\zeta \in \Omega_{\L_{i1},}$, we
will need to compare $\mu_{\L_{i1},,h}(\zeta)$ with
$\mu_{\L_{i},,h}(\zeta)$. For this purpose we introduce the notation
$\CSx = \{\s : \s(x) = 1 \}$ for the event that the spin
at the site $x$ is negative, and let
$$
\alpha = \inf_{h \leq 1}
\inf_{\xi \in \Omega} \mu_{\{0\},\xi,h}(\CS0)
\Eq(defalpha)
$$
be the largest lower bound on the probability of having in equilibrium
a spin $1$ at the origin, given any information about the other spins,
for values of $h$ in the arbitrarily chosen neighborhood $[1,+1]$ of
the origin. Clearly $\alpha >0$ for each temperature $T>0$.
With this notation,\corr{
$$
\mu_{\L_{i},,h}(\zeta) \geq
\mu_{\L_{i},,h}(\CSx) \
\mu_{\L_{i},,h}(\zeta  \CSx)
\geq \alpha \
\mu_{\L_{i1},,h}(\zeta),
$$}
$$
\mu_{\L_{i},,h}(\zeta) =
\mu_{\L_{i},,h}(\CSx) \
\mu_{\L_{i},,h}(\zeta  \CSx)
\geq \alpha \
\mu_{\L_{i1},,h}(\zeta),
$$
which can be seen as a uniform estimate on a
RadonNikodym derivative:
$$
\sup_{\zeta \in \Omega_{\L_{i1},}}
\frac{d\mu_{\L_{i1},,h}}{d\mu_{\L_{i},,h}} (\zeta)
\leq \frac{1}{\alpha}.
\Eq(RN)
$$
Using
the stationarity of the processes
$( \s_{\L,,h}^{\mu_{\L,,h}})_{t \geq 0}$,
and \equ(RN) we obtain\corr{ dot is added in useRN}
$$
\align
\int d\mu_{\L_{i1},,h} (\zeta) \P\left((G^i_{\zeta})^c\right)
& =
\int d\mu_{\L_{i1},,h} (\zeta) \P\left(
\s^{t_{i1},\zeta}_{\IP,,h;t_{i+1}}
\neq
\s^{t_{i},+}_{\IP,,h;t_{i+1}}\right) \cr
& =
\int d\mu_{\L_{i1},,h} (\zeta) \P\left(
\s^{t_{i},\zeta}_{\IP,,h;t_{i+1}}
\neq
\s^{t_{i},+}_{\IP,,h;t_{i+1}}\right) \cr
& \leq
\frac{1}{\alpha}
\int d\mu_{\L_{i},,h} (\zeta) \P\left(
\s^{t_{i},\zeta}_{\IP,,h;t_{i+1}}
\neq
\s^{t_{i},+}_{\IP,,h;t_{i+1}}\right) \cr
& =
\frac{1}{\alpha}
\int d\mu_{\L_{i},,h} (\zeta) \P\left(
\s^{\zeta}_{\L_i,,h;\exp(\delta/h)}
\neq
\s^{+}_{\L_i,,h;\exp(\delta/h)}\right) . \cr
&
\teq(useRN)
\endalign
$$
This may have seen at first sight as a minor and trivial maneuver, but
it is actually a central step in our approach towards controlling droplet
growth. We have just transformed our problem pertaining to ``growth''
into a problem pertaining to ``rapid loss of memory'' or, in other
words, ``rapid convergence to equilibrium'', since
\equ(useRN) will provide us with the aimed \equ(goal1), once we show
that
$$
\int d\mu_{\L_{i},,h} (\zeta) \P\(
\s^{\zeta}_{\L_i,,h;\exp(\delta/h)}
\neq
\s^{+}_{\L_i,,h;\exp(\delta/h)}\)
\leq C_1 \exp(C_2 / h).
\Eq(goal2)
$$
Proving
\equ(goal2) seems like a standard problem, due to the vast current
literature on this type of issue: we have a reversible Markov
process $(\s^{+}_{\L_i,,h;t})_{t\geq0}$, and we want to show
that it reaches equilibrium in a time of the order of
$\exp(\delta/h)$.
There are nevertheless still major hurdles to overcome.
The standard approach to such a problem starts with the derivation
of a lower bound on the spectral gap of the generator of the
process. The result that we are after would then follow if the
time with which we are concerned were much larger that the
inverse of the lower bound on the spectral gap. Such an approach
is nevertheless unfeasible in our case, due to the fact
that here the spectral gap is of the order of $\exp(\beta A/h)$,
so that its inverse is much larger than $\exp(\delta/h)$, when
$\delta$ is small. We will not give the full proof of this
claim on the value of the spectral gap, since it will be of
no use for us (some readers may want to take it as an exercise,
with the hint that it can be solved using techniques in this
paper), and will limit ourselves to explaining the nature of the
difficulty at the heuristic level. This difficulty lies precisely
in the sort of metastability studied in this paper. We are considering
a Glauber dynamics in the box $\L_i$, which is almost the same as a
set $\L(\frac{b}{h}W)$ with some $b \in [b_1,b_2]$.
Since $B_0 < b_1 \leq b$, in equilibrium we should have a large droplet
of the (+)phase, covering the box $\L_i$ almost entirely.
If we look at the process started from all spins down,
$(\s^{}_{\L_i,,h;t})_{t\geq0}$, then to reach equilibrium this
big droplet has to be formed, and the system has to go through the
bottleneck presented by the situation with a critical droplet.
Hence the freeenergy barrier to be overcome has height $A/h$.
The system should
then reach equilibrium in a time of order $\exp(\beta A / h)$.
(Because the linear size of the box is of the order of $1/h$,
droplet growth is of no relevance\corr{ inside of it, when estimating
the order of magnitude of the relaxation time.} for the estimate of
the order of magnitude of the relaxation time inside the box.)
Starting with all spins down should maximize the relaxation time,
since equilibrium is basically the (+)phase, and the inverse of
this relaxation time should hence give the order of magnitude of
the spectral gap.
The same heuristics above which pointed out the problem with using the
spectral gap for our purpose of proving \equ(goal2) indicates also
why we should believe that this inequality holds nevertheless. The
difficulty pointed out concerns the long time needed to relax towards
equilibrium if the process is started with all spins down. We are,
on the other hand, concerned with the case in which we start with all
spins up, much closer to equilibrium. One can talk of an heuristic
picture with a double well structure.
The configuration with all spins down is in the higher (metastable)
well, separated from the other one by the freeenergy barrier, but
the configuration with all spins up is inside the deeper (stable)
well, and our problem concerns only relaxation inside of this well.
The problem still remains at this point of how to exploit this heuristic
picture and prove \equ(goal2). The solution will be to use the
basiccoupling inequalities in order to compare our process with some
modified ones, for which the spectral gaps can be proven to be
large enough for our purposes. In doing so we were inspired by arguments
in Section 5 of [Mar].
One of the two comparison processes that we will use is
$(\s^{{\bold \cdot}}_{\L_i,+,h;t})_{t\geq0}$, in which (+)boundary conditions
are used. The other one will be denoted by \newline
$\(\s^{{\bold \cdot}}_{\L_i\backslash \Lc,(+,),h;t}\)_{t\geq0}$,
and has the following meaning. The box in which it is run is the
annulus $\L_i\backslash \Lc$, where
$\Lc = \L(\frac{B_c + B_0}{2h}W)$,
and the boundary condition
denoted by $(+,)$ refers to freezing the spins up inside the core
$\Lc$ and down outside $\L_i$.
In Section \ga, Propositions \ga.1 and \ga.2,
we will show that
for any $\delta>0$ the generators of these two processes satisfy
$$
\align
& \inf_{i=0,...,N} \
\gap (\L_i,+,h) \ \geq \ \exp\left(\frac{\delta}{2h}\right),
\teq(gaps1) \cr
& \inf_{i=0,...,N} \
\gap
\left( \L_i\backslash \Lc, (+,),h\right)
\ \geq \ \exp\left(\frac{\delta}{2h}\right),
\teq(gaps2)
\endalign
$$
for small enough h.
The intuitive reason behind these relatively large spectral
gaps, is that the extra (+)'s introduced as boundary conditions
eliminate metastability in the time evolution of these two
processes. In equilibrium these systems are again basically in the
(+)phase, and if we start them with all spins down, then there
is no need to nucleate a critical droplet in order to relax to
equilibrium. In the first case the (+)phase drifts inwards from the
(+)boundary towards the center. In the second case a supercritical
(+)droplet is frozen by hand in the center of the box, so that
the relaxation is a ``downhill'' movement on the freeenergy landscape;
the (+)phase should drift outwards from the center towards the outer
boundary of the box.
Our task now is to show how \equ(gaps1) and
\equ(gaps2) can be used to derive
\equ(goal2). We partition $\L_i$ into two sets:
$$
\align
& \L^{\text{in}} =
\L\(\frac{b_1 + B_0}{2h}W\), \cr
& \L_i^{\text{out}} =
\L_i \backslash \L^{\text{in}}.
\endalign
$$
Using the basiccoupling inequalities, we have
$$
\align
& \int d\mu_{\L_{i},,h} (\zeta) \P\left(
\s^{\zeta}_{\L_i,,h;\exp(\delta/h)}
\neq
\s^{+}_{\L_i,,h;\exp(\delta/h)}\right) \cr
& \leq
\int d\mu_{\L_{i},,h} (\zeta)\sum_{y \in \L_i}
\left\{
\P \left(\s^{+}_{\L_i,,h;\exp(\delta/h)}(y) = +1\right)
 \P \left(\s^{\zeta}_{\L_i,,h;\exp(\delta/h)}(y) = +1\right) \right\} \cr
& =
\sum_{y \in \L_i}
\left\{
\P \left(\s^{+}_{\L_i,,h;\exp(\delta/h)}(y) = +1\right)

\mu_{\L_i,,h}(\CSy+) \right\} \cr
& \leq
\sum_{y \in \L^{\text{in}}}
\left\{
\P \left(\s^{+}_{\L_i,+,h;\exp(\delta/h)}(y) = +1\right)

\mu_{\L_i,,h}\left(\CSy+\right) \right\} \cr
& +
\sum_{y \in \L_i^{\text{out}}}
\left\{
\P \left(\s^{+}_{\L_i \backslash \Lc,
(+,),h;\exp(\delta/h)}(y) = +1\right)

\mu_{\L_i,,h}\left(\CSy+\right) \right\}.
\endalign
$$
At this point we can use \equ(gaps1) and
\equ(gaps2) in a standard way. For instance,
from inequality (59) in [Sch1],\corr{ the three displays are rewritten}
$$
\multline
\left
\P \left(\s^{+}_{\L_i,+,h;\exp(\delta/h)}(y) = +1\right)

\mu_{\L_i,+,h}\left(\CSy+\right)
\right \\
\leq
\frac{e^{\exp(\delta/h) \gap(\L_i,+,h)}}{\mu_{\L_i,+,h}(+)}
\leq
\exp\( C \left(\frac{b_2}{h}\right)^2 \)
\exp \{e^{\frac{\delta}{2h}}\}
\endmultline
$$
and
$$
\multline
\left
\P \left(\s^{+}_{\L_i \backslash \Lc,
(+,),h;\exp(\delta/h)}(y) = +1\right)

\mu_{\L_i \backslash \Lc,
(+,),h}\left(\CSy+\right)
\right \\
\leq
\frac{e^{\exp(\delta/h) \gap\(\L_i \backslash \Lc,
(+,),h\)}}
{ \mu_{\L_i \backslash \Lc,
(+,),h}\left(+\right) }
\leq
\exp\( C \left(\frac{b_2}{h}\right)^2 \)
\exp \{e^{\frac{\delta}{2h}}\}.
\endmultline
$$
Combining the last three displayed inequalities, we obtain
$$
\align
\int & d\mu_{\L_{i},,h} (\zeta) \P\left(
\s^{\zeta}_{\L_i,,h;\exp(\delta/h)}
\neq
\s^{+}_{\L_i,,h;\exp(\delta/h)}\right) \cr
& \leq
\sum_{y \in \L^{\text{in}}}
\left\{
\mu_{\L_i,+,h}\left(\CSy+\right)

\mu_{\L_i,,h}\left(\CSy+\right) \right\} \cr
& +
\sum_{y \in \L_i^{\text{out}}}
\left\{
\mu_{\L_i \backslash \Lc,
(+,),h}\left(\CSy+\right)

\mu_{\L_i,,h}\left(\CSy+\right)
\right\} \cr
&
+ C' \left(\frac{b_2}{h}\right)^2
\exp\( C \left(\frac{b_2}{h}\right)^2 \)
\ \exp \{e^{\frac{\delta}{2h}}\}.
\teq(preB)
\endalign
$$
Our remaining problem concerns only equilibrium.
We will use a result from Section \dw, but observe that this
result is
intuitively natural, based on the heuristics presented immediately
before\corr{ of} the statement of Proposition \ip.1.
Set
$$
%\gather
\align
& \Cal{B}_1 = \left\{
\L^{\text{in}}
\perms
\( \L\(\frac{b_1}{h}W \) \)^c
\right\},
\teq(defB1)
\cr
& \Cal{B}_2 = \left\{
\Lc
\perms
\(\L^{\text{in}}\)^c
\right\},
\teq(defB2)
\endalign
%\endgather
$$
From Lemma \dwsp \
we know that for\corr{ any $C_2 > 0$, if $b_1$ is large enough}
every $b_1>B_0$ there exists a $C_2=C_2(b_1)>0$, $C_2\to \infty $
as $b_1 \to \infty $ and
a finite $C_1$ such that for $j=1,2$,\corr{ dot}
$$
\sup_{i=0,...,N} \mu_{\L_i,,h} (\Cal{B}_j) \leq C_1 \exp(C_2/h).
\Eq(Bexp)
$$
From \equ(psw),
for each $y \in \L^{\text{in}}$,
$$
\mu_{\L_i,+,h}\left(\CSy+\right)

\mu_{\L_i,,h}\left(\CSy+\right)
\leq
2 \mu_{\L_i,,h}\left( \Cal{B}_1 \right).
\Eq(boundB1)
$$
Similarly,
for each $y \in \L_i^{\text{out}}$,
$$
\mu_{\L_i \backslash \Lc,
(+,),h}\left(\CSy+\right)

