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\centerline{\ttlfnt Approach to Equilibrium of Glauber Dynamics }
\centerline{\ttlfnt In the One Phase Region. II: The
General Case}\vskip 1cm
\author{F. Martinelli\ddag\hskip 0.5cm and\hskip 0.5cm E.
Olivieri\dag}
\address{\ddag Dipartimento di Matematica
Universit\`a "La Sapienza" Roma, Italy \hfill\break{\rm e-mail:
martin@mercurio.dm.unirm1.it}} \address{\dag Dipartimento di
Matematica Universit\`a "Tor
Vergata" Roma, Italy \hfill\break{\rm e-mail:
olivieri@mat.utovrm.it}}
\abstract{We develop a new
method, based on renormalization group ideas (block decimation
procedure), to prove, under an
assumption of strong mixing in a finite cube $\L_o$, a \LS for
the Gibbs state of a discrete spin system. As a consequence
we derive the hypercontractivity of the
Markov semigroup of the associated Glauber dynamics and the
exponential convergence to equilibrium in the uniform norm
in all volumes $\L$ "multiples" of the cube $\L_o$.} \vskip
1cm\noindent Work partially supported by grant SC1-CT91-0695
of the Commission of European Communities
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\vskip 1cm
{\bf Section 1.}\par\noindent
\centerline{\bf Preliminaries, Definitions and Results}
\bigskip
In this paper we analyze the problem of the approach to equilibrium
for a general, not necessarily ferromagnetic,
Glauber dynamics, i.e. a single spin flip stochastic dynamics
reversible with respect to the Gibbs measure of a classical discrete spin system
with finite range, translation invariant interaction. We prove
that, if the Gibbs measure satisfies a {\it Strong Mixing Condition}
on a large enough finite cube $\L_o$, then the Glauber dynamics
reaches the
equilibrium exponentially fast in time in the {\it uniform norm},
in any finite or infinite volume $\L$, provided that $\L$
is a "multiple" of the basic cube $\L_o$. Such a result has already
been proved in our previous papers [MO1], [MO2] in the so called
"attractive case" by ad hoc methods. Here we prove the result in
greater generality by proving a \LS for the Gibbs measure of the
system. We refer to [MO2] for a general introduction to the problem
of approach to equilibrium in the one phase region for Glauber
dynamics; in particular in [MO2] one finds a critical discussion of
the various {\it finite volume } mixing conditions for the Gibbs
state and of the role played by the shape of the volumes involved
when getting near to a line of first order phase transition. We
also refer the reader to the beautiful series of papers by
Zegarlinski [Z1], [Z2], [Z3] and Zegarlinski and Stroock
[SZ1], [SZ2], [SZ3] where the theory of the \LS for Gibbs
states was developed and its role in the proof of fast convergence to
equilibrium of general, not necessarily attractive, Glauber dynamics
was clarified. We suggest in particular the interested
reader to look at the nice review in [S].\bigskip In order to
precisely state our results and for reader convenience, we recall
here the model and the notation of the first paper of this
series.\bigskip $\S\;1$\hskip 1cm {\bf The Model}\par \par We will
consider lattice spin systems with finite single spin state space
$S$. We take for simplicity $S\, =\, \{-1,+1\}$ and we will denote by
$\s\,\equiv\,\s_\L$ an element of the configuration space
$\Omega_\Lambda \, = \,S^\Lambda $ in a subset $\Lambda \subset
\Z$. The symbol $\sigma_x $ will always denote the value of the spin
at the site $x\in\Lambda $ in the configuration $\sigma$.\par The
{\it energy} associated to a configuration $\sigma\in\Omega_\Lambda
$ when the boundary condition outside $\Lambda $ is $\tau\in\Omega_
{\Lambda ^c}$ is given by : $$H^\tau_\Lambda (\sigma) \,=\,
H_\Lambda (\sigma | \tau) \,=\, \sum _{X:X\cap\Lambda
\ne\emptyset}U_{X}\, \prod_{x\in X}(\t\s )_x \Eq(1.1)$$ where, in
general, $\t\s$ denotes the configuration : $$\eqalign{(\sigma
\tau)_x \,&=\, \sigma_x\;,\quad x\in\Lambda\cr (\sigma \tau)_x
\,&=\, \tau_x\;,\quad x\in\Lambda ^c} \Eq(1.2)$$ and the potential
$U \,=\, \{U_{X},\ X \subset \subset \Z\}$, where $X \subset
\subset \Z$ means that $X$ is a finite subset of $\Z$, satisfies
the following hypotheses: \bigskip\indent {\bf H1}.{\it \ \ Finite
range : $\exists \quad r > 0 : U_{X}\equiv 0$ if diam$X > r$ (we use
Euclidean distance).} \bigskip\indent {\bf H2}.{\it \ \ Translation
invariance $$\forall X \subset \subset \Z\quad\forall k\in \Z\quad
U_{X+k} \,=\, U_{X}$$ } Because of the hypothesis {\bf H1}, $
H^{\tau}_{\Lambda}(\sigma)$ depends only on $\tau_x$ for $x$ in
$\partial^+_r\Lambda$ : $$\partial^+_r \Lambda \,=\,
\{x\not\in\Lambda : \hbox{dist} (x, \Lambda )\leq r\} \Eq(1.3)$$
With the energy function $H_\L^\t(\s )$ we construct the usual {\it
Gibbs measure} in $\Lambda $ with b.c. $\tau\in\Omega_{\Lambda ^c}$
given by: $$\mu_\Lambda ^\tau(\sigma) \,=\,
{\hbox{exp}(H^\tau_\Lambda (\sigma))\over Z^\tau_\Lambda }
\Eq(1.4)$$ where the normalization factor, or {\it partition
function}, is given by
$$Z^\tau_\Lambda \,=\,\quad\sum_{\sigma\in\Omega_\Lambda }
\hbox{exp}(H^\tau_\Lambda (\sigma)) \Eq(1.5)$$
If there exists a unique limiting
Gibbs measure for $\Lambda \to \Z,$ independent on $\tau$, it will
be denoted by $\mu$ .\bigskip
{\bf Remark} Notice that, for future notation convenience,
we have included the usual $-\beta$ factor in the
Boltzman weight \equ(1.4) directly in the energy $H^\tau_\Lambda
(\sigma)$ \bigskip
Next we define the stochastic {\it jump} dynamics, given by
a continuons time Markov process on $\Omega \,=\, S^{ \Z}$
that will be studied in the sequel. Discrete time versions can also
be considered.\par
Given $\Lambda \subset \subset \Z$ let
$$D (\Lambda) \,=\, \{f: \Omega \to \ R :
f (\eta) \,=\, f (\sigma)
\ \hbox{if}\quad
\eta_x \,=\, \sigma_x \ \forall x \in \Lambda \} $$
be the set of {\it cylindrical functions} with support $\Lambda.$
The set
$$ D\,=\, \cup_{\Lambda} D (\Lambda)$$
is the set of cylindrical functions and by $C(\Omega )$ we denote the
set of all
continuous functions on $\Omega \,=\, \Pi_x S_x$
with respect to the product topology of discrete topologies
on $S$.
\medskip
The dynamics is defined by means of its {\it generator} $L$ which is given, for
$f \in D$, by:
$$L f (\sigma) \,=\, \sum_{x,a} c_x (\sigma, a)
\bigl ( f (\sigma^{x,a})-f (\sigma) \bigr )\Eq(1.6)$$
where $\sigma^{x,a}$ is the configuration obtained from $\sigma$
by setting the spin at $x$ equal to the value $a$ and the
non-negative
quantities $c_x (\sigma, a) $ are called ``jump rates''.\par
We will also consider the Markov process associated to the above
described
jump rates
in a {\it finite volume} $\Lambda$ with boundary conditions
$\tau$
outside $\Lambda.$ By this we mean the dynamics on
$\Omega_\Lambda$
generated by $L^\tau_\Lambda$ defined as before starting from the
jump rates
$$c^{\tau, \Lambda}_x (\sigma, a) \equiv c_x (\sigma \tau, a)$$
where, given $\tau \in \Omega_{\Lambda ^c}\hbox{ and } \; \sigma \in
\Omega_\Lambda \; ,\,\sigma \tau$
has been defined in \equ(1.2) .
\medskip
The general hypoteses on the jump rates, that we shall always
assume,
are the following ones.
\bigskip
{\bf H3.}\ Finite range $r$. This means that
$\eta (y) \,=\, \sigma
(y) \quad \forall\, x,y :\; |y - x| \leq r $ implies \indent \ \
$c_x (\sigma, a) \,=\, c_x (\eta , a)$
\bigskip
{\bf H4.}\ Translation invariance. That is if $\eta (y) \,=\, \sigma
(y+x) \quad \forall \,y\;$ then $\;c_x (\sigma, a) \,=\, c_x (\eta, a)$
\bigskip
{\bf H5.}\ Positivity and boundedness. There exist two positive
constants $k_1$, $k_2$ such that
$$0\,<\, k_1\,\leq \, \inf_{\sigma, x, a} \ c_x (\sigma, a )
\,\leq\,\sup_{\sigma, x, a} \ c_x (\sigma, a )\,\leq \,k_2 $$
\bigskip
\item{-}\ \ {\bf H6.}\ Reversibility with respect to
the Gibbs measure $\mu$
(in finite
or infinite volume):
$$\hbox{exp}(\sum_{ X \ni x}U_X(\prod_{y\in X}(\s )_y))c_x(\sigma
,a)\;=\;
\hbox{exp}(\sum_{X \ni x}U_X(\prod_{y\in X}(\s^{x,a} )_y))
c_x(\sigma^{x,a} ,\sigma_x)\quad \forall\,x\in \L\Eq(1.7)$$
A similar equation holds in finite volume
$\Lambda$ with boundary conditions $\tau$, provided that we
replace in \equ(1.7) $\sigma$ with the configuration $ \sigma
\tau$.\par \bigskip
It is immediate to check that, in finite volume,
reversibility implies that
the unique invariant measure of the dynamics coincides with the Gibbs
measure
$\mu^\tau_\Lambda$. This important fact holds also in infinite volume
provided that there exists a unique Gibbs measure in the
thermodynamic limit. \par
It is well known (see $[L]$) that under the above conditions
$L$ ($L^\tau_\Lambda$) generates
a unique positive selfadjoint contraction semigroup on the space
$L^2(\O ,d\mu )$ (
$L^2(\O_\L ,d\mu_\L^\t )$) that
will
be denoted by $T_t$ or $T_t^{\Lambda ,\tau}$ .\bigskip
$\S\;2$\hskip 1cm {\bf The \LS and Hypercontractivity of
$\bf T_t$}\par In order to introduce the
\LS (LSI) we have to define the differentiation operator on the
functions of the spin configurations. We set:
$$\partial_{\s_x}f(\s)\;=\;f(\s)\,-\, {1\over 2}[f(\s^{x,+1})\,+\,
f(\s^{x,-1})]\Eq(1.7bis)$$ where $\s^{x,+1}$ is the configuration
obtained from $\s$ by setting the spin at x equal to +1 and
similarly for $\s^{x,-1}$. Given a subset A of the lattice $\Z$ the
symbol $(\nabla_A{f})^2$ will be a shorthand notation for the
expression $\sum_{x\in A}(\partial_{\s_x}f(\s))^2$.\par Finally we
define the "standard" \LSC $c_s(\nu)$ for an arbitrary measure $\nu$
on $\O_\L$ as the smallest number $c$ such that for any non negative
function f : $\O_\L\, \to \, {\bf R}$ the following inequality
holds: $$\nu(f^2log(f))\;\leq\;c\nu ((\nabla_{\L}f)^2)\;+\;\nu(f^2)
log((\nu (f^2))^{1\over 2})\Eq(1.8)$$ where $\nu (f)$ denotes the
average of the function f with respect to the measure $\nu$.\par In
the sequel we will refer to \equ(1.8) as the "standard" \LS for
$\nu$. It is very important to observe that if we denote by $${\bf
{\cal E}}_\L^\t(f,f)\;=\;-\mu_\L^\t(f\,L_\L^\t\,f)$$ the Dirichlet
form associated to the generator $L_\L^\t$ and we take in \equ(1.8)
$\nu\;=\;\mu_\L^\t$, then the term:
$$\mu_\L^\t ((\nabla_{\L}f)^2)$$ satisfies the estimate:
$$(4\max_{x,a,\s}c^{\tau, \Lambda}_x
(\sigma, a))^{-1}\,{\bf {\cal E}} _\L^\t(f,f)\leq\mu_\L^\t
((\nabla_{\L}f)^2)\leq (4\min_{x,a,\s}c^{\tau, \Lambda}_x (\sigma,
a))^{-1}\,{\bf {\cal E}} _\L^\t(f,f)$$ Therefore,
if $\mu_\L^\t$ satisfies
the "standard" \LS \equ(1.8), with standard logarithmic Sobolev
constant $c_s(\mu_\L^\t )$, then it also satisfies the \LS for the
semigroup $T_t^{\Lambda ,\tau}$ relative to the measure
$\mu_\L^\t$:
$$\mu_\L^\t(f^2log(f))\;\leq\;c_{{\bf {\cal E}}}(\mu_\L^\t ){\bf {\cal E}}
_\L^\t(f,f)\;+\;\mu_\L^\t(f^2) log((\mu_\L^\t (f^2))^{1\over
2})\Eq(1.10)$$ with logarithmic Sobolev constant
$(4k_2)^{-1}c_s(\mu_\L^\t )\,\leq \,c_{{\bf {\cal E}}}(\mu_\L^\t
)\,\leq \, (4k_1)^{-1}c_s(\mu_\L^\t )$ because of H5. Thus the
"standard" \LS and the \LS for the Dirichlet form ${\bf {\cal E}}$
are equivalent and in the sequel, whenever confusion does not arise,
we will call both of them the \LS. \bigskip
{\bf Remark} From the
above discussion it also immediately follows that, if $c^{\tau,
\Lambda}_x (\sigma,a)$ and $\tilde c^{\tau, \Lambda}_x (\sigma,
a)$ are two different jump rates satisfying H3...H6, and if ${\bf
{\cal E}}$ and $\tilde {\bf {\cal E}}$ are the corresponding
Dirichet forms, then we have: $$c_{{\bf {\cal E}}}(\mu_\L^\t
)\,\leq\, \max_{x,a,\s}{c^{\tau, \Lambda}_x (\sigma,
a)\over \tilde c^{\tau,
\Lambda}_x (\sigma, a)} c_{\tilde {\bf {\cal E}}}(\mu_\L^\t )\Eq(1.11)$$
and
$$c_{{\bf {\cal E}}}(\mu_\L^\t )\,\geq\,
\min_{x,a,\s}{c^{\tau, \Lambda}_x (\sigma,
a)\over \tilde c^{\tau,
\Lambda}_x (\sigma, a)} c_{\tilde {\bf {\cal E}}}(\mu_\L^\t
)\Eq(1.12)$$
\par
{\bf Remark} In the case when the single spin space consists of N
elements with $N>2$, the
definition of the differentiation operator is no longer so clear.
