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%\input formato.tex
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\centerline{\ttlfnt On the Two Dimensional Dynamical Ising
Model}
\centerline{\ttlfnt In the Phase Coexistence
Region}\vskip 0.5cm
\author{F. Martinelli $^{\dag}$ $^{\ddag}$}
\address{\ninerm \dag Dipartimento di Matematica,
III Universit\`a di Roma, Italy \hfill\break{\ddag Istituto per le Applicazioni
del Calcolo "Mauro Picone", CNR, Roma Italy}\hfill\break{ e-mail:
martin@mat.uniroma1.it}}
\abstract{\ninerm We consider a Glauber dynamics reversible with
respect to the two
dimensional Ising model in a finite square of side $L$, in the absence of an
external field and at large inverse temperature $\beta$. We first consider
the gap
in the spectrum of the generator of the dynamics in two
different cases: with plus and open boundary condition. We prove
that, when the symmetry under global spin flip is broken by the boundary conditions,
the gap is much larger than the case in which the symmetry is present. For this
latter
we compute exactly the asymptotics of \hskip 0.5cm $-{1\over \beta L}\log
(\hbox{gap})$\hskip 0.5cm as $L\to\infty$ and show that it coincides
with the surface
tension along one of the coordinat axes. As a consequence we are able to
study quite
precisely the large deviations in time of the magnetization and to obtain
an upper
bound on the spin-spin time correlation in the infinite volume plus phase. Our
results establish a connection between the dynamical large deviations and
those of
the equilibrium Gibbs measure studied by Shlosman in the framework of the rigorous
description of the Wulff shape for the Ising model. Finally we show that,
in the case
of open boundary conditions, it is possible to rescale the time with $L$ in such
a
way that, as $L\to \infty$, the finite dimensional distributions of the time
rescaled
magnetization converge to those of a symmetric continuous time Markov
chain on the
two state space $\{-m^*(\beta ),m^*(\beta )\}$, $m^*(\beta )$ being the spontaneous
magnetization. Our methods rely upon a novel combination of techniques
for bounding
from below the gap of symmetric Markov chains on complicate graphs, developed by
Jerrum and Sinclair in their Markov chain approach to hard computational problems,
and
the idea of introducing "block Glauber dynamics" instead of the standard single
site
dynamics, in order to put in evidence more effectively the effect of the
boundary
conditions in the approach to equilibrium.} \vskip 1cm\noindent {\eightrm Work
partially supported by grant SC1-CT91-0695 of the Commission of European
Communities}
\vfill
\eject
%\input formato.tex
\tolerance=10000
\numsec=0 \numfor=1
{\bf Section 0}\par
\centerline{\bf Introduction }\bigskip
In the last years there have been very important progresses in the rigorous
analysis of Glauber dynamics (see section 1 for a precise
definition) for classical lattice spin systems when the
thermodynamic parameters are such that the static system, described by the usual
Gibbs measure $$\mu\,=\, {\exp (-\beta H)\over Z}$$ does not undergo a phase
transition in the thermodynamic limit.\par In particular we refer the
reader to the
series of papers by Stroock and Zegarlinski (see [SZ] and references therein),
by Olivieri and myself (see [MO1], [MO2]), Olivieri, Schonmann and myself
[MOS], Yau and Shin Lin [SLY] for the proof, under various mixing conditions
on the Gibbs
measure $\mu$, of the exponential (in time) relaxation to equilibrium, represented
by
$\mu$ itself, in finite or infinite volume, of the associated Glauber dynamics,
and to the works by Schonmann (see
[Sch1] and references therein), Kotecky and Olivieri [KO1], [K02], Scoppola
[Sc] for
detail description of the metastable behaviour of Glauber dynamics for Ising
type
models close to the line of first order phase transition.\par It is important to
emphasize that some of the results in the above works cover most of the
one phase
region, going sometimes, e.g. in ferromagnetic systems, arbitrary close to the
critical point (see [MO1] and [MOS]). \par A natural
question arises as to what happens when the thermodynamic parameters are
such that we
do have a phase transition in the thermodynamic limit. To be more concrete, let
us
consider the usual Ising model in d-dimensions, $d\geq 2$, in the absence of an
external field, described by the formal (normalized) energy function: $$H(\s
)\;=\;-{1\over 2}\sum_{x,y\in {\bf Z}^d\atop \vert x-y\vert =1}(\s (x)\s
(y)-1)\qquad \s\in
\{-1,1\}^{{\bf Z}^d}\Eq(0.1)$$ and let us suppose that the inverse
temperature $\beta$ is
larger (actually in all rigorous results much larger) than the critical value
$\beta_c$.\par Then, as it is well known (see e.g. [Li]),
any associated infinite volume Glauber dynamics is not ergodic and it is rather
natural to ask how this absence of ergodicity is reflected if we look at
the dynamics
in a finite, but large cube $V_L$ of side $L$, where ergodicity is never broken.\par
A
first partial answer was provided by Thomas [T] few years ago. He showed that,
if the
symmetry of $H(\s )$ under global spin flip is not broken by the boundary
conditions
on the exterior of the cube $V_L$, then the relaxation time to equilibrium, that
in a
first approximation can be taken equal to the inverse of the gap in the spectrum
of
the generator $L_{V_L}$ of the dynamics, diverges, as $L\to \infty$, at least
as an
exponential of the {\it surface} $L^{d-1}$.\par The reason for such a
result is the following.
When the symmetry is not broken e.g. when boundary conditions are open (i.e.
absent)
or periodic, then the energy landscape determined by the function $H(\s )$ has
only
two absolute minima, corresponding to the two configurations identically equal
to
either $+1$ or to $-1$. Thus the dynamics started e.g. from all minuses, in
order to
relax to equilibrium, has to reach the neighborood of the opposite minimum by
necessarily crossing the set of configurations of zero magnetization.
Since the Gibbs
measure gives to the latter a weight of the order of a negative exponential
of the
{\it surface} (see e.g. [Sch2]) a kind of bottleneck is present and the result
follows by a rather simple argument (see the first part of the proof of theorem
4.1
below).\par The same reasoning also suggests that, if the symmetry is
broken by the
boundary conditions, e.g. by fixing equal to $+1$ all spins outside $V_L$,
then the
relaxation time should be much shorter than in the previous case since
there should
be no bottlenecks to cross. Equilibrium is, in this case, induced by the boundary
conditions by means of some sort of plus spins wave, initially attached
to the boundary and shrinking to zero as time goes on.\par The interesting but
unproven conjecture, is that, at least in two dimensions with plus boundary
conditions, the relaxation time will diverge, as $L\to \infty$, like
$L^{2}$. The proof
of the above conjecture would have some very nice consequences on the equal
site time correlations function of the infinite volume dynamics started in the
plus
phase, for which Fisher and Huse [FH] predicted, using the above conjecture, a
stretched exponential decay of the form $\exp (-\sqrt {t})$ (see also [Og] for
numerical simulations and [M] for further discussion).\par In this paper
we consider the above and other related
questions for the two dimensional Ising model at very low temperature without
external field. For some less precise results in arbitrary dimensions see
the remark after theorem 4.2\par In section 3 we prove a {\it lower} bound on the
gap in the
spectrum of the generator $L_{V_L}$ of the Glauber dynamics with $+$ boundary
conditions of the form: $$\hbox{gap}(L_{V_L})\;\geq\;\exp (-C\beta L^{{1\over
2}+\epsilon})\qquad \e\in (0,{1\over 2}]\Eq(0.2)$$ which, although
gives a bound on the relaxation time which is far from
the conjectured $L^{2}$ law, is in any case much larger than the {\it upper bound
}
obtained by Thomas without the plus boundary conditions.\par
As a consequence we derive an upper
bound of the form $$\exp (-\log (t)^\a)\qquad \a\in [0,2)$$ on the equal site
time
correlation function of the infinite volume dynamics started in the plus
phase.\par
In section 4 we compute exactly the asymptotics of the gap with open boundary
conditions. More precisely we obtain, for any $\e \in (0,{1\over 4}]$,
any $\b$ large enough and any $L$:
$$\exp (-\beta \t (\beta
)L\,-\,C\beta L^{{1\over 2}+\e})\,\leq\,\hbox{gap}(L_{V_L})\,\leq\,
\exp (-\beta \t (\beta
)L\,+\,C\beta L^{{1\over 2}+\e})$$
where $\t (\b )$ is the surface tension in the direction
of e.g. the horizontal axis. As a byproduct of the proof of this result, we show
that
the bound \equ(0.2) is valid even if the plus boundary conditions are added on
only
one side of the square $V_L$.\par
The proofs of the above two results follow two very similar steps:\bigskip\noindent
Step 1: we prove the sought result for a
generalized Glauber dynamics in which single sites are replaced by suitable
blocks.
This mean that, given apriori a covering $\{Q_i\}$ of $V_L$,
at each updating of the dynamics the spin configuration is changed in only one
block
$Q_i$ and there it is replaced by the equilibrium Gibbs
measure of the block given the configuration outside it. It turns out that a
convenient choice of the blocks in our case consists of long and thin
overlapping rectangles with basis $L$ and height $L^{{1\over
2}+\e}$, $0<\e<<1$.\medskip\noindent Step 2: we relate the gap of the single
site Glauber
dynamics to that of the generalized block dynamics in such a way that the
estimates
obtained in step 1 are not significantly changed.\bigskip The above
way to attack the problem is not entirely new; it was in fact introduced long
ago by
Holley [H] to prove exponential convergence to equilibrium in the one phase
region.
One has in fact that, if the system is away from the phase transition region
and if
the bloks are overlapping large enough (depending on the thermodynamic parameters)
cubes, the blocks Glauber dynamics behaves as a very high temperature single
site
Glauber dynamics, i.e. an almost independent systems for which the discussion
of the
approach to equilibrium is a relatively easy task (see section 4 of [MO2]).\par
While we accomplish the first step via a very natural probabilistic construction,
the second, rather
crucial, step is carried out via the application to our contest of a clever
geometric
technique introduced by Jerrum and Sinclair [JS1], [JS2], Sinclair [Si]
(see also [DS]), to estimate from {\it below} the gap in the spectrum of symmetric
Markov chain
on complicate graphs. They invented their technology while working on a stochastic
algorithm approach to compute the partition function $Z$ of the Ising model
and the
permanent of a large matrix in a time {\it polynomial} in the size of the
problem.\par Such a technique, which is illustrated in our case in a
self contained
way in section 2, gives in a very natural way a {\it lower} bound on the gap
of the
generator of the dynamics in a rectangle $R$ with shortest side $l$, with
or without
boundary conditions, of the form: $$\hbox{gap}(L_R)\;\geq\;\exp (-\beta C l)$$
Moreover, if the blocks $Q_i$ of the generalized Glauber dynamics are a
suitable translations of the rectangle $R$, then: $$\hbox{gap(Glauber)}\;\geq\;\exp
(-\beta C l)\hbox{gap(Generalized Glauber)}$$
It is worthwhile to mention that our proof of step 1 is constructive in the sense
that it indicates how actually the system reaches equilibrium: by simply propagating
the plus boundary conditions in the bulk if these are present and the initial
configuration is e.g. all minuses, or by creating inside the starting
phase, via a large fluctuation, an almost
horizontal (or vertical) interface close e.g. to the bottom side of $V_L$ which
afterwards rigidly moves to the opposite side until the other phase has
invaded the
whole volume.\par Once we have a precise control on the relaxation time with
open
boundary conditions, we can study in details the large fluctuations of the
magnetization $$m(\s_t)\,=\,{1\over L^2}\sum_x\s_t (x)$$
by considering for example the hitting time $\t_\rho$ of the set
$$M_\rho\;=\;\{\,\s ;\;m(\s )\,=\,\rho\,\}\qquad \rho\,\in\,\{-m^*(\b
),m^*(\b )\}$$
with $m^*(\b )$ the spontaneous magnetization.\par
In section 5 we show that, if the starting configuration is distributed according
to
the equilibrium measure restricted to the "phase" of positive magnetization or
if it
is identically equal to plus one, then the expected value $E(\t_\rho )$ of the
hitting
time $\t_\rho$ is of the order: $$E(\t_\rho )\;\approx\;\exp( \b L\psi (\rho\vee
0
))$$ where the rate function $\psi (\rho )$ is the same as for the static problem:
$$\mu_{V_L} (m(\s )\,=\,\rho )\;\approx\;\exp( -\b L\psi (\rho))$$
and it has been computed by Shlosman [Sh] in the framework of the rigorous
description of the Wulff shape for the Ising model carried out by Dobrushin,Koteck\'y
and Shlosman [DKS]. We also show that the hitting time
$\t_\rho$ rescaled by roughly its average converges, as $L\to\infty$, to an
exponential time of mean one.\par
It is important to outline that the typical configurations of the equilibrium
Gibbs
measure under the condition $\{m(\s )\,=\,\rho \}$ have a very precise geometric
structure related to the Wulff shape with open boundary conditions (see
[Sh]). Thus,
the fact that the rate function for $E(\t_\rho )$ is the same as in the static
problem, suggests that, when the system, started in the positively magnetized
"phase",
reaches for the first time the set $M_\rho$, it does it by forming a droplet
of the
right volume and with the correct Wulff shape. We hope to come back in a future
work
on this and related problems.\par
A key step in the discussion of the above problems is
the proof, based on the results of section 3, that the relaxation time inside a
single "phase" is much shorter than the typical values of the hitting time
$\t_\rho$
(see proposition 5.2 for a precise statement).\par
This last result indicates that the gap in the spectrum of the
generator restricted to the invariant subspace of the functions even with
respect to global spin flip is much larger than the true gap; unfortunately we do
not have any precise statement in
this direction.\par
Finally in section 6 we complete the analysis of the time evolution of the
magnetization by showing that, if the time is scaled with $L$ in such
a way that on
the new unit of time the system is likely to have jumped from one phase to the
other, then the finite dimensional distributions of the time scaled magnetization
converge, as $L\to\infty$, to those of a continuous time Markov chain on the two
states space $\{-m^*(\b ),m^*(\b )\}$ with unitary jump rate.\par
The rest of the paper contains a preliminary section, section 1, where all the
necessary definitions are given together with the required results on Wulff
shape,
cluster expansion and so forth. The proofs of various technical results for
the Ising
model have been collected in an appendix.\bigskip\noindent
{\bf Acknowledgments}\par
I am particularly in debt with Pablo Ferrari, who was involved in this work at
its early stages, for many constructive discussions and comments concerning the
material in sections 5 and 6, particularly theorems 5.2 and 6.1. I would
also like to
warmly thank A.S Sznitman, E.Bolthausen and J.Moser for their kind invitation to
E.T.H. where the part of this work was carried out, and H.Kesten, G.Grimmet
and M. Barlow for inviting
me at the Newton Institute in Cambridge, within the special program "Random Spatial
Processes", where I had the opportunity to discuss about this work and its possible
extensions with S.Shlosman, R.Kotecky, A. Mazel and
R.Schonmann.
\pagina
%\input formato.tex
\numsec=1
\numfor=1
{\bf Section 1}\par
\centerline{\bf Preliminaries}\bigskip
In this section we precisely define the model and the random
dynamics that will be the object of study in the next sections.
\bigskip $\S
1$\hskip 1cm The Ising Model in a Finite Set.\medskip\noindent Let $\Z$
be the usual two dimensional square lattice with sites
$x\,=\,(x_1,x_2)$, equipped with the norm $\norm{x}\,=\,\vert
x_1\vert \,+\,\vert x_2\vert$. We will sometime consider $\Z$ as a
graph with vertices the sites $x\in \Z$ and edges all pairs of
sites $x$ and $y$ such that $\norm{x-y}\,=\,1$. We will use the
notation $\sigma$ to denote a generic element of the set
$\O_{\Z}\;=\;\{-1,+1\}^{\Z}$; whenever $V\subset \Z$ we use the
notation $\s_V\;=\,\{\s (x),\;x\in V\}$ to denote the restriction of
$\s$ to the set $V$ and $\O_V$ to denote the set of them. \par Given
$V\subset \Z$, we define the interior and exterior boundaries of $V$
as : $$\partial_{int}V\,\equiv\,\{\,x\,\in V\,;\;\exists \,y\notin
V\,;\quad \norm{x-y}\,=\,1\}$$ $$\partial_{ext}V\,\equiv\,\{\,x\,\notin
V\,;\;\exists \,y\in V\,;\quad \norm{x-y}\,=\,1\}$$
and the boundary $\partial V$ as:
$$\partial V\;=\;\{(x,y);\;x\in
\,\partial_{int}V,\;y\,\in\,\partial_{ext}V\;\quad
\norm{x-y}\,=\,1\,\}$$ We also denote by $\vert V\vert $ the
cardinality of $V$.\par Next, for any finite subste $V$ of $\Z$, we
define the energy $H_V^{U^{\partial V},\t}(\s_V )$ of a
configuration $\s_V\,\in \,\O_V$ with boundary conditions $\tau$
outside $V$, $\t \in \{-1,+1\}^{\Z}$, and boundary coupling
$0\,\leq\,U^{\partial V}(x,y)\,\leq\,1$, $(x,y)\,\in \,\partial V$,
as: $$H_V^{U^{\partial V},\t}(\s_V )\;=\;$$ $$-{1\over
2}\sum_{\vbox{\eightpoint{\hbox{$x,y\in V$}
\hbox{$\norm{x-y}=1$}}}}(\s_V(x)\s_V(y)\,-\,1)\;-\;\sum_
{(x,y)\,\in
\,\partial V} U^{\partial V}(x,y)(\s_V(x)\t (y)\,-\,1)\Eq(1.1)$$ and
the associated Gibbs probability measure at inverse temperature
$\beta$: $$\mu_V^{U^{\partial V},\t}(\s_V)\;=\;Z(V,U^{\partial
V},\t)^{-1}\, \exp (-\beta H_V^{U^{\partial V},\t}(\s_V)) \Eq(1.2)$$
where the partition function $ Z(V,U^{\partial V},\t)$ is given by
$$Z(V,U^{\partial V},\t)\;=\;\sum_{\s_V} \exp (-\beta H_V^{U^{\partial
V},\t}(\s_V))\Eq(1.3)$$
If the boundary condition $\t$ is the special configuration $\t
(x)\,=\,1\;\forall \;x\in \Z$, then in all our notation the
superscript $\t$ will be replaced by a simple $+$. Notice that
the $-1$ appearing in the definition of the energy $H_V^{U^{\partial V},\t}(\s_V
)$ fixes equal to zero the energy with plus boundary conditions of
the configuration identically equal to plus one.\par We also set, for
any function $f\,:\,\O_V\,\to\,{\bf R}$, $$\mu_V^{U^{\partial
V},\t}(f)\,=\,\sum_{\s_V} \mu_V^{U^{\partial V},\t}(\s_V)f(\s_V)$$
Although, for technical reasons, it will be convenient to consider
cases in which the boundary coupling $U^{\partial V}(x,y)$
does depend on $x$ and $y$ and
it is for example equal to plus one along some parts of the external boundary
of $V$ and positive but very weak along some other parts of the
boundary, the most typical choices of $U^{\partial V}$
will be either $U^{\partial V}$ identically equal to one, in which
case the Gibbs measure \equ(1.2) is the usual Ising model in the set
$V$ with $\t$ boundary conditions, or $U^{\partial V}$ identically
equal to zero which corresponds to the Ising model with open boundary
conditions. In both cases the (cumbersome) notation
$\mu_V^{U^{\partial
V},\t}$, $Z(V,U^{\partial V},\t)$ will be replaced by the more
natural ones $\mu_V^{\t}$, $\mu_V^{\emptyset}$, $Z(V,\t)$,
$Z(V,\emptyset )$ respectively.\par
As a next step we recall some monotonicity properties
enjoied by the Gibbs measure $\mu_V^{U^{\partial
V},\t}$, which easily follow from the well known FKG inequalities
(see [FKG]), which will play a crucial role in the next sections.\par
Given two configurations $\t_1$, $\t_2$ in $\O_{\Z}$, we say
that $\t_1\,\leq \,\t_2$ iff $$\t_1 (x)\,\leq \,\t_2(x)\;\;\forall
\; x\in \Z$$ and similarly for $\s_V, \,\s_V'\;\in \O_V$. Then, for
any pair of finite subsets $V_1\, \subset \,V_2 $, any
pair of boundary coupling $U_1^{\partial
V_1}(x,y),\,U_2^{\partial
V_1}(x,y)$,
and boundary conditions $\t_1,\,\t_2$ such that
$$U_1^{\partial V_1}(x,y)\t_1(y)\,\leq \,
U_2^{\partial V_1}(x,y)\t_2(y)\quad \forall \;(x,y)\,\in \, \partial V_1$$
and any function
$f\,:\,\O_{V_1}\,\to\,{\bf R}$ which is increasing with respect to
the above partial order, we have: $$\mu_{V_1}^{U_1^{\partial
V_1},\t_1}(f)\,\leq\, \mu_{V_1}^{U_2^{\partial
V_1},\t_2}(f)\Eq(1.4)$$ $$ \mu_{V_2}^{U^{\partial
V_2},\t_2}(f)\,\leq\, \mu_{V_1}^{+}(f)\Eq(1.5)$$
\bigskip $\S 2$\hskip 1cm Contours and Cluster
Expansion.\medskip\noindent In this second paragraph we recall, for the
reader's convenience, a version of the cluster expansion for the
partition function $Z(V,U^{\partial V},+)$ valid under some
restrictions on the boundary coupling $U^{\partial V}$, which will
turn out to be quite essential in the next sections. The material
that follows has been adapted to our situation, in which
$U^{\partial V}$ is not necessarily identically equal to one, from
sections 3.8, 3.9 of [DKS].\par
To begin with, let us recall the definition of Peierls contours for
a generic configuration $\s$ which is identically equal to +1
outside a finite region.\par If we denote by $\Z^*$ the dual lattice
of $\Z$, we call a {\it bond} any segment in ${\bf R^2}$ connecting
two neighboring sites of $\Z^*$. Then we say that two sites $x$ and
$y$ in $\Z$ are separated by the bond $h$ if their distance (as
sites in ${\bf R^2}$) from $h$ is equal to $1\over 2$. Given $\s\in
\O_{\Z}$ we denote by $\Gamma (\s )$ the collection of all bonds
separating sites $x$ and $y$ in $\Z$ where $\s (x)\neq \s (y)$. If
moreover we use the convention that any pair of orthogonal bonds
that intersect in a given site $x^*$ of the dual lattice $\Z^*$ are
a {\it linked pair of bonds} iff they are both on the same side of
the forty-five degrees line across $x^*$, then we immediately see
that $\Gamma (\s )$ splits up in a unique way in a collection of
closed contours $\Gamma_1 (\s )\, ,\Gamma_2 (\s ),\dots\,\Gamma_i
(\s )\dots$ where a closed contour is a sequence
$e_o,\,e_1,\,e_2\,\dots e_n$ of bonds such that:\medskip \item{i)}
$e_i\,\neq \,e_j$ for all $i$ and $j$ with the exception of $i=0$
and $j=n$ for which $e_o\,=\,e_n$. \item{ii)} for all $i$ the bonds
$e_i$ and $e_{i+1}$ have a common vertex in $\Z^*$.
\item{iii)} if $e_i$, $e_{i+1}$, $e_j$, $e_{j+1}$ intersect at a
given site $x^*$ then both $e_i$, $e_{i+1}$ and $e_j$, $e_{j+1}$
are linked pairs of bonds.\medskip
The length $\vert \G\vert$ of a contour is simply the number of
bonds in $\G$. Given a contour $\G$, we denote by $\D \G$
the set of sites in $\Z$ such that either their distance (in ${\bf
R^2}$) from $\G$ is $1\over 2$ or their distance from the set of
vertices of $\Z^*$ where two non-linked pair of bonds of $\G$ meet
is equal to $1\over \sqrt{2}$. \par Since we can always identify
any finite set $V\subset \Z$ with the bounded set $\tilde V\subset
{\bf R^2}$ obtained by considering the union of all unit closed
squares centered at each site in $V$, with an abuse of
notation we will write for a generic closed contour $\G$: $\G \subset
V$ if $\G \subset \tilde V$ and $\G \cap V$ for the set of bonds
of $\G (\s )$ contained in $\tilde V$.\par Finally, given a
boundary condition $\t$ on the external boundary of a finite region
$V$, we can associate to any element $\s_V$ the configuration
$\s^{(\t +)}\in \O_{\Z}$ equal to $\s_V$ inside $V$, equal to $\t$ on
$\partial_{ext}V$ and equal to +1 outside $V\cup \partial_{ext}V $.