\mu_{\L_i,,h}\left(\CSy+\right)
\leq
2 \mu_{\L_i,,h}\left( \Cal{B}_2 \right).
\Eq(boundB2)
$$
The desired inequality \equ(goal0) and the claims about the
choice of the constant $C_2$ which appears there follow from
combining \equ(preB), \equ(boundB1), \equ(boundB2) and \equ(Bexp)
\cqd
\enddemo
%\bye
\medskip
%\input hat
\demo{Proof of Proposition \ip.2
(modulo results in Sections \dw \ and \ga) }
We will explain how the proof of Proposition \ip.1 can be adapted to
prove this proposition.
There are several extra complications, since
$\ha\mu_{\L_{i},,h}$ is not an invariant distribution for
the process $(\s^{{\bold \cdot}}_{\L_{i},,h})_{t \geq 0}$, and in
particular this
is the reason for which we will have to choose $\de_0$ small enough.
The idea is to look at this distribution instead as a
``metastable state'' for this process, and to use techniques from
Part 2 of\corr{ this} the present paper in this connection.
Using the graphical construction we can define
the following processes restricted to $\Cal{R}^c$.
For arbitrary $s \in [t_0,t_{N+1})$,
and $\eta \in \Omega_{\L_j,} \cap \Cal{R}^c$, where $j$ is
defined by $s \in [t_j,t_{j+1})$,
the process
$$
(\ha\s^{s,\eta}_{\IP,,h;t})_{t \geq s}.
$$
is obtained in the following simple way.
We freeze
all spins outside of $\IP$ as $1$ and at time $s$ set the
configuration to $\eta$.
We use then the graphical construction with
its standard rules modified by suppressing jumps which would
bring the system\corr{ to $\Cal{R}$
to update spins inside $\IP$,} to $\Cal{R}$,
to update spins inside $\IP$ after time $s$.
Bottlenecks for this dynamics are the sets\corr{
$$
\Fxp = \{\s^x : \s \in \Fx \}.
$$}
$$
\Fxp = \{\s : \s^x \in \Fx \}.
$$
Define also
$$
\Fxn = \Fx \cup \Fxp.
$$
For each $x$ the event $\Fxn$ depends only on the spins at
sites other than $x$. If $h \leq 1$, we have from the
definition \equ(defalpha) of $\alpha$, that for $i=0,...,N$ and
$x \in \L_i$,
$$
\mu_{\L_i,,h}(\Fx) \geq \alpha \
\mu_{\L_i,,h}(\Fxn).
\Eq(Fxalpha)
$$
Using this inequality in combination with the bottleneck
estimate \equ(claim1'') we have that for arbitrary $\epsilon>0$,
$$
\align
\ha\mu_{\L_i,,h}(\Fxp)
& =
\frac{ \mu_{\L_i,,h}(\Fxp) } { \mu_{\L_i,,h}(\Cal{R}^c) }
\leq
\frac{ \mu_{\L_i,,h}(\Fxn) } { \mu_{\L_i,,h}(\Cal{R}^c) }
\leq
\frac{ \frac{1}{\alpha}\mu_{\L_i,,h}(\Fx)}{\mu_{\L_i,,h}(\Cal{R}^c)}
\cr
& =
\frac{1}{\alpha} \tilde\mu_{\L_i,,h}(\Fx)
\frac{ \mu_{\L_i,,h}(\Cal{R})}{\mu_{\L_i,,h}(\Cal{R}^c) }
\leq
\frac{1}{\alpha}
\frac{C \exp\(\beta (1\epsilon)\frac{A}{h}\) }
{\mu_{\L_i,,h}(\Cal{R}^c) },
\endalign
$$
for small $h$, independent of $i$.
Since $\L\(\frac{b_1}{h}W\) \subset \L_i$
and
$b_1 > B_c$,
Lemma \dwrc \
gives us for some $\de_0 > 0$,
$$
\mu_{\L_i,,h}(\Cal{R}^c)
\geq
\exp\(\beta \frac{A}{h} + \frac{2 \de_0}{h}\),
$$
for small $h$. So, if our choice of $\epsilon$ above is made properly,
we obtain
$$
\ha\varphi = \sup_{i=0,...,N} \ \sup_{x \in \L_i} \
\ha\mu_{\L_i,,h}(\Fxp)
\ \leq \
C_1 \exp\(\frac{\de_0}{h}\),
\Eq(muFxp)
$$
for some finite $C_1$.
Define now
$$
\tau^{\zeta}_i = \inf \{t \geq t_i :
\text{the process
$(\ha\sigma^{t_i,\zeta}_{\IP,,h;t})$
has a suppressed jump at time $t$} \}.
$$
Clearly
$$
\ha\sigma^{t_i,\zeta}_{\IP,,h;t}
=
\sigma^{t_i,\zeta}_{\IP,,h;t}
\quad \text{for} \quad t_i \leq t < \tau_i^{\zeta}.
\Eq(tousetauha)
$$
From the same argument used to prove \equ(tau), we obtain now,
using \equ(muFxp), for $i=0,...,N$, and $\de < \de_0$,
$$
\align
\int d\ha\mu_{\L_i,,h} (\zeta) \P(\tau_i^{\zeta} \leq
t_{i+1})
& \leq
C_2 \exp\(C_3 e^{\de / h}\) +
C_4 \left\L\(\frac{b_2}{h}W\)\right\, e^{\de/h} \, \ha\varphi
\cr & \leq
C_5 \exp(C_6 / h),
\teq(tauha)
\endalign
$$
for small $h$.\corr{ !!} (We remind the reader that
$t_{i+1}=t_i+e^{\de/h}$.)
Before we can proceed with the adaptation of the proof of Proposition
\ip.1 to prove Proposition \ip.2, we need to derive an analogue to
\equ(RN). We will show that for
small enough $h$, for
$i=1,...,N$,
$$
\sup_{\zeta \in \Omega_{\L_{i1},}}
\frac{d\ha\mu_{\L_{i1},,h}}{d\ha\mu_{\L_{i},,h}} (\zeta)
\leq \frac{2}{\alpha}.
\Eq(RNha)
$$
To this end,
as in the argumentation for \equ(RN), we will use the
notation $x = \L_i \backslash \L_{i1}$, and
$\CSx = \{\s : \s(x) = 1 \}$.
First note now that by partitioning
$(\Fxn)^c \cap \Cal{R}^c$ according to what
the configuration in $\L_{i1}$ is and denoting by $\{E_j\}$ the
resulting parts, we have
$$
\align
\ha\mu_{\L_{i},,h}
(\CSx(\Fxn)^c)
& =
\sum_{j}
\ha\mu_{\L_{i},,h}
(E_j(\Fxn)^c) \,
\ha\mu_{\L_{i},,h}
(\CSxE_j)
\cr
& =
\sum_{j}
\ha\mu_{\L_{i},,h}
(E_j(\Fxn)^c) \,
\mu_{\L_{i},,h}
(\CSxE_j)
\cr
& \geq
\sum_{j}
\ha\mu_{\L_{i},,h}
(E_j(\Fxn)^c) \,
\alpha
=
\alpha.
\endalign
$$
Therefore, using \equ(muFxp),
$$
\multline
\ha\mu_{\L_{i},,h} (\CSx)
\geq
\ha\mu_{\L_{i},,h} ((\Fxn)^c)
\
\ha\mu_{\L_{i},,h} (\CSx(\Fxn)^c)
\\ \geq
\( 1  \ha\mu_{\L_{i},,h} (\Fxp) \)
\ha\mu_{\L_{i},,h} (\CSx(\Fxn)^c)
\geq
\frac{\alpha}{2},
\endmultline
$$
for small enough $h$, uniformly in $i$.\corr{
Finally, if $\zeta(x) = 1$,
$$
\multline
\ha\mu_{\L_{i},,h} (\zeta)
\geq
\ha\mu_{\L_{i},,h} (\CSx)
\
\ha\mu_{\L_{i},,h} (\zeta  \CSx)
\\
=
\ha\mu_{\L_{i},,h} (\CSx)
\
\ha\mu_{\L_{i1},,h} (\zeta)
\geq \frac{\alpha}{2}
\
\ha\mu_{\L_{i1},,h} (\zeta),
\endmultline
$$}
Since $\zeta(x) = 1$, we have
$$
\multline
\ha\mu_{\L_{i},,h} (\zeta)
=
\ha\mu_{\L_{i},,h} (\CSx)
\
\ha\mu_{\L_{i},,h} (\zeta  \CSx)
\\
=
\ha\mu_{\L_{i},,h} (\CSx)
\
\ha\mu_{\L_{i1},,h} (\zeta)
\geq \frac{\alpha}{2}
\
\ha\mu_{\L_{i1},,h} (\zeta),
\endmultline
$$
completing the proof of \equ(RNha).
We are now ready to explain how the proof of Proposition \ip.1
can be modified to prove Proposition \ip.2.
In place of \equ(goal1),\corr{ it is enough} we have to prove
the analogous statement:
$$
\int d\ha\mu_{\L_{i1},,h} (\zeta) \P((G^i_{\zeta})^c)
\leq C_1 \exp(C_2 / h).
\Eq(goal1ha)
$$
For this we use \equ(tousetauha), \equ(tauha),
the stationarity of
$(\ha\sigma^{t_{i1}, \ha\mu_{\L_{i1},,h}}
_{\IP,,h;t})_{t_{i1} \leq t \leq t_{i}}$,
and \equ(RNha) to\corr{ write} obtain the following replacement of \equ(useRN)
$$
\align
& \int d\ha\mu_{\L_{i1},,h} (\zeta) \P\left((G^i_{\zeta})^c\right)
=
\int d\ha\mu_{\L_{i1},,h} (\zeta) \P\left(
\s^{t_{i1},\zeta}_{\IP,,h;t_{i+1}}
\neq
\s^{t_{i},+}_{\IP,,h;t_{i+1}}\right) \cr
& \leq
\int d\ha\mu_{\L_{i1},,h} (\zeta) \P\left(
\s^{t_{i},\zeta}_{\IP,,h;t_{i+1}}
\neq
\s^{t_{i},+}_{\IP,,h;t_{i+1}}\right)
+
\int d\ha\mu_{\L_{i1},,h} (\zeta)
\P(\tau^{\zeta}_{i1} \leq
t_i) \cr
& \leq
\frac{2}{\alpha}
\int d\ha\mu_{\L_{i},,h} (\zeta) \P\left(
\s^{t_{i},\zeta}_{\IP,,h;t_{i+1}}
\neq
\s^{t_{i},+}_{\IP,,h;t_{i+1}}\right)
+
C_5 \exp(C_6/h).
\teq(useRNha)
\endalign
$$
We can show that \equ(useRNha) leads to \equ(goal1ha) by adapting
the steps used to prove that \equ(useRN) leads to \equ(goal1). The
following are the changes in the argument.
This time we take
$$
\gather
\Lc = \L \(\frac{2B_c + b_1}{3h}W \), \\
\L^{\text{in}} =
\L\(\frac{B_c + 2 b_1}{3h}W\),
\endgather
$$
and
$$
\L_i^{\text{out}} =
\L_i \backslash \L^{\text{in}}.
$$
Similarly to the derivations of
\equ(preB) and \equ(useRNha),
we can use the basiccoupling inequalities,
the spectral gap estimates in Propositions \ga.1 and \ga.2,
\equ(tousetauha), \equ(tauha), and
the stationarity of
$(\ha\sigma^{t_i, \ha\mu_{\L_i,,h}}
_{\IP,,h;t})_{t_i \leq t \leq t_{i+1}}$
to derive
$$
\align
& \int d\ha\mu_{\L_{i},,h} (\zeta) \P\left(
\s^{t_{i},\zeta}_{\IP,,h;t_{i+1}}
\neq
\s^{t_{i},+}_{\IP,,h;t_{i+1}}\right) \cr
& \leq \int d\ha\mu_{\L_{i},,h} (\zeta)\sum_{y \in \L_i}
\left\{
\P \left(
\s^{t_{i},+}_{\IP,,h;t_{i+1}}
(y) = +1\right)
 \P \left(
\s^{t_{i},\zeta}_{\IP,,h;t_{i+1}}
(y) = +1\right) \right\} \cr
& \leq
\sum_{y \in \L^{\text{in}}}
\left\{
\P \left(\s^{+}_{\L_i,+,h;\exp(\delta/h)}(y) = +1\right)

\ha\mu_{\L_i,,h}\left(\CSy+\right) \right\} \cr
& +
\sum_{y \in \L_i^{\text{out}}}
\left\{
\P \left(\s^{+}_{\L_i \backslash \Lc,
(+,),h;\exp(\delta/h)}(y) = +1\right)

\ha\mu_{\L_i,,h}\left(\CSy+\right) \right\}
\cr
& +
\L_i \ \int d\ha\mu_{\L_{i},,h} (\zeta)
\P(\tau^{\zeta}_{i} \leq
t_{i+1}) \cr
& \leq
\sum_{y \in \L^{\text{in}}}
\left\{
\mu_{\L_i,+,h}\left(\CSy+\right)

\ha\mu_{\L_i,,h}\left(\CSy+\right) \right\} \cr
& +
\sum_{y \in \L_i^{\text{out}}}
\left\{
\mu_{\L_i \backslash \Lc,
(+,),h}\left(\CSy+\right)

\ha\mu_{\L_i,,h}\left(\CSy+\right)
\right\} \cr
&
+ C_7 \left(\frac{b_2}{h}\right)^2 \left\{
C_5 \exp(C_6/h)
\ + \ \exp\( C_8 \left(\frac{b_2}{h}\right)^2 \)
\exp \{ e^{ \frac{\delta}{2h}} \}
\right\}.
\teq(preBha)
\endalign
$$
The definitions of $\Cal{B}_1$ and $\Cal{B}_2$ are the same as before
(see \equ(defB1) and \equ(defB2)),
but with the modified choices above of $\Lc$ and
$\L^{\text{in}}$. From Lemma \dwsp \
we know that for $j=1,2$,
$$
\sup_{i=0,...,N} \ha\mu_{\L_i,,h} (\Cal{B}_j)
\leq C_{9} \exp(C_{10}/h),
\Eq(Bexpha)
$$
Due to the conditioning in the definition of $\ha\mu_{\L_i,,h}$,
the derivation of the analogues of \equ(boundB1) and \equ(boundB2)
are somewhat more delicate. For $y \in \L^{\text{in}}$,
we let $\{E_j\}$\corr{ partition} denote the partition of
$(\Cal{B}_1)^c$ according to what the ($$,*)cluster of
$\(\L\(\frac{b_1}{h}W\)\)^c$ is. We obtain the following:
$$
\align
\ha\mu_{\L_i,,h} \left(\CSy+  (\Cal{B}_1)^c \right)
& =
\sum_{j} \alpha_j \
\ha\mu_{\L_i,,h} \left(\CSy+  E_j \right) \cr
& =
\sum_{j} \alpha_j \
\mu_{\L_i,,h} \left(\CSy+  E_j \right) \cr
& \geq
\mu_{\L_i,+,h} \left(\CSy+ \right)
\endalign
$$
where in the second equality we used the fact that for each $j$,
$E_j \subset \Cal{R}^c$, and in the final inequality we used the
same standard argument which gives rise to \equ(fsw) and the fact that
$\sum_j \alpha_j =1$.
From \equ(sw) it follows then that
$$
\mu_{\L_i,+,h}\left(\CSy+\right)

