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%%               Droplet dynamics for asymmetric Ising model             %%
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\topmatter



\title
Droplet dynamics for asymmetric Ising model
\endtitle
%
%
\leftheadtext{Droplet dynamics for asymmetric Ising model}
\rightheadtext{}
%
%
\author
R. Koteck\'y and E. Olivieri
\endauthor
\affil
Charles University in Prague and Universit\`a di Roma ``Tor Vergata''
\endaffil
\address
Roman Koteck\'y
\hfill\newline
Departement of
Theoretical Physics, Charles University,\hfill\newline
 V~Hole\v sovi\v
ck\'ach~2, 180~00~Praha~8, Czechoslovakia
\hfill\newline
\phantom{18.}and
\hfill\newline
Center for Theoretical Study, Charles
University,\hfill\newline
 Ovocn\'y trh~3, 116~36~Praha~1,
Czechoslovakia
\endaddress
\email
kotecky\@cspuni12.bitnet
\endemail
\address
Enzo Olivieri
\hfill\newline
 Dipartimento di Matematica, Universit\`a di Roma ``Tor
Vergata'', \hfill\newline
Via della Ricerca Scientifica, 00133 Roma, Italy
\endaddress
\email
olivieri\@mat.utovrm.it
\endemail
\keywords
Stochastic dynamics, Ising model, metastability, crystal growth, first
excursion
\endkeywords
 \abstract
Nucleation from a metastable state is studied for an anisotropic Ising
model at very low temperatures.
It turns out that the critical nucleus as well as  configurations on
a typical path to it differ from the Wulff shape of an
equilibrium droplet.
\endabstract
%\subjclass Primary 82A05; secondary 82A25\endsubjclass
\thanks
The research has been partially supported by CNR -- GNFM.
\endthanks
\endtopmatter


\document

\head 1. Introduction \endhead



We report  some new results concerning stochastic ferromagnetic Ising
models in the so called metastable region. Namely, we consider
stochastic (Glauber) dynamics whose stationary states are given by Gibbs
measures for Ising-like systems that at infinite volume, low temperature,
and zero magnetic field exhibit a phase transition with
spin-flip symmetry breaking. We study, similarly  to what Jord\~ao-Neves and
Schonmann did for the standard Ising model \cite{NS, S}, a single spin-flip
Glauber dynamics in a large but finite volume for small (positive) magnetic
fields and very low temperatures. In particular, we are interested in
asymmetric models --- models for which the Wulff shape (equilibrium droplet)
at zero temperature is not a cube. Ising models are believed to have some
relevance for a dynamical description of the crystal growth (cf\.
\cite{BCF}). The reason to consider an asymmetry stems from the fact that
these models turn out to be  simple cases where  a difference between
equilibrium and dynamical droplets  shows up.
The present note is devoted to a study of the simplest such model,
namely, an anisotropic Ising model.

 We analyze the nucleation
 of the stable plus-phase starting from the
metastable minus-phase. In particular,  we are iterested in a description of
the first passage from the configuration $-\underline 1$ (all spins in
$\Lambda$ equal  $-1$) to the configuration $+\underline 1$ (all spins  in
$\Lambda$ equal $+1$). We show that with high probability in the limit of very
low temperatures this transition takes place
\roster
\item"i)" via a formation of a critical
nucleus whose shape  may  not be  Wulff and
\item"ii)" following a path that is typically
given by a  sequence of ``non-Wulff'' configurations.
\endroster
In particular, for a two-dimensional anisotropic nearest neighbour Ising
model with coupling constants $J_1 > J_2 > 0$ along the axes, the critical
droplet is actually a square of edge $\frac{2J_2}{h}$ ($h$ is the magnetic
field),  while the Wulff shape is a rectangle with edges proportional to
$J_1$, $J_2$.

Another model suitable for a detection of a difference between equilibrium
and dynamical droplet is a ferromagnetic Ising model with isotropic nearest
neighbour and next nearest
neighbour interaction.
The critical droplet in this case turns out to have the optimal Wulff shape
--- at zero temperature it is a (non-regular) octagon with lengths of its
sides determined from the ratio of the nearest neighbour and the next nearest
neighbour coupling constants. However, the growth of a droplet
 follows (with high probability
for very low temperatures) a somehow complicated path through non-Wulff
shapes: up to a certain size it is along a sequence of regular octagons then
some edges remain constant whereas other grow up to the critical (Wulff)
shape.
A study of this model involves an additional time scale (in this respect it
reminds the  standard three-dimensional Ising model) and is discussed in
a separate publication \cite{KO 1}. The results for both models were
summarized in \cite{KO 2}.





To cover  more general cases than the standard nearest neighbour Ising
model, we    developped slightly different  arguments  and
constructions than those  by Jord\~ao-Neves and
Schonmann \cite{NS, S}.  We present here our new approach even though,
 in the particular case considered in the present paper, we could
probably have
worked out an  extension of the somehow simpler methods of Jord\~ao-Neves and
Schonmann. In their present form, however, their methods do not apply to our
situation.  We believe that our alternative is of some interest not only
due to  its more general applicability, but also
because  it clarifies some other aspects of the problem.




\head 2. Statement of the results \endhead


We consider a discrete time Metropolis dynamics for a two-dimensional
nearest neighbour ferromagnetic {\it asymmetric} Ising model.
The space of the process is $\{-1,1\}^\Lambda$ with $\Lambda$ being a
two-dimensional torus: the set $\{1,\dots,M\}^2$ with periodic boundary
conditions. A {\it configuration} $\sigma\in\{-1,1\}^\Lambda$ is a function
$$
\sigma\:\Lambda\to\{-1,1\}.
$$
The value $\sigma(x)$ is called the spin at the site $x\in\Lambda$.
The {\it energy} of a configuration $\sigma$ is
$$
H(\sigma)=-\frac{J_1}{2}\sum_{(x,y)\in\Cal H(\Lambda)}\sigma(x)\sigma(y)
-\frac{J_2}{2}\sum_{(x,y)\in\Cal V(\Lambda)}\sigma(x)\sigma(y)
-\frac{h}{2}\sum_{x\in\Lambda}\sigma(x),
\tag1
$$
where $\Cal H(\Lambda)$ is the set of horizontal nearest
neighbour pairs in $\Lambda$ and $\Cal V(\Lambda)$ is the set of vertical
nearest neighbour pairs  in $\Lambda$. We suppose that
$$
J_1\geq J_2\gg h>0 \text{ and } M\geq\Bigl(\frac{2J_1}{h}\Bigr)^3.
\tag2
$$
Further, to avoid some ``diofantine'' problems,  we assume that
$\frac{2J_1}{h}$, $\frac{2J_2}{h}$, as well as their difference, are not
integers.

The {\it dynamics} is prescribed by the following updating rule:
\proclaim\nofrills{}
 Given a configuration $\sigma$ at time $t$, we first
choose randomly a site $x\in\Lambda$ with uniform probability
 $\frac{1}{|\Lambda|}$. Then we flip the spin at the site $x$ with
probability
$$
\exp(-\beta\Delta_xH(\sigma))^+,
\tag3
$$
where
$$
\Delta_x H(\sigma))=H(\sigma^{(x)})-H(\sigma)
$$
with
$$
\sigma^{(x)}(y)=\cases &\!\!\!\!\!\sigma(y) \text{ whenever } y\neq x,\\
-&\!\!\!\!\!\sigma(y) \text{ for } y=x,
\endcases
$$
and $(c)^+=\min(c,0)$ for every $c\in\Bbb R$, $\beta$ is the inverse
temperature.
\endproclaim
Our dynamics is {\it reversible} with respect to the Gibbs measure
$$
\mu_\Lambda(\sigma)=\frac{\exp(-\beta H(\sigma))}{Z_\Lambda}
$$
with the partition function
$$
Z_\Lambda=\sum_{\sigma\in\{-1,1\}^\Lambda}\exp(-\beta H(\sigma)),
$$
in the sense that the transition probabilities  of the Markov chain
$ P(\sigma\to\sigma^\prime)$ satisfy the equalities
$$
\mu_\Lambda(\sigma) P(\sigma\to\sigma^\prime)=
\mu_\Lambda(\sigma^\prime) P(\sigma^\prime\to\sigma).
$$

The {\it space of trajectories} of the process is
$$
\Omega\equiv\bigl(\{-1,1\}^\Lambda\bigr)^{\Bbb N}.
$$
An element in $\Omega$ is denoted by $\omega$; it is a function
$$
\omega\:\Bbb N\to\{-1,1\}^\Lambda.
$$
We often write $\omega=\sigma_0,\sigma_1,\dots,\sigma_t,\dots$.

Given any set of configurations $A\subset\{-1,1\}^\Lambda$, we use
$\tau_A$ to denote the {\it first hitting time} to $A$:
$$
\tau_A=\inf\{t\geq 0\: \sigma_t\in A\}.
\tag4
$$
Sometimes we use $P_\eta(\cdot)$ to denote the probability distribution
over the process starting at $t=0$ from the configuration $\eta$.
We use $-\underline 1$, $+\underline 1$ to denote the configurations
with all spins in $\Lambda$ equal to $-1$, $+1$, respectively.

We are interested in dynamics at very low temperatures. Namely,
we will discuss the asymptotic behaviour, in the limit $\beta\to\infty$,
of typical paths of the first escape from $-\underline 1$ to
$+\underline 1$.


Having in mind a low temperature dynamics, it is natural to describe
 configurations in terms of their Peierls contours. Namely, for every
$\sigma\in\{-1,1\}^\Lambda$  we consider the union $C(\sigma)$ of all closed
unit squares centered at sites $x$ with $\sigma(x)=+1$. Connected
components of the boundary of $C(\sigma)$ are called {\it contours}.
A contour $\gamma$ is thus a polygon connecting vertices of dual lattice
$\Bbb Z^2 +(\frac12,\frac12)$ such that  any vertex is contained in
an even number (0, 2, or 4) of unit segments belonging to $\gamma$.
Often we shall identify a configuration $\sigma $ with the corresponding
set $C(\sigma)$.