Then, via the previous construction, we can associate in a unique
way to $\s_V$ the {\it finite} collection of closed contours $\G
(\s^{(\t +)} )$ that, for simplicity, will be referred to as $\G^\t
(\s_V )$. If we consider $\G^\t (\s_V )\cap V$, then it will
consists of the union of some closed contours, in the sequel
refered to as the closed contours of $\s_V$ under the boundary
condition $\t$, and some open polygonal curves that will be refered
to as the open contours of $\s_V$ under the boundary condition
$\t$, where an open polygonal line is a sequence of distinct bonds
$e_o,\,e_1,\,e_2\,\dots e_n$ satisfying ii) and iii) above.\par
Notice that, by construction, the first and last bond of an open
contour necessarily separate at least one site in
$\partial_{int}V$.\par\par Let us now assume that $$\a_V\,\equiv
\,\min_{(x,y)\,\in \,\partial V}U^{\partial V}(x,y)\,>\,0\Eq(1.6)$$
Then, if for a given closed contour $\G$ we write $\G^{\partial V}$
for the sets of bonds in $\G$ that separate two sites $(x,y)\,\in
\,\partial V$ and we set $U^{\partial V}(h )\,\equiv\, U^{\partial
V}(x,y)$ for any pair $(x,y)\,\in \,\partial V$ that are separated
by $h\in \G^{\partial V}$, a simple computation shows that:
$$H_V^{U^{\partial V},+}(\s_V )\;=\;2\sum_{\G\in \G^+ (\s_V)}\{\vert
\G\vert \,-\,\sum_{h\in \G^{\partial V}}(1-U^{\partial V}(h
))\}\Eq(1.7)$$ Thus the partition function can be written as:
$$Z(V,U^{\partial V},+)\;=\;\sum_{\s_V} \exp(-\beta (2\sum_{\G\in
\G^+ (\s_V)}\{\vert \G\vert \,-\,\sum_{h\in \G^{\partial
V}}(1-U^{\partial V}(h ))\}))\Eq(1.8)$$ We can rewrite \equ(1.8) in
a more suitable form by introducing the notion of compatibility
between different contours. \par We say that the contours
$\G_1,\dots\G_n$ in $V$ are {\it compatible} if there exists
$\s_V\in \O_V$ such that $\G^+ (\s_V)\,=\,\{\G_1,\dots\G_n\}$ and we
denote by ${\cal C}_V$ the set of them. Then, if we denote by
$z_V^{U^{\partial V}}(\G )$ the weight of a single contour $\G$:
$$z_V^{U^{\partial V}}(\G )\;=\;\exp(-\beta 2\{\vert \G\vert
\,-\,\sum_{h\in \G^{\partial V}}(1-U^{\partial V}(h ))\}))\Eq(1.9)$$
\equ(1.8) can be written as : $$Z(V,U^{\partial
V},+)\,=\,\sum_{{\cal G}\in {\cal C}_V}\prod_{\G\in {\cal
G}}z_V^{U^{\partial V}}(\G )\Eq(1.10)$$ Then the main result of the
cluster expansion that is needed in the present paper can be stated
as follows :\bigskip {\bf Proposition 1.1}\par {\it Assume that
there exists a constant $\a\in [0,1)$ such that: $$z_V^{U^{\partial
V}}(\G )\,\leq \,\exp (-2\beta \vert \G\vert (1-\a ))\quad \forall
\, \G\in {\cal C}_V$$ Then there exists $\beta_o \,=\,\beta_o(\a )$
such that
for all $\beta \geq \beta_o$ the logarithm of the partition
function $Z(V,U^{\partial V},+)$ can be written as:
$$\log (Z(V,U^{\partial V},+))\,=\,\sum_{\L\subset V}
\Phi^{U^{\partial V},+} (\L )$$
where the coefficients $\Phi^{U^{\partial V},+} (\L )$ satisfy the
following two basic properties:\medskip
\item{1)} $$\Phi^{U^{\partial V},+} (\L )\,=\,
\Phi^{+} (\L )\qquad \hbox{if } \;\partial_{ext}\L\subset V$$ where
$\Phi^{+} (\L )$ is the coefficient associated to the set
$\L\subset V$ when the boundary coupling $U^{\partial
V}$ is identically equal to one.\par
\item{2)} For all $\L\subset V$, $\vert \L\vert \geq 2$: $$\eqalign{\vert
\Phi^{U^{\partial V},+} (\phantom{\{}\L \phantom{\{})\vert \,&\leq\,\exp
(-2(1-\a )[\beta \,-\,\beta_o]d(\L ))\cr\vert\Phi^{U^{\partial V},+} (\{x\}
)\vert \,&\leq\,\exp (-8(1-\a )[\beta \,-\,\beta_o])}$$
where, for all connected (in the sense of
subgraphs of the graph $\Z$) $\L\subset V$, $d(\L )$ is the length of
the smallest connected set of bonds from $\bar \L\, \equiv\,\{$ all
bonds in $V$ that separates at least one site in $\L\,\}$ containing
all the bonds separating sites in $\partial_{int}\L$ from sites in
$\partial_{ext}\L$. If $\L$ is not connected $d(\L
)\,=\,+\infty$.
}\bigskip $\S 3$\hskip 1cm Surface Tension and Wulff
Shape.\medskip\noindent We conclude our short review of the Ising
model by recalling the definition of the surface tension
$\t_\beta({\bf \vec n })$ and of the associated Wulff shape. Again
we follow the basic reference [DKS].\par
Let us fix a direction ${\bf \vec n \,\in \, }S^1$ ($S^1$ being the
unit circle) and let us define the boundary condition $\t^{\bf \vec
n}$ as follows:
$$\eqalign{t^{\bf \vec n}(x)\,&=\, +1 \quad\hbox{if}\quad (x,{\bf
\vec n})\,>\,0\cr t^{\bf \vec n}(x)\,&=\, -1 \qquad
\hbox{otherwise}}$$ where $(x,{\bf \vec
n})$ denotes the usual scalar product in $\bf R^2$.\par
Let also
$V_{N,M}$ be the rectangle $\{x\in \Z\,;\;\;-N\leq x_1\leq
N\,;\;-M\leq x_2\leq M\;\}$. Then we define the surface tension
with respect to a surface orthogonal to the direction $\bf \vec n$,
$\t_\beta({\bf \vec n})$, as:
$$\t_\beta({\bf \vec n})\,\equiv\,\lim_{N\to \infty} \lim_{M\to
\infty}{1\over \beta d(N,{\bf \vec n})} \log ({Z(V_{N,M},t^{\bf \vec
n}) \over Z(V_{N,M},+)})\Eq(1.11)$$ where $d(N,{\bf \vec n})$ is the
length of the segment $$\{x\,;\quad (x,{\bf \vec n})\,=\,0\, ,\quad
-N\leq x_1\leq N\;\}$$
We will simply write $\tau_\beta$ to denote the surface tension
associated to the direction ${\bf \vec n}\,=\,(1,0)$.\par
For a proof
of the existence of the limit \equ(1.11) when $\beta$ is large enough
see Theorem 1.15 in [DKS].\par
We now define the Wulff shape $W\, \subset \,
{\bf R^2}$ as:
$$W\,=\,\{x\in {\bf R^2};\;\vert (x,{\bf \vec n})\vert \,\leq \,
\l\t_\beta({\bf \vec n})\quad \forall \;{\bf \vec n}\,\}\Eq(1.12)$$
where the constant $\l$ is chosen in such a way that the area of
$W$ is equal to 1. The following fundamental result has
been proved in [DKS] (see also [Pf]):\bigskip
{\bf Theorem 1.1}\par
{\it Let, for any closed, piecewise smooth curve $\g$ in
${\bf R^2}$, the Wulff functional $W_\t(\g )$ on $\g$ be given by:
$$W_\t(\g )\;=\;\int_{\g}ds\,\t_\beta({\bf \vec n({\it s})})$$
where ${\bf \vec n({\it s})}$ is the normal vector at the point $s$
on the curve $\g$. Then, if we denote by $\partial W$ the closed
curve encircling the Wulff shape $W$, we have:
$$W_\t(\g )\;\geq \;W_\t(\partial W )$$
for any closed curve $\g$ which encloses an area equal to
one, and equality holds iff $\g$ is a translate of the
curve $\partial W$.}\bigskip $\S 3$\hskip 1cm A Class of
Block-Glauber Dynamics for The Ising Model.\medskip\noindent In
this paragraph we define, for a given finite set $V\subset \Z$,
boundary condition $\t\in \O_\Z$ and boundary coupling $U^{\partial
V}$, a class of Markov processes on $\O_V$ which are all reversible
with respect to the Gibbs measure $\mu_V^{U^{\partial V},\t}$.\par
Although the main object of study in this work is any standard (e.g
Metropolis or Heath Bath) {\it single spin flip} Markov process,
reversible with respect to the Gibbs measure of the Ising model, we
found very convenient to introduce, as a technical tool, auxiliary
Markov processes for which, in each updating of the dynamics, a
whole collection of dynamical variables (i.e. spins $\s_V(x)$) are
changed instead of just one. Each one of these auxiliary Markov
processes will be indexed by a certain covering of the set $V$ by
{\it blocks} (i.e. subsets of $V$) and at a given updating only the
spins inside a particular block will be changed.\par More
precisely, let $\{Q_i\}_{i=1\dots n}$ be a covering of $V$ and let
$$\eqalign{U^{\partial Q_i}(x,y)\,&=\,1\phantom{^{\partial V}(x,y)} \qquad
\hbox{if }(x,y)\,\in
\,\partial Q_i\setminus\{\partial Q_i\cap \partial V\}\cr U^{\partial
Q_i}(x,y)\,&=\,U^{\partial V}(x,y)\qquad \hbox{if }(x,y)\,\in
\,\partial Q_i\cap \partial V}\Eq(1.12bis)$$ Then we define the
generator $L^{\{Q_i\},\t ,U^{\partial V}}$ of the Markov process
$\s_t^{\{Q_i\},\t ,U^{\partial V}}$ indexed by the covering
$\{Q_i\}_{i=1\dots n}$ by: $$(\,L^{\{Q_i\},\t ,U^{\partial V}}f\,)
(\s_V )\;=\;\sum_{i}\sum_{\eta\in \O_{Q_i}} \mu_{Q_i}^{U^{\partial
Q_i},(\t\s_V)}(\eta )\,[f(\s_V^{\eta}\;-\;f(\s_V )\,]\Eq(1.13)$$
where $(\t\s_V)$ denotes the configuration in $\O_\Z$ equal to $\t$
outside $V$ and to $\s_V$ inside $V$, while $\s_V^{\eta}$ is the
configuration in $\O_V$ equal to $\eta$ in $Q_i$ and to
$\s_{V\setminus Q_i}$ in $V\setminus Q_i$. Most of the times we will refer
to the Markov process generated by $L^{\{Q_i\},\t ,U^{\partial V}}$
as the $\{Q_i\}$-dynamics.
\par A concrete way to construct the $\{Q_i\}$-dynamics starting from a
configuration $\s\,\equiv\,\s_V$ is to choose with rate $n$, ($n$ is the
cardinality of the covering), a particular element $Q_i$ of the covering, and
to replace the restriction to $Q_i$ of the configuration $\s$ with a
configuration $\eta\,\in \,\O_{Q_i}$ with probability $\mu_{Q_i}^{U^{\partial
Q_i},(\t\s )}(\eta )$.\par The particular case in which the elements
$Q_i$ of the covering are the sites $x$ of the set $V$ is known in the
literature as the {\it Heat Bath process} (HB-dynamics in the sequel) and it is
a particular example of a Glauber dynamics for the Ising model , that is a
Markov process on $\O_V$ with generator $L^{\t ,U^{\partial V}}$ of the form:
$$(\,L^{\t ,U^{\partial V}}f\,) (\s_V )\;=\;\sum_{x\in V}\sum_{a\in \{-1,+1\}}
c_x^{\t ,U^{\partial V}}(\s_V ,a)\,[f(\s_V^{x,a}\;-\;f(\s_V
)\,]\Eq(1.14)$$ where $\s_V^{x,a}$ is obtained from $\s_V$ by
substituting the value $\s_V(x)$ with $a$ and the jump rates
$c_x^{\t ,U^{\partial V}}(\s_V ,a)$ satisfy the {\it detailed
balance condition}: $$\mu_V^{U^{\partial V},\t}(\s_V)\,c_x^{\t
,U^{\partial V}}(\s_V ,a)\;=\;
\mu_V^{U^{\partial
V},\t}(\s_V^{x,a})\,c_x^{\t ,U^{\partial V}}(\s_V^{x,a},\s_V(x))\Eq(1.14bis)$$
and a short range condition:
$$c_x^{\t ,U^{\partial V}}(\s_V ,a)\,=\,c_x^{\t ,U^{\partial
V}}(\eta_V ,a)\quad\hbox{if }\;\s_V(y)\,=\,\eta_V(y)\quad\forall
\;\norm{x-y}\,\leq\, R$$ for some finite $R$. \par
As it is easy to check, the $\{Q_i\}$-dynamics is a
(continuous time) Markov chain on $\O_V$, reversible with respect to the Gibbs
measure $\mu_V^{U^{\partial V},\t}$; in other words $L^{\{Q_i\},\t ,U^{\partial
V}}$
is symmetric in the Hilbert space $L^2(\O_V,\,d\mu_V^{U^{\partial V},\t})$ with
real
non positive eigenvalues
$$0\,=\,\l_o(\{Q_i\},\t ,U^{\partial V})\,>\,-
\l_1(\{Q_i\},\t ,U^{\partial V})\,\geq\,\dots\,\geq \, -\,
\l_{k}(\{Q_i\},\t ,U^{\partial V});\quad k\,=\,2^{\vert V\vert}$$ The
absolute value of the first negative eigenvalue, $\l_1(\{Q_i\},\t
,U^{\partial V})$, will be of special value for us and it will be
denoted by $\hbox{gap}_V(\{Q_i\},\t ,U^{\partial V})$ or by
$\hbox{gap}_V(HB,\t ,U^{\partial V})$ if the dynamics under
consideration is the Heat-Bath.\par The following variational
characterization of the gap will be particulaly useful in the
sequel. Let, for any $f\,\in \, L^2(\O_V,\,d\mu_V^{U^{\partial
V},\t})$, ${\cal E}(f,f)$ be the Dirichlet form associated to the
generator $L^{\{Q_i\},\t ,U^{\partial V}}$: $${\cal
E}(f,f)\,=\,{1\over 2}\sum_i\sum_{\s_V}\sum_{\eta\in \O_{Q_i}}
\mu_V^{U^{\partial V},\t}(\s_V)\mu_{Q_i}^{U^{\partial
Q_i},(\t\s_V)}(\eta )\,[f(\s_V^{\eta})\;-\;f(\s_V
)\,]^2\Eq(1.14tris)$$ Then:
$$\hbox{gap}_V(\{Q_i\}\,=\,\inf_{\vbox{\eightpoint{\hbox{$f\,\in
\, L^2(\O_V,\,d\mu_V^{U^{\partial V},\t})$}
\hbox{$$}}}}{{\cal E}(f,f)\over Var(f)}\Eq(1.14quatris)$$
where $$Var(f)\,=\,{1\over 2}\sum_{\s ,\h}\mu_V^{U^{\partial V},\t}(\s )
\mu_V^{U^{\partial V},\t}(\h )[f(\s )\,-\,f(\h )]^2$$
{\bf Remark} Using the above variational characterization
of the gap, it is very easy to check that, if we consider a general
Glauber dynamics defined as in \equ(1.14) with jump rates bounded
above and below uniformly in $\s_V$ and in $V$, then the
corresponding gap can be bounded from above and from below by
$\hbox{gap}_V(HB,\t ,U^{\partial V})$ multiplied by two suitable
constants. \bigskip The following simple estimate, which follows
from elementary $L^2$ consideration, illustrates the role played by
the $\hbox{gap}(\{Q_i\},\t ,U^{\partial V})$ in the approach to the
invariant measure $\mu_V^{U^{\partial V},\t}$ of the distribution
$P_{V,\eta_V}^{\{Q_i\},\t ,U^{\partial V}}(t)$ of the
$\{Q_i\}$-dynamics at time t starting from $\eta_V$ at time $t=0$:
$$\norm{P_{V,\eta_V}^{\{Q_i\},\t ,U^{\partial V}}\;-
\;\mu_V^{U^{\partial V},\t}}\;\leq \;{\exp (-t \hbox{
gap}(\{Q_i\},\t ,U^{\partial V}))\over 2[\mu_V^{U^{\partial
V},\t}(\eta_V)]^{1\over 2}}\Eq(1.15)$$ where, for two arbitrary
probability measures $\nu$ and $\mu$ on $\O_V$, $\norm{\nu\;-\;\mu}$
denotes their variation distance.\bigskip {\bf Remark} It is
worthwhile to observe that \equ(1.15) can be a very bad estimate
since the denominator $[\mu_V^{U^{\partial V},\t}(\eta_V)]^{1\over
2}$ is of order $\exp (-c\beta\,\vert V\vert )$ for some
constant $c$. There are situations, for example when $\beta$ is
smaller than the critical value $\beta_{c}$, in which the factor
$[\mu_V^{U^{\partial V},\t}(\eta_V)]^{-{1\over 2}} $ in \equ(1.15)
can be replaced by $c\vert V\vert$ for some constant $c$ (see e.g [SZ],[MO1]
and [MO2]).
However, in a phase transition regime, $\beta
\,>\, \beta_c$, the gap can be very small, something like $\exp
(-cL)$ if $V$ is a square of side $L$ with open boundary conditions
(see section 4), and therefore the possible improvement in the
denominator from $\exp (-c\beta\,\vert V\vert )$ to some negative
power of $\vert V\vert$ is negligible.\bigskip $\S 4$\hskip 1cm
Coupling for the $\{Q_i\}$-Dynamics .\medskip\noindent We conclude
this preparatory section by discussing a useful coupling for the
$\{Q_i\}$-dynamics that will be essential in the forthcoming
sections.\par
Let, for any finite set $V$,
$\t^{(1)},\,\t^{(2)}\,\dots\,t^{(N)}$ be $N\,\leq
\,2^{\vert \partial_{ext}V\vert }$ boundary conditions on
the external boundary of $V$, and let $\nu_{V}^{
\t^{(1)},\,\t^{(2)}\,\dots\,t^{(N)}}$ be the
unique invariant probability measure on $(\O_{V})^N$ ($N$ copies
of $\O_{V}$) of the following
ergodic Markov process:\bigskip \item{i)} With rate $\vert V\vert$
one chooses a site $x\,\in \,V$ and, given $x$, a random number
$\xi_x\,\in \,[0,1]$ with a uniform distribution.
$$\Eq(1.15bis)$$
\item{ii)}For $k\,=\,1\dots N$ the value of the spin at x in the
$k^{th}$ component of the initial configuration $\tilde \s_V\;\equiv
\;\{\s^{(1)}_V\dots \s^{(N)}_V\}$, $\s^{(k)}_V\,\in \, \O_{V}$, is
replaced by $+1$ if
$$\xi_x\,\leq\,\mu_{\{x\}}^{U^{\partial\{x\}},(\t^{(k)}\s^{(k)}_V)}(+1)\Eq(1.15tris)$$
and by $-1$ if the opposite inequality holds. Here
$U^{\partial\{x\}}$ is defined as in \equ(1.12bis) but with $Q_i$
replaced by $\{x\}$.\bigskip\noindent The above algorithm is of
course nothing more than an explicit way to realize on a common
probability space the HB-dynamics in $V$ with different boundary
conditions $\t^{(1)},\,\t^{(2)}\,\dotsJ\,t^{(N)}$.\par
Using this observation one can explicitely check that the measure
$\nu_{V}^{ \t^{(1)},\,\t^{(2)}\,\dots\,t^{(N)}}$ enjoyes the
following properties:
$$\sum_{\vbox{\eightpoint{\hbox{$\eta^{(1)},\dots\eta^{(k-1)}$}
\hbox{$\eta^{(k+1)}\dots\eta^{(N)}$}}}}
\nu_{V}^{ \t^{(1)},\,\t^{(2)}\,\dots \,t^{(N)}}
(\eta^{(1)},\dots\eta^{(k-1)},\eta^{(k)},
\eta^{(k+1)}\dots\eta^{(N)} )\;=\;\mu_{V}^{U^{\partial
V},(\t^{(k)})}(\eta^{(k)} )\Eq(1.16)$$
$$\nu_{V}^{ \t^{(1)},\,\t^{(2)}\,\dotsJ\,t^{(N)}}
(\eta^{(k)}\,\leq\,\eta^{(j)} )\;=\;1\quad \hbox{if }
\t^{(k)}\,\leq \t^{(j)}\Eq(1.17)$$
Given now a finite set $V$, a boundary condition $\t$ and a
covering $\{Q_i\}_{i=1}^{n}$, let, for each $i=1\dots n$,
$\t^{(1)},\,\t^{(2)}\,\dots\,t^{(N)}$ be an arbitrary enumeration of
all the possible boundary conditions on the external boundary of $Q_i$
which agree with $\t$ on $\partial_{ext}Q_i\cap \partial_{ext} V$,
and let
$$\nu_{Q_i}\,\equiv \,
\nu_{Q_i}^{ \t^{(1)},\,\t^{(2)}\,\dots\,t^{(N)}}$$
Using the measures
$\nu_{Q_i}$, we can mimick the algorithm \equ(1.15bis), \equ(1.15tris),
to realize on a
common probability space the Markov processes $\s_t^{\{Q_i\},\t
,U^{\partial V}}$ starting from an arbitrary initial condition
$\s_V$ as follows:\medskip \item{a)} With rate $n$ ($n$ is the
cardinality of the covering) we choose one of the $Q_i$'s.
$$\Eq(1.18)$$ \item{b)}
For all $k=1\,\dots N$, the configurations $\s_V$ which agree with
$\t^{(k)}$ on the external boundary of $Q_i$ are updated to
$\s_V^{\eta^{(k)}}$, $\eta^{(k)}\,\in \, \O_{Q_i}$, and the joint
probability of $\eta^{(1)},\,\eta^{(2)},\dots \eta^{(N)}$ is
$\nu_{Q_i}(\eta^{(1)},\,\eta^{(2)},\dots \eta^{(N)})$.
$$\Eq(1.19)$$ It
is clear that, because of \equ(1.16) above, a) and b) give the right
law for the evolution of any given initial configuration $\s_V$.
Moreover, because of \equ(1.17), it also follows that any ordered
set of initial conditions
$\s_{V}^1\leq\s_{V}^2\leq\dots\leq\s_{V}^k$ will remain orderded for
any future time $t$. We will refer to this last property as
monotonicity in the initial configuration.\pagina
%\input formato.tex
\numsec=2 \numfor=1
{\bf Section 2}\par
\centerline{\bf Geometric Bounds On the Gap}\bigskip In
this section we establish two basic estimates on the gap which,
besides being interesting by themselves, will play a crucial role in
the determination of the exact asymptotics in the thermodynamic
limit of the gap of the HB-dynamics in a finite square with open
boundary conditions. The first estimate relates $\hbox{gap}_V(HB,\t
,U^{\partial V})$ to $\hbox{gap}_V(\{Q_i\},\t ,U^{\partial V})$
when $V$ is a rectangle $V_{N,M}$
$$V_{N,M}\,=\,\{x;\;-N\leq x_1\leq N;\;-M\leq x_2\leq M\}$$
with, say, $M\,\leq\,N$
and the covering $\{Q_i\}$ consists of rectangles:
$$Q_i\,=\,\{\;x\in \Z;\;-N\,\leq \,x_1\,\leq\,N\quad i{l\over
2}\,\leq\,x_2\,\leq\,(i+2){l\over 2}\;\}$$ with $l\over 2$ and
$2M/l\;$ integers, $i\,=\,-{2M\over l}\dots {2M\over l}-2$. The
estimate shows that the
ratio $$\hbox{gap}_V(HB,\t ,U^{\partial V})\over
\hbox{gap}_V(\{Q_i\},\t ,U^{\partial V})$$ is bounded from below by
a suitable exponential of the {\it short} side $l$. More precisely
:\bigskip {\bf Theorem 2.1}\par {\it Let $V$ and $\{Q_i\}$ be as
above. Then for any boundary coupling $U^{\partial V}$ and any
boundary condition $\t$ we have: $$\hbox{\rm gap}_{V}(HB,\t
,U^{\partial V})\;\geq\; {1\over 2\vert Q_i\vert}{\exp
(-4\beta)\over \exp (-4\beta)+\exp (+4\beta)}\,\exp (-4\beta
(l\,+\,1))\,\hbox{\rm gap}_V(\{Q_i\},\t ,U^{\partial V})$$} \bigskip
{\bf Remark}
The above theorem remains valid also if the covering of the set V consisted
of rectangles $Q_i$ with longest side smaller than that of $V_{N,M}$.
