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non-Gibbsian measures, quenched disorder, Kac limit
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\title{Two connections between random systems and non-Gibbsian measures}
\author{
{\normalsize Aernout C.~D.~van Enter} \\[-1mm]
{\normalsize\it Centre 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@phys.rug.nl} \\[-1mm]
\\ [-1mm]
{\normalsize Christof K\"ulske} \\[-1mm]
{\normalsize\it Department of Mathematics and Computer Science} \\[-1.5mm]
{\normalsize\it Rijksuniversiteit Groningen} \\[-1.5mm]
{\normalsize\it Blauwborgje 3} \\[-1.5mm]
{\normalsize\it 9747 AC Groningen} \\[-1.5mm]
{\normalsize\it THE NETHERLANDS} \\[-1mm]
{\normalsize\tt kuelske@math.rug.nl} \\[-1mm]
{\protect\makebox[5in]{\quad}}}
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\begin{document}
\maketitle \baselineskip=14pt \noindent {\bf Abstract.} In this
contribution we discuss the role disordered (or random)
systems have played in the study
of non-Gibbsian measures. This role has two main aspects, the distinction
between which has not always been fully clear:
1) {\em From} disordered systems: Disordered systems can be used as a tool;
analogies with, as well as results and methods from the study of random
systems can be employed to investigate non-Gibbsian properties of a variety
of measures of physical and mathematical interest.
2) {\em Of } disordered systems: Non-Gibbsianness is a property of various
(joint) measures describing quenched disordered systems.
We discuss and review this distinction and a number of results related to
these issues. Moreover, we discuss the mean-field version of the
non-Gibbsian property, and present some ideas how a Kac limit approach
might connect the finite-range and the mean-field non-Gibbsian properties.
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\newblock An ultimate frustration in classical lattice-gas models.
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%\begin{document}
\maketitle
\section{Introduction}
In various situations in physics where one tries to describe many-particle
systems, a description of the system as an equilibrium (Gibbs) state
in terms of an effective Hamiltonian or an effective temperature is desirable.
It has turned out, however, that such a description is not always
available. Indeed, one may run into non-Gibbsian states, where
such a well-behaved Hamiltonian description simply does not exist.
In a classical lattice set-up, non-Gibbsian states --
which then are probability measures on (spin-or particle)
configuration spaces--
have been a subject of considerable interest for the last 15
years or so, see e.g.
\cite{EFS, Ken,Sch, LebMa,LebScho,EFHR, Isr, MaMoRe,Ferh} and also \cite{MPRF}
and references therein.
The original study of \cite{EFS} was mainly motivated by
Renormalization Group Transformation existence
questions, where the findings are that
renormalized measures can be non-Gibbsian. This
implies that in these situations
a renormalized interaction does not exist. Thus the whole
phenomenology of Renormalization Group fixed points,
critical exponents expressible in terms of eigenvalues of some Renormalization
Group operator in a space of Hamiltonians,
its domains of attraction as universality classes, may become problematical
once one tries to justify it in a rigorous manner.
Non-Gibbsian measures showed also up in other cases of
physical and/or mathematical interest, such as non-equilibrium states in
either the transient or the steady-state regime,
see e.g. \cite{EFHR,KuRe,LebScho, DR, LT}.
Yet further occurrences are for example in the study of Hidden Markov
fields, and in Fortuin-Kasteleyn random-cluster models.
In the mathematical study of these objects, at various occasions results
from disordered systems, in particular the phase transition proof of the
Random Field Ising Model and possible generalizations thereof
\cite{BK, Z} have been employed. This often is a question of mathematical
convenience, and sometimes may be avoidable.
Also, the fact that the non-Gibbsian properties are
due to a measure-zero set of ``bad'' configurations, that is
spin-configurations where an presumed effective (renormalized) interaction
diverges, shows a certain similarity to the fact,
familiar from the study of disordered systems, that various
physical properties of interest can be shown to
hold only for almost all realizations of
the disorder, but not for all of them, and that at the same time
this zero-measure set is responsible for some subtle, more
singular properties.
%{\bf It occurs in examples that the
%non-Gibbsian properties are appearing for a measure one set
%of discontinuity points (and not all of them), and
%this is analogous to the fact that in disordered
%systems physical properties of interest typically hold
%only for almost all realizations of
%the disorder (and not for all of them).
%In fact, in some examples, the first property is a mathematical
%consequence of the latter. }
This phenomenon is probably best known
from the analysis of the Griffiths singularity \cite{Gri}.
This analogy has been explored in e.g.
\cite{BCO1,BCO2,BCO3,BKL1, BKL2}.
\smallskip
A connection of different type was found first in \cite{EMSS} and then further
explored in \cite{Ku1,Ku2,Ku3, EMK, KLR}. This is the observation that
the quenched
(joint) measures of disordered systems themselves have non-Gibbsian properties
and in fact this non-Gibbsianness can be more severe that what is usually found
in the Renormalization Group setting \cite{KLR}.
Some of these considerations have been extended to a mean-field setting where
explicit computations can be made. We conjecture that in considerable
generality mean-field results correspond well to the
non-Gibbsian properties for lattice models in the limit of long-range
(Kac-)interactions.
In this paper we review a number of these aspects.
%{\bf Questionable: Do we say something on mean-field versions of non-Gibbsianness?}
% ----------------------------------------------------------------------
\section{ Notation and background}
For general background on the theory of Gibbs measures we refer
to \cite{EFS, Geo}. We will mostly consider finite-spin
models, living on a finite-dimensional lattice $\bbZ^d$. The spins
will take the values in a single-spin space $\Omega_{0}$ and
we will use small Greek letters $\si,\eta,\ldots$ to denote
spin configurations for finite or infinite sets of sites. The
Hamiltonians in a finite volume
$\Lambda \subset \bbZ^d$ with boundary conditions $\eta$ outside
will be given by
\begin{equation}
H^{\Lambda}(\sigma, \eta) = \sum_{ X \subset
\Lambda} \Phi_X([\sigma \eta] _{X})
%J(i,j)\, \sigma_i \sigma_j + \sum_{\langle i,j \rangle
%\atop i \in \Lambda, j \in \Lambda^c} J'(i,j)\, \sigma_i \eta_j
\end{equation}
For fixed boundary condition
%which is denoted by the $\eta$-variable,
these are functions on the configuration spaces.
$\Omega^{\Lambda} = \{-1,1\}^\La$.
%The boundary conditions are described by the $\eta$-variables.
