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\vskip 1 true cm
\centerline{{\gross A Remark on Differentiability of the Pressure
Functional}}
%\centerline{{\gross for Lattice Spin Systems with
%Multiparticle Interactions}}
\vskip 0.5 true in
\flushpar
\centerline{\bf Aernout van Enter}\smallskip\flushpar
\centerline{{\ss Institute for Theoretical Physics}}\flushpar
\centerline{{\ss University of Groningen}}\flushpar
\centerline{{\ss Groningen, The Netherlands}}
\bigskip\flushpar
\bigskip\flushpar
\centerline{\bf Boguslaw Zegarlinski}\smallskip\flushpar
\centerline{{\ss Department of Mathematics}}\flushpar
\centerline{{\ss Imperial College}}\flushpar
\centerline{{\ss London, UK}}
\bigskip\flushpar
\bigskip\flushpar
{\bf Abstract}:
We give a short review of results on equilibrium description and
description by stochastic dynamics for spin systems on a lattice.
We remark also that some coercive inequalities for the generators
of stochastic dynamics, as e.g. the Logarithmic Sobolev inequality,
can be used in a direct and natural way to prove
strong differentiability properties of the pressure functional
for lattice spin systems with multiparticle interactions at high temperatures.
Motivated by this, we exhibit also a class of examples of multiparticle
interactions which do not belong to the space $\B2$ of spin interactions,
but for which the Gibbs measures exist and are unique at high temperatures.
%%%%%%%%%%\endofabstract
\bigskip\bigskip\flushpar
{\bf 1. An Introduction}. \smallskip\flushpar
In recent years great progress has been made in understanding the
connection between the equilibrium description and the description
by stochastic
dynamics of lattice spin systems, see e.g. \cite{SZ1-3}, \cite{Z1-3},
\cite{MO1,2}, \cite{LY}, \cite{La}. The method used there has been based on
an indirect application of some inequalities involving a Gibbs measure and
the Dirichlet form of a generator associated to a related stochastic
dynamics.
In the present paper we would like to show that one can use them also in
a direct and natural way to obtain useful information about the strong
(or Fr\'echet) differentiability of the pressure functional.
To explain our motivation and results we would like first
to recall some known facts. We need also
to introduce some notation necessary to describe spin systems on
a lattice $\Gamma\equiv\Z^d$.
Let $\F$ be the family of all finite subsets of the lattice and let
$\F_0$ be a countable exhaustion of $\Gamma$, i.e. an increasing sequence
of finite sets invading all the lattice.
A configuration space of a spin system is by definition
a product space $(\Omega,\Sigma)\equiv(\S,\CB_\S)^\Gamma$, where
the single spin space $\S$ is either a finite set or
a (smooth, compact, connected) Riemannian
manifold and $\CB_\S$ denotes the Borel $\sigma$-algebra of subsets in $\S$.
The interaction of the spins is described by the interaction potential
$\Phi\equiv\{\Phi_X\}_{X\in\F}$, where
to a finite subset $X$ of the lattice we associate a (real) continuous
function $\Phi_X(\omega )\equiv \Phi_X(\omega _X)$, which depends only
on the spins in the set $X$. We will assume that the
interactions are translation invariant in the sense that
$\Phi_X(T_\j\omega)= \Phi_{X+\j}$, where
$(T_\j\omega)_\i\equiv\omega_{\i-\j}$.
It is convenient to classify the interactions using the following norms
$$
|| \Phi ||_\g \equiv
\sum_{X\in\F\atop {X\ni\0} }\g(X)\cdot ||\Phi _X||_u
\eqno(0.1)$$
where $\g $ is some positive translation invariant function on $\F$, and
$||\,\cdot\,||_u$ denotes the supremum norm.
The corresponding Banach spaces are denoted by $\B \g$.
In particular if $\g(X) = |X|^{n-1}$ we will denote the corresponding space
by $\B n$. Obviously we have $\B {n+1}\subset\B n$, for any $n\in\Z^+$.
\flushpar
It is known (see e.g.\cite{R2,3}) that the following
pressure functional is well defined, continuous and convex on $\B0$
$$
p(\Phi ) \equiv \vlim p_{\Lambda }(\Phi)
\eqno(0.2)
$$
where the finite volume pressure $p_\Lambda (\Phi )$ is given by
$$
p_{\Lambda }(\Phi) \equiv \frac 1{|\Lambda|} \log\mu_0^\Lambda
e^{-H_\Lambda(\Phi) }
\eqno(0.3)
$$
with $\mu _0^\Lambda $ denoting the restriction of the free measure
$\mu_0$,( which is defined as the product of uniform probability measures on
$(\S,\CB_\S)$ ), to the coordinates in the finite set $\Lambda $, and
$$
H_\Lambda(\Phi)\equiv\sum_{X\subset\Lambda }\Phi_X
\eqno(0.4)$$
and $\vlim$ denotes the limit with
a van Hove sequence $\hbox{v}\F\equiv \{\Lambda_n\in\F\}_{n\in\N}$
of finite sets invading all the lattice $\Gamma $.
Then an equilibrium state $\mu _\Phi $ of the spin system with
an interaction $\Phi\in\B0$ is defined as a tangent functional
to the pressure $p$ at the point $\Phi $, i.e. for any $\Psi\in\B0$
we have
$$
p(\Phi+\Psi)\geq p(\Phi) - \mu_\Phi(\A_\Psi)
\eqno(0.5)$$
where we have set
$$
\A_\Psi \equiv \A(\Psi )\equiv\sum_{X\in\F \atop{X\ni 0}}\frac 1{|X|}\Psi_X.
\eqno(0.6)$$
It satisfies also the following variational principle
$$
p(\Phi) = \sup\left\{- s(\nu) -\nu(\A_\Phi):\nu\in\bM_I\right\}
= -s(\mu _\Phi) - \mu _\Phi (\A_\Phi)
\eqno(0.7)$$
where $s(\nu )$ denotes the entropy of a translation invariant probability
measure $\nu \in\bM_I$ defined by
$$
s(\nu)\equiv\vlim\mu_0^\Lambda \left(
f_\Lambda \log f_\Lambda \right)
\eqno(0.8)$$
if the Radon-Nikodym derivative
$$
f_\Lambda \equiv\frac{d\nu ^\Lambda}{d\mu _0^\Lambda}
$$
of the restriction of the involved probability measures to the
$\sigma $-algebra generated by the spins in the set $\Lambda $ is finite
(and equals $+\infty$ otherwise).
%Equivalently we have also
%
%$$
%s(\mu )=\inf\{ p(\Psi)-\mu(\Psi):\Psi\in\B0\}
%\eqno(0.8')$$
\flushpar
It is known, \cite{DvE}, \cite{I2,3,4}, \cite{IP}, \cite{So}, \cite{W},
\cite{vEFS}, that the pressure
functional on the space $\B0$ has some pathological properties from
the point of view of physical applications. In particular the correspondence
between the interaction potentials and equilibrium states can be very weird,
(even if one takes into account the physical equivalence of the interactions)
in the sense that almost any measure can an equilibrium measure for various
non-equivalent interactions.
Moreover the pressure functional is nowhere Fr\'echet differentiable
in $\B0$.
%, which causes the problems with proper physical interpretation
%of the high temperature phase as well as of Gibbs phase rules in that space.
This means in particular that there is no high temperature regime.
There does not exist a ball around the origin in the space $\B0$, such that
for each interaction in this ball there exists a unique equilibrium measure.
At low temperatures all the spaces $\B n$ are too big, in the sense that in
all of them the Gibbs phase rule is generically violated and the pressure is
not Fr\'echet differentiable at any phase coexistence point in any subspace
of finite codimension.
A slightly better situation one finds in spaces $\B\g$ defined with
$\g$ such that $diam(X)\cdot\g(X) \to_{diam(X)\to\infty} \infty$, where,
whenever
$p(\Phi)$ is Gateaux differentiable at $\Phi $,
it is also Fr\'echet differentiable at this point (and the set of such
potentials is a dense $G_\delta$ set in $\B\g$),\cite{Ph}.
For the Gibbs rule to be valid one needs stronger conditions of the type
$\sum_{X\ni 0} diam (X) \cdot \g(X) < \infty$, \cite{vE}, \cite{Pa2}.