However, for reasons that will become clear in the next section,
the above choice of the covering is very sensible in the low
temperature regime. It will also become clear at the end of the
proof of the theorem that our method allows one to relate
the gap of the HB-dynamics to that of the $\{Q_i\}$-dynamics for
arbitrary geometric shapes of the elements of the covering. This
generality is however not needed in the present paper.\bigskip As a
corollary we obtain, in the same setting as above, that
$\hbox{gap}_V(HB,\t ,U^{\partial V})$ is not smaller than a negative
exponential of the shortest side $M$. More precisely we
have:\bigskip {\bf Corollary 2.1}\par {\it For any boundary coupling
$U^{\partial V}$ and any boundary condition $\t$ we have:
$$\hbox{\rm gap}_{V}(HB,\t ,U^{\partial V})\;\geq\; {1\over 2\vert
V\vert}{\exp
(-4\beta)\over \exp (-4\beta)+\exp (+4\beta)}\,\exp (-4\beta
(2M\,+\,1))$$} \bigskip {\bf Proof of the corollary}\par Let us take
in theorem 2.1 the shortest side $l$ of the elements $Q_i$ of the
covering equal to $2M$ so that the covering consists of just the
rectangle $V_{N,M}$ itself. Then the generator $L^{\{Q_i\},\t
,U^{\partial V}}$ restricted to the space of functions of mean zero
(i.e. orthogonal to the constant functions) becomes minus the
identity, so that $\hbox{gap}_V(\{Q_i\},\t ,U^{\partial V})\,=\,1$,
and the corollary follows from theorem 2.1. \bigskip {\bf Remark}
The estimate described in the corollary is a very bad one for
temperatures above the critical one (that is $\beta\,<\,\beta_c$),
since in this case it has been recently proved by Olivieri,
Schonmann and myself [MOS] that the gap is bounded away from zero
uniformly in $N$ and $M$. However, at low temperature, when the
infinite volume dynamics is not ergodic, it gives the right
dependence on the size of the set $V_{N,M}$, namely a negative
exponential of the surface and not of the volume $\vert
V_{N,M}\vert$, but the constant in the exponential is wrong by a
factor 2 even in the limit $\beta\,\to \,\infty$. A more precise
bound will be discussed in the next section.\par The proof of the
corollary represents also the first, actually rather trivial,
example of the role played by the $\{Q_i\}$-dynamics: in an approach
to a gap estimate this latter may be considerably simpler than the
single spin dynamics. In particular one may try to attack the
problem of finding a lower bound on the gap of the HB-dynamics by
first proving lower bounds on the gap of the $\{Q_i\}$-dynamics and
then, using Theorem 2.1 above, transfer the bound to the
HB-dynamics. This idea played an important
role in the analysis of the approach to equilibrium in general
Glauber dynamics in the one phase region (see e.g. [H], [SZ],
[MO1]). However its application in the phase transition region seems
to be new. \bigskip {\bf Proof of Theorem 2.1}\par The
proof is an application in our contest of some geometric techniques
developed few years ago in order to bound from below the gap of
symmetric Markov chains on complicate graphs (see e.g. [LS], [JS1],[JS2],
[Si], [DS]). We will make use in particular of some beautiful ideas
introduced by Jerrum and Sinclair in their
study of rapid mixing properties of Markov chains arising in some
hard computational problems. The way these techniques apply to spin
dynamics like the Glauber dynamics was discussed for the first time
in some unpublished notes of mine and used recently, in a slightly
different form, by Schonmann in his study of metastability for the
Ising model [Sch1]. \par
In what follows we will omit for simplicity in all the notation the
boundary condition $\t$, the boundary coupling $U^{\partial V}$ and
the volume $V$. Thus for example the Gibbs measure
$\mu_V^{U^{\partial V},\t}$ will become $\mu$, the conditional Gibbs
measure on $Q_i$, $\mu_{Q_i}^{U^{\partial Q_i},(\t\s)_V}(\eta )$,
$\mu_{Q_i}^{\s}(\eta )$ and similarly for the generators of the HB
and $\{Q_i\}$ dynamics together with their gaps. \par We start by
introducing the set of {\it canonical paths} in $\O_V$ between
configurations $\s$ and $\s '$ with $\s \,\neq\, \s '$, that are connected
by just one
single jump of the $\{Q_i\}$-dynamics, that is $\s '\,=\,\s^{\eta}$ for
some $i$ and some $\eta\in \O_{Q_i}$. We adopt the convention that, if the $\s
,\,\s
'$ can be connected by the updating either of $Q_i$ or of $Q_{i+1}$, due to their
mutual overlap, then we think of $\s '$ as arising from the updating of $Q_i$.\par
Let
us first order the sites in each rectangle $Q_i$ as follows: $$x\;< \;y\quad
\hbox{iff}\quad x_1\,<\,y_1\; \hbox{ or } x_1\,=\,y_1\; \hbox{ and
}x_2\,<\,y_2$$
Given now $\sigma\,\in \O_V$ and $\eta\in \O_{Q_i}$ we define the path $\gamma
(\sigma
,\s^{\eta})$ as the sequence of configurations obtained from $\sigma$
by adjusting
one by one, in increasing order, the values of its spins in $Q_i$ to those of
the
spins of $\s^{\eta}$. More precisely, if $x_1\,,\,.....\,x_n$ are the sites
in $Q_i$,
ordered as above, such that $\sigma (x_i)\,\neq\,\s^{\eta}(x_i)$, then we define
$\gamma (\sigma ,\s^{\eta})\,=\,\{\sigma^o\,...\,\sigma^n\}$ where $\sigma^i$,
$i=1\dots n$, is the configuration equal to: $$\eqalign{\sigma^i(x)\,
&=\,\s^{\eta}(x)\quad\hskip 1cm \hbox{iff } x\,\leq\,x_i\cr \sigma^i(x)\,&=\,\s
(x)\quad\hskip 1cm \hbox{iff } x\,>\, x_i}\Eq(2.1)$$ and $\sigma^o\,=\,\sigma$.\par
Next, for any allowed transition of the HB-dynamics
$$\bar\sigma\,\to\,\bar\s^{x,a}\quad x\,\in
\,Q_i, \quad a\,=\,-\bar\s (x)$$ we set
$$\eqalign{e\,&=\,(\bar\sigma,\bar\sigma^{x,a} ) \cr
Q(e)\,&=\,\mu(\bar\sigma)\mu_x^{\bar\sigma} (a)}\Eq(2.2)$$ and we say that the
transition $e$ belongs to the canonical path $\g$, $e\,\in \g$, if, for
some index
$i$, $(\bar \s,\bar\sigma^{x,a} )\,=\,(\s^i,\s^{i+1})$. Finally we define the
constant $\rho$ as : $$\rho \,=\,\sup_{i,e}
\sum_{\s\,,\eta\atop e\in \gamma (\sigma,\s^{\eta})} {\mu
(\sigma)\mu_{Q_i}^{\s}(\eta)\over Q(e)}\Eq(2.3)$$ Then we have:
$$\hbox{gap}_{V}(HB)\;\geq\; {1\over 2\vert Q_i\vert}{1\over \rho
}\,\hbox{gap}_V(\{Q_i\})\Eq(2.4)$$ Although the proof of \equ(2.4)
can be found in [Si], we reproduce it below because of its
simplicity.\par Using the variational principle for any
$f\,\in\,L^2(\O_V,\,d\mu )$ we have :
$$Var(f)\,\leq\,\hbox{gap}_V(\{Q_i\})^{-1}{1\over 2}\sum_{\s
,i}\sum_{\eta}\mu (\s )\mu_{Q_i}^{\s }(\eta)[f(\s^{\eta})\,-\,f(\s
)]^2\,=$$ $$=\;\hbox{gap}_V(\{Q_i\})^{-1}{1\over 2}\sum_{\s
,i}\sum_{\eta}\mu (\s )\mu_{Q_i}^{\s }(\eta)[\sum_{j=1\dots
n}(f(\s^{j}\,-\,f(\s^{j-1}))]^2\;\Eq(2.5)$$ where $\gamma
(\s\,,\s^{\eta})\,=\,\{\s^o\,\s^1\dots\s^n\}$ is the canonical path
going from $\s$ to $\s^{\eta}$.\par Using the Schwartz inequality,
the fact that the length $n$ of the path is smaller than $\vert
Q_i\vert$ and the definition of $\rho$, we can bound from above the r.h.s. of
\equ(2.5) by:
$$\hbox{gap}_V(\{Q_i\})^{-1}\rho\,\vert Q_i\vert {1\over
2}\sum_{\s,\,i}\sum_{x\in
Q_i}\sum_{a\in \{-1,1\}}\mu (\s )\mu_{x}^{\s
}(a)[f(\s^{x,a}\,-\,f(\s )]^2\;\leq\;$$
$$\leq\;2\hbox{gap}_V(\{Q_i\})^{-1}\rho\,\vert
Q_i\vert\,{\cal E}_{HB}(f,f)\Eq(2.6)$$ where
${\cal E}_{HB}(f,f)$ is the Dirichlet form of the HB-dynamics and the factor
$2$ in
the first inequality comes from the fact that most of the sites belong to two
elements ofthe covering. Thus, if we combine \equ(2.5) with \equ(2.6) and
(1.18), we
get: $$\hbox{gap}_V(HB)\,\geq\,{\hbox{gap}_V(\{Q_i\})\over 2\rho\,\vert Q_i\vert}$$
and
in order prove the theorem, we only need to estimate from above the
constant $\rho$ by: $$\rho\;\leq\;{\exp (4\beta)+\exp (-4\beta)
\over \exp (-4\beta)}\,\exp
(4\beta (l\,+\,1))\Eq(2.7)$$ uniformly in the boundary condition
$\t$ and in the boundary coupling $U^{\partial V}$.\par Apparently
this is not an easy problem since we have to count how many
canonical paths use a given allowed transition $e$ = $(\bar
\sigma,\bar\sigma^{x,a} )$. It is precisely at this stage that
Jerrum and Sinclair's lovely ideas become essential.\par Given a
transition $e\,=\,(\bar\s\,,\bar\s^{x,a})$ we define an {\it
injective $\Phi$} mapping from the set of all the canonical paths
that use the transition $e$, $\G (e)$, to $\O_V$ as follows:
$$\eqalign{\Phi(\gamma (\sigma ,\s^{\eta}))(y)\,&=\, \sigma (y)
\phantom{\h}\qquad \forall \,y\,\in \, Q_i,\quad y\,< \, x\cr \Phi(\gamma
(\sigma ,\s^{\eta}))(y)\,&=\, \s^{\eta} (y) \qquad\forall \,y\,\in \,
Q_i,\quad y\,\geq \, x\cr \Phi(\gamma (\sigma ,\s^{\eta}))(y)\,&=\,
\sigma (y)\phantom{\h} \qquad \forall \,y\,\notin \, Q_i }\Eq(2.7bis)$$ where
the
index $i$ labels the rectangle associatesd to the path $\g (\s ,\s^\h )$.\par
It is
clear that $\Phi$ is iniective. In fact the knowledege of the transition {\it
e},
that is of $x$ and $\bar \s$, {\it and } of $\xi\,\equiv\,\Phi(\gamma (\sigma
,\s^{\eta}))$ allow us to reconstruct completely the initial and final
configurations
$\sigma$ and $\s^{\eta}$ and thus the path itself, simply by observing that,
for example: $$\eqalign{\s (y)\,&=\,\bar \s (y)\quad \forall
\;y\notin Q_i\cr \s (y)\,&=\,\bar \s (y)\quad \forall \;y\in
Q_i\quad y\,\geq\,x\cr\s (y)\,&=\,\xi (y)\quad \forall \;y\in Q_i\quad
y\,< \,x}\Eq(2.8)$$ and similarly for $\s^{\eta}$.\par Let now
$c_o$ be the smallest constant such that for any canonical path
$\gamma (\sigma ,\s^{\eta})$ in $\Gamma (e)$ the following bound
holds : $$\mu_{Q_i}^{\s}(\Phi (\g ))\,Q(e)\; \geq \; {1\over c_o}\mu
(\sigma)\mu_{Q_i}^\s(\eta)\Eq(2.9)$$ Then we
have $$\rho\; \leq \; c_o\Eq(2.10)$$
Using \equ(2.9) we can in fact estimate the r.h.s. of \equ(2.3) by:
$$c_o\sup_{e,i}\sum_{\gamma\,\in \,\G (e)}\mu_{Q_i}^{\s} (\Phi (\gamma
))\Eq(2.11)$$ Since the map $\Phi$ is injective and $\mu$ is a
probability measure, the sum in \equ(2.11) is not greater than one
and \equ(2.10) follows.\par In order to estimate the constant $c_o$,
let, for $x\,\in \,Q_i$, $\partial_x$ be the set of bonds in $Q_i$
which separates sites in $Q_i$ smaller or equal than $x$ from sites in $Q_i$
larger than $x$. Clearly, by construction, $\partial_x$
consists of two vertical segments joined by a single horizontal bond
$h$ at distance $1\over 2$ from $x$ and placed below it if
$x_1\,\geq\,-N\,+\,1 $, and by a single vertical segment plus an
horizontal bond as above if $x_1\,=\,-N$. Let also, for any pair
configurations $\s_1$, $\s_2$ which agree outside $Q_i$,
$H_{\partial_x}(\sigma_1,\sigma_2)$ be the interaction through
$\partial_x$ of a configuration $\s\,\in \, \O_V$ which is equal to
$\sigma_1$ ($\sigma_2$) to the left (right) of $\partial_x$. More
precisely:
$$H_{\partial_x}(\sigma_1,\sigma_2)\;=\;
-\sum_{y\,\leq\,
x\,<\,z\atop z\in
Q_i,\,\norm{y-z}\,=\,1}(\sigma_1(y)\sigma_2(z)-1)\Eq(2.12)$$
Clearly $\vert H_{\partial_x}(\sigma_1,\sigma_2)\vert $ is bounded from above
by $2(l\,+\,1)$.
Then, by direct inspection: $${\mu (\sigma)\mu_{Q_i}^\s(\eta)\over
\mu_{Q_i}^{\s} (\Phi (\g ))\mu (\bar \s )}\,=\,$$ $$=\;\exp (-\beta
[H_{\partial_x}(\s,\s )\,+\,
H_{\partial_x}(\s^{\eta},\s^{\eta})\,-\,H_{\partial_x}(\bar\s,\bar\s
)\,-\,H_{\partial_x}(\Phi (\g ),\Phi (\g ))])\Eq(2.13)$$ for any
boundary condition $\t$ and boundary coupling $U^{\partial V}$. In
turn \equ(2.13), together with \equ(2.12) and the observation that
$$\mu_x^{\bar \s}(a)\,\geq\,{\exp (-4\beta)\over \exp (-4\beta)+\exp
(+4\beta)}\quad \forall\; x,\;\bar \s,\;\t,\;U^{\partial V}$$ implies
that the l.h.s
of \equ(2.13) is smaller than $${\exp (4\beta)+\exp (-4\beta)\over \exp
(-4\beta)}\,\exp (4\beta
(l\,+\,1))\mu_x^{\bar\s}(a)\Eq(2.14)$$ that is $$\mu_{Q_i}^{\s}
(\Phi (\g ))\,Q(e)\; \geq \;{\exp (4\beta)+\exp (-4\beta)\over \exp
(-4\beta)}\,\exp (4\beta
(l\,+\,1))
\mu (\sigma)\mu_{Q_i}^\s(\eta)\Eq(2.15)$$ Thus the constant $c_o$
can be taken equal to
$$c_o\;=\;{\exp (4\beta)+\exp (-4\beta)\over \exp
(-4\beta)}\,\exp (4\beta
(l\,+\,1))$$
Using \equ(2.7), \equ(2.10), the theorem follows.\bigskip {\bf
Remark} It is amusing to observe that, if one applies the above
construction to the one dimensional case for which the set
$\partial_x$ consists of just {\it one } bond,
$H_{\partial_x}(\sigma_1,\sigma_2)$ can be bounded by a constant
independent of $\s_1$, $\s_2$ and of the dimension of $V$, even if
the energy (1.1) of a configuration $\s_V$ is replaced by a more
general expression like: $$H(\s_V)\,=\,-{1\over 2}\sum_{x,y\in
V}J(\norm{x-y})\s_V(x)\s_V(y)\quad + \hbox{b.c.}$$ provided that
the long range potential $J(\norm{x-y})$ decays faster than
$\norm{x-y}^{-{2+\e}}$ for some $\e\,>0$. Therefore in this case the
gap of the corresponding Heat Bath dynamics in a segment of length
$L$ in $\bf Z$ has a lower bound which is only proportional to
$L^{-1}$ without any negative exponential of $L$.\par On the other
hand it is known that a long range potential $J(\norm{x-y})$ with a
fast decay as above is not able to induce any phase transition, the
reason being that the energy between two semi-infinite lines is
finite uniformly in the spin configuration.\par Thus, in some sense,
the above geometric contruction is able to capture, at least at the
level of the exponential, some (but certainly not all) of the
physical aspects of the presence (or of the absence) of a phase
transition in the Ising model at low temperature.\bigskip \pagina
%\input formato.tex
\numsec=3 \numfor=1
\tolerance =15000
{\bf Section 3}\par
\centerline{\bf A Lower Bound On the
Gap With + Boundary Conditions}
\centerline{\bf and Its Application} \bigskip In this section we consider
the HB-dynamics in a square $V\equiv V_L$: $$V_L\,=\,\{\,x\,\in \,
\Z\,;\quad 0\,\leq\,x_i\,\leq\,L\quad i=1,2\,\}$$ with full plus
boundary conditions, that is $$\eqalign{\t (x)\,&=\,+1\; \;\forall
\,x\in \Z\cr U^{\partial V_L}(x,y)\,&=\,+1\quad \forall \;(x,y)\,\in
\, \partial V_L}$$ and very large $\beta$.\par We show that, due
precisely to the presence of the plus boundary conditions, the gap
of the HB-dynamics is much larger, as $L\to \infty$, than its value
with {\it open} boundary conditions (see also the discussion in the
introduction). As a simple consequence we show that the equal site
time correlations of the infinite volume process started
in the plus phase decay faster than any inverse power of the time.\par
Before stating and discussing our main result let us fix few more
convenient notation. We will denote by $(+)$ and $(-)$ the two extreme
configurations in $\O_{V_L}$ identically equal to plus and minus one
respectively and, for any rectangle $R$, by
$\mu_R^{\t_1,\t_2,\t_3,\t_4}$, the Gibbs measure on $R$ with the boundary
conditions $\t_1,\t_2,\t_3,\t_4$ on the external boundary of its four
sides ordered clockwise starting from the bottom side. We use the usual
convention that, if one of the configurations $\t_i$ is identically equal
to +1 or -1, then we replace it by a + or a $-$ sign. Thus for example
$\t_1,+,-,+$ means $\t_1$ boundary conditions on the bottom side, plus
boundary conditions on the vertical ones and minus boundary condition on
the top one. Whenever confusion does not arise we will also omit the
subfix $V$ in the notation $\s_V$.\par
We finally denote by $\mu^+$ the infinite volume Gibbs state obtained as the limit
as $L\to \infty$ of finite volume Gibbs states $\mu_V^+$ with plus boundary
conditions, by $m^*(\b )\,=\,\mu^+( \s (0))$ the
spontaneous magnetization and by $\s_t$ the infinite volume Heat Bath dynamics
started from the (infinite volume) configuration $\s$ (see [Li] for the existence
of such process).\par
We can now state the main results : \bigskip
{\bf Theorem 3.1}\par {\it Let $\e\,\in \, (0,{1\over 2})$ be given.
Then there exists $\beta_o\,<\,+\infty$ and $C\,<\,+\infty$ such that
for any $\beta\,\geq\,\beta_o$ and any integer $L$:
$$\hbox{gap}_{V_L}(HB,\,+)\;\geq\;\exp (-C\beta L^{{1\over
2}+\e})$$}
{\bf Theorem 3.2}\par {\it Let $\a\,\in \,[0,2)$ be given.
Then there exists $\beta_o\,<\,+\infty$ and $C\,<\,+\infty$ such that
for any $\beta\,\geq\,\beta_o$
$$0\,\leq\,\int d\mu^+(\s )\s (0)E(\s_t(0))\,-\,(m^*(\b ))^2\,\leq\,C\exp
(-(log(t))^\a)\quad \forall \;t$$}
{\bf Proof of Theorem 3.1}\par Let $l\,=\,2[L^{{1\over 2}+\e}]$ and let
us suppose, without loss of generality, that $
N\,\equiv \,{2L\over l}\,-\,1$ is an integer; for
$i\,=\,1\dots N$, we define $Q_i$ to be
the rectangle: $$Q_i\,=\,\{\;x\in V_L;\,0\,\leq \,x_1\,\leq\,L,\quad
(i-1){l\over 2}\,\leq\,x_2\,\leq\,(i+1){l\over 2}\;\}$$ Then, using
theorem 2.1, we have that: $$\hbox{gap}_{V_L}(HB,\,+)\;\geq\;
{1\over \vert Q_i\vert}{\exp (-4\beta)\over \exp (-4\beta)+\exp
(+4\beta)}\,\exp (-4\beta
(l\,+\,1))\,\hbox{gap}_{V_L}(\{Q_i\},\,+)\Eq(3.1)$$ It remains to
show that the $\{Q_i\}$-dynamics has a "large" gap, where "large"
means, for example, larger than $\exp (-L^{{1+\e\over 2}})$
.\par To prove this result we will
show that, with very large probability, under the coupling for the
$\{Q_i\}$-dynamics described at the end of section 1, the two extreme
configurations, $(+)$ and $(-)$ become identical in a time smaller
than $\exp (L^{{1+\e\over 2}}) $.\par The intuitive
reason for that, which also explains our apparently strange choice
of the length $l$ of the short side of $Q_i$, is the following.\par
Let us suppose that we start with the two extreme configurations and
that we update one after the other in increasing order of $i$ the
rectangles $Q_i$. In the first updating of $Q_1$ we have to replace
$(+)_{Q_1}$ and $(-)_{Q_1}$ with two configurations $\eta_{Q_1}^+$
and $\eta_{Q_1}^-$ distributed according to $\mu_{Q_1}^{+,+,+,+}$
and $\mu_{Q_1}^{+,+,-,+}$ respectively. It is a relatively easy
matter to show (see Proposition 3.1 below) that, for large enough
$\beta$, due to our choice of $l$ and to the fact that in two
dimensions the fluctuations of an interface separating plus spins
from minus spins are of the order of the {\it square root} of the
length of the interface, it is possible to couple the two measures
$\mu_{Q_1}^{+,+,+,+}$, $\mu_{Q_1}^{+,+,-,+}$ in such a way that,
with probability much larger than $1\,-\,{1\over N}$, the two
configurations $\eta_{Q_1}^+$ and $\eta_{Q_1}^-$ are identical in a
large portion of $Q_1$, e.g. for all $x\in Q_1$ with
$x_2\,\leq\,{3l\over 4}$, and in particular on the external boundary
of the bottom side of $Q_2$. Moreover, with large probability, both
$\eta_{Q_1}^+$ and $\eta_{Q_1}^-$ will be mostly +1 on the external
boundary of the bottom side of $Q_2$. Thus the second updating in
$Q_2$ will be very similar to the first one in $Q_1$ with the
exception that now the boundary conditions on the external boundary
of the bottom side of $Q_2$ will not be identically equal to + but
only approximately. \par As we will show below this fact, with
probability much larger than $1\,-\,{1\over N}$, does not really
matter and one can, at least in a first appproximation, consider the
+ boundary conditions also on the bottom side of $Q_2$. In this
approximation the second updating will be statistically equal to the
first one and, with large probability, it will force
$(+)^{\eta_{Q_1}^+}$ and $(-)^{\eta_{Q_1}^-}$ to agree also in
$3\over 4$ of $Q_2$ without introducing any new discrepancy between
them in the previous region of agreement $$\{\,x\in
Q_1;\;x_2\,\leq\,{3l\over 4}\,\}$$ In such a way, after the first
two updatings, the evoluted of $(+)$ and $(-)$ will agree in the
set $$\{\,x\in V_L;\;\;0\leq x_2\leq {5\over 4}l\,\}$$
By iterating this procedure N times we can glue together $(+)$ and
$(-)$ in N steps with a probability of order one.\par Since the
probability of {\it not} having within time $t$ a sequence of N
updatings exactly in the order needed above is roughly of order
$$\exp (-N^{-N}{t\over N})\,<<\,1\quad \hbox{if e.g. }\quad
t\,=\,\exp (L^{{1+\e\over 2}})\quad L>>1$$ we can conclude that the
time the $\{Q_i\}$-dynamics needs to relax to equilibrium should not
be larger than $\exp (L^{{1+\e\over 2}})$ for $L$ large enough.\par
Let us start with the technicalities. Let $R$ be the rectangle
$$R\,=\,\{\;x\in \Z;\,0\,\leq \,x_1\,\leq\,L_1\quad
0\,\leq\,x_2\,\leq\,L_2\;\}$$ with $L_1\,\geq
\,L_2\,\geq\,L_1^{{1\over 2}+\e}$. \bigskip {\bf Proposition 3.1}\par
{\it Let $m\,>\,0$ and $\e\,\in \, (0,{1\over 2})$ be given. Then there exists
$\beta_o\,\equiv\,\beta_o(\e,m )$ independent of $R$
such that for all $\beta \,\geq
\,\beta_o$ and all $x\, \in \,R$ with $x_2\,\leq \,{3\over 4}L_2$, we have:
$$\mu_R^{+,+,+,+}(\s (x)=1)\;-\;\mu_R^{+,+,-,+}(\s (x)=1)\;\leq
\;\exp (-mL_1^{2\e})$$} The above result will actually be given in a
greater generality than that required here, see proposition 4.1. The proof of
proposition 4.1 has been collected with some other similar results for
the Ising model
in appendix 2.\medskip The second result that we need is an estimate on the
probability of not seeing within time $t$ a sequence of updatings of the
$\{Q_i\}$-dynamics with the correct order decribed above.\bigskip
{\bf Lemma 3.1}\par
{\it Let us call $S_N\;\equiv\;\{t_1\,\dots \,t_N\}$, $N\,=\,{2L\over l}\,-\,1$,
an
ordered sequence of updatings if for any $i=1,\dots N$:\medskip
\item{i)} at time $t_i$ the dynamics updates the rectangle $Q_i$.