Gibbs measures for an interaction $\Phi$ are probability measures
on $\Omega_{0}^{\bbZ^d}$ which have conditional probabilities of finding
$\sigma$ inside $\Lambda$, given boundary condition $\eta$ outside
$\Lambda$ which have
a continuous version, which are of the Gibbsian
form
\begin{equation}
{\frac{exp - H^{\Lambda}(\sigma, \eta)}{Z^{\Lambda}(\eta)}}.
\end{equation}
This should be true for each volume $\Lambda$ and
each boundary condition $\eta$.
One requires the uniform summability condition on the interaction
\begin{equation}\label{UNIF}
||\Phi||= \sum_{0 \in X} ||\Phi_X||_{\infty} \lneq \infty
\end{equation}
A necessary and (nearly) sufficient condition for a probability
measure $P$ to be such a Gibbs measure is that such conditional
probabilities %$P(\sigma_x| \sigma_{\bbZ^d \backslash x})$
are continuous in the product topology - it is then
said to to be quasilocal or almost Markovian. It
can be shown that being Gibbsian is topologically exceptional in the set of all
(or all translation-ergodic) probability measures \cite{Isr}.
To a Gibbs measure can be associated at most one interaction. If a
translation-invariant measure has relative entropy density zero
(it is always larger or equal than zero) with respect to a given
translation-invariant Gibbs measure, it then has the same large deviation
properties and it is Gibbs for the same interaction (variational principle).
\smallskip
To show that a particular measure is non-Gibbsian, it is sufficient to
find at least {\em one} point of discontinuity, that is at least {\em one} spin
configuration $\omega$, such that
\begin{equation}
sup_{\eta^{1}, \eta^{2}} |\mu(\sigma_0| \omega_{\Lambda}\eta^{1}_{\Lambda^c}) - \mu(\sigma_0| \omega_{\Lambda}\eta^{2}_{\Lambda^c})| \geq \epsilon,
\end{equation}
uniformly in
$\Lambda$. Such an $\omega$ one calls a {\em bad} configuration.
In many, although not all, examples, these bad configurations have measure
zero (almost Gibbs). A weaker property says that one can construct
an effective potential which is summable except on a measure zero set
(weak Gibbs) \cite{MaMoRe, KLR,EV}. If one considers transformed measures
on an image-spin space (transformed can mean deterministically or
stochastically renormalized or evolved, imperfectly observed....), one of
the typical proofs of non-Gibbsianness is to consider the original
measure {\em conditioned} on (or constrained by) an image-spin configuration.
%from the {\bf transformed measure - image space of the transformation}.
Once one finds such an image-spin
configuration for which a first-order phase transition occurs for the
conditioned system, this usually will be a bad configuration, and the
existence of such a bad configuration then implies non-Gibbsianness of the
transformed measure (see for details in particular \cite{EFS}, section 4.2).
For disordered models, where next to the spins there are disorder variables,
such that the interactions, Hamiltonians, measures, etc are themselves random
objects (they are disorder-dependent),
we will denote these disorder variables by $n$'s.
Quenched systems will be described by probability measures on the
pro\-duct space of the disorder and the spin variables. The disorder variables
are typically i.i.d. (they describe degrees of
freedom which are supposed to be
frozen after a quench from -- infinitely-- high temperature),
and the measures describing the quenched systems are
defined such that the conditional measures on the spin configurations,
given the disorder configuration, are Gibbs measures.
Whether or not a configuration is bad now turns out to depend only on the
disorder. That is to say, the configuration consists of both spin
and disorder
variables, but the goodness or badness of a
particular configuration only depends on the disorder part and the spin part
does not play any role. (This was proved in \cite{Ku1}
under the assumption that spin and disorder variables couple in a local way,
which holds for all models of interest.)
\smallskip
\section{Input from random systems for non-Gibbsianness}
% ----------------------------------------------------------------
\subsection*{ Finding bad configurations; maps random and nonrandom}
In various cases, the finding of a bad configuration for a given measure cannot be done explicitly. However,
one can show that a random choice from some Bernoulli measure will do the job. That is, conditioning on a typical random
realization of the renormalized or evolved spins can
induce a phase transition in the original system \cite{EFS,EFHR}, once
it is conditioned. The existence of this phase transition often follows
from \cite{BK,Z}. Then these typical realizations are all bad.
This strategy especially applies if one wants a result for a
continuum of parameters, such as the
magnetic field of the untransformed measure, or the bias parameter in an
asymmetric evolution. As one can continuously vary the mean of the Bernoulli
measure one draws the realization from, one is then left with a continuous
family of quenched random models.
As an example, consider a Gibbs measure for
the standard nearest neighbor Ising model in
dimension $d$ at low temperature. Apply an infinite-temperature
Glauber dynamics (independent spin flips at Poissonian times) acting on it.
Denote the spins at time $0$ by $\sigma$'s and the evolved spins at time $t$
by $\eta'$s.
%{\bf Then the }
Then, fixing the evolved spin at site $i$
at time $t$, $\eta_i(t)$ to be plus or minus, induces an extra bias,
that is an extra magnetic field in the plus or minus
direction acting on the $\sigma$ spin at site $i$. The strength
of these local (dynamical) fields
decreases to zero with increasing time \cite{EFHR}.
Choosing the $\eta$-configuration to be alternating -a chessboard
configuration- means that the study of the conditioned system means
considering an Ising system in an alternating field.
When the dynamical field strength $h(t)$ is weak enough --
which corresponds to the time being large
enough--, there is a phase transition in the conditioned system
for any dimension at least $2$.
Thus the chessboard configuration is in that case a bad configuration for the
evolved measure.
If, on the other hand, instead of a chessboard configuration
one would choose a random realization from a symmetric Bernoulli measure, we
obtain an Ising model in a random, instead of an alternating , magnetic field
and one obtains a phase transition for a measure-one set of such realizations,
but now only in dimension 3 or more \cite{BK}. In this way one obtains an
uncountable set of bad configurations.
As a side remark, note that in general any finite-volume perturbation of a
bad configuration is again a bad configuration. This also gives rise to
an infinite class of bad configurations
associated to every bad configuration. This observation, however, only
provides a countable set starting from a single configuration.
If the original system was subjected to a weak external magnetic field, one
can use a choice from a biased (in the opposite direction)
Bernoulli measure, to compensate for this, and again obtain a phase transition.
This means considering an Ising model in a non-symmetrically distributed
random field, and, as announced in \cite{Z}, the results of \cite{BK}
still hold. In this situation
one can at the same time vary the external field and the mean
of the compensating Bernoulli measure, and one obtains
in this way non-Gibbsianness for a continuum of values of
the original magnetic fields.
