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\begin{document}
\title{The (non--)Gibbsian nature of states invariant under stochastic
transformations}
{\renewcommand{\baselinestretch}{1}
\author{Christian Maes\thanks{Aangesteld Navorser N.F.W.O. Belgium.} $\;$
and Koen Vande Velde\thanks{Aspirant N.F.W.O. Belgium.} \\ Instituut voor
Theoretische Fysica \\ K.U. Leuven \\ Celestijnenlaan 200D\\ B-3001 Leuven
(Belgium) \\
e--mail: Christian=Maes\%TF\%FYS@cc3.kuleuven.ac.be \\
e--mail: Koen=Van=de=Velde\%TF\%FYS@cc3.kuleuven.ac.be }
\date{} \maketitle
\begin{abstract}
We investigate the Gibbsian nature of systems that
are invariant under some stochastic transformation. A prime example
concerns stationary measures for
interacting particle systems. We bring together various types of results
concerning this question using
entropy techniques and large deviation estimates.
\\ \\
\underline{Keywords}: entropy inequalities, non--Gibbsianess, large
deviations,
stationary states, stochastic transformations, Toom model, restrictions
of equilibrium measures.
\end{abstract} }
\newpage
\section{Introduction.}
Equilibrium statistical mechanics has to do with the study of Gibbs
states. They describe large systems in equilibrium for some given
microscopic interaction potential. A lot is known about them, in
particular for lattice spin systems, due to the rather explicit
construction of the state. In
this paper we ask for the Gibbsian nature of states on lattice
spin configurations that are invariant under certain stochastic
transformations. Understandably, invariance is a rather implicit way
of defining the physical state and yet, this happens quite often to be the
only
information we have about it. Consider for example the problem of
determining
the character of the steady state in models of non--equilibrium statistical
mechanics. We have in mind here the study of the stationary states of
certain stochastic dynamics, such as interacting particle systems or
probabilistic cellular automata.
Given are the updating rules that define the continuous time or discrete
time stochastic evolution and the question is whether a state that after
updating is transformed back into itself, is a Gibbs state for some (and
what) interaction.
It should be clear in that respect that there is more to the question of
``Gibbsianess'' than being able to take the logarithm of the probabilities.
One wants to see if in some {\it bulk} limit, the system can be
consistently described in terms of a relative energy function. This could
for example be related to the type of large deviations one sees. A
detailed account of the relevance and the consequences of these type of
considerations can be found in [1].
This paper grew out of the question how to unify various results in the
literature concerning the nature of states that are not defined in terms of
some explicit interaction potential. Crucial in
these questions is the influence of applying stochastic transformations on
the relative entropy density of two probability measures, defined on spin
systems on a lattice. We deduce that for the type of transformations that
we consider, the
translation invariant stationary measures either are all Gibbsian with
respect to the
same interaction or either are non--Gibbsian. In the context of Markov
spinflip processes on a lattice, a similar statement has been proven by
Holley [2] and K{\" u}nsch [3].
\\ This result is important not only to get a characterization of so
called {\em interacting particle systems}, but
also, to learn more about the restrictions to
a lower dimensional sublattice
of certain Gibbs measures. Indeed, some equilibrium systems
can be seen as Markov extensions of stationary measures for some
transition operator (or, having a {\em global} Markov property)
[4]. In fact, stationary measures of discrete time Markov processes
can also be obtained from a projection of corresponding
equilibrium systems on the space-time lattice [5].
A relevant question is, whether the above
stationary measures and restrictions of Gibbs measures are again
Gibbsian. \\
In the high--noise, high--temperature regime respectively, this is indeed
the case. There we do not have long range order and the weak
correlations make the probability of a configuration in a finite box
to depend continuously on the configuration far away. For probabilistic
cellular automata (PCA), discrete time Markov processes on a lattice with
local and parallel updating, it was proven in [6] that for high enough
noise, the unique stationary measure is Gibbsian. A similar statement for
continuous time spinflip processes was proven in [7]. The situation is
also quite clear in the case when the {\em detailed balance condition} is
satisfied ; see [8] for a discussion of the stochastic or kinetic
Ising model. Very little is however known about the stationary measures of
dynamics that are not reversible. A typical example is the Toom model
[9], [6], on which we will come back later.
As for the projections of Gibbs
measures, it is also straightforward to check that these restrictions are
Gibbsian at high temperature
(see e.g. [10]).
However, for the case of the two-dimensional ferromagnetic Ising model
below the critical temperature,
Schonmann [11] has proven that the restrictions of the $+$ and $-$ phase
to a one-dimensional layer are both non--Gibbsian.
\\ It would also be interesting to study fixed point measures of
real space renormalization group transformations [1],
but our
methods are not adapted to treat these kind of transformations, because the
rescaling of the lattice induces a rescaling in the entropy density as
well, making the entropy density inequalities of Section 2 useless.
We will come back to this at the end of the paper. \\
The paper is organized as follows: \\
In section 2 some definitions and notation is given. Thereafter the
main
result is presented. In Section 3 we discuss the situation for
stochastic spinflip dynamics. Section 4 is devoted to a discussion of
related problems.
\section{Invariance for stochastic transformations.}
\setcounter{equation}{0}
\subsection{Definitions and notation.}
We will consider spin systems on the regular lattice $\integ ^d$ and on
finite sublattices. For any subset $\Lambda \subset \integ ^d$
the configuration space is $\Omega _{\Lambda }=\{-1,+1\} ^{\Lambda }$
and we write $\Omega \equiv \Omega _{\integ ^d}$. The space $\Omega
_{\Lambda }$ is
equipped with the usual product topology and Borel $\sigma $-field ${\cal
F}_{\Lambda }$. The set of probability measures on $(\Omega _{\Lambda
},{\cal F}_{\Lambda })$ is denoted by ${\cal P}_{\Lambda }$ and we let
${\cal P}\equiv {\cal P}_{\integ ^d}$ denote the set of probability
measures on the full
lattice. For $\sigma ,\eta $ in $\Omega $, we say that $\sigma =\eta $ on
$\Lambda $ iff $\sigma (x)=\eta (x)$ for all $x$ in $\Lambda $.
