\documentstyle[12pt]{article}
\def\datetitle{\today}
\def\datehead{\today}
%
% Page dimensions:
%
\oddsidemargin 7mm % Remember this is 1 inch less than actual
\evensidemargin 7mm
\textwidth 15cm
\topmargin 0mm % Remember this is 1 inch less than actual
%\headsep 0.9in % Between head and body of text
\headsep 20pt % Between head and body of text
\textheight 22cm
\pagestyle{headings}
\renewcommand{\baselinestretch}{1}
\footnotesep 14pt
\floatsep 28pt plus 2pt minus 4pt % Nominal is double what is in art12.sty
\textfloatsep 40pt plus 2pt minus 4pt
\intextsep 28pt plus 4pt minus 4pt
%
% MACROS
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% change the catcode of @ (allows names containing @ after \begin{document})
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\makeatletter
%
% Equations numbered within sections
%
\@addtoreset{equation}{section}
\def\theequation{\thesection.\arabic{equation}}
% \def\@evenhead{\rm\small\thepage\hspace{1.0cm}\leftmark\hfil{\scriptsize\today}}
\def\@evenhead{\rm\small\thepage\hfil\datehead}
% \def\@oddhead{{\scriptsize\today}\hfil\rm\small\rightmark\hspace{1.0cm}\thepage}
\def\@oddhead{\datehead\hfil\rm\small\thepage}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Redeclaration of \makeatletter; no @-expressions may be used from now on
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\makeatother
%
% Symbols (Springer)
%
\def\bbbr{{\rm I\!R}} %reelle Zahlen
\def\bbbn{{\rm I\!N}} %natuerliche Zahlen
\def\bbbm{{\rm I\!M}}
\def\bbbh{{\rm I\!H}}
\def\bbbf{{\rm I\!F}}
\def\bbbk{{\rm I\!K}}
\def\bbbp{{\rm I\!P}}
\def\bbbone{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l}
{\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}}
\def\bbbc{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm C$}\hbox{\hbox
to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
{\setbox0=\hbox{$\textstyle\rm C$}\hbox{\hbox
to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
{\setbox0=\hbox{$\scriptstyle\rm C$}\hbox{\hbox
to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
{\setbox0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\hbox
to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}}
\def\bbbe{{\mathchoice {\setbox0=\hbox{\smalletextfont e}\hbox{\raise
0.1\ht0\hbox to0pt{\kern0.4\wd0\vrule width0.3pt
height0.7\ht0\hss}\box0}}
{\setbox0=\hbox{\smalletextfont e}\hbox{\raise
0.1\ht0\hbox to0pt{\kern0.4\wd0\vrule width0.3pt
height0.7\ht0\hss}\box0}}
{\setbox0=\hbox{\smallescriptfont e}\hbox{\raise
0.1\ht0\hbox to0pt{\kern0.5\wd0\vrule width0.2pt
height0.7\ht0\hss}\box0}}
{\setbox0=\hbox{\smallescriptscriptfont e}\hbox{\raise
0.1\ht0\hbox to0pt{\kern0.4\wd0\vrule width0.2pt
height0.7\ht0\hss}\box0}}}}
\def\bbbq{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm Q$}\hbox{\raise
0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}}
{\setbox0=\hbox{$\textstyle\rm Q$}\hbox{\raise
0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}}
{\setbox0=\hbox{$\scriptstyle\rm Q$}\hbox{\raise
0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}}
{\setbox0=\hbox{$\scriptscriptstyle\rm Q$}\hbox{\raise
0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}}}}
\def\bbbt{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm
T$}\hbox{\hbox to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}
{\setbox0=\hbox{$\textstyle\rm T$}\hbox{\hbox
to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}
{\setbox0=\hbox{$\scriptstyle\rm T$}\hbox{\hbox
to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}
{\setbox0=\hbox{$\scriptscriptstyle\rm T$}\hbox{\hbox
to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}}}