\item{ii)} there are no updatings between times $t_i$ and
$t_{i+1}$.\medskip Then, for any $N$ large enough (independent of
$t$): $$P(\hbox{ there exists no ordered sequence in}
\;[0,t]\,)\;\leq\;\exp (-{tN^{-N}\over 2})$$ }\bigskip {\bf
Proof}\par Given that $t_1\,\dots \,t_N$ are $N$ consecutive
updatings, the probability that $S_N\;\equiv\;\{t_1\,\dots \,t_N\}$
is an ordered sequence is clearly $N^{-N}$ since the probability of
choosing a specific rectangle is $1\over N$. Let now $\nu_t$
denotes the total number of updatings within time $t$. By
construction the process $\nu_t$ is a Poisson process of parameter
$tN$. Therefore we can estimate the probability appearing in the
Lemma by: $$P(\hbox{ there exists no ordered sequence in}
\;[0,t]\,)\;\leq\;$$ $$\leq \;\sum_{k=0}^{+\infty}{e^{-tN}(tN)^k\over
k!}(1\,-\,N^{-N})^{[{k\over N}]}\;\leq$$
$$\leq\;2e^{-tN(1\,-\,(1\,-\,N^{-N})^{1\over N})}\Eq(3.2)$$ which is
smaller, for $N$ large enough, than $$\exp (-{tN^{-N}\over
2})\Eq(3.3)$$ Let now $S_N\;\equiv\;\{t_1\,\dots \,t_N\}$ be a fixed
ordered sequence with $t_1\,=\,0$, let $\s^{\{Q_i\},+}_{t_i}$ be the
evoluted at time $t_i$ of the initial configuration $\s$, let $R_i$ be the
rectangle
$$R_i\;=\;\{ x\in \cup_{j\leq i}Q_j;\;x_2\leq (i+1){l\over
2}-[{l\over 4}]\}$$ and let, for $ i=1\dots N-1$,
$A_i(x)$, $A_i$, be the events:
$$\eqalign{A_i(x)\;&=\;\{\,(+)^{\{Q_i\},+}_{t_i}(x)\,
\neq\,(-)^{\{Q_i\},+}_{t_i}(x)\,\}\cr A_i\phantom{(x)}\;&=\;\bigcup_{\{ x\in
R_i\}}A_i(x)\cr A_N\phantom{(x)}\;&=\;\bigcup_{\{ x\in
V_L\}}A_N(x)}\Eq(3.4)$$ and let $q_i\,=\,P(A_i)$. Then we have:
$$q_{n+1}\,\leq \,q_n\;+\;P(\,A_{n+1}\cap
A_n^c\,)\,\leq\,\sum_{n=1}^{N-1}P(\,A_{n+1}\cap
A_n^c\,)\;+\;P(A_1)\Eq(3.5)$$ where $A_n^c$ is the complement set of
$A_n$.\par Then the term $P(\,A_{n+1}\cap A_n^c\,)$ in the r.h.s. of
\equ(3.5) can be estimated by: $$P(\,A_{n+1}\cap A_n^c\,)\;\leq$$
$$\sum_{\vbox{\eightpoint{\hbox{$x\in R_{n+1}\cap Q_{n+1}$}
\hbox{$\s\in \O_V$}}}}\mu_V^+(\s ) P(\,A_{n+1}(x)\cap [\cap_{y\in
R_n}\{(+)^{\{Q_i\},+}_{t_n}(y)\,=\,(-)^{\{Q_i\},+}_{t_n}(y)\,=\,\s
^{\{Q_i\},+}_{t_n}(y)\}]\;)\Eq(3.6)$$ In the derivation of \equ(3.6)
we used the fact that at time $t_{n+1}$ we update only the set
$Q_{n+1}$ and that, under the coupling described in $\S 4$ of
section 1, for any time $t$ and any configuration $\s$,
$(-)^{\{Q_i\},+}_t\,\leq\,\s_t^{\{Q_i\},+}\,\leq\,(+)^{\{Q_i\},+}_t$.\par
In turn, if we denote by $E$ the
expectation over the random configuration
$\s^{\{Q_i\},+}_{t_n}$, then a given term in the sum appearing in the r.h.s. of
\equ(3.6) can be estimated from above by: $$\mu_V^+(\s )\, E
[\mu_{Q_{n+1}}^{\s^{\{Q_i\},+}_{t_n},+,(+)^{\{Q_i\},+}_{t_n},+}(\eta
(x)=1)-
\mu_{Q_{n+1}}^{\s^{\{Q_i\},+}_{t_n},+,(-)^{\{Q_i\},+}_{t_n},+}(\eta
(x)=1)]\,=$$ $$=\,[\mu_V^+(\s )\,
E\mu_{Q_{n+1}}^{\s^{\{Q_i\},+}_{t_n},+,+,+}(\eta
(x)=1)\,-\,\mu_V^+(\s
)E\mu_{Q_{n+1}}^{\s^{\{Q_i\},+}_{t_n},+,-,+}(\eta (x)=1)]\Eq(3.7)$$
since $(+)^{\{Q_i\},+}_{t_n}$ and $(-)^{\{Q_i\},+}_{t_n}$ are, respectively,
identically equal to plus one and minus one on
the external boundary of the top of $Q_{n+1}$ because the sequence
$S_N$ is ordered.\par Let us consider the term $$\sum_{\s\in
\O_V}\mu_V^+(\s )\, E\mu_{Q_{n+1}}^{\s^{\{Q_i\},+}_{t_n},+,+,+}(\eta
(x)=1)\Eq(3.8)$$ Since the $\{Q_i\}$-dynamics is reversible with
respect to $\mu_V^+(\s )$, the distribution of
$\s^{\{Q_i\},+}_{t_n}$, given that $\s$ is distributed according to
$\mu_V^+(\s )$, will of course be again $\mu_V^+(\s )$. Therefore
\equ(3.8) will be equal to: $$\sum_{\s\in \O_V}\mu_V^+(\s )\,
\mu_{Q_{n+1}}^{\s,+,+,+}(\eta (x)=1)\,\leq\,\sum_{\s\in
\O_{R_{n+1}\cup Q_{n+1}}} \mu_{R_{n+1}\cup Q_{n+1}}^{+,+,+,+}(\s )\,
\mu_{Q_{n+1}}^{\s,+,+,+}(\eta (x)=1)\Eq(3.9)$$ where we used once
more the monotonicity (1.5).\par By the DLR property of the Gibbs
measure $\mu_{R_{n+1}\cup Q_{n+1}}^{+,+,+,+}$, the r.h.s. of
\equ(3.9) is just $$\mu_{R_{n+1}\cup Q_{n+1}}^{+,+,+,+}(\s (x)\,=\,1)\Eq(3.10)$$
Similarly we obtain that the term $$\sum_{\s\in \O_V}\mu_V^+(\s )\,
E\mu_{Q_{n+1}}^{\s^{\{Q_i\},+}_{t_n},+,-,+}(\eta (x)=1)$$ is bounded
from below by $$\mu_{R_{n+1}\cup Q_{n+1}}^{+,+,-,+}(\s (x)\,=\,1)\Eq(3.11)$$ In
conclusion, using \equ(3.10), \equ(3.11) and Proposition (3.1), we
get that, for any $n$, the r.h.s. of \equ(3.6) is bounded from above
by: $$\mu_{R_{n+1}\cup Q_{n+1}}^{+,+,+,+}(\s (x)\,=\,1)\,-\,
\mu_{R_{n+1}\cup Q_{n+1}}^{+,+,-,+}(\s (x)\,=\,1)\,\leq\, L^2\exp
(-mL^{2\e})\Eq(3.12)$$ for a suitable constant $m\,\equiv\,m(\beta )$ which diverges
as
$\beta\to \infty$ . Similarly one estimates $P(A_1)$.\par Therefore we get:
$$q_N\,\leq \,NL^2\exp (-mL^{2\e})\Eq(3.13)$$ We are now in a position to
conclude the proof of the theorem.\par Given a sequence $S_N\;\equiv\;\{t_1\,\dots
\,t_N\}$ of updatings we say that $S_N$ is a {\it good } sequence iff $S_N$ is
ordered and the event $A_N^c$ occured at the end of the sequence. Because of
\equ(3.13) we know that the probability that an ordered sequence is also a good
sequence is larger than $$1\,-\,NL^2\exp (-mL^{2\e})\,>\,{1\over 2}$$ for
$L$ large enough. Thus, using Lemma 3.1, we get that if $T\,=\,\exp (
L^{{1+\e\over 2} })$ and $L$ is large enough: $$P(\hbox{ there exists
a good sequence
in} \;[0,T]\,)\;\geq\;{1\over 3}\Eq(3.14)$$ We conclude by observing
that, if there exists a good sequence in $[0,t]$, then, by
monotonicity (see $\S 4$ section 1), the evoluted at the end of the
sequence of $(+)$ and of $(-)$ will be identical. Therefore we can
estimate $P((+)^{\{Q_i\},+}_t\,\neq\,(-)^{\{Q_i\},+}_t)$ by
$$P((+)^{\{Q_i\},+}_t\,\neq\,(-)^{\{Q_i\},+}_t)\,\leq\,({2\over
3})^{[{t\over T}]}\Eq(3.15)$$ which immediately implies that $$
\hbox{gap}_{V_L}(\{Q_i\},\,+)\,\geq\,T^{-1}log({3\over 2})\,= \,
\exp (-L^{{1+\e\over 2} })\log ({3\over 2})\Eq(3.16)$$ Clearly
\equ(3.16) together with \equ(3.1) prove the theorem.\bigskip
{\bf Proof of Theorem 3.2}\par
The first inequality, namely
$$0\,\leq\,\int d\mu^+(\s )\s (0)E(\s_t(0))\,-\,(m^*(\b ))^2\Eq(3.17)$$
follows immediately from the FKG inequality applied to $\mu^+$ and the fact that
the infinite volume Heat Bath dynamics is reversible with respect to $\mu^+$.\par
In order to obtain the upper bound we write the r.h.s. as:
$$\int d\mu^+(\s )(\s (0)+1)E(\s_t(0))\,-\,(m^*(\b ))^2\,-\,m^*(\b )\Eq(3.17bis)$$
and we observe that, by the monotonicity (1.24), (1.25), for any $L$ and any
$t$:
$$E(\s_t(0))\,\leq\, E_{V_L,+}^{+}(\s_t(0))\Eq(3.18)$$
where $E_{V_L,+}^{+}$ denotes the expectation over the HB-dynamics in $V_L$ with
plus boundary conditions starting from the configuration identically equal to
plus
one.\par
In turn, the r.h.s. of \equ(3.18) can be bounded above, using the estimate (1.19),
by:
$$E_{V_L,+}^{+}(\s_t(0))\,\leq\,\mu_{V_L}^+(\s (0))\;+\;\exp (
C\b L^2\,-\,t\hbox{gap}(HB,V_L,+))\Eq(3.19)$$
If we plug \equ(3.18) into \equ(3.17bis) and we use \equ(3.19), we obtain that
the
r.h.s. of \equ(3.17) is bounded above by:
$$(\mu_{V_L}^+(\s (0))\,-\,m^*(\b ))(m^*(\b )+1)\;+\;2\exp (
C\b L^2\,-\,t\hbox{gap}(HB,V_L,+))\Eq(3.20)$$
As it is well known
$$0\,\leq \,\mu_{V_L}^+(\s (0))\,-\,m^*(\b )\,\leq\,C_1\exp (-mL)\Eq(3.21)$$
for any large enough $\b$ where $C_1$ and $m$ are suitable constants with $m\to
\infty$ as $\b\to \infty$.\par
We now choose the size $L$ depending on $t$ as:
$$L\;=\;[{\log (t)\over 2C(\a )\b}]^\a\Eq(3.22)$$
where $C(\a )$ is the constant appearing in theorem 3.1 for the value
$\e\,=\,{2-\a\over 2\a }$ and we apply theorem 3.1 to get that the r.h.s. of
\equ(3.20) is bounded from above by:
$$C_1\exp (-m[{\log (t)\over 2C(\a )\b}]^\a )\;+\;C_2\exp (-{\sqrt (t)\over
2})\Eq(3.23)$$
for all $\b$ large enough, where $C_2$ is a suitable constant.\par
Clearly \equ(3.23) proves the theorem.\pagina
%\input formato.tex
\numsec=4 \numfor=1
{\bf Section 4}\par
\centerline{\bf Asymptotics of the Gap With Open Boundary
Conditions}\bigskip In this section we again consider the
HB-dynamics in a square $V\,\equiv\,V_L$ of side $L$ at very low
temperature, but this time with open boundary conditions, that is
$$U^{\partial V_L}(x,y)\,=\,0\quad \forall \;(x,y)\,\in \, \partial
V_L$$ In this case the two extremal configurations, $(+)$ and $(-)$,
are the only absolute minima of the energy $H_V^\emptyset(\s_V)$ and
they are related one to the other by a global spin flip.\par We show
that, due precisely to the above symmetry, the gap of the
HB-dynamics is much smaller, as $L\to \infty$, than its value with
plus boundary conditions. More precisely we obtain that the gap is
of the order of $\exp (-\beta \t_\b L)$, where $\t_\b$ is
the surface tension defined in (1.11) with respect to an interface
parallel to one of the coordinate axes.\par Since the proof of the
main result of the present section (see theorem 4.1 below) will
mimick as close as possible the proof of theorem 3.1, we will keep
the same notation of section 3 with the following modification.\par
Let $R$ be a rectangle and let us suppose that
we have a boundary coupling $U^{\partial R}$ which is constant on
each of the four components of $\partial R$ ordered clockwise
starting from the bottom. Let us denote by
$0\leq \delta_i\leq 1,\quad i=1\dots 4$, the value of the boundary
coupling on the $i\hbox{-th}$ side of $R$. Then we will write
$\mu_R^{\delta_1\t_1,\delta_2\t_2,\delta_3\t_3,\delta_4\t_4}$, to
denote the corresponding Gibbs measure on $R$ with the boundary
conditions $\t_1,\t_2,\t_3,\t_4$. As usual, if one the $\delta_i$'s
is equal to one it will be omitted in the notation, while if it
is zero the corresponding term $\delta_i\t_i$ will be replaced by
$\emptyset$. Thus for example ($\t_1,\delta +,\emptyset ,\delta +$)
means $\t_1$ boundary conditions on the bottom side, plus boundary
conditions on the vertical ones coupled to the interior of $R$ by a
constant boundary coupling equal to $\delta$ and open boundary
condition on the top one.\par As in section 3, whenever confusion
does not arise, we will omit the subfix $V$ in the notation
$\s_V$.\par Let us now state the main result : \bigskip {\bf Theorem
4.1}\par {\it Let $\e\,\in \, (0,1/4)$ be given. Then there exit
$\beta_o\,<\,+\infty$ and $C\,<\,+\infty$ such that for any
$\beta\,\geq\,\beta_o$ and any integer $L$: $$\exp (-\beta
\t_\b L\,-\,C\beta L^{{1\over 2}+\e})\,\leq\,\hbox{\rm
gap}_{V_L}(HB,\,\emptyset)\,\leq\, \exp (-\beta
\t_\b L\,+\,C\beta L^{{1\over 2}+\e})$$} {\bf Proof}\medskip {\bf Upper Bound}.
\par\noindent The idea behind the upper bound is very simple
and intuitive: when the
system starts from a typical configuration of the Gibbs measure
$\mu_V^\emptyset$ it has a magnetization $m$ approximately
equal to either $+m^*(\beta )$ or $-m^*(\beta )$, where $m^*(\beta )$ is the
value of the spontaneous magnetization at
inverse temperature $\beta$ in the infinite volume limit. Therefore, in order
to reach
the equilibrium where the expected value of the magnetization is zero by symmetry,
the
process has to hit the set of configurations of zero magnetization. Since the
probability starting at equilibrium to have at a given time $t$ zero
magnetization is
equal to $\mu_V^\emptyset(m=0)$, one expects the relaxation time to
equilibrium, which is roughly the inverse of the gap, to be at least as large as
the inverse of $\mu_V^\emptyset(m=0)$. That is actually correct and
the argument, thanks to a basic result of Shlosman (see theorem 4.2 below), gives
a
correct upper bound.\par Let us implement the above idea. Without a true
loss of
generality we may assume that $L^2$ is odd. We also denote by $m(\s )$ the total
magnetization of the configuration $\s\,\in \,\O_V$: $$m(\s )\;=\;
\sum_{x\in V}\s
(x)$$ and by $<\,;\,>$ the scalar product in $L^2(\O_V,d\mu_V^\emptyset )$.\par
If we
recall that the generator of the dynamics, $L^\emptyset$, is selfadjoint on
$L^2(\O_V,d\mu_V^\emptyset )$, we get that: $$\;\leq$$
$$\leq \;\exp(-\hbox{gap}_{V_L}(HB,\,\emptyset)t)\,\leq\,
L^4\exp(-\hbox{gap}_{V_L}(HB,\,\emptyset)t)\Eq(4.1)$$ since, by simmetry,
$\,=\,0$. \par On the other hand, again by symmetry $$(\exp
(tL^{\emptyset})m)(\s )\,=\,-(\exp (tL^{\emptyset})m)(-\s )\Eq(4.2)$$ so that:
$$\;=\;2\int_{\s ;\,m(\s )\geq 0}d\mu_V^\emptyset
(\s
)m(\s )(\exp (tL^{\emptyset})m)(\s )\Eq(4.3)$$ If we denote by $T^{(m<0)}(\s
)$ the
first hitting time of the set $\{\,m(\s )\,<\,0\,\}$ for the HB-dynamics in $V$
starting at time $t=0$ from the configuration $\s$, we get that, for configurations
$\s$ with positive magnetization, $(\exp (tL^{\emptyset})m)(\s )$ can bounded
from
below by $$(\exp (tL^{\emptyset})m)(\s )\,\geq \,P(T^{(m<0)}(\s
)\,>\,t)\;-\;L^2P(T^{(m<0)}(\s )\,\leq\,t)\,=$$ $$=\,
1\,-\,(L^2+1)P(T^{(m<0)}(\s
)\,\leq\,t)\Eq(4.4)$$ since $\inf_{\s ;m(\s )\geq 0}m(\s )\,=\,1$ in view of our
condition that $L^2$ is odd .\par A rather standard computation in the theory of
Glauber dynamics that uses the invariance of the
measure $\mu_V^\emptyset (\s )$ and the fact that
$$P(\nu_t\,\geq \,2L^2t)\;\leq\; \exp (-KL^2t)$$ for a suitable constant
$K$, where
$\nu_t$ is the number of updatings within time $t$, shows that
$$\sum_{\s}\mu_V^\emptyset (\s )P(T^{(m<0)}(\s )\,\leq\,t)\;\leq \;
2L^2t\mu_V^\emptyset (m(\s )\,=\,1)\;+\;\exp (-KL^2t)\Eq(4.5)$$ If we insert
\equ(4.4) and \equ(4.5) in \equ(4.3), we get that: $$\;\geq$$ $$\geq\; 2\mu_V^\emptyset (m(\s )\geq
0)\;-\;4(L^2+1)L^2t\mu_V^\emptyset (m(\s )\,=\,1)\,-\,2(L^2+1)\exp
(-KL^2t)\Eq(4.6)$$ By simmetry $\mu_V^\emptyset (m(\s )\geq 0)\,=\,{1\over
2}$, so
that, for all $L$ large enough and all $$1\,\leq \,t\,\leq \,
[16(L^2+1)L^2t\mu_V^\emptyset (m(\s )\,=\,1)]^{-1}\Eq(4.6bis)$$ the r.h.s of
\equ(4.6) is greater than ${1\over 4}$.\par If we combine this result with
\equ(4.1)
we obtain: $${1\over 4}\;\leq\;L^4\exp(-\hbox{gap}_{V_L}(HB,\,\emptyset)t)\quad
\forall\; 1\,\leq \,t\,\leq \, [16(L^2+1)L^2t\mu_V^\emptyset (m(\s
)\,=\,1)]^{-1}\Eq(4.6tris)$$ We use at this point a fundamental result due to
Shlosman (see Theorem 3 in [Sh]) in his study of the Wulff shape in a finite
square
with periodic or open boundary conditions:\bigskip {\bf Theorem 4.2 } (Shlosman)\par
{\it There exists $\beta_o$ such that for any $\beta\,\geq \,\beta_o$ and any
sequence of integers $\rho_L$, $L\,\in \,{\bf N}$, satisfying $$\lim_{L\to
\infty}{\rho_L\over L^2}\;=\;\rho\;\in \;(0,m^*(\beta ))\qquad
\rho_L\,-\,L^2\;=\;\hbox{ mod } \,2$$ the limit $$\psi (\rho )\;=\; \lim_{L\to
\infty}-{1\over \beta L}\log (\mu_V^\emptyset (m(\s )\,=\,\rho_L))$$
exists and it is
given by: $$\eqalign{\psi (\rho )\;&=\;{1\over 2}w\sqrt{{m^*(\beta )\,-\,\vert
\rho\vert\over 2m^*(\b )} }\quad \vert \rho\vert\,\geq\,\rho_1,\cr \psi (\rho
)\;&=\;{1\over 2}w\sqrt{m^*(\beta )\,- \,\rho_1}\quad \vert
\rho\vert\,\leq\,\rho_1}$$ where the constant $w$ is the value of the Wulff
functional $W_\t$ on the Wulff curve $\partial W$ (see theorem 1.1) and the
singularity point $\rho_1$ satisfies the equation $${1\over 2}w
\sqrt{{m^*(\beta )\,-\,\vert
\rho_1\vert\over 2m^*(\b )}}\;=\;\t_\beta$$} {\bf Warning} Due to
some misprints, the
formula for $\psi (\rho )$ in [Sh] appears with $1\over 2$ and
$\sqrt{{m^*(\beta )\,-\,\vert
\rho\vert\over 2m^*(\b )} }$ replaced by $1\over
4$ and by $\sqrt{m^*(\beta )\,-\,\vert
\rho\vert}$ respectively.\bigskip {\bf Remark} Given $\e\,\in \,(0,{1\over
4})$ and
$\beta$ large enough, it is possible to show, using the methods of [DKS],
that the
above limit is approached, as $L\to \infty$, at least as fast as $L^{-{1\over
2}+\e}$.\bigskip By plugging in \equ(4.6tris) the result of theorem 4.2 and its
strengthening mentioned in the remark above , we immediately obtain the required
upper bound on the gap.
\bigskip
{\bf Remark} Actually the above reasoning leads to an upper bound on the gap
which is a negative exponential of the surface in {\it any} dimension $d\geq 2$ if
we use the estimate of Schonmann [Sch2]:
$$\mu_V^\emptyset (m(\s )\,=\,0)\,\leq \,\exp (-c(\beta )L^{d-1})$$
for a suitable constant $c$. Moreover it is possible to show (see [CCSch]) that
in two dimensions the above estimate is valid for {\it any} $\b$ larger than the
critical value $\b_c$. Therefore, using corollary 2.1 and the above observation,
we get that in $d=2$ for any $\b\,>\,\b_c$ there exist two constants $c_1$ and $c_2$
such that for any $L$ large enough:
$$\exp (-c_1L)\,\leq\,\hbox{gap}_{V_L}(HB,\emptyset )\,\leq\,\exp (-c_2L)$$
It would be nice to show that at least one of the two constants is equal to $\b\t_\b$.\par
We finally notice that it was possible to follow a slightly different proof by
using in the variational characterization of the gap the trial function
$$f(\s )\,=\,\chi (m(\s )>0)\,-\,\chi (m(\s )<0)$$ $\chi (A)$ being the characteristic
function of the event $A$, and then exploiting Shlosman's result.\bigskip \medskip
{\bf Lower Bound} \par\noindent We start by replacing
the open boundary conditions on $\partial V$ by very weak {\it plus} boundary
conditions. More precisely, let $$\delta\;=\;L^{-{1\over 2}}$$ and let
us consider a
constant boundary coupling $U^{\partial V}(x,y)$: $$U^{\partial
V}(x,y)\,=\,\delta\qquad \forall \;(x,y)\in \partial V$$ Then, in the notation
established in section 1 $\S 3$ and at the beginning of the present section, we
trivially have for any $a\in \{-1,+1\}$: $$\eqalign{\exp (-8\beta \d L)\mu_V^{\delta
+,\delta +,\delta +,\delta +} (\s )\,&\leq\, \mu_V^\emptyset (\s
)\;\leq\;\exp (8\beta \d L)\mu_V^{\delta +,\delta +,\delta +,\delta
+}(\s )\cr \exp (-8\beta\d L)\mu_{\{x\}}^{U^{\partial x},(\s ,\delta
+)} (a)\,&\leq\, \mu_{\{x\}}^{(\s ,\emptyset)} (a)\;\leq\;\exp
(8\beta \d L)\mu_{\{x\}}^{U^{\partial x},(\s , \delta +)}
(a)\,}\Eq(4.7)$$ where $\mu_{\{x\}}^{(\s , \emptyset)}$ is the
conditional probability of having the value $a$ for $\s (x)$ given
that outside $V$ there are open boundary conditions and that the
configuration in $V\setminus \{x\}$ is $\s$. Similarly for
$\mu_{\{x\}}^{U^{\partial x},(\s , \delta +)} (a)$.\par It is
immediate to check, using the variational characterization of the
gap in term of the Dirichlet form (1.17), that \equ(4.7)
implies the following bound on $\hbox{gap}_{V_L}(HB,\,\emptyset)$ in
terms of $\hbox{gap}_{V_L}(HB,\,+,\, \delta)$:
$$\hbox{gap}_{V_L}(HB,\,\emptyset)\;\geq\; \exp (-32\beta \d
L)\hbox{gap}_{V_L}(HB,\,+,\, \delta)\Eq(4.8)$$ It is therefore
sufficient to establish the correct lower bound with "$\delta$+"
boundary conditions.\par To this purpose we proceed exactly as in
section 3, namely we consider the $\{Q_i\}$-dynamics with $Q_i$ as
in the proof of theorem 3.1 and estimate $\hbox{gap}_{V_L}(HB,\,+,\,
\delta)$ by: $$\hbox{gap}_{V_L}(HB,\,+,\, \delta)\;\geq\; {1\over
\vert Q_i\vert}{\exp (-4\beta)\over \exp (-4\beta)+\exp (+4\beta)}
\,\exp (-4\beta
(l\,+\,1))\,\hbox{gap}_{V_L}(\{Q_i\},\,+,\,\delta)\Eq(4.9))$$ where
$l\;=\;2[L^{{1\over 2}+\e}]$.\par The main difference now with the
reasoning behind the proof of theorem 3.1 is the
following.\par\noindent When we start from the two extremal
configurations $(+)$ and $(-)$ at the beginning of an ordered
sequence $S_N$ and we update the first rectangle $Q_1$, we replace
$(+)_{Q_1}$ and $(-)_{Q_1}$ with two configurations $\eta_{Q_1}^+$
and $\eta_{Q_1}^-$ distributed according to $\mu_{Q_1}^{\delta
+,\delta +,+,\delta +}$ and $\mu_{Q_1}^{\delta +,\delta +,-,\delta
+}$ respectively. Contrary to the "full" (i.e. $\delta \,=\,1$) plus
boundary conditions discussed in theorem 3.1, the measure
$\mu_{Q_1}^{\delta +,\delta +,-,\delta +}$ is {\it not} concentrated
for $\beta$ large on configurations which resemble those of the plus
phase, at least far from the top side, but instead, due precisely to
the "full" minus boundary condition on the top side, on
configurations in which the spins are mostly minus one with little
islands of plus spins.