One
possibility (see [SZ3]) is to set:
$$\partial_{\s_x}f(\s)\;=\;f(\s)\,-\, \media {f}_x\Eq(1.12bis)$$
where $\media {f}_x$ is the average with respect to the uniform
measure on $S$ of the function $f$, considered as a function of the
single spin $\s_x$. Another possibility is to order the elements
of $S$ as $s_1,...,s_N$ and to set:
$$\partial_{\s_x}f(\s)\;=\;{f(s_{i+1})\,-\,f(s_{i})\over 2}
\Eq(1.12tris)$$
if $\s_x\,=\,s_i$ with $s_{N+1}\,\equiv\,s_1$.\par
Both definitions are reasonable and equivalent in the sense that:
\item{i)} $\media {\partial_{\s_x}f}_x\;=\;0$
\item{ii)}there exists
a finite positive constant $k_o$ (in general depending on N) such
that : $$(k_o)^{-1}\,{\bf {\cal E}} _\L^\t(f,f)\leq\mu_\L^\t
((\nabla_{\L}f)^2)\leq k_o\,{\bf {\cal E}} _\L^\t(f,f)$$ where
the definition of the generator $L$ and of the Gibbs measure in
this more general setting is the obvious one.\par
Although, in some sense, the choice of the differentiation operator
reflects the choice of the dynamics for the single spin
at "infinite temperature" (a uniform sampling in the first case or
a symmetric random walk in the second case), because of ii) above,
one is free to choose whatever definition is more suited to the
methods of proof. In particular if, as we do, one wants to treat
$\partial_{\s_x}$ as much as possible as a continuous derivative,
then the second definition seems more suited.\par The above
"ambiguity" points out that the logarithmic Sobolev constant is not
intrinsically associated to the Gibbs measure but, rather, to the
pair ($\mu\,,\,\nabla$).\bigskip
As it is well known
since the basic work by L.Gross [G1] (see also [G2] ), the \LS for
the Gibbs state $\mu_\L^\t$ is strictly connected with the
hypercontractivity properties of the Markov semigroup $T_t^{\L,\t}$,
where: \bigskip \item{}{\it $T_t^{\L,\t}$ is {\it
hypercontractive} with respect to $\mu_\L^\t$ if there exists a
constant $c(\L,\t )$ such that $$\parallel
T_t^{\L,\t}(f)\parallel_{L^q(\mu_\L^\t )}\,\leq \,
\parallel f\parallel_{L^p(\mu_\L^\t )}\;\;\forall \,(p,q,t)\quad
\hbox{with } p\,\leq \, q\,\leq\, 1\,+\,(p-1)e^{2t\over c(\L,\t
)}\Eq(1.13)$$}
\par
More precisely, Gross' Theorem states that the constant
$c(\L,\t )$ in \equ(1.13) can be taken equal to the logarithmic
Sobolev constant $c_{\cal E}(\mu_\L^\t )$.\par
Besides its intrinsic interest, hypercontractivity of the Markov
semigroup $T_t^{\L,\t}$ or of $T_t$ is
a fundamental tool to transform, in the general case, exponential
convergence to equilibrium of the
Glauber dynamics in the $L^2(d\mu_\L^\t )$-sense into exponential
convergence to equilibrium in the $L^\infty$-sense. More exactly,
for an interaction $U_X$ satisfying the general hypotheses H1, H2, one
can easily prove the following theorem (see [SZ2] Lemma 2.9 and Lemma
1.8 there) :\bigskip {\bf Theorem 1.1}\par
{\it Let $\Gamma$ be a, finite or infinite, class of subsets of $\Z$
and suppose that there is a constant $c_o$ such that:
$$\sup_{\t, \L\in \Gamma}c_{\cal E}(\mu_\L^\t )\,\leq \,c_o$$
Then there exists a positive constant $m$ and for any cylindrical
(i.e. depending on finitely many spins) $f:\O_\Z\,\to \, R$ there
exists a finite constant $C_f$ such that:
$$\sup_{\t, \L\in \Gamma}\parallel P_t^{\t,\L}f\,
-\,\mu_\L^\t(f)\parallel_\infty\,\leq \,
C_f \hbox{exp}(-mt)$$}
Thus, in this way, the problem of proving exponentially fast approach
to equilibrium in the uniform norm for a Glauber dynamics is reduced
to proving a bound on the logarithmic Sobolev constant of the Gibbs
state.\par The real breakthrough in this direction was made few
years ago by Zegarlinski [Z1], [Z2], [Z3], who proved that, if the
Gibbs state satisfies a weak coupling condition similar to the old
uniqueness Dobrushin's condition (single site) [D1], then the
logarithmic Sobolev constant in a set $\L$ with boundary conditions
$\t$ is finite uniformly in $\L$ and $\t$. This result was then
extended and generalized by Stroock and Zegarlinski [SZ1],[SZ2]
,[SZ3] to spin systems with single spin space given either
by a compact Riemaniann manifold or by a finite discrete set. The
main result of the above mentioned works is the equivalence of the
existence of a finite logarithmic Sobolev constant and the
Dobrushin-Shlosman complete analyticity [DS1], [DS2] of the Gibbs
state $\mu$. Although the Stroock-Zegarlinski
result was a very important progress in the general problem
of relating the fast convergence to equilibrium of the dynamics to
"mixing" properties of the Gibbs measure, it cannot, in general, be
used concretely in order to establish results close to a line of a
first order phase transition since it requires good mixing
properties of the measure $\mu_\L^\t$ in volumes $\L$ of {\it
arbitrary shape}. As we have discussed in detail in the
first work of this series [MO2], if one is willing to produce
results really close to a line of a
first order phase transition, one should consider the Gibbs measure
only on "fat" volumes like cubes or parallelipipeds with large
enough shortest side. This is exactly the subject of
the present work.\medskip
$\S\;3$\hskip 1cm {\bf The Results}\par In
order to present the new results of the present work, we need to
recall the {\it finite volume} mixing condition that already played a
important role in the first paper of this series [MO2].\medskip We say
that a Gibbs measure $\mu_\Lambda $ on $\Omega_\Lambda $ satisfies a
{\it strong mixing} condition with constants $C, \gamma$ if for every
subset $\Delta \subset \Lambda $: $$ \sup_{\tau,\tau^{(y)} \in
\Omega _{\Lambda^c}} Var(\mu_{\Lambda , \Delta}^\tau\ \
\mu_{\Lambda , \Delta}^{\tau^{(y)}})\leq C e^{-\gamma
\hbox{dist}(\Delta, y)} \Eq(1.14)$$ where $\mu_{\Lambda ,
\Delta}^\tau$ denotes the {\it relativization} (or projection) of the
measure $\mu_\Lambda^\t$ on $\O_\D$, $Var$ is the variation distance
and $\tau^{(y)}_x = \tau_x$ for $ x\ne y$ and .\medskip We denote
this condition by $SM(\Lambda, C,\gamma)$\medskip In [MO2] a lower
bound on the gap in the spectrum of the generator $L_\L^\t$ was
derived (see Theorem 1.2 below) under the assumption
$SM(\Lambda_o, C,\gamma)$, where $\L_o$ is a cube of side $L_o$,
provided that, given $C$ and $\gamma$, $L_o$ is sufficiently
large.\par In what follows $\Gamma$ will denote the class of all
subsets of $\bf Z^d$ given by the union of translates of the cube
$\Lambda_o$ such that their vertices lay on the rescaled lattice
$L_o{\bf Z^d}$ . The constants $C$ and $\gamma$ appearing in
our mixing condition will be fixed once and for all. \bigskip {\bf
Theorem 1.2}\par {\it There exists a positive constant $\bar L$
depending only on the range of the interaction and on the dimension
d such that if {\it SM($L_o$,C,$\gamma$)} holds with $L_o\,\geq \,
\bar L$ then: \medskip \item{i)} there exists a positive constant
$m_o$ such that for any $\Lambda\,\in \,\Gamma$ and for any
function f in $L^2(d\mu_{\Lambda}^{\tau})$ : $$\vert\vert
T^{\Lambda ,\, \tau}_t(f)\, -\, \mu_{\Lambda}^{\tau}
(f)\vert\vert_{L^2(d\mu_{\Lambda}^{\tau})} \; \leq \;
\vert\vert\,f\,-\,\mu_{\Lambda}^{\tau} (f)\,\vert
\vert_{L^2(d\mu_{\Lambda}^{\tau})}\hbox{exp}(-m_ot)$$ \medskip
\item{ii)} There exist constants $C'$ and $\gamma '$ such that for
any $\Lambda\,\in \,\Gamma$, $SM(\Lambda, C',\gamma')$ holds}
\bigskip {\bf Corollary 1.1}\par {\it There exists a positive
constant $\bar L$ depending only on the range of the interaction and
on the dimension d such that if {\it SM($L_o$,C,$\gamma$)} holds
with $L_o\,\geq \, \bar L$ then there exists a positive constant
$m$ such that for any pair of cylindrical functions $f$ and $g$ with
supports $S_f$ and $S_g$ and for any $\Lambda\,\in \,\Gamma$,
with $S_f,\,S_g\, \subset \,\L$, one has:
$$\mu_\L^\t(f;g)\,\leq \,\vert f\vert _\infty\,
\vert g\vert _\infty\,\vert S_f\vert \,\vert S_g\vert _\infty
\hbox{exp}(-m\,
dist(S_f,S_g))$$}
where $\vert X\vert $ denotes the cardinality of the set $X\subset
\Z$.\par
Here we considerably strengthen part i) of Theorem 1.2 by proving
the following theorem:\bigskip
{\bf Theorem 1.3}\par {\it There exists a positive constant $\bar L$
depending only on the range of the interaction and on the dimension
d, such that if {\it SM($L_o$,C,$\gamma$)} holds with $L_o\,\geq \,
\bar L$ then there exists a positive constant $c_o$ such
that for any $\Lambda\,\in \,\Gamma$ and any boundary condition
$\t$ the logarithmic Sobolev constant $c_{{\cal E}_\L^\t}$ is bounded
by $c_o$.\par Moreover there exists a positive constant $m$
such that for any cylindrical function $f$ there exists a positive
constant $C_f$, depending only on $f$, such that:
$$\parallel P_t^{\t,\L}f\,
-\,\mu_\L^\t(f)\parallel_\infty\,\leq \,
C_f \hbox{exp}(-mt)$$ }\bigskip
The second part of the above theorem proves Theorem 4.2 of [MO2] in
the non attractive case.\medskip
$\S\;4$\hskip 1cm {\bf The Strategy of the Proofs}\par
We conclude this introduction by describing the ideas behind the
proof of Theorem 3.1 and by comparing them with the
Stroock-Zegarlinski's approach.\par
Our proof is divided into two distinct parts:
\item{1)} In this first part (see section 2) we show that any
Gibbs measure
$\nu$ on a set $\L$ which is the (finite or infinite) union of
certain "blocks" $\L_1\,\dots\,\L_j\dots$ (e.g. cubes of side $l$ or
single sites of the lattice $\Z$) has a logarithmic Sobolev constant
which is not larger than a suitable constant depending on the
maximum size of the blocks provided that the interaction (not
necessarily of finite range) {\it between} the blocks is very weak in
a suitable sense. A simple example of such a situation is
represented by a Gibbs state at high temperature, but the result is
more general since we do not assume that the interaction {\it
inside} each block is weak.\par
The result is a perturbative one since, as it is
well known [G1], if there is no interaction between the blocks then
the logarithmic Sobolev constant of $\nu$ is not larger than
$$\sup_{j,\t}c_s(\nu_{\L_j}^\t )$$
\item{2)} In the second part (see section 3) of our approach we
use renormalization group, in the form known as decimation (i.e.