\def\bbbs{{\mathchoice
{\setbox0=\hbox{$\displaystyle \rm S$}\hbox{\raise0.5\ht0\hbox
to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\hbox
to0pt{\kern0.55\wd0\vrule height0.5\ht0\hss}\box0}}
{\setbox0=\hbox{$\textstyle \rm S$}\hbox{\raise0.5\ht0\hbox
to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\hbox
to0pt{\kern0.55\wd0\vrule height0.5\ht0\hss}\box0}}
{\setbox0=\hbox{$\scriptstyle \rm S$}\hbox{\raise0.5\ht0\hbox
to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\raise0.05\ht0\hbox
to0pt{\kern0.5\wd0\vrule height0.45\ht0\hss}\box0}}
{\setbox0=\hbox{$\scriptscriptstyle\rm S$}\hbox{\raise0.5\ht0\hbox
to0pt{\kern0.4\wd0\vrule height0.45\ht0\hss}\raise0.05\ht0\hbox
to0pt{\kern0.55\wd0\vrule height0.45\ht0\hss}\box0}}}}
%
% note: changed \sans to \sf for LaTeX
%
\def\bbbz{{\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}}
{\hbox{$\sf\textstyle Z\kern-0.4em Z$}}
{\hbox{$\sf\scriptstyle Z\kern-0.3em Z$}}
{\hbox{$\sf\scriptscriptstyle Z\kern-0.2em Z$}}}}
%
% Theorems and such
%
\newtheorem{theorem}{Theorem}[section] % Numbering by sections
\newtheorem{lemma}[theorem]{Lemma} % Number all in one sequence
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{observation}[theorem]{Observation}
\def\proof{\par\noindent{\it Proof.\ }}
\def\reff#1{(\ref{#1})}
%
% Springer symbols for numeral sets
%
\let\zed=\bbbz % \def\zed{{\hbox{\specialroman Z}}}
\let\szed=\bbbz % \def\szed{{\hbox{\sevenspecialroman Z}}}
\let\IR=\bbbr % \def\IR{{\hbox{\specialroman R}}}
\let\R=\bbbr % \def\IR{{\hbox{\specialroman R}}}
\let\sIR=\bbbr % \def\sIR{{\hbox{\sevenspecialroman R}}}
\let\IN=\bbbn % \def\IN{{\hbox{\specialroman N}}}
\let\IC=\bbbc % \def\IC{{\hbox{\specialroman C}}}
\def\nl{\medskip\par\noindent}
\def\scrb{{\cal B}}
\def\scrg{{\cal G}}
\def\scrf{{\cal F}}
\def\scrl{{\cal L}}
\def\scrr{{\cal R}}
\def\scrt{{\cal T}}
\def\pfin{{\cal S}}
\def\prob{M_{+1}}
\def\cql{C_{\rm ql}}
%\def\bydef{:=}
\def\bydef{\stackrel{\rm def}{=}} %%% OR \equiv IF YOU PREFER
\def\qed{\hbox{\hskip 1cm\vrule width6pt height7pt depth1pt \hskip1pt}\bigskip}
\def\remark{\medskip\par\noindent{\bf Remark:}}
\def\remarks{\medskip\par\noindent{\bf Remarks:}}
\def\example{\medskip\par\noindent{\bf Example:}}
\def\examples{\medskip\par\noindent{\bf Examples:}}
\def\nonexamples{\medskip\par\noindent{\bf Non-examples:}}
\newenvironment{scarray}{
\textfont0=\scriptfont0
\scriptfont0=\scriptscriptfont0
\textfont1=\scriptfont1
\scriptfont1=\scriptscriptfont1
\textfont2=\scriptfont2
\scriptfont2=\scriptscriptfont2
\textfont3=\scriptfont3
\scriptfont3=\scriptscriptfont3
\renewcommand{\arraystretch}{0.7}
\begin{array}{c}}{\end{array}}
\def\wspec{w'_{\rm special}}
\def\mup{\widehat\mu^+}
\def\mupm{\widehat\mu^{+|-_\Lambda}}
\def\pip{\widehat\pi^+}
\def\pipm{\widehat\pi^{+|-_\Lambda}}
\def\ind{{\rm I}}
\def\const{{\rm const}}
\begin{document}
%
\bibliographystyle{plain}
%\title{\vspace*{-2.4cm}
\title{\vspace*{-2.4cm} On the set of pure states
for some systems \break with non-periodic long-range order}
\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}
\clearpage
%
\setcounter{page}{1}
\begin{abstract}
We discuss the set of pure states of some models which are ordered in a
non-periodic way. This includes both some results on a random model and
some results based on the Thue-Morse system. We discuss the possible
nature of non-periodic long-range order, and also the overlap
distribution between the pure states.
\end{abstract}
%\tableofcontents
\section{Introduction}
In this contribution we discuss some results and conceptual issues applying
to either disordered models, or to ``weak crystals'' \cite{vEMie1}.
