Content-Type: multipart/mixed; boundary="-------------0110190912908" This is a multi-part message in MIME format. ---------------0110190912908 Content-Type: text/plain; name="01-389.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-389.keywords" Asymmetric Ising model ---------------0110190912908 Content-Type: application/x-tex; name="mps19-10.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="mps19-10.tex" \documentclass[10pt]{article} \usepackage{amsfonts,amssymb,a4} \usepackage[dvips]{graphicx} \def\theequation{\thesection.\arabic{equation}} \newcommand{\zeq}{\setcounter{equation}{0}} \newcommand{\qed}{\hfill\rule{3mm}{3mm}} \topmargin 0cm \textheight 22.5cm \textwidth 16cm \oddsidemargin 0.5cm \renewcommand{\baselinestretch}{1.0} \newtheorem{teorema}{Theorem}[section] \newtheorem{lema}{Lemma}[section] %\input{epsf} \begin{document} %%%%%%%%%%%%%%%%% FORMATO %\magnification=\magstep1\hoffset=0.cm \voffset=-1.5truecm\hsize=16.5truecm \vsize=24.truecm \baselineskip=14pt plus0.1pt minus0.1pt \parindent=12pt \lineskip=4pt\lineskiplimit=0.1pt \parskip=0.1pt plus1pt \def\ds{\displaystyle}\def\st{\scriptstyle}\def\sst{\scriptscriptstyle} %%%%%%%%%%%%%%%% GRECO \let\a=\alpha \let\b=\beta \let\c=\chi \let\d=\delta \let\e=\varepsilon \let\f=\varphi \let\g=\gamma \let\h=\eta \let\k=\kappa \let\l=\lambda \let\m=\mu \let\n=\nu \let\o=\omega \let\p=\pi \let\ph=\varphi \let\r=\rho \let\s=\sigma \let\t=\tau \let\th=\vartheta \let\y=\upsilon \let\x=\xi \let\z=\zeta \let\D=\Delta \let\F=\Phi \let\G=\Gamma \let\L=\Lambda \let\Th=\Theta \let\O=\Omega \let\P=\Pi \let\Ps=\Psi \let\Si=\Sigma \let\X=\Xi \let\Y=\Upsilon %%%%%%%%%%%%%%%% EQUAZIONI CON NOMI SIMBOLICI %%% per assegnare un nome simbolico ad una equazione basta %%% scrivere \Eq(...) o, in \eqalignno, \eq(...) o, %%% nelle appendici, \Eqa(...) o \eqa(...): %%% dentro le parentesi e al posto dei ... %%% si puo' scrivere qualsiasi commento; per avere i nomi %%% simbolici segnati a sinistra delle formule si deve %%% dichiarare il documento come bozza, iniziando il testo con %%% \BOZZA. Sinonimi \Eq,\EQ. %%% All' inizio di ogni paragrafo si devono definire il %%% numero del paragrafo e della prima formula dichiarando %%% \numsec=... \numfor=... (brevetto Eckmannn). \global\newcount\numsec\global\newcount\numfor \gdef\profonditastruttura{\dp\strutbox} \def\senondefinito#1{\expandafter\ifx\csname#1\endcsname\relax} \def\SIA #1,#2,#3 {\senondefinito{#1#2} \expandafter\xdef\csname #1#2\endcsname{#3} \else \write16{???? il simbolo #2 e' gia' stato definito !!!!} \fi} \def\etichetta(#1){(\veroparagrafo.\veraformula) \SIA e,#1,(\veroparagrafo.\veraformula) \global\advance\numfor by 1 % \write15{@def@equ(#1){\equ(#1)} \%:: ha simbolo= #1 } \write16{ EQ \equ(#1) ha simbolo #1 }} \def\etichettaa(#1){(A\veroparagrafo.\veraformula) \SIA e,#1,(A\veroparagrafo.