\flushpar
The potentials from the space $\B1$ are called Gibbsian. For $\Phi\in\B1$
and any $\Lambda\in\F$, $\omega\in\Omega$ we can define a local Gibbs
measure $\mu_{\Phi, \Lambda}^\omega\equiv\mu_\Lambda^\omega $ as follows
$$
\mu_\Lambda^\omega(f)\equiv\delta_\omega\left(
\frac{\mu_0^\Lambda( e^{-U_\Lambda }f)}{\mu_0^\Lambda( e^{-U_\Lambda })}
\right)
\eqno(0.9)
$$
where $\delta_\omega $ is the Dirac measure concentrated at the configuration
$\omega\in\Omega $ and the interaction energy
$U_\Lambda\equiv U_\Lambda(\Phi)$ for the volume $\Lambda\in\F$ is given by
$$
U_\Lambda(\Phi)\equiv\sum_{X\cap\Lambda\neq \emptyset }\Phi_X
\eqno(0.10)$$
A Gibbs measure for an interaction $\Phi\in\B1$ is defined as
a solution of the Dobrushin-Lanford-Ruelle equations
$$
\mu\left( \mu_{\Phi, \Lambda}^{\cdot} f \right)\, = \,\mu(f)
\eqno({\DLR})$$
Since our configuration space $\Omega $ is compact and for every
$\Lambda\in\F$ the map $\omega\longmapsto
\mu_{\Phi, \Lambda}^{\omega }(\,\cdot\,)$ is continuous in the weak topology,
the set of solutions of the (\DLR) is nonempty. On physical grounds one
would expect that if the interaction is sufficiently small, then the solution
should be unique. {\it Unfortunately a result of this generality remains still
unknown.} A better situation is observed in the subspaces $\B1^{(spin)}$
and $\B1^{(gas)}$ defined
%in the case of the single spin space $\S=\{-1,+1\}$
respectively by
$$
\B1^{(spin)}
\equiv\left\{\Phi\in\B1:\,\forall X\in\F\exists J_X\in\R\quad\Phi_X(\omega)=
J_X\cdot\sigma_X \right\}
\eqno(0.11)$$
and
$$
\B1^{(gas)}
\equiv\left\{\Phi\in\B1:\,\forall X\in\F\exists j_X\in\R\quad\Phi_X(\omega)=
j_X\cdot\n_X \right\}
\eqno(0.12)$$
where $\sigma_X \equiv \prod_{\i\in X}\sigma_\i$,
with $\sigma_\i\equiv\sigma_\i(\omega )\equiv\sigma_\i(\omega_\i)$
is an affine function of the coordinate $\omega_\i$, called the spin at site
$\i\in\Gamma $, and
$n_X\equiv \prod_{\i\in X} n_\i$ is defined with the occupation number
variable $n_\i\equiv\half (1 + \sigma_\i)$. It is known, \cite{GM},
\cite{GMR}, \cite{I1}, \cite{HS2}, that in both these spaces, when the
interaction potential is sufficiently small, the Gibbs measure is not
only unique, but it is also analytic with respect to the (finite dimensional)
changes of the potential in a small neighborhood.
(A result of \cite{DM2} shows that in the general spaces $\B n$ one
can not have
analyticity even for small potentials, i.e. even at high temperatures;
see also \cite{vEF} for discussion. This answers Ruelle's question \cite{R1}
in the negative. But one always has analyticity at high
temperatures in the space $\B {exp}$ defined
with $\g (X)\equiv \exp (\alpha |X|)$ for some $\alpha >0$, \cite{DM1},
\cite{I1}, \cite{Pa}.)
\flushpar
An interesting general condition for the uniqueness of the Gibbs measure
has been introduced in \cite{D1} and since then has been known as
{\it the Dobrushin uniqueness condition}; for nice expositions of the
Dobrushin theory see also \cite{F\"o}, \cite{L}, \cite{Ge}, \cite{S}.
It has been shown in \cite{G1} that this condition alone is sufficient
for the pressure functional
on the space $\B2$ to be twice continuously differentiable in the weak sense
at a point $\Phi $, provided that
$$
||\Phi||_2'\equiv \sum_{X\in\F\atop {X\ni\0} }(|X|-1)\cdot ||\Phi _X||_u
<1
\eqno(0.13)$$
Let us stress that this fact is true without any specific assumptions about
the type of interactions or the single spin space.
The inequality (0.13) is sufficient for the Dobrushin uniqueness condition
to be true, but not necessary. Below we will give an explicit example of
a class of interactions in $\B 1\setminus\B 2$ for which the Dobrushin
condition remains true. Since the proof of Gross relies in fact
only on the Dobrushin condition and not on the inequality (0.13), our
examples extend the region of applicability of his result.\flushpar
Let us mention that the result of Gross obviously implies that
the pressure functional is once Fr\'echet differentiable in $\B 2$ for
the potentials for which the Dobrushin uniqueness condition is true.
In general it does not imply the twice Fr\'echet differentiability
of the pressure under this condition. The differentiability property
should be uniform in all directions, which is more than Gross states.
%%%%\flushpar\hfill\hskip 1 true cm
\flushpar
The nice result of \cite{G1} has been later extended in \cite{Pr}, where
the author has shown that the pressure functional is $n$-times continuously
differentiable on the space $\B{n}$, $n\in\N$, $n\geq 2$, respectively,
at high temperatures (small interactions).
\flushpar
Again the author in his proofs has in fact used a slightly weaker condition
than the inequality (0.13) plus the corresponding $\B {n}$-type conditions
and we believe that his results remains true also for our examples;
(compare \cite{G2}).
%%%%%%%%%\flushpar\hfill\hskip 1 true cm
\flushpar
(Moreover under the conditions of \cite{Pr} one should have $n-1$ Fr\'echet
differentiability.)
%%%%%\flushpar\hfill \hskip 1 true cm
\flushpar
Let us come now to the description of lattice sytems involving
stochastic dynamics.
%A stochastic dynamic is given by a Markov
%semigroup $P_t \euqiv e^{t\L}$, $t\geq 0$, with a generator $\L$ given
One defines a stochastic dynamics starting by introducing a Markov
pre-generator $\L$ defined on the set of (smooth) local functions
(i.e. dependent only on a finite number of coordinates), which is dense in the
space of continuous functions, by the following formula
$$
\L f\equiv\sum_{\j}\L_\j f
\eqno(0.14)$$
with the local generators $\L_\j$ defined in the discrete case (when $\S$ is a
finite set) as the local spin-flip operators
$$
\L_\j f(\omega)\equiv
\L_\j^Y f(\omega)\equiv \mu^\omega _{Y+\j}f - f(\omega)
\eqno(0.15)$$
with some set $Y\in\F$ and in the continuous case (when $\S$ is the Riemannian
manifold) as the local diffusion operators
$$
\L_\j f\equiv\Delta_\j f -\nabla_\j U_\j\cdot\nabla_\j f
\eqno(0.16)$$
where $\Delta_\j$ and $\nabla_\j$ denote the Laplace-Beltrami and the
gradient operators, respectively, with respect to coordinate $\omega_\j$.
Given $Y\in\F$, the corresponding spin flip generator will be denoted later
on by $\L^Y$.
A pre-generator introduced in this way satisfies the following detailed
balance condition with respect to any Gibbs measure corresponding to the same
potential
$$
\mu f_1 \cdot\L f_2 =\mu \L f_1\cdot f_2
\eqno(0.17)$$
(A similar property holds with the operators obtained by replacing the
corresponding local operators $\L_\j$ by $A_\j\L_\j$ with positive
functions $A_\j$ which are independent of the coordinates $\omega_{X+\j}$,
or by taking a convex linear combination of operators defined in this
way.) Traditionally the generators of the spin flip dynamics have been also
introduced by the formula
$$
\L_\j f(\omega ) \equiv \alpha_\j(\omega ) \partial_\j f(\omega )
\eqno(0.18)
$$
with $\partial_\j$ being the discrete gradient operator defined by
$$
\partial_\j f(\omega ) \equiv f(\omega ) - \mu _0^{\{\j\}}( f )
\eqno(0.19)
$$
and rate coefficients $\alpha_\j$ which are independent of the $\omega_\j$.
In fact, in the literature concerning the construction of the stochastic
dynamics, see e.g. \cite{Su1,2}, \cite{Li1}, \cite{HS1}, one usually
formulates the conditions for a pre-generator $\L$ to be extendible
to a Markov
generator in terms of the rate coefficients. It is well - known that
in terms of the interaction potential $\Phi $
a sufficient condition for a pre-generator $\L\equiv\L_\Phi $
as defined above to be extendible
to the generator of a Markov semigroup $P_t$, $t\geq 0$ {\it on the space of
continuous functions} $\C(\Omega)$ is the following
$$
\sup_\i \sum_\j || \nabla_\j \nabla_\i U_\i||_u < \infty
\eqno(0.20)
$$
and similarly for the discrete case with the gradients replaced by their
discrete counterparts. In particular one can see that for discrete
spins the condition (0.20) is satisfied if $\Phi\in\B 2$, (although as we will
see later the condition (0.20) is much better).
If we restrict ourselves to spin interactions, then the best result in
this case one can find in \cite{HS2}, where the stochastic dynamics has
been constructed for potentials $\Phi\in\B1^{(spin)}$ satisfying
the bound
$$
||\Phi ||_1 = \sum _{X\in \F \atop {X\ni 0}} | J_X | < \frac \pi 4
\eqno(0.21)
$$
For general interactions $\Phi\in\B1$ the existence of the stochastic dynamics
on the space of continuous functions remains an open problem and at the moment
the best that one can have for a general Gibbsian potential is
a stochastic dynamics in $L_p(\mu_\Phi)$, $p\in[1,\infty]$,
for a corresponding Gibbs state $\mu_\Phi$.
(This follows from the fact that each of our pre-generators is given on
a dense
domain in $L_2(\mu_\Phi)$ as a nonpositive and symmetric operator. Thus
it always admit a trivial Friedrichs extension and using the definition of
our pre-generator one can see that this extension generates a Markov
semigroup in $L_2(\mu_\Phi)$ which extends uniquely to the Markov semigroups
in any $L_p(\mu_\Phi)$.)
As a symmetric pre-generator may admit many different extensions, this trivial
construction is highly not satisfactory. In the non-reversible case
a condition on the rate functions similar to $\B2$ is enough for existence,
while (see an example of Gray, \cite{Li1} p.53), a $\B1$ like condition is
known to be not enough. Whether in the reversible case every interaction in
$\B1$ give rise to a stochastic dynamics as described above seems to be open.