Therefore, with large probability, the first updating of the ordered
sequence will {\it not} force $(+)$ and $(-)$ to agree in a large
portion (e.g. $3\over 4$ ) of $Q_1$.\par We notice, however, that
with very small probability the two new configurations
$(+)^{\eta_{Q_1}^+}$ and $(-)^{\eta_{Q_1}^-}$ {\it will} agree in,
say, $3\over 4$ of $Q_1$, if for example the interface
in the configuration $\eta_{Q_1}^-$
separating the minus spins on the top side of $Q_1$ from the
plus spins in the rest of the boundary
instead of being in its typical position, namely close to the
bottom side of $Q_1$, is very close to the top one.
It turns out that the probability in
question is at least of the order of $\exp (-\beta \t_\beta L )$ .
Once this rare event has occurred then, in the second updating, we
will have to consider the Gibbs measures
$\mu_{Q_1}^{\eta_{Q_1}^+,\delta +,+,\delta +}$ and
$\mu_{Q_1}^{\eta_{Q_1}^+,\delta +,-,\delta +}$ which, if we
approximate, as we did in the introduction to the proof of theorem
3.1, the boundary condition $\eta_{Q_1}^+$ with a "full" plus,
become $\mu_{Q_1}^{+,\delta +,+,\delta +}$ and $\mu_{Q_1}^{+,\delta
+,-,\delta +}$.\par Now the situation is very different from the
first updating and much more similar to the case treated in the
proof of theorem 3.1. In fact, in the Gibbs measure
$\mu_{Q_1}^{+,\delta +,-,\delta +}$, the "full" plus boundary
condition on the bottom side compensate exactly the "full" minus
boundary condition on the top one and therefore the "phase" (that is
the structure of the typical configurations) is decided by the
lateral "$\delta$+" boundary conditions. Since the typical
fluctuations of the interface separating the minus spin of the top
from the plus spins at the bottom are of order $\sqrt L\,<<\,l$, and
since $\d l\,= \,L^\e\,>>\,1$, one can conclude (see Proposition 4.1
below) that the above two Gibbs measures are very similar in, say,
$3\over 4$ of $Q_1$. Thus the second updating will, with large
probability, enlarge the region of agreement between the evoluted
of $(+)$ and $(-)$ to $$\{\,x\in V\quad 0\leq x_2\leq {5\over
4}l\,\}$$ Iterating this procedure, we see that an ordered sequence
$S_N\,=\,\{t_1,\dots \t_N\}$ will typically glue together $(+)$ and
$(-)$ with the last updating at time $t_N$, provided that in the
first one, at time $t_1$, a very rare event of probability of order
$\exp (-\beta \t_\beta L )$ has occured.\par Clearly the above
reasoning implies that the relaxation time to equilibrium for the
$\{Q_i\}$-dynamics should be at most of order $\exp (+\beta \t_\beta
L )$ and therefore, using \equ(4.9), the required lower bound would
follow.\par Let us implement the above program. We start by giving a
generalization to the case of $\delta +$ lateral boundary conditions
of proposition 3.1. As in section 3, let
$R$ be a rectangle $$R\,=\,\{\;x\in \Z;\,0\,\leq
\,x_1\,\leq\,L_1\quad 0\,\leq\,x_2\,\leq\,L_2\;\}$$ with $L_1\,\geq
\,L_2\,\geq\,L_1^{{1\over 2}+\e}$. Then we have: \bigskip {\bf
Proposition 4.1}\par {\it Let $m\,>\,0$ and $\e\,\in \, (0,{1\over
2})$ be given and let $\d\,=\,L_1^{-{1\over 2}}$. Then there exists
$\beta_o\,\equiv\,\beta_o(\e,m )$ such that for all $\beta \,\geq
\,\beta_o$ and all $x\,=\,(x_1,x_2)\, \in \,R$ with $x_2\,\leq
\,{3\over 4}L_2$ we have: $$\mu_R^{+,\delta +,+,\delta +}(\s
(x)=1)\;-\;\mu_R^{+,\delta +,-,\delta +}(\s (x)=1)\;\leq \;\exp
(-m L_1^{\e})$$ Moreover, if $R$ and $R'$ are two
rectangles as above with the same basis $L_1$ but different heights
$L_1\,\geq \,L_2\,\geq\,L_2'\,\geq \,L_1^{{1\over 2}+\e}$, then for
all $x\,=\,(x_1,x_2)\, \in \,R$ with, for example, $x_2\,\leq \,{1\over 16}L_2'$,
we have: $$\mu_{R'}^{\delta +,\delta +,+,\delta +}(\s
(x)=1)\;-\;\mu_R^{\delta +,\delta +,+,\delta +}(\s (x)=1)\;\leq
\;\exp (-m L_1^{{1\over 2}+\e})$$} For a proof see appendix
1.\par
There is an interesting corollary to the above proposition that can be viewed
as a
generalization of theorem 3.1 to the case when we have open boundary
conditions on
three sides of the square $V_L$ and full $+$ boundary conditions on the remaining
one.\bigskip
{\bf Corollary 4.1}\par
{\it Let $\e\,\in \, (0,{1\over 2})$ be given.
Then there exit $\beta_o\,<\,+\infty$ and $C\,<\,+\infty$ such that
for any $\beta\,\geq\,\beta_o$ and any integer $L$:
$$\hbox{gap}_{V_L}(HB,\,\emptyset ,\emptyset ,+,\emptyset )\;\geq\;\exp (-C\beta
L^{{1\over 2}+\e})$$} {\bf Proof}\par
We use \equ(4.8) to replace the open boundary conditions on the three sides by
$\d
+$ boundary conditions. Then we can repeat word by word the proof of theorem 3.1,
with proposition 3.1 replaced by proposition 4.1 .\bigskip
The second new result that we need is as follows.\par For
a given rectangle $R$ as above and $\s\,\in \, \O_R$, let
$\G^{+,+,-,+} (\s)$ be the family of contours of $\s$ with boundary
condition $\t$ having the constant sign $+,+,-,+$ on the external
boundary of the four sides of $R$ ordered in the usual way. As one
can immediately check, under the above boundary conditions there
exists only one open contour that will be denoted by
$\Gamma_{R,open}^{+,+,-,+}(\s )$.\par We then define the event
${\cal A}_R^{+,+,-,+}$ as: $$ {\cal A}_R^{+,+,-,+}\;=\;\{\s ;\quad
\Gamma_{R,open}^{+,+,-,+}(\s )\,\subset\, \{\;x\in
R;\;x_2\,>\,{13L_2\over 16}\}\;\} \Eq(4.10)$$ {\bf Proposition 4.2}
\par {\it In the hypotheses of proposition 4.1 there exists a
positive constant $C$ independent of $\beta$ and $L_1$ such that:
$$\mu_{R}^{\delta +,\delta +,-,\delta +}({\cal
A}_R^{+,+,-,+})\;\geq\;\exp (-\beta \t_\beta L_1\,-\,C\beta
L_1^{\e} )$$} For a proof see appendix 1.\medskip We
are now in a position to complete the proof of the lower bound.\par
As a first step and for reasons that will appear clear later in the
proof, it is convenient to modify slightly the coupling for the
$Q_i$-dynamics. More precisely we use the same algorithm described
(1.24), (1.25), but with a modified coupled measure $\tilde
\nu_{Q_1}$ for the first rectangle $Q_1$. The measure $\tilde
\nu_{Q_1}$, that will be obtained from the old one $ \nu_{Q_1}$ via
"surgery" (see for instance [DSh]) on a suitable subset of $Q_1$, will
however still enjoy the monotonicity property described at the end
section 1.\par Let $\tilde R_1$ be the rectangle: $$\tilde R_1\,=\,
\{\,x\in Q_1;\quad x_2\,\leq \,l-{3l\over 16}\,\}$$ and let
$\nu_{Q_1}^{\t^{(1)}\dots \t^{(N)}}$ be the measure on
$(\O_{Q_1})^N$, $N\,=\,2^L$,
constructed in $\S 4$ of section 1, with boundary conditions
$\delta +$ on the bottom and lateral sides of $Q_1$ and
$\t^{(1)}\dots \t^{(N)}$ on the top side of $Q_1$.\par We then
construct the new measure $\tilde \nu_{Q_1}$ on $\O_{Q_1}^{N}$ as
follows.\par Given $N$ configurations $\s^{(1)}\dots \s^{(N)}$ in
$\O_{\cal L}$, where $\cal L$ denotes the external boundary of the
top side of $\tilde R_1$, let $\nu_{Q_1\setminus {\cal
L}}^{\s^{(1)}\dots \s^{(N)};\t^{(1)}\dots \t^{(N)}}$ be the measure
constructed according to (1.20), (1.21) for the set $Q_1\setminus
{\cal L}$ and boundary conditions:\medskip \item{} $\delta +$ on
the bottom and lateral sides of $Q_1$ \item{} $\s^{(1)}\dots
\s^{(N)}$ on $\cal L$ \item{} $\t^{(1)}\dots \t^{(N)}$ on the top
side of $Q_1$, where $\t^{(1)}\dots \t^{(N)}$ are {\it all}
possible configurations on the external boundary of the top side of
$Q_1$.\medskip\noindent It is very important to notice that
$\nu_{Q_1\setminus {\cal L}}^{\s^{(1)}\dots \s^{(N)};\t^{(1)}\dots
\t^{(N)}}$ is a product of the measures $$\nu_{\tilde
R_1}^{\s^{(1)}\dots \s^{(N)}}\; \hbox{ and }\; \nu_{Q_1\setminus
\{\tilde R_1\cup {\cal L}\}}^{\s^{(1)}\dots \s^{(N)};\t^{(1)}\dots
\t^{(N)}}$$ where, for notation convenience, we have omitted to
indicate the fixed $\delta +$ boundary conditions on the bottom and
lateral sides of $Q_1$.\par Finally, given $N$ configurations
$\tilde \s^{(1)}\dots \tilde \s^{(N)}$ in $\O_{Q_1}$, we set:
$$\tilde \nu_{Q_1}^{\t^{(1)}\dots \t^{(N)}}(\tilde \s^{(1)}\dots \tilde
\s^{(N)})\,=\,\sum_{\s^{(1)}\dots \s^{(N)}}\nu_{Q_1}^{\t^{(1)}\dots
\t^{(N)}}(\s^{(1)}\dots
\s^{(N)})\,T(\s^{(1)}\dots \s^{(N)};\;\tilde \s^{(1)}\dots \tilde
\s^{(N)})\Eq(4.10bis)$$ where $$\eqalign{T(\s^{(1)}\dots
\s^{(N)};\;\tilde \s^{(1)}\dots \tilde
\s^{(N)})\;&=\;\nu_{Q_1\setminus {\cal L}}^{\s^{(1)}\dots
\s^{(N)};\t^{(1)}\dots \t^{(N)}}( \tilde \s^{(1)}_{Q_1\setminus {\cal
L}}\dots \tilde \s^{(N)}_{Q_1\setminus {\cal L}}) \quad \hbox{if }
\tilde \s^{(i)}_{\cal L}\,=\,\s^{(i)}_{\cal L}\cr T(\s^{(1)}\dots
\s^{(N)};\;\tilde \s^{(1)}\dots \tilde \s^{(N)})\;&=\;0\quad
\hbox{otherwise}}\Eq(4.10tris)$$ It is easy to see, using the DLR
equations, that if the event $A$ depends only on the $k^{th}$
configuration, $\tilde \s^{(k)}$, then $$\tilde
\nu_{Q_1}^{\t^{(1)}\dots \t^{(N)}}(A)\,=\,\mu_{Q_1}^{\delta
+,\delta +,\t^{(k)},\delta
+}(A)\Eq(4.10quatris)$$ and moreover that, if the event $A\subset
(\O_{Q_1})^N$ depends {\it only} on the values of the spins in
$Q_1\setminus {\cal L}$, then: $$\tilde
\nu_{Q_1}^{\t^{(1)}\dots \t^{(N)}}(A)\,=\,\sum_{\s^{(1)}\dots
\s^{(N)}}\nu_{Q_1}(\s^{(1)}\dots
\s^{(N)})\nu_{Q_1\setminus {\cal L}}^{\s^{(1)}\dots
\s^{(N)};\t^{(1)}\dots \t^{(N)}}( A)\Eq(4.10five)$$ Finally, it is
immediate to check, using the monotonicity (1.23) of the measures
$\nu_{Q_1}^{\t^{(1)}\dots \t^{(N)}}$ and $\nu_{Q_1\setminus
{\cal L}}^{\s^{(1)}\dots
\s^{(N)};\t^{(1)}\dots \t^{(N)}}$, that (1.23) holds true also for
$\tilde \nu_{Q_1}^{\t^{(1)}\dots \t^{(N)}}$. This fact implies, in particular,
that if we
use the coupling (1.24), (1.25) with the measures $\tilde
\nu_{Q_1}$, $ \nu_{Q_2}$, ... $ \nu_{Q_n}$, then, under this new
coupling, any ordered set of initial configurations will stay
ordered at any future time. \medskip Let now
$S_N\;\equiv\;\{t_1\,\dots \,t_N\}$ be a fixed ordered sequence with
$t_1\,=\,0$, let $A_i(x)$, $A_i$, be the events defined in (3.4) and
let $q_i\,=\,P(A_i\vert {\cal B})$ where $${\cal
B}\,=\,\{((-)^{\{Q_i\},\delta +}_{t_1} )_{Q_1}\;\in \;{\cal
A}_{Q_1}^{+,+,-,+}\,\}\Eq(4.11)$$ As in section 3 we have:
$$q_{n+1}\,\leq \,q_n\;+\;P(\,A_{n+1}\cap A_n^c\,\vert {\cal B})
\Eq(4.12)$$ Let us estimate the second term in the r.h.s. of
\equ(4.12). As in section 3 we let $$R_n\;=\;\{ x\in \cup_{j\leq
n}Q_j;\;x_2\leq (n+1){l\over 2}-[{l\over 4}]\}$$ and
$$D\,=\,\cup_{j\geq 2}Q_j$$ We observe that, since the sequence
$S_N$ is ordered, if $A_n^c$ has occurred for some $n\geq 1$ then
necessarily $(-)^{\{Q_i\},\delta +}_{t_1}$ and $(+)^{\{Q_i\},\delta
+}_{t_1}$ are both equal on the external boundary of the bottom side
of $Q_2$ to a common configuration that we call $\t$. Again because
of the ordering of the sequence $S_N$, the next updatings at time
$t_i$, $i\,\geq\, 2$, will not modify $\t$ on the external boundary
of the bottom side of $Q_2$ and therefore they will be reversible
with respect to the Gibbs measure $\mu_D^{\t,\d +,\d +,\d +}$ on
$\O_D$, .\par Thus, following section 3 (see (3.6) ... (3.9)), we
can bound $P(\,A_{n+1}\cap A_n^c\,\vert {\cal B})$ by:
$$\sum_\t\tilde \nu_{Q_1}^{\t^{(1)}\dots \t^{(N)}}((-)^{\{Q_i\},\delta +}_{t_1}=
(+)^{\{Q_i\},\delta
+}_{t_1}=\t\vert{\cal
A}_{Q_1}^{+,+,-,+} )\, F_1(\t
)\,\leq\,$$ $$\sum_\t\mu_{Q_1}^{\d +,\d +,-,\d +}(\t\vert{\cal
A}_{Q_1}^{+,+,-,+} )\, F_1(\t
)$$ where $$F_1(\t )\,\equiv\, \sum_{x\in
R_{n+1}\cap Q_{n+1}\atop\s\in \O_D}\mu_D^{\t,\d +,\d +,\d +}(\s )\,[
\mu_{Q_{n+1}}^{\s,\d +,+,\d +}(\eta (x)=1)\,-\, \mu_{Q_{n+1}}^{\s,\d
+,-,\d +}(\eta (x)=1)]\Eq(4.15)$$ As in (3.9)$\dots$(3.11) we get,
by monotonicity and the DLR equations, that $F_1(\t )$ is bounded
from above by: $$F_2(\t )\,=\, \sum_{\vbox{\eightpoint{\hbox{$x\in
R_{n+1}\cap Q_{n+1}$}}}}[ \mu_{R_{n+1}\cup Q_{n+1}\setminus
Q_1}^{\t,\d +, +,\d +}(\s )(\eta (x)=1)\,-\,\mu_{R_{n+1}\cup
Q_{n+1}\setminus Q_1}^{\t,\d +, -,\d +}(\s )(\eta (x)=1) ]\Eq(4.17)$$
In conclusion we have shown that: $$P(\,A_{n+1}\cap A_n^c\,\vert
{\cal B})\,\leq\,\sum_\t\mu_{Q_1}^{\d +,\d +,-,\d +} (\t\vert {\cal
A}_{Q_1}^{+,+,-,+} )\, F_2(\t )\Eq(4.17bis)$$ In order to prove
that \equ(4.17bis)
is very small, we need a last result on the Ising model which shows
that, conditional to the event ${\cal A}_{Q_1}^{+,+,-,+}$, the
projection (or relativization) of the measure $\mu_{Q_1}^{\d +,\d
+,-,\d +}$ on the external boundary of the bottom side of $Q_2$ is,
in some sense, very close to the same projection both of the measure
$\mu_{R_{n+1}\cup Q_{n+1}}^{\d +,\d +, +,\d +}$ and of the measure
$\mu_{R_{n+1}\cup Q_{n+1}}^{ +,\d +, -,\d +}$. More precisely:
\bigskip {\bf Proposition 4.3 } \par {\it Let $m\,>\,0$ and
$\e\,\in \, (0,{1\over 2})$ be given. Then there exists
$\beta_o\,\equiv\,\beta_o(\e ,m )$ such that for all $\beta \,\geq
\,\beta_o$ we have:\bigskip\noindent {\bf a)} $$\vert
\sum_\t\mu_{Q_1}^{\d +,\d +,-,\d +}(\t\vert {{\cal
A}_{Q_1}^{+,+,-,+}} )F_2(\t ) \,-\,
\sum_\t \mu_{R_{n+1}\cup Q_{n+1}}^{\d +,\d +, +,\d +}(\t )F_2(\t
)\vert \;\leq $$ $$\leq \;\exp (-m L^{{1\over 2}+\e})$$
\noindent {\bf b)} $$\vert \sum_\t\mu_{Q_1}^{\d +,\d +,-,\d
+}(\t\vert {{\cal
A}_{Q_1}^{+,+,-,+}} )F_2(\t ) \,-\, \sum_\t \mu_{R_{n+1}\cup
Q_{n+1}}^{+,\d +, -,\d +}(\t )F_2(\t )\vert \;\leq $$ $$\leq \;\exp
(-mL^\e)$$} For a proof see appendix 1.\par
Using proposition 4.3 and the DLR equations for $\mu_{R_{n+1}\cup
Q_{n+1}}^{+,\d +, -,\d +}$ and $\mu_{R_{n+1}\cup Q_{n+1}}^{\d +,\d +,
+,\d +}$, we get that \equ(4.17bis) is bounded from above by:
$$2\exp (-mL^\e)\;+\;\sum_{\vbox{\eightpoint{\hbox{$x\in R_{n+1}\cap
Q_{n+1}$}}}}[\mu_{R_{n+1}\cup Q_{n+1}}^{\d +,\d +, +,\d +}(\eta
(x)=1)\;-\;\mu_{R_{n+1}\cup Q_{n+1}}^{+,\d +, -,\d +}(\eta
(x)=1)]\Eq(4.18)$$ In turn, using the fact that
$$\mu_{R_{n+1}\cup Q_{n+1}}^{\d +,\d +, +,\d +}(\eta
(x)=1)\;\leq\;\mu_{R_{n+1}\cup Q_{n+1}}^{+,\d +, +,\d +}(\eta
(x)=1)$$ and applying proposition 4.1 to the rectangle $R_{n+1}\cup Q_{n+1}$,
we get that
\equ(4.18) can be bounded from above by: $$3\exp (-mL^\e)\Eq(4.19)$$ for
any given $m\,>\,0$ and
$\e\,\in\,(0,{1\over 2})$, provided that $\beta$ is large enough
depending on $m$ and $\e$.\par In conclusion we have shown that:
$$P(\,A_{n+1}\cap A_n^c\,\vert {\cal B})\,\leq \, 3\exp (-mL^{\e})
\Eq(4.20)$$ In order to conclude that $q_N$ is small, we
need to control the first term $q_1$ since
$$q_N\,\leq\,q_1\;+\;\sum_n^{N-1}P(\,A_{n+1}\cap A_n^c\,\vert {\cal
B})$$ {\bf Proposition 4.4} \par {Let $m\,>\,0$ and $\e\,\in \,
(0,{1\over 2})$ be given. Then there exists
$\beta_o\,\equiv\,\beta_o(\e,m )$ such that for all $\beta \,\geq
\,\beta_o$ we have: $$q_1\,\leq\,\exp (-mL^{{1\over 2}+\e})$$}
{\bf Proof}\par Let $\nu\,\equiv\,\tilde \nu_{Q_1}$ be the measure
on $\O_{Q_1}^{2^L}$ constructed in \equ(4.10bis). By monotonicity in
the initial configuration and by the definition of the event $A_1$,
we can estimate $q_1$ from above by: $$q_1\,\leq\,\sum_{x\in
R_1}[\nu (\s^{(N)}(x)\,=\,1\vert \{ \s^{(1)}\in {\cal
A}_{Q_1}^{+,+,-,+}\})\,-\, \nu
(\s^{(1)}(x)\,=\,1\vert \{ \s^{(1)}\in {\cal
A}_{Q_1}^{+,+,-,+}\})
]\Eq(4.21)$$ where we used the convention that $\s^{(1)}$ and
$\s^{(N)}$ are the components of a generic configuration $\tilde
\s\,\in \,\O_{Q_1}^{2^L}$ corresponding respectively to the minimal
($-$) and maximal ($+$) boundary condition on the top side of
$Q_1$.\par Let us examine separately each one of the two terms
appearing in the sum in the r.h.s. of \equ(4.21).\par Because of
\equ(4.10quatris), the second term $\nu (\s^{(1)}(x)\,=\,1\vert
\{ \s^{(1)}\in {\cal
A}_{Q_1}\} )$ is equal to: $$\nu (\s^{(1)}(x)\,=\,1\vert \{
\s^{(1)}\in {\cal A}_{Q_1}^{+,+,-,+}\})\,=\, \mu_{Q_1}^{\delta +,\delta
+,-,\delta +}(\s (x)=1\vert {\cal A}_{Q_1}^{+,+,-,+})\Eq(4.22)$$ Since the
event ${\cal A}_{Q_1}^{+,+,-,+}$ implies that the entire unique open contour
of the configuration $\s$ is {\it outside} $ R_1$, it is immediate to
check, using the monotonicity of the Gibbs measure with respect to
an increase of the boundary conditions, that: $$\mu_{Q_1}^{\delta
+,\delta +,-,\delta +}(\s (x)=1\vert {\cal A}_{Q_1}^{+,+,-,+})\,\geq \,
\mu_{Q_1}^{\delta +,\delta +,+,\delta +}(\s (x)=1)\Eq(4.23)$$ Let us
now consider the first term $$\nu (\s^{(N)}(x)\,=\,1\vert \{
\s^{(1)}\in {\cal A}_{Q_1}^{+,+,-,+}\})\,=\, {\nu (\s^{(N)}(x)\,=\,1\cap \{
\s^{(1)}\in {\cal A}_{Q_1}^{+,+,-,+}\})\over \mu_{Q_1}^{\delta +,\delta
+,-,\delta +}({\cal A}_{Q_1}^{+,+,-,+})}\Eq(4.24)$$ where we used, once more,
\equ(4.10quatris).\par We observe that the event $\{\s (x)\,=\,1\}$,
$x\in R_1$,
and ${\cal A}_{Q_1}^{+,+,-,+}$ depend {\it only} on the spins in $R_1\subset
\tilde R_1$ and $Q_1\setminus \{\tilde R_1\cup{\cal L}\}$
respectively, where $\tilde R_1$ and $\cal L$ have been defined
right after proposition 4.2 .\par Therefore, using \equ(4.10five)
and the fact that $\nu_{Q_1\setminus {\cal L}}^{\s^{(1)}\dots
\s^{(N)};\t^{(1)}\dots \t^{(N)}}$ is a product of the measures
$\nu_{\tilde R_1}^{\s^{(1)}\dots \s^{(N)}}$ and $\nu_{Q_1\setminus
\{\tilde R_1\cup {\cal L}\}}^{\s^{(1)}\dots \s^{(N)};\t^{(1)}\dots
\t^{(N)}}$, we get: $$\nu (\s^{(N)}(x)\,=\,1\cap \{ \s^{(1)}\in
{\cal A}_{Q_1}^{+,+,-,+}\})\,=\,$$ $$\sum_{\s^{(1)}\dots
\s^{(N)}}\nu_{Q_1}(\s^{(1)}\dots \s^{(N)})\nu_{\tilde
R_1}^{\s^{(1)}\dots \s^{(N)}}(\tilde \s^{(N)}(x)\,=\,1)
\nu_{Q_1\setminus \{\tilde R_1\cup {\cal L}\}}^{\s^{(1)}\dots
\s^{(N)};\t^{(1)}\dots \t^{(N)}}(\tilde \s^{(1)}\,\in\, {\cal
A}_{Q_1}^{+,+,-,+})\Eq(4.25)$$ We now observe that, because of (1.22) applied
to $\nu_{\tilde R_1}^{\s^{(1)}\dots \s^{(N)}}$, $$\nu_{\tilde
R_1}^{\s^{(1)}\dots \s^{(N)}}(\tilde \s^{(N)}(x)\,=\,1)\,=\,
\mu_{\tilde R_1}^{\delta +,\delta +,\s^{(N)},\delta +}(\s
(x)\,=\,1)\,\leq\, \mu_{\tilde R_1}^{\delta +,\delta +,+,\delta
+}(\s (x)\,=\,1)\Eq(4.26)$$ so that the r.h.s. of \equ(4.25) becomes
smaller than: $$\mu_{\tilde
R_1}^{\delta +,\delta +,+,\delta +}(\s (x)\,=\,1) \nu (\s^{(1)}\in
{\cal A}_{Q_1}^{+,+,-,+})\,=\,$$ $$\mu_{\tilde R_1}^{\delta +,\delta +,+,\delta
+}(\s (x)\,=\,1) \mu_{Q_1}^{\delta +,\delta +,-,\delta +}(\s\in
{\cal A}_{Q_1}^{+,+,-,+})\Eq(4.27)$$ where we used once more \equ(4.10quatris)
to write $$\nu (\s^{(1)}\in {\cal A}_{Q_1}^{+,+,-,+})\,=\, \mu_{Q_1}^{\delta
+,\delta +,-,\delta +}(\s\in {\cal A}_{Q_1}^{+,+,-,+})$$ In conclusion, from
\equ(4.24)...\equ(4.27), we get that $$\nu (\s^{(N)}(x)\,=\,1\vert
\{ \s^{(1)}\in {\cal A}_{Q_1}^{+,+,-,+}\})\,\leq\, \mu_{\tilde R_1}^{\delta
+,\delta +,+,\delta +}(\s (x)\,=\,1)\Eq(4.28)$$ Combining finally
\equ(4.23) and \equ(4.28) we bound from above the sum in \equ(4.21)
by: $$\sum_{x\in R_1}[\mu_{\tilde R_1}^{\delta +,\delta +,+,\delta
+}(\s (x)\,=\,1)\,-\, \mu_{ Q_1}^{\delta +,\delta +,+,\delta +}(\s
(x)\,=\,1)]\,\leq\,$$ $$\exp (-mL^{{1\over 2}+\e})\Eq(4.29)$$
for any given $m$, provided that $\beta$ is
large enough. In the derivation of the last inequality in \equ(4.29)
we use part ii) of Proposition 4.1 and the definition of $\tilde
R_1$.\bigskip If we now use proposition 4.4 together with
\equ(4.20), we get that $$P(A_N\,\vert {\cal B})\,\leq\,3N\exp
(-m L^{\e})\Eq(4.30)$$ We are now in a position to
conclude the proof of the theorem. Given a sequence
$S_N\;\equiv\;\{t_1\,\dots \,t_N\}$ of updatings we say that $S_N$
is a {\it good } sequence iff $S_N$ is ordered and the event $A_N^c$
occured at the end of the sequence. Using \equ(4.30) together
with propositions 4.2, we conclude that the probability that an ordered
sequence is also a good sequence is larger than $$[1\,-\,N\exp
(-mL^{\e})]P({\cal B})\;\geq\;{1\over 2}\exp
(-\beta L\t_\beta\,-\,C\beta L^{{1\over 2}+\e}))$$ for $L$ large
enough and some constant $C$.\par Thus, using Lemma 3.1, we get
that, if $T\,=\,\exp (+\beta L\t_\beta\,+\,2C\beta L^{{1\over
2}+\e}))$ and $L$ is large enough: $$P(\hbox{ there exists a good
sequence in} \;[0,T]\,)\;\geq\;{1\over 3}\Eq(4.31)$$ As in section 3
\equ(4.31) immediately implies that $$
\hbox{gap}_{V_L}(\{Q_i\},\,\emptyset)\,\geq\, \exp (-\beta
L\t_\beta\,-\,3C\beta L^{{1\over 2}+\e}))\Eq(4.32)$$ Clearly
\equ(4.32) together with \equ(4.8) and \equ(4.9) proves the correct
lower bound.\par The proof of the theorem is completed.