\ha\mu_{\L_i,,h}\left(\CSy+\right)
\leq
2 \ha\mu_{\L_i,,h}\left( \Cal{B}_1 \right).
\Eq(boundB1ha)
$$
Similarly we can derive, for each $y \in \L_i^{\text{out}}$,
$$
\mu_{\L_i \backslash \Lc,
(+,),h}\left(\CSy+\right)

\ha\mu_{\L_i,,h}\left(\CSy+\right)
\leq
2 \ha\mu_{\L_i,,h}\left( \Cal{B}_2 \right).
\Eq(boundB2ha)
$$
Our goal, \equ(goal1ha), follows from combining
\equ(useRNha), \equ(preBha), \equ(Bexpha), \equ(boundB1ha)
and \equ(boundB2ha).
\cqd
\enddemo
%\bye
\bigskip
%\input re
\subheading{\re. Rescaling and droplet creation}
The inverted pyramid $\IP = \IP(b_1,b_2;h;\de)$ and the event $G_+$
were conceived having in mind their use in a rescaling procedure.
To each point $k= (k_1,k_2,k_3)$
of the rescaled space time $\Z^2 \times \Z_+$ we
associate the following translate of $\IP$:
$$
\IP_k =
\(\frac{b'}{h}k_1, \frac{b'}{h}k_2, t_N k_3 \) + \IP,
$$
where as before $N$ is 1 less than the
number of elements in the sequence $\SEQ(b_1,b_2;h)$,
$t_N = N \exp(\de/h)$ and
$b'>0$ is a new parameter. We suppose that $b_1$, $b_2$ and $b'$
are such that
$$
\(\L\left(\frac{b_1}{h}W\right) + \left(\frac{b'}{h},0\right) \) \
\bigcap \
\L\left(\frac{b_1}{h}W\right)
= \emptyset,
\Eq(fit1)
$$
and
$$
\L\left(\frac{b_1}{h}W\right) + \left(\frac{b'}{h},0\right)
\ \subset \
\L\left(\frac{b_2}{h}W\right).
\Eq(fit2)
$$
It will be important that this can be done for arbitrarily large
values of $b_1$. Indeed, with $b_1$ given we can choose $b'$ for
\equ(fit1) to hold, and then choose $b_2$ for \equ(fit2) to hold.
Of course, we will sometimes
confuse inverted pyramids with their indices in the
terminology being introduced below, and no inconvenience should
arise from this.
We say that two inverted pyramids $\IP_k$ and $\IP_{k'}$ are neighbors
in case $k_3  k'_3 = 1$ and $(k_1,k_2)(k'_1,k'_2) \in
\{(0,0), (0,1), (1,0), (0,1),(1,0)\}$.
Note that $\IP = \IP_{(0,0,0)}$ has exactly 5 neighbors, all
at rescaled time 1. The bottom cylinders of these 5 inverted
pyramids will be pairwise disjoint and each will be contained
in the top cylinder of the inverted pyramid $\IP_{(0,0,0)}$.
A rescaledspacetime oriented chain will be a sequence $(k^{(1)}, ... ,
k^{(n)})$ of elements of $\Z^2 \times \Z^+$ such that
$k^{(i)}$ and $k^{(i+1)}$ are neighbors and
$k_3^{(i+1)}=k_3^{(i)}+1$, for $i=1, ..., n1$.
The start of the chain is\corr{ !!} $k^{(1)}$, and its end is $k^{(n)}$.
We will say that the inverted pyramid $\IP_k$ is open if the
corresponding event $G_+$ happens for it.
This event will be denoted by $G_{+,k}$.
More formally, $G_{+,k}$ is the set of realizations of the
graphical construction which would be in $G_+$ after space and
time were translated by the amount
$(\frac{b'}{h}k_1, \frac{b'}{h}k_2, t_N k_3)$.
The events $G_{+,k}$ have the
very nice property that they are well suited for being concatenated.
The simplest version of this idea is the following.
Suppose that $(k^{(1)}, ... , k^{(n)})$ is a
rescaledspacetime oriented chain.
We will be concerned with the process
$(\sigma^{{\bold \cdot}}_{\cup_{i} \IP_{k^{(i)}},  , h;t})$.
To fix some notation, say
that the bottom cylinder of the bottom inverted pyramid,
$\IP_{k^{(1)}}$, is
the set $\L^{\text{bot}} \times [s_0,s_1]$ and that the
top cylinder of the top inverted pyramid, $\IP_{k^{(n)}}$, is
the set $\L^{\text{top}} \times [s_3,s_4]$.
Standard applications of the basiccoupling inequalities yield
$$
\cap_{i=1,...,n} G_{+,k^{(i)}}
\ \subset \
\left\{
\s^{s_0,+}_{\cup_{i} \IP_{k^{(i)}}, ,h;s_{4}}
=
\s^{s_3,+}_{\cup_{i} \IP_{k^{(i)}}, ,h;s_{4}}
\right\}.
%\Eq(concat+)
$$
Pictorially this amounts to a droplet of the (+)phase flowing
through the open tube $\cup_{i} \IP_{k^{(i)}}$. This relation is
not yet strong enough for our purposes, because it requires a
solid blob of (+) spins at the bottom, and we will not have such
a solid droplet of (+)'s. Nevertheless the following stronger
version of this relation can be derived via the same standard use
of the basiccoupling inequalities. For an arbitrary $\eta \in
\Omega_{\L^{\text{bot}},}$
$$
\align
\left\{
\s^{s_0,\eta}_{\IP_{k^{(1)}}, ,h;s_{1}}
=
\s^{s_0,+}_{\IP_{k^{(1)}}, ,h;s_{1}} \right\}
& \bigcap \
\cap_{i=1,...,n} G_{+,k^{(i)}} \cr
& \ \subset \
\left\{
\s^{s_0,\eta}_{\cup_{i} \IP_{k^{(i)}}, ,h;s_{4}}
=
\s^{s_3,+}_{\cup_{i} \IP_{k^{(i)}}, ,h;s_{4}}
\right\}.
\teq(concat)
\endalign
$$
We will now construct a spacetime structure motivated by the
cone which appeared in the heuristics in Section \he.
Our goal is to
look at the system at time $\exp(\lambda/h)$, with
$\lambda_c \leq \lambda \leq 2\lambda_c$, and prove \equ(goalPP).
Set
$$
M = \lfloor \exp(\lambda/h)/t_N \rfloor.
$$
Moving backwards in time we choose now a set $\CCM$
of inverted pyramids
$\IP_k$. From the inverted pyramids with rescaled time coordinate
$M1$ we take only $\IP_{(0,0,M1)}$.
Inductively, once we have selected the inverted pyramids at a certain
rescaled time $m > 0$, we take at rescaled time $m1$ the inverted
pyramids which are neighbors to at least one inverted pyramid already
included in our set. The procedure stops at rescaled time 0.
Note that if we consider the indices of the selected inverted pyramids,
what we have done is precisely to construct\corr{ in the discrete
rescaled spacetime a cone,} the discrete
rescaled analog of the spacetime cone, as in the heuristics. An alternative
definition of $\CCM$ is that it is the set of inverted pyramids
from which we can start a rescaled spacetime oriented chain which
ends at $\IP_{(0,0,M1)}$.
The cardinality of the set $\CCM$ clearly satisfies
$$
C_1 M^3 \leq \CCM \leq C_2 M^3.
$$
On the other hand, the bounds on $\lambda$ and the fact that,
due to \equ(N),
$$
t_N \leq C \left( \frac{ b_2}{h} \right)^2 \exp\left(\frac{\de}{h}\right),
$$
give us, for small $h$,
$$
C' \left( \frac{h}{ b_2} \right)^2 \exp\left(\frac{\lambda\de}{h}\right)
\leq
M
\leq \exp\left(\frac{2\lambda_c}{h}\right).
$$
Therefore
$$
C_3 \left( \frac{h}{ b_2} \right)^6 \exp\left(\frac{3(\lambda\de)}{h}\right)
\leq
\CCM
\leq C_4\exp\left(\frac{6\lambda_c}{h}\right).
$$\corr{
If we make the choice} Let us choose $ \delta $ as
$$
\de = \frac{\lambda  \lambda_c}{2},
\Eq(de)
$$
which means that $\lambda  \de = \lambda_c + \de$.
Then, since $\lambda_c = \beta A / 3$, we have\corr{ !!}
$$
C_3 \left( \frac{h}{ b_2} \right)^6 \exp\left(\frac{3\de}{h}\right)
\exp\left(\frac{\beta A}{h}\right)
\leq
\CCM
\leq C_4\exp\left(\frac{6 \lambda_c }{h}\right).
\Eq(boundCCM)
$$
The lower bound in \equ(boundCCM) is central to our analysis,
and we will return to it later, when we discuss droplet creation.
The upper bound in \equ(boundCCM)
is of technical relevance in connection to droplet
growth, because we will want to have
all the events $G_{+,k}$, $k\in \CCM$ happening.
At this point recall that our goal is to prove \equ(goalPP), in
which an arbitrarily large constant $C$ is involved.
It is clear
from Proposition \ip.1 and the upper bound in \equ(boundCCM) that
given a value for the constant $C$ in \equ(goalPP)
we can choose $b_1$ so large that for some finite $C_1$
$$
\P(\cup_{k\in \CCM} (G_{+,k})^c) \leq C_1 \exp(C/h),
\Eq(allopen)
$$
for all $h>0$.
Define the spacetime region
$$
\De = \left( \bigcup_{k\in \CCM}\IP_k \right) \ \bigcup \
\sq,
$$
where
$$
\sq =
\L\left(\frac{b_2}{h}W\right)
\times \left[ M t_N, \exp\left(\frac{\lambda}{h}\right) \right].
$$
We can think of this spacetime region $\De$
as playing the role of the cone
in the heuristics.\corr{ Informally speaking, \equ(allopen) assures a large
probability for any supercritical droplet of size $(b_1/h)W$ formed
at the bottom of one of the $\IP_k$, $k\in \CCM$, to reach the top
of the uppermost inverted pyramid in $\CCM$, $\IP_{(0,0,M1)}$.
The extra cylinder $\sq$ at the top of
$\De$ allows this droplet to survive until time $\exp(\lambda/h)$.}
Informally speaking, \equ(allopen) assures that once a supercritical
droplet of size $(b_1/h)W$ is born at the bottom of some $\IP_k$, $k\in \CCM$,
then with large
probability it will reach the top
of the uppermost inverted pyramid, $\IP_{(0,0,M1)}$, of $\CCM$. If it
survives also through the top cylinder $\sq$ of $\De$, it will reach
the time $\exp(\lambda/h)$.
A further condition on $b_2$ is required to assure us that the cylinder
at the top of $\De$ is wide enough so that the aimed
conclusion \equ(goalPP)
will be proven with an arbitrarily large constant $C$.
Technically this condition is the following.
Given the constant $C$ in \equ(goalPP), we will need
to take $b_2$ large enough for there to
exist a finite $C_1$ so that
$$
\langle f \rangle_{\L(\frac{b_2}{h}W),,h}
\geq
\langle f \rangle_{h}
 C_1 \exp(C/h),
\Eq(bigb2)
$$
for all $h>0$\corr{ !!!} small enough.
That such a choice is possible is heuristically reasonable, since in
the doublewell picture of the Gibbs distribution
$\mu_{\L(\frac{b_2}{h}W),,h }$\corr{ , when $b_2$ is large} with $b_2$ large, the
mass\corr{ concentrates inside of a well which corresponds to the
(+)phase and can be made very deep by choosing $b_2$ large.}
is concentrated in the well corresponding to the
(+)phase, which can be made very deep by choosing $b_2$ appropriately large.
From the rigorous viewpoint, aside from the claim that $C$
can be arbitrarily large, a proof of \equ(bigb2) can be found
in [SS1] (see the proof of Theorem 1.b.3 there). A complete
proof of \equ(bigb2) is obtained by combining Lemma \dwsp \ with
\equ(psw) and the FKGHolley inequalities.
Observe that our choices are made in the following order.
Suppose $\lambda$ and $C$\corr{ given.} are given. First
\equ(de) gives us the value of $\de$. Then \equ(allopen) gives us
the value of $b_1$. Afterwards, \equ(fit1) gives us $b'$, and
finally \equ(fit2) and \equ(bigb2) give us $b_2$.
So far we have developed mathematically rigorous counterparts to the
notion that if
a supercritical droplet is created close to the bottom of any
of the inverted pyramids $\IP_k$, $k\in \CCM$ it is likely to
grow and bring the (+)phase
to the neighborhood of the origin at time $\exp(\lambda/h)$. What
we still need is to make mathematical
sense of the creation of such a supercritical
droplet as occurring at a rate predicted by the heuristics.
The lower bound in \equ(boundCCM) tells us that if we could
say that close to the bottom of each one of these inverted pyramids,
and independently of what happens close to the bottom of the
other inverted pyramids,
there is probability of the order of $\exp(\beta \frac{A}{h})$ of
creating such a droplet, then we would be done.
This would be akin to
saying that the rate of creation of
supercritical droplets is $\exp(\beta \frac{A}{h})$, as we expect.
Motivated by \equ(concat) we will say that supercritical
droplet creation occurs in the bottom of the inverted
pyramid $\IP_k$, which has $\L^{\text{bot}} \times [s_0,s_1]$
as its bottom cylinder, in case the following event\corr{ happens.} happens:
$$
\FF_k \ = \
\left\{
\s^{s_0,}_{\IP_{k}, ,h;s_{1}}
=
\s^{s_0,+}_{\IP_{k}, ,h;s_{1}} \right\}.
$$
The events $\FF_k$, $k \in \CCM$ are clearly mutually independent,
since they are determined by the graphical construction marks in
disjoint regions of spacetime.
The final part of this section will be concerned with proving
that for each $k$ for small $h>0$
$$
\P(F_k) = \P(F_0) \geq
\frac{1}{2} \exp\left( \frac{\beta A}{h} \right).
\Eq(rigorousrate)
$$
If for the moment we suppose that this is known, we can
complete the proof of \equ(goalPP) as follows.
Consider the event
$$
\EE =
\(\cap_{k\in \CCM} G_{+,k}\)
\ \bigcap \
\(\cup_{k\in \CCM} \FF_{k} \).
$$
From \equ(allopen), the lower bound in \equ(boundCCM),
and \equ(rigorousrate)
we have for small $h$
$$
\P(\EE^c)
\leq C_1 \exp(C/h) + \(1 
\frac{1}{2} \exp\left( \frac{\beta A}{h} \right)
\)^{\CCM}
\leq 2 C_1 \exp(C/h).
$$
Now, using basiccoupling inequalities, \equ(concat)
and \equ(bigb2) we obtain
$$
\align
\E\(f\(\s^{\nu}_{h;\exp(\lambda/h)}\)\) & \geq
\E\(f\(\s^{0,}_{\De,,h;\exp(\lambda/h)}\)\)
\\ & \geq
\E\(f\(\s^{0,}_{\De,,h;\exp(\lambda/h)}\)\ ; \ \EE\)

