Content-Type: multipart/mixed; boundary="-------------0502080847302"
This is a multi-part message in MIME format.
---------------0502080847302
Content-Type: text/plain; name="05-57.comments"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="05-57.comments"
Contribution for a Festschrift on the occasionof the 65th birtday of Mike Keane.
---------------0502080847302
Content-Type: text/plain; name="05-57.keywords"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="05-57.keywords"
metatstates, incoherent boundary conditions, Ising models.
---------------0502080847302
Content-Type: application/x-tex; name="ensk7a.tex"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="ensk7a.tex"
\documentclass[12pt]{article}
\textheight=23cm
\textwidth=16cm
\topmargin -2cm
\oddsidemargin -0.2cm
% ----------------------------------------------------------------
\usepackage[centertags]{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{newlfont}
\usepackage{stmaryrd}
\usepackage{mathrsfs}
\usepackage{pstricks,pst-node}
\usepackage{palatino} %palatcm - a missing package
\usepackage{fancyhdr}
\usepackage{graphicx}
\usepackage{fancybox}
\usepackage{multibox}
\usepackage{geometry}
\usepackage[dvips]{epsfig}
\usepackage{eepic}
\usepackage{pst-plot}
%\usepackage{makra}
% \MATHOPERATOR -----------------------------------------------------
\DeclareMathOperator{\ex}{Ex} \DeclareMathOperator{\ext}{Ext}
\DeclareMathOperator{\Int}{Int} \DeclareMathOperator{\supp}{Supp}
\DeclareMathOperator{\tr}{Tr}
\newcommand{\re}{\Re\,}
\newcommand{\im}{\Im\,}
\newcommand{\HH}{\mathsf{H}}
\newcommand{\PP}{\mathsf{P}}
\newcommand{\RR}{\mathsf{R}}
\newcommand{\EE}{\mathsf{E}\,}
\newcommand{\FF}{\mathsf{F}}
% GREEK - 2 letters ------------------------------------------------
\let\al=\alpha \let\be=\beta \let\de=\delta \let\ep=\epsilon
\let\ve=\varepsilon \let\vp=\varphi \let\ga=\gamma \let\io=\iota
\let\ka=\kappa \let\la=\lambda \let\om=\omega \let\vr=\varrho
\let\si=\sigma \let\vs=\varsigma \let\th=\theta \let\vt=\vartheta
\let\ze=\zeta \let\up=\upsilon \let\vk=\varkappa
\let\De=\Delta \let\Ga=\Gamma \let\La=\Lambda \let\Om=\Omega
\let\Th=\Theta
% \MATHCAL - \ca ----------------------------------------------------
\newcommand{\caA}{{\mathcal A}}
\newcommand{\caB}{{\mathcal B}}
\newcommand{\caC}{{\mathcal C}}
\newcommand{\caD}{{\mathcal D}}
\newcommand{\caE}{{\mathcal E}}
\newcommand{\caF}{{\mathcal F}}
\newcommand{\caG}{{\mathcal G}}
\newcommand{\caH}{{\mathcal H}}
\newcommand{\caI}{{\mathcal I}}
\newcommand{\caJ}{{\mathcal J}}
\newcommand{\caK}{{\mathcal K}}
\newcommand{\caL}{{\mathcal L}}
\newcommand{\caM}{{\mathcal M}}
\newcommand{\caN}{{\mathcal N}}
\newcommand{\caO}{{\mathcal O}}
\newcommand{\caP}{{\mathcal P}}
\newcommand{\caQ}{{\mathcal Q}}
\newcommand{\caR}{{\mathcal R}}
\newcommand{\caS}{{\mathcal S}}
\newcommand{\caT}{{\mathcal T}}
\newcommand{\caU}{{\mathcal U}}
\newcommand{\caV}{{\mathcal V}}
\newcommand{\caW}{{\mathcal W}}
\newcommand{\caX}{{\mathcal X}}
\newcommand{\caY}{{\mathcal Y}}
\newcommand{\caZ}{{\mathcal Z}}
% \MATHCAL - \sc ----------------------------------------------------
\newcommand{\scA}{{\mathscr A}}
\newcommand{\scB}{{\mathscr B}}
\newcommand{\scC}{{\mathscr C}}
\newcommand{\scD}{{\mathscr D}}
\newcommand{\scE}{{\mathscr E}}
\newcommand{\scF}{{\mathscr F}}
\newcommand{\scG}{{\mathscr G}}
\newcommand{\scH}{{\mathscr H}}
\newcommand{\scI}{{\mathscr I}}
\newcommand{\scJ}{{\mathscr J}}
\newcommand{\scK}{{\mathscr K}}
\newcommand{\scL}{{\mathscr L}}
\newcommand{\scM}{{\mathscr M}}
\newcommand{\scN}{{\mathscr N}}
\newcommand{\scO}{{\mathscr O}}
\newcommand{\scP}{{\mathscr P}}
\newcommand{\scQ}{{\mathscr Q}}
\newcommand{\scR}{{\mathscr R}}
\newcommand{\scS}{{\mathscr S}}
\newcommand{\scT}{{\mathscr T}}
\newcommand{\scU}{{\mathscr U}}
\newcommand{\scV}{{\mathscr V}}
\newcommand{\scW}{{\mathscr W}}
\newcommand{\scX}{{\mathscr X}}
\newcommand{\scY}{{\mathscr Y}}
\newcommand{\scZ}{{\mathscr Z}}
\newcommand{\sca}{{\mathscr a}}
\newcommand{\scb}{{\mathscr b}}
\newcommand{\scc}{{\mathscr c}}
\newcommand{\scd}{{\mathscr d}}
\newcommand{\sce}{{\mathscr e}}
\newcommand{\scf}{{\mathscr f}}
\newcommand{\scg}{{\mathscr g}}
\newcommand{\sch}{{\mathscr h}}
\newcommand{\sci}{{\mathscr i}}
\newcommand{\scj}{{\mathscr j}}
\newcommand{\sck}{{\mathscr k}}
\newcommand{\scl}{{\mathscr l}}
\newcommand{\scm}{{\mathscr m}}
\newcommand{\scn}{{\mathscr n}}
\newcommand{\sco}{{\mathscr o}}
\newcommand{\scp}{{\mathscr p}}
\newcommand{\scq}{{\mathscr q}}
\newcommand{\scr}{{\mathscr r}}
\newcommand{\scs}{{\mathscr s}}
\newcommand{\sct}{{\mathscr t}}
\newcommand{\scu}{{\mathscr u}}
\newcommand{\scv}{{\mathscr v}}
\newcommand{\scw}{{\mathscr w}}
\newcommand{\scx}{{\mathscr x}}
\newcommand{\scy}{{\mathscr y}}
\newcommand{\scz}{{\mathscr z}}
% \MATHBB - \bb -----------------------------------------------------
\newcommand{\bba}{{\mathbb a}}
\newcommand{\bbb}{{\mathbb b}}
\newcommand{\bbc}{{\mathbb c}}
\newcommand{\bbd}{{\mathbb d}}
\newcommand{\bbe}{{\mathbb e}}
\newcommand{\bbf}{{\mathbb f}}
\newcommand{\bbg}{{\mathbb g}}
\newcommand{\bbh}{{\mathbb h}}
\newcommand{\bbi}{{\mathbb i}}
\newcommand{\bbj}{{\mathbb j}}
\newcommand{\bbk}{{\mathbb k}}
\newcommand{\bbl}{{\mathbb l}}
\newcommand{\bbm}{{\mathbb m}}
\newcommand{\bbn}{{\mathbb n}}
\newcommand{\bbo}{{\mathbb o}}
\newcommand{\bbp}{{\mathbb p}}
\newcommand{\bbq}{{\mathbb q}}
\newcommand{\bbr}{{\mathbb r}}
\newcommand{\bbs}{{\mathbb s}}
\newcommand{\bbt}{{\mathbb t}}
\newcommand{\bbu}{{\mathbb u}}