This works as long as the dynamical field strength is neither much stronger
nor much weaker than the initial field, thus the evolved measure is
non-Gibbsian for a finite time interval.
(In physical terminology, when one tries to
heat the system fast from cold to hot -- the limit measure to which one
converges exponentially fast is a Gibbs measure, at infinite, thus very high,
temperature --,
non-Gibbsianness means that at intermediate times one has {\em no}
temperature, rather than an intermediate temperature.)
The loss of the Gibbs property has also been proved for a
diffusive time-evolution, starting from a low-temperature phase,
for a model of continuous unbounded
spins in a double-well potential \cite{KuRe}. This is a related example,
where non-Gibbsian\-ness, however, does not rely on
random field arguments, and is proved to appear in dimensions
$d\geq 2$.
Indeed, in this model one can always find a bad conditioning configuration
$\eta$ for the
time-evolved measure that is homogenous in space.
This is possible because of the continuous nature of the spins. To prove
non-Gibbsianness it suffices to consider (essentially) translation-invariant
models and avoid disordered models.
Another difference with the results for the Ising model
is that there is no recovery of the Gibbs-property for
large times, in the case of non-zero initial magnetic field. Indeed the
bad configurations are diverging with time to infinity,
which is only possible for unbounded spins.
Arguments which are mathematically very similar
to the case of the Glauber time-evolution apply also
for instance when one considers
the behaviour of Gibbs measures under a decimation transformation \cite{EFS}.
In this case, the physical conclusion reads that a
renormalized interaction does not exist.
We remark that the {\it random} Glauber evolution map (which can also be seen as a
single-site random renormalization map) and the {\it deterministic}
decimation map show very much the same kind of behaviour.
%{\bf Suggestion: Discuss Ising model with Glauber dynamics explicitly.
%Conditioning on chessboard configuration, works in $d$ at
%least $2$ for long times, conditioning on a random configuration works in $d$
%at least $3$, but the dynamical random fields can then also compensate a
%continuum of magnetic fields. Mention that same argument applies for
%decimation.}
\bigskip
\subsection*{Analogies: measure-one properties and multiscale methods}
In various models it can be shown that the set of bad configurations
has (non-Gibbsian) measure zero ("almost Gibbs").
Even if this it not true, one may in given models
construct an interaction which is defined almost surely
("weak Gibbs"). By abstract arguments almost Gibbs essentially
implies weak Gibbs \cite{MaMoRe}.
It has been observed that there is
a similarity with phenomenona one knows from disordered systems where the
physics is descibed by the behaviour of the system
for a set of measure one from the disorder realizations. One is then
interested in the typical (probability one) behaviour of the system.
In fact the similarity goes further, in the sense that certain configurations
can locally be good at different scales. E.g. in the Griffiths singularity
problem, a disorder configuration configuration consists of occupied clusters,
and one finds occupied clusters at arbitrary scales, while in the
non-Gibbsian set-up, one can find configurations where the effective
interaction reaching the origin has ranges at various scales. This observation
has led to similar techniques being applied to both types of problems.
In particular the method of multiscale cluster expansions has turned out
especially useful in both the study of disordered systems and the study of
non-Gibbsian but weak Gibbsian measures. For some examples see e.g.
\cite{BCO1,BCO2,BCO3, BKL1,BKL2,BuvL,FI, ENS}.
%When we have
\smallskip
\section{Non-Gibbsian properties of random systems}
\subsection*{Quenched measures and the Morita approach}
In \cite{EMSS}, it was discovered that if one considers the joint measure
of a site diluted Ising model, given by the Hamiltonian
\begin{equation}
-H(n, \sigma)= \sum_{*} n_i n_j \sigma_i \sigma_j
\end{equation}
where the $n_i$ are $0$ or $1$ with probability $p$ or $1-p$, and the
conditional measure on the spins, given the realization of the occupation
variables is of the Gibbsian form, this joint (quenched) measure on
the product space of the disorder
(occupation) variable space and the spin space, which is the limit of
\begin{equation} \label{666}
K(n, \sigma) = P(n) \frac{exp -H(n, \sigma)}{Z(n)}
\end{equation}
is itself not a Gibbs measure as defined above.
Informally this means that it can {\em not} be written as
\begin{equation}
K(n, \sigma) = \frac{exp -\bar H(n, \sigma)}{Z}
\end{equation}
in the thermodynamic limit with a uniform summable Hamiltonian
$\bar H(n,\sigma)$, (which should be
a function of the pair $(n,\sigma)$ of
disorder variables $n$ and spin-variables $\sigma$).
%{\bf This is very misleading. In fact, the joint measures are precisely
%an example where this is always possible in the weak sense of a summable
%potential, even though there might be a.s. non-Gibbsianness. }
In the language of disordered systems, the quenched measure
cannot be written as an annealed Gibbs measure
for a proper potential depending on the joint variables $(n,\sigma)$.
{\bf Warning:} We use the terms ``quenched'' and ``annealed'' in our paper
in the original sense, as describing either fast or slowly cooled systems, as
is the standard usage in the (mathematical) physics literature on
spin systems.
Unfortunately, in some probabilistic literature
``quenched'' is used for almost sure, and ``annealed'' for averaged
properties. To avoid confusion we stress that
this is {\em not} our convention.
\medskip
A bad configuration in this model is for
example an occupation configuration of two infinite occupied clusters,
separated by an infinite empty interval of thickness one (and spin
configuration arbitrary). To see where the nonlocality comes from, for
simplicity we first consider the $T=0$ case.
Then the ground state of the Ising model on these
two semi-infinite clusters is fourfold degenerate.
Once one connects the two clusters, it is twofold degenerate.
Adding an occupied site in this interval can thus lower
the entropy by a finite term $ln 2$ or not, depending on whether the two
clusters have another connection (which can be arbitrarily far away)
or not. At sufficiently low temperatures a similar reasoning implies
non-Gibbsianness of the quenched measure, as the extra free energy due to
adding a single site can depend nonlocally on the occupation variable far away.
Note that the goodness or badness of an $(n,\sigma)$-configuration
is due to a nonlocal behaviour of the random partition function $Z(n)$,
and only depends on the disorder variable $n$, but not on $\sigma$.
Afterwards in \cite{Ku1} criteria for the absence of the Gibbs property for
measures on the $(n,\sigma)$ product space of a general class of quenched
disordered models $\mu[n]$ depending on disorder variables
$n$ were given.
A good example for this is the random field Ising model, given
by the Hamiltonian
\begin{equation}
-H(n, \sigma)= \sum_{**}\sigma_i \sigma_j + h \sum_{i}n_i\sigma_i
\end{equation}
where the random fields $n_i$ are $1$ or $-1$ with probability $\frac{1}{2}$.