We will always restrict ourselves to classical lattice models.
In the theory of disordered spin models, in particular in the theory of
spin-glass models, there exist a number of conceptual problems, which
are raised by the Parisi approach to the Sherrington-Kirkpatrick model.
We shall discuss some of these and related issues, including the applicability
and interpretation of these ideas for short-range models. In particular we
discuss the role of boundary conditions in obtaining pure states and in
describing the state structure for random models, and give an illustrative
example \cite{CamvE}.
Up to now there are various serious problems in making Parisi's theory precise.
In the Sherrington-Kirkpatrick model the description in terms of an infinite
number of pure states suffers from the problem that up to now one is not able
to define what a pure state is. On the other hand, an (``ultrametric'')
distribution over pure states for short-range models cannot influence a free
energy density, because the free energy density is a state-independent quantity
. As the number of mathematically precise statements one can make is so rare,
even a partial implementation of the theory in some caricature of the model is
of interest.
We find, surprisingly, a non-trivial implementation of one the ideas of the
Parisi theory, namely the existence of an overlap distribution with an
ultrametric structure between the infinitely many pure states of the system, in
the realm of substitution sequences. These sequences one can consider to be the
pure states for toy models for systems which are ordered
in a non-periodic way. This long-range order can be of the quasiperiodic type,
or even less regular. We will denote such systems which break the
translation symmetry of the interaction ``weak crystals'',
cf. Aubry's \cite{Aub1, Aub2} ``weakly periodic systems'' or Ruelle's
\cite{Rue} ``turbulent crystals''.
\section{Something about Spin-Glass models}
The spin-glass models which one usually considers have Hamiltonians
containing random interactions. In the case of Ising spins, for example, one
takes
\begin{equation}
H = \Sigma_{i,j} \; J(i,j)\sigma_{i} \sigma_{j}
\end{equation}
where the coupling constants $J(i,j)$ are symmetric random variables.
When for N spins the $J(i,j)$ are chosen $1/{\sqrt N} *X(i,j)$, with the
$X(i,j)$ i.i.d. Gaussian random variables, we obtain the
Sherrington-Kirkpatrick model \cite{SheKir}. In models with a spatial
structure, such as for example the Edwards-Anderson models, one chooses the
$J(i,j)$ from a distribution which only depends on the distance between the
lattice sites $i$ and $j$. In the case where the $J(i,j)$ are independent,
almost nothing is known rigorously, and even at a physical level of rigour
there are various controversies. One of them is about the number of
``pure states'' , which one should employ in the description of the
presumed low temperature spin-glass phase. There are two main claims in the
literature. One the one hand, based on Parisi's scheme for treating the
Sherrington-Kirkpatrick (SK-) model, there is a proposal that one needs an
infinite number of states \cite{MezParVir}, on the other hand there
is a theory proposed by Fisher and Huse \cite{FisHus1,FisHus2,FisHus3,FisHus4},
that two states should suffice in any finite-dimensional model. For some
further discussion on this issue see also
\cite{BovFro,vE1,NewSte1,NewSte2,NewSte3,NewSte4}.
In the Parisi description, the infinitely many pure states are arranged
in a hierarchical (tree-like or ``ultrametric'') way. An important
quantity in the theory is the overlap distribution between different states.
For short-range models we will interpret the pure states as either extremal
Gibbs measures, or extremal ground state measures (ground state configurations)
\cite{Geo,vEFS_JSP}.
Consider two copies---replicas--- of the equilibrium state
$\mu_{J}^1$ and $\mu_{J}^2$
of the system, with the same disorder realisation.
In the Parisi theory \cite{MezParVir} the overlap
\begin{equation}
q = lim_{N \to \infty} \; 1/N \; \Sigma_{i=1... N} \sigma_i ^1 \sigma_i ^2
\end{equation}
has a distribution with respect to the product measure $\mu_{J}^1$
$\otimes$ $\mu_{J}^2$,
called $P_{J}(q)$, which is non-trivial and also non-self-averaging, that is to
say, for a fixed disorder realization it is concentrated on a countable number
of values, while after averaging over the disorder parameter the distribution
is continuous. This averaged overlap distribution occurs in the Parisi
expression for the free energy density. There are various problems in
implementing these ideas, both for the SK-model and in short-range interaction
models.
