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Ising interfaces, SOS continuous spin model, Brascamp-Lieb inequalities
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\begin{document}
\title[\,]{On entropic reduction of fluctuations}
\author{Thierry~Bodineau}
\address{Universit\'e Paris 7, D\'epartment de Math\'ematiques, Case 7012,
2
Place Jussieu, F-75251 Paris, France
}
\email{Thierry.Bodineau\@@gauss.math.jussieu.fr}
\author{GiamBattista Giacomin }
\address{Dipartimento di Matematica, Universit\`a degli studi di Milano,
(I) and Universit\'e Paris 7, D\'epartment de Math\'ematiques, Case 7012,
2
Place Jussieu, F-75251 Paris, France
}
\email{giacomin\@@math.jussieu.fr}
\author{Yvan~Velenik}
\address{
Faculty of Industrial Engineering, Technion, Haifa 32000, Israel
}
\email{velenik\@@tx.technion.ac.il}
\date{\today}
\begin{abstract}
We point out that there is no general relation between ground state
degeneracy
and finite-temperature fluctuations for tilted interfaces.
\end{abstract}
\maketitle
\section{Introduction}
In a celebrated paper \cite{D},
Dobrushin proved that at low temperature, the horizontal
interface of the three dimensional Ising model is rigid.
The uniqueness of the ground state plays a predominant role in the proof
because the rigidity comes from the fact that the fluctuations of the
interface are not strong enough to destroy the ground state.
This picture of a unique ground state slightly deformed
is expected to fail when the temperature increases : entropy should take
over, leading to a rough interface above the roughening transition.
The analysis of ground states for tilted interfaces in 3D was a long
standing open problem which has been solved recently by R. Kenyon for
the interface orthogonal to the vector $(1,1,1)$ (see \cite{Ke1}
Theorem 15).
In fact, the works of Kenyon concern the combinatorics of dimer
models and the implications of his results are going far beyond the
characterization of the 3D interfaces for Ising model at zero
temperature (see \cite{Ke2} and references therein).
The ground states associated to the interface orthogonal to the
vector $(1,1,1)$ are degenerate, and Kenyon proved in \cite{Ke2}
that a related model (domino tiling) converges in the thermodynamic
limit to a Gaussian field (the same proof should also imply a
similar statement for the ground states).
Therefore, it is tempting to argue that an increase of the temperature,
i.e.
an addition of entropy, would lead to more fluctuations.
One could believe that the fluctuations at low temperature are mainly
driven by the degeneracy of the ground states and that there should exist
some general monotonicity principle which would answer this question. For
example, one could be tempted to use correlation inequalities, in the
spirit of
the proof of rigidity of the horizontal interface due to H. van
Beijeren~\cite{vB}.\\
Nevertheless such a picture does not seem to hold in full generality.
Consider now the interface orthogonal to the vector $(1,1,0)$ obtained
by imposing the boundary conditions
%
\begin{equation*}
\overline{\gs}_i =
\begin{cases}
\phantom{-}1 & \text{if $i\cdot(1,1,0) \geq 0$ and $\abs{i_3}\leq
N$,}\\
-1 & \text{if $i\cdot(1,1,0) < 0$ and $\abs{i_3}\leq
N$,}\\
\phantom{-}0 & \text{if $\abs{i_3}> N$,}
\end{cases}
\end{equation*}
%
outside the cube $\setof{i\in\bbZ^3}{\abs{i_k}\leq N,\, k~=~1,2,3}$.
By $\overline{\gs}_i =0$ we mean that
the boundary condition at the site $i$ is free.
Notice
that the interface is tighten only on two opposite edges of the cube, and
free on
the two other sides. Then at zero temperature this model reduces to a 2
dimensional model for which the fluctuations are known to be of order
$\sqrt{N}$. However, the physical intuition would say that at finite
temperature the system should behave completely differently and fluctuate
like
$\sqrt{ \ln (N)}$.
This would say that the ground state fluctuations should play only a
limited
role in the fluctuations of the interface which, at least in this example,
should be driven only by entropy.
In fact the effect of the entropy seems to be even more drastic in
dimension 4 where no fluctuations should pertain at positive temperature
for the interface orthogonal to the vector $(1,1,0,0)$.
Notice that in dimension 4 (for a different choice of boundary
conditions), the
rigidity of the tilted interface has been derived by Messager and
Miracle-Sole \cite{MM} by means of correlations inequalities.
There are many examples of systems with infinitely degenerated
ground-states,
only a finite number of which survive at positive temperature, see
e.g.~\cite{BS}, but the mechanism at play in the situation we consider in
this
note is very different.\\
In the low temperature regime and for small tilt, the interface
of the 3D Ising model can be approximated, at least on a heuristic
level, by a gas of lines which cannot intersect.
For this model, the logarithmic fluctuations of the correlations
were derived by Pr\"ahofer and Spohn in \cite{PS}.
This would indicate that at positive temperature, the transversal
excitations reduce the amplitude of the fluctuations of the interface
(see also \cite{S1,S2} for related phenomena).