\newcommand{\bbv}{{\mathbb v}}
\newcommand{\bbw}{{\mathbb w}}
\newcommand{\bbx}{{\mathbb x}}
\newcommand{\bby}{{\mathbb y}}
\newcommand{\bbz}{{\mathbb z}}
\newcommand{\bbA}{{\mathbb A}}
\newcommand{\bbB}{{\mathbb B}}
\newcommand{\bbC}{{\mathbb C}}
\newcommand{\bbD}{{\mathbb D}}
\newcommand{\bbE}{{\mathbb E}}
\newcommand{\bbF}{{\mathbb F}}
\newcommand{\bbG}{{\mathbb G}}
\newcommand{\bbH}{{\mathbb H}}
\newcommand{\bbI}{{\mathbb I}}
\newcommand{\bbJ}{{\mathbb J}}
\newcommand{\bbK}{{\mathbb K}}
\newcommand{\bbL}{{\mathbb L}}
\newcommand{\bbM}{{\mathbb M}}
\newcommand{\bbN}{{\mathbb N}}
\newcommand{\bbO}{{\mathbb O}}
\newcommand{\bbP}{{\mathbb P}}
\newcommand{\bbQ}{{\mathbb Q}}
\newcommand{\bbR}{{\mathbb R}}
\newcommand{\bbS}{{\mathbb S}}
\newcommand{\bbT}{{\mathbb T}}
\newcommand{\bbU}{{\mathbb U}}
\newcommand{\bbV}{{\mathbb V}}
\newcommand{\bbW}{{\mathbb W}}
\newcommand{\bbX}{{\mathbb X}}
\newcommand{\bbY}{{\mathbb Y}}
\newcommand{\bbZ}{{\mathbb Z}}
\newcommand{\opunit}{\text{1}\kern-0.22em\text{l}}
\newcommand{\funit}{\mathbf{1}}
% \MATHFRAK - \fr ---------------------------------------------------
\newcommand{\fra}{{\mathfrak a}}
\newcommand{\frb}{{\mathfrak b}}
\newcommand{\frc}{{\mathfrak c}}
\newcommand{\frd}{{\mathfrak d}}
\newcommand{\fre}{{\mathfrak e}}
\newcommand{\frf}{{\mathfrak f}}
\newcommand{\frg}{{\mathfrak g}}
\newcommand{\frh}{{\mathfrak h}}
\newcommand{\fri}{{\mathfrak i}}
\newcommand{\frj}{{\mathfrak j}}
\newcommand{\frk}{{\mathfrak k}}
\newcommand{\frl}{{\mathfrak l}}
\newcommand{\frm}{{\mathfrak m}}
\newcommand{\frn}{{\mathfrak n}}
\newcommand{\fro}{{\mathfrak o}}
\newcommand{\frp}{{\mathfrak p}}
\newcommand{\frq}{{\mathfrak q}}
\newcommand{\frr}{{\mathfrak r}}
\newcommand{\frs}{{\mathfrak s}}
\newcommand{\frt}{{\mathfrak t}}
\newcommand{\fru}{{\mathfrak u}}
\newcommand{\frv}{{\mathfrak v}}
\newcommand{\frw}{{\mathfrak w}}
\newcommand{\frx}{{\mathfrak x}}
\newcommand{\fry}{{\mathfrak y}}
\newcommand{\frz}{{\mathfrak z}}
\newcommand{\frA}{{\mathfrak A}}
\newcommand{\frB}{{\mathfrak B}}
\newcommand{\frC}{{\mathfrak C}}
\newcommand{\frD}{{\mathfrak D}}
\newcommand{\frE}{{\mathfrak E}}
\newcommand{\frF}{{\mathfrak F}}
\newcommand{\frG}{{\mathfrak G}}
\newcommand{\frH}{{\mathfrak H}}
\newcommand{\frI}{{\mathfrak I}}
\newcommand{\frJ}{{\mathfrak J}}
\newcommand{\frK}{{\mathfrak K}}
\newcommand{\frL}{{\mathfrak L}}
\newcommand{\frM}{{\mathfrak M}}
\newcommand{\frN}{{\mathfrak N}}
\newcommand{\frO}{{\mathfrak O}}
\newcommand{\frP}{{\mathfrak P}}
\newcommand{\frQ}{{\mathfrak Q}}
\newcommand{\frR}{{\mathfrak R}}
\newcommand{\frS}{{\mathfrak S}}
\newcommand{\frT}{{\mathfrak T}}
\newcommand{\frU}{{\mathfrak U}}
\newcommand{\frV}{{\mathfrak V}}
\newcommand{\frW}{{\mathfrak W}}
\newcommand{\frX}{{\mathfrak X}}
\newcommand{\frY}{{\mathfrak Y}}
\newcommand{\frZ}{{\mathfrak Z}}
% \BOLDSYMBOL - \bs -------------------------------------------------
\newcommand{\bsa}{{\boldsymbol a}}
\newcommand{\bsb}{{\boldsymbol b}}
\newcommand{\bsc}{{\boldsymbol c}}
\newcommand{\bsd}{{\boldsymbol d}}
\newcommand{\bse}{{\boldsymbol e}}
\newcommand{\bsf}{{\boldsymbol f}}
\newcommand{\bsg}{{\boldsymbol g}}
\newcommand{\bsh}{{\boldsymbol h}}
\newcommand{\bsi}{{\boldsymbol i}}
\newcommand{\bsj}{{\boldsymbol j}}
\newcommand{\bsk}{{\boldsymbol k}}
\newcommand{\bsl}{{\boldsymbol l}}
\newcommand{\bsm}{{\boldsymbol m}}
\newcommand{\bsn}{{\boldsymbol n}}
\newcommand{\bso}{{\boldsymbol o}}
\newcommand{\bsp}{{\boldsymbol p}}
\newcommand{\bsq}{{\boldsymbol q}}
\newcommand{\bsr}{{\boldsymbol r}}
\newcommand{\bss}{{\boldsymbol s}}
\newcommand{\bst}{{\boldsymbol t}}
\newcommand{\bsu}{{\boldsymbol u}}
\newcommand{\bsv}{{\boldsymbol v}}
\newcommand{\bsw}{{\boldsymbol w}}
\newcommand{\bsx}{{\boldsymbol x}}
\newcommand{\bsy}{{\boldsymbol y}}
\newcommand{\bsz}{{\boldsymbol z}}
\newcommand{\bsA}{{\boldsymbol A}}
\newcommand{\bsB}{{\boldsymbol B}}
\newcommand{\bsC}{{\boldsymbol C}}
\newcommand{\bsD}{{\boldsymbol D}}
\newcommand{\bsE}{{\boldsymbol E}}
\newcommand{\bsF}{{\boldsymbol F}}
\newcommand{\bsG}{{\boldsymbol G}}
\newcommand{\bsH}{{\boldsymbol H}}
\newcommand{\bsI}{{\boldsymbol I}}
\newcommand{\bsJ}{{\boldsymbol J}}
\newcommand{\bsK}{{\boldsymbol K}}
\newcommand{\bsL}{{\boldsymbol L}}
\newcommand{\bsM}{{\boldsymbol M}}
\newcommand{\bsN}{{\boldsymbol N}}
\newcommand{\bsO}{{\boldsymbol O}}
\newcommand{\bsP}{{\boldsymbol P}}
\newcommand{\bsQ}{{\boldsymbol Q}}
\newcommand{\bsR}{{\boldsymbol R}}
\newcommand{\bsS}{{\boldsymbol S}}
\newcommand{\bsT}{{\boldsymbol T}}
\newcommand{\bsU}{{\boldsymbol U}}
\newcommand{\bsV}{{\boldsymbol V}}
\newcommand{\bsW}{{\boldsymbol W}}
\newcommand{\bsX}{{\boldsymbol X}}
\newcommand{\bsY}{{\boldsymbol Y}}
\newcommand{\bsZ}{{\boldsymbol Z}}
\newcommand{\bsal}{{\boldsymbol \alpha}}
\newcommand{\bsbe}{{\boldsymbol \beta}}
\newcommand{\bsga}{{\boldsymbol \gamma}}
\newcommand{\bsde}{{\boldsymbol \delta}}
\newcommand{\bsep}{{\boldsymbol \epsilon}}
\newcommand{\bsmu}{{\boldsymbol \mu}}
\newcommand{\bsrho}{{\boldsymbol \rho}}
\newcommand{\bsom}{{\boldsymbol \omega}}
\newcommand{\bsLa}{{\boldsymbol \Lambda}}
\newcommand{\bsGa}{{\boldsymbol \Gamma}}
\newcommand{\bsDe}{{\boldsymbol \Delta}}
% ABBREVIATION ------------------------------------------------------
\newcommand{\fig}{Fig.\;}
\newcommand{\cf}{cf.\;}
\newcommand{\eg}{e.g.\;}
\newcommand{\ie}{i.e.\;}
% MISCELLANEOUS -----------------------------------------------------
\newcommand{\un}[1]{\underline{#1}}
\newcommand{\defin}{\stackrel{\text{def}}{=}}