%
%
\midinsert
\centerline{\picture 3.92in by 3.85in (one)}
\botcaption{Fig\. 1}
\endcaption
\endinsert
%
%



With  a certain  dose of imagination, one could view an evolution of a
configuration $\sigma $ with energy  (1) as a movement
of a point  in a complicated energy landscape (in ``phase space'') --- like
that shown on Fig\. 1, simplifying, however, the multidimensional space of
configurations to a two-dimensional space --- with a natural  tendency to
follow a downhill path and an occasional, random and rather unprobable,
uphill move. An  important role is played by local minima of this landscape.
Namely, let us introduce $\Cal R(L_1,L_2)$ as the set of all configurations
(up to a a translation) with all spins $-1$ except  for those in a rectangle
$R(L_1,L_2)$  with corners on the dual lattice and horizontal (vertical)
sides of length $L_1$ ($L_2$). It is easy to verify that, for  any
$(L_1,L_2)$ with $\min (L_1,L_2)>1$,
 these configurations correspond to such local minima.
It turns out (see \cite{NS}) that  small rectangles, namely those
with small values of
$\min (L_1,L_2)$, tend to shrink, while large rectangles tend to grow.
The dynamical mechanism responsible for this behaviour has been clarified
 in \cite{NS}:
it is based on a competition between the creation of a unit square
protuberance attached to an edge of the rectangle (and consequently a growth
of a side of the rectangle by one) and an erosion of an edge.
When deciding which tendency wins, one has to realize that the typical time
for a creation of a protuberance on a vertical or horizontal edge is of the
order $\exp\{2\beta\frac{J_1-h}{h}\}$, $\exp\{2\beta\frac{J_2-h}{h}\}$,
respectively, while the typical time for eroding an edge of the length
$l$ is $\sim\exp\{\beta h (l-1)\}$ (see \cite{NS} for more details).

Notice that the anisotropy reveals itself, in addition to different growth
rates in different directions, also in an anisotropy of ``interactions''
between separate droplets. Namely, two droplets approaching each other in
the horizontal direction will coalesce, while if they approach in the
vertical direction they have to overcome an energetical barrier and the
time needed to make their coalescence probable is of the order
$\exp\{2\beta \frac{J_1-J_2}{h}\}$.
If a droplet is too small, its existence is too ephemere to participate in
a coalescence. This will be an important factor when discussing the
detailed definition of a set of configurations attracted to the
configuration $-\underline 1$.

 The configurations in $\Cal R(L_1,L_2)$ are
characterized by the point $\pmb  L\equiv(L_1,L_2)$ in $(\Bbb Z_+)^2$.
The origin $\pmb  0 \in (\Bbb Z_+)^2$ represents the configuration
$-\underline 1$. Points with $L_1$ or $L_2=M$ represent rectangles winding
around the torus. We use $\Cal R$ to denote the set of all rectangular
configurations,
$$
\Cal R=\bigcup_{L_1,L_2}\Cal R(L_1,L_2).
$$
In $(\Bbb Z_+)^2$ we introduce the distance
$$
d(\pmb  x,\pmb  y)=\max(|x_1-x_2|,|y_1-y_2|)
$$
for $\pmb  x\equiv (x_1,x_2), \pmb  y\equiv (y_1,y_2)\in (\Bbb
Z_+)^2$.
A {\it saddle point} between two neighbouring local minima, say
$(L_1,L_2-1)$ and $(L_1,L_2)$, is any configuration $\bar\sigma$
such that
$$
H(\bar\sigma)=\min_{\omega\: R(L_1,L_2-1)\to
R(L_1,L_2)}\max_{\sigma\in\omega}H.
 $$
It is easy to see that it is represented by
the set  $C(\sigma)$ consisting of
a rectangle $R(L_1,L_2-1)$ with a unit square attached to one of its sides
of   length $L_1$. We will use $\Cal P(L_1,L_2-1;L_1,L_2)$ to denote
the set of all such configurations.


A {\it global saddle point} is any configuration $\bar\sigma$
 such that
$$
H(\bar\sigma)=\min_{\omega\:-\underline 1\to +\underline
1}\max_{\sigma\in\omega}H(\sigma).
\tag5
 $$
It turns out (see  Remark at the end of  Section 3.1 below) that the set of
all global saddle points is the set $\Cal P$
 of all configurations $\bar\sigma$ giving rise to  a unique
contour $\gamma$ consisting of a rectangle with sides $L_2^\ast$,
$L_2^\ast -1$, and a unit square attached to one of its longer sides (see
Fig\. 2). Here
$$
L_2^\ast=\Bigl[\frac{2J_2}{h}\Bigr] +1,
\tag6a
$$
where $[\cdot]$ denotes the integer part. We also introduce
$$
L_1^\ast=\Bigl[\frac{2J_1}{h}\Bigr] +1.
\tag6b
$$
For any $\bar\sigma \in\Cal P$ one has
$$
H(\bar\sigma)-H(-\underline 1)\equiv \Gamma(h)=(2J_1+2J_2)
L_2^\ast-h\bigl[(L_2^\ast)^2-L_2^\ast+1\bigr]
$$
for the ``height'' of the global saddle point.


%
%
\midinsert
\centerline{\picture 3.74in by 1.38in (two)}
\botcaption{Fig\. 2}
\endcaption
\endinsert
%
%



Using this fact we shall prove that the first excursion from $-\underline 1$
 to $+\underline 1$ typically passes through a configuration from $\Cal P$
and the time needed for this to happen is of the order $\exp(\beta\Gamma)$.



\proclaim{Theorem 1}
$$
\lim_{\beta\to\infty}P_{-\underline 1}(\tau_{\Cal P}
<\tau_{+\underline 1})=1.
\tag7
$$
\endproclaim


\proclaim{Theorem 2}
$$
\lim_{\beta\to\infty}P_{-\underline 1}(\exp[\beta(\Gamma-\varepsilon)]
<\tau_{+\underline 1}<\exp[\beta(\Gamma+\varepsilon)])=1
\tag 8
$$
for every $\varepsilon > 0$.
\endproclaim

Notice, as already mentioned in Introduction, that a droplet of least
overall surface tension covering a fixed area is given  by the Wulff
construction \cite{W, H, RW, BN, DKS} and, at very low temperatures, is
close to a rectangle proportional to $R(L_1^{\ast},L_2^{\ast})$. In spite of
this, escaping trajectories are passing through $\Cal P$ ---
 a configuration close to critical nucleus $R(L_2^{\ast},L_2^{\ast})$.

Moreover, we shall see that a typical first excursion
follows a rather well specified path that visits certain growing
rectangular, almost square, configurations in well defined moments.

To state this result, we first introduce  {\it a standard
tube} (of rectangles)  as a subset $\Cal T$ of $(\Bbb Z_+)^2$ consisting of
points corresponding either  to ``almost squares'' or ``large rectangles''
(with either $x_2=L_2^\ast$ or  $x_1=M$):
$$
\Cal T = \{\pmb  x\in (\Bbb Z_+)^2\: d(\pmb  x, \Cal L_1)\leq
1\}\cup\Cal L_2  \cup\Cal L_3.
\tag9
$$
Here
$$
\aligned
&\Cal L_1=\{(x_1,x_2)\in (\Bbb Z_+)^2\: 1\leq x_1=x_2\leq L_2^\ast-1\}\\
&\Cal L_2=\{(x_1,x_2)\in (\Bbb Z_+)^2\: x_2=L_2^\ast,
 L_2^\ast\leq x_1\leq M\}\\
 &\Cal L_3=\{(x_1,x_2)\in (\Bbb Z_+)^2\: x_1=M,
 L_2^\ast\leq x_2\leq M\}.
\endaligned
\tag10
$$

We call {\it a standard sequence of rectangles} any sequence
$\pmb  x^{(1)},\dots,\pmb  x^{(2M-1)}$, $\pmb  x^{(i)}\in
(\Bbb Z_+)^2$ such that
\roster
\item  $\{\pmb  x^{(i)}\}_{i=1,\dots, 2M-1}\in\Cal
T$,
\item
$\pmb  x^{(1)}=(1,1)$ and the sequence $\{\pmb  x^{(i)}\}_{i=1,\dots, 2M-1}$
is monotonous and  consists of nearest neighbours in the sense
$$
\pmb  x^{(i+1)}\equiv (x^{(i+1)}_1,x^{(i+1)}_2) = (x^{(i)}_1,x^{(i)}_2)+
\pmb  e,
$$
where $\pmb  e$ is either $\pmb  e_1=(1,0)$ or $\pmb  e_2=(0,1)$.
\endroster


%
%
\midinsert
\centerline{\picture 3.00in by 3.03in (three)}
\botcaption{Fig\. 3}
A standard sequence of rectangles.
\endcaption
\endinsert
%
%

Now, let $\bar\tau_{-\underline 1}$ be the last instant in which $\sigma_t=
-\underline 1$ before $\tau_{+\underline 1}$:
$$
\bar\tau_{-\underline 1}=\max\{t<\tau_{+\underline 1}\:
\sigma_t=-\underline 1\}.
\tag11
 $$
Let $\tau_0, \tau_1, \dots, \tau_n,\dots$ be random times after
 $\bar\tau_{-\underline
1}$ in which $\sigma_t$ visits the set $\Cal R$ of rectangular configurations
(after a change):
$$
\aligned
&\tau_0=\bar\tau_{-\underline 1}\\
&\tau_{n+1}=\min\bigl\{t>\tau_n\:\sigma_t\in\Cal
R\setminus\{\sigma_{\tau_n}\}\bigr\}, n=0,1,2,\dots
\endaligned
\tag12
$$
We say that $\sigma_t$ is an $\varepsilon${\it-standard path} if
\roster
\item $\sigma_{\tau_0}=-\underline 1$, $\{\sigma_{\tau_n}\}_{n=0,1,\dots}$ is a
standard sequence of rectangles,
\item random times $\tau_n$ satisfy the
following conditions:
\itemitem{(a)}
$
\tau_1<e^{\varepsilon\beta},\quad \tau_2-\tau_1<e^{\varepsilon\beta},
$
\itemitem{(b)}
$
\exp\{\beta(h(L -2)-\varepsilon)\leq \tau_n-\tau_{n-1}\leq
\exp\{\beta(h(L -2)+\varepsilon)\},
$\hfill (13)\linebreak
whenever
$\sigma_{\tau_n}\in\Cal R(L,L)$ for $2\leq L \leq L_2^\ast$,
\hfil\newline
or $\sigma_{\tau_n}\in\Cal R(L-1,L)$ for $3\leq L \leq L_2^\ast$,
\itemitem{(c)}
$
\exp\{\beta(2J_2-h-\varepsilon)\leq \tau_n-\tau_{n-1}\leq
\exp\{\beta(2J_2-h+\varepsilon)\},
$
\hfill (14)\linebreak
whenever
$\sigma_{\tau_n}\in\Cal R(L,L_2^\ast)$ for $L_2^\ast+1\leq L \leq
M$, and
\itemitem{(d)}
$
\exp\{\beta(2J_1-h-\varepsilon)\leq \tau_n-\tau_{n-1}\leq
\exp\{\beta(2J_1-h+\varepsilon)\},
$
\hfill (15)\linebreak
whenever
$\sigma_{\tau_n}\in\Cal R(M,L)$ for $ L_2^\ast+1\leq L \leq
M$.
\endroster

We use $\Cal S_\varepsilon$ to denote the set of all $\varepsilon$-standard
paths.