Further we denote $\sigma (\Lambda )\equiv \prod _{x\in \Lambda }\sigma
(x)$. For $x\in \integ ^d$ we define the translation $\tau _x:\Omega
\rightarrow \Omega :\eta \mapsto \tau _x\eta $, where
$\tau _x\eta (y)=\eta (y+x)$. $\tau _x$ acts on a probability measure
$\mu \in {\cal P}$ as $\tau _x\mu (f)=\mu (f\circ \tau _{-x})$, for all
measurable functions $f$ on $\Omega $. We call $\mu \in {\cal P}$
translation invariant if $\tau _x\mu =\mu $, for all $x\in \integ ^d$.
We denote the set of translation invariant measures in ${\cal P}$ by
${\cal P}_{tr}$.
\newtheorem{Df}{Definition} \begin{Df}
For a finite set $\Lambda $ and $\mu ,\nu \in {\cal P}_{\Lambda }$
the relative entropy of $\mu $ w.r.t. $\nu $ is
\begin{equation}
I(\mu ,\nu )=\sum_{\eta \in \Omega _{\Lambda }}
\mu (\eta )
\log \frac{\mu (\eta )}{\nu (\eta )}
\end{equation}
\end{Df} (with 0log0=0).
It is not hard to verify that $I(\mu ,\nu ) \geq 0$ and $I(\mu ,\nu )=0$
iff $\mu =\nu $.
For an infinite volume the relative entropy is almost always infinite, but
for translation invariant measures we can work with the concept of relative
entropy density.
For $\mu \in {\cal P}$ and $\Lambda \subset \integ^d$, let $\mu _{\Lambda
}$ denote the restriction of $\mu $ on $(\Omega_\Lambda,{\cal F}_{\Lambda
})$. Further denote $I_{\Lambda }(\mu ,\nu )=I(\mu _{\Lambda },\nu
_{\Lambda })$. We make the following \begin{Df}
Let $\mu ,\nu \in {\cal P}_{tr}$. Consider the sequence of boxes
$\Lambda =[-L,L]^d$. The relative entropy density of $\mu $ w.r.t. $\nu $
is defined as \begin{equation}
i(\mu ,\nu )=\lim _{L \uparrow \infty }\frac{1}{L^d}I_{\Lambda }(\mu
,\nu ), \end{equation}
provided this limit exists. \end{Df}
The existence of the limit $(2.2)$ is a non--trivial problem. This
limit is known to exist in case $\nu $ is a Gibbs measure for a translation
invariant interaction [12].
We recall the definition of Gibbs measure for a translation invariant
interaction. \begin{Df}
We call $\mu \in {\cal P}_{tr}$ a Gibbs measure w.r.t. the translation
invariant interaction
$\{J_V\} _{V \subset \integ ^d}$, if
there exists a version of the local conditional
probabilities
of the form \begin{equation}
\mu [\mbox{$\sigma =\eta$ on $\Lambda | \sigma =\omega $ on
$\Lambda ^c$}]=Z_{\Lambda }^{-1}(\omega
)\exp [\sum _{V\cap \Lambda \not= \emptyset}J_V\; \eta (\Lambda )\omega
(V\backslash \Lambda )] \end{equation}
and $\sum _{V\ni o} | J_V |< \infty $. \end{Df}
With this definition, these conditional probabilities are continuous
functions of $\omega \in \Omega $.
Of course, there exist also other (non--continuous) versions, because these
conditional
probabilities are only defined $\mu $-a.s.. We denote by ${\cal G}(J)$ the
set of all translation invariant Gibbs measures with respect to
$\{J_V\} _{V \subset \integ ^d}$.
\newtheorem{LE}{Lemma}
We will use the following results:
\begin{LE} $[12]$
Suppose $\nu $ is Gibbs for a translation invariant interaction, then $\mu
$ is Gibbs for the same interaction iff $i(\mu ,\nu )=0$.
\end{LE}
\begin{LE} $[13]$
$\mu \in {\cal P}_{tr}$ is Gibbs for some translation invariant interaction
iff the one point conditional probability
$\mu [\mbox{$\sigma (o)=\eta(o)| \sigma =\omega $ on
$\{ o\} ^c$}]$ is positive and has a version continuous in $\omega \in \Omega $.
\end{LE}
Often it is more convenient to work with relative energies instead of
conditional probabilities.
\begin{Df}
Let $\mu \in {\cal G}(J)$. The relative energy $h_{\mu }$ of $\mu $
for
a spinflip in the origin is the continuous function defined by
\begin{equation}
h_{\mu }(\eta )= \sum _{V}J_V\eta ^o(V)- \sum _{V} J_V\eta (V)=-2 \sum
_{V\ni o}J_V\eta (V), \end{equation}
where $\eta ^o$ is the configuration $\eta $
flipped in
the origin, so $\eta ^o(x)=\eta (x)$ if $x\not= o$ and $\eta ^o(o)=-\eta
(o)$. \end{Df}
Combining the above definitions, we have the following relation between
the relative energy and the one point conditional probabilities:
\begin{equation}
(1+\exp h_{\mu }(\eta ))^{-1}= \mu [\mbox{$\sigma (o)=\eta(o)| \sigma
=\eta $ on $\{ o\} ^c$}].
\end{equation}
For $\mu $ not a Gibbs measure, a function $h_{\mu }$ can also be
defined $\mu $-a.s. via the relation (2.5), but it cannot be continuous.
