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%% $$ x=1 \Eq(ciccio) $$
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%% Dentro \eqalignno invece di \Eq si usa \eq.
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%% \nproclaim Proposition[peppe].
%% If bla bla, then blu blu.
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%% {\it Proof.} It is easy to check that ...
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%% Per far riferimento ad un teorema definito nel futuro: \thf
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%% di cui si dispone il .aux, includere lo statement
%% \include{file}
%% e usare \eqf o \thf
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%% Se e' presente il comando \BOZZA, viene stampato sul margine
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% \expandafter\ifx\csname introduzione\endcsname\relax%
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% \bf{1. Introduction }
\footline={\rlap{\hbox{\copy200}\ $[\number\pageno]$}\hss\tenrm
\foglio\hss}
\vskip 2cm
\centerline{\bf RELAXATION PATTERNS FOR}
\centerline{\bf COMPETING METASTABLE STATES:}
\centerline{\bf A NUCLEATION AND GROWTH MODEL}
\vskip 1cm
\centerline{F.Manzo$^{(1)}$, E.Olivieri$^{(2)}$}
\vskip 1cm
{\it (1) Dipartimento di Matematica - II Universit\`a di Roma - Tor Vergata}\par
{\it Via della Ricerca Scientifica - 00133 ROMA - Italy}\par
E-mail: MANZO@MAT.UTOVRM.IT
\bigskip
{\it (2) Dipartimento di Matematica - II Universit\`a di Roma - Tor Vergata}\par
{\it Via della Ricerca Scientifica - 00133 ROMA - Italy}\par
E-mail: OLIVIERI@MAT.UTOVRM.IT
\vskip 1cm
{\ninerm \baselineskip=11pt
{\ninebf Abstract}
We study, at infinite volume and very low temperature,
the relaxation mechanisms
towards stable equilibrium in presence of two competing metastable states.
Following Dehghanpour and Schonmann we introduce a simplified nucleation-growth
irreversible
model as an approximation for the stochastic Blume-Capel model,
a ferromagnetic lattice system with spins taking three possible values:
$-1, 0, 1$.
Starting
from the less stable state $\minus $ (all minuses) we look at a local observable.
We find that, when crossing a special line in the space of the
parameters, there is a change in the mechanism of transition towards the stable
state $\plus$: we pass from a situation: \par\noindent
1) Where
the intermediate phase $\zero$ is really observable before the final transition
with a permanence in $\zero$ typically much longer than the first hitting time to
$\zero$; \par \noindent
to the situation: \par \noindent
2) Where $\zero$ is not observable since the typical permanence in $\zero$ is much shorter than the
first hitting time to $\zero$ and, moreover, large growing $0$-droplets are
almost full of $+1$ in their interior so that there are only relatively thin
layers of zeroes between $+1$ and $-1$.
}
\vskip 3cm
{\it Key words}: metastability, nucleation, Blume-Capel model
\medskip
{\it 1991 Mathematics Subject Classification.} Primary 60K35, 82A05
\vskip 1cm
\vfill
\eject
\expandafter\ifx\csname sezioniseparate\endcsname\relax%
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\numsec=1
\numfor=1\numtheo=1\pgn=1
{\bf Section 1. Introduction. }\par
\vskip 1cm
This note concerns the study, from the point of view of mathematical
physics, of some problems arising in the context of metastability and nucleation.
Metastability is a relevant phenomenon for thermodynamic systems close
to a first order phase transition (like e.g. a vapour near its condensation
point). Suppose that initially the system is in a pure thermodynamic phase
and switch some thermodynamic parameter (like the specific volume ) to a value
corresponding to the coexistence of two or more phases. In some particular
experimental situations the system, instead of undergoing a phase transition,
persists in a pure phase which is called {\it metastable}, in an apparent (wrong)
equilibrium until an external perturbation or a spontaneous fluctuation leads to
the formation and growth of suitable (in size and shape)
nuclei of the stable phase inside the metastable phase.
It is widely accepted that metastability cannot be included in the Gibbsian
approach to the equilibrium statistical mechanics but, rather, it is a genuine
dynamical phenomenon. Since a general theory of non-equilibrium statistical
mechanical phenomena is still lacking, a particularly important role is played
by microscopic models like Glauber dynamics (also called stochastic or kinetic
Ising models) .
We refer to [PL2] and
[OS3] for a general discussion of metastability from a rigorous point of
view. Here we only say that the heuristic description based on the competition
between bulk and surface free energies leads to the introduction of the
``critical nucleus": droplets of the stable phase inside the metastable phase
which are smaller than the critical nucleus tend to shrink whereas larger
droplets tend to indefinitely grow.
In [PL1] for the
first time the dynamical and statical aspects of metastability were coupled in a
rigorous way. The point of view of [PL1] can be called the one of
``evolution of ensembles"; metastability is mainly characterized, there, as a
slow evolution of the averages over the process towards the stable equilibrium
values. The study of the Glauber dynamics in the framework of this point of view
was carried out in [CCO].
In [CGOV] a new
``pathwise approach" to metastability has been introduced where the analysis of
the behaviour of typical trajectories has a crucial importance. The approach of
[CGOV] turns out to be the starting point of many recent developments on the
rigorous theory of metastability.
In [MOS1], [MOS2], in the framework of the study of
relaxation to equilibrium of
infinite volume stochastic Ising models at low temperature and small magnetic
field, some aspects of the decay from the metastable situation were analysed in
terms of the formation of a suitable critical droplet.
A complete analysis of metastability for the kinetic Ising model in large
but finite volumes, with periodic boundary conditions, small but fixed magnetic
field, in the limit of vanishing temperature was carried out in [NSch1], [NSch2]
[Sch1]. The three-dimensional Ising model was studied in [N], [BC].
Different boundary conditions were studied in [CiL].
Different models, consisting in single spin flip markovian dynamics reversible
w.r.t. the Gibbs measure corresponding to different hamiltonians,
in the same asymptotic regime, were
studied in [KO1], [KO2], [CiO], [NO].
In particular, as we will see, the paper in [CiO]
refers to Blume--Capel model and is at the basis of the
present work.
The above described asymptotic regime can be called
``Freidlin-Wentzell regime", in analogy to the case of small random perturbations
of dynamical systems (see [FW]). It is mainly relevant for the description of the
local aspects of nucleation. As it has been pointed out in [Sch2], [Sch3],
[SchSh], a much more interesting asymptotic regime, from a physical point of view,
is the one corresponding, for the stochastic Ising model, to infinite volume, low
but fixed temperature in the limit of vanishing magnetic field.
It follows from the analysis developed by Schonmann in [Sch2] and [Sch3] that the
prescriptions on metastable behaviour introduced in [CGOV] and in particular
the asymptotic unpredictability of the transition are only suited for describing
nucleation in finite volumes. Indeed they apply to a regime of ``single
nucleation". In large enough or infinite volume we have the appearance and
coalescence of many nuclei. The mechanism of transition is definitely different
w.r.t. the one taking place in finite volume. In infinite volume the transition from the
metastable to the stable situation, say, near the origin, is driven by the
formation and subsequent long growth of supercritical droplets
sufficiently far away from the origin rather than a nucleation close to it without
a significant growth. In order for this more efficient mechanism
to take place the system must have enough room at its disposal.
In conclusion, we can say that finite and infinite volume metastable behaviors
involve physically distinct phenomena, and also for the models whose finite
volume metastability is known, the specific infinite volume features deserve
investigation.