Our results about the asymptotics of the first excursion from $-\underline
1$ to
$+\underline 1$ are then summarized in the following theorem.


\proclaim{Theorem 3}
$$
\lim_{\beta\to\infty}P_{-\underline
1}(\{\sigma_t\}_{t=1,2,\dots}\in\Cal S_\varepsilon)=1.
\tag16
$$
\endproclaim



Before indicating the proof of our crucial result, namely Theorem 2,
let us state several lemmas. First, we use the reversibility to prove
that the time necessary to reach a particular configuration is
the exponential of the concerned energy difference multiplied by inverse
temperature.

\proclaim{Lemma 1}
For every $\varepsilon >0$ and any configuarations $\eta$, $\sigma$,
such that $H(\eta)>H(\sigma)$, one has
$$
\lim_{\beta\to\infty}
P_{\sigma}(\tau_{\eta}<\exp\{\beta(H(\eta)-H(\sigma)-\varepsilon)\})
=0.
\tag17
$$
\endproclaim

\demo{Proof}
Given $T\in\Bbb N$, one has
$$
\multline
P_{\sigma}(\tau_{\eta}< T)=\sum_{s=1}^{T-1}\sum_{\bar\sigma_1,\dots,
\bar\sigma_{s-1}\neq\eta}
\!\!\!\!
P(\sigma_0=\sigma, \sigma_1=\bar\sigma_1, \dots,
\sigma_{s-1}=\bar\sigma_{s-1},\sigma_s=\eta)=\\=
\exp\{-\beta(H(\eta)-H(\sigma))\}
\sum_{s=1}^{T-1}\sum_{\bar\sigma_1,\dots,
\bar\sigma_{s-1}\neq\eta}
\!\!\!\!
P(\sigma_0=\eta, \sigma_1=\bar\sigma_{s-1}, \dots,
\sigma_{s-1}=\bar\sigma_1,\sigma_s=\sigma)\leq\\
\leq T \exp\{-\beta(H(\eta)-H(\sigma))\}.
 \endmultline
\tag18
$$
To conclude the proof we choose
$$
T= \bigl[\exp\{\beta(H(\eta)-H(\sigma)-\delta)\}\bigr]
$$
with $\delta=\min(\varepsilon,\frac{H(\eta)-H(\sigma)}{2})$
and take $\beta$ sufficiently large.
\qed
\enddemo

For any $(L_1,L_2)\in\Bbb Z^2_+$, let us denote $l=\min(L_1,L_2)$ and
$L=\max(L_1,L_2)$. The following three lemmas are direct consequences of
Theorem 1 from \cite{NS} or of the arguments used in its proof.

The first lemma claims that the size of a critical droplet is $L_2^{\ast}$
and indicates what barrier one has to pass when starting from a local
minimum.

\proclaim\nofrills{Lemma 2 [NS]$\quad$}
Using $P_{L_1,L_2}$ to denote $P_\sigma$ with $\sigma\in\Cal R(L_1,L_2)$,
we have
$$
\lim_{\beta\to\infty} P_{L_1,L_2}(\tau_{-\underline 1}<\tau_{+\underline
1})=1 \tag19
$$
and
$$
\lim_{\beta\to\infty}
P_{L_1,L_2}(\tau_{-\underline 1}>\exp\{\beta(h(l-1)+\varepsilon)\})=0,
 \tag20
$$
whenever $L_1$ and $L_2$ are such that $l=\min(L_1,L_2)<L_2^{\ast}$
and $L=\max(L_1,L_2)<M$.
Further,
$$
\lim_{\beta\to\infty} P_{L_1,L_2}(\tau_{+\underline 1}<\tau_{-\underline 1})=1
\tag21
$$
and
$$
\lim_{\beta\to\infty}
P_{L_1,L_2}(\tau_{+\underline 1}>\exp\{\beta(2J_2-h+\varepsilon)\})=0,
 \tag22
$$
whenever $L_1$ and $L_2$ are such that $\min(L_1,L_2)\geq L_2^{\ast}$.
\endproclaim

The following lemma says that subcritical shrinking is isotropical.
Namely, starting from a subcritical rectangular configuration, it is very
probable that we will first cut a shorter edge in the time given by the
height of the corresponding barrier.

\proclaim\nofrills{Lemma 3 [NS]$\quad$}
Starting from $\sigma_0\in\Cal R(L_1, L_2)$, let
$$
\tilde\tau_{\Cal R}=\inf\{t>0\:\sigma_t\in\Cal
R\setminus\{\sigma_0\}\}.
\tag23
$$
If $l=\min(L_1,L_2)<L_2^{\ast}$ and $L=\max(L_1,L_2)<M$,
then
\roster
\item"i)"
$\lim_{\beta\to\infty} P_{L_1,L_2}(\sigma_{\tilde\tau_{\Cal R}}\in\Cal
R(L_1^{\prime},L_2^{\prime}))=1 $, whenever $L_1'$ and $L_2'$ are such that
\hfill\newline
if $L_1\neq L_2$, $\max(L_1^{\prime},L_2^{\prime})=L-1$, and
$\min(L_1^{\prime},L_2^{\prime})=l$,\hfill\newline
or \hfill\newline
if $L_1=L_2=L$, $\max(L_1^{\prime},L_2^{\prime})=L$, and
$\min(L_1^{\prime},L_2^{\prime})=L-1$,
\item"ii)"
$
\lim_{\beta\to\infty}
P_{L_1,L_2}(\exp\{\beta(h(l-1)-\varepsilon)\}<\tilde\tau_{\Cal
R}<\exp\{\beta(h(l-1)+\varepsilon)\})=1 $
\hfill\newline
for any $\varepsilon >0$.
\endroster
\endproclaim

Finally, the third lemma states that a supercritical droplet first grows
in  the ``easier'' direction, and only after $L_1$ hits $M$, the side $L_2$
starts to increase.

\proclaim\nofrills{Lemma 4 [NS]$\quad$}
Whenever $l\geq L_2^{\ast} $, $L_1<M$, and  $\varepsilon>0$, one has
$$
\lim_{\beta\to\infty} P_{L_1,L_2}(\sigma_{\tilde\tau_{\Cal R}}\in\Cal
R(L_1+1,L_2))=1
\tag24
$$
and
$$
\lim_{\beta\to\infty}
P_{L_1,L_2}(\exp\{\beta(2J_2-h-\varepsilon)\}<\tilde\tau_{\Cal
R}<\exp\{\beta(2J_2-h+\varepsilon)\})=1.
\tag25
$$
For  $L_1=M$, $L_2\geq 2$, and any $\varepsilon>0$ one has
$$
\lim_{\beta\to\infty} P_{L_1,L_2}(\sigma_{\tilde\tau_{\Cal R}}\in\Cal
R(M,L_2+1))=1
\tag26
$$
and
$$
\lim_{\beta\to\infty}
P_{L_1,L_2}(\exp\{\beta(2J_1-h-\varepsilon)\}<\tilde\tau_{\Cal
R}<\exp\{\beta(2J_1-h+\varepsilon)\})=1.
\tag27
$$
\endproclaim

\remark{Remark}
It is possible to prove a stronger version of Theorem 3 giving rise to a
more accurate description of the characteristics of typical paths of the
first excursion from $-\underline 1$ to $+\underline 1$.
In particular one can prove that, with probability going to 1 as
$\beta\to\infty$, during the transition from $-\underline 1$ to a
protocritical configuration (corresponding to the part $\Cal L_1$ of the
standard tube), the pluses form a connected cluster $C$ without holes and
with a monotone boundary $\partial C$.  Here, ``monotone'' means that
$\partial C$ intersects the four edges of its rectangular envelope $R(C)$
in four intervals, and its length equals that of the perimeter $R(C)$.
All these properties follow from stronger versions of Lemmas whose proof
can again be found in \cite{NS}.
\endremark



\head 3. Proof of Theorems\endhead

The most crucial  is  the proof Theorem 2.
It will consist of two steps.

First, we define a set $\Cal A\subset\{-1,1\}^{\Lambda}$ satisfying the
following three properties:
\roster
\item
For every $\sigma\in\Cal A$ and any $\varepsilon>0$ one has
$$
\lim_{\beta\to\infty}P_\sigma(\tau_{-\underline 1}<\tau_{+\underline 1})=1
\tag28
$$
and
$$
\lim_{\beta\to\infty}P_\sigma(\tau_{-\underline
1}<\exp\{\beta(h(L_2^\ast-2)+\varepsilon)\})=1.
\tag29
$$
\item
Any path $\{\sigma_t\}_{t\in\Bbb N}$ such that $\sigma_0=-\underline 1$
and $\sigma_t=+\underline 1$ for some $t$ has to pass through
the ``boundary'' $\partial\Cal A$ of the set $\Cal A$ defined by
$$
\partial\Cal A=\{\eta=\sigma^{(x)}\text{ for some } x;\,\, \sigma\in\Cal A,
\eta\notin\Cal A\} .
\tag30
$$
Namely, there exists $s<t$ such that
$\sigma_s\in\partial\Cal A$.
\item
The minimal energy in $\partial\Cal A$ is attained for
``protocritical'' (global saddle) configurations $\sigma \in \Cal P$; namely,
$$
\aligned
\min_{\sigma\in \partial\Cal A}(&H(\sigma)-H(-\underline 1))=
H(\Cal P)-H(-\underline 1)=\Gamma
\\
\min_{\sigma\in \partial\Cal A\setminus \Cal P}(&H(\sigma)-H(\Cal P))\geq
h>0.
\endaligned
\tag31
$$
\endroster

The second step will be to prove, for any $\varepsilon>0$, that
before the time given by the upper bound from (8), one certainly reaches
the boundary of $\Cal A$; namely,
 $$
\lim_{\beta\to\infty}P_{-\underline 1}(\tau_{\partial\Cal A}\geq
T(\varepsilon))= 0
\tag32
 $$
with
$$
T(\varepsilon)=\exp\{\beta(\Gamma +\varepsilon)\}.
\tag33
$$


Once the set $\Cal A$ satisfying the conditions (1)--(3) is
constructed and the equality (32) is assured,
the proof can be easily completed.

