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\begin{document}
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%\title{\vspace*{-2.4cm}
\title{\vspace*{-2.4cm} On the possible failure of the Gibbs property \break
for measures on lattice systems }
\author{
{\normalsize Aernout C. D. van Enter} \\[-1mm]
{\normalsize\it Institute for Theoretical Physics} \\[-1.5mm]
{\normalsize\it Rijksuniversiteit Groningen} \\[-1.5mm]
{\normalsize\it Nijenborgh 4} \\[-1.5mm]
{\normalsize\it 9747 AG Groningen} \\[-1.5mm]
{\normalsize\it THE NETHERLANDS} \\[-1mm]
{\normalsize\tt AENTER@TH.RUG.NL} \\[-1mm]
}
%\date{June 7, 1994}
\date{\datetitle}
\maketitle
\thispagestyle{empty}
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\begin{abstract}
We review results which have been obtained in the last few years on the
topic whether various measures of physical interest, defined on lattice
systems, have the Gibbs property or not. Most of the measures we consider occur
either within renormalization-group theory or within models of non-equilibrium
statistical mechanics. In renormalization-group theory examples we discuss in
particular the uniqueness regime and the critical regime. Moreover we consider
the question how severely the Gibbs property is violated in various examples.
\end{abstract}
%\tableofcontents
To the memory of R.L. Dobrushin
\section{Introduction}
In \cite{vEFS_PRL,vEFS_JSP} it was shown how various real-space
renormalization-group (RG) maps acting on Gibbs measures produce
non-Gibbsian measures. In physicists' language, this means that a
``renormalized Hamiltonian'' can not be defined. The examples
presented there were valid at low temperatures or close to a first-order
phase transition. The underlying mechanism
---~pointed out first by Griffiths, Pearce and Israel
\cite{gripea78,gripea79,isr79}~--- is the fact
that for the constraints imposed by particular choices of block-spin
configurations, the resulting system exhibits a first-order phase
transition. For this to happen, it was expected that the original
system should be itself at or in the vicinity of a first-order phase
transition, cf. the title of \cite{vEFS_PRL}.
However, more recently counter-examples to this belief have been found, in that
renormalized measures for several transformations turn out to be
non-Gibbsian for arbitrarily large values of the magnetic field (at low
temperatures) \cite{vEFS_JSP, vEFK_JSP} or at arbitrarily high temperatures
\cite{vE96}.
On the other hand, there exist complementary results, either proving or
providing heuristic but plausible arguments, that renormalized Gibbsian
measures will be Gibbsian again at very high fields or very high temperatures
for ``many'' transformations, \cite{gripea78,gripea79,isr79,cam,kas},
and generally in the uniqueness region after a sufficiently often iterated
decimation (possibly combined with another transformation)
\cite{maroli93,maroli95}.
What happens in general in intervals around and including critical points
was until recently essentially unknown. On the one hand there were various
partial results which suggested that transforming a critical Gibbs measure
should result in a measure which was Gibbsian again \cite{ken92,benmaroli,
vel94,casgal}, see also \cite{ciroli96}. This would be according to the
folklore of renormalization-group
theory, which requires that a local (= short-range) Hamiltonian is mapped on
another local Hamiltonian in the same universality class. However, until very
recently, there was essentially no rigorous result
available which applied in the critical domain. Here we will discuss how
now we know of some examples in the critical regime which give rise to
either Gibbsian (see \cite{ken95, halken95}) or non-Gibbsian
(see \cite{vE96} and below) measures.
Further investigations up to 1992 about the (non-)Gibbsianness of various
measures of interest in statistical mechanics were rather extensively reviewed
in \cite{vEFS_JSP}, see also \cite{vEFS94}. Here we will give a short overview
of the more recent results
obtained since then, see e.g. \cite{lorvel94,ferpfi94,spe94,sal95,pfivel95,
maevel94,veldis95,dob95,brikup95,lordis95,loren95,maevel95,lor95,maevel96,
ciroli96}. These
include new examples as well as a more detailed description on the possible
ways of violating the Gibbs property, in particular how severe this violation
can be.
\section{Gibbs measures and quasilocality}
In this section we will describe some definitions and facts we will need about
the theory of Gibbs measures. For a more extensive treatment we refer to
\cite{geo88} or \cite{vEFS_JSP}.
We will consider spin systems on a lattice $Z^d$, where in most cases we will
take a single-spin space $\Omega_{0}$ which is finite. The configuration space
of the whole system is $\Omega =\Omega_{0}^{Z^d}$. Configurations will be
denoted by $\sigma$ or $\omega$, and their coordinates at lattice site i are
denoted by $\omega(i)$ or $\sigma(i)$. A (regular) interaction $\Phi$ is a
collection of functions $\Phi_{X}$ on $\Omega_{0}^X, X \in Z^d$ which is
translation invariant and satisfy:
\begin{equation}
\Sigma_{0 \in X} \ |\Phi_{X}|_{\infty} \leq \infty
\end{equation}
Formally Hamiltonians are given by
\begin{equation}
\ H^{\Phi} = \Sigma_{X \in Z^d} \ \Phi_{X}
\end{equation}
Under the above regularity condition these type of expressions make
mathematical sense if the
sum is taken over all subsets having non-empty intersections with a finite
volume $\Lambda$. For regular interactions one can define Gibbs measures as
probability measures on $\Omega$ having conditional probabilities which are
described in terms of appropriate Boltzmann-Gibbs factors:
\begin{equation}
{\mu(\sigma_{\Lambda}^{1}|\omega_{\Lambda^{c}})\over{{\mu(\sigma_{\Lambda}^{2}|\omega_{\Lambda^{c}})}}} =
exp - (\Sigma_{X} \ [\Phi_{X}(\sigma_{\Lambda}^{1} \omega_{\Lambda^{c}}) \ -
\ \Phi_{X}(\sigma_{\Lambda}^{2} \omega_{\Lambda^{c}})])
\end{equation}
for each volume $\Lambda$, $\mu$-almost every boundary condition
$\omega_{\Lambda^{c}}$ outside $\Lambda$ and each pair of configurations
$\sigma_{\Lambda}^{1}$ and $\sigma_{\Lambda}^{2}$ in $\Lambda$.