%%%%%%%%%%%%\flushpar \hskip 1 true cm \hfill
\flushpar
The literature concerning the ergodicity problem, i.e. the question whether
or not for a stochastic dynamics $P_t$ defined with respect to some
interaction we have a return to equilibrium
$$
P_t f\longrightarrow_{t\to\infty}\mu f
\eqno(0.22)$$
with $\mu $ being a unique Gibbs measure for the same interaction,
is very vast.
Let us mention here only some selected results; for a more comprehensive
list of references consult \cite{Li1}, \cite{Li2}.
The first sufficient condition for strong ergodicity
(with convergence to equilibrium in the uniform norm) for the single
spin flip dynamics corresponding to potentials of
finite range has been given in \cite{D2}. It naturally extended the uniqueness
condition of the author for the equilibrium description. Later this result
has been generalized in \cite{Su1} to include long range potentials as well
as multispin flip generators, \cite{Su2} (\cite{Li1}).
All these results when applied to a lattice spin system implied
the uniqueness of the $P_t$-invariant measure and, since
every Gibbs measure with the same potential is $P_t$-invariant, also the
uniqueness of the Gibbs measure. (Let us mention that in dimensions $d\geq 3$
it is still an open problem whether or not every $P_t$-invariant measure
is a Gibbs measure; for the case $d\leq 2$ see \cite{HS3}.)
\flushpar An interesting use of the stochastic dynamics to prove
analyticity properties of spin systems at high temperatures has been
made in \cite{HS2}.
\flushpar A nice extension of these ergodicity results has been
obtained in \cite{AH}. These authors have shown that if a local specification
satisfies the Dobrushin-Shlosman uniqueness condition, \cite{DS1}, in a box
$X$, (which is an extension of the Dobrushin uniqueness condition), then
every stochastic dynamics with a generator $\L^{Y}$, defined for
a box $Y\in\F$, $X\subseteq Y$, is strongly ergodic.
This implies that also any other stochastic dynamics with a generator
$\L^{Z}$, $Z\in\F$ has a spectral gap
in $L_2(\mu )$, where $\mu$ is the corresponding unique Gibbs measure,
(although it says nothing about their strong ergodicity properties).
Let us mention that in connection to this there was a conjecture,
that if a generator of a stochastic dynamics as discussed above has a spectral
gap in $L_2(\mu )$ for some Gibbs measure corresponding to a given
interaction, then this measure must be unique. We discuss this point later
in connection with the Fr\'echet differentiability.
Since the Dobrushin-Shlosman condition applies to lower temperatures
than the original Dobrushin uniqueness condition,
%(in fact in some ferromagnetic systems extends up to critical point,
%\cite{MO}),
this method gave a broader range of applications
than the previous ergodicity conditions.
However one knows that if the
temperature is lowered, but when we still remain above the critical point,
to satisfy the Dobrushin-Shlosman uniqueness condition one would have to take
larger and larger cubes $X\equiv X(\beta )$ with $|X(\beta )| \to \infty$
when the inverse temperature approaches the critical value. This means
that by that method we cannot get anything about the strong decay to
equilibrium for any stochastic dynamics $P_t^{Y}\equiv e^{t\L^Y}$ with
any fixed box $Y\in\F$. A somewhat weaker version of Dobrushin-Shlosman
theory, applying only to volumes built up from "fat cubes", gives
the ergodicity in the uniform norm all the way to $T_c$ in various models.
Whether the original Dobrushin-Shlosman theory applies in this region is
an open problem for the Ising model, but is certainly not true for Potts
models \cite {vEFK}. (If additionally one knows that the system is
ferromagnetic, the dynamic problem is solvable; cf. \cite{MO1} for
an extension of the arguments by Holley \cite{H}).\flushpar
To overcome these problems Holley and Stroock, \cite{HS4}, have invented
a very elegant strategy based on the hypercontractivity property of
the semigroup $P_t$, which means the following property
$$
|| P_t f ||_{L_q(\mu )} \leq ||f||_{L_2(\mu )}
\eqno(0.23)$$
with $q\equiv q(t) \equiv 1+ e^{\frac 2c t}$, $c\in(0,\infty)$ and $\mu $
a $P_t$-invariant measure. It is known that every Markov semigroup
is contractive in any $L_p(\mu )$ space and the property (0.23) should be
regarded as rather peculiar as it says that the semigroup is contractive
from the $L_2(\mu )$ space to the $L_q(\mu )$ space with $q\to\infty$
very rapidly as time increases. To explain why the hypercontractivity
property helps to solve the strong ergodicity problem for all dynamics
$P_t^Y$, $Y\in\F$, knowing it for one of them, let us mention that, as shown
in \cite{G3}, this property is equivalent to the following
Logarithmic Sobolev inequality with a coefficient $c\in(0,\infty)$
$$
\mu f\log f\leq 2c\mu\left( f^\half(-\L^X f^\half)\right)
\eqno(\LS)$$
for positive functions $f$ with $\mu f=1$ for which the right hand side is
finite. Now we observe that the right hand side of the Logarithmic Sobolev
inequality depends on a Markov generator through its Dirichlet form. Thus,
if it is proven for one generator $\L^X$, it automatically implies a similar
inequality for any other spin flip generator $\L^Y$, $Y\in\F$, since all
their Dirichlet forms are equivalent. (Similar arguments have been used
in \cite{AH} to get the spectral gap property for all the spin flip
generators once it is proven for one of them associated to a large box.)
\flushpar
The first proof of \LS has been obtained in \cite{CS} for the Gibbs measures
of continuous spin systems on a lattice at high temperatures by application
of the Bakry - Emery criterion. (The authors considered only finite range
interactions, although no conceptual problems would arise for a more general
case). A slight generalization of this result to similar systems
one can find in \cite{DeS}. A new approach to the \LS based on an application
of the Gibbs structure has been introduced in \cite{Z1-3} and developed
later in \cite{SZ1-3}, \cite{MO2}, \cite{LY}, \cite{La}. In particular
it has been shown in \cite{SZ2} that, for systems with finite range
interactions, the Logarithmic Sobolev inequalities for conditional measures
$\mu _{\Phi ,\Lambda }^\omega $ with a coefficient $c$ independent of
the volume $\Lambda $ and external configuration $\omega$ are equivalent
to {\it the complete analyticity} of Dobrushin and Shlosman, \cite{DS2,3},
i.e. the analyticity of the map
$\Phi \longmapsto \mu _{\Phi ,\Lambda }^\omega$ with the radius of analyticity
independent of the set $\Lambda $ and the configuration $\omega $. (Let us
remind the reader that complete analyticity is the strongest analyticity
property possible. It can fail when we are close to the critical region,
although the infinite volume Gibbs measures as well as the conditional
measures for some "fat" sets can retain their analyticity properties.)
The proof of the equivalence of \LS for all conditional measures and
the complete analyticity property involved a rather complicated route through
the corresponding strong ergodicity property of the finite volume stochastic
dynamics.
\flushpar
In our short note we would like to show that one can use \LS in a direct
and natural way to prove the strong (Fr\'echet) differentiability of the
pressure functional for spin systems with multiparticle interactions.
This is done in the next section. Later we discuss the connection of the
spectral gap of the generator with the strong differentiability.
Finally at the end we give also an example of an interesting class of
multiparticle interactions contained in $\B1\setminus\B2$, for which
we can prove \LS as well as Dobrushin uniqueness.
\bigskip\flushpar
{\bf 1. High Temperature Differentiability via Logarithmic Sobolev
Inequalities}.
\bigskip\noindent
\smallskip\noindent
As the Logarithmic Sobolev inequality can be viewed as an estimate on the
relative entropy, it is natural to expect that it should have some
thermodynamic consequences. In this section we discuss this matter
in more detail.
>From now on we will consider only translation invariant interactions with
potentials having finite $||\,\cdot\,||_1$ norm and if necessary
(in the continuous case) sufficiently smooth. We begin by noting that
every Markov generator discussed before has a Dirichlet form equivalent
to the one defined as an expectation with respect to the same Gibbs measure
of the following square of the gradient
$$
|\nabla f|^2 \equiv \sum_{\j\in\Gamma } |\nabla_\j f|^2
\eqno(1.1)$$
in the case of continuous spins and with $\partial_\j$ replacing
$\nabla_\j$ in the discrete case. Later we will need also to consider
the corresponding Dirichlet forms for the finite volume Gibbs measures.
In this case we will have to consider the square of a finite volume
gradient $\nabla_\Lambda f$ defined similarly as in (1.1), but with the
summation over $\j$'s restricted to the set $\Lambda $.
For a given interaction $\Phi $ we define a finite volume pressure
$p_{\Phi ,\Lambda}^ \omega$ as follows
$$
p_\Lambda^ \omega(\Phi)\equiv\frac 1{|\Lambda|}\log\delta_\omega
\mu_0^\Lambda e^{-U_\Lambda(\Phi) }
\eqno(1.2)$$
We have the following simple fact.