Indeed, starting from $\sigma\in\Cal P$, the probability of flipping a
spin $-1$ adjacent to the unit square proturberance in such a way that
a stable ``proturberance of length 2'' is created, is
not smaller than $1/|\Lambda|$ (see \cite{NS} for more details).
Then, for any $\varepsilon>0$, it follows from Lemmas 2 and 4 that
the probability to reach $+\underline 1$ before reaching $-\underline 1$,
and to reach it in a time needed to create a minimal proturberance,
can be bounded from below:
$$
\aligned
&P_{\Cal P}(\tau_{+\underline 1}<\tau_{-\underline 1})\geq
\frac{1}{|\Lambda|} \\
\lim_{\beta\to\infty}&P_{\Cal P}(\tau_{+\underline
1}<\exp\{2J_2-h+\varepsilon\}\mid \tau_{+\underline 1}<\tau_{-\underline 1})
=1.
 \endaligned
\tag34
$$
On the other hand, Lemma 1 and the property (3) of $\Cal A$ imply that
one needs much longer time than $T(\varepsilon)$ to reach $\partial\Cal
A\setminus\Cal P$,
$$
\lim_{\beta\to\infty}P_{-\underline 1}(\tau_{\partial\Cal
A\setminus\Cal P}< \exp\{\beta(\Gamma +h-\varepsilon)\})= 0.
\tag35
$$

Clearly,
$$
P_{-\underline 1}(\tau_{\partial\Cal
 A\setminus\Cal P}< \tau_{\partial\Cal A})\leq
P_{-\underline 1}(\tau_{\partial\Cal
 A\setminus\Cal P}< T(\varepsilon))+
P_{-\underline 1}(\tau_{\partial\Cal
 A}\geq T(\varepsilon)).
\tag36
$$
Taking $\varepsilon <\frac{h}{2}$, the equation (35) implies that the
first term on the right hand side of (36) vanishes.
Thus the relations (34), (36), and the  strong Markov property allow to
reduce the  proof of the equality
$$
\lim_{\beta\to\infty}P_{-\underline 1}(\tau_{+\underline 1}>
T(\varepsilon))= 0
\tag37
$$
to the proof of (32).

Finally, from Lemma 1 and the properties (2) and (3) of $\Cal A$ it
follows directly that one cannot reach $+\underline 1$ in a too short time,
$$
\lim_{\beta\to\infty}P_{-\underline 1}(\tau_{+\underline 1}<
\exp\{\beta(\Gamma-\varepsilon)\})= 0.
\tag38
$$

Thus, to complete the proof, it remains to construct the set $\Cal A$
and to prove the equality (32).


 \specialhead
3.1. The construction of $\Cal A$
\endspecialhead

First we introduce the notion of   acceptable
configurations $\sigma\in\{-1,1\}^\Lambda$. Any configuration
$\sigma$ can be identified with the collection $\{C_1,\dots,C_k\}$ of its
maximal connected components of plus spins (considering the union of all
closed unit squares centered at the sites occupied by plus spin). To any
such component $C$ we assign its {\it rectangular envelope} defined as the
minimal closed rectangle $R(C)$ (with edges parallel to the coordinate axes
and vertices on the dual lattice)  containing $C$. As before, we consider a
strip winding around  the torus to be a rectangle with a side of  length
$M$. If none of the rectangles $R(C_1),\dots, R(C_k)$ is winding around the
torus, we call the corresponding configuration {\it acceptable}.



For any acceptable configuration $\sigma$, there always exists a unique
component of minuses winding around the torus.
The contours touching it are {\it outer contours}.
Given any outer contour $\gamma$, we use $C(\gamma)$ to denote the
region enclosed in it and $R(\gamma)$ to denote the rectangular envelope
$R(C(\gamma))$. Notice that every edge of $R(\gamma)$ contains
at least one unit segment belonging to $\gamma$.

Now, for any acceptable $\sigma$, we shall construct a new configuration
$$
\hat\sigma = S\sigma
$$
by  ``filling up'' and ``gluing'' together  some its rectangular envelopes.
To this end we first introduce the notion of interacting rectangles and
 chains of them.
Two rectangles $R=R(L_1, L_2)$ and $R^{\prime}=
R(L_1^{\prime}, L_2^{\prime})$ are said to be
{\it interacting} if one of the following three possibilities occurs:
\roster
\item"i)"
the rectangles $R$ and $R'$ intersect, or
\item"ii)"
 there exists a unit square centered at some lattice site such that
one  its vertical edge is contained in $R$ and the other in $R^{\prime}$, or
\item"iii)" there exists a unit square centered at some lattice site such that
one  its horizontal edge is contained in $R$ and the other in $R^{\prime}$
 and, in the same time,
$\min(L_1, L_2,L_1^{\prime}, L_2^{\prime}) \geq l^{\ast}$,
where
$$
 l^{\ast}=\Bigl[\frac{2(J_1-J_2)}{h}\Bigr]+1.
$$
(Neither $R$ nor $R^{\prime}$ is {\it ephemere}.)
\endroster
A set of rectangles $R_1,\dots, R_m$ is said to form a {\it chain} $\Cal C$
if  every pair $(R_i, R_j)$ of them
can be linked by a sequence $\{R_{i_1},\dots, R_{i_n}\}$
of pairwise interacting  rectangles from $\Cal C$; $R_{i_1}=R_i$,
$R_{i_n}=R_j$, and $R_{i_{l}}$
and  $R_{i_{l+1}}$ are interacting for all $l=1,\dots, n-1$.

Given a collection of chains $\Cal C_1,\dots,\Cal C_n$ we start the
following iterative procedure:
\roster
\item
The chains $\Cal C^{(1)}_j$ of the ``first generation'' are
identical to   $\Cal C_j$,
$j=1,\dots,n$.
\item
Having defined $\Cal C^{(r)}_j$, we construct  rectangular envelopes
$R^{(r)}_j$ of the sets
$$
\bigcup_{R\in\Cal C^{(r)}_j}R
$$
and the maximal chains $\Cal C^{(r+1)}_j$ of them.
\endroster
The procedure ends once we reach a set of chains, each consisting of a
single rectangle.
Notice that every pair from the resulting set of noninteracting
rectangles
$\bar R_1,\dots,\bar R_s$ is such that either
\roster
\item"$\bullet$"
their distance is at least $\sqrt 2$, or
\item"$\bullet$"
(if their distance is $1$) they are either ``almost touching by corners''
 (see Fig\. 4A)  or  they are placed at distance $1$ in vertical direction
and at least one of the two, say $R(L_1, L_2)$, is ephemere,  $\min(L_1,
L_2) < l^{\ast}$ (see Fig\. 4B).
 \endroster

%
%
\midinsert
\centerline{\picture 3.90in by 1.22in (four)}
\botcaption{Fig. 4a \phantom{xxxxxxxxxxxxxxxxxxxxx}Fig. 4b}
\endcaption
\endinsert
%
%

Starting now from any acceptable configuration $\sigma$, we apply the above
construction on chains of rectangular envelopes of its outer contours
and define $\hat \sigma$ as the configuration obtained by placing the spin
$+1$ at all sites inside the resulting rectangles  $\bar R_1,\dots,\bar R_s$
(filling up the rectangles).

It is easy to verify that
$$
H(\sigma)\geq H(\hat\sigma).
\tag39
$$
Indeed, notice  that whenever  a configuration $\xi$
has contours $\gamma'$, $\gamma''$ with interacting rectangular envelopes
$R'=R(\gamma')$,  $R''=R(\gamma'')$, we will decrease the energy by filling
  the rectangular envelope of the union of $R'$ and $R''$.
This is evident in the case i) of the definition of interacting rectangles
(the number of horizontal and vertical bonds, separately, is nonincreasing,
the volume occupied by pluses is increasing) and in the case ii)
(flipping the minus spin in the centre of the unit square touching
$R'$ and $R''$
the energy decreases since $J_1\geq J_2$). In the case iii) we observe that
when filling the rectangular envelope of $R'\cup R''$ with pluses, one
gains at least $hl^{\ast}$ that suffices, according to the definition of
$l^{\ast}$, to compensate the loss of no more than $2(J_1-J_2)$.
   Using  this
observation  in an iterative manner, we can construct a sequence of
configurations of decreasing energy starting with $\sigma$ and ending with
$\hat \sigma$.

Now we are ready to define the set $\Cal A$.
Namely, we introduce $\Cal A$ as the set of all configurations $\sigma$
such that every resulting rectangle $\bar R(L_1,L_2)$ from the configuration
$\hat \sigma$ is subcritical, $l=\min (L_1,L_2)< L_2^{\ast}$ and
$L=\max(L_1,L_2)<M$.