\bigskip
An interesting case is the one of a kinetic Ising system in
infinite volume, small but
fixed (say positive) magnetic field in the limit of vanishing temperature. It has
been studied in [DSch1], [DSch2].
Let, for simplicity, the
dimension of the lattice be $d=2$ and suppose to start from all minuses.
The following
heuristics is at the basis of the
determination of the relaxation time $t_{rel}$, namely the typical time needed
by the spin at the origin to become plus (see [DSch1], [DSch2]):
At
very low temperature droplets of pluses with squared shape play an overwhelming
role. The growth of a squared droplet is ruled by two distinct mechanisms:
\par\noindent
1) adding a unit square protuberance (with a plus spin)
from the exterior to a
face at the rate $e^{-\b\g}$ ($\b$ being the inverse temperature and $\g$ being the
energy formation of this protuberance, namely the energy needed to flip a minus
spin with one nearest neighbor plus);
\par\noindent
2) filling up, at rate 1, of the exterior layer
to which the protuberance belongs, starting from the sites nearest neighbor to
the protuberance; the rate of this process is one since the energy needed to flip a
minus spin with two positive nearest neighbors is non-positive.
Suppose that $\G$ is the energy
formation of a critical droplet so that $e^{-\b \G}$ is the probability per unit
volume and per unit time of the formation of this critical droplet. Let $v$ be the
speed of growth of the supercritical droplets. As we will see it is possible to
show that $ v = e^{-{\b {\g\over 2}} }$.
The relaxation time $t_{rel}$ can be deduced by the following argument:
consider the space-time cone with height $t_{rel}$ and
basis given by the square centered at the origin of side $vt_{rel}$
and time coordinate $0$.
The order of magnitude of $t_{rel}$ can be deduced by:
$$
e^{-\b\G}(v t_{rel})^2t_{rel}\cong 1. \Eq (1.1)
$$
One gets:
$$
t_{rel}\cong e^{\b \frac{(\G +\g)}{3}} \Eq (1.2)
$$
Notice that a similar argument applied to the displacement of a (one-dimensional)
side yelds to the relation $ v \cong e^{-{\b {\g\over 2}} }$. Indeed now the
dimension is $d=1$ and the speed of growth of a protuberance (to complete
a new layer) is of order one and, finally, $\G$ must be replaced by the energy
formation of a unit square protuberance, namely $\g$. The factor ${1\over 3}$ in
the exponent in the r.h.s. of \equ (1.2) which for general dimension $d$ is $1/(d+1)$,
drastically reduces the lifetime w.r.t. the finite volume situation.
See [DSch] for more details.
\bigskip
The present paper concerns the study of metastability and nucleation in the
framework of a simplified dynamical Blume-Capel model.
Blume-Capel model describes a spin system on $\Z^d$ where the single spin
variable can take three possible values: $-1,0,+1$. It was
originally introduced to study the $He^3- He^4 $ phase transition.
One can think of it as a system of particles with spin. The value $\s_x = 0$
of the spin at the lattice site $x$ will
correspond to absence of particles (a ``vacancy"), whereas the values
$\s_x = +1, -1$
will correspond to the presence, at $x$, of a particle with spin $+1, -1$,
respectively.
The formal hamiltonian is given by:
$$
H(\sigma)=J\sum_{}(\sigma_{x}-\sigma_{y})^{2}-\lambda\sum_{x}
\sigma_{x}^{2}-h\sum_{x}\sigma_{x}\;\; ,
\Eq (1.3)
$$
where $\lambda$ and $h$ are two real parameters, having the meaning of the chemical
potential and the external magnetic field, respectively; $J$ is a real positive
constant (ferromagnetic interaction)
and $$ denotes a generic pair of nearest neighbour sites in $\Zdue$.
Let $ \menouno,\;\zero\;$ and $
\;\piuuno $ denote the configurations with all spins equal to$
\;-1,0,+1$, respectively. It is easy to see that for small enough $\l$ and $h$
these three configurations are local minima for the energy
In [CiO]
metastability and nucleation were studied for a two-dimensional Blume-Capel model
in the above mentioned Freidlin-Wentzell asymptotic regime:
system enclosed in a finite torus $\L$, with given $\l,h$, in the limit of zero
temperature. The interesting region of parameters that was analysed in [CiO] is
$$0 < |\l| < h \Eq (1.4)$$
with sufficiently small $h$ .
In the whole region \equ (1.4) we have (in any finite volume) :
$$
H(\menouno) > H(\zero) > H(\piuuno)
$$
In other words $\menouno$ and $\zero$ are metastable whereas $\piuuno$ is stable.
The main question that can be risen is, when starting from the highest
metastable configuration $\menouno$, whether or not the system reaches the
intermediate metastable configuration $\zero$ before relaxing to the
stable configuration $\piuuno$.
In finite volume the question is perfectly well posed . In [CiO] an answer was
found in the Freidlin-Wentzell asymptotic regime. It has been shown, in [CiO],
that there is a change in the mechanism of transition from $\menouno$ to
$\piuuno$ when crossing the line $h=2\l$. On the right side of this line ($h < 2
\l$) the transition is {\it direct} i.e. the system is driven to $\piuuno$ via the
formation of a special critical droplet: a squared ``picture frame" with
suitably large size, made by a square of pluses encircled by a unit layer of
zeroes in a sea of minuses. It is easy to see that a direct interface between
minus and plus is unstable in our region of parameters.
This explains the persistence
of a thin layer of zeroes between the pluses and the minuses.
In the other region ( $h > 2\l$) the transition is {\it indirect} in the sense
that the system first reaches the $\zero$ configuration via the formation of a
supercritical squared droplet of zeroes in a sea of minuses. Subsequently, via the
formation of a supercritical squared droplet of pluses in a sea of zeroes, the
system is driven to the final stable state $\piuuno$. In this region of parameters
$h >2\l$ the two transitions are ``Ising-like" whereas in the previous region $h <
2\l$ the mechsnism of transition and in particular the associated interface
dynamics are much more complicated.
As we argued before,
it is natural to pose the problem of the behavior of the kinetic two-dimensional
Blume-Capel model in infinite volume and, in particular, in the
Dehghanpour-Schonmann regime ($\beta \to \infty$ for small but fixed $\l$ and $h$
in infinite volume). In particular one asks himself whether the sort of
``dynamical phase transition" that has been detected in finite volume persists
and in which form in infinite volume. One
easily realizes that it is reasonable to expect a change in the mechanism of
transition over the line $\l = 0$. Indeed passing from $\l<0$ to $\l >0$ the
local energy barrier between $\menouno$ and $\zero$ becomes higher than the local
energy barrier between $\zero$ and $\piuuno$.
Moreover the speed of growth of a droplet of pluses in a
sea of zeroes becomes larger than the speed of growth of a droplet of zeroes in a
sea of minuses. In other words one expects that, starting from $\menouno$ and
looking at an observable localized close to the origin, for $\l<0$ one first sees
a large droplet of zeroes coming from a large distance and after a much larger
time one observes the arrival of a large droplet of pluses; on the contrary, for
$\l>0$ the time of the first arrival of the zeroes near the origin is much longer
than the time interval needed for the subsequent arrival of the pluses.
One expects
that, when a large droplet $\D$ of zeroes in a sea of
minuses is formed, quite soon many nuclei of pluses inside $\D$ are
formed; moreover,
due to the difference in speed, these pluses are likely to grow
much faster than the zeroes grow inside the sea of minuses so
that we expect that they rapidly almost reach the boundary of $\D$,
leaving only a thin layer of zeroes.