\bigskip \flushpar
{\bf Lemma 1.1}\smallskip\flushpar
Suppose that for an interaction $\Phi+\Psi $, we have with
some constant $c_{\Phi + \Psi}\in (0,\infty)$, independent of $\Lambda $ and
$\omega $,
$$
\mu_{\Phi+\Psi,\Lambda}^\omega f\log f\leq 2c_{\Phi+\Psi}
\mu_{\Phi+\Psi,\Lambda}^\omega |\nabla_\Lambda f^\half|^2
\eqno(1.3)$$
for every positive and normalized function $f$ for which the right
hand side is finite. Then the following inequality is true
$$
0\leq p_\Lambda^ \omega(\Phi +\Psi) - p_\Lambda^ \omega(\Phi)
- \mu_{\Phi,\Lambda}^\omega (\fraL \sum_{\j\in\Lambda}
\A_{\j,\Lambda}(\Psi))\leq \half
e^{2||\Psi ||_1} c_{\Phi+\Psi}
\mu_{\Phi,\Lambda}^\omega \fraL\sum_{\j\in\Lambda}
|\sum_{X\cap \Lambda \neq \emptyset \atop{ X\ni\j}}
\nabla_\j\Psi_X|^2
\eqno(1.4)$$
where
$$
\A_{\j,\Lambda}(\Psi) \equiv - \sum_{X\cap\Lambda\neq \emptyset \atop{X\ni\j}}
\fraX \Psi_X
\eqno(1.5)$$
In the continuous case the factor $e^{2||\Psi ||_1}$ can be omitted.
\cir
{\bf Proof}:
The lower bound is a simple consequence of the convexity of the finite
volume pressure. To prove the upper bound we use the Logarithmic Sobolev
inequality for the measure $\mu_{\Phi+\Psi,\Lambda}^\omega$
with the function
$$
f =\frac {e^{+U_\Lambda(\Psi)}}
{ \mu_{\Phi+\Psi,\Lambda}^\omega e^{+U_\Lambda(\Psi)}}
\eqno(1.6)$$
to get in the continuous case
$$
|\Lambda |\cdot p_\Lambda^ \omega (\Phi +\Psi) -
|\Lambda|\cdot p_\Lambda^ \omega(\Phi)
- \mu_{\Phi,\Lambda}^\omega (-U_\Lambda(\Psi))\leq
\half c_{\Phi+\Psi} \mu_{\Phi+\Psi,\Lambda}^\omega |\nabla_\Lambda
U_\Lambda(\Psi)|^2
\eqno(1.7)$$
In the discrete case, because of peculiar features of the discrete
differential, we get an extra factor $e^{2||\Psi ||_1}$ on the right hand
side. Now using the representation
$$
U_\Lambda (\Psi)=\sum_{\j\in\Lambda}\left(\sum_{X\cap\Lambda\neq\emptyset\atop
{X\ni\j}}\fraXL\Psi_X\right)\equiv -
\sum_{\j\in\Lambda }\A_{\j,\Lambda }(\Psi)
\eqno(1.8)$$
and dividing both sides of (1.7) by the volume of $\Lambda$, we get the right
hand side bound in (1.3). This ends the proof of the lemma.
\qed
Now we note that for (translation invariant)
interactions in $\B1$, we have
$$
\vlim p_\Lambda^\omega(\Phi+\Psi)= p(\Phi+\Psi)
\eqno(1.9)$$
and similarly with the interaction $\Phi $, independently of the configuration
$\omega $. Also for any translation invariant Gibbs measure $\mu _\Phi $
for the potential $\Phi $, we get
$$
\vlim \mu _\Phi \left(\mu_{\Phi,\Lambda}^\omega (\fraL \sum_{\j\in\Lambda}
\A_{\j,\Lambda}(\Psi))\right) =
\vlim \mu _\Phi \left(\fraL \sum_{\j\in\Lambda}
\A_{\j,\Lambda}(\Psi)\right) = \mu_{\Phi }\A_0(\Psi)\equiv\mu_{\Phi }\A(\Psi)
\eqno(1.10)$$
and
$$
\vlim \mu _\Phi \left(\mu_{\Phi,\Lambda}^\omega \fraL\sum_{\j\in\Lambda}
|\sum_{X\cap \Lambda \neq \emptyset \atop{ X\ni\j}}
\nabla_\j\Psi_X|^2\right) = \vlim \mu _\Phi \left( \fraL\sum_{\j\in\Lambda}
|\sum_{X\cap \Lambda \neq \emptyset \atop{ X\ni\j}}
\nabla_\j\Psi_X|^2\right) \equiv \mu_{\Phi}\bB^2(\Psi)
\eqno(1.11)$$
with a function $\bB$ defined by
$$
\bB^2(\Psi)\equiv |\, \sum_{X\in \F \atop{ X\ni\0}}
\nabla_\0\Psi_X \,|^2
\eqno(1.12)$$
Hence, averaging the inequality (1.4) with respect to the
Gibbs measure $\mu _\Phi $ and passing to the thermodynamic limit with the use
of (1.9) - (1.12), we conclude with the following statement in which
we use a Banach space of interactions $\H1$ defined with some norm
$||\Phi||_{\H1}$ satisfying
$$
||\Phi ||_{\H1} \geq \left(||\Phi||_1 ^2 + \bB^2(\Phi)\right)^\half
\eqno(1.13)$$
\bigskip\flushpar
{\bf Theorem 1.2}:
\smallskip\flushpar
Suppose in some neighborhood $\O_\Phi\subset\H1 $ of an interaction
$\Phi$ the finite volume measures
$\{\,\mu _{\Phi +\Psi, \Lambda}^\omega\,\}_{\Lambda\in\F_0,\omega\in\Omega }$,
defined with some countable exhaustion $\F_0$,
%and some configuration $\omega \in\Omega $,
satisfy the Logarithmic Sobolev inequality with coefficients
$c(\Phi+\Psi)\leq C(\O_\Phi)$, for some constant $C(\O_\Phi)\in(0,\infty)$
independent of $\Lambda\in\F_0$ and of the interactions from $\O_\Phi $.
Then we have
$$
0\leq p (\Phi +\Psi) - p (\Phi)
- \mu_{\Phi} \A(\Psi)\leq \half
e^{2||\Psi ||_1} c_{\Phi+\Psi}
\mu_{\Phi} \bB^2 (\Psi)
\eqno(1.14)$$
and therefore the pressure functional $\H1\ni\Xi \longmapsto p(\Xi )$
is {\bf Fr\'echet differentiable} at the point $\Phi$.
\cir
{\bf Proof}: The proof of the Fr\'echet differentiability of the
pressure functional at the point $\Phi $ clearly follows from the inequality
(1.14), since by our assumption (1.13) we have
$$
0\leq p (\Phi +\Psi) - p (\Phi)
- \mu_{\Phi} \A(\Psi)\leq \half
e^{2||\Psi ||_1} C(\O_\Phi)||\Psi||_{\H1}^2
\eqno(1.15)$$
\qed
Let us remark that in case of discrete spins we have a crude bound
$$
%\left(||\Phi||_1 ^2 +
\bB(\Phi)
%\right)^\half
\leq
%\left(
2||\Phi||_1 %%%%%%%%%%% ^2 + 4||\Phi||_2^2\right)^\half
\eqno(1.16)$$
We will show however that for some interactions the left hand side can
be actually much smaller.
%%% (and in fact finite for some interactions from
%%%% $\B1\setminus\B2$).
\flushpar One could also slightly improve the above result, by imposing
only an assumption of uniform \LS for some $\omega\in\Omega $.
\bigskip\flushpar
{\bf 2. Spectral Gap and the High Temperature Differentiability}.
\bigskip\noindent
\smallskip\noindent
In this section we discuss briefly the uniqueness hypothesis under
the assumption of a spectral gap for the generator of the stochastic
dynamics. Due to the already mentioned equivalence of the Dirichlet forms
for various dynamics, it is sufficient to consider the following
universal inequality, called {\it the spectral gap inequality}
$$
m\cdot \mu( f, f)^2\leq\mu |\nabla f|^2
\eqno(\SG )$$
with a constant $m\in(0,\infty)$ independent of the function $f$, and
$\mu(f,f)\equiv\mu(f-\mu f)^2$.
The assumption of a spectral gap is weaker than \LS and
in fact it is well known that if a probability measure $\mu $ satisfies \LS
with a coefficient $c\in(0,\infty)$, then it automatically satisfies \SG
with
$$ m \geq \frac 1 c
\eqno(2.1)$$
Now let us consider the Taylor expansion to the first order with
remainder for the finite volume pressure. (Without making
explicit the boundary conditions), we have
$$
0\leq p_\Lambda (\Phi +\Psi ) - p_\Lambda (\Phi ) -
\fraL\mu _{\Phi,\Lambda} (-U_\Lambda(\Psi ))
= \int_0^1dt\int_0^t ds\,
\fraL\mu _{\Phi+s\Psi ,\Lambda} (U_\Lambda(\Psi ), U_\Lambda(\Psi ))
\eqno(2.2)$$
If we suppose that the measures $\mu _{\Phi+s\Psi ,\Lambda}$
satisfy the Spectral Gap inequality with the corresponding spectral
gap $m(\Phi+s\Psi ,\Lambda)$, we get
$$
p_\Lambda (\Phi +\Psi ) - p_\Lambda (\Phi ) -
\fraL \mu _{\Phi,\Lambda} (-U_\Lambda(\Psi ))
\leq \int_0^1dt\int_0^t ds\,\left( m(\Phi+s\Psi ,\Lambda)\right)^{-1}
\fraL \mu _{\Phi+s\Psi ,\Lambda} |\nabla_\Lambda U_\Lambda(\Psi )|^2
\eqno(2.3)$$
One can arrange that the following limit exists and is equal to the
corresponding expectation with a (translation invariant) Gibbs measure
$\mu _\Phi $
$$
\lim_{\F_0}\fraL \mu _{\Phi,\Lambda} (-U_\Lambda(\Psi )) =
\mu _\Phi (\A(\Psi ))
\eqno(2.4)$$
Since we have
$$
\lim\sup_{\F_0}
\fraL \mu _{\Phi+s\Psi ,\Lambda} |\nabla_\Lambda U_\Lambda(\Psi )|^2
\leq \bB(\Psi )^2
\eqno(2.5)$$
after passing to the thermodynamic limit we get
$$
0\leq p (\Phi +\Psi ) - p (\Phi ) -
\mu _\Phi (\A(\Psi ))\leq
\half\left(\sup_{\F_0, s\in [0,1]} m(\Phi+s\Psi ,\Lambda)^{-1} \right)
\bB(\Psi)^2
\eqno(2.6)$$
Hence we conclude with the following
\bigskip\flushpar
{\bf Theorem 2.2}:
\smallskip\flushpar
Suppose in some neighborhood $\O_\Phi\subset\H1 $ of an interaction
$\Phi$ the finite volume measures
$\{\,\mu _{\Phi +s\Psi, \Lambda}^\omega\,\}_{\Lambda\in\F_0,\omega\in\Omega}$,
defined with some countable exhaustion $\F_0$,
%and some configuration $\omega \in\Omega $,
satisfy the Spectral Gap inequality with spectral gaps
$m(\Phi+s\Psi,\Lambda )\geq M(\O_\Phi)$, for some constant
$M(\O_\Phi)\in(0,\infty)$
independent of $\Lambda\in\F_0$ and independent of the interactions in
$\O_\Phi $.