The property 1) of the set $\Cal A$ is a direct consequence
of the definition of $\Cal A$ and of Lemma 2. Since the property 2)
is obvious, again directly from the definition,
 our next task is
to analyse the boundary $\partial \Cal A$ and to prove the property 3). Let
$\eta \in\partial\Cal A$. Then there exists $\sigma\in\Cal A$ and  $x$ such
that $\eta=\sigma^{(x)}\notin\Cal A$. It is clear that the mapping
$\xi\to\hat\xi$ is monotonous. Namely, if $\xi_1\prec \xi_2$ (i.e., by
definition, $\{x\mid \xi_1(x)=+1\}\subset \{x\mid \xi_2(x)=+1\}$), then also
$\hat\xi_1\prec \hat\xi_2$. As a consequence $\sigma(x)$ is necessarily
$-1$; otherwise $\sigma \in\Cal A$ would imply also $\eta \in\Cal A$.
Moreover, the site $x$ lies outside  of all rectangles $\bar R_1, \dots ,
\bar R_s$ corresponding to $\hat\sigma$.  Among the rectangles corresponding
to $\hat\eta$ there  exists a rectangle $\bar R(L_1,L_2)$ with the following
properties:
\roster
\item"i)" $\bar R$ is supercritical, $\min(L_1,L_2)\geq  L_2^{\ast}$
or $2\leq\min(L_1,L_2)<  L_2^{\ast}$  and $L=\max(L_1,L_2)=M$,
\item"ii)" it contains the site $x$ and several rectangles $\bar R_i$, say,
$\bar R_1, \dots , \bar R_k$, corresponding to $\hat\sigma$; the remaining
rectangles $\bar R_{k+1}, \dots , \bar R_s$ are also rectangles of
$\hat\eta$.
\endroster

Our aim now is  to prove that
$$
H(\eta)-H(-\underline 1)\geq (2J_1+2J_2)
L_2^\ast-h\bigl[(L_2^\ast)^2-L_2^\ast+1\bigr].
\tag40
$$
If $\bar R$ is winding around the torus, the inequality (40) is clearly
satisfied (the overall length of the boundaries of rectangles
$\bar R_1, \dots , \bar
R_k$ is necessarily at least $M$.)
Let us thus suppose that $\bar R$ is not wrapped around the torus.
Consider first
the configuration  $\tilde\eta$, $H(\eta) \geq H(\tilde\eta)$, whose set of
pluses  consists of the site $x$ and the rectangles $\bar R_1, \dots , \bar
R_k$ (energy decreases when skipping the subcritical rectangles
$\bar R_{k+1}, \dots , \bar R_s$).
Further, consider the set $C^{(0)}$ consisting of the union of the unit
square $q(x)$ centered at the site $x$ and those rectangles among
$\bar R_1, \dots , \bar R_k$ that intersect $q(x)$ along its edge.
Let us take now the rectangular envelope $\bar R^{(1)}$ of $C^{(0)}$
and distinguish two cases, either the rectangle $\bar R^{(1)}$ is
supercritical or not. (Notice that both, $C^{(0)}$ and $\bar R^{(1)}$
may actually coincide with $q(x)$.)

If $\bar R^{(1)}$ is supercritical, we decrease the energy of $\tilde\eta$
further by erasing all rectangles among $\bar R_1, \dots , \bar R_k$
that were not contributing to the set $C^{(0)}$ and consider the
configuration yielded by the set $C^{(0)}$. Since $\bar R^{(1)}$ is
supercritical, there exists an
$L_2^{\ast} \times L_2^{\ast}$ square $Q^{\ast}$
contained in $\bar R^{(1)}$ and containing the site $x$.
Given the fact that
the rectangles $\bar R_1, \dots , \bar R_k$ are mutually noninteracting,
there are at most two among these rectangles that contribute to
$C^{(0)}$. Moreover, if they are two, they intersect the unit square $q(x)$
along its opposite edges. Hence, always either the row or the column passing
through $x$ intersect the set $C^{(0)}$ only in $q(x)$. We decrease the
energy
 by shrinking furher the configuration corresponding to  $C^{(0)}$
to its intersection with $Q^{\ast}$ (taking into account  the fact
that the concerned rectangles are subcritical), obtaining thus the
configuration $\bar\eta$ whose boundary necessarily contains
$2L_2^{\ast}$ horizontal and $2L_2^{\ast}$ vertical bonds (as $Q^{\ast}$
does), while its pluses are covering the area that is at least by
$L_2^{\ast}-1$ sites of the mentioned row or column smaller then the area
of $Q^{\ast}$. This yield the bound (40)

Next, consider the case when $\bar R^{(1)}$ is subcritical. Take $\bar R^{(1)}$
and all rectangles among
$\bar R_1, \dots , \bar R_k$ that were not used for $C^{(0)}$ and
 construct from them  the set of chains $\Cal C_j^{(1)}$
of the first  generation.
A sequence $\Cal C_j^{(r)}, r=1,\dots , m$ of chains of following
generations is obtained from it by iteration.
Since the rectangles $\bar R_1, \dots , \bar R_k$ are mutually
noninteracting, for every generation $r$ we get a chain, say
$\Cal C_1^{(r)}$, consisting of a rectangle $\bar R^{(r)}$ containing
the site $x$ and certain subset of $\bar R_1, \dots , \bar R_k$.
The remaining chains $\Cal C_j^{(r)}, j=2,\dots $ contain each just one
rectangle from those among $\bar R_1, \dots , \bar R_k$ that have not
appeared
in $\Cal C_1^{(p)}$, $ p\leq r$, in the preceding steps.
Clearly, there is only one chain in the last generation,
$\Cal C_1^{(m)}=\{\bar R^{(m)}\}\equiv \{\bar R \}$.

Let us consider now the last rectangle $\bar R^{(p)}$ among $\bar R^{(r)}$,
$r=1,\dots,m$ that is subcritical and  take the chain $\Cal C_1^{(p)}$
with the rectangles in $\Cal C_1^{(p)}\setminus \bar R^{(p)}$
ordered in a particular way, say in  lexicographic order of their
left upper corner.
Let us unite them,  one by one in the given order, with the rectangle $\bar
R^{(p)}$ until the circumscribed rectangle is supercritical.
Cutting off the remaining rectangles from the chain $\Cal C_1^{(p)}$
we get the chain
$\tilde\Cal C_1^{(p)}\subset \Cal C_1^{(p)}$.
Let us use $\tilde R'$ to denote the last rectangle that was attached to
form the chain
$\tilde\Cal C_1^{(p)}$ and $\tilde R$ the circumscribed rectangle to
the union of rectangles from $\tilde\Cal C_1^{(p)}\setminus \{\tilde R'\}$.
Clearly, $\tilde R$ and $\tilde R'$ are subcritical interacting rectangles
with a supercritical envelope $R^{\ast}$ of their union. Hence, there exists
an $L_2^{\ast} \times L_2^{\ast}$ square $Q^{\ast}$
contained in $R^{\ast}$ such that it intersects both rectangles $\tilde R$
and $\tilde R'$
in nondegenerated rectangles $R$ and $R'$, $R=\tilde R\cap Q^{\ast}$ and
$R'=\tilde R'\cap Q^{\ast}$,
and contains the intersection $\tilde R\cap\tilde R'$ (if it is nonempty).
One has
$$
H(\tilde\eta)\geq H(R) + H(R'),
$$
where $H(R)$ and $H(R')$ denotes the energy of the configuration with
pluses in $R$ and $R'$, respectively.
To see this, it is enough to realize that the energy of $\tilde\eta$ is
certainly higher than the energy of all among original rectangles $\bar
R^{(1)}, \bar R_1, \dots , \bar R_k$ that were subsequently used in the
construction of $\tilde\Cal C_1^{(p)}\setminus \{\tilde R'\}$ plus the
energy associated to the rectangle $\tilde R'$ (notice that $\tilde R'$
is not intersecting the remaining rectangles from the original set
$ \bar R_1, \dots , \bar R_k$ used to construct $\tilde\Cal
C_1^{(p)}$ and it can only touch by its corners the set $C^{(0)}$). The first
term can be subsequently bounded from below by $H(\tilde R)$ and we get the
above inequality observing that $\tilde R$ and $\tilde R'$ are subcritical
and thus  one gains energy by reducing them to $R$ and $R'$.

Now, if we replace the rectangles $R$ and $R'$ by their
circumscribed rectangle $Q^{\ast}$, the sum on the right hand side above
decreases by at least  $(L_2^{\ast}-1)h$ leading thus to the inequality (40).


Indeed, if the rectangles $R$ and $R'$ intersect in more than one point,
there is a surplus of at least two bonds in the sum of their boundaries
yielding at least $2J_2>(L_2^{\ast}-1)h$. If $R$ and $R'$ just touch
in the corner, the boundary has the same number of horizontal and vertical
bonds  as in $Q^{\ast}$ and there is at least $2(L_2^{\ast}-1)$ minus sites
inside of $Q^{\ast}$.
If $R$ and $R'$ are interacting according to the case ii) from the definition
of interacting rectangles, the surplus of at least two vertical edges
compensates for the lack of two horizontal edges, while there is at least
$L_2^{\ast}$ minuses inside  $Q^{\ast}$.


 Finally, consider the case iii).
Notice first that since $\tilde R$ and $\tilde R'$ are interacting and
subcritical, the appearance of the case iii) necessarily means that
$L_2^{\ast}>l^{\ast}$, namely, the coupling constants satisfy the inequality
$J_1 <2 J_2$.
The rectangles $R$ and $R'$ are separated by a row of minuses and there must
exist
 a unit
square $q$ (in the concerned row) whose opposite horizontal edges intersect
$R$ and $R^{\prime}$. The column passing through this square intersects
the boundary of  $R\cup R^{\prime}$ in at least four horizontal bonds ---
two of them are the edges of $q$.
Suppose first that this is the only such column (and $q$ is the only unit
square with the property stated above), the rows below and above the
considered separating row contain together at least  $L_2^{\ast}-1$ minuses
(in addition to $L_2^{\ast}$ minuses in the concerned row)  in $Q^{\ast}$
(see Fig\. 5).
Then we see that, in the configuration with pluses at all sites inside
$Q^{\ast}$,
 the at least four
horizontal edges in the considered column are replaced by only two,
 with two new vertical edges added in the considered row.
The possible increase in energy associated with the
replacement of two ``weak'' horizontal edges by two ``strong''
vertical edges is  at most $2(J_1-J_2)$ and  is compensated
 filling up $L_2^{\ast}$ minuses
of the concerned row (recall that in the present case $J_1 < 2J_2$ and
thus
$2(J_1-J_2)<L_2^{\ast}h$ by definition of $L_2^{\ast}$).
 The additional
$L_2^{\ast}-1$ bonds found above and below  the  considered row of
minuses will then contribute to the second term on the right hand side of
(40) leading to its verification in the considered case.
If there exist at least two unit squares $q_1$ and $q_2$ with the
property formulated above, one has at least four surplus horizontal edges
to compensate for the only two additional vertical edges
(again, use $J_1 < 2J_2$). The $L_2^{\ast}$
minuses of the concerned row are then used for the additional term
$(L_2^{\ast}-1)h$ when comparing
 the right hand side of (40) with the energy of $Q^{\ast}$.
%
%
\midinsert
\centerline{\picture 2.53in by 2.53in (five)}
\botcaption{Fig\. 5}
\endcaption
\endinsert
%
%





 Thus, we are
left with the task of finding  configurations for which the energy equals
the right hand side in (40).
Thus, let us suppose that $\eta$ satisfies the equality in (40).
Then, necessarily, $\bar R^{(1)}$  in the construction above is
supercritical.
If it were not and other steps of the construction and filling of final
chains would follow, we would run into contradiction since in every of those
steps the energy strictly decreeses with, as the proof shows, the
inequality (40) maintained. This excludes a possibility to have an
equality for the starting configuration $\eta$.
Similar reasoning also shows that $\eta=\bar\eta$.
Moreover, if two rectangles from among  $\bar R_1, \dots , \bar R_k$
 were  contained in $C^{(0)}$, the bound (40) would be valid with the strict
inequality. Hence, $C^{(0)}$ consists of $q(x)$ attached to a single
rectangle and to get the equality in (40) it must be
 $(L_2^{\ast}-1) \times L_2^{\ast}$ rectangle (see Fig\. 2).
Thus, the only possibility to achieve an equality in (40) is to take
 $\eta\in\Cal P$
and we can conclude that
$$
\min_{\sigma\in\partial\Cal A}H(\sigma) = H(\Cal P)
$$
and
$$
\min_{\sigma\in\partial\Cal A\setminus \Cal P}H(\sigma) = H(\Cal P)+h,
$$
namely, the property 3) of $\Cal A$.