For the SK-model it is not clear how to define the (pure) states {\it in} the
thermodynamic limit. For short-range models, on the other hand, it is known
that the free energy density is a self-averaging quantity \cite{FigPas,Led,Vui,
vHPal}, which is state-independent \cite{vEvH} and also not dependent on
boundary conditions. Thus a quantity which is the nontrivial average
of a non-self-averaging object, describing an average over different pure
states, should not be expected to show up in the free energy density.
Moreover there is the problem of dependence on boundary conditions.
Indeed, recently Newman and Stein \cite{NewSte4} have shown that the overlap
distribution and more generally, the ``canonical'' mixture of the pure states,
(which they call the ``metastate''), when it can be defined, is self-averaging
due to its translation invariance. However, their construction still allows for
a non-trivial dependence on boundary conditions. The solution of the existence
question is non-trivial, due to the fact that the obvious approach (take say
free or plus boundary conditions in finite volumes, and take the infinite
volume limit) may not lead to a well-defined thermodynamic limit, a phenomenon
described as ``chaotic size dependence'' \cite{NewSte2}.
Here we want to illustrate some of these boundary condition problems on
a model with random site disorder, the randomly gauge transformed Potts model
\cite{CamvE}.
The first issue which we will consider is the question of {\it weak} versus
{\it strong }uniqueness of the Gibbs measure for random models. If for each
pair of disorder-independently chosen sequences of boundary conditions one
approaches the same infinite-volume pure state (which is still
disorder-dependent), for almost every realization of the
disorder, we say that {\it weak} uniqueness applies. {\it Strong} uniqueness
means that there is only one pure state for almost every realization of the
disorder. If there are other pure states, which are only obtainable via
disorder-dependent boundary conditions, this means that the uniqueness is
not strong \cite{COvE,New,Zeg}.
One example where this phenomenon is known to happen is the Ising spin-glass on
the Bethe lattice, at temperatures between the spin-glass and the ferromagnetic
critical temperatures \cite{CCST,BRZ}; another example is the high temperature
regime of extremely-long-range interactions \cite{FroZeg,GanNewSte}. Here we
give an example with a nearest neighbour interaction on a regular lattice
$Z^d$, $d \geq 2$. The Hamiltonian we will consider is
\begin{equation}
H = \Sigma_{i,j} -\delta(\xi_i \sigma_i ,\xi_j\sigma_j)
\end{equation}
where the Potts spins $\sigma_i =1...q$, and the $\xi_i$ are random
permutations of the q Potts spins, chosen independently at each site of the
lattice. This model is equivalent to a Potts ferromagnet, after applying a
random gauge transformation. The pertinent observation we make here is that any
fixed boundary condition, after undoing the random gauge transformation,
becomes random. At the transition temperature, for large q, a disordered
pure state (extremal Gibbs measure) coexists with q ordered pure states.
Choosing the spins at the boundary randomly, favours the disordered
state (in which neighbouring spins typically are different) over the q ordered
ones. This statement can in fact be made rigorous. This implies for the
Hamiltonian defined above that weak uniqueness holds, but not strong
uniqueness, {\it at} the transition temperature \cite {CamvE}.
Another simple observation about this model is that while for periodic
boundary conditions in the thermodynamic limit one obtains the
equiweighted mixture of the $q+1$ pure states \cite{BorImb}, for antiperiodic
or fixed translation invariant boundary conditions ( say $\sigma = 1$ for each
boundary spin) we obtain the pure disordered state. Thus, even though
the mixture is (trivially) translation invariant and self-averaging,
it is still dependent on boundary conditions.
The maximum minus the minimum value of the overlap, taken over all pairs of
pure states, defines the Edwards-Anderson order parameter \cite{vEGri}.
This definition in the terms introduced above is a {\it strong} definition, the
example above illustrates that the weak and the strong version of this order
parameter do not necessarily coincide, a question which was left open in
\cite{vEGri}.
\section{Some Results about Weak Crystals}
In this section we will discuss a number of issues which we will
illustrate on the (Prouhet-)Thue-Morse substitution system\cite{Kea}.
We consider the sequences generated by the substitution rule
$+ \rightarrow +-$
and
$- \rightarrow -+$,
hence the sequence
$+-\; -+\; -++-\; -++-+--+\; -++-+--++--+-++- \dots $,
its translates and limits (the orbit closure) are the sequences
we will consider. They support exactly one translation-invariant probability
measure $\mu_{TM}$ (this property is called ``unique ergodicity of the
measure'') and they form a minimal translation invariant set, that is, they
give rise to a strictly ergodic \cite{Wal} system. A set is minimal
if it is closed, translation-invariant and non-empty, and has no proper subset
with this property. Physically this means that the set does not contain
configurations with defects.