integration over a certain subset of the variables $\s_x$), to show
that, in the assumption of the theorem, the Gibbs state $\mu_\L^\t$
after a finite ($\leq \,2^d$) number of decimations becomes a new
Gibbs measure exactly of the type discussed in the part 1). It is
then a relatively easy task to derive the boundedness of the
logarithmic Sobolev constant of $\mu_\L^\t$.\par As it is well known
since the work of Olivieri [O] and Olivieri and Picco [OP], the
mixing condition $SM(\L_o,C,\gamma)$ implies that if the decimation
is done over blocks of a sufficiently large size (see for instance
[EFS] for pathologies that may occur if the size is not large
enough) then it is possible to control, e.g. by a converging cluster
expansion, the effective potential of the renormalized measure and
to show that it satisfies the weak coupling condition needed in part
1). This is however more than what it is actually needed, since the
hypotheses of part 1 are fulfilled by the renormalized measure as
soon as the truncated two point correlation functions of the {\it
original} Gibbs measure $\mu_\L^\t$ decay exponentially fast. This
is exactly the content of corollary 1.1 above; therefore the method
can avoid the lengthy computations of the cluster expansion.\par
We want to notice at this point a difference in the role
played by the DLR structure of Gibbs measures in our approach and in
the one used by Zegarlinski and Stroock-Zegarlinski. For
simplicity let us consider the case when the Dobrushin's uniqueness
condition holds true. In [Z1],[Z2],[Z3] Zegarlinski uses in a
crucial way the following property of the Gibbs local
specification operator ${\bf E}_\L$ :
$$\lim_{n\to \infty}{\bf E}_{i_n},\dots ,{\bf E}_{i_1}f\;=\;
\mu (f)$$
where $\{i_k\in \Z\}_{k\in \bf N}$ is a suitable sequence going
infinitely many times through each site of the lattice and
$\mu$ is the unique infinite volume Gibbs state. A similar
property is used in [SZ3] in the case when the
Dobrushin-Shlosman complete analitycity condition holds true.\par
In the present paper, on the contrary, we use the following simple general
property valid for any probability measure $\nu$ on a finite space
$\Omega$: $$\nu (A_1\cap\dots\cap A_n)\;=\;\nu (A_1|A_2\cap\dots\cap
A_n) \nu (A_2|A_3\cap\dots\cap A_n)\dots\nu (A_{n-1}|A_n)\nu (A_n)$$
Clearly, if the measure $\nu$ is a Gibbs measure corresponding to a
given potential, then the DLR property enters in the explicit
computation of the conditionals probabilities $\nu
(A_i|A_{i+1}\cap\dots\cap A_n)$.\par After the completion of this
work we learned that also Lu and Yau, in their work on the
gap for the Kawasaki dynamics for the Ising model [LY], obtained, by
martingale techniques, a uniform bound on the Logarithmic Sobolev
Constant of $\mu_\L^\t$, where $\L$ is an arbitrary cube of the
lattice, under the assumption that $SM(\L,C,\gamma )$ holds for all
finite cubes $\L$. \vskip 2cm
%\input formato.tex
\numsec=2\numfor=1
{\bf Section 2}\bigskip
\centerline{LOGARITHMIC SOBOLEV INEQUALITY}
\centerline{FOR WEAKLY COUPLED GIBBS MEASURES}
\bigskip
In this section we prove two results that will play a
crucial role in the derivation of the logarithmic Sobolev
inequality (LSI) for Gibbs measures satisfying a finite volume mixing
condition. In order to present our results we need to precisely
define the setting of the problem and the notation that we will
adopt in the sequel. We warn the reader that in this
section we {\it do not assume} the finite range condition (H1);
in this situation we will denote the potential by $\F$
instead of $U$. \bigskip $\S\;1$\hskip 1cm {\bf The Setting of the
Problem}\par Let $\L$ be a finite subset of the lattice $\Z$ such
that $\L\,=\,\L_1\,\cup\,\L_2\,\cup\,,\dots ,\,\cup\,\L_N$ with
$\L_i\,\cap\,\L_j\;=\;0$ if $i\,\neq \,j$ and let $\O_{\L}$ be the
space of configurations $\O_{\L}\;=\;\{-1,+1\}^{\L}$. Let also
$\F_X\;\; ,X\, \subset\, \Z$, be a "potential" such that for any
"boundary" configuration $\h \,\in\,\O_{\Z\setminus \L}$ and any
configuration $\s \,\in\,\O_{\L}$ the following Hamiltonian is
finite:
$$H^\h(\s)\;=\;\sum_{X\cap\L\,\neq\,0}\F_X (\prod_{x\in X}(\s\h )_x
) \Eq(2.1)$$
If for convenience we denote by
$H^\h_{\L_i}(\s_{\L_i})$ the sum:
$$H^\h_{\L_i}(\s_{\L_i})\;=\;
\sum_{\{X\cap\L\,\neq\,0\;;\;
X\cap\L_i\,=
\,X\cap\L\}}\,\F_X (\prod_{x\in X}(\s\h )_x ) \Eq(2.2)$$ then the
total Hamiltonian can be written as:
$$H^\h(\s )\;=\;\sum_i\,H^\h_{\L_i}(\s_{\L_i})\;+\;
W^\h(\s ) \Eq(2.3)$$
where the term $W^\h(\s )$ represents now the interaction between the
sets $\L_i$ i=1...N .\par Given the Hamiltonian $H^\h(\s)$, we will
denote by $\m^\h$ the corresponding Gibbs measure.\par In
the sequel, together with the measure $\m^\h$, we will need also
other Gibbs measures that are obtained from $\m^\h$ by integrating
out one by one the variables $\s_{\L_j}$ j=1,2..(decimation
procedure). More precisely for any i=1...N we define a new Gibbs
measure $\m_{\geq i}^\h$ on the space $$\O_{\L}^{(\geq i)}\;=\;
\{-1,+1\}^{\L_i\,\cup\,\L_{i+1}\,\cup\,,\dots ,\,\cup\,\L_N}$$ as the
relativization to $\O_{\L}^{(\geq i)}$ of the measure $\m^\h$ :
$$\m_{\geq i}^\h(\s_{\L_i},\dots ,\s_{\L_N})\;=\;
\sum_{\s_{\L_1},\dots ,\s_{\L_{i-1}}}
\m^\h(\s_{\L_1}, \dots ,\s_{\L_N})\Eq(2.5)$$ Obviously
the measure $\m_{\geq i}^\h$
is also a Gibbs measure
with a new Hamiltonian:
$$\hat H^\h_{(\geq
i)}(\s_{\L_{i}},\dots ,\s_{\L_{N}})\;=\;
log(Z^{\h,\s_{\L_{i}},\dots
,\s_{\L_{N}}}_{\L_1,\dots ,\L_{i-1}})\Eq(2.5bis)$$ where
$Z^{\h,\s_{\L_{i}},\dots ,\s_{\L_{N}}}_{\L_1,..\L_{i-1}}$ is the
partition function in $\L_1\cup..\cup\L_{i-1}$ with boundary
conditions $\h,\s_{\L_{i}},\dots ,\s_{\L_N}$.\par Finally, we will
denote by $\n_i^{\h ,\t_{i+1},\dots ,\t_{N}}$ the measure on
$\O_{\L_i}$ obtained from $\m_{\geq i}^\h$ by conditioning to the
event that the spin configurations $\s_{\L_{i+1}},\dots ,\s_{\L_N}$ are
equal to the spin configurations $\t_{i+1},\dots ,\t_{N}$. We will
write $\n_i^{\h ,\t_{i+1},\dots ,\t_{N}}$ as : $$\n_i^{\h
,\t_{i+1},\dots ,\t_{N}}\;=\;
{\hbox{exp}(W^\h_{(i)}(\s_{\L_i};\t_{\L_{i+1}},\dots
,\t_{\L_N}))\over Z_i}\Eq(2.5tris)$$ where $Z_i\,\equiv
\,Z_i(\t_{\L_{i+1}},\dots ,\t_{\L_N},\h)$ is a normalization factor
and the "effective" interaction
$$W^\h_{(i)}(\s_{\L_i};\t_{\L_{i+1}},\dots
,\t_{\L_N})\,\equiv\,W^\h_{(i)}(\s )$$ is given by:
$$W^\h_{(i)}(\s_{\L_i};\t_{\L_{i+1}},\dots ,\t_{\L_N})\;=\; \hat
H^\h_{(\geq i)}(\s_{\L_i};\t_{\L_{i+1}},\dots ,\t_{\L_N})\;-\;\hat
H^\h_{(\geq
i)}(\tilde \t_{\L_{i}},\t_{\L_{i+1}}\dots ,\t_{\L_N})\Eq(2.5iv)$$
where $\tilde \t_{\L_{i}}$ is a given reference configuration in
$\L_i$ (e.g. all spins up).\par \bigskip $\S
\;2$\hskip 1cm {\bf Assumptions and Results}\par We are now in a
position to precisely state the hypotheses on the "potential" $\F_X$
that we need in order to prove the main results of this section.
\bigskip {\bf Assumptions} \item{\bf a)}
There exists a
positive constant $\e$ such that: $$\sup_{(\h ,i,N)}
\sum_{j=1}^{i-1}\sup_{x\in\L_i}\supnorm{\partial_{\s_x}
W_{(j)}^\h(\s)}\;\leq\;\e
\Eq(2.6)$$ and $$\sup_{(\h ,k,N)}
\sum_{j=k+1}^{N}\sup_{x\in\L_j}\supnorm{\partial_{\s_x}
W_{(k)}^{\h}(\s)}\;\leq\;\e
\Eq(2.6bis)$$ \item{\bf b)} There exists a positive constant $k_o$
such that: $$\sup_{(\h ,N)}
\sum_{j=1}^{N}\sup_{x\in\Z\setminus\L}\supnorm{\partial_{\h
(x)}W_{(j)}^\h(\s)}\;\leq\;k_o\Eq(2.7) $$ and
$$\sup_{(\h ,N)}\sup_k
\sum_{x\in
\Z\setminus\L}\supnorm{\partial_{\h
(x)}W_{(k)}^\h(\s)}\;\leq\;k_o\sup_i|\L_i| $$ \item{\bf c)} There
exists a positive constant $k_1$ such that for any N, any $\h$, any
i=1,\dots ,N, any f $\in\; L^2(\O_{\L_i},d\n_i^{\h
,\t_{i+1},\dots ,\t_{N}})$ and any value of the conditioning spins
$\t_{i+1},\dots ,\t_{}$ one has: $$\sum_{\s_{\L_i},\hat\s_{\L_i}}
\n_i^{\h ,\t_{i+1},\dots ,\t_{N}}(\s_{\L_i}) \n_i^{\h
,\t_{i+1},\dots ,\t_{N}}(\hat\s_{\L_i})
[f(\s_{\L_i})\,-\,f(\hat\s_{\L_i})]^2\;\leq$$ $$\leq k_1^2\n_i^{\h
,\t_{i+1},\dots ,\t_{N}}((\grad{i}{f})) \Eq(2.8)$$
{\bf Remark}\par
Assumptions a) and b) are clearly expressing some decay property of
the effective interaction of the Gibbs measures $\m_{\geq i}^\h$ .
