Content-Type: multipart/mixed; boundary="-------------9811080852788" This is a multi-part message in MIME format. ---------------9811080852788 Content-Type: text/plain; name="98-701.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="98-701.keywords" Gibbs measures, quasilocality, almost Gibbs, weakly Gibbs ---------------9811080852788 Content-Type: application/x-tex; name="contrex.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="contrex.tex" \documentstyle[12pt]{article} \setlength{\textheight}{8.9in} \setlength{\textwidth}{6.2in} \topmargin= -1.2cm \hoffset -1cm \raggedbottom \renewcommand{\baselinestretch}{2.0} \newcommand{\be}{\begin{equation}} \newcommand{\en}{\end{equation}} \newcommand{\ba}{\begin{eqnarray}} \newcommand{\ea}{\end{eqnarray}} \newcommand{\bea}{\begin{eqnarray*}} \newcommand{\eea}{\end{eqnarray*}} \newcommand{\bee}{\begin{enumerate}} \newcommand{\ene}{\end{enumerate}} \newcommand{\non}{\nonumber} \newcommand{\no}{\noindent} \newcommand{\vs}{\vspace} \newcommand{\hs}{\hspace} \newcommand{\e}{é} \newcommand{\D}{\dagger} \newcommand{\ef}{è} \newcommand{\bC}{{\bf C}} \newcommand{\Bbb}{\bf} \newcommand{\p}{\partial} \newcommand{\ha}{{1\over 2}} \newcommand{\un}{\underline} \newcommand{\ov}{\overline} \newcommand{\var}{\varphi} \newtheorem{Th}{Theorem} \newtheorem{Def}{D\'efinition} \newtheorem{Lem}{Lemma} \newtheorem{Pro}{Proposition} \pagestyle{plain} \title{{\bf Weakly Gibbsian Measures and Quasilocality : a long range pair-interaction example.}} \author{R. Lefevere\\ UCL, Physique Th\'eorique, B-1348, Louvain-la-Neuve, Belgium\\lefevere@fyma.ucl.ac.be\\} \begin{document} \maketitle \begin{abstract} We exhibit an example of a measure on a discrete and finite spin system whose conditional probabilities are given in terms of an almost everywhere absolutely summable potential but are discontinuous almost everywhere. \end{abstract} \no{\bf Keywords:} Gibbs measures, quasilocality, almost Gibbs, weakly Gibbs \section{Introduction.} This note is intended to provide a simple example of the non-equivalence of different notions introduced recently in order to generalize the standard Gibbs theory. We want to consider the difference between {\it weakly} Gibbsian measures and {\it almost} Gibbsian measures. We shall work on a spin system whose configuration space is given by $\Omega=\{0,+1\}^{{\Bbb Z}^d}$. We denote by $s$ an element of $\Omega$ and by $s_A$ an element of $\Omega_A=\{0,+1\}^{A}$, for $A\subset {\Bbb Z}^d$. Consider potentials (interactions) $\Phi=(\Phi_X)$ which are families of functions \be \Phi_X:\Omega_X\rightarrow{\Bbb R}, \en indexed by $X\subset{\cal L}$, $|X|<\infty$. We say that a potential is $\overline{\Omega}$-pointwise absolutely summable, with $\ov \Omega \subset \Omega$, if \be \sum_{X \ni x} | \Phi_X (s_{X }) | < \infty\;\;\; \forall x \in {\Bbb Z}^d, \forall s \in {\overline \Omega}. \label{1.1} \en Then, if $\ov {\Omega}$ is in the tail-field, one may define a Hamiltonian in a finite volume $V$ with boundary conditions $\overline{s}\in \ov \Omega$ by the usual formula : \be H (s_V | \bar s_{V^c}) = \sum_{X \cap V \neq \emptyset} \Phi_X (s_{X \cap V} \vee \bar s_{X \cap V^c}) \en where $s_V \in \Omega_V $, $ \bar s_{V^c}$ is the restriction to $V^c$ of $\bar s \in \overline\Omega$ and, for $X \cap Y = \emptyset$, $s_X \vee s_Y$ denotes the obvious configuration in $\Omega_{X\cup Y}$. We say that a measure on $\Omega$ is {\it weakly} Gibbsian if there exist a translation-invariant set $\overline{\Omega}$, and a $\overline{\Omega}$-pointwise absolutely summable interaction $\Phi$ such that $\mu(\overline{\Omega})=1$ and for $\mu$ there exists a version of the conditional probabilities that satisfy $\forall V\subset {\Bbb Z}^d$, $|V|$ finite, $\forall s_{V} \in \Omega_{V}$ \be \mu(s_V | \bar s_{V^c}) \;=\; \cases{ Z^{-1} (\bar s_{V^c}) \exp (-H (s_V | \bar s_{V^c})) & for $\bar s \in \overline{\Omega}$ \cr 0 & for $\bar s \not\in \overline{\Omega}$ \cr } \label{1.2} \en A function $f$ on $\Omega$ is said to be quasilocal at a certain $s$, if $s$ is a point of continuity of $f$ in the product topology on $\Omega$, i.e. \be \forall \epsilon>0,\,\exists V(s)\,such\,that\,if\,s'_{V(s)}=s_{V(s)} ,\,we\,have,\,|f(s)-f(s')|<\epsilon. \label{1.3} \en A function $f$ is {\it essentially} discontinuous at $s$ if \be \exists\delta>0,\,\forall V,\,\exists s'\,s.t.