As long as $\Omega_{0}$ is compact, there always exists at least one
Gibbs measure for every regular interaction; the existence of more than one
Gibbs measure is one definition of the occurrence of a first-order phase
transition of some sort. Every Gibbs measure has the property that (for one of
its versions) its conditional probabilities are continuous functions of the
boundary condition $\omega_{\Lambda^{c}}$. It is a non-trivial fact that this
continuity, which goes by the name ``quasilocality'' or ``almost
Markovianness'', in fact characterizes the Gibbs measures \cite{sul,koz}, once
one knows that all the conditional probabilities are bounded away from zero
(that is, the measure is {\em nonnull} or has the {\em finite energy}
property).
To show that a particular measure is non-Gibbsian it is thus sufficient to find
{\em one} point in configuration space ({\em one} boundary condition) and {\em
one} conditional probability which has an essential, that is non-removable,
discontinuity at that point. The main step in such a proof is to show that when
this special configuration is viewed as a constraint, the constrained system
has a phase transition.
A second characterization of Gibbs measures uses the variational principle
expressing that in equilibrium a system minimizes its free energy. A
probabilistic formulation of this fact naturally occurs in terms of the theory
of large deviations. A (third level) large deviation rate function is up to a
constant and a sign equal to a free energy density.
To be precise, let $\mu $ be a translation invariant Gibbs measure, and
let $\nu $ be an arbitrary translation invariant measure.
Then the relative entropy density $i( \nu |\mu )$ can be defined as the limit:
\begin{equation}
i(\nu |\mu ) = lim_{\Lambda \rightarrow Z^d} \ {1 \over |\Lambda |} \ I_{\Lambda}(\nu |\mu)
\end{equation}
where
\begin{equation}
I_{\Lambda}(\nu |\mu)= \int log ({d\nu_{\Lambda} \over d\mu_{\Lambda}}) d\nu_{\Lambda}
\end{equation}
It has the property that $i( \nu |\mu )= 0$ if and only if the measure $\nu$ is
a Gibbs measure for the same interaction as the base measure $\mu$.
We can use this result in applications if we know for example that a known
measure $\nu$ cannot be a Gibbs
measure for the same interaction as some measure $\mu$ we want to investigate.
For example, if $\nu$ is a
point measure, or if it is the case that $\nu$ is a product measure and $\mu$
is not, we can conclude from the statement: $i( \nu |\mu )= 0$, that $\mu$
lacks the Gibbs property.
For another method of proving that a measure is non-Gibbsian because of having
the ``wrong'' type of (in this case too small) large deviation probabilities,
see \cite{sch87}.
\section{Examples of renormalized measures}
We consider the standard nearest neighbor Ising model with (formal)
Hamiltonian
\begin{equation}
\ H = \Sigma_{*} \; -\sigma(i) \sigma(j)
\end{equation}
at inverse temperature $\beta$. The dimension $d$ has to be at least 2.
We will consider various renormalization or block-spin transformations which
act on the Ising Gibbs measures. We divide the lattice into a collection of
non-overlapping blocks.
A renormalization group transformation will be a probability kernel
\begin{equation}
T(\omega'| \omega)= \Pi_{blocks}T(\omega'(j)|\omega(i);\ i \ \in block_j)
\end{equation}
This means that the distribution of the block-spin associated to a block
depends only on the spins in this block, in other words the transformation
is local in real space. The case of a deterministic transformation is included,
by having a T which is either zero or one.
Renormalization group methods are widely in use to study phase transitions
and in particular critical phenomena of various sorts (see for example
\cite {kogwil,ma,domgre}). Some good more recent references in which the theory
is explained, mostly at a physical level of rigour, but with careful statements
about what actually has and has not been proven are \cite{gol,bengal}.
\bigskip
1) The first class of examples we will consider are (linear) block-average
transformations. In \cite{vEFS_JSP} it was shown that at low temperatures for
a choice of cubic blocks of even length the renormalized measures
will be non-Gibbsian. The constraint which acts as a point of discontinuity
is taking all the averages in the blocks to be zero. Then in each block one
forces an interface, which tends to be in the middle of the block. Minimizing
the (free) energy density forces these interfaces to form a collection of
parallel lines (c.q. planes or hyperplanes). As these interfaces can be in $d$
directions, and because even if we are given the interfaces, there are still
two possible choices where to put the pluses and minuses, this means that the
constrained system has $2d$ low-temperature Gibbs measures. Because the
constraint: ''zero magnetization in each block'' removes the field-dependence,
the argument which provides the non-quasilocality is valid in arbitrarily
strong external magnetic fields. This result in \cite{vEFS_JSP} applies at very
low temperatures. In \cite{benmaroli} it was numerically investigated in $d=2$
up to which temperatures
one should expect the lack of quasilocality to persist. The result was that
for 2 by 2 blocks the critical point is outside the region of pathologies,
although the maximal temperature for which pathologies occur is only a few
percent below the critical temperature.
Here we show on the other hand the new result that if one takes anisotropic
blocks of size
1 by 2, the critical point is included in the region of pathologies.
To be definite, let us take blocks of length 2 in the vertical direction and
unit length in the horizontal direction. We take the (unnormalized)
block-average:
\begin{equation}
\sigma'_{i_{1},i_{2}} = \sigma_{i_{1},2i_{2}} + \sigma_{i_{1},2i_{2}+1}
\end{equation}
The block-spin configuration which
will be the point of discontinuity, is again the configuration which is zero in
each block, $\sigma'_{i_1,i_2}=0$ for all $i_{1},i_{2}$. The constrained system
thus allows for 2 configurations compatible with the constraint in each block,
namely plus up and minus below (configuration A), and minus up and plus below
(configuration B). This means that the constrained block-spin system is
equivalent to a standard nearest
neighbour Ising model, with an antiferromagnetic bond in the vertical direction
and a ferromagnetic bond of twice the original strength in the horizontal
direction. Such a model trivially has its critical point at a higher
temperature than the original model. The arguments how to translate the phase
transition for the constrained system into a violation of quasilocality for
the renormalized system are the same as in \cite{vEFS_JSP}, sections 4.1 to
4.3, in particular compare the proofs of Theorems 4.6 and 4.8.
