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Asymmetric Ising model
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%\BOZZA
%\input Srctex.sty
{\centerline{\bigfnt Gas Phase of Asymmetric Nearest Neighbor }}
{\centerline{\bigfnt Ising Model}}

\vglue1.5truecm {\centerline{ Alexander Mazel}}
{\centerline{Discus Data Solutions}} {\centerline{505 Eighth Ave,
Suite 802}} {\centerline{ New York, NY 10018 USA}} {\centerline{
AMazel@discusdata.com}}

\vglue1.truecm {\centerline{ Aldo Procacci}} {\centerline{
Departamento de Matematica \footnote{{{\arm Partially supported by
CNPq}}} }} {\centerline{ ICEx - UFMG}} {\centerline{ Belo
Horizonte, Brazil}} {\centerline{aldo@mat.ufmg.br}}

\vglue1.truecm {\centerline{ Benedetto Scoppola}} {\centerline{
Dipartimento di Matematica \footnote{{\arm Partially supported by
CNR, G.N.F.M.}} }} {\centerline{  Universit\'a ``La Sapienza'' di
Roma}} {\centerline{ Piazzale A. Moro 2}} {\centerline{00185 Roma,
Italy}}{\centerline{Benedetto.Scoppola@roma1.infn.it}}



\vglue1.truecm

{\centerline{ABSTRACT}}
\vglue.5truecm
\\{\it We consider an asymmetric $d$-dimensional, $d > 1$,
Ising model with the pair interaction $I$ in one direction
different from the pair interaction $J$ in all other directions.
We show that for any inverse temperature $\b$ the system is in the
gas phase as soon as $|J| < C\b^{-1}d^{-2} (1-\tanh (\b |I|))$
with $C > 0$ being a small numeric constant.}

\numsec=1\numfor=1
\vglue0.5truecm

\\Asymmetric or anisotropic bounded spin systems are classical models
in statistical mechanics. A general idea, suggested originally in
[DS2], is to study the system in which the interaction is strong
in one direction and weak in other directions in terms of a
perturbation of one-dimensional systems. Since the latter do not
exhibit phase transitions at any finite temperature it should be
possible to find for any fixed value of the strong interaction  an
orthogonal interaction sufficiently small to ensure the uniqueness
of the Gibbs state. Recently this problem has been discussed in
various context, see e.g. [NOZ] and [BK], since it is a basic tool
to solve different models both in equilibrium and in
non-equilibrium statistical mechanics.

In this note we consider an asymmetric nearest neighbor Ising
model described by the following partition function

$$Z_{\L}=2^{-|\L|} \sum_{\{{\bf\s}\}}
\exp (\b\sum_{\{x,y\}\in\L}J_{xy}\s_{x}\s_{y}) \Eq(1.1)$$

\\where $\L\subset {\bf Z}^d$ is  a cubic box of linear size $L$
containing $|\L|=L^{d}$ lattice points, the sum
$\sum_{\{{\bf\s}\}}$ runs over {\it spin configurations}
$\{{\bf\s}\}\in \{-1,1\}^{|\L|}$, $\b>0$ is the inverse
temperature, $\{x,y\}$ denotes unit bonds of $Z^d$ and the
coupling $J_{xy}$ is given by

$$
J_{xy}= \cases{J &if $|x-y|=1$ and $x^{(d)}-y^{(d)}=0$\cr I &if
$|x-y|=1$ and $|x^{(d)}-y^{(d)}|=1$ \cr 0 &otherwise}\Eq(1.1.1)
$$

\\Here $x=(x^{(1)}, \dots ,x^{(d)})$ and $y=(y^{(1)}, \dots
,y^{(d)})$ are points of $Z^d$ and $|x-y|$ is the Euclidean
distance between them.

Without loss of generality we suppose from now on that $I > J > 0$
and an empty boundary conditions are imposed on $\L$. The gas
phase is characterized by the fact that $|\L|^{-1} \log Z_{\L}$ is
analytic in $\b$ uniformly in the volume $\L$. For $d = 2$ an
exact solution, see e.g. [B], shows that for large $\b I$ the
model given by \equ(1.1) has a unique limit Gibbs state as soon as
$$\b J < C e^{-2\b I}\Eq(1.1.3)$$
where $C$ is a numerical constant.