\remark{Remark}
It is easy to convince oneself that, for every $\sigma$ in $\Cal P$,
there exists a path $\bar\omega$, starting in $\sigma$ and ending in
$+\underline 1$, such that it first passes through
$\Cal R(L_2^{\ast},L_2^{\ast})$, then it visits the whole sequence of
rectangles $R(L_1,L_2)$ in the sets $\Cal L_2$ and $\Cal L_3$ of the
standard tube (see the equation (10)) as well as the saddle points
$P(L_1,L_2^{\ast};L_1+1,L_2^{\ast})$ and $P(M,L_2;M,L_2+1)$ between them,
and in the same time
$$
\max_{\sigma\in\bar\omega}H(\sigma)=H(\Cal P).
$$
In other words, along this path the energy is almost monotonously
decreasing with some uphill jump over barriers
that never overcome the initial height.

It follows from the properties (2) and (3) of the set $\Cal A$ and from the
existence of the above mentioned path
$\bar\omega$ that
$$
H(\Cal P)=\min_{\omega\: -\underline 1\to +\underline
1}\max_{\sigma\in\omega}H(\sigma),
$$
so that, indeed, any $\sigma$ in $\Cal P$ is a global saddle point. On the
other hand, again by the property (3) of $\Cal A$, it follows that any such
a global saddle point is in $\Cal P$.
\endremark



\specialhead
3.2. Proof of the equality (32)
\endspecialhead


The strategy of the proof is as follows:
we introduce  an event $\Cal E_\sigma$
consisting of a set of trajectories starting from $\sigma \in \Cal A$,
taking place over an interval of time $T_1 $, and such that
\roster
\item
if $\Cal E_\sigma$ takes place,  one necessarily reaches the set
$\partial \Cal A$ (in a particular manner) before the time $T_1$.
\endroster
Moreover, we suppose that there is a lower bound on the probability
of this event such that, attempting many times the event
$\Cal E_\sigma$, we can conclude with the help of strong Markov property
that after a time $T_2>T_1$ it happens with a high probability.
Namely, we are assuming that
\roster
 \item"(2)"
one has a uniform lower bound for the probability
$\inf_\sigma P(\Cal E_\sigma)\geq \alpha(T_1)$
such that
$$
\lim_{\beta\to\infty}(1-\alpha(T_1))^{\frac{T_2}{T_1}}=0,
\tag 41
$$
\endroster
Hence, if we succeed in choosing times $T_1, T_2$ and the event $\Cal
E_\sigma$ so that the conditions (1) and (2) are satisfied, we will be able
to conclude  that, with probability approaching 1
as $\beta\to\infty$, one has to reach $\partial \Cal A$ before $T_2$.



Next we pass to the construction of the event $\Cal E_\sigma$.
 It can be quite special, once a correct lower bound on its probability is
satisfied.

The first portion of $\Cal E_\sigma$ is an essentially downhill path from
$\sigma$ to $-\underline 1$. Namely,
for every $\sigma\in\Cal A$, $t_1\in\Bbb N$, we define
$$
\Cal E^{(1)}_{\sigma, t_1}=\{\omega\in\Omega\:\sigma_0= \sigma,
\tau_{-\underline 1}=t_1\}.
\tag 42
$$
Next portion of the event means simply that one is staying in the
configuration $-\underline 1$;
 for every $t_2 > t_1$, $t_2\in\Bbb N$, we
set
$$
\Cal E^{(2)}_{t_1,t_2}=\{\omega\in\Omega\:\sigma_t= -\underline
1,t_1\leq t\leq t_2 \}.
  \tag 43
$$
Now comes a very particular growth to $\Cal P$ starting from $-\underline
1$.
Namely, our aim is to consider a set of paths passing through a standard
sequence of rectangles, reaching the rectangles in random times of
particular orders. The orders of random times are chosen so that,
roughly speaking, at every basin of attraction of a particular rectangle
one is allowed to stay for a time proportional to the exponent of the
product of inverse temperature and the height of the energetical barrier
that prevents an erosion and after that one reaches in a shortest possible
time the local saddle point toward an enlarged rectangle. This saddle point
is higher than the saddle toward the eroded rectangle and the exponent of
the difference of the energies of these two barriers will be the main
ingredient for the lower bound on the probability of the event
$\Cal E_{\sigma}$. To be more precise,
 simplifying the
notation and writing $L^{\ast}$ for $L^{\ast}_2$ and  $Q_{L_1,L_2}$ for
the rectangle with horizontal edge $L_1$ and vertical edge $L_2$ and with the
upper left corner in the point $(-\frac{1}{2},+\frac{1}{2})$ (the origin of
$\Bbb Z^2$ is the first site $x$ in $Q$; the edges of $Q$ lie on the dual
lattice),
for every $t_2\in\Bbb N$ we set
$$
\Cal E^{(3)}_{t_2}=\{\omega\in\Omega\:\sigma_{t_2}= -\underline
1,\sigma_{t_2+1}=Q_{1,1},  \sigma_{t_2+2}=Q_{1,2}, \sigma_{t_2+4}=Q_{2,2}\}.
  \tag 44
$$
This is the portion of the path starting with $-\underline 1$ and growing
to $Q_{2,2}$.  Further, for every $T_0 < t_{2,2} < t_{2,3} < t_{3,3} < \dots
 < t_{L^{\ast}-1,L^{\ast}}$ ($t_{L_1,L_2}\in\Bbb N$) we set
$$
\Cal
E^{(4)}_{t_2, t_{2,2},t_{2,3},\dots,t_{L^{\ast}-1,L^{\ast}}}
= \Cal E_{2,2}\cap \Cal E_{2,3}\cap\dots\cap
\Cal E_{L,L}\cap \Cal E_{L,L+1}\cap\dots\cap
\Cal E_{L^{\ast}-1,L^{\ast}-1}\cap
\Cal E_{L^{\ast}-1,L^{\ast}},
  \tag 45
$$
where, for every $2\leq L\leq L^{\ast}-1$,
$$
\multline
\Cal E_{L,L}=\{\omega\in\Omega \: \sigma_{T_{L,L}}=
Q_{L,L}, \sigma_t\in \Cal B(Q_{L,L}) \text{ for every }\\
t\in [T_{L,L},T_{L,L}+t_{L,L}-T_0],
\sigma_{T_{L,L}+t_{L,L}}=Q_{L,L+1}\},
\endmultline
\tag 45$'$
$$
$$
\multline
\Cal E_{L,L+1}=\{\omega\in\Omega \: \sigma_{T_{L,L+1}}=
Q_{L,L+1}, \sigma_t\in \Cal B(Q_{L,L+1}) \text{ for every }\\
t\in [T_{L,L+1},T_{L,L+1}+t_{L,L+1}-T_0],
\sigma_{T_{L,L+1}+t_{L,L+1}}=Q_{L+1,L+1}\},
\endmultline
\tag 45$''$
$$
 and
$$
\multline
\Cal E_{L^{\ast}-1,L^{\ast}}=\{\omega\in\Omega \:
\sigma_{T_{L^{\ast}-1,L^{\ast}}}=
Q_{L^{\ast}-1,L^{\ast}}, \sigma_t\in \Cal B(Q_{L^{\ast}-1,L^{\ast}})
 \text{ for
every }\\ t\in
[T_{L^{\ast}-1,L^{\ast}},T_{L^{\ast}-1,L^{\ast}}+t_{L^{\ast}-1,L^{\ast}}-T_0],
\sigma_{T_{L^{\ast}-1,L^{\ast}}+t_{L^{\ast}-1,L^{\ast}}}=S_{L^{\ast}}\}.
\endmultline
\tag 45$'''$
$$
Here
$$
T_{2,2}=t_2+4,
$$
$$
T_{L,L+1}=t_2 +4 +t_{2,2}+t_{2,3}
+\dots + t_{L,L}
$$
for every $2\leq L \leq L^{\ast}-1$, and
$$
T_{L,L}=T_{L-1,L}+ t_{L-1,L}
$$
for every $3\leq L \leq L^{\ast}-1$.
The set $S_L$ is for every $L\leq L^{\ast}$ obtained by adding a unit square
to the vertical right hand edge of $Q(L-1,L)$, $S_L\in \Cal P(L,L-1;L,L)$.
The time $T_0$ is chosen so that
 $\frac{T_0}{2}$ is an upper bound on the time needed to
monotonously decrease the energy from any configuration $\sigma$ to any
other (say $\eta$) through a path of ``nearest neighbour
configurations'',
$$
T_0=\Bigl[\frac{2}{h}\bigl(\max_{\sigma\in\{-1,1\}^{\Lambda}} H(\sigma)
-\min_{\sigma'\in\{-1,1\}^{\Lambda}} H(\sigma')\bigr)\Bigr]+1.
\tag 46
$$
Of course, $S_{L^{\ast}}\in\Cal P$.