The complete proof of these conjectures appears quite complicated; in the present
paper, following Dehghanpour and Schonmann, we introduce a simplified model which
is nothing but an adaptation of the Ising-like simplified model introduced in
[DSch1] to our Blume-Capel case. Only nucleation and growth of two species of
particles (representing the zeroes and the pluses, respectively) will be allowed.
Our dynamics is irreversible since the spins can olny grow: a zero cannot turn
into a minus and a plus, once created, cannot be destroyed.
To mimic the behavior
of Blume-Capel model we only allow nucleation of pluses in sites which are
surrounded by all nearest neighbors zero so that direct minus-plus interfaces are
never permitted. As in the Dehghanpour-Schonmann model we take for simplicity the
size of the critical droplets as 1.
In the framework of our simplified irreversible model,
using mainly results of references [DS], [KS],
we are able to prove the
above conjectures in a quite weak form, as it will appear clear from the precise
statements contained in the theorems.
We indeed prove (Theorems 1, 2 and 3) a change in the asymptotic behavior of the
ratio between the time of first appearance of a zero at the origin, denoted by
$\t_0$, and the time interval, denoted by $\t_{0+}$, between $\t_0$ and the first
appearance of a $+$ at the origin.
When $\t_{0+}<<\t_0$, we also give information on the shape
of large droplets:
we show, in Theorem 4, that large
droplets of zeroes tend to be invaded by pluses in their interior so that,
asymptotically, they become completely full of pluses with only a relatively thin
layer of minuses between the internal pluses and the sea of minuses. Theorem 5
deals with the ``pursuit" of the zeroes made by the internal pluses as seen from
the origin at time $\t_0$.
The rest of the paper is organized in the following way:
in Section 2 we will give definitions and precise results. In Section 3 we will
prove Theorems 1, 2 and 3; in Section 4 we will prove Theorems 4 and 5.
Section 5 contains a discussion of the results.
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\vskip 1cm
{\bf Section 2. Definitions and results. }
\bigskip
In what follows we use $\lfloor a \rfloor$
to denote
the integer part of
the real number $a$ (i.e. the largest integer smaller than or equal to $a$)
and by $\lceil a \rceil$ the smallest integer larger than or equal to $a$.
\bigskip
We construct our model using ``exponential clocks".
To each site $x \in \Z^2$, we associate six independent exponential
random variables $\zGo(x)$, $\zgo(x)$, $\zunozero(x)$, $\zGuno(x)$,
$\zguno(x)$, and $\zunouno(x)$,
of rates $\Gro$, $\gro - \Gro$, $1 - \gro$,
$\Gruno$, $\gruno - \Gruno$, and $1 - \gruno$ respectively.
The parameters $\G_i,\g_i$ have the meaning of activation energies of
critical droplets and protuberances respectively.
We will always assume
\par \bigskip \noindent
{\bf Hypothesis $H_0$:}
\bigskip
$$
\matrix{
\G_0 & > \;\g_0 \;>\;0\cr
\G_1 & > \;\g_1 \;>\; 0\cr
}
\Eq(H0)
$$
The meaning of \equ (H0) is that the formation rate of a protuberance is,
exponentially in $\b$, larger than that of a new critical droplet.
The choice of parameters corresponding to Blume--Capel model is made in the
following way (see [CiO]):\par
Let $\d':=\frac{2J}{h-\l}- \lfloor \frac{2J}{h-\l} \rfloor$ and
$\d'':=\frac{2J}{h+\l}- \lfloor \frac{2J}{h+\l} \rfloor$.
We have:
$$\G_{0}^{BC}:=\frac{4 J^2}{h-\lambda} +2J -h+\l +h(\d'-{\d'}^2),$$
$$\Guno^{BC}:=\frac{4J^2}{h+\lambda} +2J -h-\l +h(\d''-{\d''}^2),$$
$$\g_{0}^{BC}:=2J-(h-\lambda),\Eq (2.0)$$
$$\guno^{BC}:=2J-(h+\lambda).$$
\bigskip
Note that $\G^{BC}_{0} \magg 5 \g^{BC}_{0}$,
$\G^{BC}_{1} \magg 5 \g^{BC}_{1}$.
\bigskip
Supposing to start from the $- \underline 1$ configuration, we
recursively define the times
of first appearance of $0$ and $+$ spin at the site $x$
as follows:
let $V^{(i)}(x), i=1,2,$ be the first time when the minus site $x$ has
at least $i$ $0$-neighbors (all other neighbours being $-1$)
and let $U^{(i)}(x), i=1,2,$ be the first time when the zero site $x$ has
at least $i$ ($+1$)-neighbors and no ($-1$)-neighbors.
Then the transition time to $0$ of a minus site $x$ is
simply defined to be
$$V(x)=\min \{\zGo(x), V^{(1)}(x) + \zgo(x),
V^{(2)}(x) + \zunozero(x) \}; \Eq (2.1)$$
the transition time of a zero site to $+1$ is:
$$U(x)=\max_{y : |x-y|\minug 1} (V(y)) +
\min \{\zGuno(x), U^{(1)}(x) + \zguno(x),
U^{(2)}(x) + \zunouno(x) \}. \Eq (2.2)$$
\bigskip
We call {\it acceptable} a configuration $\eta \in \{-1,0,+1\}^{\Z^2}$
with no pairs of nearest neighbours $-1,+1 $. It is easily seen that, starting
from $\menouno$, with the above updating rule we only reach acceptable
configurations. Using \equ (2.1), \equ (2.2), we can define the times of first
appearance of zero and $+1$ starting from any acceptable configuration as well.
\par
Note that, by definition, the growth of a droplet of $0$ does not depend
on the possible $+1$ inside the droplet.
\bigskip
We denote by $\s^\eta_{t}$ the
process at time $t$ starting from the acceptable configuration $\eta$. Sometimes we
omit the initial condition $\eta$ from notation if $\eta=-\underline 1$.
Given the acceptable configuration $\eta \neq \menouno$,
we denote by $\tilde {\s}^\eta_{t}$ the process, starting from $\h$, in which
nucleation of isolated zeroes is not allowed, namely the
transition time to zero of a negative site $x$ is:
$$\tilde V(x)=\min \{V^{(1)}(x) + \zGo(x),V^{(1)}(x) + \zgo(x),
V^{(2)}(x) + \zunozero(x) \}
\Eq(tilde)$$
Note that we introduce the time $\zGo$ in \equ(tilde) in order
to have the right transition rates
and that this is not related with nucleation, i.e. with the appearance of an
isolated zero which is forbidden.
Finally, given a configuration $\eta$ free of minuses, $\eta \in \{0,1\}^\Zdue$, we
denote by $\bar {\s}^\eta_{t}$ the Ising-like process on $\{0,1\}^\Zdue$,
starting from $\h$, in which
nucleation of isolated pluses is not allowed, namely the transition time
to $+1$ of a zero site $x$ is:
$$\bar U(x)=
\min \{U^{(1)}(x) + \zGuno(x),U^{(1)}(x) + \zguno(x),
U^{(2)}(x) + \zunouno(x) \}.
\Eq (barra)$$
Beyond $\menouno$, we will consider two classes of auxiliary (acceptable) initial
conditions $\eta_0^y$, $\eta_1^y$ ($y \in \bZ ^2$) defined, respectively, as:
$$
\eta_0^y(x)=-1 \ \forall \;x \in \Zdue \backslash \{y\}, \
\eta_0(y)=0,
\Eq(2.2o)
$$
$$
\eta_1^y(x)=0 \ \forall \;x \in \Zdue \backslash \{y\}, \
\eta_1^y(y)=1.