Thus we conclude:
\bigskip
{\em Theorem:}
Let $\mu_{\beta,h}$ be a Gibbs measure for the two-dimensional Ising
model. In an infinite half-plane in the $(\beta,h)$-plane, which contains the
critical point, the measures obtained by taking an block-average transformation
on 1 by 2 blocks are non-Gibbsian.
\bigskip
2) Majority rule and Kadanoff transformations.
In the case of majority rule transformations \cite{nievle} applied to odd
blocks, in a strong plus field, the proof of the non-Gibbsianness uses as a
constraint the configuration in which the majority is minus, that means it
opposes the field in each block. Thus at very low temperatures the majority is
minimal, which implies that the total spin is close to zero, and the
constrained system has a phase transition of similar type as the block-average
equal to zero constrained system of the first example had \cite {vEFK_JSP}.
On the othere hand there are strong indications \cite{ken92,ciroli96} that
there is a region around the critical point in $d=2$ which is free from
pathologies under majority rule transformations.
The Kadanoff transformation is a soft version (a proper example of a
stochastic transformation) of the majority rule:
\begin{equation}
T(\sigma'(j)|\sigma(i); \ i \in block_{j}))={C \ exp \ [p \sigma'(j) \Sigma_{i \in
block_{j}} \ \sigma(i)]}
\end{equation}
In \cite{vEFS_JSP} it was shown that at low temperatures, and for small
external fields, the transformed measures become non-Gibbsian. In
\cite{halken95} for the Ising model on a triangular lattice it is shown that
there is an interval around the critical point for which the transformed
measures are Gibbsian, hence free of pathologies.
\bigskip
3) a highly non-linear transformation.
We consider the following transformation :
Divide the lattice $Z^d$ into blocks $B_j^L$ of linear size $L$ and define
block-spins $\sigma'(j)$
by
\begin{equation}
\sigma'(j) = \Sigma_{ i \in B_j^L} \; \sigma(i)
\end{equation}
except in the case where all spins in the block are minus, in which case
\begin{equation}
\sigma'(j) = +L^d
\end{equation}
%This induces a map $T_L$
%on the measures on the space of spin-configurations.
%For more details we refer to \cite{vEFS_JSP}.
Now we have the following theorem:
\bigskip
{\em Theorem}: Let $\mu_{\beta}$ be the Gibbs measure of the nearest neighbor
Ising model in dimension $d \ge 2$, at inverse temperature $\beta > 0$.
Then there exists a constant $L_{0}$
such that for all $L > L_{0}$, $\mu'_{L}=\mu \circ T_{L}$ is non-Gibbsian.
\bigskip
{\em Proof}: The proof follows again the scheme of \cite{vEFS_JSP}, section
4.2. The special block-spin configuration which will be the point of
(essential) discontinuity for a suitably chosen conditional probability will be
the choice $\sigma'(j)$=$L^d$ in each block.
This means that within each block all spins are unanimous, either in the plus
or in the minus-direction. Thus imposing the special block-spin configuration
as a constraint means that the sum of all spins in each block can take only
two values (either unanimously positive or unanimously negative). Hence the
thus constrained system is again a nearest neighbour Ising model, but now at
an inverse temperature $\beta' = L^{d-1} \times \beta$.
For $L$ large enough, $\beta'$ becomes so large that the constrained system
is in the phase transition (low temperature) regime (step 1 of the proof).
The block-spin boundary conditions selecting the plus and minus measures
are $\sigma'(j) = + (L^d -2)$ or $- ( L^d - 2)$ in each boundary block.
This means that in every block there is only one spin opposing all the others.
When $L > 2$, a standard Peierls argument proves the phase transition (step 2
of the proof). As the expectation of $\sigma'(0)$ (unfixed according to step
3 of the proof) is different in the plus and minus phases, we see that the
scheme of proof of \cite{vEFS_JSP}, section 4.2, can again be straightforwardly
applied.
%Average block-spin transformations:Average (linear) block spin transormations
%have the property that te renormalized spins assume more values than the
%original spins.
\bigskip
4)Decimations and projections.
We will call a ``decimation'' taking the marginal of a measure restricted to
the spins on a sublattice of the same dimension as the original system, and a
``projection'' taking the marginal to a lower-dimensional sublattice.
In some sense these transformations are the simplest, and the most is known
about them. In the region of strong uniqueness (complete analyticity for
regular volumes), repeating the transformation maps a Gibbs measure again onto
a Gibbs measure, even though after one or a finite number of transformations
the transformed measure can be non-Gibbsian, and the same is true if one
composes a sequence of decimations with a finite number of applications of any
other transformation \cite{vEFS_JSP,maroli93,maroli95,MarOli1,MarOli2}.
Recently Haller and Kennedy showed that there is an interval around the
critical temperature where the once decimated measures on the sublattice with
even coordinates are Gibbsian \cite{halken95}.
For Potts models applying a decimation slightly above the transition
temperature results in a non-Gibbsian measure \cite{vEFK_JSP}, while at
sufficiently high temperatures the transformed measure is Gibbsian
\cite{lordis95,lor95}.
As for projections, it is known that on the phase transition line of the
2-dimensional Ising model the projection to $Z$ of any Gibbs measure is
non-Gibbsian \cite{sch87}. In the whole uniqueness regime, except possibly at
the critical point, this projection results in a Gibbsian measure
\cite{maevel92,lordis95,lor95}. In three dimensions the projected measures
to two-dimensional planes are
again non-Gibbsian on the transition line \cite{ferpfi94,maevel94}, however,
now, due to the presumably existing surface (layering) transition between
different Basuev states, one expects that the projected
measures also in a small field will be non-Gibbsian \cite{lordis95,lor95}.