\P(\EE^c)
\\ & =
\E\(f\(\s^{Mt_N,+}_{\sq,,h;\exp(\lambda/h)}\) \ ; \ \EE\)

\P(\EE^c)
\\ & \geq
\E\(f\(\s^{Mt_N,+}_{\sq,,h;\exp(\lambda/h)}\)\)

2 \P(\EE^c)
\\ & \geq
\langle f \rangle_{\L(\frac{b_2}{h}W),,h}
 4 C_1 \exp(C/h)
\\ & \geq
\langle f \rangle_h  C'_1 \exp(C/h).
\endalign
$$
All that remains to be done in this section is to
show \equ(rigorousrate).
For this purpose we will insert another inverted
pyramid inside the bottom cylinder of
each one of our inverted pyramids
$\IP_k$, $k\in \CCM$.
To distinguish the new inverted pyramids
from the ones that
we have been discussing so far (parametrized by $b_1$, $b_2$, $\de$ and
$h$), we will call the old ones ``growth inverted pyramids'' and the
ones which we are introducing now ``creation inverted pyramids''.
The creation inverted pyramid which is inserted inside the bottom
cylinder of $\IP = \IP_0$ is described next.
It will be of the form
$$
\IPc = \bigcup_{i=0}^{\Nc} \
[u_i,u_{i+1}] \times \Lcr_i,
$$
with the following features. As with the growth inverted pyramids,
for \newline $i = 0, ...,\Nc1$,
the set $\Lcr_{i+1}$ will be obtained from $\Lcr_i$ by adding one site to it.
In particular we will have $\Lcr_0 \subset \Lcr_1 \subset ...
\subset \Lcr_N$.
These sets $\Lcr_i$ will all be $l_0$quasiWulffshaped.
We will take $\Lcr_0 = \L\(\frac{b_0}{h}W\)$, where
$b_0 \in (B_c, b_1)$ is a new parameter. At the other end,
$\Lcr_{\Nc} = \L\(\frac{b_1}{h}W\)$, where $b_1$ is the same
one used for the growth inverted pyramids. The bottom
cylinder of $\IPc$ will have height $u_1  u_0 =
\exp\(\frac{\de}{2h}\)$, while all its other cylinders will have
height $u_{i+1}  u_{i} = \exp\(\frac{\de'}{h}\)$, for $i=1,...,\Nc$,
where $\de' < \de$ is also a new parameter.
With the parameters
$b_1$, $\de$, $b_0$ and $\de'$ fixed and satisfying the
conditions above, it is clear that for
small enough $h>0$, there is an inverted pyramid
as described above which fits inside the bottom cylinder of
$\IP$, and has its top $\Lcr_{\Nc} \times u_{\Nc + 1}$ coinciding
with the top of the bottom cylinder of $\IP$, i.e., such that
$u_{\Nc + 1} = \exp(\de/h)$. This is the inverted pyramid that we
will denote by $\IPc$.
The basiccoupling inequalities imply that
$$
\left\{
\s^{u_0,}_{\IPc,,h;u_{\Nc+1}}
=
\s^{u_{\Nc},+}_{\IPc,,h;u_{\Nc+1}}
\right\}
\ \subset \
\left\{
\s^{0,}_{\IP, ,h;\exp(\de/h)}
=
\s^{0,+}_{\IP, ,h;\exp(\de/h)} \right\}.
$$
Therefore the next lemma implies \equ(rigorousrate).
\proclaim{Lemma \PP.2}\corr{ 3.3.1 !!}
Given $b_1 > B_c$ and $\de>0$ there are
$b_0 \in (B_c, b_1)$ and $\de' > 0$
such that for small $h>0$
$$
\P\left(
\s^{{u_0},}_{\IPc,,h;u_{\Nc+1}}
=
\s^{u_{\Nc},+}_{\IPc,,h;u_{\Nc+1}}
\right)
\geq
\frac{1}{2} \exp\left( \frac{\beta A}{h} \right).
$$
\endproclaim
\demo{Proof (modulo results in Sections \dw \ and \ga)}
Before starting the rigorous proof we will motivate it heuristically.
Consider first the bottom cylinder of $\IPc$.
Intuitively, when $b_0$ is close to $B_c$,
we can see the system in the box $\L\(\frac{b_0}{h}W\)$
as having a double well structure with the deeper well corresponding to
the ($$)phase, and the higher well
corresponding to the presence of a supercritical droplet of the
(+)phase.
The barrier between these
wells is given by the configurations with a critical droplet.
Note that if we were in equilibrium at time $u_1$ inside the box
$\L\(\frac{b_0}{h}W\)$
with ($$) boundary conditions, then the quantity
$\frac{1}{2} \exp\left( \frac{\beta A}{h} \right)$
would indeed be a lower bound on the probability of
being in the higher well and hence having a supercritical droplet.
We would therefore like to say that inside this bottom cylinder the
system started at time $u_0$
with all spins down should reach equilibrium at
the top of this cylinder, i.e., at time $u_1 =
u_0 + \exp\(\frac{\de}{2h}\)$.
In other words, we would like to say that the relaxation time for
the process $\(\s^{{\bold \cdot}}_{\L\(\frac{b_0}{h}W\),,h;t}\)$ is shorter than
$\exp\(\frac{\de}{2h}\)$.
For this to be true it should be
enough to take $b_0$ close enough to $B_c$,
so that from the higher well there is a very small barrier to overcome
to reach equilibrium.\corr{ The} This freeenergy barrier can be made smaller
than $\frac{\de}{2h}$, so\corr{ that} the available time should be enough
to equilibrate the system.
The heuristics in the last paragraph can be made rigorous by considering
the spectral gap of the generator of the process. This will be done in
Section \ga. From Proposition
\ga.3 we have that if $b_0$ is chosen close enough to $B_c$, then
$$
\gap \(\L\(\frac{b_0}{h}W\),,h\)
\geq \exp\left(\frac{\delta}{4h}\right),
\Eq(gapb0)
$$
for small enough $h$.
We start now the rigorous proof of the lemma. First we break things
down according to what happens at time
$u_1 = u_0 + \exp\(\frac{\de}{2h}\)$.
$$
\multline
%\align
\P\left(
\s^{u_0,}_{\IPc,,h;u_{\Nc+1}}
=
\s^{u_{\Nc},+}_{\IPc,,h;u_{\Nc+1}}
\right)
%& =
= \\
\sum_{\zeta} \P\(
\s^{u_0,}_{\IPc,,h;u_{1}} = \zeta \)
\P\(
\s^{u_1,\zeta}_{\IPc,,h;u_{\Nc+1}}
=
\s^{u_{\Nc},+}_{\IPc,,h;u_{\Nc+1}} \).
\endmultline
$$
Combining \equ(gapb0) with the standard inequality (59)
in [Sch1], we have
$$
\multline
%\align
\left \sum_{\zeta} \P\(
\s^{u_0,}_{\IPc,,h;u_{1}} = \zeta \)
\P\(
\s^{u_1,\zeta}_{\IPc,,h;u_{\Nc+1}}
=
\s^{u_{\Nc},+}_{\IPc,,h;u_{\Nc+1}} \)
\right.  \\
\left.
\sum_{\zeta} \mu_{\L\(\frac{b_0}{h}W\),,h} \( \zeta \)
\P\(\s^{u_1,\zeta}_{\IPc,,h;u_{\Nc+1}}
=
\s^{u_{\Nc},+}_{\IPc,,h;u_{\Nc+1}} \)
\right \\
\leq
\frac{e^{\exp(\frac{\de}{2h})
\gap\(\L\(\frac{b_0}{h}W\),,h\)}}
{\mu_{\L\(\frac{b_0}{h}W\),,h}()}
\leq
\exp \( C \left(\frac{b_0}{h}\right)^2 \) \
\exp \{ e^{\frac{\delta}{4h}} \}.
\endmultline
$$
But using Lemma \dwrc \
and Proposition \ip.2, we can take $\de'$ small enough so that
$$
\multline
\sum_{\zeta} \mu_{\L\(\frac{b_0}{h}W\),,h} \( \zeta \)
\P\(\s^{u_1,\zeta}_{\IPc,,h;u_{\Nc+1}}
=
\s^{u_{\Nc},+}_{\IPc,,h;u_{\Nc+1}} \) \\
\geq
\mu_{\L\(\frac{b_0}{h}W\),,h} \( \Cal{R}^c\)
\sum_{\zeta \in \Cal{R}^c}
\ha\mu_{\L\(\frac{b_0}{h}\),,h} \( \zeta \)
\P\(\s^{u_1,\zeta}_{\IPc,,h;u_{\Nc+1}}
=
\s^{u_{\Nc},+}_{\IPc,,h;u_{\Nc+1}} \) \\
\geq
\(\exp\left( \frac{\beta A}{h} \right)\) \cdot
\frac{3}{4},
\endmultline
$$
for small $h$.
The three displays above combined give us the lemma.
\cqd
\enddemo
At this point the proof of part (ii) of Theorem 1 has been reduced
to proving the claims in Sections \dw \ and \ga.
%\bye
\bigskip
%\input dw
%@@ RENUMBER C_... in last lemma of dw.
%% END OF PART 3
%% BEGINNING OF PART 4
\subheading{\dw. Double well structure of equilibrium distributions}
In this section we will study some of the features of the
Gibbs distributions on finite simplyconnected sets
with ($$)boundary conditions.
We will extend
and strengthen results contained in Theorem 1 of [SS1];
the purpose being the use of these stronger results in
several other sections of the current paper.
While the main results that we will derive in this
section and use in other ones
could be derived using the same approach as in [SS1],
we will nevertheless introduce an alternative method for proving
them, based on results and techniques from
Section \CCC \ of this
paper. Basically, in [SS1] we started from results on large
deviations\corr{ on} of the average spin (i.e., the magnetization) inside
of a box, under no external field. We could see the external field
then as tilting the distribution. Such an approach is appealing
and even natural, but since in the current paper our interest lies
primarily on contours and not on the magnetization, it seemed
even more natural to search for a direct approach to the problems, in
which the magnetization need not to be mentioned.
Having in mind that we are interested in the contours of
configurations in $\Omega_{}$,
it is natural to regard $\Cal{W}$ and $\bV$, defined
in\corr{ Section \CCC \ } Section \CCC ,
as random variables.
Given a configuration $\eta \in \Omega_{}$, the associated values
of these random variables are, respectively, $\Cal{W}(S)$ and
$\bV(S)$, where $S$ is the collection of skeletons corresponding
to the external contours of $\eta$.
Basically we will show that in a sense the function
$$
\phi(b) = wb  m^*b^2 \qquad b \geq 0,
$$
plays the role of a largedeviation rate function for the random
variable $\bV$. The title of this section derives from the shape
of this function. The reader will realize that in the lemmas below
the results are stated and proven with a certain amount of uniformity
over the allowed sets $\L$; this uniformity is needed in some of
our applications of the lemmas, since for instance in Section \ip \
we need uniformity over all the sets which are bases of cylinders
of inverted pyramids.
Recall\corr{ also} that $B_c$ is the value of $b$ which maximizes $\phi$,
with $\phi(B_c) = A$,
while $B_0 = 2B_c$ is the value of $b$ above which this function
becomes negative.
Recall also that given a finite set of vertebrate
contours $G$ we denote by $\SG$ the set of configurations
which belong to $\Omega_{}$ and which have as their collection of
external vertebrate contours the set $G$.
In some of the lemmas below, we will use as a ``reference'' the
set of configurations with no external vertebrate contours,
$\Sempty$. Our next lemma shows that this is essentially the
same as using the set $\Cal{R}$, defined by \equ(defR), as a
``reference''.
We will use the notation
$o_h(1)$ to represent some function\corr{ of $h > 0$
which has} %\newline
of $h > 0$, satisfying the property
$\lim_{\hg} o_h(1) = 0$.
\proclaim{Lemma \dwr}
For any $p > 0$
there is a function $o_h(1)$ such that
for any simplyconnected set
$\Lambda \subset \Z^2$ which satisfies
$\Lambda \leq 1/h^p$,
$$
%\align
1 \leq
\frac{Z_{\Lambda,,h}\( \Cal{R} \)}
{Z_{\Lambda,,h}\(\Sempty\)}
\leq 1 + o_h(1),
%\endalign
$$
%where $o_{h}(1)$ is a function of $h$ and $p$ which vanishes as
%$h \searrow 0$.
\endproclaim
\demo{Proof}
From Lemmas \CCC.3 and \CCC.7 we have, for small enough $h$
depending on $p$ and some finite $C$ also
depending on $p$,\corr{
$$
\tilde\mu_{\Lambda,,h}\left(\(\Sempty\)^c\right) \leq
C \exp \left( \beta \frac{w}{4\sqrt{2} h^b} \right) .
$$}
$$
\tilde\mu_{\Lambda,,h}\left(\(\Sempty\)^c\right) \leq
C \exp \left( \beta \frac{w}{4\sqrt{2}
h^{ \widetilde b}} \right) .
$$
(Here we use $\widetilde b$ to denote the parameter entering the definition
of the vertebrate contour.)
The thesis follows immediately from\corr{ this.} the last estimate.
\cqd
\enddemo
\proclaim{Lemma \dwub}
For any $p > 0$
there is $h_0>0$ and
a function $o_h(1)$ such that
given $D_0 > 0$ there is a finite constant $C$ so that
for any $D>D_0$, any $b>0$,
any $0 < h \leq h_0$,
and any simplyconnected set
$\Lambda \subset \Z^2$ which satisfies
$\Lambda \leq 1/h^p$,
$$
%\align
\frac{Z_{\Lambda,,h}\(
\Cal{W} \geq \frac{D}{h} \ , \ \bV \leq \(\frac{b}{h}\)^2
\)}
{Z_{\Lambda,,h}\(\Sempty\)}
\leq
C \exp \left( \frac{\beta}{h}
(D  m^* b^2) (1 + o_h(1))
\right).
%\endalign
$$
%where $o_{h}(1)$ is a function of $h$ which vanishes as
%$h \searrow 0$.
\endproclaim
\demo{Proof}
The proof of Lemma \CCC.6 and \equ(volume'')
show that
we can choose $h_0$ and $C$ so as to have
for each collection of
skeletons $S$ such that
\corr{ $\Cal{W}(S) \geq \frac{D}{h}$ and} $\bV(S) \leq \(\frac{b}{h}\)^2$\corr{
$$
\sum_{G \in \CShelp} \frac{Z_{\L,,h}(\SG)}
{Z_{\L,,h}(\Sempty)}
\leq
C \exp \left( \frac{\beta}{h}
(D  m^* b^2) (1 + o_h(1))
\right).
$$}
$$
\sum_{G \in \CShelp} \frac{Z_{\L,,h}(\SG)}
{Z_{\L,,h}(\Sempty)}
\leq
\exp \left( \frac{\beta}{h}
(\Cal{W}(S)  m^* b^2) (1 + o_h(1))
\right).
$$
Summing then\corr{ over $S$,} over $S$ with $\Cal{W}(S) \geq \frac{D}{h}$,
using the entropy estimate in the proof
of Lemma \CCC.7,
we obtain the desired conclusion.
\cqd
\enddemo
In what follows, with $b>0$ and $0<\rho<1$ given,
$\Ebr$ will denote the event that there is an
external contour which surrounds $\frac{b(1\rho)}{h}W$
and is contained in $\frac{b(1+\rho)}{h}W$, and that
moreover this is the only external vertebrate contour.
\proclaim{Lemma \dwlb}
For any $p > 0$ and $ \gamma <1$
there are finite positive constants\corr{ $h_0$, $C_1$ and $C_2$} $h_0$ and $C_1$
and a function $o_h(1)$ such that
for any $0 < h \leq h_0$,
any $b>0$,\corr{ any $\rho>0$} any $\rho=\rho(h)>h^\gamma$ and any
simplyconnected set
$\Lambda \subset \Z^2$ which satisfies
$\Lambda \leq 1/h^p$ and contains
$\L\(\frac{b(1+\rho)}{h}W\)$,\corr{
$$
\frac{Z_{\Lambda,,h}\(
\Ebr
\)}
{Z_{\Lambda,,h}\(\Sempty\)}
\geq
C_1 \exp \left( \frac{\beta}{h} \Big( \phi(b) (1+o_{h}(1))
+ C_2 \rho b \Big)\right).
$$}
$$
\frac{Z_{\Lambda,,h}\(
\Ebr
\)}
{Z_{\Lambda,,h}\(\Sempty\)}
\geq
C_1 \exp \left( \frac{\beta}{h} \phi(b) (1+o_{h}(1))\right).
$$
In particular, $\rho$ can be a positive constant.
\endproclaim
\demo{Proof}\corr{ !!!} Without loss of generality we can suppose
that $\rho=h^\gamma$ with $ \gamma <1$.
Partition $\Ebr$ according to what the vertebrate external contours
are, as
$$
\Ebr = \bigcup_{\G \in \Gbr} \SG.
$$
By the definition\corr{ of $\Ebr$} of $\Ebr$, each $\G \in \Gbr$ is a singleton.
Exactly as in \equ(remh), we have
$$
%\align
\multline
\frac{Z_{\Lambda,,h}\( \Ebr \)}
{Z_{\Lambda,,h}\(\Sempty\)}
=
\sum_{\G \in \Gbr } \frac{Z_{\L,,h}(\SG)}
{Z_{\L,,h}(\Sempty)} \\
=
\sum_{\G \in \Gbr} \frac{Z_{\L,,0}(\SG)}
{Z_{\L,,0}(\Sempty)}
\exp \left( \frac{\beta}{2} \int_0^h \sum_{x \in \L} \left[
\langle \sigma(x) \rangle_{\L,,h'}^{G,h}