In this note we show by elementary cluster expansion techniques
that an estimate of the form \equ(1.1.3) is also true for $d > 2$
with the constant $C$ depending on $d$.

\vskip.5cm
\\{\bf Theorem 1}. {\it Consider an asymmetric Ising model given
by  \equ(1.1) and \equ(1.1.1). Then for any $I$ and  $\b$ there
exists $J_0 = \b^{-1}C d^{-2} (1-\tanh (\b I))$, $C > 0$ such that
$|\L|^{-1}\log Z_{\L}$ can be written as an absolutely convergent
series uniformly in $|\L|$ for all $J < J_0$}.

\medskip\noindent
{\it Proof}. Perform a standard high temperature expansion for the
partition function
$$
 Z_{\Lambda} = 2^{-|\Lambda|} \sum_{\sigma_{\Lambda}}
\exp ( \beta \sum_{\{x,y\} \in \Lambda} J_{xy} \sigma_x \sigma_y)=
$$
$$
=2^{-|\Lambda|} \sum_{\sigma_{\Lambda}} \prod_{\{x,y\} \in
\Lambda} \cosh(\beta J_{xy})(1 +\s_x\s_y \tanh(\beta J_{xy}))
$$
Denote by $|B_w(\L)|$ the number of weak ($=$orthogonal to
$e^{(d)}=(0, \dots, 0, 1)$) nearest neighbor bonds in $\L$ and by
$|B_s(\L)|$ the number of strong ($=$parallel to $e^{(d)}=(0,
\dots, 0, 1)$) nearest neighbor bonds in $\L$. Then

$$
\prod_{\{x,y\} \in
\Lambda} \cosh(\beta J_{xy})=\cosh(\beta J)^{|B_w(\L)|}~
\cosh(\beta I)^{|B_s(\L)|}
$$
Moreover
$$
\sum_{\s_\L}\prod_{\{x,y\} \subset
\Lambda} (1 +\s_x\s_y \tanh(\beta J_{xy}))=
\sum_{\s_\L}\prod_{b \in
B(\Lambda)} (1 +\tilde\s_b \tanh(\beta J_{b}))
$$
where $b$ denotes a nearest neighbor pair  $\{x,y\}$, $\tilde
\s_b=\s_x\s_y$, $J_b =J_{xy}$ and $B(\Lambda)$ denotes the set of
all unit bonds in $\Lambda$.

\def\sG{{{\rm supp\,}g}}\def\sg{{{\rm supp\,}g}}\def\sGa{{{\rm supp\,}\G}}
\def\sga{{{\rm supp\,}\g}}\def\sd{{{\rm supp\,}\d}}

A {\it connected graph} $g\equiv\{b_1, \dots ,b_n\}$ on $\L$ is by
definition a non empty connected set of distinct pairs
$b_i=\{x_i,y_i\}\subset \L$ called {\it links of the graph}. Given
$g\equiv\{b_1, \dots ,b_n\}$ we denote by $|g|=n$ the number of
links in $g$  and by ${\rm supp \,}g=\cup_{i=1}^n b_i =
\cup_{i=1}^n (x_i \cup y_i)$ the support of $g$. Elements of $\sG$
are called {\it vertices} of the graph. Given $g$ and $x\in \sG$
the coordination number $d_x$ of the vertex $x$ is defined as the
number of links $b$ in $g$ such that $b\cap\{x\}\neq \emptyset$.