Then we have
$$
0\leq p (\Phi +\Psi) - p (\Phi)
- \mu_{\Phi}(\A(\Psi))\leq
\half M(\O_\Phi) \bB^2 (\Psi)
\eqno(2.7)$$
and therefore the pressure functional $\H1\ni\Xi \longmapsto p(\Xi )$
is {\bf Fr\'echet differentiable} at the point $\Phi$.
\cir
We see that the statement in the present case is similar to that when
we have assumed \LS. We believe that in fact \LS may have stronger
consequences (e.g. in some cases the analyticity with respect to the potentials in $\H1$),
although at the moment we do not see whether one could get them directly
from it. (As we have already mentioned the uniform - in volume and external
configurations - \LS inequality is equivalent to
Dobrushin-Shlosman complete analyticity.)
%%%%\flushpar\hfill\hskip 1 true cm
%%% \flushpar
\bigskip\flushpar
{\bf 3. An Example of Interactions in $\B1\setminus \B2$}.
\bigskip\noindent
\smallskip\noindent
In this section we would like to exhibit examples of some peculiar Gibbsian
interactions which do not belong to $\B2\cup\B1^{(spin)}$, but for which
we can show \LS as well as the Dobrushin uniqueness condition.
For this, let us consider a configuration space $\Omega \equiv\bM^\Gamma $
defined with $\bM = S^{N-1}$, where $S^{N-1}$ denoting a unit sphere in the
Euclidean space $\R^N$. If $N\geq 2$, for two unit vectors $\sigma $ and
$\eta$ representing
the points in $\bM$, the expression $\sigma \cdot \eta$ will denote their
Euclidean scalar product.\flushpar
Let $\Phi \equiv \{ \Phi_X\}_{X\in\F}$ be an interaction
on $\Omega$ given by
$$
\Phi_X \equiv \rho_{X} \cdot cos \left(
\frac {\sum_{\j\in X} \eta_\j\cdot\sigma_\j}
{|X|^{\alpha }}\right)
\eqno(3.1)$$
with some unit vectors $\eta_\j\in\bM$, $\alpha \in[0,\infty)$ and
coefficients $\rho_{X}\equiv\rho(|X|)\in\R$ such that
$$
\sup_{\i\in\Z^d}\sum_{X\in\F\atop{X\ni \i}}
|\rho_X|\, < \,\infty
\eqno(3.2)$$
and
$$
\sup_{\i\in\Z^d}\sum_{X\in\F\atop{X\ni \i}}
|X|\cdot|\rho_X|\, =\,\infty
\eqno(3.3)$$
Then clearly we have
$$
||\Phi||_1 =
\sup_{\i\in\Z^d}\sum_{X\in\F\atop{X\ni \i}}
||\Phi_X||_{\infty}\, < \,\infty
\eqno(3.4)$$
i.e. $\Phi\in\B1$, but
$$
||\Phi||_2 =
\sup_{\i\in\Z^d}\sum_{X\in\F\atop{X\ni \i}}
|X|\cdot ||\Phi_X||_\infty\, = \,\infty
\eqno(3.5)$$
i.e. $\Phi\notin\B2$. Clearly similar properties are possessed by
the interactions defined with the function $\cos(x)$ replaced by $\sin(x)$.
Moreover for $\alpha <1$ the interaction given above
cannot be transformed into a Gibbsian interaction from the space
$\B1^{(spin)}$. This can be seen by simple computations similar to the one
given for a special case (of $e^{ix}$ type complex interaction) with
$\alpha =\half$ considered in \cite{vEF}, when discussing the example
given in \cite{DM2} of complex (Gibbsian) interactions corresponding
to $\alpha=0$, which define a spin system failing to be analytic even in
the high temperature region.
\par\noindent
We have the following result showing that the applicability of our method
presented earlier extends far beyond the space $\B2$.
\bigskip\flushpar
{\bf Theorem 3.1}:
\smallskip\flushpar
Suppose that $N\geq 2$ and $\alpha\geq\half$. Then there is
$\beta_0\in(0,\infty)$ such that for any
$|\beta| < \beta _0$ the Gibbs measure corresponding to the potential
$\beta \Phi $ satisfies \LS.
\cir
{\bf Proof}: Since for $N\geq 3$ the unit spheres have positive Ricci
curvature, we can use the method based on the criterion of \cite{BE},
(see also \cite{CS}). Then we need only to show that there is a constant
$C\in(0,\infty)$ such that
$$
\sum_{\i,\j\in\Z^d\atop{l,k=1,..,N}}\left(
\sum_X \nabla_\i^l\nabla_\j^k \Phi_X \right)
\nabla_\i^l f\nabla_\j^k f
\leq C |\nabla f|^2
\eqno(3.6)$$
Since we have
$$
\sum_{\i,\j\in\Z^d\atop{l,k=1,..,N}}\left(
\sum_X \nabla_\i^l\nabla_\j^k \Phi_X \right)
\nabla_\i^l f\nabla_\j^k f
\leq
\sum_{\i\in\Z^d\atop{l=1,..,N}}\left(\sum_{\j\in\Z^d\atop{k=1,..,N}}
\sum_X |\nabla_\i^l\nabla_\j^k \Phi_X| \right)
|\nabla_\i^l f|^2
\leq
$$
$$
\leq
\sup_{\i',l'}\left(\sum_{\j\in\Z^d\atop{k=1,..,N}}
|\sum_X \nabla_{\i'}^{l'}\nabla_\j^k \Phi_X| \right)
\sum_{\i\in\Z^d\atop{l=1,..,N}} |\nabla_\i^l f|^2
\eqno(3.7)$$
and
$$
\sup_{\i',l'}\left(\sum_{\j\in\Z^d\atop{k=1,..,N}}
\sum_X |\nabla_{\i'}^{l'}\nabla_\j^k \Phi_X| \right)
=
\sup_{\i',l'}\left(\sum_{\j\in\Z^d\atop{k=1,..,N}}
\sum_{X\ni\i,\j} |\nabla_{\i'}^{l'}\nabla_\j^k \Phi_X| \right) =
$$
$$
=
\sup_{\i',l'}\left(\sum_{X\ni\i}
\sum_{\j\in X\atop{k=1,..,N}}
|\nabla_{\i'}^{l'}\nabla_\j^k \Phi_X| \right)
\leq
N\sup_{\i',l'}\left(\sum_{X\ni\i}
|X|
\sup_{\j,k}|\nabla_{\i'}^{l'}\nabla_\j^k \Phi_X| \right) ,
\eqno(3.8)$$
using the explicit form of the interaction $\Phi$
we get
$$
\sup_{\i',l'}\left(\sum_{X\ni\i}
|X|
\sup_{\j,k}|\nabla_{\i'}^{l'}\nabla_\j^k \Phi_X| \right)
\leq
\sup_{\i',l'}\left(\sum_{X\ni i}
|X|\cdot |\rho_X|
\sup_{\j,k}\frac{|\eta_{\i'}^{l'}\eta_{\j}^k|}{(|X|^{\alpha })^2} \right)
\eqno(3.9)$$
Hence, provided that $\alpha\geq\half$, the last term can be bounded by
$$
\leq
\sup_\i \sum_{X\ni\i} |\rho_X|
= ||\Phi||_1 \, < \, \infty
\eqno(3.10)$$
This implies (3.6). Hence we conclude that for $\beta\in[0,\beta_0)$, with
some sufficiently small $\beta_0\in(0,\infty)$, the corresponding
conditional expectations $\mu_{\beta \Phi,\Lambda }^\omega $
satisfy Logarithmic Sobolev inequalities with a coefficient
independent of the volume $\Lambda $ and the external configuration $\omega $.
The case $N=2$ (for which one cannot apply the Bakry-Emery condition, because
the corresponding Ricci curvature equals zero) follows by checking
the condition (0.13) from \cite{Z3}.
This ends the proof of Theorem 3.1.
\qed
A similar result should be true also for discrete spins,
(although the details of the proof might be much more complicated).
\flushpar
After we have checked that \LS is true, a natural question arises
whether or not one can also prove that the Dobrushin uniqueness condition
is satisfied. As one could see in the above considerations only the condition
on the second derivative of the interaction entered. It obviously looks
similar to the sufficient condition for the Dobrushin uniqueness condition
given in Chapter V of \cite{S}. However, there the assumption that the
interaction is affine in the spin variables, (i.e. that it belongs to
a space $\B{}^{(spin)}$), has been explicitly used. We will show
that in fact it is not essential and we have the following result.