We will say that $\mu_{TM}$ has long-range order because at
large distances correlations do not factorize:
\begin{equation}
lim_{ n\to \infty} \mu_{TM}(\sigma_0 \sigma_n) \neq {\mu_{TM} (\sigma_0)}^2 = 0
\end{equation}
This can be seen easily because of the fact that the correlations
are scale-invariant in the sense that
\begin{equation}
\mu_{TM} (\sigma_0 \sigma_n) = \mu_{TM} (\sigma_0 \sigma_{n*2^p})
\end{equation}
for all $n$ and $p$. In general these correlations are non-zero.
The long-range order is non-periodic, in the sense that one cannot
write $\mu_{TM}$ as an average (mixture) of periodic pure states.
Each measure in which the translation symmetry of the interaction is broken in
the sense that it is not a mixture of pure translation invariant states
and moreover has some positional long-range order in the sense that some
correlation functions do not factorize at large distances, we will call a
``(weak) crystal'' \cite{vEMie1}.
(It is possible to have
uniquely ergodic measures which have very good mixing properties, but are not
pure states. Therefore the notions of long-range order in the sense of having
long-range correlations and in the sense of being a mixture of pure states
are not necessarily equivalent. Such ``unique disordered'' measures
which also can be unique translation invariant ground states
have been discussed in \cite{Rad2}.)
The measure $\mu_{TM}$ is the {\em only} translation invariant ground state
measure of some fast-decaying 4-spin interaction $V(i,j,k,l)$ with Hamiltonian
\begin{equation}
H= \Sigma V(i,j,k,l)(\sigma_i+\sigma_j)^2 (\sigma_k + \sigma_l)^2
\end{equation}
As an aside, we remark that both the properties of being a strictly ergodic
system and having non-perodic long-range order are typical for ground states of
summable interactions \cite{Aub1,Rad1,Rad2,Mie1}.
The construction of these interactions is based on the
observation \cite{GarRadvEMie} that pairs of equal neighboring spins occur
only at even distances from each other in any Thue-Morse sequence, combined
with scale invariance. What then is needed to complete the argument is to show
that the above property actually fully characterizes the Thue-Morse system.
As for positive temperatures, it can be shown that a similar non-periodic
long-range order occurs, but for slowly decaying 4-body interactions
\cite{vEMie2}. This slow decay is unavoidable in a one-dimensional model,
it is hoped that in three or more dimensions a fast decaying interaction
will lead to the same result \cite{vEMieHolZah}.
Another way of studying non-periodic long-range order is to concentrate
on the spectral properties of the Thue-Morse system. In the physics
literature it is often claimed that there occurs purely singular
spectrum, see for example \cite{CheSav}, although mathematically the
description of the Thue-Morse system, considered as a dynamical system in the
sense of ergodic theory, gives rise to a mixed discrete/singular spectrum. The
explanation in physics terms of this apparent contradiction is in that if one
considers the Thue-Morse system as a lattice gas model there is a distinction
between the atomic spectrum which is singular and the molecular spectrum which
can be discrete. To be precise, the Fourier transform of
$\mu_{TM}(\sigma_0 \sigma_n)$ is a singular continuous measure ( the atomic
diffraction spectrum or structure factor), while the Fourier transform of
$\mu(X_0 X_n)$ is countable discrete (the molecular diffraction spectrum), if
for example $X_i = \sigma_0 \sigma_{i+1}$. Thus the ``molecule'' consists
of 2 equal neighbouring atoms.
As remarked before, the Thue-Morse sequences are pure ground
states for some translation-invariant interaction. The translation-invariant
measure on them is the ``canonical'' mixture ( in the language of \cite
{NewSte4} the metastate). It is hence possible to study the overlap
distribution with respect to this measure between the different pure states
\cite{vEHofMie}.
The first result is for the Thue-Morse overlap distribution:
\begin{equation}
lim_{N\to \infty} (1/N \ \Sigma_i \sigma_{i}^1 \sigma_{i}^2) =0
\end{equation}
almost surely with respect to the product measure $\mu_{TM} \otimes \mu_{TM}$.