The reason why in this section we {\it do not } require finite
range of the interaction is that after few steps of our decimation
procedure, even a Gibbs state corresponding to a finite range
interaction will be transformed into a new Gibbs measure
corresponding to an effective interaction with unbounded range at
least in some directions.\par Assumption c) looks somewhat more
mysterious but nevertheless plays an important role. In some sense
c) is a hypothesis of rapid approach to equilibrium for a heat bath
or Metropolis dynamics in $\L_i$, reversible with respect to the
Gibbs measure $\n_i^{\h ,\t_{i+1},\dots ,\t_{N}}$. Using the
arguments of section 1 $\S \,2$, the constant $k_1^2$ becomes in fact
proportional to the inverse of the gap of the generator of the
dynamics, i.e. $k_1^2$ is proportional to the relaxation time in the
"block" $\L_i$. In the perturbation argument given below, the
constant $\epsilon$, which expresses the weak coupling between the
blocks $\L_1\dots \L_N$, will always appear multiplied by the
costant $k_1^2$ and therefore the true "small" parameter of the
analysis becomes the "coupling among the blocks $\times$ the
relaxation time in a single block".\par We will
see in the next section that all the above assumptions follow from
the finite volume mixing condition $SM(C,\gamma ,L)$ defined in
section 1 provided that $L$ is large enough and that the set $\L$
consists of a union of sufficiently "fat" subsets of the lattice
$\Z$.\bigskip
Under the above assumptions
the following two theorems hold.\bigskip {\bf Theorem 2.1} \par
{\it Given the constant $k_1$, for
any $\d\,>\,0$ there exists $\e_o\,=\,\e_o(k_1,\d)$ such that if
$\e\sup_i|\L_i|\,\leq \,\e_o$ then:
$$\sup_\h c(\m^\h)\,\leq\,(1\,+\d)\,
\sup_i\sup_{\h,\t_{i+1},\dots ,\t_{N}}c(\n_i^{\h ,\t_{i+1},\dots ,\t_{N}})$$}
\bigskip
{\bf Theorem 2.2} \par
{\it Given the constants $k_o$ and $k_1$ there exists
$\e_o\,=\,\e_o(k_0,k_1)$ such that if $\e\sup_i|\L_i|\,\leq \,\e_o$ then
there exist two constants $k_2$, $k_3$ depending on $k_o$ and $k_1$
such that for any function f : $\O_\Z\,
\to \, R$ the following inequality holds:
$$(\nabla_{\Z\setminus\L}(\m^\h(f^2))^{1\over 2})^2\;
\leq\; k_2\m^\h((\nabla_{\Z\setminus\L}f)^2)\;+\;
k_3\sup_i|\L_i|\m^\h((\nabla_{\L}f)^2)$$}\bigskip
{\bf Remark } Notice that in theorem 2.2 the function f is a function
from $\O_\Z$ to $R$ and thus it may depend also on the boundary
spins $\h$. Therefore the expression $(\m^\h(f^2))^{1\over 2}$
may depend on the spins $\h$ which are involved in the derivatives
$\nabla_{\Z\setminus\L}$ in two ways: through the Gibbs measure
$\m^\h$ and through the function f.\bigskip
$\S \;3$\hskip 1cm {\bf Proof of Theorem 2.1}\par
If there was no interaction among the sets $\L_i$ then the total
Gibbs measure $\m^\h$ would have been a product measure and the proof
of the theorem would be a very simple exercise . However our
hypotheses say that the mutual interaction among the sets $\L_i$ is
very weak in a suitable sense and it is therefore natural to try to
make some perturbation theory around the non interacting case.
Because of the structure of the LSI, we found convenient first of
all to write the average $\m^\h(f)$ of an arbitrary function f in a
form that
resembles as much as
possible that of the average of f over a product measure. This
form is as follows : $$\m^\h(f)\;=\; \n_N^\h(\n_{N-1}^{\h
,\t_N}(,\dots ,(\n_1^{\h ,\t_2,\dots ,\t_N}(f)..)\Eq(2.9)$$ If we
now apply this representation of $\m^\h(f)$ to the function
$\log{f}$ we get :
$$\m^\h(\log{f})\;=\;
\n_N^\h(\n_{N-1}^{\h ,\t_N}(,\dots ,(\n_1^{\h
,\t_2,\dots ,\t_N}(\log{f})..)\;\leq $$ $$\leq \;
c_1\n_N^\h(\n_{N-1}^{\h ,\t_N}(....(\n_1^{\h ,\t_2,\dots ,\t_N}(
\grad{1}{f}))\;+\;$$
$$\n_N^\h(\n_{N-1}^{\h ,\t_N}(....(\n_2^{\h ,\t_3,\dots ,\t_N}(
\logg{\n_1^{\h ,\t_2,\dots ,\t_N}(f^2)}))\Eq(2.10)$$ where :
$$c_1\;=\;\sup_{\t_2,\dots ,\t_N,\,\h}c(\n_1^{\h ,\t_2,\dots ,\t_N})$$
Next we define the new function $$g_1\,=\,(\n_1^{\h
,\t_2,\dots ,\t_N}(f^2)) ^{1\over 2}$$ and, more generally:
$$g_i\,=\,(\n_i^{\h ,\t_{i+1},\dots ,\t_N}(g_{i-1}^2))^{1\over
2}\Eq(2.11)$$ With these notation if we iterate \equ(10) we obtain :
$$\m^\h(\log{f})\;\leq\;\sum_ic_i\m^\h(\grad{i}{g_{i-1}})\;+\;
\logg{\m^\h(f^2)}\Eq(2.12)$$ where
$$c_i\;=\;\sup_{\t_{i+1},\dots ,\t_N,\,\h}c(\n_i^{\h
,\t_{i+1},\dots ,\t_N})$$ We are thus left with the estimate of the
term: $$\sum_ic_i\m^\h(\grad{i}{g_{i-1}})\Eq(2.13)$$ This is done in
the next proposition where we show that, by paying a small price if
the constant $\e$ is small enough, one can safely replace in
\equ(2.13) the functions $g_{i-1}$ with the function f.\bigskip {\bf
Proposition 2.1}\par {\it Given the constant $k_1$, for
any $\d\,>\,0$ there exists $\e_o\,=\,\e_o(k_1,\d)$ such that if
$\e\sup_i|\L_i|\,\leq \,\e_o$ then:
$$\sum_i\m^\h(\grad{i}{g_{i-1}})\;\leq\;(1\,+\,\d)
\sum_i\m^\h(\grad{i}{f})$$}\bigskip Before giving the proof of the
proposition let us finish the proof of the theorem. If we use the
result of the propostion we see that for any $\d\,>\,0$ there
exists $\e_o\,=\,\e_o(k_1,\d)$ such that
if $\e\sup_i|\L_i|\,\leq \,\e_o$
then the r.h.s. of \equ(2.12) can be bounded above by:
$$(1\,+\,\d)\sup_ic_i\,\sum_i\m^\h(\grad{i}{f})\;+\;
\logg{\m^\h(f^2)}\Eq(2.14)$$ \ie
$$\sup_\h c(\m^\h)\,\leq\,(1\,+\d)\,
\sup_i\sup_{\h,\t_{i+1},\dots ,\t_{N}}c(\n_i^{\h
,\t_{i+1},\dots ,\t_{N}})$$ and the theorem follows.\bigskip {\bf Proof
of Proposition 2.1}\par One technical complication of working with
discrete spins and discrete derivatives is that the latter do not
enjoy exactly the same properties of the usual continuous
derivatives like Leibniz rule and so forth.\par Therefore before
entering into the details of the proof let us give some elementary
results concerning discrete derivatives that will be frequently used
later on. Properties a),b),c),d),e) follow only from the definition
of $\partial_{\s_x}$ and are true in general whereas property f) is
a consequence of assumption {\bf c)} above on the interaction.
A proof can be found in
the Appendix.\par In what follows $\media{f}_x$ will denote the
average of the function f($\s_x$) with respect to the measure
${1\over 2}(\d_{+1}\;+\;\d_{-1})$ and, for any given x
$\in \;\L_j$ with $j\,>\,i$ $$\l^x_i\;=\;
\supnorm{{d\n_i^{\h ,\t_{i+1}^{x,+1},., \t_N^{x,+1}}\over d\n_i^{\h
,\t_{i+1}^{x,-1},.\t_N^{x,-1}}}}\,\vee\,\supnorm{{d\n_i^{\h
,\t_{i+1}^{x,-1},., \t_N^{x,-1}}\over d\n_i^{\h
,\t_{i+1}^{x,+1},.\t_N^{x,+1}}}}$$ Then we have:
\bigskip \item{a)}
$\partial_{\s_x}f^2\;=\;2\media{f}_x\partial_{\s_x}f$
\item{b)}
$\n_i^{\h ,\t_{i+1},.,
\t_N}(\media{|f|}_x)\;\leq\;\l_i^x\media{\n_i^{\h ,\t_{i+1},.,
\t_N}(|f|)}_x$ \item{c)} If f is
non negative then $f(\s_x)\,\leq\,2\media{f}_x$. \item{d)} Given x
$\in \;\L_j$ with $j\,>\,i$:
$$\vert\partial_{\t_x} [\n_i^{\h ,\t_{i+1},\dots ,
\t_N}(....(\n_1^{\h ,\t_2,\dots ,\t_N}(f)..)]\vert\;\leq $$
$$|[\partial_{\t_x}( \n_i^{\h ,\t_{i+1},\dots ,\t_N}(....(\n_1^{\h
,\t_2,\dots ,\t_N}](f)|\;+$$ $$\prod_{j=i}^1\l^x_j \n_i^{\h
,\t_{i+1},\dots ,\t_N}(....(\n_1^{\h ,\t_2,\dots ,\t_N}
(|\partial_{\t_x}f|)..) $$ where $|[\partial_{\t_x} \n_i^{\h
,\t_{i+1},\dots ,\t_N}(....(\n_1^{\h ,\t_2,\dots ,\t_N}](f)|$ is a
convenient way to denote the expression: $${|\n_i^{\h
,\t_{i+1}^{x,+1},.,\t_N^{x,+1}} (.(\n_1^{\h
,\t_2^{x,+1},.,\t_N^{x,+1}}(f(\s\t)..)\,-\, \n_i^{\h
,\t_{i+1}^{x,-1},.,\t_N^{x,-1}} (.(\n_1^{\h
,\t_2^{x,-1},.,\t_N^{x,-1}}(f(\s\t)..)|\over 2}$$ \item{e)}
("Leibniz rule"). Let f be non negative. Then : $$|[\partial_{\t_x}
(\n_i^{\h ,\t_{i+1},\dots ,\t_N}(....(\n_1^{\h ,\t_2,\dots
,\t_N})](f)|\;\leq$$ $$|[\partial_{\t_x}\n_i^{\h ,\t_{i+1},\dots
,\t_N}]( \n_{i-1}^{\h ,\t_{i},\dots ,\t_N}(...(\n_1^{\h ,\t_2,\dots
,\t_N}(f)..)|\;+\;$$ $$\l^x_i\n_i^{\h ,\t_{i+1},\dots ,\t_N}(
|[\partial_{\t_x}( \n_{i-1}^{\h ,\t_{i},\dots ,\t_N}(....(\n_1^{\h
,\t_2,\dots ,\t_N})](f)| $$
\item{f)}
$$|[\partial_{\t_x}\n_i^{\h ,\t_{i+1},\dots ,\t_N}](f^2)|\;\leq$$
$$2k_1\l^x_i\supnorm{\partial_{\t(x)}W_{(i)}^\h(\s\t)}
[\n_i^{\h ,\t_{i+1},\dots ,\t_N}(f^2)]^{1\over 2}
[\n_i^{\h ,\t_{i+1},\dots ,\t_{N}}(\grad{i}{f})]^{1\over 2}$$
\bigskip
We are now ready to start our computations.\par Since we have to
estimate terms like $\grad{i}{g_{i-1}}$, we start to estimate
the following quantity:
$$|\partial_{\t_x}((\n_{i-1}^{\t_{i},\dots ,\t_N}(g^2_{i-2}))^{1\over
2})|$$ where x is an arbitrary site in the set $\L_i$.