\bigskip\flushpar
{\bf Theorem 3.2}:
\smallskip\flushpar
Suppose that $N\geq 1$ and $\alpha\geq\half$. Then there is
$\beta_0\in(0,\infty)$ such that for any
$|\beta| < \beta _0$ the Gibbs measure corresponding to the potential
$\beta \Phi $ satisfies the Dobrushin uniqueness condition.
\cir
{\bf Proof}:
Let
$$
\gamma_{\i\k}\equiv\sup\left\{|\mu_{\beta \Phi,\i}^\sigma (f) -
\mu_{\beta\Phi,\i}^{\tilde\sigma} (f)|\,
:\,\sigma_\j=\tilde\sigma_\j,\hbox{ for }
\j\neq\k,\i\hbox{ and }||f||_{Lip}=1\right\}
\eqno(3.11)$$
where $f\equiv f(\sigma_\i)$ and $||\,\cdot\,||_{Lip}$ denotes the Lipschitz
norm associated to a given metric $d(\cdot\,,\,\cdot)$ on $\bM$, i.e. the norm
defined by
$$
||f||_{Lip}\equiv\sup_{x\neq y}\frac {|f(x)-f(y)|} {d(x,y)}
$$
We want to show that there is $\beta _0\in(0,\infty)$ such that for
all $\beta \in (-\beta _0,+\beta _0)$ we have
$$
\sup_{\i\in\Gamma }\sum_{\k\neq\i}\gamma_{\i\k}\,<\, 1
\eqno(\DU)$$
To estimate $\gamma_{\i\k}$ for a given $\k\neq\i$ let us define
an interpolating potential
$$\Phi (s,\omega ) \equiv
s\Phi(\omega_{\Gamma \setminus {\k}}\bullet\sigma_\k) +
(1-s) \Phi(\omega_{\Gamma \setminus {\k}}\bullet\tilde\sigma_\k)
\eqno(3.12)$$
Now we have
$$
|\mu_{\beta \Phi,\i}^\sigma (f) - \mu_{\beta\Phi,\i}^{\tilde\sigma} (f)|
=|\int_0^1ds\,\frac d{ds} \mu_{\beta \Phi(s),\i}^\sigma (f)|
=
|\beta \int_0^1ds\, \mu_{\beta \Phi(s),\i}^\sigma
(f,\frac d{ds} U_\i(\Phi(s)))|
\eqno(3.13)$$
Using the Schwartz inequality for the covariance we get
$$
|\mu_{\beta \Phi,\i}^\sigma (f) - \mu_{\beta\Phi,\i}^{\tilde\sigma} (f)|
\leq |\beta |\int_0^1ds\,
\left(\mu_{\beta \Phi(s),\i}^\sigma (\frac d{ds} U_\i(\Phi(s))) ,
\frac d{ds} U_\i(\Phi(s)))
\right)^\half\cdot
\left(\mu_{\beta \Phi(s),\i}^\sigma (f , f)\right)^\half
\eqno(3.14)$$
Now we note that for any function $F$ we have
$$
0\leq\mu_{\beta \Phi(s),\i}^\sigma (F , F)=
\half\mu_{\beta \Phi(s),\i}^\sigma \otimes\tilde\mu_{\beta \Phi(s),\i}^\sigma
(F(\omega_\i) - F(\tilde\omega_\i ) )^2
\leq\half e^{4||\beta \Phi||_1}
\mu_0 \otimes\tilde\mu_0 (F(\omega_\i) - F(\tilde\omega_\i) )^2
\eqno(3.15)$$
where the tilded measure denotes the isomorphic copy of the untilded one.
>From this one can see that, for any function $f$ such that $||f||_{Lip}=1$,
we have
$$
\left(\mu_{\beta \Phi(s),\i}^\sigma (f , f)\right)^\half
\leq
2^{-\half} e^{2||\beta \Phi||_1}
\left(\mu_0 \otimes\tilde\mu_0 (d(\omega_\i,\tilde\omega_\i )^2)\right)^\half
\eqno(3.16)$$
On the other hand since the free measure $\mu _0$ satisfies \SG with a
mass gap $m_0>0$, we have
$$
\mu_{\beta \Phi(s),\i}^\sigma (F , F)=
\leq m_0^{-1}e^{4||\beta \Phi||_1}
\mu_0 |\nabla_\i F|^2
\eqno(3.17)$$
(respectively with a discrete gradient if the spins are discrete.)
Applying this to the first factor in the integrand on the right hand side
of (3.14), we get
$$
\left(\mu_{\beta \Phi(s),\i}^\sigma
(\frac d{ds} U_\i(\Phi(s))) , \frac d{ds} U_\i( \Phi(s)))
\right)^\half
\leq m_0^{-\half}e^{2||\beta \Phi||_1}
\left(\mu _0|\nabla_\i \frac d{ds} U_\i( \Phi(s)))|^2\right)^\half
\eqno(3.18)$$
Finally, using the definition of the potential $\Phi $, we observe that
$$
|\nabla_\i \frac d{ds} U_\i( \Phi(s)))|\leq
\sum_{X\ni\i,\k}|\rho_X|\cdot\frac1{|X|^{2\alpha }}
\eqno(3.19)$$
Combining (3.13)-(3.19), we obtain
$$
\gamma _{\i,\k} \leq D \cdot
\sum_{X\ni\i,\k}|\rho_X|\cdot\frac 1{|X|^{2\alpha }}
\eqno(3.20)$$
with the constant
$$
D\equiv |\beta| 2^{-\half}m_0^{-\half}e^{4||\beta \Phi||_1}\cdot
\left(\mu_0 \otimes\tilde\mu_0 (d(\omega_\i,\tilde\omega_\i )^2)\right)^\half
\eqno(3.21)$$
Thus we have
$$
\sum_{\k\neq\i}\gamma_{\i,\k} \leq
D\cdot\sum_{\k\neq\i} \sum_{X\ni\i,\k}|\rho_X|\cdot\frac 1{|X|^{2\alpha }} =
D\cdot\sum_{X\ni\i}\frac {|X|-1}{|X|^{2\alpha }}\cdot |\rho_X|
\eqno(3.22)$$
If $\alpha\geq\half$ we obtain
$$
\sum_{\k\neq\i}\gamma_{\i,\k} \leq D\cdot||\Phi||_1
\eqno(3.23)$$
As by our assumption we have $||\Phi||_1<\infty$ and
$D\equiv D(|\beta|)\to 0$,
when $|\beta|\to 0$, there is $\beta_0\in(0,\infty)$ such that
for every $\beta\in(-\beta_0,+\beta_0)$ the right
hand side of (3.23) is smaller that $1$, i.e. {\DU } is satisfied.
This ends the proof of Theorem 3.2.
\qed
\flushpar
{\it Remark} Let us note that for the special case of $\bM =\{-1,+1\}$
we could get better estimates with $D=|\beta|\cdot||\Phi||_1$.
\cir
\flushpar
Finally let us remark that for the potentials $\Phi$ defined with
$\alpha \geq \half$ one can construct the stochastic dynamics. Then \LS
proven above (together with an appropriate approximation property) should
allow us to prove the exponential decay to equilibrium in the uniform norm
(with the rate governed by the spectral gap of the generator) similarly
as in \cite{SZ2}.
Alternatively, due to \DU, one can use the method of \cite{AH}
(see also \cite{SZ2}) to get a similar result. Thus at high temperatures
we have again a very nice correspondence
between dynamics and "equilibrium" uniqueness conditions
for the spin systems described by the discussed interactions with
$\alpha \geq\half$.
\flushpar
For the moment we leave open the intriguing problem concerning
the high temperature behavior of systems with $\alpha <\half$ potentials.
As one can expect from the form of the interactions $\Phi $, it should
connected to some interesting large deviations theory.
We believe that solving this problem would be important for a better
understanding of Gibbsian description, as well as its nonequilibrium
counterpart, for lattice spin systems.
\bigskip%%\hfill \hskip 1 true cm
\bigskip\bigskip\flushpar
{\bf Acknowledgements}: A part of this work arose from the discussion
of the authors at the Newton Institute during a workshop organized
by G. Grimmett. We would like to thank him, as well as the staff and
visitors of the Newton Institute, for the stimulating and nice atmosphere.
We would like also to thank London Center for Mathematics %% and .........
for financial support. %% which make our collaboration possible.
Part of the research of ACDvE was made possible by a fellowship of the
KNAW (Royal Dutch Academy of Arts and Sciences). This work was also partially
supported by contract EU CHRX - CT93 - 0411. He also thanks
the Mathematics Department of Imperial College for its hospitality.
%%........................................................................
%\vfill\break\eject
\bigskip\flushpar\flushpar
\bigskip\flushpar\flushpar
%\bigskip\flushpar
%\font\ss=cmss10 %scaled % \magstep1
%\font\bss=cmssdc10 %scaled %\magstep1
\def\flushpar{{\par\noindent}}
%\def\bigpagebreak{{\bigskip\noindent}}
\def\bigpagebreak{{\smallskip\noindent}}
\def\ref{}
\def\paper{{, {\sl }}}
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\def\ed{{{, Ed.}}}
\def\eds{{{, Eds.}}}
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\def\key #1{{[{\bf #1}]}}
\def\by{{ {\rm }}}
\def\vol#1#2#3{{ {\bf #1#2#3} }}
\def\yr #1#2#3#4{{{ ({\rm #1#2#3#4}) }}}
\def\pages{{ {\rm }}}
\def\bysame{{{--------------}}}
\bigskip\noindent
{\bf References}
\bigskip\noindent
%\vfill\break\eject
%\vskip.5in
%\Refs
\def\ke #1#2{{{\hskip -1 true cm [{#1}]\hskip #2 pt }}}
\bigpagebreak\ref\key{AH}\by Aizenman M. and Holley R.A.