\pagina
\numsec=5 \numfor=1
{\bf Section 5}\par
\centerline{\bf Rare Excursions of the Magnetization }\bigskip
In this section we apply the results obtained in the previous sections
to study in detail the time evolution of the magnetization
$m(\s_t)$ of the process. In particular we will analyze the large fluctuations
of
the observable $m(\s_t)$ and prove some asymptotic results close in spirit to
the
results obtained by Shlosman for the static problem (see Theorem 4.2).\par The
setting will be that of section 4, namely the HB-dynamics in a square $V\equiv
V_L$
of side L with open boundary conditions. Although the case with $+$ boundary
conditions could be treated as well without any significant modification,
we decided
to omit it in order not to burden too much the reader.\par Let $\rho_L$, $L\in
{\bf
N}$, be a sequence of integers such that:
$$\lim_{L\to \infty}{\rho_L\over L^2}\;=\;\rho\;\in \;(-m^*(\beta ),m^*(\beta
))\qquad
\rho_L\,-\,L^2\;=\;\hbox{ mod } \,2$$ where, as usual, $m^*(\beta )$ denotes the
spontaneous magnetization, and let $\t_{\rho_L}$ be the stopping time:
$$\t_{\rho_L}\,=\,\inf \{t\geq 0;\quad m(\s_t
) \,\leq\,\rho_L\,\}\Eq(5.1)$$ Then our two main results can be stated as
follows:\bigskip {\bf Theorem 5.1}\par {\it There exists $\beta_o$
such that for any
$\beta\,\geq \,\beta_o$ and any $\rho_L$ as above: $$\lim_{L\to
\infty}{1\over \beta L}\log (\sum_{\s \atop m(\s )>0}\mu_V^\emptyset (\s
)E_\s(\t_{\rho_L}))\; =\;\psi (\rho\vee 0 )$$ where the symbol
$E_\s$ denotes the expectation over the
HB-dynamics starting from the configuration $\s$ and the function $\psi
(\rho )$ is
given in theorem 4.2 .\par The same asymptotics holds if instead of starting
from equilibrium with positive magnetization we start from the configuration
identically equal to all pluses .}\bigskip {\bf Theorem 5.2}\par {\it
There exists
$\beta_o$ such that for any $\beta\,\geq \,\beta_o$ and any $\rho_L$ as
above, there
exist numbers $\{a_L\}_{L\in {\bf N}}$ such that for any $t\,>\,0$ : \item{{\bf
a})}
$$\lim_{L\to \infty}{1\over \beta L}\log (a_L)\;=\;\psi (\rho\vee 0
)$$ \item{{\bf b})}$$
\lim_{L\to\infty}\sum_{\s \atop m(\s )>0}\mu_V^\emptyset (\s
)P_\s(\t_{\rho_L}\,>\,ta_L)\;=\;\exp (-t)$$
An analogous result holds if instead of
starting from equilibrium we start from the configuration identically equal
to all
pluses .}\bigskip {\bf Remark} Theorem 5.2 says that, under a suitable rescaling
determined by the numbers $a_L\,\approx\,\exp (\beta L\psi (\rho ))$,
the stopping
time $\t_{\rho_L}$ started at equilibrium with positive magnetization becomes
essentially {\it unpredictable} i.e. it can be thought of as the (random)
number of
independent attempts, each of which has a probability of success of the order of
$\exp (-\beta L\psi (\rho ))$, that one has to make before seeing a success.
\par
For results with the magnetization density $\rho$ outside the region
$(-m^*(\beta ),+m^*(\beta ))$ we refer the reader to the paper by Lebowitz and
Schonmann [LSch]).\bigskip {\bf Proof of Theorem 5.1}\medskip We start
by proving a
lower bound of the right order when we start from the measure $\mu_V^\emptyset$
restricted to the configurations of positive magnetization.\par Clearly for such
class of configurations: $$\t_{\rho_L}\,\geq\,\t_{\rho_L\vee 0}$$ so that it is
enough to prove a correct lower bound only for $\rho\,\in\,[0,m^*(\beta ))$.\par
For
any positive $T$ we can write: $$\sum_{\s \atop m(\s )>0}\mu_V^\emptyset (\s
)E_\s(\t_{\rho_L})\;\geq\; T\sum_{\s \atop m(\s )>0}\mu_V^\emptyset (\s
)(1\,-\,P_\s(\t_{\rho_L}\,\leq\,T))\,\geq\,$$ $${T\over 2}\,-\, T\sum_{\s
\atop m(\s )>0}\mu_V^\emptyset (\s )P_\s(\t_{\rho_L}\,\leq\,T)\Eq(5.2)$$ where
we
used the symmetry of the Gibbs measure under global spin flip.\par As in section
4
(see (4.4) and (4.5)) the sum in the r.h.s. of \equ(5.2) can be estimated from
above by: $$2L^2T\mu_V^\emptyset(m(\s )\,=\,\rho_L)\,+\,\exp (-KL^2T)\Eq(5.3)$$
for a suitable constant $K$.\par
We now take the time $T$ of the form
$$T\,=\,\exp (\beta L(\psi (\rho )-\d))\Eq(5.4)$$
where $\d$ is any fixed small number independent of $L$. If we recall theorem
4.2,
we get that, with this choice of $T$ \equ(5.3) goes to zero as $L$ gets large.
This fact together with \equ(5.2) implies that $$\lim_{L\to
\infty}{1\over \beta L}\log (\sum_{\s \atop m(\s )>0}\mu_V^\emptyset (\s
)E_\s(\t_{\rho_L}))\;\geq\;\psi (\rho )-\d\quad \forall \;\rho\in
\,[0,m^*_\beta )\Eq(5.5)$$ Since $\d$ can be taken arbitrarily small (after the
limit $L\to \infty$) the required lower bound follows. It is also clear that
the same
lower bound applies also to $E_+(\t_{\rho_L})$, that is when the starting
configuration is identically equal to plus one, since, because of monotonicity
in
the initial configuration,
$$E_+(\t_{\rho_L})\,\geq\,\sum_{\s \atop m(\s )>0}\mu_V^\emptyset (\s
)E_\s(\t_{\rho_L})$$ In order to prove an upper bound we have to distinguish
between two cases: $$\eqalign{\hbox{Case 1}\qquad\rho\;&\in\;(-m^*(\beta
),\rho_1]\cr\hbox{Case 2}\qquad\rho\;&\in\;(\rho_1,m^*(\beta
))}$$ where $\rho_1$ is
the singularity point of the function $\psi (\rho )$ defined in theorem 4.2 .\par
Let us begin with the first one.\par Clearly
$$\sum_{\s \atop m(\s )>0}\mu_V^\emptyset (\s
)E_\s(\t_{\rho_L})\,=\,\sum_{\s \atop m(\s )>0}\mu_V^\emptyset (\s
)\sum_nP_\s(\t_{\rho_L}\,\geq\,n)\,\leq\,\sum_nP_+(\t_{\rho_L}\,\geq\,n)\Eq(5.6)$$
since, by monotonicity in the initial configuration,
$$P_\s(\t_{\rho_L}\,\geq\,n)\,\leq\,P_+(\t_{\rho_L}\,\geq\,n)\quad
\forall\;n\quad \forall \;\s\Eq(5.6bis)$$
In turn, for any integer $N$, it follows from the Markov property and
\equ(5.6bis) that:
$$P_+(\t_{\rho_L}\,\geq\,n)\,\leq\,P_+(\t_{\rho_L}\,\geq\,N)^{[{n\over
N}]}\Eq(5.7)$$
Let us therefore estimate $P_+(\t_{\rho_L}\,\geq\,N)$. We write
$$P_+(\t_{\rho_L}\,\geq\,N)\,=\,P_+(\int_0^Ndt \chi
(m(\s_t)\,\geq\,\rho_L)\,=\,N)\,\leq\,{\int_0^Ndt
P_+(m(\s_t)\,\geq\,\rho_L)\over
N}\Eq(5.8)$$
where $\chi (A)$ denotes the characteristic function of the event $A$ and we
used
the generalized Chebyshev inequality in order to get the last inequality.\par
Let us now choose the integers $N$, $N_o$ equal to
$$\eqalign{N\;&=\;\exp (\beta L(\psi (0)\,+\,2\d ))\cr N_o\;&=\;
\exp (\beta L(\psi
(0)\,+\,\d ))\qquad \d\,<<\,1}$$
Then we can estimate the integral in the r.h.s. of \equ(5.8) by
$${\int_0^Ndt P_+(m(\s_t)\,\geq\,\rho_L)\over
N}\,\leq\,$$
$${N_o\over N}\;+\;\mu_V^\emptyset (m(\s )\,\geq\,\rho_L )\;+\;
{\int_{N_o}^Ndt [P_+(m(\s_t)\,\geq\,\rho_L)
\,-\,\mu_V^\emptyset (m(\s )\,\geq\,\rho_L )]\over
N}\Eq(5.9)$$
Let us examine separately each one of the three terms in the r.h.s. of
\equ(5.9) in the limit as $L\to \infty$. The first term goes to zero by
construction. The second term converges to $1\over 2$ because of theorem
4.2 . The
third term also goes to zero for $\beta$ large enough, if we use theorem 4.1,
the
basic estimate (1.19), the fact that $\psi (0)\,=\,\t_\beta$ and our choice
of the
integer $N_o$.\par
In conclusion we have shown that, for all $\beta$ large enough and all large enough
$L$
$$P_+(\t_{\rho_L}\,\geq\,N)\,\leq\, {2\over 3}\Eq(5.10)$$
Clearly \equ(5.10) together with \equ(5.6), \equ(5.7) prove that
$$\sum_{\s \atop m(\s )>0}\mu_V^\emptyset (\s
)E_\s(\t_{\rho_L})\,\leq\,E_+(\t_{\rho_L})\,\leq\,3N\;=\;3\exp (\beta L(\psi
(0)\,+\,2\d ))\Eq(5.11)$$
which establishes the correct upper bound in the limit $L\to\infty$ due
to the arbitrariness of $\d$ also for the case when the starting configuration
is
identically equal to plus one.\par Let us now treat the (more difficult)
second case $\rho\;\in\;[\rho_1,m^*(\beta ))$.\par First of all we bound $$\sum_{\s
\atop m(\s )>0}\mu_V^\emptyset (\s )E_\s(\t_{\rho_L})$$ by the same quantity but
computed for the HB-dynamics in $V$ with extra $+$ boundary conditions on
the top
horizontal side and starting from all pluses:
$$\sum_{\s \atop m(\s )>0}\mu_V^\emptyset (\s
)E_\s(\t_{\rho_L})\,\leq\,
E_+^{\emptyset,\emptyset,+,\emptyset}(\t_{\rho_L})\Eq(5.12)$$
with self explanatory notation. The reason for introducing on only one side
of $V$
extra $+$ boundary conditions is the following. For large L, the relaxation
time to
equilibrium ($\equiv \hbox{gap}(HB,\emptyset,\emptyset,+,\emptyset )^{-1}$)
with the
indicated boundary conditions is of the
order of $\exp (C\beta L^{{1\over 2}+\e})$ (see Corollary 4.1); therefore the
relaxation time is much smaller than the inverse of the equilibrium measure
of the hitting set $\{\s;\; m(\s )\leq \rho_L\}$,
$$\mu_V^{\emptyset,\emptyset,+,\emptyset}(m(\s )\leq \rho_L)^{-1}\,\geq\,
\exp (\beta c(\rho ) L)\; ;\quad c(\rho )>0$$ It thus follows from a standard
argument (see e.g. [A]) that:
$$E_+^{\emptyset,\emptyset,+,\emptyset}(\t_{\rho_L})\,\leq\,
\mu_V^{\emptyset,\emptyset,+,\emptyset}(m(\s )\leq \rho_L)^{-1}\exp (\beta \d
L)\qquad \d<<1\Eq(5.13)$$ Thus one needs, in strict analogy with theorem 4.2, to
estmate from below $$\mu_V^{\emptyset,\emptyset,+,\emptyset}(m(\s )\leq \rho_L)$$
as $L\to \,\infty$. This is the content of the next proposition:\bigskip
{\bf Proposition 5.1}\par
{\it Let $\rho_L$ be as above. Then there exists $\beta_o$ such that for any
$\beta\,\geq \,\beta_o$ and any given positive $\d$
$$\mu_V^{\emptyset,\emptyset,+,\emptyset}(m(\s )\leq \rho_L)\,\geq\,
\exp(-\beta (\psi (\rho)+\d) L)$$
for all $L$ large enough, where $\psi (\rho )$ is as in theorem 4.2 .}\bigskip
The proposition can be proved by exactly the same methods developed in [DKS]
(see
also [PF]) and employed by Shlosman in [Sh] in his proof of Theorem 4.2; the
proof
is however lengthy and therefore it is not included in
this work.\par It is possible to give a convincing explanation why the extra $+$
boundary conditions on the top side of $V$ do not affect the aymptotics
(or at least
a lower bound) of $$\mu_V^{\emptyset,\emptyset,+,\emptyset}(m(\s )\leq \rho_L)$$
In
[Sh] (see [DKS] for full details in the case of periodic boundary conditions and
[Pf] in the case of plus boundary conditions) the asymptotics of
$\mu_V^{\emptyset,\emptyset,\emptyset ,\emptyset}(m(\s )= \rho_L)$ is derived by
proving that the typical structure of the set of configurations under the event
$\{m(\s )= \rho_L\}$ is as follows:\medskip \item{1)} In case $\rho \,>\,\rho_1$
there is a bubble $W^{o}_\rho$ of the minus phase close to one of the four
corners
of $V$, while in $V\setminus W^{o}_\rho$ one has the plus phase. The shape of
$W^{o}_\rho$ is that of the intersection with $V$ of the rescaled Wulff shape
$2\sqrt{{m^*_\beta -\rho\over 2m^*_\beta}}W$
of total volume $4{m^*_\beta
-\rho\over 2m^*_\beta}$ and centered at one of the corners
of $V$. It is clear from the results in [DKS] (see ch. 5) that the probability
(with open
boundary conditions) for the above situation to occur is of the order $$\exp
(-\beta
L{1\over 2} \sqrt{{m^*_\beta -\rho\over 2m^*_\beta}}w)\;=\;
\exp(-\beta \psi (\rho ) L)$$
where $w$ is the Wulff functional computed on the Wulff curve $\partial W$.
\item{2)} In case $\rho\,\leq\, \rho_1$, where $\rho_1$ is as in theorem 4.2,
it is more convenient to divide the volume $V$ into roughly two rectangles, with
the
correct volumes determined by $\rho$, by means of a (roughly) straight horizontal
line. It is clear that in this other case the probability is of the order of
$$\exp (-\beta \t_\beta L)$$
for any $\rho\,\leq\,\rho_1$.\medskip
In the first case, we can impose to our configuration to have a unique "large"
contour exactly like the one described above, at distance greater than $cL$
from the
top side of $V$, where $c>{1\over 2}$ is a suitable constant depending on $\rho$.
Since the
coefficients $\Phi^{\emptyset,\emptyset,+,\emptyset}(\L )$ of the cluster
expansion
of the partition function decay exponentially fast in the "size" of the set $\L$,
it
is not difficult to see that in this way one obtains a lower bound on \hfill
$\mu_V^{\emptyset,\emptyset,+,\emptyset}(m(\s
)\leq \rho_L)$ which, apart from minor
corrections that are adsorbed in the $\d$ appearing in the proposition, is like
the one obtained without the extra plus boundary condition on the top
side.\bigskip It is clear that if we plug the statement of the proposition into
\equ(5.13) we get the required upper bound. The proof of the theorem is
complete.\bigskip {\bf Proof of Theorem 5.2}\par Let us define
the numbers $a_L$ by
the following condition: $$\sum_{\s ;\,\atop m(\s )>0}\mu_V^{\emptyset}(\s
)P_\s(\t_{\rho_L}\,\geq\,a_L)\;=\;\hbox{e}^{-1}\Eq(5.14)$$
and let $f_L(t)$ be given
by: $$f_L(t)\,=\,\sum_{\s ;\,\atop m(\s )>0}\mu_V^{\emptyset}(\s
)P_\s(\t_{\rho_L}\,>\,a_Lt)\Eq(5.15)$$
In order to prove the theorem it is enough, using the normalization \equ(5.14),
to show that:
$$\lim_{L\to \infty}\vert f_L(t+s)\,-\,f_L(t)f_L(s)\vert\;=0\Eq(5.16)$$
and that the asymptotics of the number $a_L$, as $L\to \infty$, is the right
one.
Because of \equ(5.2) applied to $T\,=\,a_L$, we immediately get that:
$$a_L\;\leq\;e\sum_{\s ;\,\atop m(\s )>0}\mu_V^{\emptyset}(\s
)E_\s(\t_{\rho_L})\Eq(5.17)$$
In order to obtain a lower bound on $a_L$ we observe that, using the argument
employed in section 4 (see e.g. (4.4), (4.5)): $$1-e^{-1}\,=\,\sum_{\s\atop
m(\s )>0}\mu_V^{\emptyset}(\s )P_\s(\t_{\rho_L}\,<\,a_L)\,\leq$$
$$2L^2(a_L\vee 1)\mu_V^\emptyset(m(\s )\,=\,(\rho_L\vee 0))\,+\,\exp
(-KL^2(a_L\vee 1))\Eq(5.18)$$ for a suitable constant $K$. Thus
$$a_L\,\geq\,{1-e^{-1}\over 4L^2\mu_V^\emptyset(m(\s )\,=\,(\rho_L\vee
0))}\Eq(5.19)$$ for large $L$. Clearly \equ(5.17) and \equ(5.19) together with
theorems 5.1, 4.2 prove the first part of the theorem.\par
Let us turn to the proof of \equ(5.16). We observe that, because of the definition
of the stopping time $\t_{\rho_L}$, it trivially follows that:
$$f_L(t)\,=\,\sum_{\s }\mu_V^\emptyset(\s )P_\s(\t_{\rho_L}>a_Lt)
\,-\,\sum_{\s \atop \rho_L\leq m(\s )\leq 0}\mu_V^\emptyset(\s )P_\s(\t_{\rho_L}>a_Lt)\,\equiv
\,\bar
f_L(t)\,-\,\e_L$$
Clearly,using theorem 4.2 $\e_L$ goes to zero exponentially fast in $L$.\par
Using the reversibility of the dynamics with respect to the Gibbs measure
$\mu_V^\emptyset$, we can write $\bar
f_L(t+s)$ as:
$$\bar
f_L(t+s)\,=\,\sum_{\s }\mu_V^\emptyset(\s )
P_\s(\t_{\rho_L}>a_Lt) P_\s(\t_{\rho_L}>a_Ls)\,=\,$$
$$
\sum_{\s ;\atop m(\s )>0 }\mu_V^\emptyset(\s )
P_\s(\t_{\rho_L}>a_Lt) P_\s(\t_{\rho_L}>a_Ls)\,+\,\e_L$$
so that the difference $
\vert f_L(t+s)\,-\,
f_L(t)
f_L(s)\vert$ can be estimated from above by:
$$\vert \sum_{\s , \h \atop\,m(\s )>0,m(\h )>0}\mu_V^\emptyset(\s )\mu_V^\emptyset(\h
)
P_\s(\t_{\rho_L}>a_Lt)[ P_\s(\t_{\rho_L}>a_Ls)\,-\,
P_\h(\t_{\rho_L}>a_Ls)]\vert \,+\,2\e_L\Eq(5.20)$$
If we now couple the HB-dynamics starting from $\s$ and $\h$ together in the way
described in section 1, we can estimate the first term in \equ(5.20) by
$$\sum_{\s , \h \atop\,m(\s )>0,m(\h )>0}\mu_V^\emptyset(\s )\mu_V^\emptyset(\h
)
P(\t_{\rho_L}(\s )\neq \t_{\rho_L}(\h ))\Eq(5.21)$$
where, with an abuse of notation, $P$ denotes the probability measure of the
coupled process, $\t_{\rho_L}(\s )$ and $\t_{\rho_L}(\h )$ the stopping times
starting from $\s$ and $\h$ respectively.\par
The idea behind the estimate of $P(\t_{\rho_L}(\s )\neq
\t_{\rho_L}(\h ))$ (see below) is at this point very natural: when
the two starting
configurations $\s$ and $\h$ are both chosen at random with respect to the Gibbs
measure $\mu_V^\emptyset$ restricted to the "phase" $\{m\,>\,0\}$, then in
a time
scale $T_o$, which is much shorter than the typical time scale of $\t_{\rho_L}(\s
)$
and $\t_{\rho_L}(\h )$, the two configurations become identical with very large
probability. The reason for this quick loss of memory inside the "phase"
$\{m\,>\,0\}$,
in contrast to the smallness of the gap (see Theorem 4.1), has to be found in
the
fact (see the proof of proposition 5.2 below) that, starting in equilibrium with
positive magnetization, with large probability the HB-dynamics in $V$ with open
boundary conditions cannot be distinguished, at a given site $x\in V$, from the
HB-dynamics in $V$ with an extra $+$ boundary condition on one of the
sides of $V$.