\newcommand{\bound}{\partial}
\newcommand{\sbound}{\hat{\partial}}
\newcommand{\rel}{\,|\,}
\newcommand{\pnt}{\rightsquigarrow}
\newcommand{\pa}{_\bullet}
\newcommand{\nb}[1]{\marginpar{\tiny {#1}}}
\newcommand{\pair}[1]{\langle{#1}\rangle}
\newcommand{\0}{^{(0)}}
\newcommand{\1}{^{(1)}}
\newcommand{\con}{_{\text{con}}}
\newcommand{\diam}{\text{diam\,}}
%\newcommand{\dom}{\text{dom}\,}
\newcommand{\support}{\text{supp}\,}
\newcommand{\triple}{|\hspace{-0.6mm}|\hspace{-0.6mm}|}
\newcommand{\id}{\mathrm{d}}
\newcommand{\idv}{\id p \id q\;}
\newcommand{\dL}{\partial\La}
\newcommand{\sm}{_{\text{small}}}
\DeclareMathOperator{\dist}{dist}
\DeclareMathOperator{\Inn}{Int}
\DeclareMathOperator{\inn}{int} \DeclareMathOperator{\Ext}{Ext}
\DeclareMathOperator{\Ran}{Ran}
\DeclareMathOperator*{\limi}{\ul{lim}}
\DeclareMathOperator*{\lims}{\ol{lim}}
\DeclareMathOperator{\co}{con} \DeclareMathOperator{\free}{free}
\DeclareMathOperator{\card}{card}
\DeclareMathOperator{\var}{Var}
\DeclareMathOperator{\cn}{Con}
\DeclareMathOperator{\dom}{Dom}
\DeclareMathOperator{\cst}{const}
\newcommand{\print}[1]{${#1} \quad
{\mathcal {#1}} \quad
{\mathfrak {#1}} \quad
{\mathbb {#1}} \quad
{\boldsymbol {#1}}$ \newline}
\DeclareMathOperator{\prb}{Prob}
%\DeclareMathOperator{\Exp}{\mathsf{E}}
\DeclareMathOperator{\sign}{Sign}
\DeclareMathOperator{\pref}{Pref}
%\newtheorem{lemma}{Lemma}[section]
%\newtheorem{proposition}{Proposition}[section]
%\newtheorem{theorem}{Theorem}[section]
%\newtheorem{definition}{Definition}[section]
%\newtheorem{corollary}{Corollary}[section]
\newcommand{\bde}{\begin{definition}}
\newcommand{\ede}{\end{definition}}
\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\ben}{\begin{enumerate}}
\newcommand{\een}{\end{enumerate}}
\newcommand{\ble}{\begin{lemma}}
\newcommand{\ele}{\end{lemma}}
\newcommand{\bpr}{\begin{proof}}
\newcommand{\epr}{\end{proof}}
\newcommand{\bet}{\tilde\be}
\newcommand{\lra}{\leftrightarrow}
\newcommand{\comp}{\sim}
\newcommand{\inc}{\not\sim}
\newcommand{\st}{\star}
\renewcommand{\pa}{\partial}
\renewcommand{\1}{\mbox{{\bf 1}}}
\DeclareMathOperator{\cov}{Cov}
\newcommand{\kn}{\textbf{\emph{KN\ }}}
% ---------------------------------------------------------------
\title{Incoherent boundary conditions and metastates}
\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 Hendrikjan G.~Schaap} \\[-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 H.G.Schaap@phys.rug.nl} \\[-1mm]
\\ [-1mm] \and
{\normalsize Karel Neto\v{c}n\'y}\\[-1.5mm]
{\normalsize\it Institute of Physics} \\[-1.5mm]
{\normalsize\it Academy of Sciences of the Czech Republic} \\[-1.5mm]
{\normalsize\it Na Slovance 2} \\[-1.5mm]
{\normalsize\it 182 21 Prague 8} \\[-1.5mm]
{\normalsize\it CZECH REPUBLIC} \\[-1mm]
{\normalsize\tt netocny@fzu.cz} \\[-1mm]
{\protect\makebox[5in]{\quad}}}
\pagenumbering{arabic}
\begin{document}
\maketitle \baselineskip=14pt \noindent {\bf Abstract.} In this
contribution we discuss the role which incoherent boundary
conditions can play in the study of phase transitions. This is a
question of particular relevance for the analysis of disordered
systems, and in particular spin glasses. For the moment our
mathematical results only apply to ferromagnetic models which have
an exact symmetry between low-temperature phases.
%We discuss
%them, and what it might imply for the situation where many pure
%states can coexist. We will also discuss our results in the
%Newman-Stein language of metastates. \emph{\textbf{How about
%this:}
We give a survey of these results and discuss possibilities
to extend them to some situations where many pure states can coexist.
An idea of the proofs as well as the reformulation of our results in
the language of Newman-Stein metastates are also presented.
\newtheorem{theorem}{Theorem} % Numbering by sections
\newtheorem{lemma}{Lemma} % Number all in one sequence
\newtheorem{proposition}[lemma]{Proposition}
\newtheorem{corollary}[lemma]{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})}
\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^{+|-_\La\bibitem{mi}
\newblock An ultimate frustration in classical lattice-gas models.
\newblock To appear in {\em J. Stat. Phys.}
mbda}}
\def\ind{{\rm I}}
\def\const{{\rm const}}
\bibliographystyle{plain}
%\begin{document}
\maketitle
\section{Introduction}
In the theory of Edwards-Anderson (short-range, independent-bond Ising-spin)
spin-glass models, a long-running controversy
exists about the nature of the spin-glass phase, and in particular about
the possibility of infinitely pure states coexisting. Whereas on the one hand
there is a school which, inspired by Parisi's \cite{par2, par1} famous
(and now rigorously justified \cite{tal1, tal2}) solution of the
Sherrington-Kirkpatrick equivalent-neighbour model, predicts that
infinitely many pure states (= extremal Gibbs measures, we will use both
terms interchangably) can coexist, the other extreme, the droplet model of
Fisher and Huse, predicts that only two pure states can exist at low
temperature, in any dimension \cite{FH}.
An intermediate, and mathematically more responsible, position was developed
by Newman and Stein, who have analyzed a number of properties which
a situation with infinitely many pure states should imply
\cite {N,NS1,NS2,NS3,NS4,NS5}.
One aspect which is of particular relevance for interpreting
numerical work is the fact that the commonly used periodic or
antiperiodic boundary conditions might prefer different pairs of pure states.