\, s'_{V}=s_{V}\, and\, \exists V',\,V'\supset V\,s.t.\, \forall s'' \,s.t.\, s''_{V'}=s'_{V'},\, |f(s'')-f(s)|>\delta. \label {1.4}. \en The difference between discontinuity and essential discontinuity is important in our context. Indeed, the points of essential discontinuity of a system of conditional probabilities can not be tranformed into points of continuity by modifying the conditional probabilities on a set of measure zero. A measure $\mu$ on $\Omega$ is said to be {\it almost} Gibbsian \cite{MaRe} if there exists a version of its conditional probabilities that is quasilocal (as a function of the conditioning) $\mu$-almost everywhere. The quasilocality {\it everywhere} of the conditional probabilities is an important characterization of the measure due to a theorem by Kozlov and Sullivan \cite{Ko,Su}. This theorem states that if a measure has a system of conditional probabilities that is quasilocal everywhere then, modulo a positivity condition on the conditional probabilities, the measure is a Gibbs measure in the standard sense and the converse is also true. The relationship between the two definitions introduced above (i.e. weakly and almost Gibbsian) is not clear a priori. A generalization of the result of Kozlov and Sullivan was obtained in \cite{MaRe} for almost Gibbsian measures. Almost Gibbsian measures are also weakly Gibbsian. But, for example, the conditional probabilities of some measures obtained after a Renormalization Group Transformation on the Ising model at low temperature have a non-empty set of points of essential discontinuity \cite{vefs}. While the size measure of this set is unknown, it is shown, as in the Schonman example \cite{Scho,Mavdv}, that those measures are weakly Gibbsian \cite{BKL}. We provide here an example of measure $\mu$ on $\Omega$ that is weakly Gibbsian but whose conditional probabilities are essentially discontinuous on a set of $\mu$-measure 1. Our example is similar to the one given in \cite{MaRe} but different in the sense that we are able to prove that the set of points of discontinuity is of measure 1 with respect to the measure under study, the form of our interaction being extremely simple (pair-interactions). Besides our model is defined in any dimension $d$ (but is not translation invariant, as in \cite{MaRe}). \section{Results} Consider the measure on $\Omega$ \be \mu(ds)=\frac{\exp-H(s)}{Z}\mu_0(ds) \label{2.1} \en with $\mu_0$ the product measure, $H(s)$ is formally given by \be H(s)=\sum^{\infty}_{i,j\in {\Bbb Z}^d} 2^{|i|+|j|}s_i s_j \label{2.2} \en and obviously $Z=\sum_{s}\exp-H(s)\mu_0(ds)$. $Z$ is well-defined because $\exp-H(s)$ is a limit of measurable functions and and it is uniformly bounded on $\Omega$, besides it is obviously non-zero. {\bf Remark.} Our argument in this note can be easily generalized to a Hamiltonian defined like in (\ref{2.2}) but with $2^{|i|+|j|}$ replaced by $\phi_{ij}$ with $\phi_{ij}$ such that $\phi_{ij}\geq 0$ and $\phi_{ii}\rightarrow\infty$ as $|i|\rightarrow\infty$. It is easy to see that $H(s)$ is finite on the set \be \Omega_g=\{s\in \Omega| \exists \ov V(s)\,a\,cube\,, \, \forall n\notin\ov V(s)\, s_n=0\}. \label{2.3} \en \begin{Pro} \be \mu(\Omega_g)=1 \en \end{Pro} {\it Proof.} Let us prove that $\mu(\Omega^c_g)=0$. One has, $\Omega^c_g=\cap_{N=0}^{\infty}\Omega_N$ with, \be \Omega_N=\{s\in \Omega|\exists n \,s.