\\With this definitions we can write

$$
2^{-|\Lambda|} \sum_{\sigma_{\Lambda}}\prod_{b \in
B(\Lambda)} (1 +\tilde\s_b \tanh(\beta J_{b}))=
1+ \sum_{{\{g_1, \dots ,g_n\}\atop g_i ~{\rm connected\; graph}}
\atop~\sg_i \cap \sg_j=\emptyset}
\r(g_1)\dots \r(g_n)
$$
where
$$
\r(g)=2^{-|\sg|} \sum_{\sigma_{\sg}}
\left[\prod_{b\in g}\tilde\s_b \tanh(\beta J_{b})\right]
$$
For any connected graph $g$ we have
$$
\prod_{b\in g}\tilde\s_b = \prod_{x\in \sg}\s_x^{d_x}
$$
Hence
$$
\sum_{\s_{\sg}}\prod_{b\in g}\tilde\s_b=\cases{0 &if $d_x$ is odd
for some $x\in \sg$\cr\cr 2^{|\sg|} &if $d_x$ is even for all
$x\in \sg$ }\Eq(pi)
$$
and
$$
\prod_{b\in g} \tanh(\beta J_{b}) = \prod_{x\in \sg} \tanh(\beta
|J_{b}|) \Eq(pi1)
$$
if $d_x$ is even for all $x\in \sg$.


Define a {\it contour} $\gamma$ as a connected graph in $\L$ such
that $d_x$ is even for all $x\in \sga$. Then
$$
2^{-|\Lambda|} \sum_{\sigma_{\Lambda}}\prod_{b \in B(\Lambda)} (1
+\tilde\s_b \tanh(\beta J_{b}))= 1+ \sum_{{\{\g_1, \dots
,\g_n\}\atop \g_i ~{\rm contour}} \atop~{\rm supp} \g_i \cap {\rm
supp} \g_j=\emptyset} \r(\g_1)\dots \r(\g_n)
$$
where by \equ(pi) and \equ(pi1) we simply have
$$
\r(\g) = \prod_{b\in \g} \tanh(\beta |J_{b}|)
$$
In terms of contours one has a representation
$$
Z_{\Lambda}= \cosh(\beta J)^{|B_w(\L)|}~
\cosh(\beta I)^{|B_s(\L)|}~~\Xi_\L
$$
where
$$
\Xi_\L= 1+ \sum_{{\{\g_1, \dots ,\g_n\}\subset B(\L)\atop \g_i
~{\rm contour}} \atop~{\rm supp} \g_i \cap {\rm supp}
\g_j=\emptyset} \r(\g_1)\dots \r(\g_n)
$$
is the grand canonical partition function of a hard core gas of
contours $\g$ with activity $\r(\g)$. The infinite volume pressure
of the system is
$$
p= \lim_{\L\to \infty}{1\over |\L|}\log Z_{\Lambda}= \lim_{\L\to \i}
\left[{|B_w(\L)|\over |\L|}\log \cosh(\beta J)+ {|B_s(\L)|\over
|\L|}\log \cosh(\beta I)\right]+ \lim_{\L\to \i}{1\over |\L|}\log
\Xi_{\Lambda}
$$
Clearly $|\L|^{-1} |B_w(\L)| \log \cosh(\beta J)+ |\L|^{-1}
|B_s(\L)| \log \cosh(\beta I)$ is analytic for all positive $\b$
uniformly in $\L$. Therefore to prove the analyticity of $p$ one
has simply to check the analyticity of the pressure of the hard
core polymer gas. By the standard cluster expansion (see [KP]) the
analyticity of $\lim_{\L\to\i}|\L|^{-1}\log  \Xi_{\Lambda}$
follows from the fact that for some $a>0$
$$
\sup_{x\in  \mathbb{Z}^d}\sum_{\gamma:\ x\in \sga} \r(\gamma)
 e^{a|\sga|} \le
{a}\Eq(1.2)
$$
where the sum is taken over all contours whose support contains a
given lattice site $x$. In our case this estimate is a simple
exercise in contour summation.

Observe that each contour consists of even number of {\it elbows},
where an {\it elbow}, $\d$, is a connected subgraph of $\g$
consisting of a maximal in $\g$ segment of unit lattice bonds
along one direction followed by a maximal in $\g$ segment of unit
lattice bonds along an orthogonal direction. More precisely, a
connected subgraph $\d$ of $B(\L)$ is an elbow if, for some $1\le
j<k\le d$
$$
\d= \{b_1, \dots b_ s, b'_1,\dots ,b'_\ell\}
$$
with $b_i = \{x+(i-1)e^{(j)}, x+ ie^{(j)}\}$, (for $i=1, 2, \dots
, s$) and $b'_{i'} = \{x'+(i'-1)e^{(k)}, x'+ i'e^{(k)}\}$, (for
$i'=1, 2, \dots , \ell$) with $x' = x+ se^{(j)}$. Given an elbow
$\d$ denote by $\{x^\d,y^\d\}$ the end-points of the elbow, i.e.
the only two vertices of the elbow with coordination number equal
to one. 