\langle \sigma(x) \rangle_{\L,,h'}^{\emptyset,h}
\right] dh' \right) .
\endmultline
%\endalign
$$
Arguments analogous to the ones which led to \equ(diffs) provide
also the following bound, in the opposite direction. For small enough
$h_0$, and some finite positive $C_2$,
if $\G \in \Gbr$ and $0 0$ and any $b > B_c$
%there is $\epsilon > 0$ such that
there are finite positive constants $h_0$ and $\epsilon$
such that for any $0 < h \leq h_0$
and any simplyconnected set
$\Lambda \subset \Z^2$ which satisfies
$\Lambda \leq 1/h^p$ and contains
$\L\(\frac{b}{h}W\)$,
$$
\mu_{\Lambda,,h}\( \Cal{R}^c \)
\geq
\exp \left( \frac{\beta}{h} A (1 \epsilon) \right).
$$
\endproclaim
\demo{Proof}
From Lemmas \dwr \ and \dwlb \
we have, for small\corr{ enough $h_0$, $\epsilon$
and $\rho$} enough $h_0$, $\epsilon$,
small constant $\rho$ and some positive finite $C_1$ and $C_2$,\corr{
$$
\align
\frac{Z_{\Lambda,,h}\( \Cal{R}^c \)}
{Z_{\Lambda,,h}\(\Cal{R}\)}
& \geq
\frac{ Z_{\Lambda,,h} \( E^h_{b(1\rho),\rho} \) }
{2 \ Z_{\Lambda,,h}\(\Sempty\)}
\\ & \geq
C_1 \exp \left( \frac{\beta}{h} \Big( \phi(b(1\rho)) (1+o_{h}(1))
+ C_2 \rho b (1\rho) \Big)\right)
\\ & \geq
2 \exp \left( \frac{\beta}{h} A (1\epsilon) \right).
\endalign
$$}
$$
\align
\frac{Z_{\Lambda,,h}\( \Cal{R}^c \)}
{Z_{\Lambda,,h}\(\Cal{R}\)}
& \geq
\frac{ Z_{\Lambda,,h} \( E^h_{b(1\rho),\rho} \) }
{2 \ Z_{\Lambda,,h}\(\Sempty\)}
\\ & \geq
C_1 \exp \left( \frac{\beta}{h} \phi(b(1\rho)) (1+o_{h}(1)) \right)
\\ & \geq
2 \exp \left( \frac{\beta}{h} A (1\epsilon) \right).
\endalign
$$
The conclusion is now immediate.
\cqd
\enddemo
\proclaim{Lemma \dwpv}
For any $p > 0$,
there are finite positive constants
$h_0$, $C_1$ and $C_2$
and a function $o_h(1)$ such that
given also $0 < b_1 < b_2$
there is a finite constant $C'_1$ so that
for any $0 < h \leq h_0$,
any $b \in [b_1,b_2]$, any\corr{ small $\epsilon > 0$  but in
Lemma 3.4.8 we need it big. The statement and the proof is changed}
$ \kappa \in (0,1)$, any integer $k$ and any
simplyconnected set
$\Lambda \subset \Z^2$ which satisfies
$\Lambda \leq 1/h^p$ and contains
$\L\(\frac{b}{h}(1 \kappa ^2/4)W\)$,
$$
%\align
\multline
C_1 \exp \left( \frac{\beta}{h} \Big(
\min _{b(1 \kappa)< \widetilde b0$ is small,
$\rho = \epsilon/2$,
then for small $h$ for all $\widetilde b>0$,
$$
E^h_{ \widetilde b(1\rho),\rho} \ \subset \ \left\{
\(\frac{ \widetilde b}{h}(1\epsilon)\)^2 \leq
\bV \leq \(\frac{ \widetilde b}{h}\)^2
%+ep \bV \leq \(\frac{b}{h}(1+\epsilon)\)^2
\right\},
$$
and also
$\L\(\frac{b(1\rho)(1+\rho)}{h}W\) \subset
%\L\(\frac{b}{h}W\) \subset \L$.
\L$.
Hence the claim follows from Lemma \dwlb.
For the second inequality,
note that from the variational result for families of curves
presented in Section 2.9 of [DKS]
we know that if
$\bV \geq \(\frac{ b}{h}\)^2$
then
$\Cal{W} \geq w \frac{b}{h}$.
We can therefore apply Lemma \dwub \ to each of the $k$ events
$\(\frac{b}{h}(1i \frac{ \kappa }{k} )\)^2 \leq
\bV \leq \(\frac{b}{h}(1(i1) \frac{ \kappa }{k} )\)^2$, $i=1,2,...,k$,
to conclude that
$$
\multline
\frac
{Z_{\Lambda,,h}\( \(\frac{b}{h}(1i \frac{ \kappa }{k} )\)^2 \leq
\bV \leq \(\frac{b}{h}(1(i1) \frac{ \kappa }{k} )\)^2\)}
%\bV \leq \(\frac{b}{h}(1+\epsilon)\)^2 \)}
{Z_{\Lambda,,h}\(\Sempty\)}
\\ \leq
\frac{Z_{\Lambda,,h}\( \Cal{W} \geq w\frac{b}{h}(1 i \frac{ \kappa }{k})
\ , \ \bV \leq \(\frac{b}{h}(1(i1) \frac{ \kappa }{k} )\)^2
%+ep \ , \ \bV \leq \(\frac{b}{h}(1+\epsilon)\)^2
\)}
{Z_{\Lambda,,h}\(\Sempty\)}
\\ \leq
C'_1 \exp \left( \frac{\beta}{h} \Big(
\phi(b (1(i1) \frac{ \kappa }{k}) (1+o_{h}(1))  C_2 \frac{ \kappa }{k}b
\Big) \right).
\endmultline
$$
\cqd
\enddemo
The previous lemma basically contains the promised characterization
of $\phi(\cdot)$ as a largedeviation rate function for the random
variable $\bV$. Informally it tells us that under appropriate conditions
on $\L$,
$$
\frac
{Z_{\Lambda,,h}\( \bV \approx
\(\frac{b}{h}\)^2 \)}
{Z_{\Lambda,,h}\(\Sempty\)}
\ \sim \
\exp \left( \frac{\beta}{h} \phi(b) \right).
$$
There is a technical difficulty in applying Lemma \dwpv, nevertheless,
and this is the motivation for the next lemma.
The issue is\corr{ that
as stated above} that,
as stated above, the lemma cannot be use to estimate
$ {Z_{\Lambda,,h}\( \bV \in (I/h)^2\)}/ {Z_{\Lambda,,h}\(\Sempty\)}$,
if $I$ is, e.g., an interval of the form $[0,b]$,
for\corr{ some $b>0$.} some $b>0$, since $C_1'= C_1'(b_1,b_2)$ can
explode as $b_1\to 0$.
\proclaim{Lemma \dwpvv}
For any $p > 0$,
there is $h_0>0$
and a function $o_h(1)$ such that
for any $0 < h \leq h_0$,
any $0 0$ and
$\rho > 0$
there are finite positive\corr{ constants $\epsilon$  we need slightly
more general statement in 3.5.1} constants $\epsilon_0$ and
$C_2$ such that
given also
%$\bar b > 0$
$0 0$. Scaling
lengths back to their original value, we have obtained the following
lower bound for the Wulff functional of any configuration in the event
with which we are concerned:
$$
\Cal{W} \geq w \frac{b}{h}
(1\epsilon) (1+G(\rho,\epsilon)).
$$
If\corr{ !!!} $\epsilon_0$ is chosen small enough
so that $(1\epsilon_0) (1+G(\rho,\epsilon_0))>1$,
and then $h_0$ is chosen small enough, based on Lemma \dwub, we
obtain\corr{ rewritten} for all $ \epsilon \le \epsilon _0$
$$
\align
& \frac{Z_{\Lambda,,h} \(
(\bEbr)^c \cap \left\{
\(\frac{b}{h}(1\epsilon)\)^2 \leq
\bV \leq \(\frac{b}{h}\)^2 \right\}
\)}
{Z_{\Lambda,,h}\(\Sempty\)} \\
& \leq
\frac{Z_{\Lambda,,h}\(
\Cal{W} \geq w \frac{b}{h}
(1\epsilon) (1+G(\rho,\epsilon))
\ , \ \bV \leq \(\frac{b}{h}\)^2
\)}
{Z_{\Lambda,,h}\(\Sempty\)} \\
& \leq
C_1 \exp \left( \frac{\beta}{h} \Big(
w b (1\epsilon) (1+G(\rho,\epsilon))(1+o_h(1)) 
m^* b^2 (1+o_h(1))
\Big) \right) \\
& \leq
C_1 \exp \left( \frac{\beta}{h} \phi(b)(1+o_h(1)) 
\frac{C'_2 b}{h} \right),
\endalign
$$
where $C'_2 = \beta w ( (1\epsilon) G(\rho,\epsilon)  \epsilon )/2$.
The result now follows from the comparison between this estimate
and the first inequality in Lemma \dwpv.
In this fashion,\corr{ with $h_0$ small enough, we can take % \newline
$C_2 = \beta w ( (1\epsilon) (1+G(\rho,\epsilon)  1)/3$.}
with both $h_0$ and $ \epsilon_0 $ small enough, we can take
$C_2 = \beta w G(\rho,0) /3$.
\cqd
\enddemo
As motivation for the next lemma, we recall that by controlling
how deeply the ($$,*) cluster of the boundary penetrates in a set
$\L$, we can obtain estimates similar to \equ(psw) for the expected
value of observables. Define
$$
\Bems = \bigcup_{b' \in (\epsilon b, b]} \
\left\{ \L\(\frac{b'\epsilon b}{h}W\) \perms
\(\L\(\frac{b'}{h}W\)\)^c \right\}.
$$
Recall that a
set $\L \subset \Z^2$ is said to be $l_0$quasiWulffshaped with
linear parameter $l$ in case it is simplyconnected and
$$
\L((ll_0)W) \ \subset \ \L \ \subset \ \L((l+l_0)W).
$$
Recall also that
$\ha \mu_{\L,,h} (\ \cdot \ )
=
\mu_{\L,,h} (\ \cdot \ \ \Cal{R}^c)$.
\proclaim{Lemma \dwsp}
For any $l_0 > 0$ and any
$\epsilon > 0$
there is $C_2 > 0$ such that
%finite constants $C_1$ and $C_2$ such that
given also
$B_c < b_1 \leq b_2$
there are positive finite constants $h_0$ and $C_1$ such that
if $0 < h \leq h_0$
then for any $l_0$quasiWulffshaped set $\L$ which has linear size
parameter $b/h$ with $b_1 \leq b \leq b_2$,
$$
\ha \mu_{\L,,h} (\Bems) \leq C_1 \exp \(\frac{C_2 (b_1B_c)}{h} \).
\Eq(goaldwsp1)
$$
In case $b_1 > B_0 $ the same holds also for the measure
$\mu_{\L,,h}$:
$$
\mu_{\L,,h} (\Bems) \leq C_1 \exp \(\frac{C_2 (b_1B_0)}{h} \).
\Eq(goaldwsp2)
$$
\endproclaim
\demo{Proof}
We start by introducing more terminology.
The event that a certain contour $\Gamma$ is present is
equivalent to the statement that
the spins in a certain set of sites $S_1(\Gamma)$
all have the same sign and
the spins in a certain other set of sites $S_2(\Gamma)$
all have the opposite sign. Exactly one of the two sets,
$S_1(\Gamma)$ or $S_2(\Gamma)$ is surrounded by the contour,
while the other is completely outsite of the contour.
The one which is surrounded by $\Gamma$ will be denoted
by $\partial_{\text{int}}\Gamma$, while the one which is
outside of $\Gamma$ will be denoted by
$\partial_{\text{ext}}\Gamma$.
%We will say that $\Gamma$ is a negativeoutside (same as
%positiveinside) contour of
%a configuration $\eta$ in case $\eta$ is identically $1$
%on $\partial_{\text{ext}}\Gamma$
%and identically $+1$ on
%$\partial_{\text{int}}\Gamma$. External contours of any
%configuration in $\Omega_{}$ are negativeoutside.
%First we will obtain an upper bound on the probability of the event
%$$
%\Peb =
%\left\{
%\L\(\frac{b(1\epsilon)}{h}W\)
%\perm
%\(\L\(\frac{b(1+\epsilon)}{h}W\)\)^c \right\}.
%$$
Let $\Deb$ be the event that for some contour $\Gamma$ which
surrounds
$\frac{b(1\epsilon/2)}{h}W$
%and is contained in $\frac{b(1+\epsilon/2)}{h}W$
all spins in
$\partial_{\text{int}}\Gamma$ are $+1$.
Our first goal is to prove that
$$
\ha \mu_{\L,,h} \(\Peb\) \leq C_1 \exp \(\frac{C_2 (b_1B_c)}{h} \).
\Eq(Pha')
$$
\corr{ I rewrote the proof, without changing it. } In order to do it let
us consider the event $\bV \geq \(\frac{b'}{h}\)^2$, which for $b' < B_c$
and $h$ small enough is bigger than $\Cal{R}^c$, that is
$\Cal{R}^c \ \subset \
\left\{ \bV \geq \(\frac{b'}{h}\)^2 \right\}$. If $b'$ is just slightly smaller
than $B_c$, then the bigger event is a good approximation to the smaller one.
So instead of considering the conditional distribution $\ha \mu_{\L,,h}$,
we begin with the distribution
$$
\bar \mu_{\L,,h} ( \ \cdot \ ) =
\mu_{\L,,h} ( \ \cdot \  \ \bV \geq (b'/h)^2).
$$
The choice of $b'$ is immaterial. The only thing we need is that $b' < B_c$ and
that it is close enough to $B_c$,
so that $\phi(b') > \phi(b_1)$. Under such a choice this value $b'$ would not
even appear in our estimates.
Elementary geometric considerations show that for small enough $\rho$,
dependent on $\epsilon$ but not on $b$ and $h$,
$$
\Omega_{\L,} \cap \Peb \subset \(\bEbr\)^c.
$$
And obviously every configuration in
$\Omega_{\L,}$
has $\bV \leq \(\frac{b}{h}\)^2 + \frac{C}{h}$, for some
finite constant $C$,\corr{ !!!} which depends on $b$.
So we can use Lemma \dweb \ to conclude that
we can choose
an appropriate $b''$ (larger than but close enough to $b$), and
small $\epsilon' > 0$, $\epsilon''$ and $h_0>0$ so as
to have
$$
\align
\bar \mu_{\L,,h} & \( \Peb \left
\bV \geq \(\frac{b}{h}(1\epsilon')\)^2
\right. \) \cr
& =
\mu_{\L,,h} \( \Peb \left