Following Zegarlinski and using a) above, we observe that it is
sufficient to estimate
$$|\partial_{\t_x}((\n_{i-1}^{\t_{i},\dots ,\t_N}(g^2_{i-2}))|$$
by :
$$2\media{(\n_{i-1}^{\t_{i},\dots ,\t_N}(g^2_{i-2}))^{1\over
2}}_x[....]$$
to get that
$$|\partial_{\t_x}(((\n_{i-1}^{\t_{i},\dots ,\t_N}(g^2_{i-2}))^{1\over
2})|\,\leq\,
[...]\Eq(2.15)$$
Using d) and e) above with $f$ replaced by $f^2$, we get:
$$\vert\partial_{\t_x}
[\n_{i-1}^{\t_{i},\dots ,
\t_N}(....(\n_1^{\h ,\t_2,\dots ,\t_N}(f^2)..)]\vert\;\leq\;
$$
$$(\prod_{j=i-1}^1\l_j^x)\n_{i-1}^{\t_{i},\dots ,
\t_N}(....(\n_1^{\h ,\t_2,\dots ,\t_N}(|(\partial_{\t_x}f^2)|)..)\;+$$
$$\sum_{k=1}^{i-1}(\prod_{j=i-1}^{k+1}\l_j^x)
\n_{i-1}^{\t_{i},\dots ,\t_N}(..\n_{k+1}^{\t_{k+2},\dots ,\t_N}
(|[\partial_{\t_x}\n_{k}^{\t_{k+1},\dots ,\t_N}]
\n_{k-1}^{\t_{k},\dots ,\t_N}
(..\n_1^{\h ,\t_2,\dots ,\t_N}(f^2)...)|)\Eq(2.16)$$
The first term in the r.h.s of \equ(2.16), using the Schwartz
inequality, and properties a) and b), is bounded from above by:
$$2(\prod_{j=i-1}^1\l_j^x)^{3\over 2}\media{(\n_{i-1}^{\t_{i},\dots
,\t_N} (...(\n_1^{\h ,\t_2,\dots ,\t_N} (f^2)..)^{1\over
2}}_x (\n_{i-1}^{\t_{i},\dots ,\t_N} (...(\n_1^{\h ,\t_2,\dots ,\t_N}
((\partial_{\t_x}f)^2)..)^{1\over
2}\Eq(2.17)$$
Using f) above, the second term in
the r.h.s of \equ(2.16) is bounded
from above by: $$\sum_{k=1}^{i-1}(\prod_{j=i-1}^{k+1}\l_j^x)
2k_1\l^x_k\supnorm{\partial_{\s(x)}W_{(k)}^{\h}(\s\t)}$$
$$\n_{i-1}^{\t_{i},\dots ,\t_N}(..(\n_{k+1}^{\t_{k+2},\dots ,\t_N}(
[\n_k^{\t_{k+1},\dots ,\t_N}(g_{k-1}^2)]^{1\over 2}
[\n_k^{\t_{k+1},\dots ,\t_{N}}((\grad{k}{g_{k-1}}))]^{1\over 2}
\Eq(2.18)$$
Using the definition of the function $g_{k-1}$ we have :
$$[\n_k^{\t_{k+1},\dots ,\t_N}(g_{k-1}^2)]^{1\over 2}\;=\;
[\n_k^{\t_{k+1},\dots ,\t_N}(...(\n_1^{\h ,\t_2,\dots ,\t_N}
(f^2)..)]^{1\over2}$$
and thus, using again a) and the Schwartz inequality, we get that
\equ(2.18) is bounded above by:
$$2\media{[\n_{i-1}^{\t_{i},\dots ,\t_N} (...(\n_1^{\h ,\t_2,\dots ,\t_N}
(f^2)..)]^{1\over
2}}_x\sum_{k=1}^{i-1}[(\prod_{j=i-1}^{k+1}\l_j^x)
\supnorm{\partial_{\s(x)}W_{(k)}^{\h}(\s\t)}
2k_1\l^x_k]$$
$$\{\n_{i-1}^{\t_{i},\dots ,\t_N} (...(\n_1^{\h ,\t_2,\dots ,\t_N}
((\grad{k}{g_{k-1}}))\}^{1\over2}\Eq(2.19)$$
We now define:
$$V_{i,k}\;=\;\sup_{x\in\L_i}
[(\prod_{j=i-1}^{k+1}\l_j^x)
\supnorm{\partial_{\s(x)}W_{(k)}^{\h}(\s\t)}
2k_1\l^x_k]$$
and
$$B_{i-1}\;=\;\sup_{x\in \L_i}(\prod_{j=1}^{i-1}\l_j^x)^{3\over 2}$$
With this notation, if we put together \equ(2.17) and \equ(2.19), we
obtain that :
$$\vert\partial_{\t_x}
[\n_{i-1}^{\t_{i},\dots ,
\t_N}(....(\n_1^{\h ,\t_2,\dots ,\t_N}(f^2)..)]\vert\;\leq\;$$
$$2\media{(\n_{i-1}^{\t_{i},\dots ,\t_N}
(...(\n_1^{\h ,\t_2,\dots ,\t_N} (f^2)..)^{1\over
2})}_x \{B_{i-1}(\n_{i-1}^{\t_{i},\dots ,\t_N} (...(\n_1^{\h ,\t_2,\dots ,\t_N}
((\partial_{\t_x}f)^2)..)^{1\over
2}\;+\;
$$
$$+\;\sum_{k=1}^{i-1}V_{i,k}[\n_{i-1}^{\t_{i},\dots ,\t_N}
(...(\n_1^{\h ,\t_2,\dots ,\t_N}
((\grad{k}{g_{k-1}}))]^{1\over2}\}\Eq(2.20)$$
\ie , using \equ(2.15),
$$ \vert\partial_{\t_x}
[\n_{i-1}^{\t_{i},\dots ,
\t_N}(....(\n_1^{\h ,\t_2,\dots ,\t_N}(f^2)..)]^{1\over 2}\vert\;\leq\;$$
$$B_{i-1}(\n_{i-1}^{\t_{i},\dots ,\t_N} (...(\n_1^{\h ,\t_2,\dots ,\t_N}
((\partial_{\t_x}f)^2)..)^{1\over
2}\;+\;$$
$$+\; \sum_{k=1}^{i-1}V_{i,k}[\n_{i-1}^{\t_{i},\dots ,\t_N}
(...(\n_1^{\h ,\t_2,\dots ,\t_N}
((\grad{k}{g_{k-1}}))]^{1\over2}\Eq(2.21)$$
Thus, using the above bound, we get that
$$\sum_{x\in\L_i}\vert\partial_{\t_x}
[\n_{i-1}^{\t_{i},\dots ,
\t_N}(....(\n_1^{\h ,\t_2,\dots ,\t_N}(f^2)..)]^{1\over 2}\vert^2\;\leq\;$$
$$pB_{i-1}^2(\n_{i-1}^{\t_{i},\dots ,\t_N} (...(\n_1^{\h ,\t_2,\dots ,\t_N}
((\partial_{\L_i}f)^2)..)\;+\;$$
$$+\;
|\L_i|q(\sum_{k=1}^{i-1}V_{i,k})
\sum_{k=1}^{i-1}V_{i,k}[\n_{i-1}^{\t_{i},\dots ,\t_N}
(...(\n_1^{\h ,\t_2,\dots ,\t_N}
((\grad{k}{g_{k-1}}))]\Eq(2.22)$$
for any $p\,>\,1$, $q\,>\,1$ with ${1\over p}\,+\,{1\over
q}\;=\;1$.\par In conclusion, by summing \equ(2.21) over the index i,
we obtain for the initial expression $\sum_i\m^\h(\grad{i}{g_{i-1}})$
the bound :
$$\sum_i\m^\h(\grad{i}{g_{i-1}})\;\leq\;$$
$$\sum_i\{
pB_{i-1}^2\m^\h((\grad{i}{f})^2)\;+\;
|\L_i|q(\sum_{k=1}^{i-1}V_{i,k})
\sum_{k=1}^{i-1}V_{i,k}\m^\h(
((\grad{k}{g_{k-1}}))\}\Eq(2.23)$$
At this point we use our decay assumption {\bf a)} on the
interaction in order to estimate the numbers $B_i$ and
$V_{i,k}$.\par
>From the definition of $\l_j^x$ one has immediately that for
$j\,\leq \,i$ :
$$\sup_{x\in \L_i}\l_j^x\;\leq
\;\hbox{exp}(4\sup_{x\in \L_i}
\supnorm{\partial_{\t_x}W_{(j)}^\h})\Eq(2.24)$$ Therefore :
$$B_{i-1}\;\leq \;\hbox{exp}(8\sum_{j=1}^{i-1}\sup_{x\in \L_i}
\supnorm{\partial_{\t_x}W_{(j)}^\h})\;\leq\;
\hbox{exp}(8\e)\;\leq\;1\,+\,10\,\e\Eq(2.25)$$
if $\e$ is small enough.\par
Similarly :
$$\eqalign{&\sum_{k=1}^{i-1}V_{i,k}\;\leq \;C\e\cr
&\sum_{i=k+1}^{N}V_{i,k}\;\leq \;C\e}$$ for a suitable
constant C depending on $k_1$ and any $\e$ sufficently small.\par
Thus the second term in the r.h.s. of \equ(2.23) is smaller than:
$$\sup_i|\L_i|q(C\e)^2\sum_{k=1}^{N}\m^\h(
((\grad{k}{g_{k-1}}))\}\Eq(2.26)$$
Therefore if $\sup_i|\L_i|q(C\e)^2\;<\;1$ we
get from \equ(2.23), \equ(2.25) and
\equ(2.26) that:
$$\sum_i\m^\h(\grad{i}{g_{i-1}})\;\leq\;$$
$$\leq \;{p(1\,+\,10\e)\over
1\,-\,\sup_i|\L_i|q(C\e)^2}\sum_i\{
\m^\h((\grad{i}{f}))\;\leq\;(1\,+\,\d)\sum_i\{
\m^\h((\grad{i}{f}))$$
if we choose for example $q\,=\,{1\over \e}$ and $\e$ sufficiently
small. The proposition is proved.
\bigskip
$\S \;4$\hskip 1cm {\bf Proof of Theorem 2.2}\par
We proceed very similarly to the proof of proposition 2.1. By doing
the same kind of computations as in \equ(2.15),\dots ,\equ(2.21) we
obtain that:
$$ \{\vert\partial_{\h_x}
[\n_{i-1}^{\t_{i},\dots ,
\t_N}(....(\n_1^{\h ,\t_2,\dots ,\t_N}(f^2)..)]^{1\over
2}\vert\}^2\;\leq\;$$
$$\{2(\,\sup_{x\in
\Z\setminus\L}\prod_{j=1}^N(\l_j^x)^{3\over
2}\,)^2(\n_{i-1}^{\t_{i},\dots ,\t_N} (...(\n_1^{\h ,\t_2,\dots
,\t_N}
((\partial_{\h_x}f)^2)..)\;+\;$$
$$+\; 2(\sum_{k=1}^{N}\hat V_{x,k}[\n_{N}^{\h }
(...(\n_k^{\h ,\t_{k+1},\dots ,\t_N}
((\grad{k}{g_{k-1}}))]^{1\over 2}\,)^2\,\}\Eq(2.27)$$
where
$$\hat V_{x,k}\;=\;
[(\prod_{j=i-1}^{k+1}\l_j^x)
\supnorm{\partial_{\h(x)}W_{(k)}^{\h}(\s\t)}
2k_1\l^x_k]$$
It is important to observe at this point that in general $\hat
V_{x,k}$ is not small. Assumption a) (see \equ(2.6) or
\equ(2.6bis)) in fact concerns only the interaction between
different blocks while in some sense $\hat
V_{x,k}$ measures the interaction between one
block $\L_k$ and the boundary spin $\h_x$. However, thanks
to assumption b) (see \equ(2.7)) and using \equ(2.24), we
have that : $$\sup_{\h ,k}\sum_{x\in\Z\setminus\L}\hat
V_{x,k}\;\leq \; 2k_1k_o\sup_i|\L_i|\hbox{exp}
(\sum_{j=1}^N4\supnorm{\partial_{\h(x)}W_{(j)}^{\h}(\s\t)})\;\leq$$
$$\leq \;\sup_i|\L_i|2k_1k_o\hbox{exp}(4k_o)\Eq(2.28)$$
Thus if we sum over $x\in\Z\setminus\L$ the second term in the
r.h.s. of \equ(2.27), we get, after a Schwartz inequality :
$$\sum_{x\in\Z\setminus\L}\,2(\sum_{k=1}^{N}\hat
V_{x,k})[\sum_{k=1}^{N}\hat
V_{x,k}\n_{N}^{\h } (.(\n_k^{\h ,\t_{k+1},\dots ,\t_N}
((\grad{k}{g_{k-1}}))]\;\leq \; $$
$$\leq \;
\sup_i|\L_i|\,2k_o^2k_1^2\hbox{exp}(8k_o)(\sum_{k=1}^{N}\n_{N}^{\h }
(...(\n_k^{\h ,\t_{k+1},\dots ,\t_N}
((\grad{k}{g_{k-1}}))\Eq(2.29)$$
Using the identity:
$$\n_{N}^{\h }
(...(\n_k^{\h ,\t_{k+1},\dots ,\t_N}
((\grad{k}{g_{k-1}}))\;=\;\m^\h((\grad{k}{g_{k-1}})$$
and proposition 2.1 , we see that if $\e$ is small enough there
exists a constant $k_3$ such that:
$$\sum_{x\in\Z\setminus\L}\,2(\sum_{k=1}^{N}\hat
V_{x,k})[\sum_{k=1}^{N}\hat
V_{x,k}\n_{N}^{\h } (...(\n_k^{\h ,\t_{k+1},\dots ,\t_N}
((\grad{k}{g_{k-1}}))]\;\leq \;$$
$$\sup_i|\L_i|\,k_3\m^\h(\grad{\L}{f})\Eq(2.30)$$ Analogously if we sum
over $x\in\Z\setminus\L$ the first term in the r.h.s. of \equ(2.27) we
get : $$\sum_{x\in\Z\setminus\L}\,2(\,\sup_{x\in
\Z\setminus\L}\prod_{j=1}^N\l_j^x\,)^2 (\n_{i-1}^{\t_{i},\dots ,\t_N}
(...(\n_1^{\h ,\t_2,\dots ,\t_N}
((\partial_{\h_x}f)^2)..)\;\leq\;$$
$$\leq \;2\hbox{exp}(8k_o)\m^\h(\grad{\Z\setminus\L}{f})\Eq(2.31)$$
If we finally combine \equ(2.31) and \equ(2.30) we obtain the theorem
with $k_2\;=\;2\hbox{exp}(8k_o)$.\vskip 2cm
%\input formato.tex
\numsec=3\numfor=1
\bigskip
{\bf Section 3}\bigskip
\centerline{DECIMATION APPROACH TO THE}
\centerline{LOGARITHMIC SOBOLEV CONSTANT}\bigskip
$\S\;1$\hskip 1cm {\bf Proof of theorem 1.1}\par In this section we
prove our main result namely theorem 1.1 .\par The result holds in
any dimension but for the sake of simplicity of the exposition we