The fact that the overlap distribution is trivial follows from a well-known
argument in ergodic theory, which needs a slight modification in our case.
The original result (see for example Walters \cite{Wal}) shows that if for a
measure $\mu$ the translation operator has no point spectrum, that is, the
measure is weak mixing, the product measure $\mu \otimes \mu$ is ergodic, and
hence the overlap distribution, because it is translation invariant, is a
constant. As a side remark we note that the property of weak mixing is in fact
compatible with the existence of long-range order, the absence of long-range
order is a strictly stronger mixing property. The Thue-Morse system behaves
like a weakly mixing system on
observables which are odd under a global spin-flip \cite{Kea}.
The second result is for the ``molecular'' overlap distribution of the
``molecules'' $X_i$ = $\sigma_i \sigma_{i+1}$ . These molecules form a
so-called Toeplitz system \cite{JacKea}.
A Toeplitz sequence can be constructed by first taking half of the lattice
sites (with probability 1/2 the sites with even coordinates, with probability
1/2 the sites with odd coordinates) and occupying them with $-1$.
>From the as yet unoccupied sublattice we occupy one of the two period-four
sublattices with pluses; again, which one of the two sublattices becomes
occupied, is chosen randomly. We repeat this procedure, putting in pluses and
minuses alternatingly at each step. The Toeplitz system is known to be
almost periodic in the sense that the translation operator has
pure point spectrum, with infinitely many generators \cite{JacKea}.
The overlap distribution is concentrated on a countable number of values, and
the pure states (the Toeplitz sequences) are organized in an ultrametric way.
The overlap between two sequences depends only on which is the first step at
which the choice of the sublattice to be occupied
is different. This result is fully consistent with the observation of
Newman and Stein \cite{NewSte4} that the (non-)triviality of the overlap
distribution has nothing to do with non-self-averaging; indeed, because
there is no randomness here, self-averaging is trivially satisfied,
but the self-averaging overlap distribution is non-trivial, and in fact
even ultrametric.
As a side remark we mention that also for the Fibonacci system (which is
quasiperiodic) the overlap distribution can be computed, and that it has a
continuous part \cite{vEHofMie}.
%In \cite{vEFS_PRL,vEFS_JSP,vEFK_JSP} it was shown how various
%\begin{equation}
%\ H = \Sigma_{*} \; -\sigma(i) \sigma(j)
%\end{equation}
%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 $T_L$
%{\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.
%{\em Proof}: The proof follows the scheme of \cite{vEFS_JSP}, section 4.2.
{\em Acknowledgements}: I thank the organizers, in particular J. Jedrzejewski,
for inviting me to a very enjoyable conference. Various parts of this research
were done in collaboration with Massimo Campanino, Clifford Gardner, Bob
Griffiths, Leo van Hemmen, Bert Hof, Jacek Mi{\c e}kisz, Enzo Olivieri and
Charles Radin. I am very grateful to them all for these collaborations and for
all they taught me. I thank Jacek Mi{\c e}kisz also for useful suggestions on
the manuscript.
Moreover I thank Chuck Newman for many useful discussions and suggestions, and
for making his work with Dan Stein available before publication. Part of this
work was supported by a fellowship of the KNAW (Royal Dutch Academy of Arts and
Sciences). 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{Aub1}
S.~Aubry.
\newblock Devil's staircase and order without periodicity in classical
condensed matter.
\newblock {\em J. Phys (Paris)}, 44:147-162, 1983.
\bibitem{Aub2}
S.~Aubry.
\newblock Weakly periodic structures and example.
\newblock {\em J. Phys. (Paris) Coll.} 50 {\bf C} 3: 97-106, 1989.
\bibitem{BRZ}
P.~M.~Bleher, J.~Ruiz and V.~A.~Zagrebnov.
\newblock On the purity of the limiting {G}ibbs state for the {I}sing model
on the Bethe lattice.
\newblock {\em J. Stat. Phys.}, 79:473-482, 1995.
\bibitem{BorImb}
C.~Borgs and J.~Z.~Imbrie.
\newblock A unified approach to phase diagrams in field theory and statistical
mechanics.
\newblock {\em Comm. Math. Phys.}, 123:305-328, 1989.
\bibitem{BovFro}
A.~Bovier and J.~Fr\"ohlich.
\newblock A heuristic theory of the spin glass phase.
\newblock {\em J. Stat. Phys.}, 44:347-391, 1986.