\paper Rapid Convergence to Equilibrium of Stochastic Ising Models in the
Dobrushin-Shlosman Regime \inbook Percolation Theory and Ergodic Theory
of Infinite Particle Sytems\ed Kesten H.\publ Springer-Verlag\yr 1987
\pages 1-11
\endref
\bigpagebreak\ref\key{BE}\by Bakry D. and Emery M.
\paper Hypercontractivit\` e de semi - groupes des
diffusion \jour C.R.Acad.Sci. Paris Ser.\vol I 299 \yr 1984
\pages 775-777 ,%\endref %\bigpagebreak
%\ref\key{ }\by\ke{}
\paper Diffusions hypercontractives \pages pp 177-206 \inbook Sem. de
Probabilites XIX, Azema J. and Yor M.(eds.) \vol LNM 1123
\endref
\bigpagebreak\ref\key{CS}\by Carlen E.A. and Stroock D.W.
\paper An application of the Bakry - Emery
criterion to infinite dimensional diffusions\jour
Sem.de Probabilites XX, Azema J. and Yor M. (eds.) \vol LNM 1204
\pages 341-348
\endref
%\bigpagebreak\ref\key{DaGSi}\by
% Davies E.B., Gross L. and Simon B.
%\paper Hypercontractivity :
% A bibliographical review \jour in proceedings of the Hoegh
% -Krohn Memorial Conference
%\endref
%\bigpagebreak\ref\key{DS}
%\by Deuschel J.-D. and Stroock D.W.
%\paper Large Deviations
%\jour Academic Press 1989
%\endref
\bigpagebreak\ref\key{DeS}
\by Deuschel J.-D. and Stroock D.W.
\paper Hypercontractivity and Spectral Gap of Symmetric Diffusions
with applications to the Stochastic Ising Model
\jour J. Func. Anal. \vol{}92 \yr 1990 \pages 30-48
\endref
\bigpagebreak\ref\key{D1}\by Dobrushin R.L.
\paper The description of Random Fields by Means of Conditional
Probabilities and Conditions of its Regularity
\jour Theor. Prob. its Appl. \vol {}13 \yr 1968\pages 197-224
\endref
\bigpagebreak\ref\key{D2}\by Dobrushin R.L.
\paper Markov Processes with a Large Number of Locally Interacting
Components
\jour Problems of Inf. Trans. \vol {}{}7 \yr 1971\pages 149-164 and
235-241
\endref
\bigpagebreak\ref\key{DM1}\by Dobrushin R.L. and Martirosyan M.R.\paper Nonfinite Perturbations of Gibbs Fields
\jour Theor. Math. Phys. \vol{}74 \yr 1988\pages 10-20
\endref
\bigpagebreak\ref\key{DM2}\by Dobrushin R.L. and Martirosyan M.R.\paper Possibility of the High Temperature
Phase Transitions Due to the Many-Particle Nature of the Potential
\jour Theor. Math. Phys. \vol{}75 \yr 1988\pages 443-448
\endref
\bigpagebreak\ref\key{DS1}\by Dobrushin R.L. and Shlosman S.B.
\paper Constructive criterion for the uniqueness of Gibbs field\
\inbook Statistical Physics and Dynamical Systems, Rigorous Results\eds Fritz,
Jaffe, and Szasz \publ Birkh\"auser\yr 1985\pages 347--370\endref
\bigpagebreak\ref\key{DS2}\by Dobrushin R.L. and Shlosman S.B.
\paper Completely analytical Gibbs fields \pages 371--403\inbook
Statistical Physics and Dynamical Systems, Rigorous Results\eds Fritz, Jaffe,
and Szasz \publ Birkh\"auser\yr 1985\endref
\bigpagebreak\ref\key{DS3}\by Dobrushin R.L. and Shlosman S.B.
\paper Completely analytical interactions: constructive description
\jour J. Stat. Phys.\vol
{}46\yr1987\pages 983--1014 \endref
\bigpagebreak\ref\key{DvE}\by Dani\"els H.A.M. and van Enter A.C.D.\paper Differentiability Properties of
the Pressure in Lattice Systems\jour Commun. Math. Phys. \vol{}71 \yr 1980\pages 65-76
\endref
\bigpagebreak\ref\key{vE}\by van Enter A.C.D.\paper A Note on the Stability of Phase Diagram in Lattice Systems
\jour Commun. Math. Phys. \vol{}79 \yr 1981\pages 25-32
\endref
\bigpagebreak\ref\key{vEF}\by van Enter A.C.D. and Fern\'andez R.\paper A Remark on Different Norms and
Analyticity for Many-Particle Interactions\jour J. Stat. Phys. \vol{}56\yr 1989
\pages 965-972
\endref
\bigpagebreak\ref\key{vEFK}\by van Enter A.C.D., Fern\'andez R. and
Koteck\'y R.
\paper Pathological Behavior of Renormalization - Group Maps at High Fields
and above the Transition Temperature
\jour Groningen Preprint, submitted to J. Stat. Phys.
%\vol{} \yr 1994\pages
\endref
\bigpagebreak\ref\key{vEFS}\by van Enter A.C.D., Fern\'andez R. and Sokal A.D.\paper Regularity Properties
and Pathologies of Position - Space Renormalization - Group
Transformations: Scope and Limitations of Gibbsian Theory
\jour J. Stat. Phys. \vol{}72 \yr 1993\pages 879-1167
\endref
\bigpagebreak\ref\key{F\"o}\by F\"ollmer H.
\paper A Covariance Estimates for Gibbs Measures
\jour J. Func. Anal. \vol{}46\yr 1982\pages 387-395
\endref
\bigpagebreak\ref\key{GM}\by Gallavotti G. and Miracle-Sole S.\paper Correlation Functions of Lattice
Systems\jour Commun. Math. Phys. \vol{}{}7 \yr 1968\pages 274 - 288
\endref
\bigpagebreak\ref\key{GMR}\by Gallavotti G., Miracle-Sole S. and Robinson D.W.\paper Analyticity
Properties of a Lattice Gas\jour Phys. Lett.\vol A 25 \yr 1967\pages 493-494
\endref
\bigpagebreak\ref\key{G1}\by Gross L.\paper Absence of Second-Order Phase
Transitions in the Dobrushin
Uniqueness Region\jour J. Stat. Phys. \vol{}25 \yr 1981\paper 57-72
\endref
\bigpagebreak\ref\key{G2}\by Gross L.\paper
Thermodynamics, Statistical Mechanics and Random Fields
\inbook Ecole d'Et\'e de Probabilit\'es de Saint-Flour Phys. \vol{}{}X \yr
1980 \vol LNM 929 \publ Springer - Verlag \publaddr Berlin 1982
\endref
\bigpagebreak\ref\key{G3}\by Gross L.\paper Logarithmic Sobolev Inequalities
\jour Amer. J. Math. \vol{}97 \yr 1976\paper 1061-1083
\endref
\bigpagebreak\ref\key{Ge}\by Georgii H.O.\book Gibbs Measures and Phase
Transitions
\publ Walter-de-Gruyter \yr 1988
\endref
\bigpagebreak\ref\key{GrGr}\by Gray L. and Griffeath D.
\paper On the Uniqueness of Certain Interacting
Particle Systems\jour Z. Wahr. v. Geb. \vol{}35\yr 1976\pages 75-86
\endref
\bigpagebreak\ref\key{H}\by Holley R.A.
\paper Possible Rates of Convergence in Finite Range Attractive Spin Systems
\jour Contemp. Math. \vol{}41 \yr 1985\pages 215-234
\endref
\bigpagebreak\ref\key{HS1}\by Holley R.A. and Stroock D.W.
\paper A martingale Approach to Infinite Systems
of Interacting Processes\jour Ann. Prob. \vol{}{}4 \yr 1976\pages 195-228
\endref
\bigpagebreak\ref\key{HS2}\by Holley R.A. and Stroock D.W.
\paper Applications of the Stochastic Ising Model
to the Gibbs States\jour Commun. Math. Phys. \vol{}48 \yr 1976\pages 249-265
\endref
\bigpagebreak\ref\key{HS3}\by Holley R.A. and Stroock D.W.\paper
In One and Two Dimensions, Every Stationary Measure for a Stochastic Ising
Model is a Gibbs State
\jour Commun. Math. Phys.\vol {}55\yr 1977\pages 37-45\endref
\bigpagebreak\ref\key{HS4}\by Holley R.A. and Stroock D.W.\paper
Logarithmic Sobolev inequalities and stochastic Ising models\jour J. Stat.
Phys.\vol{}46\yr 1987\pages 1159--1194\endref
% \bigpagebreak\ref\key{HS5}\by Holley R.A. and Stroock D.W.\paper
%Uniform and $L_2$ Convergence in One Dimensional Stochastic Ising models
%\jour Commun. Math. Phys.\vol 123\yr 1989\pages 85 - 93
%\endref
\bigpagebreak\ref\key{I1}\by Israel R.B.