will discuss explicitely only the two dimensional case. As already
announced in section 1, our proof is based upon ideas coming from
rigorous renormalization group in classical statistical mechanics in
the form known as {\it decimation}. We begin therefore by
illustrating our decimation procedure.\par For any odd integer $L_o$
let us consider the renormalized lattice
$\Z(L_o)\;=\;L_o\Z\,\subset \Z$ and let us define for any $x\,\in
\,\Z(L_o)$ the block $Q_{L_o}(x)$ as the square in the original
lattice, centered at x and of side $L_o$. We will collect the blocks
$Q_{L_o}(x)$ into four different families, denoted in the sequel by
the letters A, B, C, D, according to whether the coordinates of
their centers x are (even, odd), (even, even), (odd, even) or (odd,
odd). We will also order in some way the blocks belonging to the
same family so that they will be denoted by
$A_1,\,A_2\,...,A_n,...$ etc.\par Let now $\L(L_o)$ be a finite
subset of $\Z(L_o)$, let $\L\,=\,\bigcup_{x\in \L(L_o)}Q_{L_o}(x)$
and let $\m^\h$ be the Gibbs state in $\L$ corresponding to the
Hamiltonian (1.1) with some fixed boundary condition $\h$ outside
$\L$ . Given $\m^\h$ we will consider new
measures, denoted by:
$$\n_{A},\;\n_{B}^{\t_A},\;\n_{C}^{\t_A,\t_B},\;
\n_{D}^{\t_A,\t_B,\t_C}\Eq(3.1)$$
on the finite sets $\O_{A\cap\L}$, $\O_{B\cap\L}$, $\O_{C\cap\L}$,
$\O_{D\cap\L}$. Such a measures are defined in analogy with the
measures $\n_i^{\t_{i+1}..\t_N}$ of section 2 as follows: \par
$\n_{D}^{\t_A,\t_B,\t_C}$ is simply obtained from the Gibbs
measure $\m^\h$ by conditioning the spins in
${A\cap\L},\;{B\cap\L},\;{C\cap\L}$ to have the prescribed
values $\t_A,\,\t_B,\,\t_C$. To construct $\n_{C}^{\t_A,\t_B}$ we
first integrate out the spins $\s_D$ in $\m^\h$ and then we
condition to the spins in
${A\cap\L},\;{B\cap\L}$ to have the prescribed
values $\t_A,\,\t_B$. Similarly to construct $\n_{B}^{\t_A}$ we first
integrate our the spins $\s_D$ and $\s_C$ in $\m^\h$ and then we
condition to the spins in
${A\cap\L}$ to have the
values $\t_A$. $\n_{A}$ is simply the relativization of $\m^\h$ to
$A\cap\L$.\bigskip
{\bf Remark} We observe that by construction the intersection
between the family of blocks of type A with the set $\L$ consists of
a finite collection of blocks say $A_1,\,A_2\,...,A_{N_A}$ and the
same for the other families.\par
In the notation for the measures $\n_{D}^{\t_A,\t_B,\t_C}$
etc. obtained after the decimation, we have omitted for convenience
the supmaxerscript $\h$ .\bigskip
We are
now in a position to start our calculations. Given an arbitrary
function f : $\O_\L\,\to \, R$ we write, following [OP]:
$$\m^\h(\log{f})\;=\;\n_{A}(\n_{B}^{\t_A}(\n_{C}^{\t_A,\t_B}(
\n_{D}^{\t_A,\t_B,\t_C}(\log{f})..)\Eq(3.1bis)$$
Let us now define $c(L_o)$ to be the largest among the logarithmic
Sobolev contant (LSC) of the measures
$$\n_{A},\,\n_{B}^{\t_A},\,\n_{C}^{\t_A,\t_B}
,\,\n_{D}^{\t_A,\t_B,\t_C}$$ more precisely :
$$c(L_o)\;=\;\sup_{\t_A,\t_B,\t_C}\ max \{c(\n_{A})\,,\,
c(\n_{B}^{\t_A})\,,\, c(\n_{C}^{\t_A,\t_B})\,,\, c(
\n_{D}^{\t_A,\t_B,\t_C})\}$$ With this notation and if we apply the
logarithmic Sobolev inequality to $\n_{D}^{\t_A,\t_B,\t_C}$ we
obtain that the r.h.s. of \equ(3.1bis) is bounded above by:
$$
c(L_o)\n_{A}(\n_{B}^{\t_A}(\n_{C}^{\t_A,\t_B}
(\n_{D}^{\t_A,\t_B,\t_C}(\grad{D}{f})..)\;+\;$$
$$\n_{A}(\n_{B}^{\t_A}(\n_{C}^{\t_A,\t_B}
(\logg{\n_{D}^{\t_A,\t_B,\t_C}(f^2)}))\Eq(3.2)$$
Next we define the new functions :
$$\eqalign{g_D\,&=\,\n_{D}^{\t_A,\t_B,\t_C}(f^2) ^{1\over 2}\cr
g_C\,&=\,\n_{C}^{\t_A,\t_B}(g_{D}^2))^{1\over
2}\cr g_B\,&=\,\n_{B}^{\t_A}(g_{C}^2))^{1\over
2}\cr g_A\,&=\,\n_{A}(g_{B}^2))^{1\over
2}}$$
With these notation, if we iterate \equ(3.2), we obtain :
$$\m^\h(\log{f})\;\leq\;$$
$$\leq \;c(L_o)[\m^\h(\grad{A}{g_{B}})\;+\;
\m^\h(\grad{B}{g_{C}})\;+\;
\m^\h(\grad{C}{g_{D}})\;+\;
\m^\h(\grad{D}{f})]\;+\;
$$
$$\logg{\m^\h(f^2)}\Eq(3.3)$$
The main idea at this stage is to show
that, in the hypotheses of the theorem and provided that the
parameter $L_o$ is large enough, each one of the measures
$$\n_{A},\,\n_{B}^{\t_A},\,\n_{C}^{\t_A,\t_B}
,\,\n_{D}^{\t_A,\t_B,\t_C}$$ satisfies the conclusions of theorems
1.2 and 2.2 of section 2. Let us state this as a theorem: \bigskip
{\bf Theorem 3.1} {\it Let $\{*,\,\t^*\}$ be one of the pairs $\{A,\,\h\}$ $\{B,\,\t_A\}$
$\{C,\,\t_A\t_B\}$ $\{D,\,\t_A\t_B\t_C\}$ and let $\n_{*}^{\t_{*}}$ be the corresponding Gibbs measure.
There exists a constant
$\bar L$ such that if $SM(L,C,\gamma )$ holds for some $L\,\geq
\,\bar L$, then there exists $\bar L_o\;>\;\bar L$
such that if $L_o\,\geq \,\bar L_o$ then there exist
two constants $a_o$ and $a_1$ such that
:\item{i)}
$$c(L_o)\;<\;\infty$$
\item{ii)}
$$
\grad{\t(x)}{(\n_{*}^{\t_{*}}(f^2))^{1\over
2}}\;\leq \; a_o\n_{*}^{\t_{*}}(\grad{\t(x)}{f})\;+\;a_1
\n_{*}^{\t_{*}}(\grad{*}{f})$$
for any x $\notin\;*$.}\bigskip
Before giving the proof of the above crucial
result, let us first complete the proof of the main theorem.\par
If we apply ii) of theorem 3.1 to $\m^\h(\grad{A}{g_{B}})$ we get
that :
$$\m^\h(\grad{A}{g_{B}})\;\leq \;a_o\m^\h(\grad{A}{g_C})\;+\;a_1
\m^\h(\grad{B}{g_C})$$
We have thus succeeded in moving the gradient from the function
$g_B$ to the function $g_C$. If we continue this procedure two
more times we end up with all the gradients acting on the original
function f. More explicitely, after three repeated applications of
ii) of theorem 3.1, we have that :
$$
\m^\h(\grad{A}{g_{B}})\;\leq \;a_2\m^\h((\nabla_{\L}f)^2)\Eq(3.4)$$
for a suitable constant $a_2$. The same estimate of course applies
also to the terms:
$$\m^\h(\grad{B}{g_{C}})\quad\hbox{and }\quad
\m^\h(\grad{C}{g_{D}})$$. In conclusion we have shown that:
$$c(L_o)[\m^\h(\grad{A}{g_{B}})\;+\;
\m^\h(\grad{B}{g_{C}})\;+\;
\m^\h(\grad{C}{g_{D}})\;+\;
\m^\h(\grad{D}{f})]\;\leq $$
$$\leq\;c'(L_o)
\m^\h((\nabla_{\L}f)^2)\Eq(3.5)$$
provided that SM(L,C,$\gamma$) holds for some L large enough and
the size of the blocks of the decimation was also sufficiently
large.\par
The theorem (1.1) is proved.\bigskip
$\S\;2$\hskip 1cm {\bf Proof of Theorem 3.1}\par
The idea of the proof is very simple: we will verify that, in the
hypotheses of the theorem, all the assumptions a), b) and c) of
section 2 are satisfied for any one of the measures \equ(3.1) with
constants $k_o,\;k_1$ uniformly bounded in the side $L_o$ of the
blocks of the decimation and with the constant $\e,\;$ going
exponentially fast to zero as $L_o\;\to\;\infty$.\par In order to
verify a) and b) we first need to write in a convenient way the
derivative with respect to a conditioning spin of the effective
potential appearing in any of the measures \equ(3.1). One
possibility is to use a cluster expansion to write down the effective
potential; there is however another way in which the derivative with
respect to a conditioning spin of the effective potential becomes
essentially a truncated correlation function of a suitable pair of
local observables computed with respect to the {\it original} Gibbs
measure $\m^\h$. That is of course very convenient since (see
Theorem 1.2) it has been proved that in the hypotheses of the
theorem the truncated correlations functions of the measure $\m^\h$
decay exponentially fast.\par So let us discuss the second way in a
rather general setting.\par Suppose that we are given a subset
$\L\;=\;\L_1\,\cup\,\L_2\,\cup\,\L_3$ of the lattice $\Z$ and a
Gibbs measure $\m$ on $\{-1,+1\}^{\L}$ with Hamiltonian
$H(\s_{\L_1},\,\s_{\L_2},\,\s_{\L_3})$ corresponding to a interaction $\Phi$
with finite norm $$||\Phi||\;=\;\sum_{O\in X}|\Phi (X)|$$ Let
$$\hat
H(\s_{\L_1},\,\s_{\L_2})\;=\;\hbox{log}
(Z_{\L_3}^{\s_{\L_1},\s_{\L_2}})\Eq(3.6)$$ be the effective
Hamiltonian after the integration of the $\s_{3}$ variables in
$\L_3$ and let $$W_1^{\s_{\L_2}}(\s_{\L_1})\;\equiv \;\hat
H(\s_{\L_1},\,\s_{\L_2})\,-\,\hat H(\t_{1},\,\s_{\L_2})\Eq(3.7)$$ be
the effective interaction entering in the conditional Gibbs measure
of the spins $\s_{\L_1}$ given the spins $\s_{\L_2}$ after the
decimation of the spins $\s_{\L_3}$ (see (2.6) ). In \equ(3.7) $\t_1$
is an arbitrary reference configuration of the spins $\s_{\L_1}$
e.g. all the spins up. Then we have:\bigskip {\bf Lemma 3.1}\par
{\it For each $x\,\in\,\L_2$ and $y\in \L_1$ there exist two
functions $f_x^{\s_{\L_1},\s_{\L_2}}(\s_{\L_3})$ and
$g_y^{\s_{\L_1},\s_{\L_2}}(\s_{\L_3})$ with the following
properties: \item{i)} f and g, as functions of the spins
$\s_{\L_3}$, have support in a ball centered at x and y
respectively with radius equal to the range of the interaction
$\Phi$.