\Eq(2.2oo)
$$
The configurations are naturally partially ordered as follows:
$$\eta \le \eta' \text{ if } \eta(x) \le \eta'(x) \text{ for all } x \in \Z^2.$$
Note that the processes
$\s^{\eta}_{t}$, $\s^{\eta '}_{t}$ are defined on the same
probability space (of exponential clocks), so by attractiveness, if $\eta \leq
\eta'$ are any two acceptable configurations, we have the following inequality
for arbitrary $t \magg 0$:
$$
\s^{\eta}_{t} \leq \s^{\eta'}_{t}.
\Eq(fkg1)
$$
Moreover:
$$
\tilde \s^{\eta}_{t} \leq \s^{\eta}_{t}
\Eq(fkg2)
$$
$$
\bar \s^{\eta}_{t} \leq \s^{\eta}_{t},
\Eq(fkg3)
$$
where in the last inequality $\eta$ is a configuration in
$\{0,1\}^\Zdue$.
Inequalities analogous to \equ(fkg1) also hold for the processes
$\tilde\s^{\eta}_{t}$ and $\bar\s^{\eta}_{t}$ .
For $x \in \Z^2$ we set $||x||_\infty := \max_{j=1,2} |x_j|$. Given an integer $i$
and $x \in \Z^2$, let $$Q_i(x) = \{y \in \Z^2: ||x-y||_\infty \leq i\}$$
be the squared box of side--length $2i+1$ centered at $x$ and $Q_i:=Q_i(0)$.
For the process with initial condition $\minus$, let
$$
\t_0:=V(0)
$$
$$
\t_1:=U(0)
$$
$$
\t_{0+}:=\t_1-\t_0.
$$
We also set
$$
k_0:=\max \left\{\g_0,\frac{\G_{0}+\g_{0}}{3} \right\}.
\Eq(k0)
$$
$$
k_{1}:=\max \left\{\g_{1},\frac{\G_{1}+\g_{1}}{3} \right\}.
\Eq(k1)
$$
As it has been
shown by [DSch1] these are the logarithms, divided by $\b$, of the typical
occupation times of the origin by
the $0,+1$ spins starting from
$-\underline {1}$, $\underline {0}$, respectively.\par
We introduce, at this point, the quantity:
$$
k^+:=\max \left\{k_1, \frac{\G_1+\g_0}{3},\g_0 \right\} \Eq (2.2')
$$
We will show (see Proposition 1 below) that $e^{\b k^+}$ is an upper bound for
the typical value of $\t_{0+}$ in the limit of
large $\b$.\par
\bigskip
In the following Theorem DS, we give a reformulation, in the specific context
of our model, of the main result obtained by Dehghanpour and Schonmann for
their Ising-like nucleation and growth model.
This theorem concerns the asymptotic behaviour, for large $\b$, of the
first
appearance of a zero (plus) at the origin starting from all minuses (zeroes).
\bigskip
{\bf Theorem DS}
\bigskip
{\it
Let hypotesis $H_0$ be satisfied.
then
$$ \forall \; kk_0 \ \
\lim_{\b \to \infty}
\P \left( \s^{- \underline 1}_{e^{\b k}}(0)>-1 \right)=1
\Eq (>k0)
$$
$$ \forall \; kk_1 \ \
\lim_{\b \to \infty}
\P \left( \s^{\underline 0}_{e^{\b k}}(0) = 1 \right)=1
\Eq (>k1)
$$
}
\bigskip
{\bf Remark 1}:\par
{\it Notice that equation} \equ(0$ such that
$$
\lim_{\b \to \infty}
\P \left(\frac{\t_{0+}}{\t_0}\magg e^{\b C} \right) = 1
\Eq (th1)
$$
}
Theorems 2, 3, 4 and 5 refer to the other region $ k_0 > k_1$
possibly with some further restrictions.
\bigskip
{\bf Theorem 2}
\bigskip
{\it
Let hypothesis $H_0$ be satisfied.
If
$$
k_0 > k_1
$$
then $\forall \; k>k_0$
$$
\lim_{\b \to \infty}
\P \left( \s^{- \underline 1}_{e^{\b k}}(0)=1 \right)=1
\Eq (th2)
$$
}
Theorem 2, together with Theorem 1 and Remark 1, asserts that there is a sort of
``dynamical phase transition" for $k_0=k_1$.
Indeed
in the region $k_0k0) and equation \equ (k1)
$\forall \; k > k_1$
$$
\lim_{\b \to \infty}
\P \left( \s_{e^{\b k}}(0)=1 \right)=1;
\Eq (k0k1), for $k_1 < k$, it is $+1$.
On the contrary, in the region $k_0>k_1$, the zero phase
is ``not observable" in the sense that, as a consequence of \equ(0$, on a
time-scale $e^{\b k}$, with high probability for large $\b$, we do not see the value
zero for the spin at the origin:
\noindent $\forall \; k \neq k_0$
$$
\lim_{\b \to \infty}
\P \left( \s_{e^{\b k}}(0)=0 \right)=0;
\Eq (k1k_1 \cr
\G_0 & > \G_1 \cr
\G_0 & \magg 2\g_0 \cr
}
\right. ,
\Eq(param1)
$$
then $\exists \;C'>0$ such that
$$
\lim_{\b \to \infty}
\P \left(\frac{\t_{0+}}{\t_0}\minug e^{-\b C'} \right)=1
\Eq(th3)
$$
}
Theorem 1 and Theorem 3 imply (at least when $\G_0 > \G_1, \;\; \G_0 > 2 \g_0$) a
stronger statement about the above described dynamical phase transition. Indeed
they say that when going from
$k_0 < k_1$ to $k_0 > k_1$, not only $\t_{0+}$ passes from a value
typically larger than $\t_0$ to a value typically smaller than
$\t_0$,
but that this happens {\it exponentially in } $\b$.\par
Notice that if $h>-\l>0$, $\G^{BC}_0,\G^{BC}_1,\g^{BC}_0$ and $\g^{BC}_1$
(see \equ(2.0) )
fulfil the hypoteses of Theorem 1 whereas if $h>\l>0$
they fulfil hypotheses of Theorems 2 and 3.
In the following Theorem 4 we will consider the process $\tilde \s^{\eta_0^0}$
(see \equ (2.2o) ). Since nucleation of zeroes is not allowed, there is, in
$\tilde \s^{\eta_0^0}$, only one droplet of zeroes (possibly containing pluses in
its interior).
\bigskip
{\bf Theorem 4}
\bigskip
{\it
Let $\G_0>\G_1$, $\g_0>\g_1$, $\G_{0}
\magg 2 \g_{0}$, $\G_{1} \magg 2 \g_{1}$.
Let
$R^0_{ext}(t)$ be the smallest rectangle containing the unique droplet
of zeroes in $\tilde \s^{\eta_0^0}$
at time $t$.
Consider the process at time $t = e^{\b k}$ where $ k > k_1+\frac{\g_0}{2}$.
Then there is a rectangle $R^1_{int}(t)$ inside $R^0_{ext}(t)$ with
$\tilde\s^{\eta_0^0}_t(x)=1$
$\forall \; x \in R^1_{int}(t)$ and such that
$\forall \; r < \min \{\frac{\g_0}{2},k-k_1-\frac{\g_0}{2} \}$,
$$
\lim _{\b \to \infty} \P\left(
\frac{\text{diam} (R^0_{ext}(t))}{\text{diam} (R^1_{int}(t))} \minug
1+C'' e^{- \b r}
\right)\;=\;1
\Eq (2.10)
$$
where $C''$ is a suitable positive constant.