Further we define
$$
\Cal E^{(4)}_{t_2}=\bigcup_{n_{2,2}=1}^{\bar n_{2,2}}
\bigcup_{n_{2,3}=1}^{\bar n_{2,3}}\cdots
\bigcup_{n_{L^{\ast}-1,L^{\ast}}=1}^{\bar n_{L^{\ast}-1,L^{\ast}}}
\tilde\Cal E_{n_{2,2},\dots,n_{L^{\ast}-1,L^{\ast}}},
\tag 47
$$
where
$$
\tilde\Cal E_{t_2,n_{2,2},\dots,n_{L^{\ast}-1,L^{\ast}}}=
\Cal
E^{(4)}_{t_2,t_{2,2}=n_{2,2}T_0,\dots,t_{L^{\ast}-1,L^{\ast}}=
n_{L^{\ast}-1,L^{\ast}}T_0},
\tag 47$'$
$$
and
$$
\bar n_{L-1,L-1}=\bar n_{L-1,L}=\bigl[\exp\{\beta[h(L-2)+\delta]\}\bigr]
\tag 47$''$
$$
for every $L=1,\dots,L^{\ast}$.
Choosing now the times
$$
\bar t_1=\bar t_2 = \bigl[\exp\{\beta[h(L^{\ast}-2)+\delta]\}\bigr],
\tag 48
$$
we define
$$
\Cal E_{\sigma}= \bigcup_{t_1=1}^{\bar t_1}\Cal E^{(1)}_{\sigma,t_1}
\bigcap \bigl(\bigcup_{t_2=t_1+1}^{t_1+\bar t_2}\bigl[\Cal
E^{(2)}_{t_1,t_2}\cap\Cal E^{(3)}_{t_2}\cap\Cal E^{(4)}_{t_2} \bigr]
\bigr).
\tag 49
$$
The constant $\delta$ will be fixed later when also the reason for this
particular choice of the constants $\bar n$ will be apparent..


We use $\Cal B(Q)$ to denote the set of all connected clusters $C$ of pluses
whose rectangular envelope is $Q$ and such that $\partial C$ contains at
least a segment of length not shorter than $2$ in any edge of $Q$.
The set  $\Cal B(Q)$  is a subset of the basin of attraction of $Q$ in the
sense that any sequence of spin flips decreasing the energy and leading to
some rectangle $R$ necessarily is such that $R\equiv Q$.

The crucial point in the lower bound on $P(\Cal E_{\sigma})$ will be the
following inequality:
$$
P(\{\omega\:\sigma_0=Q_{L-1,L}; \sigma_s\in\Cal B(Q_{L-1,L}) \text{ for
every } s\leq t\})\geq
\bigl(  1-e^{\varepsilon\beta}e^{-h(L-2)\beta}\bigr)
^t
\tag50
 $$
and, similarly,
$$
P(\{\omega\:\sigma_0=Q_{L,L}; \sigma_s\in\Cal B(Q_{L,L}) \text{ for
every } s\leq t\})\geq
\bigl(  1-e^{\varepsilon\beta}e^{-h(L-1)\beta}\bigr)
^t.
\tag51
 $$

To get the estimate (50) ((51) is completely analogous), we introduce,
following   Freidlin and Wentzell\cite{FW}, an auxilliary Markov chain whose
space of states is
$$
\Cal X =Q\cup \partial \Cal B
$$
(to simplify the notation we write $Q$ for $Q_{L-1,L}$ and $\Cal
B$ for $\Cal B(Q_{L-1,L})$).
The boundary $\partial\Cal B$ of $\Cal B$ is given by
$$
\partial \Cal B= \{\eta=\sigma^{(x)} \text{ for some } x\:\eta\notin \Cal
B, \sigma\in\Cal B\}.
\tag 52
 $$
We introduce a sequence of times
$$
v_0 <u_0 \leq v_1<u_1\leq v_2 <\dots
$$
with $v_0=0$, $u_i, v_i\in \Bbb N$,
$$
\aligned
&u_n=\inf\{t> v_n\:\sigma_t\neq \sigma_{t-1}\}\\
&v_n=\inf\{t\geq u_n\:\sigma_t\in\partial\Cal B \cap Q\}.
\endaligned
\tag53
$$
We set
$$
\xi_n =\sigma_{v_n}, \xi_0=\sigma_0=Q,
\tag54
$$
$$
\nu=\inf\{n\: \xi_n\in\partial\Cal B\}.
\tag55
$$

For every $s\in\Bbb N$ one has
$$
P_Q(\tau_{\partial \Cal B}> s)\geq P_Q(\nu> s)= P(Q\to Q)^s = \bigl[
1- P(Q\to \partial \Cal B)\bigr]^s,
\tag56
 $$
where
$$
P(Q\to Q)= P(\xi_1=Q\mid \xi_0=Q)
\tag57
$$
and
$$
P(Q\to \partial \Cal B)= \sum_{\rho\in\partial \Cal
B}P(\xi_1=\rho\mid\xi_0=Q). \tag58
$$
For every $\varepsilon >0$ we have
$$
\multline
P(Q\to \partial \Cal B)\leq \sum_{s=1}^{[e^{\varepsilon\beta}]}
\sum_{\bar\sigma_1,\dots,\bar\sigma_{s-1}}P(\sigma_0=Q,
\sigma_1=\bar\sigma_1,\dots,\sigma_{s-1}=\bar\sigma_{s-1},\sigma_s\in
\partial\Cal B)+\\+P_Q(\sigma_t\notin\Cal M \text{ for every }
t\in[1,[e^{\varepsilon\beta}]]),
\endmultline
 \tag59
$$
where
$$
\Cal M =\{\sigma\in\{-1,1\}^{\Lambda}\: \sigma \text{ is a local minimum
for } H\}.
 $$
Of course $\Cal M \supset \Cal R$.

We have
$$
P_Q(\sigma_t\notin\Cal M \text{ for every }
t\in[1,[e^{\varepsilon\beta}]])\leq (\frac{1}{2})^{[e^{\varepsilon\beta}]}
\tag60
$$
for $\beta$ sufficiently large.
Indeed, one can see immediately that
$$
\inf_{\sigma\in\{-1,1\}^{\Lambda}}P_{\sigma}(\tau_{\Cal M} <
T_0)> (\frac{1}{|\Lambda|})^{T_0}.
\tag61
$$
Hence, by strong Markov property, we get
$$
\inf_{\sigma\in\{-1,1\}^{\Lambda}}P_{\sigma}(\tau_{\Cal M} <
[e^{\varepsilon\beta}])>\frac{1}{2}
\tag62
$$
for all $\varepsilon>0$ and all $\beta$ sufficiently large, and thus,
again by strong Markov property, the bound (60) is implied.

To estimate the first sum on the right hand side of the inequality (59),
we first observe that if $\eta\in\partial \Cal B$, then either
\roster
\item the recangular envelope of $\eta$ is $Q'\supset Q$ with $\eta\cap
(Q'\setminus Q)= \{x \}$ if $\eta=\sigma^{(x)}$, $\sigma\in\Cal B$, and $x$
is adjacent, from exterior, to $Q$, or
\item $\eta$ is contained in $Q$, but it is not connected, or, finally,
\item $\eta$ is a connected cluster whose rectangular envelope is $Q$,
but at least on one edge it intersects $\eta$ on a single unit square.
\endroster

We claim that, given  $\bar Q=Q_{L_1,L_2}$ with $L_1 \leq L_2<L_2^{\ast}$,
then
$$
\min_{\sigma\in\partial\Cal B(\bar Q)}H(\sigma)= (2J_1 L_2 + 2J_2 L_1)-h L_1
L_2 + h(L_1 -1) = H(Q_{L_1,L_2})+h(L_1-1)=H(S_{L_1,L_2}),
\tag63
$$
where $S_{L_1,L_2}$ is obtained from $Q_{L_1,L_2-1}$ by adding to it a unit
square adjacent to its (short) vertical edge.
Indeed, it is easy to see that for $\eta$ like in case 1) it is
$$
H(\eta)- H(Q) \geq 2J_2 -h,
$$
whereas in case 2) one has
$$
H(\eta)- H(Q) > h(L_1 +L_2 -1).
$$
Hence, since $L_2<L_2^{\ast}$,
 we are left with the case 3) that is easily treated and leads to (63). By
exploiting the reversibility like in Lemma 1, and using the inequalities
(59) and (60), it is easy to see that  $$
P(Q\to \partial \Cal B)\leq e^{\varepsilon\beta}\exp\{-h\beta(L-2)\}
\tag64
$$
for all $\varepsilon >0$ and all $\beta$ sufficiently large.
>From the inequalities (56) and (64) we get the estimate (50).
The bound (51) follows in a similar way.

>From the estimates (50) and (51) it is easy to deduce that for every
$t_{L-1,L},t_{L,L} \in\Bbb N$, all $\varepsilon>0$, and all $\beta$
sufficiently large  one has
$$
P(\Cal E_{L-1,L})\geq
\bigl(1-e^{\varepsilon\beta}\exp\{-h\beta(L-2)\}\bigr)^{t_{L-1,L}}
\frac{1}{|\Lambda|^{T_0}}\exp\{-(2J_2-h)\beta\}
 \tag 65
$$
and
$$
P(\Cal E_{L,L})\geq
\bigl(1-e^{\varepsilon\beta}\exp\{-h\beta(L-1)\}\bigr)^{t_{L,L}}
\frac{1}{|\Lambda|^{T_0}}\exp\{-(2J_1-h)\beta\}.
 \tag 66
$$
To get the bound (65) we consider, for every $\sigma \in\Cal B_{L-1,L} $,
the following event
$$
\multline
\bar \Cal E_{L-1,L}(\sigma)=\{\sigma=\sigma_0, \tau_{Q_{L-1,L}}\leq T_0-L,
\sigma_{\tau_{Q_{L-1,L}}+1}=S_L,\tau_{Q_{L,L}}=\tau_{Q_{L-1,L}}+L,\\
\sigma_t=Q_{L,L} \text { for every } t\in
[\tau_{Q_{L,L}},\tau_{Q_{L,L}}+T_0-L-\tau_{Q_{L-1,L}}]\}.
\endmultline
 \tag67
$$
To put this definition into words: every path in $\bar \Cal
E_{L-1,L}(\sigma)$ starts from $\sigma\in\Cal B(Q_{L-1,L})$.
In a time shorter than $T_0-L$ it reaches $Q_{L-1,L}$. For every
$\sigma\in\Cal B(Q_{L-1,L})$ there exists such a path along which the
energy is decreasing. Then a unit square protuberance is attached to the
vertical right edge of $Q_{L-1,L}$, this occurs  with probability
$\frac{1}{|\Lambda|}\exp\{-(2J_2-h)\beta\}$. After that follows a sequence of
spin flips, decreasing energy, on contiguous sites adjacent from the exterior
to $Q$ starting near the proturberance and leading to $Q_{L,L}$.
The rest of the time up to $T_0$ is spent in  $Q_{L,L}$.