This latter looses memory of the initial condition much faster than the dynamics
with open boundary conditions (see corollary 4.1) and the result follows.\par
Let us start with the technicalities. Let $\e\in (0,{1\over 2})$ be given and
let
$T_o\,=\,\exp (\beta L^{{1\over 2}+\e })$. Then we estimate \equ(5.21) by:
$$2\sum_{\s , \atop m(\s )>0}\mu_V^\emptyset(\s )P_\s(\t_{\rho_L}\,<\,T_o)\;+\;
2\sum_{\s , \atop m(\s )>0}\mu_V^\emptyset(\s )P(\s_{T_o}\neq (+)_{T_o})\Eq(5.22)$$
where $(+)_{T_o}$ is the evoluted at time $T_o$ of the configuration identically
equal to $+1$.\par
We know already (see \equ(5.18)) that the first term in \equ(5.22) goes to zero
as
$L\to \infty$ provided that $\beta$ is large enough. The second term is controled
by the following new result:\bigskip
{\bf Proposition 5.2}\par {\it Let $\e\,\in \, (0,{1\over 2})$ and $m\,>\,0$ be
given. Then there exist $\beta_o\,<\,+\infty$ and $C\,<\,+\infty$ such that
for any $\beta\,\geq\,\beta_o$, any integer $L$ and any time
$t\;\geq\;\exp (C\beta L^{{1\over 2}+\e })$:
$$\sum_{\s \atop m(\s )>0}\mu_V^\emptyset(\s )P((+)_t\neq\s_t)\;\leq\;\exp
(-mL)$$
}\bigskip
It is clear that the above proposition concludes the proof of Theorem 5.2 in the
case the starting configuration is distributed according to the restriction
of the
Gibbs measure to the set $\{\s;\;m(\s )>0\}$. A similar argument can be repeated
if
the starting configuration is identically equal to $+1$.\bigskip {\bf Proof of
proposition 5.2}\par Since $$P((+)_{t+s}\neq\s_{t+s})\,\leq\,P((+)_t\neq\s_t)\qquad
\forall \;s\,\geq \,0$$ it is sufficient to prove the result for the fixed time
$t_o\,=\,\exp (C\beta L^{{1\over 2}+\e })$. We first estimate $P((+)_{t_o}\neq\s_{t_o})$
by $$P((+)_{t_o}\neq\s_{t_o})\;\leq\;\sum_{x\in V}P((+)_{t_o}(x)\neq\s_{t_o}(x))$$
Given now $x\in V$ let us uppose, without loss of generality, that the top
horizontal side of $V$ is such that its distance from $x$ is greater or
equal than
$L\over 2$. Let also $(+)_{t_o}^{\emptyset ,\emptyset ,+,\emptyset
}$ be the evoluted at time ${t_o}$ of the configuration $(+)$ under the HB-dynamics
in
$V$ with $(\emptyset ,\emptyset ,+,\emptyset )$ boundary conditions on
$\partial_{ext} V$. Then, by monotonicity, we have:
$$P((+)_{t_o}(x)\neq\s_{t_o}(x))\;\leq\;P((+)_{t_o}^{\emptyset
,\emptyset ,+,\emptyset
}(x)\neq\s_{t_o}(x))\;=\;$$
$$P((+)_{t_o}^{\emptyset ,\emptyset ,+,\emptyset}(x)\,=\,+1)\,-\,
P(\s_{t_o} (x)=+1)\Eq(5.23)$$
Thus
$$\eqalign{&\phantom{\sum_{x\in V}}\sum_{\s \atop m(\s )>0}\mu_V^\emptyset(\s
)P((+)_{t_o}\neq\s_{t_o})\;\leq\;\cr\sum_{x\in
V}&[\sum_{\s \atop m(\s )>0}\mu_V^\emptyset(\s
)(P((+)_{t_o}^{\emptyset ,\emptyset ,+,\emptyset}(x)\,=\,+1)\,-\,
P(\s_{t_o} (x)=+1))]\,=\,\cr\sum_{x\in V}&[{1\over 2}P((+)_{t_o}^{\emptyset
,\emptyset
,+,\emptyset}(x)=+1)\;-\;\sum_{\s \atop m(\s )>0}\mu_V^\emptyset(\s )P(\s_{t_o}
(x)=+1)\,]}\Eq(5.23bis)$$ Let us first treat the term
$P((+)_{t_o}^{\emptyset ,\emptyset ,+,\emptyset}(x)=+1)$.
Using (1.19), Corollary 4.1
and our choice of the time ${t_o}$, we get that: $$0\,\leq\,P((+)_{t_o}^{\emptyset
,\emptyset ,+,\emptyset}(x)\,=\,+1)\,-\, \mu_V^{\emptyset ,\emptyset ,+,\emptyset}(\s
(x)=+1)\,\leq\,$$ $$\exp (C'L^2\,-\,{t_o}\exp (-C\beta L^{{1+\e\over
2}}))\,\leq\,{1\over 3}\exp (-mL)\Eq(5.24)$$ for any given $m>0$ and any $L\in
{\bf N}$, provided that $\beta$ is large enough.\par
As far as the second term in the square parenthesis in the r.h.s. of \equ(5.23bis)
is concerned, we write:
$$\sum_{\s \atop m(\s )>0}\mu_V^\emptyset(\s )P(\s_{t_o}
(x)=+1)\;\geq\;\sum_{\s \atop m(\s )>0}\mu_V^\emptyset(\s )P(\s_{t_o}
(x)=+1\cap m(\s_{t_o} )>0)\;=\;$$
$$\sum_{\s}\mu_V^\emptyset(\s )P(\s_{t_o}
(x)=+1\cap m(\s_{t_o} )>0)\;-\;\sum_{\s ;\;m(\s )\leq 0}\mu_V^\emptyset(\s
)P(\s_{t_o}
(x)=+1\cap m(\s_{t_o} )>0)\;=\;$$
$$\mu_V^\emptyset(\s (x)=1\cap m(\s )\geq 0)\;-\;
\sum_{\s ;\;m(\s )< 0}\mu_V^\emptyset(\s )P(\s_{t_o}
(x)=+1\cap m(\s_{t_o} )\geq 0)\Eq(5.25)$$
where we used the invariance of the measure $\mu_V^\emptyset$.\par
The last term in the r.h.s. of \equ(5.25) can be bounded from above by:
$$\sum_{\s }\mu_V^\emptyset(\s )P(\hbox{there exists }s\leq {t_o};\;
m(\s_s )=0)\,\leq\,2L^2{t_o}\mu_V^\emptyset(m(\s )\,=\,0)\,+\,\exp (-KL^2{t_o})\Eq(5.26)$$
for a suitable constant $K$ and large enough $\beta$, by the argument illustrated
in
section 4 (see (4.5)).\par
Clearly, because of our choice of ${t_o}$ and of theorem 4.2, the r.h.s.
of \equ(5.26)
is smaller than ${1\over 3}\exp (-mL)$ for any given $m$ provided $\beta$ is large
enough.\par
In conclusion we have shown that:
$${1\over 2}P((+)_{t_o}^{\emptyset ,\emptyset
,+,\emptyset}(x)\;-\;\sum_{\s \atop m(\s )>0}\mu_V^\emptyset(\s )P(\s_{t_o}
(x)=+1)\,\leq\,$$
$${1\over 2}\vert\mu_V^{\emptyset ,\emptyset ,+,\emptyset}(\s
(x)=+1)\,-\,\mu_V^\emptyset(\s (x)=1\vert m(\s )\geq 0)\vert \;+\;{2\over 3}
\exp (-mL)\Eq(5.27)$$
for any $L$, provided that $\beta$ is large
enough.\par
In order to complete the proof we need a last, rather obvious result on the Ising
model, whose
proof is an exercise in the cluster expansion and it is therefore omitted.\bigskip
{\bf Lemma 5.1}\par
{\it Given $m>0$ there exists $\beta_o$ such that for all $\beta \geq \beta_o$
and
all $L$:
$$\vert\mu_V^{\emptyset ,\emptyset ,+,\emptyset}(\s
(x)=+1)\,-\,\mu_V^\emptyset(\s (x)=1\vert m(\s )\geq
0)\vert\;\leq {1\over 3}\exp
(-mL)$$}\bigskip
If we apply the lemma to \equ(5.27) we obtain:
$$\sum_{x\in V}\sum_{\s \atop m(\s )>0}\mu_V^\emptyset(\s
)P((+)_t\neq\s_t)\;\leq\;L^2\exp
(-mL)$$
for any given $m>0$ and any $L\in
{\bf N}$, provided that $\beta$ is large enough. The proposition is proved.
\pagina
%\input formato.tex
\def\muo{\mu_{V_L}^\emptyset}
\def\Pr{ P^{\mu}}
\def \rh#1{\rho (\tilde\s_{#1})}
\tolerance=10000
\numsec=6 \numfor=1
{\bf Section 6}\par
\centerline{\bf Markov Chain Description of the Time Rescaled
Magnetization}\bigskip In this final section we work in the same setting and
notation of the previous two sections and we consider the normalized magnetization
$$\rho (\s_t)\,=\,{1\over \vert V_L\vert}\sum_{x\in V_L}\s_t (x)$$
of the process started at equilibrium (or from one of the two extreme configurations
$(+)$ or $(-)$).\par
We show that it is possible to rescale the time $t$ by a multiplicative factor
$t_L$ depending on the side $L$ of the square $V_L$, in such a way that, as
$L\to
\infty$, the finite dimensional distributions of the rescaled process
$$ \rho (\tilde \s_{t})\,=\,\rho (\s_{t_Lt})\Eq(6.1.0)$$
converge to those of a continuous time Markov chain on the set $\{-m^*(\b
),m^*(\b
)\}$ with unitary jump rate for both states.\par
>From what we just said, it is clear that the speeding factor $t_L$ must be determined
essentially by the condition that:
$$\int_{m(\s )>0}d\muo (\s )P(\rho (\s_{t})\,\approx \,-m^*(\b
))\;\approx\; {p\over 2}$$
where
$$p\;=\;{1\over 2}(1-\exp (-2))\Eq(6.1)$$
is the probability that a continuous time Markov chain with unitary jump rate on
$\{-1,1\}$ starting at time $t=0$ in $+1$ is, at time $t=1$, in the state
$-1$.\par
It is also clear from the results of section 4 and section 5 that
$$t_L\;\approx\;\exp (\b \t_\b L)$$
for large $L$.\par
Let us state more precisely our result. We denote by $M$ the two state space
$\{-m^*(\b ),m^*(\b
)\}$ and by $Y_t$ a continuous time Markov chain on $M$ with unitary jump rate
for
both states. Clearly the invariant measure $\nu$ of the chain $Y_t$ is uniform
over
$M$. Let also, for any given $\e\,\in\,(0,{m^*(\b )\over 2})$, $t_L\,\equiv
\,t_L(\e
)$ be the such that: $$\Pr (\rh{0}\,\geq\, m^*(\b )-\e\,;\,\rh{1}\,\leq\, -m^*(\b
)+\e)\;=\;{p\over 2}\Eq(6.2)$$ where $p$ is given by \equ(6.1)
and $\Pr$ denotes the
probability over the HB-dynamics started from the equilibrium distribution
$\muo$.
Then we have :\bigskip
{\bf Theorem 6.1}\par
{For any $\b$ large enough, any $\e$ as above and for any choice of times
$t_1\,<\,t_2\,<\, \dots\,<\,t_k$ and numbers $m_i\,\in \, M$, $i=1\dots k$ :
$$\lim_{L\to \infty}\Pr (\vert \rh{t_1}-m_1\vert <\e,\dots
,\vert \rh{t_k}-m_k\vert <\e )\;=\;P^{\nu }(Y_{t_1}=m_1,\dots ,Y(t_k)=m_k)\Eq(6.3)$$
where $P^{\nu
}$ denotes the probability of the chain $Y_t$ with initial distribution the
invariant measure $\nu$. Moreover
$$\lim_{L\to \infty}{1\over \b L}\log (t_L)\,=\,\t_\b$$}\bigskip
{\bf Proof}\par
The second part follows immediately from the results of section 4 and section
5.\par
As far as the first part is concerned, it is well known
that $$\lim_{L\to \infty}\muo (\vert\rho (\s )\,-\,m^*(\b )\,\vert\,\leq \,
\d)\;=\;{1\over 2}\quad \forall \,\d>0\Eq(6.2bis)$$ and similarly, by the
symmetry under global spin flip, for $m^*(\b )$ replaced by $-m*(\b )$.
Hence, if the
limit in the l.h.s. of \equ(6.3) exists for fixed $t_1\,<\,t_2\,<\, \dots\,<\,t_k$
and arbitrary choice of $m_i\,\in \, M$, $i=1\dots k$, it must be a probability
measure on $M^k$.\par We will prove \equ(6.3) by showing that the limit along
any
convergent subsequence is equal to the r.h.s. of \equ(6.3). The key step in our
argument is to prove that, asympotically as $L\to \infty$, the variables
$\rh{t_1},\dots ,\rh{t_k}$ enjoy the Markov property. This is the content of the
following proposition. For notation convenience we denote by $A_{t_i}(m_i)$
the event
$\vert \rh{t_i}-m_i\vert <\e$.\bigskip {\bf Lemma 6.1}\par {\it In the same
hypotheses of theorem 6.1 the difference $$\Pr (A_{t_1}(m_1),..
,A_{t_k}(m_k))-\Pr (A_{t_1}(m_1),..
,A_{t_{k-1}}(m_{k-1})) 2\Pr (A_{t_{k-1}}(m_{k-1}),A_{t_k}(m_k))$$
tends to zero as $L\to \infty$.}\bigskip
Before giving the proof of the above key result we complete the proof of theorem
6.1\par
Using the lemma and \equ(6.2bis) it is clearly enough
to prove that:
$$\lim_{L\to \infty}\Pr (\vert \rh{0}+m^*(\b )\vert \leq \e;\vert\rh{t}-m^*(\b
)\vert \leq \e)\,=$$
$$=\, P^{\nu }(Y_{0}=-m^*(\b );Y_t=m^*(\b
))\,=\,(1-(1-2p)^t)\Eq(6.4)$$ Let us first consider times $t$ of the form
$t\,=\,{1\over m},\;m\in {\bf N}$ and let us define by $a({1\over
m})$ any limit of
the l.h.s. of \equ(6.4) computed for such $t$. From the lemma applied to times
$t_i\,=\,{i\over m}\;i=1\dots m$ and the fact that, by construction
$$a(1)\;=\;{p\over 2}$$ one immediately gets $$a({1\over m})\,=\,(1-(1-2p)^{1\over
m})\Eq(6.5)$$ Once we know the value of $a({1\over m})$ we can repeat the same
argument to show that \equ(6.4) holds also for rational times of the form
$t\,=\,{n\over m}$. In order to extend \equ(6.4) to all times $t$,
it is sufficient to
prove for example that, if $\bar a(t)$ and $\underline a(t)$ denote the $\limsup$
and
$\liminf$ of the l.h.s. of \equ(6.4), then both of them are non decreasing
function of
$t$.\par For this purpose and denoting by $a_L(t)$ the l.h.s. of
\equ(6.4), we immediately obtain from the lemma and \equ(6.2bis) that
$a_L(t+s)$ satisfies the equation: $$a_L(t+s)\,=\,a_L(t)\,
+\,a_L(s)(1-4a_L(t))\,+\,r_L\Eq(6.6)$$ where
$\lim_{L\to\infty}r_L\;=\;0$.\par We now observe that
$$a_L(t)\;\leq\;{1\over 4}\;+\;r'_L\Eq(6.7)$$
where, as before, $\lim_{L\to\infty}r_L'\;=\;0$.\par
In fact, again because of \equ(6.2bis)
$$\Pr (\vert
\rh{t}-m^*(\b )\vert \leq \e;\,\vert \rh{0}+m^*(\b )\vert \leq
\e)\,=\,{1\over 2}\,-\,\Pr(\rh{t}>0;\,\rh{0}>0)\;+\;r'_L$$
and, by the F.K.G. property of the measure $\Pr$ (see e.g. [Li]):
$$\Pr(\rh{t}>0;\,\rh{0}>0)\,\geq\,\Pr(\rh{t}>0)\Pr(\rh{0}>0)\,=\,{1\over 4}$$
If we insert \equ(6.7) we obtain:
$$a_L(t+s)\,\geq\,a_L(t)\,+\,r_L\,-\,4r'_L\Eq(6.8)$$
Clearly \equ(6.8) shows that $\bar a(t)$ and $\underline a(t)$ are non decreasing
function of $t$ and thus \equ(6.4) holds for all $t$.\bigskip
{\bf Proof of Lemma 6.1}\par
Using the reversibility we can write:
$$\Pr (A_{t_1}(m_1)
\,\dots\,A_{t_k}(m_k))\,=$$
$$=\;\int_{A_0(m_{k-1})} d\muo (\s)P_\s
(A_{\vert t_k-t_{k-1}\vert}(m_k))P_\s(A_{\vert t_{k-1}-t_1\vert}(m_1),\dots
,A_{\vert
t_{k-2}-t_{k-1}\vert}(m_{k-2}))\Eq(6.9)$$
where $P_\s$ denotes the probability measure on the HB-dynamics starting from
$\s$.\par
We now compare the r.h.s. of \equ(6.9) with the quantity:
$$(\muo (A_0(m_{k-1}))^{-1}\Pr (A_{t_1}(m_1),..