}
\par
Again the Hypotheses of Theorem 4 are fulfilled by Blume--Capel parameters
in the region $h>\l>0$.\par
The following Theorem 5 asserts that at time $\t_0$ the pluses are close
to the origin more than $e^{\b k^+}$.
This is by far an upper bound
(since we can argue, using heuristic arguments, that coalescence
between large droplets should not change too much the situation w.r.t.
the case of a single droplet) but still shows that
the first droplet that reaches the origin contains
a wide droplet of pluses in its interior.
\bigskip
{\bf Theorem 5}
\bigskip
{\it
Let
$\G_0>\G_1+\g_0$, $\g_0>\g_1$, $\G_{0} \magg 2 \g_{0}$, $\G_{1} \magg 2 \g_{1}$,
and $\forall \; \e>0$ let $\check L := e^{\b (k^+ +\e)}$; then
$$
\P \left(
\exists \; x \in Q_{\check L} :
\s_{\t_0}(x)=1
\right) \to 1 \text{ as } \b \to \infty
\Eq (2.11)
$$
}
The hypoteses of Theorem 5 are fulfilled by Blume--Capel parameters
in the region
$h>\l>0$
in the vicinities of the ``triple point" $h=\l=0$.
In this case, which is the most
interesting one, we have that $k^+$ is
strictly smaller than $k_0$. In this case Theorem 5 says that,
for sufficiently small $h$ and $\l$, when the first zero
hits the origin the pluses are at a much smaller distance than the
typical size of a zero droplet.\par
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\vskip 1cm
{\bf Section 3. Proof of Theorems 1, 2 and 3.}
\vskip 1cm
{\it \bf Proof of theorem 1.}
\bigskip
This is a corollary of Theorem DS:
By \equ(fkg1), we get
$$\t_1\geq \min \{t : \s^{\underline 0}_{t}(0)=1\}.
\Eq(3-3)$$
By Theorem DS (see \equ(>k0), \equ(0$,
$$
\P \left( \t_0 < e^{\b(k_0 +\e)} \right) \to 1
$$
while
$$
\P \left( \t_1 > e^{\b(k_1 -\e)} \right) \to 1
$$
Let us choose $\e < \frac{k_1-k_0}{2}$. We have:
$$
\P \left( \frac{\t_{0+}}{\t_0}>
{ e^{\b(k_1-\e)} - e^{\b(k_0+\e)}\over e^{\b(k_0+\e)}
}\right) \to 1 \;\;\hbox {\rm as } \;\;\b \; \to \; \infty
\Eq (2.11')
$$
This concludes the proof.
\QED
\bigskip
We will sometimes make use of super-exponential
estimates in $\b$ in the following way:
we tile a square having exponential sidelength $e^{\b M}$ with smaller squares
with exponential sidelength $e^{\b m}$.
If $p k(M)$ and sufficiently large $\b$:
$$\P(\t^o_{g}(L) > e^{\b k} )
\minug e^{-C_1
%\frac{1}{\b}
e^{\b (k-k(M))} }
\Eq (2.12)$$
}
{\it Proof:}
\bigskip
We set $$s:=e^{\b k} \Eq (2.13)$$
and
$$D:= \lceil e^{\beta \frac{\g_0}{2}} \rceil.$$
In order to estimate $\t^o_{g}$, we use a construction analogous
to the one introduced in [DSch1] in the proof of Theorem DS:
we fill the square one shell at a time, using a suitable number of
sites becoming zero at rate $e^ {-\b \g_0}$ and then
filling the other sites of the
shell at rate $1$. The main differences with the work of
Dehghanpour and Schonmann are that we consider
the filling of squares larger than $D$ and that
we obtain estimates decayng super-exponentially fast in $\b$.
Let $\partial Q_i:=Q_i \backslash Q_{i-1}$ and denote by ${\cal C}_i$ the
set of four sites at
the corners of $Q_i$. In order to estimate the time of growth of $Q_i$
we split $\partial Q_i$ into
suitable intervals $A^h_i$ (horizontal or vertical).
An efficient way to fill a shell $\partial Q_i $ larger than $D$
is to divide it into
intervals $A_i^h$ of length of order
$D$
and wait until in each of them a first site is occupied so that all other
sites can be subsequently occupied at rate $1$.
Let
$n_i:= 4\lceil \frac{2i-1}{D}\rceil$.
Starting from the upper left corner,
we split the upper side of $\partial Q_i$ into $n_i/4$
intervals $A_i^{h}$:
if $2i-1 > D$ we take the first $n_i/4-1$ intervals of length
$D$ and the last one of length
larger or equal to $D$ but
less than
$2 D+1$; we continue clockwise by sequentially partitioning, with the same criterion,
the other sides without the corners that we have previously considered.
With this construction, if $2i-1 \minug D$, then $n_i=4$ and
$A_i^h$ can be seen as the four sides of $Q_i$
(possibly without some of the corners in ${\cal C}_i$).
Let us set $\hat A_i^h:=A_i^h \backslash {\cal C}_i$.
Let $\Psi_L$
be the subset of $\bN ^L$ given by:
$$\Psi_L:= \left\{ \underline{h} = (h_1,\dots , h_L)\; :\; {h}_1
\minug n_1,...,
{h}_L \minug n_L \right\}$$
Let
$$
{\check \t}_i^{h}:= \min_{x \in \hat A_i^{h}}
\left\{ \zeta_0^{\gamma}(x) \right\}
\Eq (2.13')
$$
$$
{\hat \t}_i^{{h}}:=
({\check \t}_i^{{h}} +\sum_{y \in A^h_i} \zeta_0^{1}(y))
\Eq (2.14)
$$
using inequality \equ(fkg1),
we estimate from above $\t^o_{g}(L)$ as
$$
\hat \t^o_g:= \max_{\underline h \in \Psi_L}\left\{
\sum_{i=1}^{L}
{\hat \t}_i^{{h}_i} \right\}
\Eq (2.14')
$$
Note that in \equ(2.13') we use $\hat A^h_i$ instead $A^h_i$
because we need that every site that we consider has at least one zero
neighbour.
In what follows we will refer to the ``exponential Chebychev's inequality''
for a random variable $\xi$ as
$$\P(\xi >s) \minug e^{-\theta s} \E(e^{\theta \xi}) \ \forall \;\theta >0,
\ \ \ s \in \R$$
We split the proof into two cases:
\item {1)} case $M = \frac{\g_0}{2}$ (so that $L=D$ and $k(M) = \g_0$).
Consider $A_i^{h_i}$, $h_i=1,..,4$. We have:
$$
\P \left(
\sum_{i=1}^{D} \max_{{h}_i \minug 4} \left\{
{\check \t}_i^{{h}_i} +
\sum_{y \in A_i^{{h}_i}} \zeta_0^1(y)
\right\}
> s \right) \minug
$$
$$
\minug 4 \P \left(\sum_{i=1}^{D}
\min_{j \minug 2i+1}
\left\{\zeta_0^{\g}(x_j) \right\} >
\frac{s}{2} \right) +
\P \left( \sum_{j=1}^{(2 D+1)^2} \zeta^1_0 (x_j)
> \frac{s}{2} \right)\Eq (2.15)
$$
where in the r.h.s. of \equ(2.15) $\{\zeta^1_0 (x_j)\}_j$
as well as $\{\zeta^1_0 (x_j)\}_j$ are families of i.i.d. variables.