\subsection{Main result.}
\paragraph {Finite volume inequalities}
Fix the finite volumes $\Lambda ,\Lambda '$ and suppose we have a
pointwise strictly
positive stochastic matrix $P$, defined by its transition
probabilities \begin{eqnarray}
&& p(\cdot |\cdot ):\Omega _{\Lambda }\times \Omega _{\Lambda '}\rightarrow
[0,1] \nonumber \\
&& \sum _{\sigma \in \Omega _{\Lambda }}p(\sigma |\cdot )=1 \nonumber \\
&& p(\cdot |\cdot )>0
\end{eqnarray}
$P$ acts on a probability measure $\mu \in {\cal P}_{\Lambda '}$ as
\begin{equation}
\mu P(\sigma )=\sum _{\eta \in \Omega _{\Lambda '}}p(\sigma |\eta )\mu
(\eta ), \hspace{1cm}\sigma \in \Omega _{\Lambda }. \end{equation}
It is also convenient to define the measure $\mu \times P$ on
$\Omega _{\Lambda }\times \Omega _{\Lambda '}$ by
\begin{equation}
\mu \times P(\sigma ,\eta )=\mu (\eta )p(\sigma |\eta )
\end{equation}
and let $\mu \times P[\sigma ]$ denote the conditional probability of $\mu
\times P$
given $\sigma $ on $\Omega _{\Lambda }$, i.e. \[
\mu \times P[\sigma](\eta)=\frac{\mu \times P(\sigma,\eta)}{\sum _{\eta'}
\mu \times P(\sigma,\eta')}. \]
We remind the reader of the following \begin{LE}
Let $\mu ,\nu \in {\cal P}_{\Lambda '}$, $P$ as in (2.6), then
\begin{equation}
I(\mu ,\nu )=I(\mu P,\nu P)+\sum _{\sigma \in \Omega _{\Lambda }}\mu
P(\sigma )\; I(\mu\times P[\sigma ] ,\nu \times P[\sigma ]).
\end{equation} \end{LE}
\underline{Proof:} \\ \\
In case $\mu $ is not absolutely continuous w.r.t. $\nu $, both sides
equal $+\infty $. Otherwise
let $f(\eta )=\mu (\eta )/ \nu (\eta )$ and $g(\sigma )=\mu P(\sigma
)/ \nu P(\sigma )$ for $\eta \in \Omega _{\Lambda '},\sigma \in \Omega
_{\Lambda }$. Then
the identity follows from the observation that \\ $\mu \times P[\sigma
](\eta ) /\nu \times P[\sigma ](\eta )=f(\eta )/ g(\sigma
)$. \\ \\ \hspace*{10cm} $\Box $
\paragraph{Local stochastic operators}
In this paragraph we deal with stochastic operators $P$ on the full
lattice $\integ ^d$. They are defined by their kernel $P(\cdot |\cdot )$.
That is, $P(\cdot|\eta )\equiv P^{\eta }(\cdot )\in {\cal P}$ for every
$\eta \in \Omega $ and $P(A|\cdot ) \in {\cal F}$ for every $A\in {\cal
F}$. Let $\Lambda $ be any translation of $[-L,L]^d$, i.e.
$\Lambda =[-L,L]^d+x$ for some $x\in \integ^d$.
We suppose that $P$ satisfies the following conditions:
\begin{enumerate}
\item $P_{\Lambda }(\eta |\omega )\equiv P(\sigma =\eta$ on $\Lambda
|\omega )$ depends on $\omega $
only through its coordinates in $\Lambda '=[-L-R,L+R]^d+x$, where $R>0$ is
a constant. We can thus look upon $P_{\Lambda }$ as a
transition
kernel $P_{\Lambda }:\Omega _{\Lambda }\times \Omega
_{\Lambda '}\rightarrow [0,1]$.
\item $P_{\Lambda }$ satisfies all of (2.6).
\item \begin{equation}
\lim _{L\uparrow \infty }\frac{P_{\Lambda
}(\eta |\omega ^o)}{P_{\Lambda }(\eta |\omega )}
\mbox{ is continuous in } \omega \hspace{2.5mm} \mbox{for every }
\eta \in \Omega . \end{equation} \end{enumerate}
We have the following result.
\newtheorem{TH}{Theorem}
\begin{TH}
Suppose we have a stochastic operator $P$
satisfying the above conditions
1,2 and 3. If $P$ has an invariant (or stationary) measure $\nu $ (i.e. $\nu
P=\nu $
), which is Gibbs
for some translation invariant interaction, then any other translation
invariant stationary measure
is also Gibbs with respect to the same interaction. \end{TH}
\newtheorem{COR}{Corollary}
\begin{COR}
The translation invariant stationary measures of $P$
are either all
Gibbsian with respect to the same potential or either they are all
non--Gibbsian.
Moreover, if two of them have non--vanishing relative entropy density,
then they are all non--Gibbsian.
\end{COR}
\underline {Remarks:} \begin{itemize} \item Theorem 1 can easily be
extended to stochastic
operators $P$ that do not satisfy condition 1 of (2.10), but can be
suitably approximated by
a sequence of operators satisfying this condition. (The reason is that
(2.14) below is uniform in $R$.) \item
For the same reason, Theorem 1 can be applied to continuous time spinflip
processes with
positive and local rates. Such a process can be approximated by
a sequence
of discrete time Markov processes with parallel updating satisfying all of
(2.10) (see e.g. [14], [15]). Also positive, continuous rate functions
that are not strictly local pose no
problem because of our first remark.
\item
Condition 2 avoids degeneracies and is certainly necessary for the
statements above.
\item
Condition 3 is very similar to condition 1 but is independent of it : in
a sense, it requires good locality properties of the reversed
transformation. It implies that if you condition a Gibbs measure on what
state you get after
transforming it via $P$, then this conditioned measure is still
Gibbsian.
\end{itemize} \underline{Proof of Theorem 1:} \\
\\ Take $\mu ,\nu \in {\cal P}_{tr}$. Suppose that $\nu P=\nu $, $\nu \in
{\cal G}(J)$. For $\eta \in \Omega
_{\Lambda }$, we have that \begin{equation}
(\mu P)_{\Lambda }(\eta )=\mu _{\Lambda '}P_{\Lambda }(\eta ).