\item{ii)}$$\sup_{\s_{\L_1},\s_{\L_2},\s_{\L_3}}|f_x^{\s_{\L_1},\s_{\L_2}}(\s_{\L_3})|
\;\leq\;\exp (2||\Phi||)$$
$$\sup_{\s_{\L_1},\s_{\L_2},\s_{\L_3}}|g_y^{\s_{\L_1},\s_{\L_2}}(\s_{\L_3})|
\;\leq\;\exp (2||\Phi||)$$ \item{iii)}
$$\eqalign{\sup_{\s_{\L_1},\s_{\L_2}}||\partial_{\s_{\L_2}(x)}W_1^{\s_{\L_2}}(\s_{\L_1})|
\;&\leq\; \sum_{y\in \L_1}\exp (4||\Phi||)
\sup_{\s_{\L_1},\s_{\L_2}}|\media{f_x^{\s_{\L_1},\s_{\L_2}}\,;
\,g_y^{\s_{\L_1},\s_{\L_2}}}^{\s_{\L_1},\s_{\L_2}}_{\L_3}|\cr
\;&+\;\sum_{y\in\L_1}\sup_{\s_{\L_1},\s_{\L_2}}|\hbox{\rm log} ({
\media{g_y^{\s_{\L_1},\s_{\L_2}}}^{\s_{\L_1},\s_{\L_2}}_{\L_3}\over
\media{g_y^{\s_{\L_1},\s_{\L_2}^x}}^{\s_{\L_1},\s_{\L_2}}_{\L_3} })|}$$
where $\media{f}^{\s_{\L_1},\s_{\L_2}}_{\L_3}$ is the conditional average of
the observable f with respect to the original Gibbs state given that
the spins in $\L_1$ and $\L_2$ are equal to $\s_{\L_1}$ and $\s_{\L_2}$ and
$\media{f\,;\,g}$ denotes the usual truncated expectation.}
\bigskip
{\bf Proof}\par Let for $x\,\in\,\L_2$ and $y\in \L_1$
$$\eqalign{f_x^{\s_{\L_1},\s_{\L_2}}(\s_{\L_3})\;&=\;
\hbox{exp}(H^{\s_{\L_1},\s_{\L_2}^x}(\s_{\L_3})\,-\,H^{\s_{\L_1},\s_{\L_2}}(\s_{\L_3}))\cr
g^{\s_{\L_1},\s_{\L_2}}(\s_{\L_3})\;&=\;
\hbox{exp}(
H^{\s_{\L_1}^y,\s_{\L_2}^x}(\s_{\L_3})\,-\,H^{\s_{\L_1},\s_{\L_2}^x}(\s_{\L_3}))}\Eq(3.7bis)$$ If we use
\equ(3.6), \equ(3.7) and the definition of $\partial_{\s(x)}$ we
obtain:
$$\eqalign{\sup_{\s_{\L_1},\s_{\L_2}}|\partial_{\s_{\L_2}(x)}W_1^{\s_{\L_2}}(\s_{\L_1})|
\;&\leq\;\cr 2\sum_{y\in
\L_1}\sup_{\s_{\L_1},\s_{\L_2}}|\partial_{\s_{\L_2}(x)} (\hat
H(\s_{\L_1},\,\s_{\L_2})\,&-\,\hat
H(\s_{\L_1}^y,\,\s_{\L_2}))|\;=\;\cr \sum_{y\in
\L_1}\sup_{\s_{\L_1},\s_{\L_2}}|\hbox{\rm log} (
\media{f_x^{\s_{\L_1},\s_{\L_2}}}^{\s_{\L_1},\s_{\L_2}}_{\L_3}\,&-\,
\hbox{\rm log}
({\media{g_y^{\s_{\L_1},\s_{\L_2}}f_x^{\s_{\L_1},\s_{\L_2}}}^{\s_{\L_1},\s_{\L_2}}_{\L_3}\over
\media{g_y^{\s_{\L_1},\s_{\L_2}^x}}^{\s_{\L_1},\s_{\L_2}}_{\L_3}})|}\Eq(3.8)$$
Using $\hbox{\rm log} (1\,+\,x)\;\leq \;x$ if $x\;>\;0$ we get
immediately that the r.h.s. of \equ(3.8) is bounded by: $$\sum_{y\in
\L_1}\hbox{exp}(4||\Phi||)
\sup_{\s_{\L_1},\s_{\L_2}}|\media{f_x^{\s_{\L_1},\s_{\L_2}}\,;
\,g_y^{\s_{\L_1},\s_{\L_2}}}^{\s_{\L_1},\s_{\L_2}}_{\L_3}|\;$$
$$\;+\; \sum_{y\in \L_1}\sup_{\s_{\L_1},\s_{\L_2}}|\hbox{\rm log} ({
\media{g_y^{\s_{\L_1},\s_{\L_2}}}^{\s_{\L_1},\s_{\L_2}}_{\L_3}\over
\media{g_y^{\s_{\L_1},\s_{\L_2}^x}}^{\s_{\L_1},\s_{\L_2}}_{\L_3}
})|\Eq(3.9)$$
The Lemma is proved.\bigskip
One can now apply the above lemma to each one of the measures \equ(3.1)
by conveniently choosing the sets $\L_1,\,\L_2,\,\L_3$. Since the
discussion is the same for anyone of the measures \equ(3.1), let us
treat in detail only one of them, say $\n_B^{\t_A}$.\par
Thus let us suppose that we have fixed a block of type B, say
$B_j$, and let us set
$$\L_1\;=\;B_j\quad \L_3\;=\;C\,\cup\,D\Eq(3.9bis)$$ and
$$\L_2\;=\;A\,\cup\,\cup_{i>j}B_i\Eq(3.9tris)$$
Let also x be a site in $B_i$, with $i\,>\,j$, and let us estimate:
$$|\partial_{\s(x)}W^{\s_{\L_2}}_{(1)}(\s)|\Eq(3.10)$$
In order to apply Lemma 3.1, we first observe that if $L_o$ is
greater than the range of the interaction then
the above defined function
$g_y^{\s_{\L_1},\s_{\L_2}}$ does not depend on $\s(x)$
if $y\in B_j$ and $x\in B_i$. Therefore in this case the second term
in the r.h.s. of iii) of the Lemma is zero and we get:
$$\sup_{\s_{\L_1},\s_{\L_2}}||\partial_{\s(x)}W_1^{\s_{\L_2}}(\s)|
\;\leq\; \sum_{y\in B_j}\hbox{exp}(4||\Phi||)
\sup_{\s_{\L_1},\s_{\L_2}}|\media{f_x^{\s_{\L_1},\s_{\L_2}}\,;
\,g_y^{\s_{\L_1},\s_{\L_2}}}^{\s_{\L_1},\s_{\L_2}}_{\L_3}|\Eq(3.11)$$
Using now theorem 2.1 we see that there exist constants C and $m$,
depending only on the norm of the original interaction and on its
range $r$, such that for any sufficiently large $L_o$ the
r.h.s. of \equ(3.11) is smaller than:
$$C\sum_{y\in
B_j}\hbox{exp}(-m\,|x\,-\,y|)\Eq(3.12)$$
which implies that:
$$\sup_{(\h,\s_A,j,N_B,)}
\sum_{i=1}^{j-1}\sup_{x\in B_i}\supnorm{\partial_{\s
(x)}W_{(1)}^{\s_{\L_2}}(\s)}\;\leq\;\e \Eq(3.13)$$
with e.g. $\e\;=\;\hbox{exp}(-{m\over 2}\,L_o)$ for
any sufficiently large $L_o$.\par
In a very similar way one checks the bounds (2.7) and (2.8).\bigskip
{\bf Remark} The conclusion of the above discussion is that the
effective potential between any two sites x and y, defined for
example as $\partial_{\s
(x)}\partial_{\s
(y)}\hat H$ with $\hat H$ the effective hamiltonian of anyone of the
measures \equ(3.1), decays exponentially fast in the distance
between x and y. This implies in particular that in the second
inequality in b) of section 2 we can replace
$\sup_i|B_i|\,=\,L_o^d$ with $L_o^{d-1}$.\bigskip
In order to complete the proof of theorem 3.1 we are left with the
problem of verifying assumption c) of section 2. The idea at this
point is to show that assumption c) is equivalent to a {\it lower
} bound on the gap of the generator of the Glauber
dynamics defined in section 1, reversible with respect to
$\nu_*^{\t_*}$. In turn, such a lower bound will follow from theorem
1.2 i) . As before we do the computations only for $\nu_B^{\t_A}$
the other cases being analogous.\par
We keep the notation \equ(3.9bis), \equ(3.9tris) and we denote by
$\nu_1$ the conditional distribution on $\O_{\L_1}$ of the
relativization to $\L_1\cup\L_2$ of the original Gibbs measure
$\mu^\h$. For simplicity, in $\nu_1$, we have omitted to specify
the boundary conditions $\t_1$ since all our estimates will hold
uniformly in $\t_1$. First of all we observe that,
from assumption H5
on the jump rates and Theorem 1.2 i), it follows, for any
function $f\in L^2(\Omega_{\L_1},d\nu_1 )$ with
$\nu_1(f)\,=\,\mu^\h(f)\,=\,0$, that:
$$\nu_1(f^2)\,=\,\mu^\h(f^2)\,\leq\,{k_2\over
m_o}\mu^\h((\nabla_{\L_1}f)^2)\,=\,{k_2\over
m_o}\nu_1((\nabla_{\L_1}f)^2)\Eq(3.15)$$
Therefore we get immediately that:
$$\sum_{\s_{\L_1},\hat\s_{\L_1}} \n_1(\s_{\L_1})
\n_1(\hat\s_{\L_1}) [f(\s_{\L_1})\,-\,f(\hat\s_{\L_1})]^2\;\leq$$
$$\leq {2k_2\over
m_o}\nu_1((\nabla_{\L_1}f)^2) \Eq(3.16)$$
i.e. assumption c) holds true with the constant $k_1\,=\,
({2k_2\over m_o})^{1\over 2}$.\par
The Theorem is proved.\vskip 2cm
%\input formato.tex
\numsec=4\numfor=1
\tolerance=10000
\vskip 1cm
{\bf Section 4}\par\noindent
\centerline{\bf An Application to a Non-Ferromagnetic Model}
\bigskip
We conclude this paper by considering a non-ferromagnetic model
in two dimensions, obtained by adding a small
antiferromagnetic next nearest-neighbor coupling to the standard
ferromagnetic Ising model with small positive external field.\par
We will show that if the antiferromagnetic n.n.n. coupling is small
with respect to the n.n. ferromagnetic one, then it is possible to
find constants $C$ and $\gamma$ such that for all large enough $L$
and all low enough temperatures, depending on $Lo$, $C$ and $\gamma$,
the system satisfies our mixing condition $SM(\L,C,\gamma)$. It is
very likely (see e.g. the examples in section 1 of [MO2]) that the
system, in the same range of the parameters, does not satisfies the
condition of [SZ3].\par If $\L$ denotes the square of side $L$
($L$ odd) centered at the origin of $\bf Z^2$, then our
Hamiltonian reads as follows:
$$
({-1\over \beta})H(\sigma)=-{J\over 2}\sum_{\subset
\Lambda}\sigma(x)\sigma(y)+{K\over 2}\sum_{<\!\!\!\!>\subset\Lambda}
\sigma(x)\sigma(y)-{h\over 2}\sum_{x\in\Lambda}\sigma(x) +\,\hbox{b.c.}
\Eq(4.1)
$$
where $\sum_{\subset \L}$ runs
over the nearest neighbors pairs in $\L$, $\sum_{<\!\!\!\!>\subset \L}$
runs over the next nearest neighbors pairs in $\L$ and b.c. contains
the interaction with the boundary configuration $\t$.
Notice that, in
order to follow our convention (see section 1), we have inserted the
factor $-\beta$ directly into the Hamiltonian.\bigskip
{\bf Theorem 4.1}\bigskip
{\it There exist $C$, $\gamma$, $\bar L$ such that for any
$0\,\leq \,K\,<\,{J\over 4}$ and for any
$L\,\geq \,\bar L$, $L$ odd, there exists $\beta_o$ such that
for any $\beta \,\geq \,\beta_o$ the mixing condition
$SM(\L,C,\gamma )$ holds.}\bigskip {\bf Proof}
\par
Let us denote by
dist' the following distance on $\bf Z^2$:
$$\hbox{dist'}(x,y)\;=\;\max_i|x_i\,-\,y_i|\quad x\,,\,y\;\in \Z$$
Let
$$
{\bar {\cal C}} = \{ x \in \L ;\; \hbox {dist'} (x,\hbox {corners of }
\L) \geq l_o\} $$
with
$$
l_o = [2(J-2K) +8K] / h
$$
Thus ${\bar {\cal C}}$ looks like a cross.