We estimate both terms of \equ(2.15) as large deviations. Using the fact that
the minimum among $N$ exponential variables of rate $\omega$ is an
exponential variable with rate $\omega N$ and using exponential Chebychev's
inequality
with $\theta={e^{-\b \g_0}}$, we obtain
$$
\hbox {\rm r.h.s. of \equ (2.15) } \minug
4 e^{- \theta \frac{s}{2}} \prod_{i=1}^{D}
\left (\frac{1}{1-\frac{\theta}{(2i+1)e^{-\b \g_0}}}\right )
+e^{-\theta \frac{s}{2}} \left( \frac{1}{1-\theta}
\right)^{(2 D+1)^2}
\minug
$$
$$
\minug
e^{-c_1 e^{-\b \g_0} s}
\left( c_2 e^{-\sum_{i=1}^{D} \log (1-\frac{1}{(2i+1)})}
+ c_3 e^{c_4 \b}
\right)
\minug
$$
$$
\minug
c_5 e^{-c_1 e^{-\b \g_0} s}
\left( e^{{c_6} \sum_{i=1}^{D} \frac{1}{2i+1}}
+ e^{c_4 \b} \right)
\minug
$$
$$
\minug c_5 e^{-c_7 e^{\b ( k-k(M) )} }
$$
where in the last inequality we used \equ (2.13) and
$\sum_1^D 1/(2i+1)\sim \hbox
{\rm const} \ \b$.
\item 2) case $M > \frac{\g_0}{2}$
(so that $k(M) >\g_0$).
Recalling \equ (2.14') we have:
$$\P \left(\t_{g}^o(L) > s \right) \leq
\P \left( \max_{\underline h \in \Psi_D} \left\{
\sum_{i=1}^D
{\hat \t}_i^{h_i} \right\}
> \frac{s}{2} \right)+
\P\left(\max_{\underline h \in \Psi_L}
\left\{ \sum_{i=D+1}^{L}
{\hat \t}_i^{h_i}\right\}
> \frac{s}{2}\right)
\Eq(3.9')
$$
The first term in the r.h.s. of \equ(3.9') can be bounded as in case 1).
To estimate the second one, we use exponential Chebychev's inequality with
$\theta = \frac{1}{2}e^{-\beta \frac{{\gamma_0}}{2}}$. We have, for
sufficiently large $\beta$:
$$
\P
\left( \sum_{i=D+1}^{L}
{\hat \t}_i^{{h}_i} > e^{\b k}
\right)=
\P \left( \sum_{i=D+1}^{L}
({\check \t}_{i}^{{h}_i} +\sum_{y \in A_i^{{h}_i}}\zeta_0^1(y))
> e^{\b k} \right) \leq
$$
$$
\leq e^{-c_8 e^{\beta (k-\frac{\g_0}{2})}}
\prod_{i=D+1}^{L}
\left \{ \E \left(
e^{\frac{1}{2}e^{-\beta \frac{{\gamma_0}}{2}}{\check \t}_i^{{h}_i}}
\right)
\left [ \E \left(
e^{\frac{1}{2}e^{-\beta \frac{\gamma_0}{2}}\zeta_0^1}
\right)\right ]^{\vert A_i^{{h}_i} \vert} \right \}=
$$
$$
= e^{-c_8 e^{\beta (k-\frac{\g_0}{2})}}
\prod_{i=D+1}^{L} \left\{
\frac{e^{-\beta \gamma_0} \vert A_i^{{h}_i} \vert}
{e^{-\beta \gamma_0} \vert A_i^{{h}_i} \vert -
\frac{1}{2}e^{-\beta \frac{{\gamma_0}}{2}}}
\left(\frac{1}{1- \frac{1}{2}e^{-\beta \frac{{\gamma_0}}{2}}} \right)^{2
D+1}
\right\}
\leq \Eq(prod)
$$
$$
\leq e^{-c_9 e^{\beta (k-\frac{\g_0}{2})}}.
$$
where, to get the last inequality we have bounded the generic term of the product
in \equ(prod) by a constant independent of $\b$ and used that
$k-\frac{\g_0}{2} > M$.\par
So we get
$$
\P \left( \max_{\underline h \in \Psi_L}
\left( \sum_{i=D+1}^{L}
{\hat \t}_i^{h_i} \right)
> \frac{s}{2} \right)
\leq
\left(
4 \prod_{i=1}^{L}\lceil \frac{2i-1}{D}\rceil
\right)
e^{-c_9 e^{\beta (k-\frac{\g_0}{2})}}
\leq
$$
$$
\leq
4 e^{-c_9 e^{\beta (k-\frac{\g_0}{2})}}
e^{e^{\b M} (\b M - \b \frac{\g_0}{2} + c_{10})}
\minug
4 e^{-c_{11} e^{\beta (k-\frac{\g_0}{2})}}
\minug
4 e^{-c_{11} e^{\beta (k-k(M))}}
$$
The Lemma is then proved.
\QED
\bigskip
{\bf Proof of Theorem 2}
\bigskip
Let us take $M = k_0 - {\g_0 \over 2}$ so that $k(M) = k_0$. We
have, on one hand, for $t=\frac{1}{2} e^{\b k}$ and $k > k_0$:
$$
\P \left( \lbrace
\s_{t}(x)= - 1 \ \forall \; x \in Q_{L_0}
\rbrace \right) \leq
\left( \P \left( \zeta_{o}^{\G}(0) \minug t
\right) \right)^{{L_0}^2}
=
e^{
-t
e^{
- \beta \Gamma_0
}
{L_0}^2
}
$$
$$
= e^{-{1 \over 2}({e^{\b k} e^{-\b \G_0} e ^{-\b (2 k_0 - \g_0 )}})}
\minug
e^{-{1 \over 2} e^{\b(k-k_0)} }.
\Eq (stimatauzerob)
$$
On the other hand, from Lemma 1 we have $\forall \; y \in Q_{L_0}$:
$$
\P \left ( \exists \; x \in Q_{L_0} : \tilde \s^{\h^y_0}_{t} (x) = -1 \right)
\leq e^ { - C_1 e^{\b (k - k_0) }}
\Eq (3.13')
$$
>From \equ (fkg1), \equ (fkg2), \equ (stimatauzerob), \equ (3.13') we have that
$\exists
\; \bar C_1 >0$ :
$$
\P \left ( \exists \; x \in Q_{L_0} : \s_{e^{\b k}} (x) = -1 \right) \leq e^
{ - \bar C_1 e^{\b (k - k_0) }}.
\Eq (3.13'')
$$
Now, if $L_1 = e^{\b( k_1-\frac{\g_1}{2})} > L_0$ we can use the
super-exponentially fast decay in $\b$ of the above estimate by tiling $Q_{L_1}$
with the translates of $Q_{L_0}$; if $L_{max}$ is the maximum between $L_0$ and
$L_1$ we have that the probability that
$Q_{L_{max}}$ contains minuses at time $t$ goes to $0$ as $\b \to \infty$.
We can repeat the above arguments based on inequalities
\equ (stimatauzerob), \equ (3.13') for the transition $ 0 \to 1$ to conclude the
proof.
\QED
\bigskip
Let, now $\g := \g_1 \vee \g_0$.
In order to prove Theorem 3,
we give an upper bound for $\t_{0+}$ in the following
way: starting from a single zero at the origin,
we consider a square of suitable side $L^+$ and
we wait until no one of the spins of $Q_{L^+}$ is minus.
Then we wait until a plus appears and finally
we wait until the droplet originated by this plus grows and fills up of
pluses the whole $Q_{L^+}$.