\end{equation}
This just means that \begin{equation}
\int P(\sigma = \eta \hspace{2mm} {\rm on } \hspace{2mm} \Lambda |
\omega)\,\mu(d\omega) = \sum_{\omega\in\Omega_{\Lambda'}}
P_{\Lambda}(\eta |\omega )\,\mu_{\Lambda'}(\omega)
\end{equation}
For $\sigma _{\Lambda }\in \Omega _{\Lambda }$, define \[
(\mu \times P[\sigma _{\Lambda }]) (d\omega )=
{\cal N}(\sigma _{\Lambda }) \mu(d\omega )P_{\Lambda }(\sigma_{\Lambda
}|\omega )
\]
where ${\cal N}(\sigma _{\Lambda })$ is a normalizing factor.
Then, for $\omega \in \Omega
_{\Lambda '}$ we have \[
(\mu \times P[\sigma _{\Lambda }])_{\Lambda '}(\omega )=
\mu _{\Lambda '}\times P_{\Lambda }[\sigma _{\Lambda }](\omega).
\]
For each $L$
we can apply the identity (2.9) to the
stochastic matrix $P_{\Lambda }$:
\begin{eqnarray}
&& \frac{1}{L^d}I_{\Lambda '}(\mu ,\nu )
= \frac{1}{L^d}I_{\Lambda }(\mu P
,\nu P)+ \\
&& \hspace{3cm} \frac{1}{L^d}\sum _{\sigma _{\Lambda }\in \Omega _{\Lambda
}} (\mu P)_{\Lambda }(\sigma _{\Lambda })
I_{\Lambda '}(\mu \times P[\sigma_{\Lambda }],\nu \times P[\sigma _{\Lambda
}]). \nonumber
\end{eqnarray}
Since $R$ is fixed, it will not matter if we write $L^d$ instead of
$(L+R)^d$ when we let $L\uparrow \infty $.
Now note that $P_{\Lambda }$ obeys condition 3 above.
Therefore $\nu \times P[\sigma _{\Lambda }]$ is Gibbs because $\nu $ is.
Moreover, \begin{equation}
h_{\nu \times P[\sigma _{\Lambda }] }
(\eta )= h_{\nu }(\eta )+\log \frac{P_{\Lambda }
(\sigma _{\Lambda }
|\eta ^o)}{P_{\Lambda }(\sigma _{\Lambda }|\eta )}, \hspace{1cm} \eta
\in \Omega . \end{equation}
Suppose now that $\mu $ is not Gibbs with respect to the same
interaction as $\nu $, then $i(\mu ,\nu )>0$ (see Lemma 2) or equivalently
$I_{\Lambda }(\mu ,\nu )={\cal
O}(L^d)$
(this means that for any strictly positive real numbers $c_1,c_2,d_1,d_2$
with $d_1L_0$
$c_1L^{d_1}0$ such that $h_{\mu
}(\eta )\not= h_{\nu }(\eta )$, for $\eta \in {\cal A}$.
Using (2.13), we get that $\mu $-a.s.
\begin{equation}
h_{\mu \times P[\sigma _{\Lambda }]}-h_{\nu \times P[\sigma
_{\Lambda }]}=h_{\mu }-h_{\nu },
\end{equation}
because the $P_{\Lambda }$ just drops out. \\
But then, $\mu \times P[\sigma _{\Lambda }] $ cannot be Gibbs for the same
interaction
as $\nu \times P[\sigma _{\Lambda }] $. \\ So,
$ I_{\Lambda '}(\mu \times P[\sigma _{\Lambda }]
,\nu \times P [\sigma _{\Lambda }]
)={\cal
O}(L^d)$ .
Taking $L\uparrow \infty $, we obtain now that \begin{equation}
i(\mu ,\nu )>i(\mu P,\nu )
\end{equation}
and $\mu $ cannot be stationary for $P$. \\
\hspace*{10cm} $\Box $
\underline{Proof of Corollary 1 :} \\
Suppose $P$ has a translation invariant stationary Gibbs measure.
Then Theorem 1 together with Lemma 1 give that for any two translation
invariant stationary measures $\mu,\nu$, $i(\mu,\nu)=0$. If, on the
contrary, there exist two translation invariant stationary measures with
non--zero relative entropy density, it follows that all translation
invariant stationary measures are non--Gibbsian. \\
\hspace*{10cm} $\Box $
Theorem 1 provides a common framework to understand various results in the
literature having to do with the question Gibbs versus non--Gibbs. In the
following Sections we illustrate this point more clearly.
\section{Spinflip dynamics.} \setcounter{equation}{0}
By a spinflip dynamics we mean a stochastic time evolution for infinite
configurations of spins on a lattice in which every individual spin flips
with a certain rate or a certain probability according to the
configuration of spins nearby. Such continuous time (sequential)
interacting spin systems are thus determined via a spinflip rate $c(x,\eta
)$ giving the probability per unit time that the spin at site $x$ flips
if the present configuration is $\eta $. A well known example is the so
called Glauber dynamics, where the rates $c(x,\eta )\sim \exp [-\beta
h_x(\eta )]$ are (essentially) proportional to the Boltzmann factor of the
relative energy $h_x(\eta )=H(\eta ^x)-H(\eta )$ for some given Hamiltonian
given of the spin system.
More generally, such dynamics that are governed by an energy function
and that satisfy the so called detailed balance condition, go under the
name
of stochastic Ising models. Quite a lot is known about their stationary
states. In particular, the Gibbs states $\mu \sim e^{-\beta H}$ are
time invariant and the dynamics is reversible w.r.t. $\mu $.
However, there are many other physically relevant examples of spinflip
dynamics, which do not satisfy the detailed balance condition and for
which in general very litlle is known about their stationary states.