It is now clear that any contour $\g$
can be partitioned
(generally not in a unique way) into a union of elbows
$$
\g = \cup_{i=1}^k\d_i
$$
Here $(\d_1, \dots , \d_k)$ is a $k$-tuple of ordered distinct
elbows connected in the sense that  $y^{\d_i}=x^{\d_{i+1}}$ and
$x^{\d_1}=y^{\d_k}$. The existence of this partition
is provided by Euler theorem, which states that for any graph 
with even coordination
numbers at all 
its vertices it exists a path which passes for every link only once
and end in the starting point. 


If $(\d_1, \dots , \d_k)$ is a connected $k$-tuple of ordered
distinct elbows and $\g=\cup_{i=1}^k\d_i$ is the corresponding
contour then with
$$
\r(\d) = \prod_{b\in \d} \tanh(\beta |J_{b}|)
$$
we have
$$
\r(\g)=\prod_{i=1}^k \r(\d_i)
$$
and also
$$
e^{a|\sga|} \le \prod_{i=1}^k e^{a|{\rm supp}\, \d_i|}
$$
Now we can reorganize the sum over all contours whose support
contains a fixed point $x$ into a sum over connected
$(2k-1)$-tuples of elbows starting at $x$. Such a $(2k-1)$-tuple
can be uniquely completed to a contour by adding an additional
elbow which connects the beginning of the first elbow in the
$(2k-1)$-tuple with the end of the last elbow in the
$(2k-1)$-tuple. Therefore
$$
\sup_{x\in  \mathbb{Z}^d}\sum_{\gamma:\ x\in \sga} \r(\gamma)
 e^{a|\sga|} \le  \sup_{\d}\r(\d)e^{a|{\rm supp}\, \d|}
\sum_{k=1}^{\i} \sum_{(\d_1, \dots , \d_{2k-1}) \atop 0\in {\rm
supp}\,\d_1 } \prod_{i=1}^{2k-1} \r(\d_i)e^{a|{\rm supp}\, \d_i|}
\le
$$
$$
\le \sup_{\d}\r(\d)e^{a|{\rm supp}\, \d|} \sum_{k=1}^{\i}
\left[\sum_{\d:\ 0\in {\rm supp}\,\d}\r(\d) e^{a|{\rm supp}\,
\d|}\right]^{2k-1}\Eq(AA)
$$
\noindent To get an upper bound for the term in square brackets in
\equ(AA) we make the worst hypothesis that the elbow has an arm in
the strong coupling direction. Let $m$ be the number of unit bonds
in the elbow along the strong coupling direction and $n$ be the
number of unit bonds in the elbow along the other direction. Note
that the support of the elbow contains $m+n+1$ lattice sites. Then
$$
\sum_{\d:\ 0\in {\rm supp}\,\d}\r(\d) e^{a|{\rm supp}\, \d|} <
e^a\Big(2d \sum_{m = 1}^{\infty} \tanh(\beta I)^m e^{a m}\Big)
\Big( 2(d-1) \sum_{n=1}^{\infty} \tanh(\beta J)^n e^{a
n}\Big)\Eq(star)
$$
where $2d$ is the number of possibilities to select a direction of
the first leg of the elbow, $2(d-1)$ is a number of possibilities
to select a direction of the second leg of the elbow and in the
worst situation the first leg goes along the strong coupling
direction.