Again we look at the large-volume limits of
\begin{equation}\label{RFIMJM}
K(n, \sigma) = P(n) \frac{exp -H(n, \sigma)}{Z(n)}
\end{equation}
for different spin-boundary conditions.
The class of systems allowed in the analysis of \cite{Ku1}
includes also the site-diluted Ising model for arbitrary dilutions $p$,
(while \cite{EMSS} was restricted to the small-$p$ regime),
bond disordered models etc. It was shown in considerable generality
that the failure of this Gibbs property in product space occurs
whenever there is a discontinuity in the quenched expectation $\mu[n]$
of the spin-observable in the Hamiltonian that is conjugate to
the disorder variable $n$.
In the case of the random field Ising model, this is just the expectation
of the spin $\sigma_i$ taken w.r.t. the random-field dependent Gibbs measure
$\mu[n](\sigma_i)$, viewed as a function of the random fields $n$.
In 3 dimensions a typical random field configuration
allows both a plus and a minus state (as proved by \cite{BK}), for small
enough $h$ and small enough temperature. As changing
the field outside some volume $\Lambda$ to either plus or minus picks out one
of the two possibilities, here in fact the set of bad configurations (points of
essential discontinuity) has full measure, w.r.t. the joint measure.
It consists of a full measure set
of fields combined with any spin configuration.
In the physics literature, such an
annealed description of quenched disordered systems
in terms of an effective (so-called ``grand'')
potential had been introduced by Morita long ago,
and has been reinvented and studied at various occasions, see e.g
\cite{Mor, Kue,KM} and also \cite{Kue2, Ku3}.
As a uniformly convergent
grand potential does not exist in many examples,
a controlled application of the Morita method is thus quite problematical.
What happens, however, if one gives up {\it any} assumptions on
the speed of convergence of the potential
depending both on spin and disorder variables, and asks
only for convergence, possibly arbitrarily slow, on a set of full measure?
In \cite{Ku2} the existence of such a potential was shown
by soft (martingale) arguments for disordered general models.
The proof however exploits the product structure of the model, so that
this it would not generalize (say) to renormalized measures.
To summarize, for quenched models
one loses in general any control when one truncates
the Hamiltonian, but it least there is a well-defined Hamiltonian
to talk about.
To get any bounds on the speed of convergence of the "grand potential",
even on a restricted set of realizations of the disorder, is difficult and
model-dependent work is needed that can be hard.
In the case of the random field Ising model,
this can however be done \cite{KLR},
building on the renormalization group arguments of \cite{BK}.
It shows the decay of the potential like a stretched exponential
for almost any random field configuration $n$.
%{\bf VAGUE IDEA: explain difference of RG for "renormalizing the rg pathologies"
%and application of RG to show existence and decay of Morita
%potential ?? Well, in quenched systems the bad configurations are disorder
%configurations as opposed to spin configurations (actually joint
%configurations, but goodness or badness is only dependent on the
%$n$-variable.)}
%It turns out
\subsection*{Almost versus weak Gibbs, violating the variational principle
%the colors of H\"aggstr\"om, fuzzy(?)Potts
}
We just saw that although the quenched measures of the RFIM are weak
Gibbsian with an almost surely rapidly decaying potential,
they have a measure-one set of bad configurations.
(This holds in $3$ dimensions,
at low temperature, and small random fields.)
The possibility of having a measure-one set of bad configurations,
can have severe consequences:
%This means that one can write the conditional
%probabilities in Gibbsian form with an interaction which converges
%on a full measure set.
A particularly surprising fact is
that in this situation the variational principle can be violated \cite{KLR}.
This means that the almost surely
defined potentials for the plus measure and the minus measure are not the same,
even though the relative entropy density between these two
translation-invariant measures is zero.
In more physical terms,
the different phases have different (almost surely defined) grand potentials.
This is a sharp contradiction to classical Gibbs formalism built
around the notion of a uniformly summable potential (\ref{UNIF}).
Here the relative entropy density between two measures
vanishes if and only if they have the same conditional probabilities
(or in other words: equivalent interaction potentials).
The random field Ising model thus clearly shows that
the proposed class of weak Gibbs measures is too broad for a variational
principle to hold.
One needs to strive for a smaller class. Partial, but not final, results
have been obtained in this direction:
On the positive side, \cite{EV}
%\cite{KLR}
showed the validity of the
classical variational principle assuming concentration properties
on certain "nice" configurations (invoking assumptions which are weaker than
almost Gibbsianness -- where it follows from \cite{MaMoRe}--,
in the spirit of, but stronger than, weak Gibbsianness).
%{\bf Corrected}
%I get confused now.
%I thought that the weakest conditions for a VP are in \cite{EV}, and they
%are weaker than almost Gibbs. Why do you say that one needs MORE than almost
%Gibbs?}
\section{Mean field}
A related analysis has turned out to be possible for mean-field models
of Curie-Weiss type.
Product states are trivially Gibbs, and non-trivial combinations
thereof have a full set of discontinuity points, and thus are non-Gibbs
in a strong sense \cite{EL}.
However, the interesting approach turns out to
replace continuity of the conditioning in the product topology
by continuity properties of conditional probabilities as a
function of {\em empirical averages} \cite{Ku3,HK,KL}. One looks at conditional probabilities in
finite volumes and then takes the limit, rather
than immediately considering the infinite-volume measures.
{\em One} value of the empirical average for
the magnetization now corresponds with a whole collection of spin
configurations.
The results are often of similar nature as in the short-range situation,
but are more complete in the sense that in many cases the whole phase diagram can be treated.
% by reducing the problem to the bifurcation analysis
%of a rate function of one magnetization-variable depending on the model-parameters.
The parallel holds,
to the extent that the "hidden" phase-transitions that are responsible
for the discontinuities that cause non-Gibbsianness on the lattice
occur in the same way as for the
mean-field counterpart, assuming e.g. large enough lattice dimensions. Remember however that non-Gibbsianness
is expressed in different topologies.
Properties like the full-measure set of discontinuities in the quenched
measures in the random field Ising model reappear as large-volume asymptotics
of finite-volume probabilities of "bad sets" \cite{Ku3,Ku4}.
Mean-field systems can serve as an illustration but also
as a source of heuristics, suggesting new and sometimes unexpected
mechanisms of non-Gibbsianness, as in the example of time-evolved measures
\cite{KL}, where a symmetry-breaking in the set of bad configurations appears.