Could it be that there is still some (effective) Hamiltonian out there
for which the stationary states are Gibbsian? We discuss this question
first by commenting on some work of K\"{u}nsch ([3]) and a paper of
Garrido and Marro ([16]). We end this section by a discussion of the
situation for discrete time spinflip dynamics, examples of probabilistic
cellular automata. We must however start with a disclaimer. We know that
in general the answer to the above question is no. (Non--trivial)
counter examples are provided by voter models in 3 or more dimension
[17]. \paragraph{
K\"{u}nsch (1984)} is studying the local conditional distributions of a
stationary measure for a spinflip process with strictly positive and
continuous flip rates. He shows among other things that either all
translation invariant stationary measures are Gibbsian with the same
potential or no translation invariant stationary Gibbs measure exists.
This has to be compared with our Theorem 1 (Remarks).
Moreover, he constructs an example with infinitely many stationary
measures (all of which are Gibbsian), but no reversible measures. In fact,
the general tone of the paper is that, again for strictly positive and
continuous rates, reversible or not, all stationary measures should
be Gibbsian. In particular, this would mean that the local conditional
distributions of a single spin given all other spins are the same for all
stationary measures. Although this last statement is formulated as a
conjecture in the paper, we wish to point out here that local conditional
distributions as such, carry very little information unless assumed to have
continuous versions. More precisely, if $\mu $ and $\nu $ are both
stationary measures, we can always find a function $h$ on $\Omega $ for
which \begin{eqnarray*}
&& \mu [\sigma (o)=\eta (o)|\sigma (x)=\eta (x), x\not=o]=h(\eta )
\hspace{6mm} \mu -\mbox{a.s.} \\
&& \nu [\sigma (o)=\eta (o)|\sigma (x)=\eta (x), x\not=o]=h(\eta )
\hspace{6mm} \nu -\mbox{a.s.}
\end{eqnarray*}
and $\mu $ and $\nu $ need not be Gibbsian.
\paragraph{Effective Hamiltonian description?}
It has been suggested in Garrido and Marro (1989) that strictly positive
and local flip rates for the dynamics always give rise to Gibbs states as
stationary states. In fact they give an explicit form of the interaction
potential (they call it the effective Hamiltonian). It basically amounts to
writing down the equation ($c(x,\eta )$ can be for example
a sum of Glauber rates with different temperatures) \begin{equation}
\exp [-2\beta \sum _{A\ni x}J_A\eta (A)]=\frac{c(x,\eta )}{c(x,\eta ^x)}
\end{equation}
for the coupling constants $\{ J_A\} $ which would appear in the
(effective) Hamiltonian \begin{equation}
``H(\eta )= -\sum _{A}J_A\eta (A)\mbox{''.}
\end{equation}
Then, the Gibbs measures with respect to this Hamiltonian are of course
invariant under the dynamics. There is however only a consistent solution
to (3.1) if the rates satisfy the condition \begin{equation}
\sum _{\eta}\; \eta (A)\log \frac{c(x,\eta )}{c(x,\eta ^x)}
\frac{c(y,\eta )}{c(y,\eta ^y)}=0
\end{equation}
for every finite set $A$ and all $x,y\in A$. It is easy to see that this
condition implies in fact that we are really dealing with a stochastic
Ising model and (3.3) can never be satisfied by a dynamics which is not
reversible.
\paragraph{Discrete time}
In this paragraph, we deal with probabilistic cellular automata
(PCA). They can be viewed as discrete time versions of spinflip
dynamics.
PCA are discrete time Markov processes on $\integ ^d$, with local and
parallel updating of Ising type spins. The finite volume transition
probabilities can be written as \begin{equation}
P_{\Lambda }(\sigma |\eta )=\prod _{x\in \Lambda }P _{\{ x\} }
(\sigma (x)|\eta )
\end{equation}
and we take the $P_{\{ x\} }(\sigma (x)|\eta )$ positive and local in
$\eta $ (cf. conditions 1 and 2). It is easy to check that also our
condition 3 is verified. Theorem 1 can thus
be applied to them. In the reversible case, where the PCA satisfies the
detailed balance condition (see [6]) with respect to some Gibbs measure for
some translation
invariant interaction $\{ J(V)\} _{V\subset \integ ^d}$, it is a
consequence of Theorem 1 that the translation invariant stationary measures
are precisely all Gibbs measures for $\{ J(V)\} _{V\subset \integ ^d}$. \\
At high noise (i.e. when the dependence for a spin on the value of its
neighboring spin on the previous timestep is small) it is known that there
is a unique stationary Gibbs measure [6]. \\ In the non--reversible
case,
practically nothing is known about the nature of the stationary measures in
the regime where there is more than one stationary measure.
Perhaps it is possible that these stationary
measures are non--Gibbsian. To prove such a statement, it would be
sufficient (because of Theorem 1) to prove that the relative entropy
density between two different stationary measures is nonzero.
A typical example of a non--reversible PCA (for which it is conjectured
that the stationary measures at low noise are non--Gibbsian [1]) is the
Toom model.
The simplest version of the Toom model is a PCA which can be seen as a
North--East--Center majority
vote model on the square lattice. That is, in (3.19) one takes
\begin{equation}
P_{\{x\}}(\sigma (x)|\eta )=1/2(1-(1-2\epsilon )\sigma (x)\mbox{sgn}
[\eta (x)+\eta (x')+\eta (x'')]) \end{equation}
with $x'=x+(1,0)$ and $x''=x+(0,1)$ the ``eastern'' and ``northern''
nearest
neighbor of $x$ on $\integ^2$. Equivalently, we can define the Toom model
on the triangular lattice, but arrange it so that, for the space-time
lattice, each site is the center of a triangle on the previous layer.