\bibitem{CamvE}
M.~Campanino and A.~C.~D.~van Enter.
\newblock Weak versus strong uniqueness of {G}ibbs measures: a regular
short-range example.
\newblock {\em J. Phys. A, Math. Gen.}, 28:L45-47, 1995.
\bibitem{COvE}
M.~Campanino, E.~Olivieri and A.~C.~D.~van Enter.
\newblock One-dimensional spin-glasses with decay $1/r^{(1 + \epsilon)}$.
\newblock {\em Comm. Math. Phys.}, 108:241-255, 1987.
\bibitem{CCST}
J.~T.~Chayes, L.~Chayes, J.~P.~Sethna and D.~J.~Thouless.
\newblock A mean field spin glass with short-range interactions.
\newblock {\em Comm. Math. Phys.}, 106:41-89, 1986.
\bibitem{CheSav}
Z.~Cheng and R.~Savit.
\newblock Structure factor of substitutional sequences.
\newblock {\em J. Stat. Phys.}, 60:383-393, 1990.
\bibitem{vE1}
A.~C.~D.~van Enter.
\newblock Stiffness exponent, number of pure states and Almeida-Thouless line
in spin-glasses.
\newblock {\em J. Stat. Phys.}, 60:275-279, 1990.
\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 Gibbsian
theory.
\newblock {\em J. Stat. Phys.}, 72:879-1167, 1993.
\bibitem{vEGri}
A.~C.~D.~van Enter and R.~B.~Griffiths.
\newblock The order parameter in a spin glass.
\newblock {\em Comm. Math. Phys.}, 90:319-327, 1983.
\bibitem{vEHofMie}
A.~C.~D.~van Enter, A.~Hof and J.~Mi\c ekisz.
\newblock Overlap distributions for deterministic systems with many states.
\newblock {\em J. Phys. A, Math. Gen.}, 25:L1133-1137, 1992.
\bibitem{vEMie1}
A.~C.~D.~van Enter and J.~Mi\c ekisz.
\newblock How should one define a (weak) crystal?
\newblock {\em J. Stat. Phys.}, 66:1147-1153, 1992.
\bibitem{vEMie2}
A.~C.~D.~van Enter and J.~Mi\c ekisz.
\newblock Breaking of periodicity at positive temperatures.
\newblock {\em Comm. Math. Phys.}, 134:647-651, 1990.
\bibitem{vEMieHolZah}
A.~C.~D.~van Enter, J.~Mi\c ekisz, P.~Holick\'y and M.~Zahradn\'ik.
\newblock In preparation.
\bibitem{vEvH}
A.~C.~D.~van Enter and J.~L.~van Hemmen.
\newblock Statistical mechanical formalism for spin glasses.
\newblock {\em Phys. Rev. A}, 29:355-365, 1984.
\bibitem{FigPas}
A.~L.~Figotin and L.~A.~Pastur.
\newblock Theory of disordered spin systems.
\newblock {\em Theor. Math. Phys.}, 35:403-414, 1978.
\bibitem{FisHus1}
D.~S.~Fisher and D.~A.~Huse.
\newblock Absence of many states in realistic spin glasses.
\newblock {\em J. Phys. A, Math. Gen.}, 20:L1005-1010, 1987.
\bibitem{FisHus2}
D.~S.~Fisher and D.~A.~Huse.
\newblock Pure states in spin glasses.
\newblock {\em J. Phys. A, Math. Gen.}, 20:L997-1003, 1987.
\bibitem{FisHus3}
D.~S.~Fisher and D.~A.~Huse.
\newblock Ordered phase of short-range {I}sing spin-glasses.
\newblock {\em Phys. Rev. Lett.}, 51:1601-1604, 1986.
\bibitem{FisHus4}
D.~S.~Fisher and D.~A.~Huse.
\newblock Equilibrium behavior of the spin-glass ordered phase.
\newblock {\em Phys. Rev. B}, 38:386-411, 1988.
\bibitem{FroZeg}
J.~Fr{\"o}hlich and B.~Zegarlinski.
\newblock The high-temperature phase of long-range spin glasses.
\newblock{\it Comm. Math. Phys.}, 110:121-155, 1987.
\bibitem{GanNewSte}
A.~Gandolfi, C.~M.~Newman and D.~L.~Stein.
\newblock Exotic states in long-range spin glasses.