Clearly
$$
\multline
\Cal E_{L-1,L}\supset \bigcup_{\sigma\in\Cal B(Q_{L-1,L})}
\bigl\{ \sigma_0=Q_{L-1,L},\sigma_s\in\Cal B(Q_{L-1,L})\\ \text { for every }
s\leq t_{L-1,L}-T_0, \sigma_{t_{L-1,L}-T_0}=\sigma \bigr\}\bigcap
\bigl\{  G_{t_{L-1,L}-T_0} \bar \Cal E_{L-1,L}(\sigma) \bigr\}.
\endmultline
 \tag68
$$
Here $G_s $ is  the time translation operation by $s$ acting in a natural
way on paths.
Since also, directly from the definition (67), one gets
$$
P(\bar \Cal E_{L-1,L}(\sigma))\geq \frac{1}{|\Lambda|^{T_0}}
\exp\{-(2J_2-h)\beta\}
\tag69
$$
for every $\sigma\in\Cal B(Q_{L-1,L})$, the bound (50) implies the bound
(66). In a similar way one obtains the bound (66).
Directly from the definitions (45$'$), (45$''$), (45$'''$), (47), (47$'$), and
(47$''$) it is seen that the events
$\tilde\Cal E_{n_{2,2},\dots,n_{L^{\ast}-1,L^{\ast}}}$ are mutually
disjoint. Hence, using (65), (66), and the Markov property, we get
$$
\multline
P(\Cal E^{(4)})\geq \sum_{n_{2,2}=1}^{\bar n_{2,2}}\dots
\sum_{n_{L^{\ast}-1,L^{\ast}}=1}^{\bar n_{L^{\ast}-1,L^{\ast}}}
\bigl( 1- e^{(\varepsilon - h)\beta}\bigr)^{n_{2,2} T_0}
\frac{1}{|\Lambda|^{T_0}}
\exp\{-(2J_1-h)\beta\}\dots \\ \dots
\bigl( 1- e^{\varepsilon \beta}
e^{-h L^{\ast} -2}\bigr)^{n_{L^{\ast}-1,L^{\ast}} T_0}
\exp\{-(2J_2-h)\beta\}
\endmultline
\tag70
$$
for every sufficiently small
$\varepsilon>0$, and all $\beta$ sufficiently large.
Given the choice (47$''$) of the constants $\bar n$ and the values of the
quotients in the geometric series above, the   sums in (70)
turns out to run effectively to $\infty$.
Given
$\delta$ in the equation (47$''$), we can choose $\varepsilon
$ sufficiently small to get
$$
P(\Cal E^{(4)})\geq \exp\bigl\{-[H(\Cal P)-H(Q_{2,2})-2\delta]\beta \bigr\}.
\tag71
$$

Now, since the events $\Cal E^{(1)}_{\sigma,t_1}$ with different $t_1$'s
are mutually disjoint, and similarly for  $\Cal E^{(2)}_{t_1, t_2}$,
we have, for $\beta$ sufficiently large,
$$
P(\Cal E_{\sigma})\geq \sum_{t_1=1}^{\bar t_1}\sum_{t_2=t_1+1}^{t_1 +\bar
t_2} P(\Cal E^{(1)}_{\sigma,t_1}\cap\Cal E^{(2)}_{t_1, t_2}\cap \Cal
E^{(3)}_{t_2} )\exp\bigl\{-[H(\Cal P)-H(Q_{2,2})-2\delta]\beta \bigr\}.
 \tag72
$$
Suppose that, for all $\varepsilon>0$ and all $\beta$ sufficiently large,
we are able to prove that
$$
\inf_{\sigma\in\Cal A}\sum_{t_1=1}^{\bar t_1}P(\Cal E^{(1)}_{\sigma,t_1})
\geq e^{-\varepsilon\beta}.
\tag73
$$
Now, since  for all $\varepsilon>0$, from Lemma 1 one has
$$
\lim_{\beta\to \infty}
P_{-\underline 1}(\tau_{Q_{1,1}}<\exp (2J_1+2J_2-h-\varepsilon)\beta)=0
\tag74
$$
and
$$
P(\Cal E^{(3)}) \geq
\frac{1}{|\Lambda|^4}\exp\{-(H(Q_{2,2})-H(-\underline 1))\beta\}.
 \tag 75
$$
>From (72),  (74), (75),   and (48) one gets
$$
\sum_{t_2=t_1+1}^{t_1 +\bar
t_2} P(\Cal E^{(2)}_{t_1, t_2}\cap \Cal
E^{(3)}_{t_2} )\geq \exp\{\beta h
(L^{\ast}-2)-\beta(H(Q_{2,2})-H(-\underline 1)-\delta)\}
\tag76
$$
and then, from (73) and (76), one has
$$
P(\Cal E_{\sigma})\geq e^{-3\delta\beta}\exp\{-\Gamma \beta +
\beta h (L^{\ast}-2)\}.
\tag77
$$

To get (73) we use the following argument:

\flushpar
in a time shorter than $T_0$ and with a probability larger than
$\frac{1}{|\Lambda|^{T_0}}$ we go, starting from any $\sigma\in\Cal A$
to a configuration given by a set of noninteracting subcritical rectangles.
Then, from Lemmas 2, 3, and the definition of $\bar t_2$ (see the equation
(47$'$)), with large probability one goes to $-\underline 1$ before
$\bar t_2$. We leave the details of this argument to the reader.

Now let
$$
T_1= \exp\{ \beta [h(L^{\ast}-2)+\delta_1]\}
\tag78
$$
and
$$
T_2= \exp\{ \beta (\Gamma+\delta_2)\}.
\tag79
$$
Further, let us divide the time interval $T_2$ into $m$ subintervals of
length $T_1$ with $m=\frac{T_2}{T_1}$ supposed to be an integer.
Let
$$
U_i=i T_1, \quad i=1, \dots, m-1.
$$
We have ($\Cal E^c$ denotes the complementary set of $\Cal E$)
$$
\multline
P_{-\underline 1}(\tau_{\partial \Cal A}> T_2)=
\sum_{\bar\sigma_1,\dots,\bar\sigma_{m-1}\Cal A}
P_{-\underline 1}(\tau_{\partial \Cal A}> T_2, \sigma_{U_i}=\bar\sigma_i,
i=1,\dots,m-1)\leq\\
\leq \sum_{\bar\sigma_1,\dots,\bar\sigma_{m-1}\Cal A}
P_{-\underline 1}( \sigma_{U_i}=\bar\sigma_i,
i=1,\dots,m-1,
 \Cal E^c(-\underline 1)\cap G_{U_1}\Cal E^c(\bar\sigma_1)
\cap \dots \cap G_{U_{m-1}}\Cal E^c(\bar\sigma_{m-1})
)\leq\\ {}\\
\leq \bigl( 1- \inf_{\sigma\Cal A}P(\Cal E(\sigma)) \bigr)^m\leq
\exp\bigl\{  e^{-3\delta\beta -\Gamma \beta + h(L^{\ast}-2)\beta}
e^{\beta (\Gamma +\delta_2-\delta1)}e^{-h(L^{\ast}-2)} \bigr\}.
\endmultline
\tag80
$$
If $\delta_2 > \delta_1 + 3\delta$, we get
$$
\lim_{\beta\to \infty}
P_{-\underline 1}(\tau_{\partial \Cal A}> T_2) = 0.
$$
Since $\delta$, $\delta_1$ are  arbitrarily small,
this concludes the proof of the inequality (32) and thus also of Theorem 2.
\qed

Theorem 1 is now a corollary of Theorem 2 --- it follows from the
properties 2) and 3) of the set $\Cal A$ whose existence was established
during the proof of Theorem 1.

Finally, Theorem 3 directly follows from the results  in \cite{S}, Lemmas
2,3,4, and
Theorem 1.




\vfill\newpage


\Refs

\widestnumber\key{CGOV}

\ref\key BCF
\by W. K. Burton, N. Cabrera, and F. C. Frank
\paper The Growth of Crystals and the Equilibrium Structure of their Surfaces
\jour Phil\. Trans\. Roy\. Soc.
\vol 243 A
\pages 40--359
\yr1951
\endref


\ref\key BN
\by H. van Beijeren and I. Nolden
\paper The roughening transition
\inbook Topics in Current Physics \vol 43
\eds W. Schommers and P. von Blackenhagen
\publ Springer
\publaddr Berlin
\yr1987
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\ref\key CGOV
\by M. Cassandro, A. Galves, E. Olivieri, M. E. Vares
\paper Metastable behaviour of stochastic dynamics: A pathwise
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\vol 35\pages 603--34 \yr1984
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\ref\key DKS\by  R. L. Dobrushin, R. Koteck\'y, S. Shlosman
\book The Wulff construction: a global shape from local interactions
\finalinfo{in print by AMS.}
\endref

\ref\key FW
\by M. I. Freidlin, A. D. Wentzell
\book Random Perturbations of Dynamical Systems
\publ Springer Verlag
\publaddr New York
\yr1979
\endref



\ref\key H
\by R. Holley
\paper Possible rates of convergence in finite range, attractive
spin systems
 \jour Contemp\. Math\.
\vol 41\pages 215--34 \yr1985
\endref


\ref\key He
\by C. Herring
\paper Some Theorems on the Free Energies of Crystal Surfaces
\jour Phys\. Rev\.
\vol 82\pages 87--93 \yr1951
\endref

\ref\key  KO1
\by R. Koteck\'y, E. Olivieri
\paper  Shapes of growing droplets --- a model of escape from a
metastable phase \jour
in preparation
 \yr 1992
\endref


\ref\key  KO2
\by R. Koteck\'y, E. Olivieri
\paper  Stochastic models for nucleation and crystal growth
\jour
Preprint \yr 1992
\endref


\ref\key MOS
\by F. Martinelli, E. Olivieri, E. Scoppola
\paper Metastability and Exponential Approach to Equilibrium for
Low-Temperature Stochastic Ising Models
 \jour Jour\. Stat\. Phys\.
\vol 61\pages 1105--19 \yr1990
\endref


\ref\key  NS1
\by E.J. Neves, R.H. Schonmann
\paper  Critical droplets and metastability for a
Glauber dynamics at very low temperatures
\jour
Preprint \yr
1989\endref


\ref\key  NS2
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\ref\key  S
\by  R.H. Schonmann
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\ref\key W
\by G. Wulff
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\endRefs


\enddocument
\end



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