For any $L^+$, we have
$$\t_{0+} \minug \t^o_{g}(L^+) + \t^1_{b}(L^+) +\t^1_{g}(L^+),
\Eq(3.12')$$
where $\t^o_{g}(L^+)$
is defined by
\equ(tauzerog) while
$$
\t^1_{b}(L^+):=
\min\{t: \exists \;x \ \hbox {\rm such that} \ \s_t^{\underline{o}}(x)=+1\}
$$
is the time needed by a $+1$ to appear in $Q_{L^+}$ if $Q_{L^+}$
is initially full of zeroes, and
$$
\t^1_{g}(L^+):=\max_{y \in Q_{L^+}}
\min \{t; \bar\s_t^{\eta_1^y}(x) =+1\ \forall \; x \in Q_{L^+}\}
$$
is the maximum over $y$ of the time needed by the $+1$ droplet emerging from $y$
to invade the whole $Q_{L^+}$.
Since
$\t^0_{g}(L^+)$ and $\t^1_{g}(L^+)$ are typically smaller than
$\max\{ L^+ e^{\b\frac{\g}{2}},e^{\b \g}\}$
(see Lemma 1)
and
$\t^1_{b}(L^+)$ is typically smaller than
$ e^{\b \G_1}/(L^+)^2$ (see \equ(stimatauzerob)),
the minimum of $\t^o_{g}(L^+) +\t^1_{b}(L^+) +\t^1_{g}(L^+)$
is achieved when
$$
L^+ = e^{\b M^+} \text{ where } M^+ :=
\left\{ \matrix{
\frac{\G_1 - \frac{\g}{2}}{3} & \text{ if } \G_1 \magg 2 \g \cr
\g /2 & \text{ if } \G_1 < 2 \g \cr}
\right.
\Eq(L+)
$$
\bigskip
By \equ(L+) and \equ(2.2'), we have that:
$$
k^+ = M^+ +\frac{\g}{2}=
\left\{ \matrix{
\frac{\G_1 +\g}{3} & \text{ if } \G_1 \magg 2 \g \cr
\g & \text{ if } \G_1 < 2 \g \cr}
\right.
$$
On the basis of \equ(3.12'),
in the following proposition we will give an upper bound on $\t_{0+}$.
\bigskip
{\bf Proposition 1}
\bigskip
{\it
Let hypothesis $H_0$ be satisfied.
$\forall \; k > k^+ \ \ \exists \; C,C'>0:$
$$
\P \left( \sigma_{\t_0+e^{\b k}}(0) \neq +1 \right)
\minug C e^{-C' e^{\b (k-k^+)} }
$$
}
{\it Proof}
Using inequality \equ(fkg1)
$$
\P(\t_{0+} > e^{\b k}) \minug
\P(\t^o_{g}(L^+) > \frac{1}{3} e^{\b k}) +
\P(\t^1_{b}(L^+) > \frac{1}{3} e^{\b k}) +
\P(\t^1_{g}(L^+) > \frac{1}{3} e^{\b k})
$$
The first and the third terms are estimated by using Lemma 1 and its analogous
for the zero to plus transition, while the second one is estimated
in the same way as in
\equ(stimatauzerob)
\QED
\bigskip
{\bf Proof of Theorem 3}
We can conclude the proof of
Theorem 3 by observing that this is a corollary of Proposition 1 since
$k_0 > k^+$ if and only if the hypotheses of Theorem 3 are satisfied.\par
\QED
\vskip 1cm
\expandafter\ifx\csname sezioniseparate\endcsname\relax%
\input macro \fi
\numsec=4
\numfor=1\numtheo=1\pgn=1
{\bf Section 4. Proof of theorems 3 and 4.}
\bigskip
To prove Theorem 4 we
%will show that a droplet of zeroes larger than
%$e^{\b (k_1-\frac{\g_1}{2})}$
%and not interacting with any other droplet, contains, in its interior, a
%droplet of pluses
%so large that the ratio between the sides of the two droplets goes to 1
%exponentially fast in $\b$.
%
%We
will make use of the following
\bigskip
{\bf Lemma 2}
\bigskip
{\it
Let $R_{a,b}$ be a rectangle of sides
$\inte{e^{\b a}}$ and $\inte{e^{\b b}}$
with $a \magg b>0$, and let
us consider an initial configuration only containing zeroes and minuses,
such that there exists a
\noindent {*-connected} set
of zeroes
(in the sense of site percolation)
inside $R_{a,b}$ connecting both pairs of opposite sides of the rectangle.\par
Let
$$
\t_R := \min \{t : \s_t(x) \not = -1 \ \forall \; x \in R_{a,b}\}
$$
Then $\forall \; k>a$ there is a constant $C$ such that
$$
\P \left( \t_R > e^{\b k} \right) \minug e^{-C e^{\b k}}
$$
Proof
}
It is sufficient to consider the Ising-like system with only minuses and zeroes
obtained by neglecting the zero-plus transitions and freezing the spins outside
$R_{a,b}$ to be minus
(state space $\equiv \{ -1,0 \}^{R_{a,b}}$).
By hypothesis there exists a *-connected path
of zeroes that divides $R_{a,b}$ into (at most) four regions.
Let $SE$ be the South-Eastern of these regions.
The time to fill $SE$ is the same as the one needed to fill $SE$
starting from the configuration $\eta$ given by
$\{\eta(x)=0 {\text{ for }} x \in R_{a,b}\backslash SE ,\
\eta(x)=-1 {\text{ elsewhere}}\}$;
by \equ (fkg1), this is less
than the time needed to fill the whole $R_{a,b}$ starting from the configuration
in which the sides $N$ and $W$ are occupied by zeroes
while the rest of $R_{a,b}$ is full of minuses.
Using again \equ (fkg1) we start from the $NW$ corner and allow only
``diagonal growth":
\par\noindent
given $i\in \Z$, we consider sets of the form
$$
A_i:=\left\{ x : x_2=x_1+i \right\} \cap R_{a,b}
$$
Let $i'$ and $i''$, respectively, be the smallest and the largest $i$ such
that $A_i$ is non-empty (so that $i''=i'+\inte{e^{\b a}} +\inte{
e^{\b b}}-1$)
and define
$$
\xi_i:=\max_{x \in A_i} \z_0^1(x)
$$
if $A_{i-1}$ is full of zeroes, all sites of $A_i$ have two zero neighbours
and can be filled at rate $1$, so that
taking the sum over $i$ of $\xi_i$ corresponds to
fill $R_{a,b}$ by filling the $A_i$ consecutively.
Then
$$
\P \left( \t_R > e^{\b k} \right) \minug
\P \left( 4 \sum_{i=i'}^{i''} \xi_i > e^{\b k} \right) \minug
$$
$$
\minug
\left(e^{\b b} \right)^{2 e^{\b a}}
\P \left( \sum_{i=1}^{\inte{e^{\b a}} + \inte{e^{\b b}}} \z_0^1(x_i) >
{1 \over 4} e^{\b k} \right)
\minug
e^{-C e^{\b k}}
$$
where
we have used arguments similar to the ones used to prove Lemma 1
%to take the maximum out of the sum,
and the fact that $|A_i|\minug e^{\b b}$.