Thus mean-field computations can be a fertile source,
motivating further research.
\subsection*{Mean field - via Kac limits }
Let us discuss the relation between the notion of
non-Gibbsianness on the lattice and in mean field in some more
detail.
A priori it might not be clear that there should be a close
connection, since different topologies are involved
when looking at the continuity of various conditional
probabilities.
To draw a link between lattice and mean-field properties (also) on the level
of the present discussion of non-Gibbsianness it should be
very interesting to investigate Kac-models.
Kac-models, going back to \cite{KUH}, are defined in terms of
long-, but finite range interactions of the form $J_\g(r)\equiv \g^dJ(\g r)$.
Here $J(x)\geq 0$ is a nice function, rapidly decaying or of bounded support
with $\int J(x)d^d x=1$.
A main example is the ferromagnetic Kac-Ising model of the form
\begin{equation} \label{KacIsing}
-H_{\gamma}(\sigma)= \frac{1}{2}\sum_{i,j} J_\gamma(i-j) \sigma_i \sigma_j+h\sum_{i}\sigma_i
\end{equation}
We shall argue that in many cases
the Gibbsian/non-Gibbsian properties
should be compatible in a nice way
via the Kac limit.
More precisely, consider
a translation-invariant lattice model with long-range (Kac-)
potential, depending
on a number of parameters describing
the interaction, such as e.g. temperature and field for an Ising model.
Look at a stochastic (or deterministic)
transformation of the Gibbs measures, and identify
the sets $\text{nG}(\g)$ of the parameter space (such as the space spanned by
the temperature and field variables) for
which the image measure is non-Gibbs as a lattice-measure,
in the sense of the product topology.
%{\bf How do you get here from a spin configuration to a parameter value?, this
%needs some explaining. Empirical averages seem to be involved, but as
%I mentioned before, in d=2 the chessboard configuration is bad,
%the Bernoulli configurations are good, and both have the same magnetisation.
%Sorry I misunderstood this at first.}
Define $\text{nG(MF)}$ as the parameter set for
which the (corresponding) image measure
of the (corresponding) mean-field model is
non-Gibbs. Then, we expect the following,
expressing Kac-compatibility of non-Gibbsianness.
\begin{conjecture}
"Usually"
\begin{equation}
\text{nG}(\g) \rightarrow \text{nG(MF)}
\end{equation}
in the Kac-(long range)-limit $\g \downarrow 0$.
\end{conjecture}
Such a statement is reminiscent of
the stability of the phase-diagram at low temperatures
(proved in the framework of Pirogov-Sinai theory),
structural stability in bifurcation theory etc.
A precise statement will depend on the transformation
and on the precise definition of the models under consideration.
In particular one may need
%{\bf instead of ``the validity will assume''}
sufficiently high lattice dimensions.
To develop full proofs presumably will be non-trivial and at present
the above conjecture is more a research program then a theory.
% {\bf leave out ``,and cannot be given here.''}
We will however describe in the examples below what the theorem means more
precisely and which steps at this point are missing.
\subsubsection*{Kac-limit background}
Let us start by reviewing what is the relation between
translation-invariant Kac-models and mean-field models, see also
\cite{BoKu05}.
A general motivation for the introduction of Kac-models is the hope that by
taking first the infinite-volume limit, and then the Kac-limit
$\gamma\downarrow 0$,
one obtains the corresponding mean-field model. Historically this
was motivated first by the desire to justify the Maxwell construction
(equal-area rule) from a microscopical model.
The first fundamental result relating Kac and mean field and expressing
a "Kac-compatibility"
is the Lebowitz-Penrose theorem \cite{LP}.
It says that the free energy of the lattice-model
converges to the convex (envelope of the) free energy of the corresponding
mean-field model.
The Lebowitz-Penrose theorem holds in considerable generality for various
lattice spin models with short-range interactions, in any dimension.
%generically, although there are counterexamples {\bf Gates-Penrose ???}.
%This is equivalent to the more usual
%convergence of the free energy density.
However, as the example of ferromagnetic Kac-model
in 1d shows, convergence to a model with flat mean-field
rate-function (a free energy with a phase transition)
does not need to be accompanied
by non-uniqueness of the infinite-volume Gibbs measure on the lattice.
Indeed, to understand the Gibbs measures is
a much more subtle (and dimension-dependent) question.
While uniqueness
holds in one dimension for any finite $\g$,
the Gibbs measure behaves in a non-trivial way and concentrates at a
mesoscopic scale on profiles with jumps between values close to
the corresponding (positive or negative)
mean-field magnetisation
(\cite{COP},\cite{CaOrPi99}).
In lattice dimensions $d\geq 2$ however, Kac-compatibility is expected to hold
for translation-invariant ferromagnetic models also on the level of the phase
diagram, that is, the set of Gibbs measures.
In the special case of the standard Kac-Ising model
(\ref{KacIsing}) this problem is indeed settled.
The existence of ferromagnetically
ordered low-temperature states in the Kac-model, and moreover
the convergence of the critical inverse temperature $\beta_c(\g)\rightarrow \beta_c(\text{MF})=1$
was proved independently in (\cite{BoZa1} and \cite{CMP}). Both proofs are
based on spin-flip symmetry
and don't generalize to models without symmetry between the phases, see however
\cite{BoZa2}. We stress that the Kac-model might behave different
from a nearest neighbour model. An example
of this is the new result of \cite{GoMe} on two-dimensional
three-state Potts models.
For a recent general description of Kac-limit results we refer to \cite{Pres}.
\subsubsection*{Decimation of standard Kac-Ising model to a sublattice}
As our first illustration we
consider now the decimation transformation of the ferromagnetic Kac-Ising
model to a sublattice $S$ that is kept fixed,
independently of the Kac-parameter
$\gamma$. Denote by $1-p$ the density of $S$ in $\bbZ^d$.
Think of $d=2$ and the sub-lattice $S=(2\bbZ)^2$ for concreteness, so that $p=\frac{3}{4}$.
Denote by $nG(\g)$ the range of critical inverse temperatures
for which the projected measure is non-Gibbsian.
We will argue that
\begin{equation}\label{KID}
\text{nG}(\g) \rightarrow [\frac{1}{p},\infty)=\text{nG}(\text{MF})
\end{equation}
as $\g \downarrow 0$.
To analyze Gibbsianness of the transformed Gibbs
measures before the Kac-limit, and analyze
badness of a configuration $\eta$, we go through the standard program for
lattice systems.