Which one then would depend on the
disorder realization and on the (realization-dependent) volume,
a scenario described by them as ``chaotic pairs'' (see also \cite{vE}).
This is a particular example of their notion of ``chaotic size-dependence'',
the phenomenon that the Gibbs measures on an increasing
sequence of volumes may fail to converge almost surely (in the weak topology)
to a thermodynamic limit. Such a situation may occur when the boundary
conditions are not biased towards a particular phase (or a particular
set of phases), that is they are not {\em coherent}.
If such pointwise convergence fails, a weaker type of probabilistic
convergence, e.g. convergence in distribution, may still be
possible. Such a convergence has as its limit objects ``metastates'',
distributions on the set of all possible Gibbs measures. (These objects
are measures on all possible Gibbs measures, including the non-extremal ones.
The metastate approach should be distinguished from the more commonly known
fact that all Gibbs measures are convex combinations -mixtures- of extremal
Gibbs measures.) For a recent description of the
theory of metastates see also \cite{bov1,bov2}. Although our understanding
of spin glasses is still not sufficient to have many specific results,
the metastate theory has been worked out for a number of models, mostly
of mean-field type \cite{BEN,BG,ES,Ku1,Ku2}.
Recently we have developed it for the simple case of the ferromagnetic
Ising model with random boundary conditions \cite{EMN, ENS, Sch}.
This analysis is in contrast to most studies of phase transitions of lattice
models which consider special boundary conditions,
such as pure, free or (anti-)periodic ones.
In this paper we review these results as well as discuss some
possible implications for more complex and hopefully more realistic
situations of disordered systems, compare \cite{fw}. We feel especially
encouraged to do so by the recent advice from a prominent
theoretical physicist that:
`` Nitpickers ... should be encouraged in this field.... \cite{fis}''.
% ----------------------------------------------------------------------
\section{ Notation and background}
For general background on the theory of Gibbs measures we refer
to \cite{EFS, Geo}. We will here always consider Ising spin
models, living on a finite-dimensional lattice $\bbZ^d$. The spins
will take the values $1$ and $-1$ and
%will be denoted by small Greek letters, e.g. $\tau,\sigma,\eta$.
we will use small Greek letters $\si,\eta,\ldots$ to denote
spin configurations in finite or infinite sets of sites. The
nearest-neighbour Hamiltonians in a finite volume
$\Lambda \subset \bbZ^d$ will be given by
\begin{equation}
H^{\Lambda}(\sigma, \eta) = \sum_{\langle i,j \rangle \subset
\Lambda} 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}
Fixing the 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.
We allow sometimes that the boundary bonds $J'$ take a different
value (or are drawn from a different distribution) from the bulk
bonds $J$. Our results are based on considering the ferromagnetic
situation with random boundary conditions, where the $J$ and the
$J'$ are constants ($J < 0$ for ferromagnets), and the disorder
is only in the
$\eta$-variables which are chosen to be symmetric i.i.d.
Associated to these random Hamiltonians $H^{\Lambda}(\sigma,
\eta)$ are random Gibbs measures
\begin{equation}
\mu^{\Lambda}_{\eta}(\sigma)= {1 \over Z^{\Lambda}(\eta)} \exp [-H^{\Lambda}(\sigma, \eta)]
\end{equation}
%Our interest is in
We analyze the limit behaviour of such random Gibbs measures
at low temperatures (\ie $|J| \gg 1$), in the case of dimension at
least two, so that the set of extremal infinite-volume Gibbs
measures contains more than one element. It is well
known that in two dimensions there exist exactly two pure
states, the plus state $\mu^+$ and the minus state
$\mu^-$. In more than two dimensions also translationally
non-invariant Gibbs states (e.g.\ Dobrushin interfaces) exist.
Intuitively, one might expect that a random boundary condition
favours a randomly chosen pure translation-invariant state. Such
behaviour in fact is expected to hold in considerable generality,
including the case where an infinite number of ``similar''
extremal Gibbs states coexist.
%\textbf{AvE I disagreed with the
%original version of what follows, for example in the RFIM the plus
%and minus state are similar but not related by a symmetry.
%Similarly in some of the Fisher-White examples. After some
%adaptations a Pirogov-Sinai statement along these lines should be
%true too, so please
%check again the rewritten version.
%AvE I agree with your criticism, added therefore a sentence that we will only
%consider models where some symmetry is present. In my paper with Campanino
%we considered symmetric i.i.d for Potts, which produced the disordered
%pure state, but this is of course a more trivial regime. :}\\
%\emph{
In our ferromagnetic examples all these
%By the similarity we mean here that
pure states are related by a (\eg spin-flip) symmetry of the
interaction and the distribution of the boundary condition is also
invariant under this symmetry. In more general cases one
%on \textbf{on$->$ one KN}
might have ``homogeneous'' pure phases not related by a
symmetry. In such a case one might need to consider non-symmetric,
and possibly even volume-dependent distributions for the boundary
conditions to obtain chaotic size-dependence, but for not
specially chosen distributions one expects to obtain a single one
of these pure states. We will consider here only situations where
a spin flip symmetry is present, at least at the level of the
disorder distribution.
%This condition seems
%to be the main restriction of our approach.
Note further, that in some of these more general situations, in particular
situations with many pure states, other types of boundary conditions
(\eg the periodic and the free ones, in contrast to what we are used to)
may be not coherent, and they would pick out a random
-chaotic- pair of Gibbs states, linked by the spin-flip symmetry.
%\textbf{\emph{KN: Yes, I agree with what is written. However, I do
%not find the general case - phases not related by symmetry and
%non-symmetric distributions - too interesting(?) The problem is
%that this would require a very fine tuning of the asymmetry of the
%boundary field, possibly in a dependent on $\La$ way (on its size,
%geometry,...). Along the whole sequence of cubes, the plus and
%minus free energies would have to be essentially identical (note
%these are quantities proportional to volume.) This is why I do not
%believe in any interesting generalization of our approach dealing
%with random b.c. to non-symmetric models. Simply, I see nothing
%generic when the detailed symmetry is ralaxed...?\\
%I think, the RFIM and other models with a bulk-disorder is a
%slightly different problem: Indeed, the plus and the minus states
%are not related by the spin-flip symmetry for a fixed disorder
%configuration, however, their distributions are. So, a bit
%generalized but again a very detailed symmetry between both
%states? After all, the very same symmetry is present in our random
%b.c. Ising between the two random finite-volume Gibbs measures?}}
The physical intuition for the prevalence of the boundary
conditions picking out an extremal Gibbs measure is to some extent
supported by the result of \cite{geo}, stating that
for any Gibbs measure $\mu$ --pure or not--,
$\mu$-almost all boundary conditions will give rise to a pure state.
However, choosing some prescribed Gibbs measure to weigh the boundary
conditions has some built-in bias, and a fairer question would be to ask
what happens if the boundary conditions are symmetric random i.i.d.
We will see that the above intuition,
%\textbf {AvE
that then both interfaces and
mixtures are being suppressed, is essentially correct, but that
the precise statements are somewhat weaker than one might naively expect.
As a side remark we mention that biased, e.g. asymmetric, random
boundary conditions are known to prefer pure states. This is
discussed in \cite{CE,HY} for some examples. A similar behaviour
pertains for boundary conditions interpolating between pure and
free, as is for example shown for some particular cases in
\cite{BKM,LP}.
\section{Results on Ising models with random boundary conditions}
In this section we describe our results on the chaotic size
behaviour of the Ising model under random boundary conditions in
more detail. We consider the sequences of finite-volume Gibbs
states $\mu^\La_\eta$ along a sequence of concentric cubes
$\La_N$ with linear size $N$, for any configuration
$\eta \in \Om = \{-1,1\}^{\bbZ^d}$ sampled from the symmetric i.i.d.\
distribution with the marginals
$\prb(\eta_i=-1) = \prb(\eta_i=1) = \frac{1}{2}$.