The composition of a projection and a decimation in the phase transition region
gives rise to a new phenomenon, namely the possibility of a state-dependent
result. The transformed plus and minus measures are Gibbs measures for
different interactions, while the transformed mixed measures are non-Gibbsian
\cite{lorvel94}.
\section{Further results on non-Gibbsian measures}
Further investigations about non-Gibbsian measures have been done about the
random-cluster models of Fortuin and Kasteleyn \cite{pfivel95,gri95,borcha95,
Hag5}, about some image analysis model \cite{maevel95}, and about some
non-equilibrium models \cite{spe94,veldis95,marsco,maroli95}.
\bigskip
1) Random-cluster models
About the random-cluster model, it was shown in \cite{vEFS_JSP}, following
\cite{ACCN}, that all measures which are consistent with the specification are
not quasilocal when $q > 1$. We did not know at the time whether the wired and
free boundary condition limit states are in fact consistent with the
random-cluster specification. Due to the lack of quasilocality of the
specification this is not immediate from standard arguments, but
this assumption was shown to hold in \cite{pfivel95,borcha95} for the free b.c.
state and in \cite{gri95} for both free (disordering) and wired (ordering)
boundary conditions. In the proofs essential use is made of the attractivity
(or FKG-)property. By considering the Fortuin-Kasteleyn-Swendsen-Wang
representation \cite{edwsok} one has, similarly to the renormalization-group
examples, again that the marginal of a Gibbsian ---~quasilocal~--- measure
turns out to be non-quasilocal.
\bigskip
2) Contracting the single-spin space.
Various results about measures obtained by contracting single-spin spaces
have been obtained, following the first results on ``projected Gaussians''
of \cite{lebmae87}.
In \cite{maevel95} the so-called fuzzy Potts model was considered, in which the
q Potts states are grouped together in $q'= q/n$ groups. This together grouping
can give rise to non-quasilocal conditional probabilities. This model is of
interest for image analysis. It is remarkable
that a similar procedure applied to the maximal entropy measures for subshifts
of finite type (SOFTs), which one
can view as ground states, sometimes results in Gibbsian measures \cite{Hag1,
Hag2,Hag3,Hag4}.
\bigskip
3)Non-equilibrium models.
It is known that the invariant measures of the voter model and the
Martinelli-Scoppola cluster dynamics are non-Gibbsian \cite{vEFS_JSP,
marsco,lebsch}. More recently it was shown that the invariant measures for the
two-species TASEP (totally asymmetric exclusion process) is non-Gibbsian
because its conditional probabilities are neither quasilocal \cite{spe94}
nor bounded away from zero \cite{veldis95}. In this model there are particles
of type 1, particles type 2 and holes. Both type of particles can jump to a
hole on the right, and a type 1 particle can exchange its position with a type
2 particle neighbouring it on its right. The result applies to measures which
have a positive density of both types of particles and also a positive density
of holes.
\section{Robustly non-Gibbsian measures?}
The non-Gibbsian character of the various measures considered comes
often as an unwelcome surprise. A description in terms of effective
or coarse-grained or renormalized potentials is often convenient, and
seems for some applications even essential. The fact that such a description is
not available thus is a severe drawback. There have been attempts to
tame the non-Gibbsian pathologies, and here we want to give a short
description of how far one gets with some of those attempts.
\bigskip
1) The fact that the constraints which act as points of discontinuity
often are configurations which are very untypical for the measure under
consideration, suggests a
notion of a {\em weakly} non-Gibbsian measure as a measure which has
conditional probabilities which are continuous on a set of full measure. This
notion was first suggested by Dobrushin and Shlosman, and first
appeared in print in \cite{lorwin92}. It has been developed and was applied to
various models in \cite{ferpfi94}, and later in \cite{maevel95} and
\cite{pfivel95}. Examples of measures which are at the worst weakly
non-Gibbsian measures in this sense are decimated or projected Gibbs
measures in an external field, random-cluster measures on regular lattices,
and low temperature fuzzy Potts measures. In the random-cluster measures one
can actually identify explicitly all bond configurations which give rise to the
non-quasilocality. They are precisely those configurations in which
(possibly after a local change) more than one infinite cluster coexist.
On a tree, because of the possible coexistence of infinitely many infinite
clusters with positive probability, the random-cluster measure can violate the
weak non-Gibbsianness condition and be strongly non-Gibbsian \cite{Hag5}.
In a related approach Dobrushin \cite{dob95,brikup95,maevel96}
showed that for a projected pure phase on the coexistence line of the
2-dimensional Ising model it is possible to find an almost everywhere defined
interaction. This approach, which is via low-temperature expansions, provides
a way of obtaining good control for the non-Gibbsian projection. For some
renormalization examples an investigation via low temperature expansions into
the possibility of recognizing non-Gibbsianness was started in \cite{sal95}.
Another simple counter-example of a strongly non-quasilocal measure, where
each configuration can act as a point of
discontinuity, is a mixture of two Gibbs measures for different interactions
\cite{loren95}.
\bigskip
2) Stability under decimation (and other transformations).
In \cite{maroli93,maroli95} it was shown how decimating once renormalized
non-Gibbsian measures results in Gibbs measures again after a sufficiently
large number of iterations. These often decimated measures are in the
high-temperature regime, in which the usually applied renormalization-group
maps are well-defined (this does not hold true for all maps though, in view of
the highly non-linear example given before).
On the other hand, in examples
where the non-Gibbsian property is associated with large deviation properties
which are not compatible with a Gibbsian character (this holds for example
for projected Gaussians, invariant measures of the voter and the
Martinelli-Scoppola model) the non-Gibbsian property survives all sort of
transformations \cite{maroli95,loren95}. The argument is that when some
obviously non-Gibbsian measure has a rate function zero with respect to the
measure under
consideration, this property is generally preserved (\cite{vEFS_JSP}
Section 3.2 and 3.3).