The theorem will
immediately follow (see also section 5 of [MO2]) if we can show that
for any $L$ large enough, any boundary configuration $\t$ and any
$0\,\leq \,K\,<\,{J\over 4}$ the ground state of $H_{\L}^\t(\s)$
is equal to $+1$ at all the sites $x$ in ${\bar {\cal C}}$ .
Let in
fact $C$ and $\gamma$ be such that:
$$C\exp (-\gamma \sqrt 2 l_o)\,\geq\,
1\Eq(4.2)$$
If, for any boundary condition $\t$, the ground state has the
structure described above, then, because of the screening effect of
the plus spins in $ \bar {\cal C}$ the ground states in each connected
component (square) $Q_i , \, i=1,\dots,4$ of $\L \setminus \bar {\cal C}
$ is only affected by a change of a boundary spin $\t _y,\; y
\in \partial^+_{\sqrt 2} Q _i \cap \partial^+_{\sqrt 2} \Lambda$.
Therefore we can always estimate the quantity
$$
\sup_{\tau,\tau^{(y)} \in \Omega _{\Lambda_o^c}} Var(\mu_{\Lambda_o
, \Delta}^\tau\ \ \mu_{\Lambda_o , \Delta}^{\tau^{(y)}}), \quad
\Delta\subset \,\L \Eq(4.3)$$
appearing in $SM(\L,C,\gamma )$ by $1$
when for some $i=1,\, \dots , 4$
$$
y \in \partial^+_{\sqrt 2}
Q_i \cap \partial^+_{\sqrt 2} \Lambda ,\quad \D \cap Q_i \not=
\emptyset$$
or by $2\mu_{\L}^\t(\s_x \neq \,+1 \hbox{ for some }x\in {\bar {\cal C}}\;)$ otherwise.
In both cases we get an estimate smaller
than: $$C\exp (-\gamma \hbox { dist} (\Delta ,y))$$ for large enough $\beta$
because of our choice of $C$ and $\gamma$ and the fact that
$$\lim_{\beta \to \infty}\,2\mu_{\L}^\t(\s \neq \,\hbox{ ground state}\;)\;=\;0\Eq(4.4)$$
In order to prove the above structure of the set of all the ground states we will
use the following "rules" :\bigskip
\item{i)} Let $N^-$ be the number of minus spins in a ground state
configuration. Then: $$N^-\;\leq \; {4(J+2K)L\,+\,16K\over h}$$
\item{ii)} In any ground state configuration there exists no
horizontal or vertical segment of minus spins ( thought as a thin rctangle)
surronded on two
adjacent sides (one of which a "long" one) by plus spins and with at
least a plus spin along a third side. \item{iii)} In any ground state
configuration there exists no Peierl's contour with a horizontal or
vertical segment of length $l\,\geq\, l_o$ . \item{iv)} In any
ground state configuration if there exists a
Peierl's contour $\gamma$ with a right angle at the site
$(x_1+{1\over 2},x_2+{1\over 2})$ of the dual lattice,
$x\equiv (x_1,x_2)\,\in \,\L$, such that the plus spins lay along
the exterior of the angle then, starting from
$(x_1+{1\over 2},x_2+{1\over 2})$, the contour $\gamma$ has to reach
the vertical and horizontal boundary of $ \L$ without
bending.\bigskip
Rules i), ii), iii) are easily verified by simple energy
arguments if $4K\,<\,J$. Rule iv) is slightly more complicate. The
proof goes as follows.\par
Without loss of generality let us suppose that the angle has the
plus spins at its right and bottom and let us suppose that
the contour $\gamma$ has another right angle at the site
$(x_1\,-\,n\,+{1\over 2},x_2\,+\,{1\over 2})$ with $x_1\,-\,n\,
\geq \,-{(L-1)\over 2}$. Using ii) the contour $\gamma $ at the new
angle can only bend down; moreover, again by ii), the minus spins
above $\gamma$ at the sites $(x_1-j,x_2+1)\;j=1\dots n$ have to be
surrounded from the left and from above by minus spins. It is easy
to see that if the spin at the site $(x_1+1,x_2+2)$ is minus then
it is energetically convenient to flip to plus one all the minus
spins at the sites $(x_1-j,x_2+1)\;j=1\dots n$ irrespectively
of the value of the spin at the site $(x_1-n,x_2+2)$, and the
same if the spin at $(x_1-n,x_2+2)$ is minus. If
the spins at $(x_1-n,x_2+2)$ and at $(x_1+1,x_2+2)$ are both
plus then it is energetically
convenient to flip to plus all the minus spins at the sites
$(x_1-j,x_2+1)\;j=1\dots n$, $(x_1-j,x_2+2)\;j=1\dots n$. In any
case the original configuration was certainly not a ground state.
Similar arguments cover all the other situations.\par
It is easy to show, at this point, that , for every $\t$, the
structure of the ground state is the one depicted above.
\par
If $L$ is
taken large enough,
$$
L^2 > 4(3l_o)^2 [(4J+2K)L +16K]/h ,
$$
then, using i), for any ground state it is
possible to find a square $Q^*$ of side $l\,=\,3l_o$ completely
filled up with pluses and strictly contained in ${ \bar {\cal C}}$ .
Using ii), iii),
we know that on top of each face of $Q^*$ there exists a segment of
length $l_1$ larger than $l_o$ of plus spins.
If $l_1 < l$ then there is at least one right angle with exterior +
spins at one end of the concerned segment. If not, all the spins
adjacent from the exterior to that face of $Q^*$ are plus and we
can repeat the argument. Then, continuing in this way, starting
from any face of $Q^*$, either we get to $\partial \L$ on a
parallel segment of equal length $l$ or, at a given step, we find a
right angle inside $\L$. Using iv), by further decreasing the
energy, we obtain a configuration containing a cross ${\cal C}$ of
plus spins centered inside $Q^*$ with width at least $l_o$ .
The complement of ${\cal C}$ in $\L$ splits into four disjoint rectangles
$R_1, R_2,R_3,R_4$.
Each $R_i$, by construction, contains a square $ Q_i$ of side
$l_o$ having a vertex coinciding with one of the four vertices of
$\L$.
By applying to the faces of $R_i$, internal to $\L$, a construction
similar to the one leading to ${\cal C}$ it is easy to show that it
is energetically convenient to fill of pluses the sets $R_i
\setminus Q_i$ so that we end up with a configuration where
$\bar {\cal C}$ is full of pluses.
%\input formato.tex
\numsec=1\numfor=1
\tolerance=10000
\vskip 1cm
{\bf Appendix}\par\noindent
\bigskip
We prove formulae a)$\dots$ f) given at the beginning of
the proof of proposition 1.2.\par
The first three ones, a) b) c), are trivially verified. In order to
derive d) we assume, without loss of generality, that $\t_x\,=\,+1$
and we let $g_i^x\,=\,{d\n_i^{\h ,\t_{i+1}^{x,-1},.,
\t_N^{x,-1}}\over
d\n_i^{\h
,\t_{i+1}^{x,+1},.\t_N^{x,+1}}}$. Then we write:
$$\vert\partial_{\t_x}[ \n_i^{\h ,\t_{i+1},\dots ,
\t_N}(....(\n_1^{\h
,\t_2,\dots ,\t_N}(f)..)]\vert\;=\;$$
$$\vert \n_i^{\h ,\t_{i+1}^{x,-1},\dots , \t_N^{x,-1}}(....(\n_1^{\h
,\t_2^{x,-1},\dots ,\t_N^{x,-1}}(\partial_{\t_x} f)..)\;-\;
[\partial_{\t_x}( \n_i^{\h
,\t_{i+1},\dots ,\t_N}(....(\n_1^{\h ,\t_2,\dots
,\t_N}](f)\vert\;=$$
$$= \;\vert \n_i^{\h ,\t_{i+1},\dots , \t_N}(....(\n_1^{\h
,\t_2,\dots ,\t_N}(\prod_{j=i}^1g_j^x\partial_{\t_x} f)..)\;-\;
[\partial_{\t_x}( \n_i^{\h
,\t_{i+1},\dots ,\t_N}(....(\n_1^{\h ,\t_2,\dots
,\t_N}](f)|\;\leq$$
$$\leq \;|[\partial_{\t_x}( \n_i^{\h
,\t_{i+1},\dots ,\t_N}(....(\n_1^{\h ,\t_2,\dots ,\t_N}](f)|\;+$$
$$\prod_{j=i}^1\l^x_j \n_i^{\h ,\t_{i+1},\dots ,\t_N}(....(\n_1^{\h
,\t_2,\dots ,\t_N} (|\partial_{\t_x}f|)..) $$ where $|[\partial_{\t_x}
\n_i^{\h ,\t_{i+1},\dots ,\t_N}
(....(\n_1^{\h ,\t_2,\dots ,\t_N}](f)|$ denotes the expression:
$${|\n_i^{\h ,\t_{i+1}^{x,+1},.,\t_N^{x,+1}}
(.(\n_1^{\h ,\t_2^{x,+1},.,\t_N^{x,+1}}(f(\s\t)..)\,-\,
\n_i^{\h ,\t_{i+1}^{x,-1},.,\t_N^{x,-1}}
(.(\n_1^{\h ,\t_2^{x,-1},.,\t_N^{x,-1}}(f(\s\t)..)|\over
2}\Eqa(a1.1)$$
Clearly \equ(a1.1) proves d).\par
The "Leibniz rule" e) follows by essentially the same argument.\par
In order to prove f) we follow Zegarlinski [Z1] and we write:
$$4|[\partial_{\t_x}\n_i^{\h ,\t_{i+1},\dots ,\t_N}](f^2)|$$
as :
$$2|\{\n_i^{\h ,\t_{i+1},\dots ,\t_N}(f^2)\,-\,
\n_i^{\h ,\t_{i+1},\dots ,\t_N}(g_i^xf^2)\}|\,=\,$$
$$|\n_i^{\h ,\t_{i+1},\dots ,\t_N}\times
\n_i^{\h ,\t_{i+1},\dots ,\t_N}([f^2(\sigma_{\L_i})\,-\,f^2(\tilde
\sigma_{\L_i})] [g_i^x(\sigma_{\L_i})\,-\,g_i^x(\tilde
\sigma_{\L_i})])|\Eqa(a1.2)$$
where $\sigma_{\L_i}$ and $\tilde \sigma_{\L_i}$
are two independent replicas in $\Omega_{\L_i}$.\par
If we assume, without loss of generality, that
$g_i^x(\sigma_{\L_i})\,\geq\,g_i^x(\tilde \sigma_{\L_i})$ then,
from the definition of $g_i^x$, we get:
$$g_i^x(\sigma_{\L_i})\,-\,g_i^x(\tilde \sigma_{\L_i})\;\leq \;
g_i^x(\sigma_{\L_i})4|\partial_{\t_x}W_{(i)}^\h(\s\t
)|_\infty \Eqa(a1.3)$$
The Schwartz inequality gives that:
$$|\n_i^{\h ,\t_{i+1},\dots ,\t_N}\times
\n_i^{\h ,\t_{i+1},\dots ,\t_N}([f^2(\sigma_{\L_i})\,-\,f^2(\tilde
\sigma_{\L_i})])|\;\leq $$
$$\leq \;(\n_i^{\h ,\t_{i+1},\dots ,\t_N}\times
\n_i^{\h ,\t_{i+1},\dots ,\t_N}([f(\sigma_{\L_i})\,-\,f(\tilde
\sigma_{\L_i})]^2))^{1\over 2}\,2 (\n_i^{\h ,\t_{i+1},\dots
,\t_N}(f^2))^{1\over 2}\Eqa(a1.4)$$
which, in turn, implies, using (2.11) and \equ(a1.3) above, that
$$4|\partial_{\t_x}\n_i^{\h ,\t_{i+1},\dots ,\t_N}](f^2)|\;\leq $$
$$\leq \;8k_1\l^x_i\supnorm{\partial_{\t(x)}W_{(i)}^\h(\s\t)}
[\n_i^{\h ,\t_{i+1},\dots ,\t_N}(f^2)]^{1\over 2}
[\n_i^{\h ,\t_{i+1},\dots ,\t_{N}}(\grad{i}{f})]^{1\over 2}$$
i.e. the inequality f).\vskip 2cm
%\input formato.tex
\def\refj#1#2#3#4#5#6#7{\parindent 2.2em
\item{[{\bf #1}]}{\rm #2,} {\it #3\/} {\rm #4} {\bf #5} {\rm #6}
{(\rm #7)}}
\tolerance=10000
\vskip 1cm
{\bf Acknowledgments}\bigskip\noindent
We would like to thank B. Zegarlinski for
some useful discussions concerning the differences between the
results and the methods of the present paper and those developed
in his joint work with D. Stroock; we are grateful to H.T.Yau for
informing and explaining to us his work prior to publication.
\bigskip
\centerline{\bf References}\bigskip\noindent
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\end
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