,A_{t_{k-1}}(m_{k-1})) \Pr (A_{t_{k-1}}(m_{k-1}),A_{t_k}(m_k)) \Eq(6.10)$$
Using the stationarity of the measure $\Pr$ and reversibility, we can write
their
difference as: $$(\muo (A_0(m_{k-1}))^{-1}\int\int_{A_o(m_{k-1})\atop
A_o(m_{k-1})}
d\muo (\s )d\muo (\h )G(\h ,\s )
\Eq(6.11)$$
where
$$\eqalign{G(\h ,\s )&=\cr [P_\h (A_{\vert t_k-t_{k-1}\vert}(m_k))-P_\s
(A_{\vert
t_k-t_{k-1}\vert}(m_k))]&
P_\s(A_{\vert t_{k-1}-t_1\vert}(m_1),\dots ,A_{\vert
t_{k-2}-t_{k-1}\vert}(m_{k-2}))}\Eq(6.12)$$
Using the coupling described in section 1
and the symmetry under global spin flip, the absolute value of \equ(6.12) can be
estimated from above by:
$$2\int\int_{m(\s )>0\atop m(\h )>0}
d\muo (\s )d\muo (\h )P(\tilde\s_{t_k-t_{k-1}}\neq
\tilde\h_{t_k-t_{k-1}})\Eq(6.12bis)$$ which
tends to zero as $L\to \infty$ because of
proposition 5.2 . We have in fact that $$t_L[t_k-t_{k-1}]\,>>\,\exp (C\b
L^{{1\over
2}+\e})$$ because of the second part of theorem 6.1.\par
The statement of the lemma now follows from \equ(6.2bis) since, in \equ(6.10)
we can
safely replace the factor $(\muo (A_0(m_{k-1}))^{-1}$ with $2$. The proof is
complete.\pagina
%\input formato.tex
\numsec=1 \numfor=1
\centerline{\bf Appendix 1}\bigskip In this appendix we prove
propositions 4.1, 4.2, 4.3. Since the proof of proposition 3.1 is
very similar, although much simpler, than that of proposition 4.1, we
decided, for shortness, to omit it.\medskip\noindent
{\bf Proof of Proposition 4.1}\medskip Let us fix $\e\,\in \,
(0,{1\over 2})$ and a rectangle $R$: $$R\,=\,\{\;x\in \Z;\,0\,\leq
\,x_1\,\leq\,L_1\quad 0\,\leq\,x_2\,\leq\,L_2\;\}$$ with $L_1\,\geq
\,L_2\,\geq\,L_1^{{1\over 2}+\e}$.\par If ${\cal A}_R^{+,+,-,+}$ is
the event described in (4.12): $$ {\cal A}_R^{+,+,-,+}\;=\;\{\s ;\quad
\Gamma_{open}(\s )\,\subset\, \{\;x\in R;\;x_2\,\geq\,{13L_2\over
16}\,\}\;\}$$ we can write: $$\eqalign{\mu_R^{+,\delta +,-,\delta
+}(\s (x)=1)\;&=\cr \mu_R^{+,\delta +,-,\delta +}(\s (x)=1\vert
{\cal A}_R^{+,+,-,+})\mu_R^{+,\delta +,-,\delta +}({\cal
A}_R)\,&+\,\mu_R^{+,\delta +,-,\delta +}(\s (x)=1\cap
({\cal A}_R^{+,+,-,+})^c)}$$ where $ ({\cal A}_R^{+,+,-,+})^c$ is
just the complement event.\par Since $$ \mu_R^{+,\delta +,-,\delta
+}(\s (x)=1\vert {\cal A}_R)\,\geq \, \mu_R^{+,\delta +,+,\delta
+}(\s (x)=1)\Eqa(1)$$ (see 4.27), we obtain that the difference
$$\mu_R^{+,\delta +,+,\delta +}(\s (x)=1)\;-\;\mu_R^{+,\delta
+,-,\delta +}(\s (x)=1)$$ can be bounded from above by
$$\mu_R^{+,\delta +,-,\delta +}(({\cal A}_R^{+,+,-,+})^c)\Eqa(2)$$ In order to
estimate the above probability, we first observe that the event
$({\cal A}_R^{+,+,-,+})^c$ is a decreasing event (in the sense that
its
characteristic function is a non increasing function of the
configuration). Therefore, if ${\hat R}$ is the new rectangle
$${\hat R}\,=\,\{\;x\in \Z;\,0\,\leq \,x_1\,\leq\,L_1,\quad -{L_2\over
16}\,\leq\,x_2\,\leq\,L_2\;\}$$ $\t$ is the configuration:
$$\eqalign{\t (x)\,&=\,+1\quad \forall \,x\in
\partial_{ext}{\hat R}\quad\hbox{with }x_2\,\leq\,L_2-{L_2\over 16}\cr \t
(x)\,&=\,-1\quad \hbox{otherwise}}$$ and $$\eqalign{
U^{\partial {\hat R}}(x,y)\,&=\,\delta \qquad \forall \,(x,y)\in \partial
{\hat R}\quad \hbox{with }x_1\,=\,0\;\hbox{or }L_1,\;\hbox{and
}\,0\,\leq\,x_2\,\leq\,L_2-{L_2\over 16}\cr U^{\partial
{\hat R}}(x,y)\,&=\,1\qquad \hbox{otherwise}}$$ then $$\mu_R^{+,\delta
+,-,\delta +}(({\cal A}_R^{+,+,-,+})^c)\,\leq\,
\mu_{{\hat R}}^{U^{\partial {\hat R}},\t}(({\cal
A}_R^{+,+,-,+})^c)\Eqa(3)$$
If we denote by $\G^\t_{{\hat R},open}(\s )$ the (unique) open contour
of $\s\in \O_{{\hat R}}$ under the $\t$ boundary conditions described
above, it is immediate to check that:
$$({\cal A}_R^{+,+,-,+})^c\, \subset \,({\cal
A}^\t_{{\hat R}})^c\,\equiv\,\{\s ;\;\G^\t_{{\hat R},open}(\s )\cap \{x\in
{\hat R};\;x_2<{13\over 16}L_2\}\neq \emptyset\}$$
so that
$$\mu_{{\hat R}}^{U^{\partial {\hat R}},\t}(({\cal A}_R^{+,+,-,+})^c)\,\leq\,
\mu_{{\hat R}}^{U^{\partial {\hat R}},\t}(({\cal
A}^\t_{{\hat R}})^c))\Eqa(3bis)$$
For simplicity in the sequel we will denote the measure $
\mu_R^{U^{\partial {\hat R}},\t}$ by $P$.\par
Let us now order the bonds in $\G_{{\hat R},open}^\t(\s )$ from left to right
and let us denote by $e_{k_1}$,
$e_{k_1+n}$, the smallest, respectively the largest, bond in
$\G_{{\hat R},open}^\t(\s )$ such that no site in the portion of the
exterior boundary of the left, respectively right, lateral side of ${\hat
R}$ where the
boundary coupling is $\d$ is separated by one of the bonds $e\in\G_{{\hat
R},open}^\t(\s )$
with $e\,\geq\,e_{k_1}$, respectively $e\,\leq\,e_{k_1+n}$.\medskip
We will denote by $\gamma\,=\,e_{k_1}\dots
e_{k_1+n}$ the portion of the open contour
$\G_{{\hat R},open}^\t(\s )$, $\s\in \O_{{\hat R}}$, between $e_{k_1}$
and $e_{k_1+n}$ and by ${\cal F}$ the set of them.\par
Notice
that, by construction, $\g$ is itself an open polygonal line and that the first,
respectively the last, bond in $\g$ separates at least one site in the internal
boundary of the left, respectively right vertical side of ${\hat R}$. Moreover,
if we
denote by $h_\g$ be the horizontal line in $\bf R^2$ containing the middle
point of
the first bond $e_1$ of $\g$, then $h_\g$ is at distance at least ${L_2\over
16}-1$
from the horizontal portion of the boundary of ${\hat R}$ . Let also $$d(\g
)\;=\;\hbox{dist}(\g,h_\g )$$ We now define the event $\cal C$ as: $${\cal
C}\;=\;\{\s\, ;\; d(\g )\;\leq\;{L_2\over 32}\}\Eqa(4)$$ Then we estimate
\equ(2) by
$$P(({\cal A}_{{\hat R}}^{\t})^c)\,\leq\, P({\cal C}^c)\,+\, P(({\cal
A}_{{\hat R}}^{+,+,-,+})^c\cap {\cal C})\Eqa(5)$$ {\bf Lemma A.1}\par {\it
Given $m\,>\,0$ there exists $\beta (\e,m)$ such that for all
$\beta\,\geq \,\beta (\e,m)$ $$ P({\cal C}^c)\,\leq\, \exp
(-mL_1^{2\e})$$} {\bf Proof}\par Given $\gamma$, the set
${\hat R}$ can be written as the disjoint union of three sets: $${\hat R}\,=\,
\D\g\cup R^+_\g\cup R^-_\g$$
where $\D\g$ has been defined in section 1 and $R^+_\g$, $R^-_\g$
lay, in a natural way, below and above $\g$ respectively.\par
Associated to the set $R^+_\g$ we consider the partition function
$Z(R^+_\g,U^{\partial {\hat R}},\t )$ with $\t$ boundary condition and
boundary coupling $U^{\partial {\hat R}}$ on $\partial_{ext}R^+_\g\cap
\partial_{ext}{\hat R}$ and $+$ boundary condition on
$\partial_{ext}R^+_\g\cap\D\g$; similarly for $Z(R^-_\g,U^{\partial
{\hat R}},\t )$.\par We can now write: $$P({\cal C}^c)\,=\,
{Z({\hat R},U^{\partial {\hat R}},+)\over Z({\hat R},U^{\partial {\hat R}},\t)}\sum_{\g
;d(\g )\geq {L_2\over 32}\atop \g \in {\cal F}}\exp (-2\beta \vert \g \vert )
{Z(R^+_\g,U^{\partial {\hat R}}, \t)Z(R^-_\g,U^{\partial {\hat R}},\t )\over
Z({\hat R},U^{\partial {\hat R}},+)}\Eqa(5.a)$$ Unfortunately we cannot yet
use the cluster expansion described in section 1 to simplify the
above ratio of partition functions since, although in
$Z(R^+_\g,U^{\partial {\hat R}},\t)$ the boundary condition is constantly
equal to $+1$ because of a) in the definition of $\g$, so that
$$Z(R^+_\g,U^{\partial {\hat R}},\t)\;=\;Z(R^+_\g,U^{\partial {\hat R}}, +)$$
in $Z(R^-_\g,U^{\partial {\hat R}},\t )$ the lateral boundary condition
may change sign. However a trivial and rough comparison shows that:
$$\exp (-4\beta \d L_2)\,\leq\,{Z(R^-_\g,U^{\partial {\hat R}},\t )\over
Z(R^-_\g,U^{\partial {\hat R}},- )}\,\leq\,\exp (+4\beta \d
L_2)\Eqa(5bis)$$ Therefore the r.h.s. of \equ(5.a) is bounded from
above by: $$\exp (+8\beta \d L_2){\sum_{\g ;d(\g )\geq {L_2\over 32}\atop \g
\in {\cal F}}
\exp
(-2\beta \vert \g \vert) {Z(R^+_\g,U^{\partial {\hat R}},
+)Z(R^-_\g,U^{\partial {\hat R}},- )\over Z({\hat R},U^{\partial
{\hat R}},+)}\over \sum_{\g\subset {\hat R}\atop \g \in {\cal F}} {Z(R^+_\g,U^{\partial
{\hat R}},
+)Z(R^-_\g,U^{\partial {\hat R}},- )\over Z({\hat R},U^{\partial
{\hat R}},+)}}\Eqa(6)$$ We observe at this point that each one of the
partition functions $$Z({\hat R},U^{\partial {\hat R}},+)\quad
Z(R^+_\g,U^{\partial {\hat R}}, +)\quad Z(R^-_\g,U^{\partial {\hat R}},- )$$
can be written as in (1.10), with weights that satisfy the condition
of proposition 1.1 with constant $\a\,=\,{1\over 2}$. Therefore,
following [DKS], we can apply proposition 1.1 to write:
$${Z(R^+_\g,U^{\partial {\hat R}}, +)Z(R^-_\g,U^{\partial {\hat R}},- )\over
Z(R,U^{\partial {\hat R}},+)}\,=\,\exp (-\sum_{\vbox{\eightpoint{
\hbox{$\L\subset {\hat R}$}\hbox{$\L\cap \D\g\neq
\emptyset$}}}}\Phi^{U^{\partial {\hat R}}, +}(\L ))\Eqa(7)$$
so that \equ(6)becomes:
$$\exp (16\beta \d L_2) {\sum_{\g ;d(\g )\geq
{L_2\over 32}\atop \g\in {\cal F}}\exp (-2\beta \vert \g\vert ) \exp
(-\sum_{\L\subset {\hat R}\atop \L\cap
\D\g\neq \emptyset}\Phi^{U^{\partial {\hat R}}, +}(\L ))\over \sum_{\g\subset
{\hat R}\atop \g\in {\cal F}} \exp
(-2\beta \vert \g\vert ) \exp (-\sum_{\L\subset {\hat R}\atop \L\cap \D\g\neq
\emptyset}\Phi^{U^{\partial {\hat R}}, +}(\L )) }\Eqa(9)$$
We can use at this point two
basic results in [DKS] (see the proposition and the theorem in
section 4.14 and section 4.16 respectively) to conclude that, since $ L_1\,\geq\,
L_2\,\geq\,L_1^{{1\over 2}+\e}$,
for any given
$m\,>\,0$, the ratio between the two sums in \equ(9) is smaller than
$$\exp (-m{L_2^2\over L_1})\,=\,\exp (-mL_1^{2\e})$$ provided that $\beta$
is large enough.\par\noindent
The lemma is proved.\bigskip
We now turn to the estimate of the second term in the r.h.s. of
\equ(5), $P((({\cal A}_{{\hat R}}^{\t})^c\cap {\cal C})$.\par Let $l\,=\,{L_2\over
16}$ (we are assuming for simplicity that $L_2\over 16$ is an
integer) and let, for $i\,=\,0\dots N\,=\,32$, $R_i$ be the
rectangle: $$R_i\,=\,\{\;x\in R;\,0\,\leq \,x_1\,\leq\,L_1,\quad
-{L_2\over 16}+i{l\over 2}\,\leq\,x_2\,\leq\,-{L_2\over
16}+(i+2){l\over 2}\;\}$$ Then we define $P_i$ as:
$$P_i\,=\,P(\{\g\,\subset\,R_i\}\cap {\cal C})$$ and we estimate from
above $P((({\cal A}_{{\hat R}}^{\t})^c\cap {\cal C})$ by: $$P((({\cal A}_{{\hat
R}}^{\t})^c\cap {\cal
C})\,\leq\,\sum_{i=1\dots N-5}P(\{\g\subset R_i\}\cap {\cal
C})\Eqa(9.1)$$ In \equ(9.1) we used the fact that, if the event
$(({\cal A}_{{\hat R}}^{\t})^c\cap {\cal C}$ occurs, then, by construction,
$\g$ is
entirely contained in some $R_i$ because of ${\cal C}$, with the
index $i\,\neq\, N,\,\dots N-4$ because of $(({\cal A}_{{\hat R}}^{\t})^c$
and $i\neq 0$ again because of $\cal C$.\par In
order to estimate each term in the r.h.s. of \equ(9.1) we proceed in a slightly
different way
depending whether $i=1$ or $i\,>\,1$ the reason being that in the case
$i=1$ the poygonal
line $\g$ is very close to the discontinuity point of the lateral boundary coupling.\par
Let us first consider the case $i\geq 2$. In this case we bound from above the
ratio:
${P_i\over P_{N-2}}$ uniformly in $i\,=\,2\dots N-5$. If we use the
representation \equ(5.a) for the probability of a given $\g$, we may
write: $${P_i\over P_{N-2}}\,=\,{\sum_{\g\subset R_i;\, d(\g )\leq
{L_2\over 32}}\exp (-2\beta \vert \g \vert ) Z(R^+_\g,U^{\partial
{\hat R}}, \t)Z(R^-_\g,U^{\partial {\hat R}},\t ) \over \sum_{\g\subset
R_{N-2};\, d(\g )\leq {L_2\over 32}}\exp (-2\beta \vert \g \vert )
Z(R^+_\g,U^{\partial {\hat R}}, \t)Z(R^-_\g,U^{\partial {\hat R}},\t
)}\Eqa(10)$$ Given $\g\subset R_i$, let $F_i(\g )$ be its image
under a vertical translation in $\bf R^2$ by an amount
$(N-2-i){l\over 2}$.\par Then clearly $F_i$ establish a bijection
between the $\g$ in $R_i$ and those in $R_{N-2}$, so that the r.h.s.
is estimated from above by: $$\sup_{\g\subset R_i;\, d(\g )\leq
{L_2\over 32}}{Z(R^+_\g,U^{\partial {\hat R}}, \t)Z(R^-_\g,U^{\partial
{\hat R}}, \t)\over Z(R^+_{F_i(\g )},U^{\partial {\hat R}}, \t) Z(R^-_{F_i(\g
)},U^{\partial {\hat R}}, \t)}\Eqa(11)$$ which we write as:
$$\sup_{\g\subset R_i;\, d(\g )\leq {L_2\over
32}}{Z(R^+_\g,U^{\partial {\hat R}}, +)Z(R^-_\g,U^{\partial {\hat R}}, -)\over
Z(R^+_{F_i(\g )},U^{\partial {\hat R}}, +) Z(R^-_{F_i(\g )},U^{\partial
{\hat R}}, -)}\, {Z(R^-_\g,U^{\partial {\hat R}}, \t)Z(R^-_{F_i(\g
)},U^{\partial {\hat R}}, -)\over Z(R^-_\g,U^{\partial {\hat R}},
-)Z(R^-_{F_i(\g )},U^{\partial {\hat R}}, \t)}\Eqa(12)$$ Let us consider
the first ratio $${Z(R^+_\g,U^{\partial {\hat R}}, +)Z(R^-_\g,U^{\partial
{\hat R}}, -)\over Z(R^+_{F_i(\g )},U^{\partial {\hat R}}, +) Z(R^-_{F_i(\g
)},U^{\partial {\hat R}}, -)}\Eqa(13)$$ If we divide numerator and
denominator by $Z({\hat R},U^{\partial {\hat R}},+)$ and we use \equ(7), we
get that \equ(13) is equal to: $$\exp (-\sum_{\vbox{\eightpoint{
\hbox{$\L\subset {\hat R}$}\hbox{$\L\cap \D\g\neq
\emptyset$}}}}\Phi^{U^{\partial {\hat R}}, +}(\L
)\,+\,\sum_{\vbox{\eightpoint{ \hbox{$\L'\subset {\hat R}$}\hbox{$\L'\cap
\D F_i(\g )\neq \emptyset$}}}}\Phi^{U^{\partial {\hat R}}, +}(\L'
))\Eqa(14)$$ Notice that, for any pair $\L\subset {\hat R}$, $\L'\subset
{\hat R}$ that intersect neither the horizontal part of $\partial
{\hat R}$ nor the lateral portion where $U^{\partial {\hat R}}\,=\,1$ and are
one the translated of the other: $$\L'\,=\,F_i(\L )$$ we have: $$
\Phi^{U^{\partial {\hat R}}, +}(\L' )\;=\;\Phi^{U^{\partial {\hat R}}, +}(\L
)\Eqa(15)$$ by the very definition of the coefficients
$\Phi^{U^{\partial {\hat R}}, +}(\L )$.\par Therefore the difference
between the two sums appearing in \equ(14) becomes simply
$$\sum^\g_{\L,\L'} \Phi^{U^{\partial {\hat R}}, +}(\L'
)\,-\,\Phi^{U^{\partial {\hat R}}, +}(\L )\Eqa(16)$$ where $
\sum^\g_{\L,\L'}$ is a shorthand notation for the sum over all pairs
$\L$, $\L'$ which intersect $\D\g$ and $\D F_i(\g )$ respectively,
and are such that one of the above two requirements is violated by $\L$
or $F_i( \L )$ and by $\L'$ or $F_i^{-1}( \L' )$, where
$F_i^{-1}$ is the inverse of $F_i$.\par Since $\g\subset R_i$,
$F_i(\g )\subset R_{N-2}$ and $d(\g )\,\leq \,{L_2\over 32}$, we can
bound from above \equ(16), uniformly in $i=2\dots N-5$, by:
$$\sum_{\L\cap R_i\neq
\emptyset\atop \L\cap \{\partial {\hat R}\setminus \partial
R_i\}\neq \emptyset }\vert \Phi^{U^{\partial {\hat R}}, +}(\L
)\vert \;+\; \sum_{\L'\cap R_{N-2}\neq \emptyset \atop \L'\cap \{ \partial
{\hat R}\setminus \partial R_{N-2}\}\neq \emptyset }\vert
\Phi^{U^{\partial {\hat R}}, +}(\L' )\vert \;\leq \;C \Eqa(17)$$ for a
suitable constant $C$ independent of ${\hat R}$.\par In \equ(17) we used
the exponential decay of $\Phi^{U^{\partial {\hat R}}, +}(\L )$ in the
"size" $d(\L )$ of $\L$, the fact that the distance between the
horizontal part of the boundaries of ${\hat R}$, $R_i$, $R_{N-2}$ is, by
construction and because $ d(\g )\leq {L_2\over 32}$, at least
$L_2\over 32$ and the fact that the boundary coupling $U^{\partial {\hat
R}}$ is equal to $\d$ on the lateral boundary
of $R_i$, $i=2\dots N-5$ by construction.\par Let us consider the second ratio
in \equ(12)
$${Z(R^-_\g,U^{\partial {\hat R}}, \t)Z(R^-_{F_i(\g )},U^{\partial {\hat R}},
-)\over Z(R^-_\g,U^{\partial {\hat R}}, -)Z(R^-_{F_i(\g )},U^{\partial
{\hat R}}, \t)}$$ Using Jensen inequality we obtain:
$$\eqalign{{Z(R^-_\g,U^{\partial {\hat R}}, \t)\over Z(R^-_\g,U^{\partial
{\hat R}}, -)}\;&\leq \; \exp (2\beta\delta\sum_{(x,y)\in \partial
R^-_\g;U^{\partial {\hat R}}(x,y)=\d}<\s (x)>^\t)\cr {Z(R^-_{F_i(\g
)},U^{\partial {\hat R}}, -)\over Z(R^-_{F_i(\g )},U^{\partial {\hat R}},
\t)}\;&\leq \; \exp (-2\beta\d\sum_{(x,y)\in \partial R^-_{F_i(\g
)};U^{\partial {\hat R}}(x,y)=\d}<\s (x)>^-)}\Eqa(18)$$ where $<\s
(x)>^\t$ is a shorthand notation for the average of the spin $\s
(x)$ in the Gibbs measure $\mu_{R^-_\g}^{U^{\partial {\hat R}},\t}$ and
similarly for $<\s (x)>^-$.\par A simple Peierls argument shows that
$$<\s (x)>^\t\;\leq\;-1\,+\,k\qquad \forall\;x\in \,\partial_{int}R^-_\g;\;U^{\partial
{\hat R}}(x,y)=\d $$ with $k\,\to\,0$ as $\beta\,\to \,
\infty$, so that, from \equ(18) we obtain that
$${Z(R^-_\g,U^{\partial {\hat R}}, \t)Z(R^-_{F_i(\g )},U^{\partial {\hat R}},
-)\over Z(R^-_\g,U^{\partial {\hat R}}, -)Z(R^-_{F_i(\g )},U^{\partial
{\hat R}}, \t)}\,\leq\,\exp (-\beta \d l\,+\,2\beta\d k L_2)\Eqa(19)$$
Finally, combining \equ(17) and \equ(19), we obtain that:
$$P_i\;\leq\;C\exp (-\beta \d l\,+\,2\beta\d k L_2)\quad \forall \;
i=2\dots N-5\Eqa(20)$$
Let us now treat the case $i=1$ by estimating from above the ratio $P_1\over
P_{N-1}$. We define the map $F_1$ to be simply the clockwise rotation of $\pi$
around the center of the rectangle ${\hat R}$ and we proceed as before. In
this case, by symmetry, the ratio \equ(13) with
$F_i(\g )$ replaced by $F_1(\g )$ is equal to
one and the rest of the argument does not change.\par
In conclusion, since $l$ is proportional to $L_2$ and $k$ is
very small for large $\beta$, we get that, for any $m\,>\,0$
$$P({\cal A}^c_{{\hat R}}\cap {\cal C})\,\leq \,\exp (-m L_1^{\e})\Eqa(21)$$
Thus, combining together \equ(3), \equ(3bis), Lemma A.1 and \equ(21) we get the
first part of the proposition. \medskip
The second part follows immediately by a standard Peierls argument.\par
The proposition is proved.\bigskip
{\bf Proof of Proposition 4.2}\medskip
Following the proof of proposition 4.1 let $\t$ be the configuration: $$\eqalign{\t
(x)\,&=\,+1\quad \forall \,x\in \partial_{ext}R\quad\hbox{with
}x_2\,\leq\,L_2-{L_2\over 16}\cr \t (x)\,&=\,-1\quad \hbox{otherwise}}$$ and
let $$\eqalign{ U^{\partial R}(x,y)\,&=\,\delta \qquad \forall \,(x,y)\in \partial
R\quad \hbox{with }\,0\,\leq\,x_2\,<\,L_2\cr U^{\partial
R}(x,y)\,&=\,1\qquad \hbox{otherwise}}$$
Let also $S$ be the cigar-shaped neighborhood of the segment of the
horizontal line at height $L_2-{L_2\over 16}$ and joining the two vertical
sides of $R$: $$S\,=\,\{(x_1,x_2)\in R;\;\vert x_2-(L_2-{L_2\over 16})\vert
\,\leq\,({x_1(L_1-x_1)\over L_1})^{{1+\e\over 2}}\}\Eqa(22)$$
Notice that, for large values of $L_1$, the region $S$ is at distance at least
$L_2\over 32$ from the upper horizontal side of $R$.\par
Then it is immediate to see
that $$\mu_R^{\delta +,\delta +,-,\delta +}(({\cal A}_R^{+,+,-,+}))\,\geq\,
\mu_{R}^{U^{\partial R},\t}({\cal S}_R^{\t})\Eqa(23)$$
where
$${\cal S}_R^{\t}\,=\,\{\s ;\;\G_{R,open}^{\t}\,\subset\,S\}$$
Let ${\cal F}_R$ be the set of all possible configurations of $\G_{R,open}^{\t}$.
As
in \equ(5.a) we write: $$\mu_{R}^{U^{\partial R},\t}({\cal
S}_R^{\t})\,=\,
{Z(R,U^{\partial R},-)\over Z(R,U^{\partial R},\t)}\sum_{\G\in {\cal
F}_R;\,\G\subset S }\exp (-2\beta \vert \G \vert ) {Z(R^+_\G,U^{\partial R},
\t)Z(R^-_\G,U^{\partial R},\t )\over Z(R,U^{\partial R},-)}\Eqa(24)$$
where $R^+_\G$ and $R^-_\G$ are defined as in the proof of proposition 4.1 .\par
The ratio ${Z(R,U^{\partial R},-)\over Z(R,U^{\partial R},\t)}$ is clearly
bounded from below by
$$\exp (-\delta (2L_2+L_1)\beta )\Eqa(24bis)$$
Notice that, since the polygonal line $\G$ is contained in the region $S$, the
boundary conditions in the partitions functions $Z(R^+_\G,U^{\partial R},
\t)$ and $Z(R^-_\G,U^{\partial R},
\t)$ are, by construction, $+$ and $-$
respectively. Therefore, using the representation (1.10), the ratio
in the second
factor in \equ(24) can be written as:
$${Z(R^+_\G,U^{\partial R},
\t)Z(R^-_\G,U^{\partial R},\t )\over Z(R,U^{\partial R},-)}\,=\,
\exp (-\sum_{\L\subset R\atop\L\cap \D\g\neq
\emptyset}\Phi^{U^{\partial R}, +}(\L ))\Eqa(25)$$
Using proposition 1.1 and the fact that the polygonal line $\G$ is contained in
the region $S$, it is easy to see that:
$$\sum_{\L\subset R\atop \L\cap \D\g\neq
\emptyset}\Phi^{U^{\partial R}, +}(\L )\,\leq\,
\sum_{\L\cap \D\g\neq
\emptyset}\Phi^{+}(\L ) \,+\,C_o$$
for a suitable constant $C_o$ independent of $L_1$.\par
Thus \equ(24) can be estimated from below by
$$\exp (-C_o\,-\,3\d \b L_1)\sum_{\G\in {\cal
F}_R;\,\G\subset S }\exp (-2\beta \vert \G \vert -\sum_{\L\subset
R\atop \L\cap \D\g\neq
\emptyset}\Phi^{ +}(\L ))\Eqa(26)$$
We use at this point the fundamental result of [DKS] (see section 4.16 ) which
says
that:
$$ \sum_{\G\in {\cal
F}_R;\,\G\subset S }\exp (-2\beta \vert \G \vert -\sum_{\L\subset
R\atop\L\cap \D\g\neq
\emptyset}\Phi^{U^{\partial R}, +}(\L ))\,\geq\,$$
$$ \exp (- \beta L_1\t_\beta- C(\log (L_1))^{max(6,{2\over \e})})\Eqa(27)$$
If we combine together \equ(23) \equ(24), \equ(24bis), \equ(26) and \equ(27) we
finally get the result.\bigskip
{\bf Proof of Proposition 4.3}\medskip
It is easy to show that the expression appearing in part a) is bounded from above
by:
$$\vert F_2\vert _\infty \sum_{x\in \partial_{ext}(\hbox{bottom side of }{Q_2})}
\mu_{Q_1}^{\d +,\d +,-,\d +}(\t (x)=1\vert {\cal A}_{Q_1}^{+,+,-,+})\,-\,
\mu_{R_{n+1}\cup Q_{n+1}}^{\d +,\d +,+,\d +}(\t (x)=1) \Eqa(28)$$
By monotonicity
$$\mu_{Q_1}^{\d +,\d +,-,\d +}(\t (x)=1\vert {\cal A}_{Q_1}^{+,+,-,+})\,\leq\,
\mu_{R_1}^{\d +,\d +,+,\d +}(\t (x)=1)$$
A standard Peierls argument shows that, for each $x\in \partial_{ext}
(\hbox{bottom side of }{Q_2})$
and any given positive $m$ $$0\,\leq \,\mu_{R_1}^{\d +,\d +,+,\d +}(\t (x)=1)\,-\,
\mu_{R_{n+1}\cup Q_{n+1}}^{\d +,\d +,+,\d +}(\t
(x)=1)\,\leq\, \exp (-m L^{{1\over 2}+\e})\Eqa(29)$$
provided that $\beta$ is large enough.\par
Clearly \equ(29) proves part a) since
$$\vert F_2\vert _\infty \,\leq\,2L^2$$
Part b) follows immediately from part a) and proposition 4.1 applied to the
rectangle $R_{n+1}\cup Q_{n+1}$. The proposition is proved.\pagina
%\input formato.tex
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\end
ENDBODY