\QED
\bigskip
{\it \bf Proof of Theorem 4}
\bigskip
We will consider the process at times $\hat t$, $t'$, $t''$
defined as
$\hat t :=t- e^{\b (k-r)}$,
$t'=\hat{t} + c_1 e^{\b (k-\frac{\g_0}{2}+\e)}$,
$t'':=t'+c_2 e^{\b (k_1+\e)}=
\hat{t} + c_1 e^{\b (k-\frac{\g_0}{2}+\e)} +c_2 e^{\b (k_1+\e)}$
(notice that $\hat t0$ such that
$$\P (\text{diam} (R^0_{ext}(\hat t))
\magg c_3 e^{\b (k-\frac{\g_0}{2})})\to 1
$$
and
$$
\P \left(\text{\rm diam} (R^0_{ext}( t)) -
\text{\rm diam} (R^0_{ext}(\hat t)) \minug
c_4 e^{\b(k-\frac{\g_0}{2}-r)}\right) \to 1 $$
Using Lemma 2,
we can show that $\forall \;\e>0$, looking at the process
at time $t'$, we have
$$\P (\forall \;x \in R^0_{ext}(\hat t) : \s_{t'}(x) \not= -1) \to 1$$
We now tile $R^0_{ext}(\hat t)$
with squares of side $e^{\b (k_1-\frac{\g_1}{2})}$.
These tiles are less than $ e^{2\b k}$.
Using the same argument as in the proof of
Proposition 1, we can show that $ \exists c_5,c_6>0$ such that $\forall \;\e>0$,
looking at the process at time
$t''$, we have:
$$
\P \left( \exists \;x \in R^0_{ext}(\hat t) : \s_{t''}(x) \not= 1 \right) \minug
c_5 e^{2 \b k} e^{-c_6 e^{\b \e} } \to 0
$$
By choosing $\e=\frac{\hat{r}-r}{2}$
such that $t'' < t$, Theorem 4 is then proven.
\QED
\bigskip
{\it \bf Proof of theorem 5}
\bigskip
Let us consider the system at time $\t':=(\t_0 - e^{\b (k^+ +\frac{\e}{2})})\vee 0$.
Recalling that $\check L = e^{\b(k +\e)}$ let
$$\Xi:=\{x \in Q_{{\check L}} :\s_{\t'}(x)
\not=
-1\}$$
With high probability as $\b \to \infty$ we have:
since the speed of growth of a droplet of $0$ is at most one,
the droplet that reaches the origin is not coming from outside $Q_{{\check L}} $
and
since $\G_0 \magg \G_1 + \g_0$, it is not originated
inside $Q_{{\check L}} $
after $\t'$, so that
$\P\{\Xi \not= \emptyset \} \to 1$.
On the other hand, by using the same argument
as the one used in the proof of Theorem
3, we can show that if $x \in \Xi$ at time $\t'$, then
$$\P \left( \s_{\t'+e^{\b(k^+ +\frac{\e}{3})}}(x) =1 \right) \to 1 $$
that concludes the proof.
\QED
\bigskip
{\bf Remark.}
We can also prove that
calling $\hat \Xi := \bigcup_{x \in \Xi} Q_{L^+}(x) ,$
because of ferromagnetic inequality \equ(fkg1)
$$
\P \left(
\exists x \in \hat \Xi :
\s_{\t'+e^{\b(k^+ +\frac{\e}{3})}}(x) \not= 1
\right) \minug
$$
$$
\minug
c_1 e^{2\b k^+} e^{-c_2 e^{\b \frac{\e}{2}} }
\to 0
$$
We remark once more that this estimate works only if
$\k^+ \minug \k_0-\frac{\g_0}{2}$,
but this is the case in the neighbourhood of
the triple point of Blume-Capel model.
\vskip 1cm
\expandafter\ifx\csname sezioniseparate\endcsname\relax%
\input macro \fi
\numsec=5
\numfor=1\numtheo=1\pgn=1
{\bf Section 5. Discussion.}
\bigskip
Let us make some further remarks on the hypotheses of Theorems 3 and 5.
Blume-Capel parameters $\G^{BC}_0,\G^{BC}_1,\g^{BC}_0$ and $\g^{BC}_1$
either fulfil hypoteses of theorem 1 for $h>-\l>0$ or hypotheses of theorems 2, 3 ,4
for $h>\l>0$;
hypoteses of theorem 5 are fulfilled in the interesting case $h>\l>0$, $h \to 0$.
Indeed for Blume-Capel parameters
$\G^{BC}_0>\G^{BC}_1$ iff $\g^{BC}_0>\g^{BC}_1$.
Equation \equ(k0 2 \g_0$ of theorem 3 (see \equ(param1)) is only needed to have
a strict inequality $C'>0$ since otherwise the droplet of zeroes that
reaches the origin does not achieve its asymptotic speed of growth.
%(ricostruire particolari)
%We can ask ourselves on what happens if the second condition
%in \equ(param1)
%is violated and whether or not it is possible that
%$\t_{0+}$ is not far less than $\t_0$ though $\t_1 \sim \t_0$.
The physical meaning of the condition $\G_0>\G_1$
in \equ(param1) is that a single droplet of zeroes
becomes large enough to allow nucleation of the pluses
in its interior before $\t_0$.
Indeed the volume of the space-time droplet of zeroes at time
$\t_0$ is by definition (see \equ(1.1) and \equ(k0) ) the inverse of the
nucleation rate of the zeroes so that the probability of nucleation
of pluses inside the droplet of zeroes is bounded by
$e^{-\b[\G_1-\G_0]} \wedge 1$.
In the other case $\G_0<\G_1$, even if
the speed of growth of the pluses is very large, there is no way
for a plus droplet to reach the origin by a further time
$\t_{0+}$ typically smaller than $e^{\b k_0}$, unless a
large scale coalescence of the zeroes occurs connecting with the origin
a nucleus of pluses that is very far from it.
The only way for a plus to be originated in the cluster of zeroes containing the origin
at time $\t_0+e^{\b k^+}$ is that this cluster becomes large enough from the
coalescence of many droplets of typical size $e^{\b (k_0 +\frac{\g_0}{2})}$.
Our difficulties in taking into account the coalescence of the droplets of zeroes at time
$\t_0$ arise from the fact that we are exactly at the percolation threshold
in the sense of [AL]
and we cannot use the methods of Aizenman and Lebowitz as it was done in [DSch1].
Indeed even if we know from Theorem DS that for $kk_0$, coalescence happens on super-exponential scale, we
cannot say anything about the case $k=k_0$.
>From heuristic analysis we can argue that the probability that the cluster of zeroes
including the origin has a large enough volume to allow the nucleation of the pluses
is separated from zero and one and hence
the second of \equ(param1) is necessary for Theorem 3.
A similar problem we have in order to prove Theorem 5, where to give the right
distance from the origin of the pluses we would like to rule out the
possibility
that the origin becomes occupied because of a large-scale bootstrap
percolation phenomenon (see [AL] and [DSch1]).
\bigskip
Finally we remark that the estimate on $\t_{0+}$ given by Theorem 3 is far from being optimal.
In order to give the right asymptotics for $\t_{0+}$, it is necessary
to study the interface
between zeroes and minuses from a ``mesoscopic" point of view.
Indeed to deal with the pursuit of the pluses it is crucial to have this
interface regular on scale $e^{\b k_1}$ in order for the droplet of pluses
inside the droplet of zeroes to have enough room to
reach its asymptoptic speed of growth.
We can argue, on heuristic grounds, that this interface is regular on scale
$e^{\b \gmezzi}$.
This result would also improve the estimates of Theorems 4 and 5.
\bigskip
{\bf Aknowledgements}
\bigskip
We want to express thanks to Roberto Schonmann who made available his results with
P. Dehghanpour before publication.
This work has been partially
supported by the grant CHRX-CT93-0411 and CIPA-CT92-4016 of the Commission
at European Communities.
\vfill
\eject
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\end