The family of projected Gaussians include scaling
limits for majority-rule transformations in high dimensions \cite{dorvE}.
Transforming those scaling limits corresponds to making a move from a fixed
point in what is usually called a ``redundant'' direction (cf. Wegner's
contribution to \cite{domgre}) in some space of Hamiltonians, although here of
course the whole point is that such Hamiltonians do not exist.
\bigskip
3) The two criteria mentioned above are distinct. Maes and Vande Velde
\cite{maevel96} have a simple one-dimensional example of a
one-dependent measure which has a set of discontinuity points of full measure,
but due to the one-dependence the measure becomes after decimation independent
and therefore trivially Gibbsian.
\section{Final Comments, Conclusions and Some Open Problems}
1.The highly non-linear block-spin map defined above is admittedly artificial
from a physical point of view. However, the heuristic considerations of
\cite{benmaroli,casgal}, which were designed for the block-average
transformation formally equally well apply to this map, but lead
to an incorrect conclusion. This illustrates the danger of relying
on formal as opposed to fully rigorous arguments. Also, the fact that the
question whether
the critical point is free of pathologies for block-average transformations
has different answers depending on the block shape, illustrates that a
general argument justifying the use of RG-algoritms may not exist.
In view of the many numerically sucessful and intuitively appealing
applications of the theory this situation is a rather unsatisfactory.
\bigskip
2.If we consider the sequence $\mu \circ T_L$ for the highly non-linear example
mentioned above, starting from a high temperature state, as a corollary to the
(local) central limit theorem \cite{iagsou,carsof,cam,new80},
it is easy to see that, with a proper rescaling of the block spins, this
sequence converges weakly to an independent Gaussian zero-mean measure.
The measures in the sequence are almost all (that is, all except
for possibly a finite number) non-Gibbsian.
\bigskip
3.If we repeat the 1 by 2 block-average transformation, in sufficiently
high dimension (presumably $ d \geq 3 $), acting on the critical Ising Gibbs
measure, a central limit theorem holds \cite{new80}, and the critical
measure shows ``high temperature'' behaviour as opposed to the occurrence
of non-Gibbsianness which is more like a ``low-temperature'' behaviour.
%4.The set of discontinuity points has measure zero (PERCOLATION
%ARGUMENT).
%4.Applying a decimation transformation often enough will make
%the measure Gibbsian again \cite{maroli95}. This seems to mean that
%there are maps which reinforce, as well as maps which weaken, the Gibbsianness
%of measures in the uniqueness region.What the behaviour of a ``typical''
%map - whatever that might be- is, remains open. In this sense, how robust
%the phenomenon of renormalisation-group pathologies is, remains an issue
%of debate.Our result shows that the pathologies are more widespread than
%was known before, while the results of \cite{maroli93,maroli95,benmaroli}
%go in the opposite direction, indicating how one might make some of these
%pathologies disappear.
\bigskip
4.One approach which might look more hopeful is to consider renormalization
transformations which are non-local in some sense. One could either think
of momentum-space transformations, or try to renormalize non-local objects
like contours. It seems that the momentum-space maps suffer from the same
type of problems as real-space transformations \cite{hashas, vEF96}, but
an implementation of renormalization-group ideas on contour models looks
more promising, at least for the description of first-order phase transitions
\cite{gkk,bovkul}.
\bigskip
5.In non-equilibrium statistical mechanics there are many open questions
about the occurrence of non-Gibbsian measures. The term non-Gibbsian or
non-reversible is often used for invariant measures in systems in which there
is no detailed balance \cite{lig,eylebspo,ern95}. It is an open question when
such measures are non-Gibbsian in the sense we have described here.
It has been conjectured that such measures for which there is no detailed
balance are quite generally non-Gibbsian in systems with a
stochastic dynamics, see for example \cite{lebsch} or \cite{eylebspo},
Appendix 1; on the other hand it has been predicted
that non-Gibbsian measures are rather exceptional (\cite{lig}, Open problem
IV.7.5, p.224) at least for non-reversible spin-flip processes under the
assumptions of rates
which are bounded away from zero. The examples we have are for the moment too
few to develop a good intuition on this point.
\bigskip
Thus, although the number of examples of non-Gibbsian measures is on the
increase, a good general description or classification is still an outstanding
problem.
Indeed, non-Gibbsian measures occur in quite different areas of statistical
mechanics, and can have quite different properties. By now it seems somewhat
surprising that it took so long to realize the fact that the Gibbs property is
rather special, in particular in view of Israel's \cite{isr92} result that in
the set of all translation invariant (ergodic, nonnull) measures, Gibbs
measures are exceptional.
\bigskip
{\em Acknowledgments}: My contribution reflects mainly joint work with Roberto
Fern{\'a}ndez and Alan Sokal, and also partly with Roman Koteck{\'y} and
J{\'o}zsef L{\"o}rinczi. I had very useful conversations with the late R. L.
Dobrushin, who first asked me about the possible occurrence of pathologies at
high and possibly very high temperatures during a workshop in les Houches,
and at a workshop at the Erwin Schr{\"o}dinger Institute
in Vienna and who presented his ideas and results on robustness of the Gibbs
property at a workshop in Renkum the Netherlands. I am grateful for the
occasions I had to profit from his always stimulating and challenging
discussions and questions. His scientific contributions have been of immense
value and were matched by a wonderful personality. I thank Geoffrey Grimmett
and Enzo Olivieri, who explained to me how imposing constraints can raise, as
well as lower, critical temperatures. At the
Budapest meeting I had useful discussions with Enzo Olivieri and Tom Kennedy
about the behaviour of renormalization group maps around critical points.
I also thank Jean Bricmont, Krist Maes and Koen Vande Velde and Marinus Winnink
for very helpful discussions and correspondence.
This research has been supported by EU contract CHRX-CT93-0411.
%\section{Basic Set-up}
\addcontentsline{toc}{section}{\bf References}
%\bibliography{potts}
\begin{thebibliography}{10}
\bibitem{ACCN}
M.~Aizenman, J.~T.~Chayes, L.~Chayes and C.~M.~Newman.