To any such sequence of states we assign the collection of its
weak-topology limit points, which can in general be non-trivial
and $\eta$-dependent. However, in our simple situation
we show that the set of limit points has a simple structure:
with probability 1, it contains exactly two
elements -- the Ising pure states $\mu^+$ and $\mu^-$. This is
proven to be true, provided that a sufficiently sparse (depending
on the dimension) sequence of cubes is taken and for sufficiently
low temperatures ($-J \gg 1$), at least in certain regions in the
$(J,J')$-plane which will next be described. Although one gets an
identical picture in all the cases under consideration, these
substantially differ in the complexity of the analysis required in
the proof.
%\textbf{Next some more suggestions for changes -- for simplicity, marks
%over the whole paragraphs where they have been made:}\\
% ----------------------------------------------------------------
\subsection*{Ground state with finite-temperature boundary
conditions}
As a warm-up problem, following \cite{EMN}, we consider
%the simplified version in
the case where $-J= \infty$ and $J'$ is finite. Then all spins
inside $\Lambda$ take the same spin value, either plus or minus.
Our choice of coupling parameters has excluded interface
configurations. The total energy for either the plus or the minus
configuration of the system in a cube $\La_N$,
%a cube of linear size $N$
\begin{equation}
H^N(\pm,\eta) = \pm J'\sum_{i:\, d(i,\La_N) = 1} \eta_i
\end{equation}
is a sum of $\caO(N^{d-1})$ 2-valued random variables, which are
i.i.d.\ and of zero mean.
%as is their difference.%
Obviously,
\begin{equation}
\mu^N_\eta(\pm) = \bigl\{1 + \exp[\pm 2H^N(+,\eta)]\bigr\}^{-1}
\end{equation}
and the possible limit states are the plus configuration, the
minus configuration, or a statistical mixture of the two,
depending on the limit behaviour of the energy $H^N(+,\eta)$.
According to the local limit theorem, the probability of
%the difference between the plus and minus energy
this energy being in some finite interval decays as
$N^{-{d-1\over 2}}$.
Summing this over $N$ gives a finite answer if either
$d$ is at least 4, or if one chooses a sufficiently sparse sequence
of increasing volumes $\Lambda_{N_k}$ in $d=2$ or $d=3$. A
Borel-Cantelli argument then implies that almost surely, that is
for almost all boundary conditions, the only possible limit points
are the plus configuration and the minus configuration. On the
other hand, without the sparsity assumption on the growing volumes
in dimension 2 and 3, again via a Borel-Cantelli argument, a
countably infinite number of statistical mixtures is seen to occur
as limit points. However, as these mixtures occur with decreasing
probabilities when the volume increases,
the metastate of our system is concentrated only on
the plus and minus configuration, and the mixtures do not show up
(they are null-recurrent). We expect a similar distinction
between almost sure and metastate behaviour in various other
situations.
\bigskip
\subsection*{Finite low temperatures with weak boundary conditions}
In the case where $J$ and $J'$ are both finite, but $|J'|
\ll -J$, the spins inside $\La$ are no more frozen and thermal
fluctuations have to be taken into account. Yet, one can expect
that for the bulk bond $-J$ being large enough, the behaviour does
not change dramatically and the model can be analyzed as a small
perturbation around the $-J = \infty$ model of the last section. A
difference will be that the frozen plus and minus configurations
are to be replaced with suitable plus and minus ensembles, and the
energies with the free energies of these ensembles.
%the difference
%between what are now the free energies (the logarithms of
%appropriate partition functions) of plus and minus states is no
%longer writable as a sum of independent terms.
A technical complication is that these free energies can no longer
be written as sums of independent terms. This prevents us from
having a precise local limit theorem. However, physically the
situation should be rather similar, and one can indeed prove a
related, but weaker result.
To describe the perturbation method in more detail, we need to go
to a contour description. Every pair of spin configurations
related by a spin-flip corresponds to a contour configuration.
We will distinguish the ensemble of plus-configurations
$\Omega^{\Lambda}_{+}$ in which the spins outside of the exterior contours
are plus and similarly the ensemble of minus-configurations
$\Omega^{\Lambda}_{-}$. When a contour ends at the boundary and separates one
corner from the rest, this corner is defined to be in the interior, and for
contours separating
%{\textbf AvE
at least two corners from at least two other corners
%the rest of the volume}
%\emph{two corners from the other two corners} {\bf AvE What about the
%higher-dimensional staement of our paper with Igor?}
(these are interfaces of some sort) we can make a choice for what we call
the interior,
%}\\
see \cite{EMN,ENS}. It will turn out that our results do not
depend on our precise choice. We consider the measures restricted to
the plus and minus ensembles:
\begin{equation}
\mu^{\Lambda}_{\eta,+}(\sigma)= \frac{1}{Z^{\Lambda}(\eta,+)} \exp
[-H^{\Lambda}(\sigma, \eta)]\,\funit_{[\si\in\Om^\La_+]}
\end{equation}
%on $\Omega^{\Lambda}_{+}$ \kn
and similarly for the minus ensemble.
%We can show that for almost all $\eta$, these ensemble measures will
%converge weakly to the plus and minus measures.
Under the weakness assumption $|J'| \ll -J$, the Gibbs probability
of any interface in these ensembles is damped exponentially in the
system size $N$ uniformly for all $\eta$. Actually, we can prove
the asymptotic triviality of both ensembles in the following
strong form:
\begin{proposition}\label{prop: as.triv.}
Let $-J \geq \max\{J^*,\De|J'|\}$ with large enough constants
$J^*, \De > 0$. Then $\mu^{\La_N}_{\eta,\pm} \stackrel{N}{\longrightarrow}
\mu^\pm$ in the weak topology. (The convergence is exponentially
fast uniformly in $\eta$.)
\end{proposition}
%\textbf{KN: Maybe to add a sentence explaining where the $\De$
%comes from? Something like: Note that for $d|J'| > -J$, $d \geq
%3$, there exist translationally non-invariant Gibbs states
%(Dobrushin interfaces) and the statement of the proposition can no
%longer be true for all boundary conditions $\eta$. However, a
%weaker statement is true, see the next section...AvE, the statement should
%already be about two dimensions. Proposal:
\noindent
{\bf Remark:} Note that for Dobrushin boundary conditions,
if $\tilde\De|J'| > -J$ for some sufficiently small $\tilde\De$,
%$|J'| > -\De J$ for some sufficiently large $\De$,
these boundary conditions favour an interface. However,
such boundary conditions are exceptional, and in the next section we
will prove a weaker statement, based on this fact, in $d=2$.
The convergence properties of the full, non-restricted measures are based
on an estimate for the random free energies:
%}
%\\ \textbf{\emph{KN: I agree, this is better. Only, to keep consistency, one could
%use the inequality
%$\De|J'| > -J$ for some sufficiently small $\De$ instead.
%}}\\ For the random free energies
\begin{equation}
F^{\Lambda}_{\pm}(\eta) = \log Z^{\Lambda}(\eta,\pm)
\end{equation}
%\begin{equation}
%F^{\Lambda}_{-}(\eta) = \log Z^{\Lambda}(\eta,-)
%\end{equation}
%Then, if $\Lambda$ is a cube of linear size N,
namely, we prove the following weak local limit type upper bound:
\begin{proposition}\label{prop: F-difference}
%Assuming Let the inverse temperature
%$\beta$ be large enough, and let $J'\leq J^{*}$ for some $J^{*}$
%ufficiently smaller than J. Then the inequality
Under the assumptions of Proposition~\ref{prop: as.triv.}, the
inequality
\begin{equation*}
\prb (|F^{\La_N}_{+}(\eta) - F^{\La_N}_{-}(\eta)| \geq
N^{\varepsilon}) \leq \cst N^{-(\frac{d-1}{2} - \varepsilon)}
\end{equation*}
holds for any $\ve > 0$ and $N$ large enough.