\,t.\, |n|\geq N\,s_n=1\} \label{2.4} \en and also, \be \Omega_N\subset\cup_{n:|n|\geq N}\ov \Omega_n \label{2.5} \en with, \be \ov\Omega_n=\{s\in\Omega|s_n=1\}. \label{2.6} \en Now, we write, \be \mu(\ov \Omega_n)=\mu(s_n=1)=\sum_{s:s_n=1}\frac{\exp -H(s)}{Z}, \label{2.7} \en and using the Hamiltonian (\ref{2.2}), we see that this expression is bounded from above by \be e^{-2^{2|n|}}\sum_{s:s_n=0}\frac{\exp -H(s)}{Z}=e^{-2^{2|n|}}\mu(s_n=0)\leq e^{-2^{2|n|}}, \label{2.8} \en because we have $H(s)-H(\ov s)\geq 2^{2|n|}$ if $s$ is such that $s_n=1$ and $\ov s$ such that $\ov s_n=0$, $\ov s_{{\Bbb Z}^d \backslash\{n\}}=s_{{\Bbb Z}^d \backslash\{n\}}$. From the bound (\ref{2.8}) it is easy to conclude that $\mu(\Omega_N)\leq cN^{d-1}e^{-2^{2N}}$ and thus that $\mu(\Omega^c_g)=0$, which concludes the proof. \vspace*{5mm} We can then easily compute the conditional distribution of the spin at the origin and get \be \mu(s_0|\ov s_{\{0\}^c})=\frac{1}{1+\exp(-(1-2s_0)\sum_{k\in {\Bbb Z}^d}2^{|k|}\ov s_k )} \label{2.9} \en Obviously, this conditional probability is expressed in terms of long-range two-body interactions and we may define the relative energy function of the spin at the origin. \be h_0(s)=(1-2s_0)\sum_{k\in {\Bbb Z}^d}2^{|k|} s_k. \label{2.10} \en Obviously, as $H(s)$, it is (absolutely) summable on $\Omega_g$. To conclude, we need to show that this function is not only discontinuous but essentially discontinuous on $\Omega_g$. \begin{Pro} $h_0$ is essentially discontinuous on $\Omega_g$ in the product topology on $\Omega$. \end{Pro} {\it Proof.} From the definition of $h_0$ in (\ref{2.10}) and $\Omega_g$ in (\ref{2.3}), it is easy to see that if $s\in \Omega_g$ and $ s'\in\Omega_g$ are such that $s_{\ov V(s)}=s'_{\ov V(s)}$ and $s_k\neq s'_k$, then $|h_0(s)-h_0(s')|\geq2^{|k|}$ ( $k\in \ov V^c(s)$). It is then easy to see that $h_0$ is essentially discontinuous on $\Omega_g$; for any $s\in \Omega_g$, in (\ref{1.4}) take $\delta=1$ and $\forall V$, choose $V'$ and $s'$ as follows, take $V'$ a finite (connected) set such that $V'\supset (V\cup\ov V(s))$ and $s'$ a configuration such that $s'_{V\cup\ov V(s)}=s_{V\cup\ov V(s)}$ and $s'_{V'\backslash (V\cup\ov V(s))}\neq s_{V'\backslash (V\cup\ov V(s))}$. Then for any configuration $s''$ such that $s''_{V'}= s'_{V'}$, it is clear that one has $|h_0(s'')-h_0(s)|>1$. \vs{10mm} \no{\bf Acknowledgments.} I would like to thank J.Bricmont for his advices and A.van Enter for discussions. \vs{10mm} \addcontentsline{toc}{section}{\bf References} \begin{thebibliography}{10} \bibitem{BKL} J.~Bricmont A.~Kupiainen and R.Lefevere. \newblock Renormalization Group pathologies and the definition of Gibbs states. \newblock{\em Commun.Math.Phys.}, {\bf 194}:359-388, 1998. \bibitem{vefs} A.C.D.~van Enter R.~Fernandez and A.D.~Sokal. \newblock Regularity properties and pathologies of position-space renormalization transformations : Scope and limitations of Gibbsian theory. \newblock{\em J.Stat.Phys.}, {\bf 72}:879-1167, 1993. \bibitem{Ko} O.K.~Kozlov. \newblock Gibbs description of a system of a random variables \newblock{\em Problems Info Trans.}, {\bf 10}:258-265, 1974. \bibitem{MaRe} C.~Maes F.~Redig and A.~Van Moffaert. \newblock Almost Gibbsian versus Weakly Gibbsian \newblock{\em to appear in Stoch.Proc. Appl.} \bibitem{Mavdv} C.~Maes and K.~Vande Velde. \newblock Relative energies for non-Gibbsian states \newblock{\em Commun.Math.Phys.}, {\bf 189}:277-286, 1997. \bibitem{Scho} R.H.~Schonmann. \newblock Projections of Gibbs measures may be non-Gibbsian. \newblock{\em Commun.Math.Phys.}, {\bf 124}:1-7, 1989. \bibitem{Su} W.G.~Sullivan. \newblock Potentials for almost Markovian random fields. \newblock{\em Commun.Math.Phys.}, {\bf 33}:61-74, 1973. \end{thebibliography} \end{document} ---------------9811080852788--