This leads us to analyze the quenched model with Hamiltonian for
the a spin-model on the "three-quarter lattice" $S^c=\bbZ^d\backslash S$ given by
\begin{equation} \label{quenchedKI}
-\frac{1}{2}\sum_{i,j\in S^c} J_\gamma(i-j) \sigma_i \sigma_j
-\sum_{i\in S^c}\Bigl(\sum_{j\in S}J_\gamma(i-j)\eta_j\Bigr)\sigma_i
\end{equation}
Consider a checkerboard-configuration $\eta=\eta_{\text{spec}}$. We claim
that it is a bad configuration for low temperatures at sufficiently
small $\g$. Indeed, the $\eta$-dependent
term in the Hamiltonian is neutral and translation-invariant,
so that we can expect phase-coexistence
for the quenched model (\ref{quenchedKI}).
Note that the site-dependent effective
magnetic field $\sum_{j\in S}J_\gamma(i-j)\eta_j$ acting on the spin $\sigma_i$
becomes small uniformly in the Kac-limit, and so
a mild modification of the contour-construction given in
\cite{BoZa1} or \cite{CMP} should prove that there is indeed
phase coexistence at low temperature.
What about the critical inverse temperature $\beta_c(\g,S^c,\eta_{\text{spec}})$?
We see that the effective
interaction-strength on the three-quarter lattice $S^c$ is reduced
by the knocking out of $S$, in this way
reducing the effective inverse temperature of the model to $p\beta$.
Again, using (the modification of) \cite{BoZa1,CMP} up to the critical point should prove indeed
$\beta_c(\gamma,S^c,\eta_{\text{spec}})\rightarrow \frac{1}{p}=\frac{4}{3}$.
If we further assume that the "worst" bad configuration is indeed given by the checkerboard
configuration, we have indeed proved the l.h.s. of (\ref{KID}).
Finally consider the corresponding mean-field set-up at zero external field.
Here a Curie-Weiss model of size $N$ is projected
to a subset of size $(1-p)N$, and continuity of the single-site
conditional probabilities of the projected measure at fixed magnetisation of the conditioning
is investigated. A computation shows
that the range of inverse temperatures
for which non-Gibbsianness holds is indeed
given by $[\frac{1}{p},\infty)$ (see \cite{Ku3}).
In this example, we do not need the disordered model as a tool
for the mean-field analysis, in contrast to what happens on the
finite-dimensional and Kac models. This is the case because conditioning
on a finite fraction of the spins being fixed does not depend on the location
of these spins; indeed in mean-field models talking about the location of
spins on a lattice does not make sense.
Let us contrast this with a result from \cite{KuJSP2001}.
Here the following opposite result was proved: Start with a Kac-Ising
model, with or without site-dependent
magnetic fields. Take block-averages over blocks with width $l$ (sufficiently)
smaller than the range $\frac{1}{\gamma}$ of the Kac-potential.
Then the resulting measure is Gibbs.
\subsubsection*{Decimation of Kac models without symmetry}
Let us consider more generally Hamiltonians of the form
\begin{equation}
\begin{split}
&H(\s)=\frac{1}{2}\sum_{i,j}\Phi^\gamma_{i-j}(\s_{i},\s_{j})+\sum_{i}U(\s_i)
\label{1.3}
\end{split}
\end{equation}
with $\s_i$ taking values in a finite set, the interaction kernel $\Phi^\gamma_{i-j}$
has finite range $1/\gamma$, is "smooth enough" and
$\sum_k\|\Phi^\gamma_{k}\|_\infty =O(1)$.
No symmetry of the interaction under permutation of the phases is assumed.
Project a Gibbs-state of this model to a $\gamma$-independent sublattice $S$.
What can we say about non-Gibbsianness? Do we expect convergence of
the region of inverse temperatures and fields for which the decimated
model is non-Gibbs to the corresponding mean-field values as $\gamma\downarrow 0$?
To start with, we review what is known about the translation-invariant phases.
The work of (\cite{BoZa2},\cite{Z2}) proves the existence of
ordered phases at low-temperatures, uniformly in (sufficiently small) $\gamma$.
It provides essential steps of
a Kac-Pirogov-Sinai theory, and gives a good
understanding of the structure of the phases.
What is lacking however is the control up to the critical temperature of
the corresponding mean-field model. So, Kac-compatibility is only partially proved
already on the level of translation-invariant phases.
In order to analyze badness of a configuration $\eta$ of the
projected lattice we need to understand whether resp. for what values
of parameters the quenched model
\begin{equation} \label{quenched1.3}
\frac{1}{2}\sum_{i,j\in S^c}\Phi^\gamma_{i-j}(\s_{i},\s_{j})+\sum_{i\in S^c}\Bigl( U(\s_i)
+\sum_{j\in S}\Phi^\gamma_{i-j}(\s_{i},\eta_{j})\Bigr)
\end{equation}
has a phase transition.
This should be possible when we reverse the order of choices, taking a periodic
$\eta$ first, and looking for values of the parameters for which there is coexistence, using
the method of (\cite{BoZa2},\cite{Z2}). A difficulty here is that the range of temperature for which
one is able to prove coexistence will in general be non-uniform in the period.
Such an argument would show the existence
of bad configurations at a discrete $\gamma$-dependent set
of parameter-values of the model.
So, this method would provide a proof of non-Gibbsianness
that is not exhaustive over the whole
presumed region of non-Gibbsianness.
To improve this and prove non-Gibbsianness for a continuum of parameter values,
again randomizing to choose bad configurations would be necessary.
To prove this rigorously one would need a fully developed nonsymmetric
random Kac-Pirogov-Sinai-theory.
In such a generality such a theory does not (yet) exist.
For an outline of corresponding results in the random field Kac Ising model (with symmetric random field distribution)
see \cite{BoKu05}.
Finally, to study a.s. Gibbsianness such a theory would have to allow
for non-independent Gibbs-quenching, too.
All of this would be highly non-trivial. As in the short-range situation
the steps involving randomized bad configurations are expected
to work only in $d\geq 3$, whereas the arguments before should be valid
in $d\geq 2$.
\subsubsection*{Non-Gibbsian quenched measures in mean field}
As remarked before, the non-Gibbsianness of the quenched joint measures $K(n,\sigma)$ (cf. (\ref{666},\ref{RFIMJM}))
has a mean-field analogue, as first shown in \cite{Ku3}. Physically, this again
expresses the fact that an annealed description of such measures
in terms of a Morita ``grand potential'' does not work.
The analogy goes quite far, in that the set of bad configurations
has measure zero or full measure in the mean-field situations according to the behaviour of the analogous model
in finite dimensions (Zero measure but
non-empty for dilution, full measure in the phase transition regime of the
random field model).