\newblock Discontinuity of the magnetization in the one-dimensional
$1 \over{|x-y|^{2}}$ {I}sing and {P}otts models.
\newblock {\em J. Stat. Phys.}, 50:1--40, 1988.
\bibitem{bengal}
G.~Benfatto and G.~Gallavotti.
\newblock {\em Renormalization group}, Physics Notes, Vol.1.
\newblock Princeton University Press, Princeton, 1995.
\bibitem{benmaroli}
G.~Benfatto, E.~Marinari and E.~Olivieri.
\newblock Some numerical results on the block spin transformation for the $2d$
{I}sing model at the critical point.
\newblock {\em J. Stat. Phys.}, 78:731--757, 1995.
\bibitem{borcha95}
C.~Borgs and J.~T.~Chayes.
\newblock The covariance matrix of the Potts model: a random cluster analysis.
\newblock {\em J. Stat. Phys.}, 82:1235-1297, 1996.
\bibitem{bovkul}
A.~Bovier and C.~K{\"u}lske.
\newblock A rigorous renormalization group method for interfaces in random
media.
\newblock {\em Rev. Math. Phys.}, 6:413-496, 1994.
\bibitem{brikup95}
J.~Bricmont and A.~Kupiainen.
\newblock Private communication, 1995.
\bibitem{cam}
C.~Cammarota.
\newblock The large block spin interaction.
\newblock {\em Nuovo Cimento B}, 96:1--16, 1986.
\bibitem{carsof}
E.~A.~Carlen and A.~Soffer.
\newblock Entropy production by block spin summation and central limit
theorems.
\newblock {\em Comm. Math. Phys.}, 140:339--371, 1992.
\bibitem{casgal}
M.~Cassandro and G.~Gallavotti.
\newblock The Lavoisier law and the critical point.
\newblock {\em Nuovo Cimento B}, 25:691--705, 1975.
\bibitem {ciroli96}
E.~N.~M.~Cirillo and E.~Olivieri.
\newblock Renormalization-group at criticality and complete analyticity of
constrained models: a numerical study.
\newblock Bari-Roma preprint, 1996.
\bibitem{dob95}
R.~L.~Dobrushin.
\newblock Lecture given at the workshop ``Probability and Physics'', Renkum,
Holland, 1995.
\bibitem{dobshl87}
R.~L.~Dobrushin and S.~B.~Shlosman.
\newblock Completely analytical interactions: constructive description.
\newblock {\em J. Stat. Phys.}, 46:983--1014, 1987.
\bibitem{domgre}
C.~Domb and M.~S.~Green (Eds.).
\newblock {\em Phase transitions and critical phenomena}, Vol.6.
\newblock Academic Press, New York, 1976.
\bibitem{dorvE}
T.~C.~Dorlas and A.~C.~D.~van Enter.
\newblock Non-Gibbsian limit for large-block majority spin transformations.
\newblock {\em J. Stat. Phys.}, 55:171-181, 1989.
\bibitem{edwsok}
R.~G.~Edwards and A.~D.~Sokal.
\newblock Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation
and Monte Carlo algorithm.
\newblock {\em Phys. Rev. D}, 38:2009--2012, 1988.
\bibitem{ern95}
M.~H.~Ernst and H.~J.~Bussemaker.
\newblock Algebraic spatial correlations in lattice gas automata violating
detailed balance.
\newblock {\em J. Stat. Phys.}, 81:515-536, 1995.
\bibitem{eylebspo}
G.~L.~Eyink, J.~L.~Lebowitz and H.~Spohn.
\newblock Hydrodynamics and fluctuations outside of local equilibrium: driven
diffusive systems.
\newblock preprint, 1995, Austin math. phys. archives, 95-168.
\bibitem{ferpfi94}
R.~Fern\'andez and C.-Ed.~Pfister.
\newblock Non-quasilocality of projections of {G}ibbs measures.
\newblock EPFL preprint, 1994, Austin math. phys. archives, 94-113.
\bibitem{gkk}
K.~Gaw{\c e}dzki, R.~Koteck{\'y} and A.~Kupiainen.
\newblock Coarse graining approach to first order phase transitions.
\newblock {\em J. Stat. Phys.} 47:701-724, 1987.
\bibitem{geo88}
H.-O.~Georgii.
\newblock {\em Gibbs Measures and Phase Transitions}.
\newblock Walter de Gruyter (de Gruyter Studies in Mathematics, Vol.\ 9),
Berlin--New York, 1988.
\bibitem{gol}
N.~Goldenfeld.
\newblock {\em Lectures on phase transitions and the renormalization group.}
\newblock Addison-Wesley, Frontiers in Physics 85, 1992.
%\bibitem{gri72}
%R.~B. Griffiths.
%\newblock Rigorous results and theorems.
%\newblock In C.~Domb and M.~S. Green, editors, {\em Phase Transitions and
% Critical Phenomena, {V}ol.\ 1}. Academic Press, London--New York, 1972.
\bibitem{gripea78}
R.~B.~Griffiths and P.~A.~Pearce.
\newblock Position-space renormalization-group transformations: {S}ome proofs
and some problems.
\newblock {\em Phys. Rev. Lett.}, 41:917--920, 1978.
\bibitem{gripea79}
R.~B.~Griffiths and P.~A.~Pearce.
\newblock Mathematical properties of position-space renormalization-group
transformations.
\newblock {\em J. Stat. Phys.}, 20:499--545, 1979.
\bibitem{gri95}
G.~Grimmett.
\newblock The stochastic random-cluster process and the uniqueness of
random-cluster measures.
\newblock Cambridge University preprint, to appear in
{\em Ann. Prob.}, 23, 1995, see also
\newblock The Random-cluster Model.
\newblock Chapter 3 in: Probability, Statistics and Optimisation, Ed. F.~P.~
Kelley, Wiley and Sons Ltd., 1994.