\paper High Temperature Analyticity in Classical Lattice Systems
\jour Commun. Math. Phys. \vol{}50 \yr 1976\pages 245-257
\endref
\bigpagebreak\ref\key{I2}\by Israel R.B.\book Convexity in the Theory of
Lattice Gases\publ Princeton Univ.
Press \yr 1979
\endref
\bigpagebreak\ref\key{I3}\by Israel R.B.\paper Existence of Phase
Transitions for Long-Range Interactions
\jour Commun. Math. Phys. \vol{}43 \yr 1975\pages 59-68
\endref
\bigpagebreak\ref\key{I4}\by Israel R.B.\paper Generic Triviality
of Phase Diagrams in Spaces of Long-Range Interactions
\jour Commun. Math. Phys. \vol 106 \yr 1986\pages 459-466
\endref
\bigpagebreak\ref\key{IP}\by Israel R.B. and Phelps R.R.
\paper Some Convexity Questions Arising in
Statistical Mechanics\jour Math. Scand. \vol{}54\yr 1984\pages 133-156
\endref
\bigpagebreak\ref\key{L}\by Lanford III O.E.\paper Entropy and Equilibrium
States in Classical Statistical
Mechanics\inbook in Statistical Mechanics and Mathematical Problems\ed
Lenard A., \vol{LNP } 20 \yr 1973 \publ Springer-Verlag
\endref
\bigpagebreak\ref\key{La}\by Laroche E.\paper Sur des In\'egalit\'es de
Corr\'elation et sur les In\'egalit\'es de Sobolev Logarithmiques en
M\'ecanique Statistique\inbook Th\'ese Toulouse 1993
\endref
\bigpagebreak\ref\key{Li1}\by
Liggett Th. M.
\book Infinite Particle Systems\publ
Springer - Verlag \bookinfo Grundlehren Series \# 276\publaddr New York
\yr 1985\endref
\bigpagebreak\ref\key{Li2}\by
Liggett Th. M.
\book Survival and Coexistence in Interacting Particle Systems\pages 209-226
\inbook
Probability and Phase Transition\ed Grimmett G., \publ Kluwer Academic Pub.
\yr 1994\endref
\bigpagebreak\ref\key{LY}\by
ShengLin Lu and Horng - Tzer Yau
\paper Spectral Gap and Logarithmic Sobolev Inequality for Kawasaki and
Glauber Dynamics\jour Commun. Math. Phys. \vol 156 \yr 1993 \pages 399-433
%publ Preprint 1993
\endref
\bigpagebreak\ref\key{MO1}\by
Martinelli F. and Olivieri E.
\paper Approach to Equilibrium of Glauber Dynamics in the One Phase
Region: I. The Attractive case \jour Commun. Math. Phys.\vol 161\yr 1993
\pages 447-486
\endref
\bigpagebreak\ref\key{MO2}\by
Martinelli F. and Olivieri E.
\paper Approach to Equilibrium of Glauber Dynamics in the One Phase
Region: II. The General Case \jour Commun. Math. Phys.\vol 161\yr 1993
\pages 487-514
%\publ Preprints 1992 /1993
\endref
%\bigpagebreak\ref\key{MM}\by
% Malyshev W.A. and Minlos R.A.
%\book Gibbsian Random Fields : The Method of Cluster Expansions
%\publ Moscow, Nauka\yr 1985
%\endref\bigpagebreak
%
%\ref\key{P}
% Preston C.
%\book Random fields\publ Lec. Notes in Math. 534,
%Springer \yr 1976 \endref
\bigpagebreak\ref\key{Pa1}\by Park Y.M.
\paper Cluster Expansion for Classical and Quantum Lattice Systems,
\jour J. Stat. Phys. \vol{}27 \yr 1982\pages 553-576
\endref
\bigpagebreak\ref\key{Pa2}\by Park Y.M.\paper Extension of Pirogov-Sinai
Theory of Phase Transitions to Infinite Range Interactions:
I Cluster Expansion, II Phase Diagram
\jour Commun. Math. Phys. \vol 114 \yr 1988\pages 187-218, 219-241
\endref
\bigpagebreak\ref\key{Ph}\by Phelps R.R.
\paper Generic Fr\'echet Differentiability of the Pressure in
Certain Lattice System
\jour Commun. Math. Phys. \vol{}91 \yr 1983\pages 557-562
\endref
\bigpagebreak\ref\key{Pr}\by Prakash Ch.
\paper High-Temperature Differentiability of Lattice Gibbs States
by Dobrushin Uniqueness Techniques
\jour J. Stat. Phys \vol{}31 \yr 1983\pages 169-228
\endref
\bigpagebreak\ref\key{R1}\by Ruelle D.
\paper Must Thermodynamic Functions Be Piecewise Analytic
\jour J. Stat. Phys. \vol{}26 \yr 1981\pages 397-399
\endref
\bigpagebreak\ref\key{R2}\by Ruelle D.
\book Statistical Mechanics: Rigorous Results\publ W.A. Benjamin Inc.
\yr 1969
\endref
\bigpagebreak\ref\key{R3}\by Ruelle D.\book Thermodynamic Formalism
\inbook Encyclopedia of Mathematics and
its Applications \vol{}{} 5\publ Addison-Wesley Pub. Company \yr 1978
\endref
%\bigpagebreak\ref\key{S1}\by Simon B.\paper A Remark on Dobrushin's
%Uniqueness Theorem
% \jour Commun. Math. Phys. \vol{} 68\yr 1979\pages 183-185
%\endref
\bigpagebreak\ref\key{S}\by Simon B.
\book The Statistical Mechanics of Lattice Gases
\publ Princeton Univ. Press \yr 1993
\endref
\bigpagebreak\ref\key{So}\by Sokal A.D.
\paper More Surprises in the General Theory of Lattice Systems
\jour Commun. Math. Phys. \vol{}86\yr 1982\pages 327-336
\endref
\bigpagebreak\ref\key{Su1}\by Sullivan W.G.\paper A Unified Existence and
Ergodic Theorem for Markov Evolution of Random Fields
\jour Z. Wahr. verw. Geb. \vol{}31\yr 1974\pages 47-56
\endref
\bigpagebreak\ref\key{Su2}\by Sullivan W.G.\paper Processes with Infinitely
Many Jumping Particles
\jour Proc. AMS \vol{}54\yr 1976\pages 326-330
\endref
\bigpagebreak\ref\key{SZ1}
\by Stroock D.W. and Zegarlinski B.
\paper The Logarithmic Sobolev Inequality for Continuous Spin Systems
on a Lattice
\jour J. Func. Anal.\vol 104 \yr 1992\pages 299 - 326 \endref
\bigpagebreak\ref\key{SZ2}
\by Stroock D.W. and Zegarlinski B.
\paper The Equivalence of the Logarithmic Sobolev Inequality and the
Dobrushin--Shlosman Mixing Condition \jour
Commun. Math. Phys. \vol 144\yr 1992\pages 303 - 323 \endref
\bigpagebreak\ref\key{SZ3}
\by Stroock D.W. and Zegarlinski B.
\paper The Logarithmic Sobolev Inequality for Discrete Spin Systems
on a Lattice \jour Commun. Math. Phys.\vol 149 \yr 1992\pages 175 - 193
\endref
% \bigpagebreak\ref\key{Th}
%\by Thomas L.E.
%\paper Bound on the Mass Gap for Finite Volume Stochastic Ising Models
%at Low Temperature \jour Commun. Math. Phys.\vol 126\yr 1989\pages 1 - 11
%\endref
\bigpagebreak\ref\key{W}\by Walters P.\paper Differentiability Properties of
the Pressure
of a Continuous Transformation on a Compact Metric Space
\jour J. London Math. Soc. \vol{}46 \yr 1992\pages 471-481
\endref
\bigpagebreak\ref\key{Z1}\by Zegarlinski B.
\paper On log-Sobolev inequalities for
infinite lattice systems\jour Lett. Math. Phys.\vol{}20\yr 1990\pages 173--182
\endref
\bigpagebreak\ref\key{Z2}\by Zegarlinski B.
\paper Log-Sobolev inequalities for infinite one-dimensional lattice systems
\jour Comm. Math. Phys.\vol 133\yr 1990\pages 147--162 \endref
\bigpagebreak\ref\key{Z3} \bysame
\paper Dobrushin uniqueness
theorem and logarithmic Sobolev inequalities \jour J. Func. Anal.
\vol 105\yr 1992\pages 77--111\endref
% \bigpagebreak\ref\key{Z4}\bysame
%\paper
%Strong exponential decay to equilibrium for the Markov hypercontractive
%semigroups associated to unbounded spin systems on a lattice
%\jour MIT Preprint 1992
%\endref
% \bigpagebreak\ref\key{Z,5}\bysame
%\paper
%Hypercontractive Markov Semigroups
%\jour unpublished lecture notes 1992
%\endref
% \bigpagebreak\ref\key{Z,6}\bysame
%\paper
%Gibbsian Description and Description by Stochastic Dynamics
%in the Statistical Mechanics of Lattice Spin Systems with Finite
%Range Interactions
%\jour in Proc. of IIIrd International Conference: Stochastic
% Processes, Physics and Geometry, Locarno, Switzerland,
% June 24 - 29, \yr 1991
%\endref
% \bigpagebreak\ref\key{Z,6}\bysame
%\paper
% Logarithmic Sobolev Inequalities for Gibbs Measures and Applications
%\jour in Proc. of Prague Conference 1992
%\endref
%\endRefs
%\enddocument
\end