Thus for these quenched measures the connection between disorder and
non-Gibbsianness again is ``intrinsic'', whereas disordered methods
as a tool may or may not occur in the mean-field analysis of systems
without disorder (in fact it occurs for the Glauber-evolved
situation \cite{KL}, but not for decimation \cite{Ku3}).
Furthermore, for mean-field quenched models
one can perform a rather complete analysis of the Morita
approach and where and how it breaks down and what one still can learn from it
(see \cite{Kue2,Ku4} and references mentioned there).
Again, it would be of interest to see if one can connect the analyses
of the mean-field and the lattice models via a Kac limit, although again
we expect that some of the same complicated
technical problems we mentioned before
would need to be solved (in particular developing a general
non-symmetric random Pirogov-Sinai theory).
%{\bf STOP READING HERE}
%\vfill\eject{}
%A class of non-symmetric Kac-models could be treated in \cite{BoZa2} by
%a different contour method, proving low-temperature order at temperatures
%that are uniform in the range of the interaction, however not extending
%to the true mean field critical temperature.
%However, one has the following Conjecture:
%\begin{itemize}
%\item The phase diagram
%of the lattice model with finite $\g$ is a small deformation
%of the mean field phase diagram.
%\end{itemize}
%This has been proved for various models with pair interactions,
%assuming some restrictions, in particular spatial
%regularity of the interaction and low-enough temperatures.
%It hass not been proved in such generality around the critical temperature.
%{\bf Results due to Bovier-Zahradnik, up to critical T I thought,
%Potts 1st order, Biskup-L.Chayes-Crawford, Bidineau-Merola}
%\subsubsection*{Gibbs vs. non-Gibbs in Kac and mean-field for decimation}
%To analyze Gibbsianness in Kac-models we must
%essentially look at a conditioned model, which in many cases will be a
%quenched model.
%Let us fix a sublattice $\La'$, put $\La_0:=\La\backslash \La'$
%and look at the conditional system
%\begin{equation}
%H_{\gamma}(\sigma)= \sum_{i,j\in \La_0}J_{\gamma}(i-j)U(\sigma_i, \sigma_j)
%+ \sum_{q\in \La_0}h(q) |\Lambda_q|
%+ \sum_{i\in \La_0, j\in \La'}J_{\gamma}(i-j)U(\sigma_i, \eta_j)
%\end{equation}
%To have non-Gibbsian behavior we must
%have a first-order phase-transition
%on $\La_0$, in the sense of existence
%of two different translation-invariant Gibbs measures.
%Further it must be possible to
%select the stable phases by choosing a boundary
%condition of $\eta's$.
%\subsubsection*{Kac-limit at fixed sublattice}
%At first we fix $\La'= b \bbZ^d$ with $b$ fixed and let $\g$ tend
%to zero. Then the corresponding Lebowitz-Penrose
%theorem should state the following: The limit $\g\downarrow 0$
%of the rate function of the empirical distribution $\frac{1}{|\La|}\sum_{i\in \La}\delta_{\s_i}$
%converges to the convex
%hull
%{\bf envelope?}
%of the mean-field rate-function
%\begin{equation}
%I(\nu)= \Bigr( 1- \frac{|\La'|}{|\La|}\Bigl)^2(\nu\times \nu) (U) + \nu(h)
%+ s(\nu|\rho)-\text{Const}
%\end{equation}
%PREFACTOR MIGHT BE WRONG!
%Extending the previous conjecture about
%the connection between Kac and mean-field to volumes
%having
%that are complements of sub-lattices,
%we should arrive that the internal system of the Kac-model
%has a phase diagram that is a small perturbation of the
%mean-field phase diagram.
%A simple special case is the symmetric Kac-Ising model.
%Here an example of a bad configuration should be provided by the checker-board
%configuration.
%{\bf leave out or movesomewhere else what is written now, add some more about
%the decimation at three-quarter lattice. Next we must check that it is possible to choose
%a boundary condition on $\La'$ that steers the system
%at the coexistence point. Indeed, choosing periodic
%configurations, this will in general, i.e. when we do not
%assume symmetry, depend on $\g$.
%Again, to have a balancing situation for a continuum
%of values of parameters, as well as of $\g$,
%we would need a Zahradnik-type asymmetric Pirogov-Sinai
%theory, but this time also for Kac-models.
%}
%\subsubsection*{Gibbs vs. non-Gibbs in Kac and mean-field
%for independent symmetric time-evolution}
%Let us recall
%things for
%properties of
%the mean-field model. Start
%with a mean-field Ising-model with Hamiltonian $H(m)$
%at time $t=0$.
%Divide spins into the sites $\La_+$ and $\La_-$ of sites with positive (resp. negative)
%$\eta_i$'s.
%Denote the corresponding magnetization by
%$m_+$ and $m_-$.
%Then the rate-function for the pair of magnetizations
%$(m_+,m_-)\in [-1,1]^2$ is given by
%\begin{equation}\label{RFRF}
%\begin{split}
%&I(m_+,m_-)\cr
%&= H(p_+ m_+ + p_- m_-) - h_t p_+ m_+ + h_t p_- m_- +p_+ I(m_+)
%+p_- I(m_-)-\text{Const}
%\end{split}
%\end{equation}
%where the rate is $N$.
%Fixing (say) $m_-$ yields the rate function $I(m_+|m_-)$
%of an Ising model for $m_+$ with Hamiltonian
%$m_+\mapsto H(p_+ m_+ + p_- m_-)$. The rate function for
%the total magnetization $m$ is obtained from (\ref{RFRF})
%by taking the infimum over the pair $(m_+, m_-)$
%under the constraint $p_+ m_+ + p_- m_-=m$.
%Think in particular of the case of the standard
%quadratic Hamiltonian
%$H(m)=-\frac{\beta}{2} m^2$.
%So, in this particular case we have for the conditional
%rate-function the expression
%\begin{equation}\label{RFRF1} \begin{split}
%&I(m_+|m_-)\cr
%&=- \frac{\beta}{2} p^2_+ m^2_+ + \beta p_+p_- m_- m_+
%- h_t p_+ m_+ +p_+ I(m_+) -\text{Const}\cr
%&=p_+\Biggl( -\frac{\beta p_+}{2} m^2_+ - \Bigl(
%\beta p_- m_- + h_t \Bigr) m_+ + I(m_+) \Biggr)-\text{Const}\cr
%\end{split}
%\end{equation}
%with rate $N$.
%So the conditional distribution of
%$m_+$ is that of an Ising model at inverse
%temperature $\beta p_+$ in the presence of
%a magnetic field $\beta p_- m_- + h_t $.