\veraformula) \global\advance\numfor by 1\write16{ EQ \equ(#1) ha simbolo #1 }} \def\BOZZA{\def\alato(##1){ {\vtop to \profonditastruttura{\baselineskip \profonditastruttura\vss \rlap{\kern-\hsize\kern-1.2truecm{$\scriptstyle##1$}}}}}} \def\alato(#1){} \def\veroparagrafo{\number\numsec}\def\veraformula{\number\numfor} \def\Eq(#1){\eqno{\etichetta(#1)\alato(#1)}} \def\eq(#1){\etichetta(#1)\alato(#1)} \def\Eqa(#1){\eqno{\etichettaa(#1)\alato(#1)}} \def\eqa(#1){\etichettaa(#1)\alato(#1)} \def\equ(#1){\senondefinito{e#1}$\clubsuit$#1\else\csname e#1\endcsname\fi} \let\EQ=\Eq %%%%%%%%%%%%%%%% GRAFICA \def\bb{\hbox{\vrule height0.4pt width0.4pt depth0.pt}}\newdimen\u \def\pp #1 #2 {\rlap{\kern#1\u\raise#2\u\bb}} \def\hhh{\rlap{\hbox{{\vrule height1.cm width0.pt depth1.cm}}}} \def\ins #1 #2 #3 {\rlap{\kern#1\u\raise#2\u\hbox{$#3$}}} \def\alt#1#2{\rlap{\hbox{{\vrule height#1truecm width0.pt depth#2truecm}}}} \def\pallina{{\kern-0.4mm\raise-0.02cm\hbox{$\scriptscriptstyle\bullet$}}} \def\palla{{\kern-0.6mm\raise-0.04cm\hbox{$\scriptstyle\bullet$}}} \def\pallona{{\kern-0.7mm\raise-0.06cm\hbox{$\displaystyle\bullet$}}} %%%%%%%%%%%%%%%% PIE PAGINA \def\data{\number\day/\ifcase\month\or gennaio \or febbraio \or marzo \or aprile \or maggio \or giugno \or luglio \or agosto \or settembre \or ottobre \or novembre \or dicembre \fi/\number\year} \setbox200\hbox{$\scriptscriptstyle \data $} \newcount\pgn \pgn=1 \def\foglio{\number\numsec:\number\pgn \global\advance\pgn by 1} \def\foglioa{a\number\numsec:\number\pgn \global\advance\pgn by 1} %\footline={\rlap{\hbox{\copy200}\ $\st[\number\pageno]$}\hss\tenrm \foglio\hss} %%%%%%%%%%%%%%% DEFINIZIONI LOCALI \def\sqr#1#2{{\vcenter{\vbox{\hrule height.#2pt \hbox{\vrule width.#2pt height#1pt \kern#1pt \vrule width.#2pt}\hrule height.#2pt}}}} \def\square{\mathchoice\sqr34\sqr34\sqr{2.1}3\sqr{1.5}3} \let\nin\noindent \let\ciao=\bye \def\fiat{{}} \def\pagina{{\vfill\eject}} \def\\{\noindent} \def\bra#1{{\langle#1|}} \def\ket#1{{|#1\rangle}} \def\media#1{{\langle#1\rangle}} \def\ie{\hbox{\it i.e.\ }} \let\ii=\int \let\ig=\int \let\io=\infty \let\i=\infty \let\dpr=\partial \def\V#1{\vec#1} \def\Dp{\V\dpr} \def\oo{{\V\o}} \def\OO{{\V\O}} \def\uu{{\V\y}} \def\xxi{{\V \xi}} \def\xx{{\V x}} \def\yy{{\bf y}} \def\kk{{\bf k}} \def\zz{{\V z}} \def\rr{{\V r}} \def\zz{{\V z}} \def\ww{{\V w}} \def\Fi{{\V \phi}} \let\Rar=\Rightarrow \let\rar=\rightarrow \let\LRar=\Longrightarrow \def\lh{\hat\l} \def\vh{\hat v} \def\ul#1{\underline#1} \def\ol#1{\overline#1} \def\ps#1#2{\psi^{#1}_{#2}} \def\pst#1#2{\tilde\psi^{#1}_{#2}} \def\pb{\bar\psi} \def\pt{\tilde\psi} \def\E#1{{\cal E}_{(#1)}} \def\ET#1{{\cal E}^T_{(#1)}} \def\LL{{\cal L}}\def\RR{{\cal R}}\def\SS{{\cal S}} \def\NN{{\cal N}} \def\HH{{\cal H}}\def\GG{{\cal G}}\def\PP{{\cal P}} \def\AA{{\cal A}} \def\BB{{\cal B}}\def\FF{{\cal F}} \def\tende#1{\vtop{\ialign{##\crcr\rightarrowfill\crcr \noalign{\kern-1pt\nointerlineskip} \hskip3.pt${\scriptstyle #1}$\hskip3.pt\crcr}}} \def\otto{{\kern-1.truept\leftarrow\kern-5.truept\to\kern-1.truept}} \def\arm{{}} \font\bigfnt=cmbx10 scaled\magstep1 %\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 $1\le j 0$ 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. \end{document} ---------------0110190912908--