\end{proposition}
The proofs of both propositions are based on convergent cluster
expansions for the measures $\mu^\La_{\eta,+}$ and the
characteristic function of the random free energy difference,
respectively. Combining Proposition~\ref{prop: F-difference} with
a Borel-Cantelli argument we get
\begin{equation}
\lim_\La |F^\La_{+}(\eta) - F^\La_{-}(\eta)| = \infty
\end{equation}
provided that the limit is taken along a sparse enough sequence of
cubes (unless $d \geq 4$). By symmetry, this reads that the
random free energy difference
has $+\infty$ and $-\infty$ as the only limit points. Since the
full Gibbs state is a convex combination of the plus and the minus
ensembles with the weights related to the random free energy
difference,
\begin{equation}
\mu^\La_\eta = \Bigl[1 +
\frac{\caZ^\La_{\eta,-}}{\caZ^\La_{\eta,+}}\Bigr] \mu^\La_{\eta,+}
+ \Bigl[1 + \frac{\caZ^\La_{\eta,+}}{\caZ^\La_{\eta,-}}\Bigr]
\mu^\La_{\eta,-}
\end{equation}
this immediately yields the spectrum of the limit Gibbs states
\cite{EMN}.
\subsection*{Finite low temperatures in $d=2$, with strong boundary
conditions}
%When we have
In the case $-J = |J'|$, due to exceptional (e.g.
Dobrushin-like, all spins left minus, all spins right plus)
boundary conditions, we are to expect no uniform control anymore
over the convergence of the cluster expansion. Indeed, one checks
that for the contours touching the boundary the uniform lower
bounds on their energies cease holding true. In order to control
the contributions from these contours, we need to perform a
multiscale analysis, along the lines of \cite{FI}, with some large
deviation estimates on the probability of these exceptional
boundary conditions. We obtain in this way the following Propositions,
corresponding to
Propositions~\ref{prop: as.triv.}-\ref{prop: F-difference}
\cite{ENS}:
\begin{proposition}\label{prop: as.triv.2}
Let $d = 2$ and $|J'| = -J \geq J^{**}$, with the constant
$J^{**} > 0$ being large enough. Then
$\mu^{\La_N}_{\eta,\pm} \stackrel{N}{\longrightarrow}
\mu^\pm$ weakly for almost all $\eta$.
\end{proposition}
\begin{proposition}
Under the same assumptions as in Proposition~\ref{prop: as.triv.2},
\begin{equation}
\prb (|F^{\Lambda}_{+}(\eta) - F^{\Lambda}_{-}(\eta)| \geq \tau)
\leq \cst(\tau)\, N^{-(\frac{1}{2} - \varepsilon)}
\end{equation}
for any $\tau,\ve > 0$ and $N$ large enough.
\end{proposition}
Observe that the limit statement of Proposition~\ref{prop:
as.triv.2} holds true only on a probability $1$ set of the
boundary conditions (essentially the ones not favouring
interfaces). The construction of this set is based on a
Borel-Cantelli argument.
These two Propositions imply the following Theorems on
the almost sure behaviour and the metastate behaviour respectively:
\smallskip
\begin{theorem}
Let the conditions of either Proposition~\ref{prop: as.triv.} or
Proposition~\ref{prop: as.triv.2} be satisfied, and take a
sequence of increasing cubic volumes, which in $d=2$ and
$d=3$ is chosen sufficiently sparse. Then for almost all boundary
conditions $\eta$ it holds that the weak limit points of the
sequence of finite-volume Gibbs states are the plus and minus
Ising states. Almost surely any open set in the set of Gibbs
states, from which the extremal Gibbs measures have been removed,
will not contain any limit points.
\end{theorem}
\begin{theorem}
Let the conditions of either Proposition~\ref{prop: as.triv.} or
Proposition~\ref{prop: as.triv.2} be satisfied, and take a
sequence of increasing cubic volumes. Then the metastate equals
the mixture of two delta-distributions:
${1 \over 2} (\delta_{\mu^{+}} +\delta_{\mu^{-}})$.
\end{theorem}
\begin{remark}
This metastate is a different one from the one obtained with
free or periodic boundary conditions which would be
$\delta_{{1\over 2}(\mu^{+} + \mu^{-})}$.
%\textbf{subscript $->$ superscript, to keep a consistency of the notation \kn}
In simulations it would
mean that for a fixed realization and a fixed finite volume one
typically sees the same state (either plus or minus, which one
depending on the volume), the other one being invisible. For
periodic or free boundary conditions both plus and minus states
are accessible for any fixed volume.
\end{remark}
\begin{remark}
We expect that, just as in the ground state situation, for a
non-sparse increasing sequence of volumes, mixtures will be
null-recurrent in dimensions 2 and 3. In this case, any
mixture, not only a countable number of them, could be a
null-recurrent limit point. However, as our Propositions only
provide upper bounds, and no lower bounds, on the probabilities of
small (and not even finite) free energy differences, such a result
is out of our mathematical reach.
\end{remark}
Note that although the metastate description of Theorem 2
gives less detailed information than the almost sure statement of Theorem 1,
it encapsulates the physical intuition actually better.
\begin{remark}
It also follows from our arguments that, for almost all
boundary conditions $\eta$, large (proportional to
$N^{d-1}$) contours are suppressed for large systems, so neither
rigid, nor fluctuating interfaces will appear anywhere in the
system. Fluctuating interfaces also would produce mixed states,
so even in 2 dimensions, where interface states do not occur, this
is a non-trivial result.
\end{remark}
%\textbf{AvE, new remark} \\
\begin{remark}
Similar results can also be obtained for the case of very strong
boundary conditions, $|J'| \gg -J \gg 1$, \cite{Sch}.
%\textbf{KN: Maybe to add here that 1) o
Our method can easily be adapted to get
the same result for all $|J'| \leq -J$, and we believe in fact the
result to be true for all $J' \neq 0$, $-J \gg 1$.
%\kn:
%\textbf{changed $|J|$ to $-J$ to indicate that we still deal with
%a ferromagnetic situation)}
%THm 1 almost sure statement
\end{remark}
%Thm 2 metastate statement
%Short remarks re proofs
\section{Spin glasses, Fisher-White type models, many states, space versus time}
%\textbf{
Although our original motivation was due to our interest in
the spin-glass model, our results for spin-glass models are still very modest.
We note, however, that by a gauge transformation the ferromagnet
with random boundary conditions is equivalent to a nearest-neighbour
Mattis \cite{mat} spin-glass ($=$ one-state Hopfield) model
with fixed boundary conditions, the behaviour of which we can thus analyze by our methods.
%}
In a recent paper by Fisher and White {\cite{fw}}, they discuss possible
scenarios of infinitely many states, mostly based on constructions of stacking
lower-dimensional Ising models. The motivation is to investigate what might
happen in the
putative infinite-state spin-glass scenario, for some (high-dimensional
and{/}or long-range) Edwards-Anderson type model.
One of their simplest models is based on ``stacking'' lower-dimensional
Ising models, and we can derive some properties for such models
from our results in the last section.
For example one can consider $N$ 2-dimensional Ising models in
squares of size $N$ by $N$ in horizontal planes,
decoupled in the vertical direction.
There are now $2^N$ ground states for free boundary conditions,
corresponding to the choice of plus or minus in each plane. To
avoid reducing the problem to studying the symmetry group of the
interaction, one can randomize the problem some more, as follows:
By choosing in every plane the bonds across a single line through the
origin (in that plane) randomly plus
or minus, one obtains that in every plane in the thermodynamic limit
%\textbf{symmetric around zero I guess? Yes, AvE, KN},
%ground \textbf{plane ground $->$ plane? KN}
four ground states appear, (plus-plus, plus-minus, minus-plus and
minus-minus). By imposing random
%finite-temperature
boundary conditions, independently in each plane, one now expects
an independent choice out of these four in each plane.