More precisely,
if we denote by ${\cal L}_n$ the layer at
time $t=-n$, then under projection perpendicular to the lattice plane, the
sites of ${\cal L}_{n-1}$ coincide with the
centers of alternate triangles in ${\cal L}_n$, i.e. those with some fixed
orientation. The sites of ${\cal L}_{n-2}$ coincide with the centers of the
similarly oriented triangles in ${\cal L}_{n-1}$ and finally ${\cal
L}_{n-3}$ coincides with ${\cal L}_n$. \\
The deterministic Toom rule ($\epsilon =0$) assigns to the spin at site $x$ the
majority of
the spins at the corners of the triangle directly above. The evolution in
the stochastic Toom model follows the deterministic rule with
probability (at least) $1-\epsilon $ and does something else with
probability (at most) $\epsilon $. \\
The deterministic dynamics has $\sigma (x) \equiv +1$ and $\sigma (x)
\equiv -1$ as stationary states. Moreover, these states are stable
against finite excitations of the opposte sign. In fact, such
excitations shrink at a constant rate and hence disappear in a finite time.
This is the so called eroder property. For $\epsilon $ small but
different from zero, there are still (at least) two stationary measures,
in one of which most spins are $+1$, in the other of which most are $-1$,
obtained from $+$ and $-$ begin conditions respectively
(they will be called $\rho _+$ and $\rho _-$ in the following). For
more information, see [6].
We take as basis vectors to generate our three dimensional space-time
lattice, the vectors $e_1,e_2,e_3$ connecting the origin $o$ with the
corners of the triangle directly above $o$. The time coordinate $t(x)$
of a space--time point $x$ is given by $t(x)=x.(e_1+e_2+e_3)$. \\
As mentioned before, to prove that $\rho _+, \rho _-$ are non--Gibbsian,
one should show that $i(\rho _+,\rho _-)>0$. It is
equivalent to show the probability for $\rho _+$ of having a large cluster
of minusses is (at least) exponentially small in the volume of the cluster.
(see for example [1]). This would be true if
the probability to find $\sigma _x\equiv -1$ in the triangle
$\Delta \equiv
\{ x |t(x)=-K; x.e_1, x.e_2, x.e_3 \leq K \}$, starting from the all $+$
configuration at time $t=-N$, is smaller than $cte \; \epsilon
^{|\Delta |}$, for all $K>0$ and for all $N>K$; ($|\Delta |$ is the
cardinality of the set $\Delta $). \\
In [9], it is shown that \begin{equation} \mbox{Prob}[\sigma
(o)=-1]\leq 2\epsilon
\end{equation}
by making an expansion in graphs for this probability.
However, this expansion cannot be used to prove our statement.
In fact, there is a model of Domany (see [6]) for which Toom's expansion
can be used
to prove non--ergodicity, but for which the stationary measures are Gibbsian.
The transition function for this model is \begin{equation}
P_{\{ x\} }(\sigma (x)|\eta )=1/2[1+\sigma (x)\tanh \beta (\eta (x)+
\eta (x')+\eta (x'')].
\end{equation}
It is not hard to see that, if one sets $1-2\epsilon =1/2(\tanh 3\beta
+\tanh \beta )$, this transition function appoaches that of the Toom
model (3.5) if $\beta\uparrow\infty, \epsilon \downarrow 0$. So, it is not
clear that one
should expect the stationary measures of the Toom model at low noise to be
non--Gibbsian. \\
For this reason we are trying to extract the necessary information from
simulations on the Toom model. However, so far this has not given
conclusive
evidence. Although we are inclined to interpret the simulation data as
showing Gibbsian behavior for the stationary states,
a further and more systematic search needs to be done before drawing
any conclusions.
One can also see from here that the question of Gibbsianess is more than an
academic question. Of course it is an important characterization of the
stationary states but more than that, here we see that the question is
equivalent to an estimate of the efficiency of the
repeated transformation in removing mistakes :
saying that the relative entropy density of the two stationary states is
non--zero means that you need to create a volume fraction of
``mistakes'' to see in one stationary state, an island of the opposite
stationary state --- for equilibrium dynamics relaxing towards one of the
extremal stationary
Gibbs states, it suffices to essentially introduce a boundary of the
``wrong'' phase for the dynamics to automatically fill up the enclosed
volume with this (for example, opposite) phase. It may also happen that
you
need less than a boundary (such as in the voter model, see [17]) and then
the stationary states may again be non--Gibbsian even though their relative
entropy densites are zero. The picture that emerges is therefore that the
stationary states are Gibbsian whenever, when the dynamics is started from
a particular stationary state, by introducing {\em on the boundary}
of a volume a configuration that is ``typical'' of another stationary
state, after a time of the order of the diameter of that volume, one sees
this ``wrong'' state {\em in the interior} of the volume.
\section{Relation with other problems.}
\setcounter{equation}{0}
\subsection{Restriction of $+$ and $-$ phases of the Ising model.}
Consider the standard ferromagnetic Ising model in $d$ dimensions.
In finite boxes $\Lambda=[-L,L]^d$ we consider $+$ and $-$ boundary
conditions :
the corresponding Gibbs measures $\mu^{\pm }_{\Lambda}\sim \exp (-\beta
H^{\pm }_{\Lambda})$ converge as $L\uparrow \infty$ to Gibbs measures
$\mu ^{\pm }$.
At low temperatures ($\beta \uparrow \infty$) $\mu^+\not= \mu^-$ are two
extremal, translation invariant Gibbs measures. Both are satisfying the
global Markov property [4]. For both measures,
we can consider the restrictions $\nu ^{\pm}$ to a $(d-1)$-dimensional
hypersurface ($\cong \integ ^{d-1}$). Schonmann [11] proved that in
$d=2$
dimensions and below the critical temperature, both these measures $\nu
^{\pm}$ are non--Gibbsian
(i.e. (2.3) is never satisfied). We wish to add here two remarks:
\begin{enumerate}
\item The higher dimensional situation ($d>2$) (open question 6 in
[1]):
Schonmann's analysis consists of two main steps: a large deviation estimate
and the study of a wetting problem. The large deviation estimate boils
down to the statement that the relative entropy density of the projected
phases is non--zero: $i(\nu^+,\nu^-)>0$. If we allow to look at very small
temperatures, this result is valid in any dimension $d\geq 2$, from
standard arguments [18].