\newblock{\em Comm. Math. Phys.}, 157:371-387, 1993.
\bibitem{GarRadvEMie}
C.~Gardner, J.~Mi\c ekisz, C.~Radin, A.~C.~D.~van Enter.
\newblock Fractal symmetry in an Ising model.
\newblock{\em J. Phys. A, Math. Gen.}, 22:L1019-1023, 1989.
\bibitem{Geo}
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{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{vHPal}
J.~L.~van Hemmen and R.~G.~Palmer.
\newblock The thermodynamic limit and the replica method for short range
random systems.
\newblock {\em J. Phys. A, Math. Gen.}, 15:3881-3890, 1982.
\bibitem{JacKea}
K.~Jacobs and M.~Keane.
\newblock $0-1$ Sequences of Toeplitz type.
\newblock {\em Z. Wahrs. u. verw. Geb.}, 13:123-131, 1969.
\bibitem{Kea}
M.~Keane.
\newblock Generalized Morse sequences.
\newblock {\em Z. Wahrs. u. verw. Geb.}, 10:335-353, 1968.
\bibitem{Led}
F.~Ledrappier.
\newblock Pressure and variational principle for random {I}sing model.
\newblock {\em Comm. Math. Phys.} 56:297-302, 1977.
\bibitem{MezParVir}
M.~M{\'e}zard, G.~Parisi and M.~A.~Virasoro.
\newblock {\em Spin Glass Theory and Beyond}.
\newblock World Scientific, Singapore, 1987.
\bibitem{Mie1}
J.~Mi{\c e}kisz.
\newblock {\em Quasicrystals}.
\newblock Leuven Lecture Notes in Mathematical and Theoretical Physics,
Vol.5, 1993.
\bibitem{New}
C.~M.~Newman.
\newblock Ising systems and random cluster representations.
\newblock In G.~Grimmett, editor, {\em Probability and Phase Transition,
Cambridge, 1993}, pages 247-260. Kluwer, Dordrecht, 1994.
\bibitem{NewSte1}
C.~M.~Newman and D.~L.~Stein.
\newblock Multiple states and thermodynamic limits in short-range {I}sing
spin-glass models.
\newblock {\em Phys. Rev. B}, 46:973-982, 1992.
\bibitem{NewSte2}
C.~M.~Newman and D.~L.~Stein.
\newblock Ground state structure of a highly disordered spin glass model.
\newblock {\em J. Stat. Phys.} to appear, 1996, see also {\em Phys. Rev. Lett.}
72:2286-2289, 1994.
\bibitem{NewSte3}
C.~M.~Newman and D.~L.~Stein.
\newblock Non-mean-field behavior of realistic spin glasses.
\newblock Preprint, 1995.
\bibitem{NewSte4}
C.~M.~Newman and D.~L.~Stein.
\newblock Spatial inhomogeneity and thermodynamic chaos.
\newblock Preprint, 1995.
\bibitem{Rad1}
C.~Radin.
\newblock Correlations in classical ground states.
\newblock {\em J. Stat. Phys.}, 43:707-712, 1986.
\bibitem{Rad2}
C.~Radin.
\newblock Disordered ground states of classical lattice models.
\newblock {\em Rev. Math. Phys.}, 3:125-135, 1991.
\bibitem{Rue}
D.~Ruelle.
\newblock Do turbulent crystals exist?
\newblock {\em Physica}, 113A:619-623, 1982.
\bibitem {SheKir}
D.~Sherrington and S.~Kirkpatrick.
\newblock Solvable model of a spin glass.
\newblock {\em Phys. Rev. Lett.}, 35:1792-1795, 1975.
\bibitem{Vui}
P.~A.~Vuillermot.
\newblock Thermodynamics of quenched random spin sysyems, and application
to the problem of phase transitions in magnetic (spin) glasses.
\newblock {\em J. Phys. A}, 10:1319-1333, 1977.
%\bibitem{lorwin}
%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, Dordrecht, 1993.
% Kluwer.
\bibitem{Wal}
P.~Walters.
\newblock {\em An introduction to ergodic theory}.
\newblock Springer Graduate texts in mathematics 79; Springer,
New York-Heidelberg-Berlin, 1982.
%\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{Zeg}
B.~Zegarlinski.
\newblock Spin systems with long-range interactions.
\newblock {\em Rev. Math. Phys.} 6:115-134, 1994.
\end{thebibliography}
\end{document}
*