\bV \geq \(\frac{b}{h}(1\epsilon')\)^2
\right. \) \cr
& =
\mu_{\L,,h} \( \Peb \left
\bV \geq \(\frac{b''}{h}(1\epsilon'')\)^2
\right. \) \cr
& =
\mu_{\L,,h} \( \Peb \left
\(\frac{b''}{h}(1\epsilon'')\)^2
\leq
\bV
\leq
\(\frac{b''}{h}\)^2
\right. \) \cr
& \leq
\mu_{\L,,h} \( \(\bEbr\)^c \left
\(\frac{b''}{h}(1\epsilon'')\)^2
\leq
\bV
\leq
\(\frac{b''}{h}\)^2
\right. \) \cr
& \leq
\mu_{\L,,h} \( \( \bar E^h_{b'',\rho/2}\)^c \left
\(\frac{b''}{h}(1\epsilon'')\)^2
\leq
\bV
\leq
\(\frac{b''}{h}\)^2
\right. \) \cr
%\mu_{\L,,h} \( \Peb \left
%\bV \geq \(\frac{b}{h}(1\epsilon')\)^2
% \right. \)
%\\ & \leq
& \leq
C_3 \exp\(\frac{C_4 b}{h}\),
\teq(Pha1)
\endalign
$$
for some finite positive $C_3$ and $C_4$, which depend on $\rho$,
and hence on $\epsilon$, but with $C_4$ not depending
on $b$, $b_1$ and $b_2$.
Lemma \dwpv \ implies\corr{ here we need the Lemma \dwpv \
with its $ \epsilon $ quite large, so that is why I changed it}
that given $\epsilon''' >0$,
after possibly readjusting the value of $h_0$,
we will also have
$$
\align
\bar \mu_{\L,,h} \(
\bV \leq \(\frac{b}{h}(1\epsilon''')\)^2 \)
& \leq
C_5 \exp\(\frac{\beta}{h} \frac{\phi(b(1\epsilon'''))  \phi(b)}{2}\)
\cr & \leq
C_5 \exp\(\frac{C \epsilon''' (b_1B_c)}{h}\).
\teq(Pha2)
\endalign
$$
The last step above is based on elementary calculus, and the resulting
$C>0$ is a constant which depends only on the temperature.
From \equ(Pha1) and \equ(Pha2) (with $\epsilon' = \epsilon'''$)
we have
$$
\bar \mu_{\L,,h} \(\Peb\) \leq C_6 \exp \(\frac{C_7 (b_1B_c)}{h} \).
\Eq(Pha)
$$
To go back from $\bar \mu_{\L,,h}$ to
$\ha \mu_{\L,,h}$, observe that if we choose an appropriate $\epsilon'''$,
from \equ(Pha2) and
an application of Lemma \dweb \ similar to the one above we obtain \corr{ !!!}
$$
\bar\mu_{\L,,h}(\Cal{R}) \leq C_8 \exp \(\frac{C_9b_1}{h} \).
$$
Because\corr{ !!!}
$\Cal{R}^c \
\subset \
\left\{ \bV \geq \(\frac{b'}{h}\)^2 \right\}
$, we have
$$
\multline
\ha \mu_{\L,,h} \(\Peb\) =
\ha \mu_{\L,,h} \(\Peb\ \left \ \bV \geq \(\frac{b'}{h}\)^2 \right. \)
\\ =
\bar \mu_{\L,,h} \(\left. \Peb \ \right \ \Cal{R}^c \)
\leq
\frac{ \bar \mu_{\L,,h} \(\Peb\)}{\bar \mu_{\L,,h}(\Cal{R}^c)}.
\endmultline
$$
From the last three displays we obtain \equ(Pha')
%$$
%\ha \mu_{\L,,h} \(\Peb\) \leq 2 C_6 \exp \(\frac{C_7 b_1}{h} \).
%\Eq(Pha')
%$$
Let now $\{E_j\}$ partition
$\Deb$ according to what the outermost contour $\Gamma$ in its
definition is.
In this fashion we obtain the following:\corr{ exponent is 4}
$$
\align
\ha\mu_{\L,,h} \left( \left. \Bems \ \right \ \Deb \right)
& =
\sum_{j} \alpha_j \
\ha\mu_{\L,,h} \left(\left. \Bems \ \right \ E_j \right) \cr
& =
\sum_{j} \alpha_j \
\mu_{\L,,h} \left(\left. \Bems \ \right \ E_j \right) \cr
& \leq
%\mu_{\L,+,h} \left(\Bems \right) \cr
\mu_{\L,+,h} \left(\Cal{B}_{h,b}^{ *}(\epsilon/2)\right) \cr
& \leq
C_{10} \(\frac{b_2}{h} \)^4 \exp\(\frac{C_{11} b}{h}\) \cr
& \leq
C_{12} \exp\(\frac{C_{13} b_1}{h}\),
\teq(Bha)
\endalign
$$
where in the second equality we used the fact that for each $j$,
$E_j \subset \Cal{R}^c$, in the first inequality we used the
%same standard argument which gives rise to \equ(fsw)
the Markov property of Gibbs distributions, the FKGHolley inequalities
and the fact that
$\sum_j \alpha_j =1$, and in the second inequality we used \equ(CCS).
The first claim that we wanted to prove, \equ(goaldwsp1),
follows from \equ(Pha')
and \equ(Bha). Regarding the second claim, \equ(goaldwsp2),
which refers to the
case in which $b_1 > B_0$, note that then Lemmas \dwpv \
(lower bound), \dwpvv
\ and \dweb \ imply
$$
\multline
\mu_{\L,,h} \left(\Cal{R} \right)
\leq
C_{14} \exp \left( \frac{\beta}{h} \frac{\phi(b)}{2} \right)
+ C_1 \exp\(\frac{C_2 b}{h}\)
\\ \leq
C_{14} \exp \left( \frac{\beta}{h} \frac{\phi(b_1)}{2} \right)
+ C_1 \exp\(\frac{C_2 b_1}{h}\)
\leq
C_{15} \exp\(\frac{C_{16} (b_1B_0)}{h}\).
\endmultline
$$
This shows that\corr{ this} our second claim follows from the first one,
already proven, since
$$
\mu_{\L,,h} ( \ \cdot \ ) \leq
\ha \mu_{\L,,h} ( \ \cdot \ ) +
\mu_{\L,,h} ( \Cal{R} ).
$$
\cqd
\enddemo
%\bye
\bigskip
%\input ga
\subheading{\ga. Spectral gap estimates}
In this section we will prove three propositions which provide
lower bounds on the spectral gap of the generator of the kinetic
Ising models on some finite sets. A basic technique to be used
comes from the fundamental paper [Mar], where for the first time
(to our knowledge) a rigorous mathematical relation was established
between relaxation times of kinetic Ising models and the equilibrium
surface tension. This was done in a setting in which there is no
external applied field, and the system\corr{ is considered to be} was taken in a
square box of size $l \times l$
with free boundary conditions and at low temperature (in that paper
only very low temperatures where considered, but the main results were
later extended up to $T_c$ in [CGMS]). The ``time to jump between the
(+)phase and the ($$)phase'' was shown in that paper to behave as
$\exp(\beta\ttau l)$, as $l \to \infty$, where $\ttau = \tau((1,0))$
is the surface tension in a coordinate direction.
For each finite $\L \subset \Z^2$, $\eta \in
\Omega$ and $h$, the process $(\s^{{\bold \cdot}}_{\L,\eta,h;t})_{t \geq 0}$
is a finitestatespace reversible irreducible Markov process
and its generator has its (discrete) spectrum contained in
the interval $(\infty, 0]$, with $0$ being in the spectrum.
The spectral gap, denoted by $\gap(\L,\eta,h)$,
is then simply\corr{ the} the absolute value of the largest nonnull
number in the spectrum.
We will prove the three propositions below, the first two of
them being needed in Section \ip \ and the third one being
needed in Section \re. The heuristics behind these three
propositions was explained in those sections.
For a recap of the terminology used in these lemmas, see the
begining of Section \dw \ and the paragraph in that section
which preceds Lemma \dwsp.
\proclaim{Proposition \ga.1}\corr{ How to make ``gap'' not to appear in italics?}
For any $l_0 > 0$, any $\epsilon > 0$ and any $\bar b>0$
there is $h_0 > 0$ such that if $0 < h \leq h_0$
then for any $l_0$quasiWulffshaped set $\L$ which has linear size
parameter $b/h$ with $ b \leq \bar b$,
$$
\gap (\L,+,h) \ \geq \ \exp\left(\frac{\epsilon}{h}\right).
$$
\endproclaim
\proclaim{Proposition \ga.2}
For any $l_0 > 0$, any $\epsilon > 0$ and any $B_c<\bc<\bar b$
there is $h_0 > 0$ such that
if $0 < h \leq h_0$
then for any $l_0$quasiWulffshaped set $\L$ which has linear size
parameter $b/h$ with $ \bc < b \leq \bar b$,
$$
\gap
\left( \L\backslash \Lc, (+,),h\right)
\ \geq \ \exp\left(\frac{\epsilon}{h}\right),
$$
where $\Lc = \L\(\frac{\bc}{h}W\)$, and the boundary condition
$(+,)$ refers to freezing the spins up inside the core
$\Lc$ and down outside $\L$.
\endproclaim
\proclaim{Proposition \ga.3}
For any $\epsilon > 0$ there are $b>B_c$
and $h_0 > 0$ such that if $0 < h \leq h_0$
then
$$
\gap \(\L\(\frac{b}{h}W\),,h\) \
\geq \ \exp\left(\frac{\epsilon}{h}\right).
$$
\endproclaim
We will explain how certain results and techniques in [Mar]
can be used to prove the three propositions above.
The specific problem in [Mar] which is close to ours is the subject
of Section 3 in that paper. This problem concerns the spectral gap
for the process with no
external field, in a square box of size $l \times l$
with \newline
(+)boundary conditions. It is shown that
given $\epsilon \in (0,1/2)$, at low enough temperature there is
$C > 0$ so that
$$
\gap(\L(L),+,0) \geq \exp(C L^{1/2 + \epsilon}).
\Eq(Mar)
$$
Heuristically speaking, in this problem (contrary to the case of
freeboundary conditions, which is the main concern in [Mar]),
there is no freeenergy barrier to overcome for the system either
starting with all spins down or all spins up to reach equilibrium.
One simply expects the (+)phase to drift inwards from the boundary.
(This indicates that the result \equ(Mar) is far
from optimal, and that the corresponding gap should be much larger
than this lower bound.) This situation is similar to our problems,
as stated in the propositions above. In the first two propositions
we are dealing with situations without freeenergy barriers, and in
the third one the freeenergy barrier can be made as small as needed,
by adjusting the value of $b$.
The technique used in [Mar] to prove \equ(Mar) consists in comparing
the spectral gap for the kinetic Ising model with the one for a
blockdynamics. The estimate on the spectral gap
for the blockdynamics is reduced to equilibriumstatisticalmechanics
problems, which are then solved.
By a blockdynamics the following is meant. Suppose that
$\{\L^0,...,\L^J \}$ is a finite
collection of finite subsets of $\Z^2$ and
that $\L \subset \cup_{j=0,...,J} \L^j$ is another
finite subset of $\Z^2$.
The blockdynamics in $\L$ with blocks
$\{\L^0,...,\L^J \}$ and
with boundary condition $\eta \in \Omega_{\L,}$ will be denoted by
$$
(\sigma^{{\bold \cdot}}_{\L,\{\L^0,...,\L^J \},\eta,h; t})_{t \geq 0}.
$$
It is defined by updating each block $\Lambda^j \cap \L$
at rate 1, independently
of the other blocks, and at each update of $\Lambda^j \cap \L$
replacing the configuration inside this block
with a configuration chosen at random
according to the Gibbs distribution
\newline $\mu_{\L^j \cap \L,\sigma,h}$, where
$\sigma$ is the current configuration. The corresponding generator is
given by\corr{ better Cal l}
$$
( \Cal Lf)(\sigma) = \sum_{j = 0,...,J}
\mu_{\L^j \cap \L, \sigma, h}(\sigma')(f(\sigma')  f(\sigma)).
$$
To state an inequality which compares the spectral gap
$\gap(\L,\eta,h)$
with the spectral gap $\gap(\L,\{\L^0,...,\L^J \},\eta,h)$
of the block dynamics, we need to introduce some notation.
Set
$$
\gather
L_j = \max_{k\in \Z} \{(x_1,x_2) \in \L^j : x_2 =k \}, \\
L= \max_{j = 0,...,J} L_j,
\endgather
$$
and
$$
V = \max_{j = 0,...,J} \L^j.
$$
Theorem 2.1 in [Mar] provides us with the following\corr{ bound.} bound:
$$
\gap(\L,\eta,h) \ \geq \
\frac{C_1 \exp(C_2L)}{V} \
\gap(\L,\{\L^0,...,\L^J \},\eta,h),
\Eq(blockgap)
$$
where $C_1$ and $C_2$ are finite positive constants which depend
only on the temperature. The proof of this result in [Mar] is
restricted to the case in which the blocks are of a certain type,
adapted to the needs in that paper, but this restriction
is clearly not relevant