As for the wetting problem, one has to prove that for each $n$, there exists
an $N(n)$ such that \[ \mbox{$\nu^+[\cdot | \tilde{\sigma}=-1$ on $A_{n,N}]
\rightarrow \nu^-$ as $N\uparrow \infty$}, \]
where $\tilde{\sigma}$ is a configuration on the $(d-1)$-dimensional annulus
\[ A_{n,N}=\{ x=(i,0), i\in \integ ^{d-1}:n\leq |i_k| \leq N, k=1,\ldots,
d-1
\}. \] The proof of this statement is a consequence of the recent work of
Holick\'{y} and Zahradn\'{\i}k [19] on the problem of entropic repulsion
in low temperature Ising models. Therefore, for any dimension
($d\geq 2$) the projection of the Ising phases on a hypersurface is
non--Gibbsian at low temperatures.
\item The transfer matrix \\
A well known tool in the analysis of the Ising model is the machinery of
the transfer matrix formalism.
In some sense one could say that the projections on hypersurfaces are
invariant measures for this transfer matrix. Therefore, Corrolary 1
(with P= transfer matrix) very much resembles the statement of Proposition
2 in [11] that if $\nu^+$ is Gibbsian, then $\nu^-$ is Gibbsian for the
same potential.
We show however that there exists no stochastic operator $P$,
satisfying the conditions of (2.10), for which both $\nu ^+$ and $\nu ^-$
are invariant. \\
Consider a box $B=[-K,K]\times V_L; V_L=[-L,L]^{d-1}$.
Let the (finite volume) transfer matrix $a_L$ be defined by its matrix
elements
$a_L(\tau ,\tau '), \tau ,\tau ' \in \Omega _{V_L}$:\begin{equation}
a_L(\tau ,\tau ')=\exp [\beta /2(\sum_{\subset V_L}(\tau (x)\tau (y)
+\tau '(x)\tau '(y))+\beta \sum_{x\in V_L}\tau (x)\tau '(x)].
\end{equation}
(here we sum over nearest neighbor pairs)\\
>From the Perron--Frobenius theorem, we have that the largest eigenvalue
$\lambda _L$ of $a_L$ is non--degenerate. Let us call the corresponding
normalized eigenvector $\psi _L$ and its components $\psi _L(\tau ),
\tau \in \Omega _{V_L}$.
The matrix $a_L$ can be normalized to become a stochastic matrix $\pi _L$:
\begin{equation}
\pi _L(\sigma |\eta )=\frac{a_L(\sigma ,\eta )\psi _L(\sigma )}{\lambda _L
\psi _L(\eta )} \hspace{1cm} \sigma ,\eta \in \Omega _{V_L}.
\end{equation}
Note that the combination \[
\pi _L(\sigma |\eta ) \psi_L^2(\eta) \]
is symmetric in $(\eta,\sigma)$ which is the analogue of the detailed
balance condition for PCA.\\ Consider now $\mu _{K,L}$, the Gibbs state on
the finite box
$B$, corresponding to the Hamiltonian $H_B$ with free boundary conditions:
\begin{equation}
H_B(\sigma )=-\sum_{\subset B}\sigma (x)\sigma (y).
\end{equation}
We have that \begin{equation}
\mu _{K,L}[\sigma \mbox{ on } \{0\}\times V_L|\eta
\mbox{ on } \{ -1\} \times V_L]=a_L(\sigma ,\eta )\frac{\sum_{\zeta }
a_L^K(\zeta ,\sigma )}{\sum_{\zeta }a_L^{K+1}(\zeta ,\eta )}
\end{equation} and thus
\begin{equation}
\lim_{K\uparrow \infty} \mu _{K,L}[\sigma \mbox{ on } \{0\}\times V_L|\eta
\mbox{ on } \{ -1\} \times V_L]=\pi _L(\sigma |\eta ).
\end{equation} Hence,
\begin{equation}
\mu [\cdot \mbox{ on } \{0\}\times \integ^{d-1}|\eta
\mbox{ on } \{ -1\} \times \integ^{d-1}]=\lim _{L\uparrow \infty}
\pi _L(\cdot|\eta ) \hspace{0.6 cm} \mbox{for $\mu $-a.e. $\eta $}.
\end{equation}
The left hand side is the probability on the zero'th layer (the
hypersurface
passing through the origin) in the measure $\mu = 1/2(\mu^+ + \mu^-)$
conditioned on the ``previous'' layer.
Any $P$ that leaves $\nu ^+$ and $\nu ^-$ invariant also leaves
$\nu =1/2(\nu ^++\nu ^-)$ invariant. Therefore its kernel should satisfy
\begin{equation}
P[\cdot|\eta ]=\mu [\cdot \mbox{ on } \{0\}\times \integ^{d-1}|\eta
\mbox{ on } \{ -1\} \times \integ^{d-1}]
\end{equation}
for $\mu $-a.e. $\eta $. \\
We claim that the right-hand side (and so the left-hand side) of (4.7)
cannot have any version satisfying condition 3 of (2.10).
We apply the following Lemma (the proof follows from [1], Prop. 4.12)
\begin{LE}
Suppose that $\nu=1/2(\nu ^++\nu ^-)$ is a Gibbs measure for some
interaction, then $\nu ^+$ and $\nu ^-$ are also Gibbs with respect to
the same interaction and so $i(\nu ^+,\nu ^-)=0$.
\end{LE}
Since $i(\nu ^+,\nu ^-)>0$, we conclude that the measure $\nu $ is
non--Gibbsian. In particular (see Lemma 1), $\nu [\sigma (o)=\eta (o)|
\sigma (x)=\eta (x), \hspace{3 mm} x\not=o]$ is an essential discontinuous
function of $\eta $, i.e. no modification on a set of $\nu $-measure zero
can change it into a continuous function. Using (4.3), we calculate that for
$\nu $-a.e. $\omega $
\begin{equation}
\frac{P_{V_L}[\eta |\omega ]}{P_{V_L}[\eta |\omega ^o]}=
\lim_{N\uparrow \infty}\{\frac{\psi _N(\omega ^o)}{\psi _N(\omega )}\}
\times \exp[\beta \sum_{x:}(\eta (x)\eta (o)+\omega (x)\omega (o))+
2\beta \eta (o)\omega (o)].