In order to emphasize the lack of a general monotonicity property for
the amplitude of the fluctuations, we are going to prove this entropic
stabilization phenomenon for an effective interface model in
dimension $d+1 \geq 3$ which shares features similar to the Ising model
(including FKG inequalities ...).
For any finite domain $\gL \subset \bbZ^d$, we define the Hamiltonian
\begin{equation}
H_\gL^{\gP_{\gL^c}} (\gP_\gL) = \sum_{i \sim j} \nu(\gp_i -\gp_j),
\end{equation}
where $\nu: \bbR \to \bbR^+$ is a convex function, $\gP_\gL = \{
\gp_i\}_{i \in \gL}$.
The heights $\gP_{\gL^c}$ outside $\gL$ are fixed.
The Gibbs measure associated to this Hamiltonian will be denoted by
\begin{equation}
\mu_{\gL,\gb}^{\gP_{\gL^c}} (d \gP_{\gL}) =
\frac{1}{Z_{\gL,\gb}^{\gP_{\gL^c}}}
\exp \big( - \gb H_\gL^{\gP_{\gL^c}}
(\gP_\gL) \big) \, \Pi_{i \in \gL} \, d \gp_i
\,,
\label{eq_measure}
\end{equation}
%
where the parameter $\gb$ plays the role of the inverse of
the temperature and $d \gp$ denotes the Lebesgue measure.
As explained before, we are mainly interested in the interface orthogonal
to the vector $(1,1,0, \dots, 0)$.
We denote by $\mu_{\gb,N}$ the Gibbs measure on the domain
$\gL_N = \{-N+1, \dots, N-1\}^d$
with tilted boundary conditions on the sides $|i_1| =N$
\begin{eqnarray*}
\forall (i_2, \dots, i_d), \qquad \gp_{N,i_2, \dots, i_d} = N,
\quad \text{and} \quad
\gp_{-N, i_2, \dots, i_d} = -N.
\end{eqnarray*}
and free (or periodic) boundary conditions on the other sides.
To give core to the previous heuristic, the most natural effective model
should have been the SOS model ($\nu(x) = |x|$).
However,
only few results have been obtained about the fluctuations of this model,
because
the singularity of the interaction does not allow to use the techniques
based
on strict convexity of the potential.
In the case of 0 boundary conditions, Bricmont, Fontaine, Lebowitz
\cite{BFL} analyzed the fluctuations by means of
infrared bounds, but the latter estimates rely on a transfer matrix method
which does not seem to be suitable for our choice of boundary conditions.
Therefore, we will consider an alternative model which has the same
features
but with a quadratic potential at infinity.
Let $\nu$ be
\begin{equation}
\nu(x) =
\begin{cases}
|x|, & \qquad \text{if } x \in [-2,2], \\
x^2 /2 , & \qquad \text{if } x \in [-2,2]^c.
\end{cases}
\label{nu}
\end{equation}
\\
The ground state of this model is described by the limit
of~\eqref{eq_measure}
as $\gb \to \infty$, i.e. by the uniform measure on all configurations of
the
field compatible with the boundary conditions, and satisfying
%
\begin{itemize}
\item $\gp_i=\gp_j$ when $i_1 = j_1$;
\item $\gp_i-2 \leq \gp_j \leq \gp_i$, when $i_1=j_1+1$.
\end{itemize}
%
Before stating the result we observe that, by the symmetries of the model
$
\mu_{\gb,N} (\varphi_0) =0
$ for any $\gb \in (0,\infty)$.
\\
\begin{thm} \label{th1}
\noindent
1. For $\gb = \infty$, in any dimension greater or equal to 2,
$$
\mu_{\infty ,N} (\varphi_0{}^2) = \frac{N}{2}+O(\sqrt{N}).
$$
2. For any $0<\gb<\infty$, there exists $C_1>0$ and $C_2=C_2(d)<\infty$
such that
$$
\mu_{\gb,N} (\varphi_0{}^2) \leq C_1\log N\,,
$$
in dimension 2, and
$$
\mu_{\gb,N} (\varphi_0{}^2) \leq C_2\,,
$$
in higher dimensions.
\end{thm}
%
\bigskip
\noindent
{\bf Acknowlegments:}
We are indebted to H. Spohn for sharing many insights on the
problem of interface fluctuations.
We also would like to thank R. Kenyon for his comments on the manuscript.
GG has been partially supported by M.U.R.S.T. (Cofin99) and by GNAFA.
YV is supported by a Swiss National Science Foundation grant
\#8220-056599.
Finally, TB and YV acknowledge the kind hospitality of Technion, where
part of this work was made.
%
\section{Zero-temperature fluctuations}
%
We prove here part 1 in Theorem \ref{th1}.
In this case the only coordinate along which a nontrivial
behavior appears is the first one and
we will employ the reduced description of the model
$\{ \s _i \}_{i=-N, \ldots, N}$, interpreted as a
collection of random variables (under the uniform measure
introduced above).