in the proof. Somewhat more importantly, the rates of the kinetic
Ising models in [Mar] are not as general as ours, with only a
special case of rates, which satisfy detailedbalance and
our assumptions (H1)  (H4) being considered.
This is not a problem, and indeed, once
\equ(blockgap) is established for one choice of rates satisfying
detailedbalance and (H4), it holds for
all such rates, thanks to the fact that the spectral gap is bounded
below and above by, respectively, the same equilibrium quantity
multiplyed by $c_{\min}(T)$ and $c_{\max}(T)$ (for this see, e.g.,
equation (61) in [Sch1]).
In order to prove Propositions \ga.1  \ga.3,
our blocks will be Wulffannuli, defined as follows.
Given $\rho >0$ set
$$
\gather
A^0_{\rho} = 2 \rho W \\
A^j_{\rho} = (j+2)\rho W \ \backslash \ j\rho W \quad \text{for} \quad
j=0,1,...
\endgather
$$
and for $h >0$, and $j = 0, 1, ...$
$$
\L^j = \L\( \frac{1}{h} A^j_{\rho} \),
$$
where the value of $\rho$ is chosen in a fashion that we describe next,
and which depends on the value of $\epsilon$ and $\bar b$ in
the propositions that we are proving (in the case of
Proposition \ga.3
%we will write \bar b = b, so as to keep the notation unified).
we can choose some arbitrary $\bar b > B_c$).
Note that
for each
$\rho>0 $,
$j = 0, 1, ...$,
and $r \in \R$, the set
$A^j_{\rho} \cap \{(x_1,x_2) \in \R^2 : x_2 =r \}$
if not empty consists of
either an interval or the union of two intervals, and that
its Lebegue measure satisfies
$$
\max_{r \in \R} \max_{j = 0,...,\lfloor \bar b / \rho\rfloor}
 A^j_{\rho} \cap \{(x_1,x_2) \in \R^2 : x_2 =r \}
\to 0
\quad \text{as} \quad
\rho \to 0.
$$
Therefore we can choose $\rho$ small enough and
take $J = \lfloor \bar b / \rho \rfloor$,
so that the corresponding collection of blocks,
$\{\L^0,...,\L^J \}$,
satisfies for each $h > 0$,
$$
L \leq \frac{\epsilon}{2C_2h},
\Eq(boundL)
$$
where $C_2$ comes from
\equ(blockgap).
Since $V$ grows only as a power of $1/h$, from \equ(blockgap) and
\equ(boundL) we see that all that remains is to show that with the
choices above and the pertinent $\L$ and $\eta$,
$ \gap(\L,\{\L^1,...,\L^J \},\eta,h)$ can be assured to
be large enough. In the case of Proposition \ga.1 and \ga.2 this
amounts to showing that this quantity can be bounded below by a
positive constant which does not depend on $h$. In the case of
Proposition \ga.3, we need to show that by taking $b$ sufficiently
close to $B_c$ this will also be the case. (The way we set things up
above, some blocks $\L^j$ may not intersect the set $\L$ in
some situations. This, of course, is not important, since such
blocks have no effect on the dynamics. The setup above was chosen
for notational convenience.)
To study
$ \gap(\L,\{\L^0,...,\L^J \},\eta,h)$
one can couple the processes \newline
$(\sigma^{}_{\L,\{\L^0,...,\L^J \},\eta,h; t})_{t \geq 0}$ and
$(\sigma^{+}_{\L,\{\L^0,...,\L^J \},\eta,h; t})_{t \geq 0}$
in such a way that the first marginal never lies above the second one,
and after they hit each other they coalesce and remain together.
This can be done, for instance, via a graphical construction, in
which to each block $\L^j$ we associate a rate 1 exponential Poisson
process. The occurrence times of this Poisson process then determine
the moments of update of $\L^j \cap \L$
in both marginal processes, and the
updates are coupled in a way that preserves the order, which is
possible due to the FKGHolley inequalities. In what follows we
will use $\P$ to denote the probabilities associated to this coupling.
The goal is now to show that there are positive finite
constants $t_0$, and $h_0$ so that for all $0 < h \leq h_0$ and all
$\L$ with which we are concerned, and corresponding $\eta$,
$$
\P
\(\sigma^{}_{\L,\{\L^0,...,\L^J \},\eta,h; t_0} \neq
\sigma^{+}_{\L,\{\L^0,...,\L^J \},\eta,h; t_0}\)
\leq
\frac{1}{2}.
\Eq(goalt0)
$$
From this inequality it then follows that for
all $t > 0$,
$$
\P
\(\sigma^{}_{\L,\{\L^0,...,\L^J \},\eta,h; t} \neq
\sigma^{+}_{\L,\{\L^0,...,\L^J \},\eta,h; t}\)
\leq
\(\frac{1}{2}\)^{\lfloor t/t_0 \rfloor}
\leq
C \exp\(\frac{\log(1/2)}{t_0} \ t \).
%\Eq(goalt)
$$
In particular then we have
$$
\gap(\L,\{\L^0,...,\L^J \},\eta,h)
\geq
\frac{\log(1/2)}{t_0},
%\Eq(goalt)
$$
as needed to complete the proofs.
Concerning the proof of \equ(goalt0), we start by observing that we
have only a finite number of blocks. Therefore, if $t_0$ is chosen
large enough it will be likely that before time $t_0$ a sequence of
updates will have occurred in the coupled processes in which the blocks
were updated in a particular, predetermined order. In the case of
Propositions \ga.1 and \ga.3 the good order is the decreasing one,
from $J$ down to 0, while in the case of Proposition \ga.2 it is the
increasing order, from 0 up to $J$. The point is that at the end of
a sequence of updates produced in the good order it is very likely,
when $h$ is small, that the two marginal processes will have hit each
other. To show this, we will use an equilibrium estimate
on how much the boundary conditions can influence the
Gibbs distributions inside the annular blocks. This is the
contend of the following lemma.\corr{ !?!} After stating it and before going
into the proof of it we will explain the idea of how it has to be used in
order to prove \equ(goalt0). In the lemma we will use the notation
$$
\PEb =
\left\{ \L\(\frac{b(1\epsilon)}{h}W\) \perm
\(\L\(\frac{b(1+\epsilon)}{h}W\)\)^c \right\},
$$
and
$$
\PEp =
\left\{ \L\(\frac{b(1\epsilon)}{h}W\) \perp
\(\L\(\frac{b(1+\epsilon)}{h}W\)\)^c \right\}.
$$
\proclaim{Lemma \ga.1}
For any $l_0>0$, any $\epsilon > 0$
and any $00$. And if also $x \in \(\L\(\frac{b_1 + \epsilon}{h}W\)\)^c$
then
$$
 \langle \s(x) \rangle _{\L_2 \backslash \L_1,(+,),h}
 \langle \s(x) \rangle _{\L_2 \backslash \L_1,(,),h} 
\leq C_1 \exp(C_2/h),
$$
for all $h>0$.
\item"(b)" If $\phi(b_1) > \phi(b_2)$ then for all $\L_1$ and
$\L_2$ which are $l_0$quasiWulff shaped with respective
linearsizeparameters
$\frac{b_1}{h}$ and $\frac{b_2}{h}$,
$$
\mu_{\L_2 \backslash \L_1,(+,),h} (\PEn2) \leq C_1 \exp(C_2/h),
$$
for all $h>0$. And if also $x \in \L\(\frac{b_2  \epsilon}{h}W\)$
then
$$
 \langle \s(x) \rangle _{\L_2 \backslash \L_1,(+,),h}
 \langle \s(x) \rangle _{\L_2 \backslash \L_1,(+,+),h} 
\leq C_1 \exp(C_2/h),
$$
for all $h>0$.
\item"(c)"
For all $\L_1$ and
$\L_2$ which are $l_0$quasiWulff shaped with respective
linearsizeparameters
$\frac{b_1}{h}$ and $\frac{b_2}{h}$,
$$
\mu_{\L_2 \backslash \L_1,(,+),h} (\Pen1) \leq C_1 \exp(C_2/h),
$$
for all $h>0$. And if also $x \in \(\L\(\frac{b_1 + \epsilon}{h}W\)\)^c$
then
$$
 \langle \s(x) \rangle _{\L_2 \backslash \L_1,(,+),h}
 \langle \s(x) \rangle _{\L_2 \backslash \L_1,(+,+),h} 
\leq C_1 \exp(C_2/h),
$$
for all $h>0$.
\endroster
\endproclaim
\demo{Proof of \equ(goalt0)} Here we explain the idea of deriving
the estimate \equ(goalt0) from the above lemma. We
will do it for the case of the Proposition \ga.1, for which the
statement (c) of the lemma is used. The main observation is
very simple. As the reader remembers, we are concerned with the
event that the updates of the blocks happen in decreasing
order, that is first the block $ \Lambda ^J$ is updated, then the
block $ \Lambda ^{J1}$ is, and so on. Note, that the
boundary condition on the outer boundary of $ \Lambda ^J$ is $(+)$.
As we learn from the lemma, after the update of the
block $ \Lambda ^J$, with overwhelming probability it becomes
almost completely filled with $(+)$phase, no matter what the
boundary condition on the inner boundary of $ \Lambda ^J$ is.
In particular, its middle line  which is the outer boundary of
the block $ \Lambda ^{J1}$  is in the $(+)$phase. So the
argument can be repeated. For the complete argument the
reader is referred to [Mar], Theorem 3.1.
\enddemo
In the final remark before going into the
proof of the Lemma \ga.1 \ let us explain the relations
between different parts of the lemma and Propositions 3.5.1  3.5.3:
part (a) has to be used for the Proposition 3.5.3, part (b) 
for the Proposition 3.5.2, and part (c)  for the Proposition 3.5.1.
The choice of $b$ in
Proposition 3.5.3 has to be done in such a way that
$\phi(b)> \phi(b\rho)$ with $\rho$ chosen before (3.51).
\demo{Proof of Lemma \ga.1}
In each one of the three parts of the lemma, the\corr{ !!} second claim, about
the expected value of the spin at a site $x$, follows from the
first claim in the same part of the lemma, by
arguments analogous to \equ(psw).
Regarding the first claim in each part of the lemma, we
could in principle develop a machinery similar to the
one developed in Section \dw, to deal with the Gibbs
distribution inside Wulff annuli. Some technical complications
would arise from the fact that such sets are not
simplyconnected.
It turns out, nevertheless, that for our purposes we can avoid
this lengthy approach, and rather use the results for
simplyconnected sets
(more specifically, for quasiWulffshaped sets)\corr{
in} of Section \dw, combined with some tricks involving
conditioning. To stress that our approach is to some extent
natural, we observe that
the hypothesis in our lemma refer the values of $\phi(b_1)$ and
$\phi(b_2)$, and hence somehow we should use our knowledge
about quasiWulffshaped boxes to prove these results on the
annuli. And to make the conditioning below appear less of a trick,
observe that, say in part (a), we are interested in (+)boundary
conditions in the center, something that is akin to having a
droplet of the (+)phase placed there;
the conditioning places such a droplet in the center.
We turn now to the proof of the first claim in part (a) of the
lemma.
Recall the definitions in the first paragraph of the proof of
Lemma \dwsp. Using that notation,
we will say that $\Gamma$ is a negativeoutside (same as
positiveinside) contour of
a configuration $\eta$ in case $\eta$ is identically $1$
on $\partial_{\text{ext}}\Gamma$
and identically $+1$ on
$\partial_{\text{int}}\Gamma$. External contours of any
configuration in $\Omega_{}$ are negativeoutside.
Let $E$ be the event that there is a contour which surrounds
$\L_1$ and\ccor{ CC} $E' \subset E$ be the event that
such a contour is contained
in $\frac{b_1(1+\epsilon/2)}{h}W$
and is negativeoutside.\ccor{ more details}
We are going to argue that if $E$ happens, then with very high probability
$E'$ happens as well.