%\textbf{The
%last claim I do not see entirely... Why finite-temperature b.c.?
%AvE You're right this we did not need, I rephrased. What role is
%played by the random bonds across the splitting line in this
%argument; how if they are e.g. no bonds there? KN AvE, No bonds
%displays the same behaviour, basically left and right half in each
%plane are plus or minus, which one dependent both on the boundary
%conditions and the bonds on the line in between, and the
%configuration from the other half which also act as random
%boundary conditions. Because the ground states in the other half
%plane, are either plus or minus we compare a finite number of
%draws from the bond distribution, which produces the same
%asymptotics as before.}
%\\ \textbf{\emph{KN: I understand now and agree.}}\\
By similar considerations as in the last section, we find that
although this is true in most planes, in
$O(\sqrt N)$ of them (that is a fraction of
$O({1 \over {\sqrt N}})$ of them) mixtures will appear.
%\textbf{
If our boundary conditions are finite-temperature, this happens if
the difference between the number of pluses and minuses stays bounded,
otherwise this happens when there is a tie, and at the same time not such a
strong spatial fluctuation that an interface ground state appears.
%}
The same remains true if one connects the planes by a sufficiently weak
(one-dimensional) random coupling, such as for example having a random choice
for all the vertical bonds on lines, each line connecting a pair
of adjacent planes.
If one would choose the values of {\em all} vertical bonds
randomly from some symmetric distribution, we would obtain something like a
collection of two-dimensional random field Ising models. Indeed,
the effect of one plane on the next one would be like that of a
random field, which would prevent a phase transition in any plane,
and our arguments break down. For further discussion see \cite{fw}.
%\textbf{However, this we cannot
%prove, or can we? KN AvE, I think we can, the one-dimensional coupling should
%act as an extra random term, similarly to the boundary conditions. }
%\\ \textbf{\emph{KN: In the case you mean the coupling via a single bond or something like that
%then I agree. However, if the interaction is via all vertical
%bonds then it is more like the RFIM I guess? Maybe one could specify more the nature
%of the coupling in the sentence above?}}\\
%{\emph
Similar properties hold for models in which on one
periodic sublattice one stacks in a horizontal direction, and in another one
in a vertical direction.
%}
As the topology of weak convergence is a topology of convergence
of {\em local} observables, a statement along the lines of Theorem
1 still holds, however the interpretation that one almost surely
%one \textbf{one twice KN}
avoids mixtures now is incorrect. In
fact, in these circumstances, for almost all boundary conditions,
the system will be in a {\em mixed} state. Note that, although such mixed
states are less stable than interface states (one can change the weights in the
mixture by a finite-energy perturbation), they are much more
likely to occur than interface states.
%\textbf{I feel this is correct, yet I do not understand entirely... KN AvE
%The probability of a mixture scales as $1 \over \sqrt N$, due to local
%limit behaviour, interfaces are suppressed either infinitely strong or as
%$\exp - cst N$. The result is somehow counterintuitive indeed.}
%\\ \textbf{\emph{Well, this comparison I understand but... I always tend to think
%of the mixtures as of something `nonphysical', something which
%only quantifies our ignorance about which physical state from a
%list is actually chosen by the equilibrium system. Simply, when I
%prepare a system in equilibrium then via a small number of
%macro-measurements I can uniquely determine in what phase it is.
%Possibly, it can also be in a translationally non-invariant state
%(which is less stable, in some sense less probable, etc.).
%Nevertheless, the mixtures and extremal tr.\ non-invariant states
%seem to me qualitatively incomparable - the latter represents a
%macrostate of a single system (in a sense
%deterministic/reproducible), while the former takes into account
%purely statistical considerations. Of course, this is only my
%curiosity to understand the point better; I agree with the text as
%it is written now.}}
As the set of sites which are influenced by
the measure being a mixture has a density which approaches zero,
in the thermodynamic limit the Parisi overlap distribution
\cite{par1} will be trivially concentrated at 1.
Similarly to theorem 2, also the metastate still
will be concentrated on the symmetric mixture of the delta-distributions on
the pure states in each plane.
We obtain similar results at sufficiently low temperatures, although again,
now we cannot prove the occurrence of mixtures.
It has been conjectured that similar to the chaotic size-dependence
we have discussed, for random (incoherent) initial conditions
a related phenomenon of chaotic time-dependence (non-convergence)
might occur in disordered systems. If even the Cesaro average fails
to converge, this phenomenon also has been called ``historic''
behaviour \cite{ru1}.
A similar distinction in the spatial problem also occurs. Although
for ferromagnets with random boundary conditions the Cesaro
average of the magnetisation at the origin, taken over a sequence
of linearly increasing volumes, will exist, one does not expect
this for the random field Ising model with free or periodic
boundary conditions, essentially for the same reason as in the
mean-field version of the problem \cite{Ku1,Ku2}. One would only
expect a convergence in distribution to the metastate ${1 \over 2}
(\delta_{\mu^{+}} +\delta_{\mu^{-}})$.
%\textbf{subscript $->$ superscript (changed) KN}
One could both scenarios (chaotic size dependence and chaotic time dependence)
describe as examples of what Ruelle \cite{ru2} calls ``messy'' behaviour.
However, chaotic time dependence for stochastic dynamics of for example Glauber
type requires the system to be infinitely large.
Otherwise, for finite systems either one has a finite-state Markov chain
with a unique invariant measure (for positive temperature Glauber dynamics),
while for zero-temperature dynamics the system would get trapped,
up to zero-energy spin flips.
Concluding, we have illustrated how the notion of chaotic size-dependence
naturally occurs, already in the simple ferromagnetic
Ising model, once one allows for incoherent boundary conditions.
As in many situations the choice of incoherent boundary conditions
seems a physically realistic one, our results may be helpful in
explaining why in general one expects experimentally to observe pure phases.
%Time dependence, random initial vs random boundary conditons?
%Remark that chaotic time dependence is historic dependence?
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%% MODIFIED SECTION STARTS HERE, 13/09/2004 %%%%%%
%%%%%% %%%%%%
%%%%%% VVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVV %%%%%%
%%%%%% AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA %%%%%%
%%%%%% %%%%%%
%%%%%% MODIFIED SECTION ENDS HERE, 13/09/2004 %%%%%%
%%%%%% %%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{ Discussion}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\addcontentsline{toc}{section}{\bf References}
%\begin{thebibliography}{10}
%\begin{bibsection}
%\begin{biblist}
%}
%\bibliographystyle{abbrv}
%\bibliography{univervp}
%\end{document}
\bigskip
\noindent {\em Acknowledgements}:
This research has been supported by FOM-GBE.
We thank Christof K\"ulske for helpful advice on the manuscript.
A.~C.~D.~v.~ E. thanks Pierre Picco and Veronique Gayrard for useful dicussions
on metastate versus almost sure properties.
He would like to thank Mike Keane for all the probabilistic
inspiration he has provided to Dutch mathematical physics over the years.
\addcontentsline{toc}{section}{\bf References}
\begin{thebibliography}{10}
\bibitem{BKM} C.~Borgs, R.~Koteck\'y and I.~Medved'.
\newblock Finite-Size Effects for the Potts Model with Weak Boundary
Conditions.
\newblock {\em J.~Stat.~Phys.}, 109:67--131, 2002.
\bibitem{bov1}
A.~Bovier.
\newblock {\em Statistical Mechanics of Disordered Systems}.
\newblock MaPhySto Lectures 10, Aarhus, 2001.
\bibitem{bov2}
A.~Bovier.
\newblock {\em Statistical Mechanics of Disordered Systems, a mathematical perspective}.
\newblock Cambridge University Press, to appear.
%\newblock Springer, Berlin, 1994.
\bibitem{BEN}
A.~Bovier, A.~C.~D.~van Enter and B.~Niederhauser.