Let $\{ X_i \}_{i\in {\bbZ}}$ be an IID sequence of
variables which are uniformly distributed on $[-1,+1]$
and for $n >-N$ set $S_n=\sum_{i=-N}^n X_i$ and $S_{-N}=0$.
Let us denote by $f_I$,
$I$ a finite subset of $\{-N+1, -N+2, \ldots\}$,
the density of $\{ S_n \}_{n \in I}$.
The random vector $\{ S_n\}_{n=-N+1, \ldots, N-1}$
admits a regular conditional density given $S_N$:
it is
\begin{equation}
f(s_{-N+1}, \ldots,
s_{N-1} \vert s_N)\equiv {\bf 1}_{\{f_N (s_N) \not=0 \} }
\frac{
f_{-N+1,\ldots , N}(s_{-N+1}, \ldots s_N)}{
f_{N} (s_N)}
.
\end{equation}
We now observe that $f(s_{-N+1},\ldots , s_{N-1}\vert 0)$
is the density of the random vector $\{ \s_i -i \}_{i=-N+1,
\ldots, N-1}$.
By the Markov property and by the symmetry of the $X_i$'s we obtain that
the density of $\s_0$ is
equal to $[f_0 (\cdot )]^2/f_N (0)$.
We now apply a Local Limit Theorem for densities (\cite{P}, page 214)
to obtain that $\sqrt{N}f_N(0)= (1/2 \sqrt{\pi}) +O(1/\sqrt{N})$
and that there exists $c$ such that
\begin{equation}
\left\vert
\sqrt{N} f_0(x\sqrt{N}) -g(x)
\right\vert \le \frac{c}{\sqrt{N} (1+x^2)},
\end{equation}
where $g$ is the density of the standard Gaussian.
This in particular implies
that the variance of $\s_0$ is $N/2+O(\sqrt{N})$.
\section{Upper Bound at finite $\gb$}
We adapt the decimation procedure introduced in \cite{BLL} in the
case of the interface orthogonal to the vector $(1,1,0, \dots, 0)$.
The key step is contained in the following Theorem based on Brascamp-Lieb
inequality \cite{BL}
\begin{thm}\cite{BLL}
\label{thmBLL}
Let $\nu:\bbR \to \bbR^+$ be a convex function such that there are
positive
constants $A,B,C,D,E$
\begin{enumerate}
\item $0< A \leq \nu''(x) \leq B < \infty, \qquad
\forall |x| \ge E$.
\item $|\nu(x) - C x^2| \leq D < \infty, \qquad \forall x$.
\end{enumerate}
Then the function $g$ defined by
\begin{equation}
g(y_1,\dots,y_k) = - \ln \left(
\int_\bbR \exp \left(- \sum_{i=1}^k \nu( y_i -z) \right) \, dz
\right)
\end{equation}
is of the form
\begin{equation}
g(y_1,\dots,y_k) = a \sum_{i,j} ( y_i - y_j)^2 + h (y_1,\dots,y_k),
\end{equation}
where $a$ is a positive constant and the function $h$ is convex.
Notice that the quadratic interaction involves every pair $(i,j)$.
\end{thm}
The potential $\nu$ in \eqref{nu} has been designed to satisfy the
previous assumptions.
Following \cite{BLL}, we partition the domain $\gL_N$ into 2 sets
$\gL_N^{\text{even}}, \gL_N^{\text{odd}}$ containing the even and the odd
sites. A site $i = (i_1, \dots, i_d)$ is said to be odd (resp. even)
if $\sum_{k = 1}^d i_k$ is odd (resp. even).
We will also denote by $Y_{\gL_N^{\text{even}}} =
\{ y_i \}_{\gL_N^{\text{even}}}$
the heights $\gP_{\gL_N}$ restricted to the even lattice.
A straight application of Theorem \ref{thmBLL} implies that the measure
restricted to even sites is a Gibbs measure with Hamiltonian
\begin{equation*}
H_{\gL_N}^{\text{even}} (Y_{\gL_N^{\text{even}}}) = a
\sum_{i \sim j} (y_i - y_j)^2 + F(Y_{\gL_N^{\text{even}}}) \, ,
\end{equation*}
where the constant $a$ is positive and $F$ is a convex function.
The interactions concern only the nearest neighbors on the sublattice
$\gL_N^{\text{even}}$, i.e. those which are $\star$-connected on
$\bbZ^d$.
One can choose $F$ such that the boundary conditions remain unchanged:
just include in $F$ the interactions between points of the even lattice
and the boundary.
We now use the Brascamp--Lieb inequality to dominate the
variance $\mu_{\gb,N} (\varphi_0{}^2) $
with the variance at the origin of the Gaussian model on even sites
with Hamiltonian $H_{\gL_N}^{\text{even}} (Y_{\gL_N^{\text{even}}}) = a
\sum_{i \sim j} (y_i - y_j)^2$ and boundary conditions which are
$y_{\pm N, i_2, \ldots, i_d}=0$ (when $(\pm N, i_2, \ldots, i_d)$ is even)
and free
(or periodic) on the other sides. The variance of this
Gaussian field has of course the
stated behavior.
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\end{document}
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