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Gibbs measures, quasilocality, almost Gibbs, weakly Gibbs
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\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}
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