\bibitem{Hag1}
O.~H{\"a}ggstr{\"o}m.
\newblock On the relation between finite-range potentials and subshifts of
finite type.
\newblock {\em Prob. Th. Rel. Fields}, 101:469-478, 1995.
\bibitem{Hag2}
O.~H{\"a}ggstr{\"o}m.
\newblock A subshift of finite type that is equivalent to the {I}sing model.
\newblock {\em Erg. Th. Dyn. Sys.}, 15:543-556, 1995.
\bibitem{Hag3}
O.~H{\"a}ggstr{\"o}m
\newblock On phase transitions for subshifts of finite type.
\newblock {\em Isr. J. Math.}, to appear, 1996.
\bibitem{Hag4}
O.~H{\"a}ggstr{\"o}m.
\newblock Aspects of spatial random processes.
\newblock Chalmers University of G{\"o}teborg thesis, 1994.
\bibitem{Hag5}
O.~H{\"a}ggstr{\"o}m.
\newblock Almost sure quasilocality fails for the random-cluster model on a
tree.
\newblock Chalmers University of G{\"o}teborg preprint, to appear in {\em J.
Stat. Phys.}, 1996.
\bibitem{halken95}
K.~Haller and T.~Kennedy.
\newblock Absence of renormalization group pathologies near the critical
temperature--two examples.
\newblock University of Arizona preprint, Austin Archives 95-505, 1995.
\bibitem{hashas}
A.~Hasenfratz and P.~Hasenfratz.
\newblock Singular renormalization group transformations and first order
phase transitions (I).
\newblock {\em Nucl. Phys. B}, 295[FS21]:1-20, 1988.
\bibitem{iagsou}
D.~Iagolnitzer and B.~Souillard.
\newblock Random fields and limit theorems.
\newblock In J.~Fritz, J.~L.~Lebowitz, and D.~Sz{\'a}sz, editors, {\em Random
Fields (Esztergom, 1979), Vol.\ II}, pages 573--592. North-Holland,
Amsterdam, 1981.
\bibitem{isr79}
R.~B.~Israel.
\newblock Banach algebras and {K}adanoff transformations.
\newblock In J.~Fritz, J.~L.~Lebowitz, and D.~Sz{\'a}sz, editors, {\em Random
Fields (Esztergom, 1979), Vol.\ II}, pages 593--608. North-Holland,
Amsterdam, 1981.
\bibitem{isr92}
R.~B.~Israel.
\newblock Private communication, 1992.
\bibitem{kas}
I.~A.~Kashapov.
\newblock Justification of the renormalization-group method.
\newblock {\em Theor. Math. Phys.}, 42:184--186, 1980.
\bibitem{ken92}
T.~Kennedy.
\newblock Some rigorous results on majority rule renormalization group
transformations near the critical point.
\newblock {\em J. Stat. Phys.}, 72:15--37, 1993.
\bibitem{ken95}
T.~Kennedy.
\newblock Budapest meeting on disordered systems and statistical
physics, rigorous results, august 1995.
\bibitem{koz}
O.~K.~Kozlov.
\newblock Gibbs description of a system of random variables.
\newblock {\em Prob. Inf. Transmission}, 10:258--265, 1974.
\bibitem{lebmae87}
J.~L.~Lebowitz and C.~Maes.
\newblock The effect of an external field on an interface, entropic repulsion.
\newblock {\em J. Stat. Phys.}, 46:39--49, 1987.
\bibitem{lebsch}
J.~L.~Lebowitz and R.~H.~Schonmann.
\newblock Pseudo-free energies and large deviations for non-Gibbsian FKG
measures.
\newblock {\em Prob. Th. and Rel. Fields}, 77:49-64, 1988.
\bibitem{lig}
T.~M.~Liggett.
\newblock Interacting particle systems.
\newblock Springer, Berlin, 1995.
\bibitem{lor94}
J.~L{\"o}rinczi.
\newblock Some results on the projected two-dimensional {I}sing model.
\newblock In M.~Fannes, C.~Maes, and A.~Verbeure, editors, {\em Proceedings
NATO ASI Leuven Workshop ``On Three Levels''}, pages 373--380, Plenum Press,
1994.
\bibitem {lordis95}
J.~L{\"o}rinczi.
\newblock On limits of the Gibbsian formalism in thermodynamics.
\newblock University of Groningen thesis, 1995.
\bibitem{lor95}
J.~L{\"o}rinczi.
\newblock Quasilocality of projected Gibbs measures through analyticity
techniques.
\newblock University of Groningen preprint, 1995, to appear in {\em Helv. Phys.
Acta}, 1996.
\bibitem{lorvel94}
J.~L{\"o}rinczi and K.~Vande Velde.
\newblock A note on the projection of {G}ibbs measures.
\newblock {\em J.\ Stat.\ Phys.}, 77:881--887, 1994.
\bibitem{lorwin92}
J.~L{\"o}rinczi and M.~Winnink.
\newblock Some remarks on almost {G}ibbs states.
\newblock In N.~Boccara, E.~Goles, S.~Martinez, and P.~Picco, editors, {\em
Cellular Automata and Cooperative Systems}, pages 423--432, Kluwer,
Dordrecht, 1993.
\bibitem{ma}
S.~K.~Ma.
\newblock {\em Modern theory of critical phenomena}.
\newblock Benjamin, Reading, Massachusetts, 1976.
\bibitem{maevel92}
C.~Maes and K.~Vande Velde.
\newblock Defining relative energies for the projected Ising measure.
\newblock {\em Helv. Phys. Acta}, 65:1055--1068, 1992.
\bibitem{maevel94}
C.~Maes and K.~Vande Velde.
\newblock The (non-)Gibbsian nature of states invariant under stochastic
transformations.
\newblock {\em Physica A}, 206:587--603, 1994.
\bibitem{maevel95}
C.~Maes and K.~Vande Velde.
\newblock The fuzzy Potts model.
\newblock {\em J. Phys. A, Math. Gen} 28:4261--4271, 1995.