%The conditional distribution of
%$m_-$ is that of an Ising model at inverse
%temperature $\beta p_-$ in the presence of
%a magnetic field $\beta p_+ m_+ - h_t $.
%To explain the occurrence of different states,
%consider a situation of $p_+\approx \frac{1}{2}$.
%Now, suppose that there is ferromagnetic
%ordering individually on each of the
%subsets $\La_{\pm}$.
%Suppose that $h_t\approx 0$ is very small. Then the ferromagnetic
%coupling between the two subsets dominates the $h_t$-influence
%and we see a symmetric mixture of
%$(m_+,m_-)\approx (+,+),(-,-)$. However, by making $p_+$
%slightly bigger than $\frac{1}{2}$ we can choose the $(+,+)$-state.
%Suppose next that $h_t$ gets bigger, but not so big as to destroy
%the ferromagnetic ordering on the subsets.
%Then we see a state
%$(m_+,m_-)\approx (+,-)$. This state we call a "zero-like state".
%The magnetizations average out
%to be zero when both sets are equal, and we see a zero-like
%state, that is still an ordered state.
%A first-order transition between the $(+,-)$-state and
%the $(+,+)$-state is thus produced by a transition on
%the set $\La_-$.
%\subsubsection*{Kac-analogue}
%Start with a Kac-Ising model at $t=0$.
%Then the quenched model we need to look at is
%\begin{equation}
%H_{\gamma}(\sigma)= - \sum_{i,j }J_{\gamma}(i-j)\sigma_i \sigma_j
%-h_t \sum_{i\in \La_+}\sigma_i
%+ h_t \sum_{i\in \La_-}\sigma_i
%\end{equation}
%where $ \La_+$ has the same meaning, but is a set of lattice sites now.
%The question whether to see nG with symmetry-breaking
%in the Kac-model can be now formulated in this way:
%Given finite but small $\g>0$ and (suitable) $\beta$ and $t$,
%can we choose $\La_+$ such that
%there is phase coexistence between a $(+,+)$-state
%and a $(+,-)$-state?
%For general random sets $\La_+$
%this poses the problem of establishing the existence
%of a $(+,-)$-state in the random field Kac-model.
%In the classical random field Kac model we would choose
%the distribution of the set $\La_+$ independently over the sites.
%For the discussion of a.s. Gibbsianness we would choose
%$\La_+$ to be chosen from a Gibbs-distribution itself, that is consider
%a Gibbs-quenching.
%However, for proving non-Gibbsianness, that is show the
%existence of particular bad configurations, the situation
%is simpler, since we may be satisfied with a specific form
%of $\La_+$ for which the corresponding quenched model
%can be analyzed.
%To start we may proceed as follows. We reverse
%the order of choices and
%take $\La_+$ to be periodic such that its density in the whole
%lattice amounts to a empirical magnetization $\alpha$.
%This $\alpha$ has to be in a range of values such that
%we can find a triple $\alpha$ (chosen first), $\beta$ and $h_t^0$ for which we have
%nGSB in the mean-field model. These values can be read off from the phase
%diagram found in [KuLN05].
%Then we get non-Gibbsianness with symmetry breaking in a Kac model
%if we can find $h_t=h_t(\gamma)$ (which will be a small perturbation of $h_t^0$
%when $\gamma$ is small) such that there is phase-coexistence
%between the $(+,+)$-state and the $(+,-)$-state at finite (small enough) $\gamma$.
%That such a choice can be made is a plausible but non-trivial statement
%and needs to be proved by a suitable Kac-Pirogov-Sinai theory.
%The proof would proceed roughly as follows:
%We need to look at block-magnetizations taken over the plus- resp.
%minus-sites in fixed blocks $l<<\g^{-1}$ in the lattice
%$(m_+(x),m_-(x))$. Then we define restricted ensembles,
%defined in terms of the $(+,+)$ and $(+,-)$-magnetizations (taken from
%the mean-field model)
%within given error thresholds. This defines an ensemble
%of interacting contours on the scale $l$ where the restricted ensembles
%will be expanded and provide interactions for the contours.
%It needs to be verified that the resulting Peierls-constants are
%good enough, but this seems plausible.
%It might even be that we are lucky and the model
%satisfies the assumption of Bovier-Zahradnik Kac-PS (for non-symmetric models).
%Maybe this already works, in the sense that it gives coexistence, for suitable values of $h_t$.
%Finally we must ensure that by putting boundary conditions
%in the whole lattice we are able to distinguish between
%the $(+,+)$-state and the $(+,-)$-state.
%This seems an easier task to do assuming that the first step could be done.
%Assuming now that both steps works,
%we get, for fixed $\beta$ a discrete, $\gamma$-dependent,
%set of times $t$ for which the model shows nGSB.
%To get nGSB for an interval of $t$ we would need to randomize
%over $\La$, just like in the short-range model for non-vanishing
%external magnetic field.
%To prove this rigorously one would need a random-Kac-Pirogov-Sinai-theory
%(allowing non-symmetric models).
%Finally, to study a.s. Gibbsianness such a theory would have to allow
%for non-independent Gibbs-quenching, too. This last step is expected
%to work only in $d\geq 3$, whereas the arguments before should be valid
%in $d\geq 2$.
%}
\section{Summary and Conclusions}
We have reviewed two different connections which exist between the study of
non-Gibbsian measures and of disordered systems. It turns out that there are
two main types of connections:
\noindent
1) Results and insights developed in the study of
disordered systems often provide useful tools to study non-Gibbsianness.
\noindent
2) Disordered systems give rise to non-Gibbsian measures.
We have discussed these in some detail, mentioning various examples
and also the various physical meanings of these non-Gibbsian results,
including in renormalization group theory, non-equilibrium
questions and the Morita approach to disordered systems.
Also we mentioned how these ideas are developed in a mean field setting, and
we have suggested how a link may be developed in the Kac limit.
%Time dependence, random initial vs random boundary conditons?
%Remark that chaotic time dependence is historic dependence?
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%\section{ Discussion}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\addcontentsline{toc}{section}{\bf References}
%\begin{thebibliography}{10}
%\begin{bibsection}
%\begin{biblist}
%}
%\bibliographystyle{abbrv}
%\bibliography{univervp}
%\end{document}
\bigskip
%\end{equation}
\noindent {\em Acknowledgements}:
We thank all our colleagues with whom we have worked on or discussed
these topics for all they have taught us, and
Joel Lebowitz and Pierluigi Contucci for their
invitation to contribute to this volume.
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