To ensure a convergence of the sum over $m$ we need to choose $a$
in such way that
$$
e^a \tanh \b I < 1 \Eq(bol)
$$
For the definiteness we set
$$
a = - {1\over 2}\log \tanh \b I \Eq(bul)$$ The sum over $n$ in
\equ(star) also converges because $J<I$. With the choice \equ(bul)
the right hand side of  \equ(star) can be estimated from above by
$$
4d(d-1) {\tanh(\beta I) e^{2a} \over 1 - \tanh(\beta I) e^{a}}
\cdot {\tanh(\beta J) e^{a} \over 1 - \tanh(\beta J) e^{a}} =
{4d(d-1) \over 1 -[ \tanh(\beta I)]^{1\over 2}} \cdot {\tanh(\beta
J)\over [\tanh(\beta I)]^{1\over 2}  - \tanh(\beta J) } \Eq(BB)
$$
Also the contribution of a single elbow $\d$ may be bounded by
$$
\r(\d)e^{a|{\rm supp}\, \d|} \le [\tanh(\b J) e^{a}]~[\tanh(\b I)
e^{a}]~ e^a\le \tanh\b J( \tanh\b I)^{-{1\over 2}}\Eq(fixed)
$$
Inserting now \equ(BB) and \equ(fixed) in \equ(AA) we complete the
proof of the theorem by the following estimate
$${\tanh\b J\over (
\tanh\b I)^{1 \over 2}}\sum_{k=1}^{\infty} \left[ {4d(d-1) \over 1
-[ \tanh(\beta I)]^{1\over 2}} \cdot {\tanh(\beta J)\over
[\tanh(\beta I)]^{1\over 2}  - \tanh(\beta J) }\right]^k < -
{1\over 2}\log \tanh( \beta I)\Eq(CC)
$$
which is true for $\beta J < C d^{-2} (1-\tanh( \beta I))$ and $C
> 0$ small enough.



\vglue.5truecm
\\{\it Aknowledgements} This work was supported by Ministero dell'Universita' e della
Ricerca Scientifica e Tecnologica (Italy) and Conselho Nacional de
Desenvolvimento Científico e Tecnológico  - CNPq, a Brazilian
Governamental agency promoting Scientific and tecnologic
development (grant n. 460102/00-1).



We thank Marzio Cassandro for many useful suggestions  and
discussions on a first version of this note, Giovanni Gallavotti
for useful discussions and Senya Shlosman for bibliographic
 suggestions.
\vglue.5truecm


\section*{References}


\vskip.3cm
\\[B]
R. J. Baxter: {\it  Exactly Solved Models in Statistical
Mechanics} Academic Press, 1982

\vskip.5cm
\\[BK]
J. Bricmont and A. Kupiainen: {\em  High temperature expansions
and dynamical systems} Comm. Math. Phys. {\bf 178}, 703-732 (1996)

\vskip.5cm
\\[CO]  M. Cassandro and E. Olivieri:
{\em Renormalization group and analiticity in
one dimension: a proof of Dobrushin's theorem},
Comm. Math. Phys.  {\bf 80}, 255-269 (1981)


\vskip.5cm
\\[DM]  R. L. Dobrushin and M. R. Martirosyan:
{\em Possibility of high temperature phase
transitions due to the many-particle nature
of the potential}, Theor. Math Phys. {\bf 75},
443-448 (1989)

\vskip.5cm
\\[DS1] R. L. Dobrushin,  and S. B.  Shlosman: {\it Completely
analytical Gibbs fields. Statistical physics and dynamical
systems}. (${\rm K\ddot{o}szeg}$, 1984), 371--403, Progr. Phys., 10, Birkhäuser Boston, Boston, MA, 1985.

\vskip.5cm
\\[DS2]  R. L. Dobrushin,  and S. B.  Shlosman: {\it Constructive criterion
for the uniqueness of Gibbs field. Statistical
physics and dynamical systems} (${\rm K\ddot{o}szeg}$, 1984), 347--370, Progr. Phys., 10,
Birkhäuser Boston, Boston, MA, 1985.

\vskip.5cm
\\[KP]  R. Koteck\'y and D. Preiss, {\it Cluster expansion 
for abstract Polymer models}, Comm. Math Phys., {\bf 103} (1986), 
491--498.

\vskip.5cm

\\[NOZ] F.R. Nardi, E. Olivieri and M. Zahradnik:
{\it On the Ising model with strongly anisotropic external field}, J.
Statist. Phys. {\bf 97} (1999), no. 1-2, 87--144.











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