\end{equation}
Since
$\psi_N^2(\cdot)$ gives, for every $N$, the probabilities in the finite
volume measure $\nu_{V_N}$,
with $\nu_N\rightarrow \nu$, and $\nu$ is not--Gibbsian, it is clear that
the right--hand side of (4.8) cannot
have a continuous version. The first factor of that right-hand side is
essentially discontinuous because of the non--Gibbsianess of $\nu $.
\end{enumerate}
\subsection{Difference with renormalization transformations.}
Theorem 1 is not adapted to treat real space renormalization
transformations, since they necessarily involve a rescaling of the lattice
at each application of the transformation. Thus, these transformations
don't satisfy property 1 of (2.10), because for every $t$, the spins on
a line segment of length $L$ at time $t+1$ depend on the spins on a line
segment of length $3L$ at time $t$.
Vasilyev [20] has given an
example of a transition kernel, involving the same kind of rescaling as in
renormalization transformations, for which two different Bernoulli measures
are stationary (i.e. two Gibbs measures for different interaction).
There is thus an essential difference between these renormalization
transformations and the other stochastic transformations considered in the
paper. For related questions in renormalization group techniques, we refer
to [1]. However, these two kinds of transformations become very similar
in the so called mean field limit. In many cases, a first approach to
study (complicated) PCA dynamics consists in replacing the space--time
lattice (say, $\integ ^{d+1}$) by a Cayley graph or tree, for which the
neighborhoods of any pair of sites in one time layer are non--overlapping.
Applying the PCA rules in this situation yields a closed relation for the
magnetization $\langle \sigma _x \rangle $ in a stationary state. An example of
this for
the Toom model can be found in [6]. \\ \\
\underline{Acknowledgment:} We thank J. Bricmont, A.C.D. van Enter,
R. Fern{\'a}ndez, H.-O. Georgii, T. Matsui and M. Zahradn\'{\i}k for
helpful discussions and comments.
\begin{thebibliography}{99}
\bibitem[1]{3} A.C.D. van Enter, R. Fern{\' a}ndez, A.D. Sokal:
{\em Regularity properties and pathologies of position-space
renormalization transformations}, to appear in J. Stat. Phys..
\bibitem[2]{10} R. Holley: {\em Free energy in a Markovian model of a
lattice spin system}, Commun. Math. Phys. {\bf 23}, 87-99 (1971).
\bibitem[3]{11} H. K{\" u}nsch: {\em Non reversible stationary measures
for
infinite interacting particle systems}, Z. Wahrsch. Verw. Gebiete {\bf 66},
407-424 (1984).
\bibitem[4]{55} S. Goldstein, R Kuik and A.G. Schlijper: {\em Entropy
and
Global Markov properties}, Commun. Math. Phys. {\bf 126}, 496--482 (1990).
\bibitem [5]{9} S. Goldstein, R. Kuik, J.L. Lebowitz and C.Maes:
{\em From PCA to equilibrium systems and back}, Commun. Math. Phys.
{\bf 125}, 71-79 (1989).
\bibitem[6]{13} J.L. Lebowitz, C. Maes and E.R. Speer: {\em Statistical
mechanics of probabilistic cellular automata}, J. Stat. Phys. {\bf 59},
117-170 (1990).
\bibitem[7]{19} C. Maes and K. Vande Velde: {\em The interaction
potential
of the stationary measure of a high-noise spinflip process}, to
appear in J. Math. Phys..
\bibitem[8]{12} T.M. Liggett: {\em Interacting paricle systems},
Springer, Berlin (1985).
\bibitem[9]{28} A. Toom: {\em Nonergodic multidimensional systems of
automata}, Prob. Inform. Transm. {\bf 10}, 239-246 (1974).
\bibitem[10]{21} C. Maes and K. Vande Velde: {\em Defining relative
energies for the projected Ising measure}, Helv. Phys. Acta {\bf 65},
1055-1068 (1992).
\bibitem[11]{23} R.H. Schonmann: {\em Projections of Gibbs measures may
be non--Gibbsian}, Commun. Math. Phys. {\bf 124}, 1-7 (1989).
\bibitem[12]{7} H.-O. Georgii: {\em Gibbs measures and phase
transitions}, de Gruyter, Berlin (1988).
\bibitem[13]{27} W.G. Sullivan: {\em Potentials for almost Markovian
random fields}, Commun. Math. Phys. 33, 61-74 (1973).
\bibitem[14]{26} J.E. Steif: {\em The ergodic structure of interacting
partical systems}, Stanford Mathematics department Ph.D. thesis (1988).
\bibitem[15]{17} C. Maes and S.B. Shlosman: {\em When is an interacting
paricle system ergodic?}, Commun. Math. Phys. {\bf 151}, 447-466 (1993).
\bibitem [16]{29} P.L. Garrido and J. Marro: {\em Effective Hamiltonian
description of non--equilibrium spin systems}, Phys. Rev. Letters
{\bf 62}, 1929-1932 (1989).
\bibitem[17]{15} J.L. Lebowitz and R. Schonmann: {\em pseudo--free
energies and
large deviations for non--Gibbsian FKG measures}, Probab. Th. Rel. Fields
{\bf 77}, 49-64 (1988).
\bibitem[18]{2} J. Bricmont: private communication.
\bibitem [19]{41} P. Holick\'y and M.
Zahradn\'{\i}k: {\em On entropic repulsion in low temperature Ising
models}, Preprint.
\bibitem[20]{29} N. Vasilyev: {\em Bernoulli and Markov stationary
measures
in discrete local interactions} in {\em Locally interacting systems and
their applications in biology}, Lecture notes in math. {\bf 653},
Springer, Berlin (1978).
\end{thebibliography}
\end{document}