So we need an estimate on $\mu_{\L_2,,h} \(E \)$ from below, and an
estimate on
$\mu_{\L_2,,h} \( E \cap (E')^c \)$ from above.
From Lemma 3.4.3 we know that the first probability is at least of the
order of $\exp \left( \frac{\beta}{h} \phi(b_1) \right)$.
To obtain the second estimate, let us introduce the number $b'>b_1$, which
is close enough to $b_1$, so that $\delta=1  {b_1\over b'} $ is small
enough. The choice of $\delta$ will be made later.
One sees immediately that
$$\aligned
\mu_{\L_2,,h} &\((E')^c \ \cap \ E \)\\
=&
\mu_{\L_2,,h} \((E')^c \cap E \cap \left[ \
\(\frac{b_1}{h}\)^2 \leq
\bV \leq \(\frac{b_2}{h}\)^2 \right] \)\\
\le&
\mu_{\L_2,,h} \(( E')^c \cap E\left \ \(\frac{b'}{h}(1
\delta)\)^2 \leq
\bV \leq \(\frac{b'}{h}\)^2 \right. \) \times\\
&\ \ \times
\mu_{\L_2,,h}\( \(\frac{b'}{h}(1
\delta)\)^2 \leq
\bV \leq \(\frac{b'}{h}\)^2 \)\\
+&
\mu_{\L_2,,h} \(\(\frac{b'}{h}\)^2 \leq
\bV \leq \(\frac{b_2}{h}\)^2\)
.\endaligned
$$
Note now, that if both events $(E')^c \cap E$ and $
\bV \leq \(\frac{b'}{h}\)^2$ happen, and $\delta$ is small enough, then
we can claim that the following three properties hold:
\noindent
there is an exterior contour surrounding $ \Lambda _1$;
\noindent
this contour can not be shifted so as to fit inside $ {
b'(1+\epsilon/4)\over h}W$;
\noindent
there are no other exterior contours which
can surround $ \Lambda _1$ even after being shifted.
\noindent
In other words, we
have the inclusion $(E')^c \cap E\cap \(\bV \leq \(\frac{b'}{h}\)^2\)
\subset ({\bar E}^h_{b',\epsilon/4})^c $.
So from the hypothesis that
$\phi(b_1) < \phi(b_2)$ and
Lemmas \dwlb, \dwpv \ and \dweb \ we have
$$
\mu_{\L_2,,h} \((E')^c \  \ E \)
\ \leq \ C_1 \exp(C_2/h),
\Eq(fromlemmasdw)
$$
for some positive finite constants $C_1$ and $C_2$.
Set
$$
F=
\left\{ \L\(\frac{b_1(1+\epsilon/2)}{h}W\) \perp
\(\L\(\frac{b_1(1+\epsilon)}{h}W\)\)^c \right\},
$$
Partitioning $E'$ according to what the innermost
negativeoutside contour
around $\L_1$ is and using the FKGHolley inequalities
we obtain
$$
\mu_{\L_2,,h} \(F \  \ E' \)
\leq
\mu_{\L_2,,h} \(F \).
$$
In order to have
$\phi(b_1) < \phi(b_2)$
we must have $b_2 < B_0$.
Clearly also, for some $\epsilon'>0$ which depends on $\epsilon$
and $b_1$,
$F \cap \Omega_{\L_2,}
\ \subset \ \left\{\Cal{W} \geq \frac{\epsilon'}{h} \right\}$.
Therefore, if we choose a conveniently small $\epsilon''>0$ and
use Lemmas \dwub, and \dwpv \ we obtain
$$
\align
\mu_{\L_2,,h} \(F \)
& \leq
\mu_{\L_2,,h} \(\Cal{W} \geq \frac{\epsilon'}{h}
\ , \ \bV \leq \(\frac{\epsilon''}{h}\)^2 \)
+
\mu_{\L_2,,h} \( \bV \geq \(\frac{\epsilon''}{h}\)^2 \)
\cr & \leq
C_1 \exp(C_2/h).
\endalign
$$
Combining the various inequalities displayed above, we have
$$
\align
\mu_{\L_2,,h} \(F \  \ E \)
& \leq
\mu_{\L_2,,h} \((E')^c \  \ E \)
+
\mu_{\L_2,,h} \(F \  \ E' \)
\cr & \leq
C_1 \exp(C_2/h).
\teq(lemmagap1)
\endalign
$$
To proceed, we have to introduce a new notion. We will call
a (*)circuit $ \gamma =x_1,\dots,x_n$ a $c$circuit iff there
exists a configuration $ \sigma \in \Omega_$ with one
contour, $ \Gamma $, such that $ \partial _+ \Gamma = \{
x_1,\dots,x_n \} $. (Note that our definition depends on the
choice of the splitting rules, used in connection with
transforming of contours into closed curves.) The interior
$\text{Int } \gamma $ of a $c$circuit $ \gamma $ is by
definition the interior of the corresponding contour $ \Gamma $.
Let $ \gamma _1, \gamma _2$ be two $c$circuits;
we define the intersection $c$circuits $ \delta_k $ in the following way.
Let $I=\text{Int } \gamma_1 \cap \text{Int } \gamma_1$ and $I_k$ be
connected components of $I$ in the sense of the above mentioned
splitting rules. Since $I$ is simplyconnected, so are the $I_k$s.
Let $ \delta _k$ be $c$circuits,
such that $\text{Int } \delta _k=I_k$. For immediate
use we need the following property of
$c$circuits: let $ \gamma
_1$ and $ \gamma _2$ be two $c$circuits,
$ \delta $ be one of the intersection $c$circuits,
and a configuration $ \sigma $ be given, such that both $ \gamma
_1$ and $ \gamma _2$ are $(+*)$circuits of it.
Then so is the circuit $ \delta $. To see this, let us introduce the
corresponding contours $ \Gamma_1$, $\Gamma_2$ and
$ \Delta $, and let first a site $x$ of the
$c$circuit $
\delta $ be at the distance $ {1\over 2} $
from some bond $b$ of $ \Delta $. But this bond $b$ evidently belongs to at
least one of the contours $ \Gamma_1$, $\Gamma_2$, while $x$ is inside this
contour, hence $ \sigma (x)=+1$. In the
remaining case the distance $ \text{dist}(x, \Delta )= {\sqrt 2\over 2} $,
and in such a situation there are two adjacent
bonds $b_1, b_2$ of $ \Delta $, with $ \text{dist}(x, b_1 )=
\text{dist}(x, b_2 )={\sqrt 2\over 2} $, and two nearest
neighbors $x_1, x_2$ of $x$, belonging to $ \delta $, such that
$ \text{dist}(x_i, b_i )= {1\over 2}, i=1,2$. Consider
another nearest neighbor $y$ of the sites $x_1, x_2$. It stays
outside $ \Delta $, hence it also stays outside at least one of
the contours $ \Gamma_1$, $\Gamma_2$,
while all three sites $x,x_1,x_2$ are inside both of them.
Hence both bonds $b_1,
b_2$ belong to that contour, which again implies that $ \sigma (x)=+1$.
Let $E''$ be the event that some $c$circuit $ \gamma $ which surrounds
$\L_1$ and is contained in
$\frac{b_1(1+\epsilon/2)}{h}W$
is a $(+*)$circuit.
Let $\{E''_j\}$ be the partition of $E''$
according to what the innermost such circuit $ \gamma $ is.
(The preceding paragraph ensures the existence of
such an innermost $(+)$$c$circuit.)
Using the fact that for each $j$ we have
$\Omega_{\L_2,} \cap E''_j \ \subset \ E$,
the FKGHolley inequalities
and \equ(fromlemmasdw)
we obtain
$$
\align
\mu_{\L_2,,h} \(F \  \ E \)
& \geq
\sum_j
\mu_{\L_2,,h} \(F \  \ E''_j \cap E \) \
\mu_{\L_2,,h} \(E''_j \  \ E \)
\cr & =
\sum_j
\mu_{\L_2,,h} \(F \  \ E''_j \) \
\mu_{\L_2,,h} \(E''_j \  \ E \)
\cr & \geq
\mu_{\L_2 \backslash \L_1,(+,),h} (F)
\
\sum_j
\mu_{\L_2,,h} \(E''_j \  \ E \)
\cr & =
\mu_{\L_2 \backslash \L_1,(+,),h} (F)
\
\mu_{\L_2,,h} \(E'' \  \ E \)
\cr & \geq
\mu_{\L_2 \backslash \L_1,(+,),h} (F)
\
\mu_{\L_2,,h} \(E' \  \ E \)
\cr & \geq
\frac{1}{2} \ \mu_{\L_2 \backslash \L_1,(+,),h} (F),
\teq(lemmagap2)
\endalign
$$
for small $h$.
Comparing \equ(lemmagap1) with \equ(lemmagap2) we obtain
$$
\mu_{\L_2 \backslash \L_1,(+,),h} (\Pep1)
\leq
\mu_{\L_2 \backslash \L_1,(+,),h} (F)
\leq 2 C_1 \exp(C_2/h).
$$
This completes the proof of part (a) of the lemma, and we turn to
to the proof of the first claim in part (b) of the lemma.
Let this time $E$ be the event that there is a contour which surrounds
$\frac{b_1(1\epsilon)}{h}W$
and $E'$ be the event that there is
a contour which surrounds
$\frac{b_2(1\epsilon/2)}{h}W$.
With no loss of generality, we will suppose that $\epsilon$ is
small enough so that $E' \subset E$.
From the hypothesis that
$\phi(b_1) > \phi(b_2)$ and
Lemmas \dwpv \ and \dweb \ we have
$$
\mu_{\L_2,,h} \((E')^c \  \ E \)
\ \leq \ C_1 \exp(C_2/h),
\Eq(fromlemmasdwb)
$$
for some positive finite constants $C_1$ and $C_2$.
Therefore also
$$
\mu_{\L_2,,h} \(\left. \PEn2 \ \right \ E \)
\ \leq \ C_1 \exp(C_2/h).
\Eq(lemmagap1b)
$$
Let this time $E''$ be the event that there is a
(+)circuit which surrounds
$\frac{b_1(1\epsilon)}{h}W$.
and is
contained in
$\frac{b_1(1\epsilon/2)}{h}W$.
Let $\{E''_j\}$ partition $E''$
according to what the innermost such (+)circuit is.
By partitioning $(\PEn2)^c$ according to what the ($$)cluster of
$(\L_2)^c$ is and using the FKGHolley inequalities combined with
\equ(CCS) and the graphtheoretic duality between connectivity and
(*)connectivity, in addition to \equ(lemmagap1b), we obtain
$$
\align
\mu_{\L_2,,h} \(\left. (E'')^c \ \right \ E \)
& \leq
\mu_{\L_2,,h} \( \left. \PEn2 \ \right \ E \)
+
\mu_{\L_2,,h} \((E'')^c \ \left \ \(\PEn2\)^c \right. \)
\cr & \leq
C_1 \exp(C_2/h).
\teq(lemmagap1bb)
\endalign
$$
Using the fact that for each $j$ we have
$\Omega_{\L_2,} \cap E''_j \ \subset \ E$,
the FKGHolley inequalities
and \equ(lemmagap1bb) we obtain
$$
\align
\mu_{\L_2,,h} \(\left. \PEn2 \ \right \ E \)
& \geq
\sum_j
\mu_{\L_2,,h} \(\left. \PEn2 \ \right \ E''_j \cap E \) \
\mu_{\L_2,,h} \(E''_j \  \ E \)
\cr & =
\sum_j
\mu_{\L_2,,h} \(\left. \PEn2 \ \right \ E''_j \) \
\mu_{\L_2,,h} \(E''_j \  \ E \)
\cr & \geq
\mu_{\L_2 \backslash \L_1,(+,),h} \(\PEn2\)
\
\sum_j
\mu_{\L_2,,h} \(E''_j \  \ E \)
\cr & =
\mu_{\L_2 \backslash \L_1,(+,),h} \(\PEn2\)
\
\mu_{\L_2,,h} \(E'' \  \ E \)
\cr & \geq
\frac{1}{2} \ \mu_{\L_2 \backslash \L_1,(+,),h} \(\PEn2\),
\teq(lemmagap2b)
\endalign
$$
for small $h$.
Comparing \equ(lemmagap1b) with \equ(lemmagap2b) we obtain
$$
\mu_{\L_2 \backslash \L_1,(+,),h} \(\Pep1\)
\leq 2 C_1 \exp(C_2/h).
$$
This completes the proof of part (b) of the lemma, and we turn to
to the proof of the first claim in part (c) of the lemma.
Because in Section \dw \ we did not study systems with
(+) boundary conditions, we will use a somewhat artificial
approach to part (c), by reducing it to part (a), studied
above. (In doing so we will proceed
as someone who first heats up cold water in order to freeze
it later, because he knows how to freeze hot water, but has never
frozen cold water). First note that by reversing all signs and
then using the FKGHolley inequalities, we obtain, for any
$h' \geq h$,
$$
%\align
\mu_{\L_2 \backslash \L_1,(,+),h} \(\Pen1\)
%& =
=
\mu_{\L_2 \backslash \L_1,(+,),h} \(\Pep1\)
%\cr & \leq
\leq
\mu_{\L_2 \backslash \L_1,(+,),h'} \(\Pep1\).
%\endalign
$$
Suppose that $h' > 0$ and set
$$
b'_1 = \frac{h'}{h} b_1 \qquad \text{and} \qquad
b'_2 = \frac{h'}{h} b_2.
$$
If $h'$ is small enough we have $0 < b'_1 < b'_2 < B_c$ and hence
$\phi(b'_1) < \phi(b'_2)$. But $\L_1$ and
$\L_2$ are $l_0$quasiWulff shaped with respective
linearsizeparameters
$\frac{b_1}{h} = \frac{b'_1}{h'}$ and
$\frac{b_2}{h} = \frac{b'_2}{h'}$.
So we can quote part (a) of the lemma to conclude the proof of
part (c).
\cqd
\enddemo
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\bigskip
R.H.Schonmann
Mathematics Department
University of California at Los Angeles
Los Angeles, CA 90024, U.S.A.
\bigskip
S.B.Shlosman
Mathematics Department
University of California at Irvine
Irvine, CA 92697, U.S.A.
\medskip
and
\medskip
Institute for the Information Transmission Problems
Russian Academy of Sciences
Mos\corr{ kow,}cow, Russia
\bye