\bibitem{maevel96}
C.~Maes and K.~Vande Velde.
\newblock Relative energies for non-Gibbsian states.
\newblock Leuven preprint, 1996.
\bibitem{maroli93}
F.~Martinelli and E.~Olivieri.
\newblock Some remarks on pathologies of renormalization-group transformations.
\newblock {\em J. Stat. Phys.}, 72:1169--1177, 1993.
\bibitem{maroli95}
F.~Martinelli and E.~Olivieri.
\newblock Instability of renormalization-group pathologies under decimation.
\newblock {\em J. Stat. Phys.}, 79:25--42, 1995.
\bibitem{MarOli1}
F.~Martinelli and E.~Olivieri.
\newblock Approach to equilibrium of Glauber dynamics in the one phase region
I.
\newblock {\em Comm. Math. Phys.}, 161:447--486, 1994.
\bibitem{MarOli2}
F.~Martinelli and E.~Olivieri.
\newblock Approach to equilibrium of Glauber dynamics in the one phase region
II.
\newblock {\em Comm. Math. Phys.}, 161:487--515, 1994.
\bibitem{MOS}
F.~Martinelli, E.~Olivieri and R.~H.~Schonmann.
\newblock For 2-D lattice spin systems weak mixing implies strong mixing.
\newblock {\em Comm. Math. Phys.}, 165:33--48, 1994.
\bibitem{marsco}
F.~Martinelli and E.~Scoppola.
\newblock A simple stochastic cluster dynamics: rigorous results.
\newblock {\em J. Phys. A, Math. Gen.}, 24:3135--3157, 1991.
\bibitem{new80}
C.~M.~Newman.
\newblock Normal fluctuations and FKG inequalities.
\newblock {\em Comm. Math. Phys.}, 74:119-128, 1980.
\bibitem{nievle}
Th.~Niemeijer and J.~M.~J.~van Leeuwen.
\newblock Renormalizaton Theory for Ising-like spin systems.
\newblock In {\em Phase Transitions and Critical Phenomena}, Vol.6,
pp.425--505, C.~Domb and M.~S.~Green, editors, Academic Press, New York, 1976.
\bibitem{pfivel95}
C.-Ed.~Pfister and K.~Vande Velde.
\newblock Almost sure quasilocality in the random cluster model.
\newblock {\em J. Stat. Phys.}, 79:765--774, 1995.
\bibitem{sal95}
J.~Salas.
\newblock Low temperature series for renormalized operators: the ferromagnetic
square {I}sing model.
\newblock {\em J. Stat. Phys.}, 80:1309--1326, 1995.
\bibitem{sch87}
R.~H.~Schonmann.
\newblock Projections of Gibbs measures may be non-Gibbsian.
\newblock {\em Comm. Math. Phys.}, 124:1--7, 1989.
\bibitem{SchShl}
R.~H.~Schonmann and S.~B.~Shlosman.
\newblock Complete analyticity for 2D Ising completed.
\newblock {\em Comm. Math. Phys.}, 170:453--482, 1995.
\bibitem{spe94}
E.~R.~Speer.
\newblock The two species totally asymmetric simple exclusion process.
\newblock {\em Proceedings NATO ASI Leuven Workshop ``On three levels''},
M.~Fannes, C.~Maes and A.~Verbeure, editors, 91--103, Plenum Press, 1994.
\bibitem{sul}
W.~G.~Sullivan.
\newblock Potentials for almost Markovian random fields.
\newblock {\em Comm. Math. Phys.}, 33:61--74, 1973.
\bibitem{vE96}
A.~C.~D.~van Enter.
\newblock Ill-defined block-spin transformations at arbitrarily high
temperatures.
\newblock {\em J. Stat. Phys.}, 83:761--765, to appear, 1996.
\bibitem{vEF96}
A.~C.D.~van Enter and R.~Fern{\'a}ndez.
In preparation.
\bibitem{vEFK_JSP}
A.~C.~D. van Enter, R.~Fern{\'a}ndez, and R.~Koteck{\'y}.
\newblock Pathological behavior of renormalization group maps at high fields
and above the transition temperature.
\newblock {\em J. Stat. Phys.}, 79:969--992, 1995.
\bibitem{vEFS_PRL}
A.~C.~D. van Enter, R.~Fern{\'a}ndez, and A.~D. Sokal.
\newblock Renormalization transformations in the vicinity of first-order phase
transitions: {W}hat can and cannot go wrong.
\newblock {\em Phys. Rev. Lett.}, 66:3253--3256, 1991.
\bibitem{vEFS_JSP}
A.~C.~D. van Enter, R.~Fern{\'a}ndez, and A.~D. Sokal.
\newblock Regularity properties and pathologies of position-space
renormalization-group transformations: Scope and limitations of {G}ibbsian
theory.
\newblock {\em J. Stat. Phys.}, 72:879--1167, 1993.
\bibitem{vEFS94}
A.~C.~D.~van Enter, R.~Fern{\'a}ndez, and A.~D.~Sokal.
\newblock Renormalization transformations: Source of examples and problems
in probability and statistics.
\newblock {\em Resenhas do Instituto de Matematica e Estatistica da
Universidade de Sao Paulo}, 1:233-262, 1994.
\bibitem{loren95}
A.~C.~D.~van Enter and J.~L\"orinczi.
\newblock Robustness of the non-Gibbsian property: some examples.
\newblock University of Groningen preprint, 1995, {\em J. Phys. A, Math. and
Gen.}, to appear, 1996.
\bibitem{vel94}
K.~Vande Velde.
\newblock Private communication, 1994.
\bibitem {veldis95}
K.~Vande Velde.
\newblock On the question of quasilocality in large systems of locally
interacting components.
\newblock K.U. Leuven thesis, 1995.
\bibitem{kogwil}
K.~G.~Wilson and J.~Kogut.
\newblock The renormalization group and the $ \epsilon $-expansion.
\newblock {\em Phys. Rep.}, 12C:75-200, 1974.
\end{thebibliography}
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