\newblock Stochastic Symmetry Breaking in a Gaussian Hopfield Model.
\newblock {\em J.~Stat.~Phys.}, 95:189--213, 1999.
\bibitem{BG}
A.~Bovier and V.~Gayrard.
\newblock Hopfield Models as Generalized Random Mean-Field Models.
\newblock in {\em Mathematical Aspects of Spin Glasses and Neural Networks},
Eds. A.~Bovier and P.~Picco, pp. 243--287, Birkh\"auser, Boston, 1998.
\bibitem{CE}
M.~Campanino and A.~C.~D.~van Enter.
\newblock Weak Versus Strong Uniqueness of Gibbs Measures: A Regular Short-Range Example.
\newblock {\em J.~Phys.~A, Math.Gen.}28:L45--L47, 1995.
\bibitem{vE}
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{EFS}
A.~C.~D. van Enter, R. Fern\'andez and A.~D. Sokal.
\newblock Regularity Properties of Position-Space Renormalization-Group
Transformations: Scope and Limitations of Gibbsian Theory.
\newblock {\em J.~Stat.~Phys.} 72:879--1167, 1993.
\bibitem{EMN}
A.~C.~D. van Enter, I.~Medved' and K.~Neto\v{c}n\'y.
\newblock Chaotic Size Dependence in the Ising Model with Random
Boundary Conditions.
\newblock {\em Markov Proc.~Rel.~Fields.}, 8:479--508, 2002.
\bibitem{ENS}
A.~C.~D. van Enter, K.~Neto\v{c}n\'y and H.~G.~Schaap.
\newblock On the Ising Model with Random Boundary Condition.
\newblock {\em J.~Stat.~Phys.} 118, 2005, to appear.
\bibitem{ES}
A.~C.~D. van Enter and H.~G.~Schaap.
\newblock Infinitely many states and stochastic symmetry breaking in a
Gaussian Potts-Hopfield model.
\newblock {\em J.~Phys.~A, Math.~Gen.}, 35:2581--2592, 2002.
\bibitem{fis}
D.~S.~Fisher.
\newblock Equilibrium States and Dynamics of Equilibration: General Issues
and Open Questions.
\newblock Les Houches Summer School 2002, \emph{Slow Relaxations and
Nonequilibrium Dynamics in Condensed Matter}, Editors J.-L.Barrat,
M.~Feigelman, J.~Kurchan et al, pp 523--553, Springer, 2004.
\bibitem{FH}
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{fw}
D.~S.~Fisher and O.~L.~White.
\newblock Spin Glass Models with Infinitely Many States.
\newblock cond-mat{/}0412335, 2004.
\bibitem{FI}
J.~Fr\"ohlich and J.~Z.~Imbrie.
\newblock Improved perturbation expansion for disordered systems: Beating
Griffiths singularities.
\newblock {\em Comm.~Math.~Phys.}, 96:145--180, 1984.
\bibitem{geo}
H.~-O.~Georgii.
\newblock Two remarks on extremal equilibrium states.
\newblock {\em Comm.~Math.~Phys.} 32:107--18, 1973.
\bibitem{Geo}H.~-O.~Georgii.
\newblock {\em Gibbs Measures and Phase Transitions}, de Gruyter, Berlin, 1988.
\bibitem{HY}
Y.~Higuchi and N.~Yoshida.
\newblock Slow Relaxation of Stochastic Ising Models with Random
and Non-Random Boundary Conditions. In K.~Elworthy, S.~Kusuoka and
I.~Shikegawa (Eds.), {\em New Trends in Stochastic Analysis}, pp
153--167, World Scientific, Singapore, 1997.
\bibitem{Ku1} C.~K\"ulske.
\newblock Metastates in Disordered Mean-Field Models: Random Field and
Hopfield Models.
\newblock {\em J.~Stat.~Phys.}, 88:1257--1293, 1997.
\bibitem{Ku2} C.~K\"ulske.
\newblock Metastates in Disordered Mean-Field Models II: the Superstates.
\newblock {\em J.~Stat.~Phys.} 91:155--176, 1998.
\bibitem{LP} J.~L.~Lebowitz and O.~Penrose.
\newblock Thermodynamic Limit of the Free energy and Correlation
Functions of Spin Systems.
\newblock {\em Acta Physica Austriaca, Suppl.~XVI:}, 201--220, 1976.
\bibitem{par2}
E.~Marinari, G.~Parisi, F.~Ricci-Tersenghi, J.~J.~Ruiz-Lorenzo and F.Zuliani.
\newblock Replica Symmetry Breaking in Short-Range Spin Glasses: Theoretical
Foundations and Numerical Evidences.
\newblock {\em J.~Stat.~Phys.} 102:973--1074, 2000.
\bibitem{mat} D.~C.~Mattis.
\newblock Solvable Spin Models with Random Interactions.
\newblock {\em Phys. Lett. A} 56:421--422, 1976.
\bibitem{par1}
M.~M\'ezard, G.~Parisi and M.-A.~Virasoro.
\newblock {\em Spin Glass Theory and Beyond},
\newblock World Scientific, Singapore 1987.
\bibitem{N}
C.~M.~Newman.
\newblock {\em Topics in Disordered Systems}.
Lectures in Mathematics, ETH-Z\"urich,
\newblock Birkh\"auser, Basel, 1997.
\bibitem{NS1}
C.~M.~Newman and D.~L.~Stein.
\newblock Multipe States and Thermodynamic Limits in Short-Ranged Ising
Spin-Glass Models.
\newblock {\em Phys.~Rev.~B}, 46:973--982, 1992.
\bibitem{NS2} C.~M.~Newman and D.~L.~Stein.
\newblock Metastate Approach to Thermodynamic Chaos.
\newblock {\em Phys.~Rev.~E} 55:5194--5211, 1997.
\bibitem{NS3} C.~M.~Newman and D.~L.~Stein.
\newblock Thermodynamic Chaos and the Structure of Short-Range Spin Glasses.
\newblock in {\em Mathematical Aspects of Spin Glasses and Neural Networks},
Eds. A.~Bovier and P.~Picco, pp.~243--287, Birkh\"auser, Boston,
1998.
\bibitem{NS4} C.~M.~Newman and D.~L.~Stein.
\newblock The state(s) of replica symmetry breaking: Mean field theory versus
short-ranged spin glasses
\newblock {\em J.~Stat.~Phys.} 106:213--244, 2002, formerly known as ``Replica Symmetry Breaking's New Clothes''.
\bibitem{NS5} C.~M.~Newman and D.~L.~Stein.
\newblock Ordering and broken symmetry in short-ranged spin glasses.
\newblock {\em J.~Phys.: Cond.~Matter} 15:1319--1364, 2003.
\bibitem{ru1} D.~Ruelle.
\newblock Historical Behaviour in Smooth Dynamical Systems.
\newblock In {\em Global analysis of dynamical systems},
(Takens Festschrift), Eds. H.~W.~Broer,
B.~Krauskopf and G.~Vegter, Institute of Physics, London, 2001.
\bibitem{ru2} D.~Ruelle.
\newblock Some Ill-Defined Problems on Regular and Messy Behavior
in Statistical Mechanics and Smooth Dynamics for which I Would Like the
Advice of Yasha Sinai.
\newblock {\em J.~Stat.~Phys.} 108:723--728, 2002.
\bibitem{Sch} H.~G.~Schaap.
\newblock {\em Ising models and neural networks}.
\newblock Groningen thesis, 2005, to appear.
\bibitem{tal1}
M.~Talagrand.
\newblock The Generalised Parisi Formula.
\newblock {\em Comptes Rendus Academie des Sciences, Mathematique}, 337:111--114, 2003.
\bibitem{tal2}
M.~Talagrand.
\newblock The Parisi Formula.
\newblock {\em Annals of Mathematics}, 2005, to appear.
\end{thebibliography}
\end{document}
---------------0502080847302--