Content-Type: multipart/mixed; boundary="-------------1010120433707" This is a multi-part message in MIME format. ---------------1010120433707 Content-Type: text/plain; name="10-168.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="10-168.keywords" Parisi overlap distribution, ultrametric, paperfolding sequences ---------------1010120433707 Content-Type: application/x-tex; name="egoverlap3.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="egoverlap3.tex" \documentclass[12pt]{article} \input{epsf.sty} \newlength{\Taille} \def\1{{\mathchoice {1\mskip-4mu\mathrm l} {1\mskip-4mu\mathrm l} {1\mskip-4.5mu\mathrm l} {1\mskip-5mu\mathrm l}}} \usepackage{hyperref} \usepackage{amsmath, amssymb} %\usepackage[pdftex]{graphicx} \usepackage{amsmath, amssymb} \textheight=23cm \textwidth=16cm \oddsidemargin 7mm \evensidemargin -1mm \topmargin -4mm \topmargin -2cm \oddsidemargin -0.2cm \newtheorem {thm}{Theorem}[section] \newtheorem {prop}[thm]{Proposition} \newtheorem {lem}[thm]{Lemma} \newtheorem {cor}[thm]{Corollary} \newtheorem {defn}[thm]{Definition} \newtheorem {conj}[thm]{Conjecture} \newtheorem {rem}[thm]{Remark} \newtheorem {cond}[thm]{Condition} \newenvironment{remark}[1][Remark:]{\begin{trivlist} \item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}} \def\Cox{\hfill \Box} \def\del{\partial} \def\N{{\mathbb N}} \def\Z{{\mathbb Z}} \def\R{{\mathbb R}} \def\P{{\mathbb P}} \def\Q{{\mathbb Q}} \def\E{{\mathbb E}} \begin{document} \title{An ultrametric state space with a dense discrete overlap distribution: \\ Paperfolding sequences} \author{ Aernout C. D. van Enter \\[-1mm] {\normalsize\it Johann Bernoulli institute} \\[-1.5mm] {\normalsize\it Rijksuniversiteit Groningen} \\[-1.5mm] {\normalsize\it Nijenborgh 9} \\[-1.5mm] {\normalsize\it 9747 AG Groningen} \\[-1.5mm] {\normalsize\it THE NETHERLANDS} \\[-1mm] {\normalsize\tt A.C.D.van.Enter@.rug.nl} \\[-1mm] %\\ [-1mm] {\normalsize Ellis de Groote} \\[-1mm] {\normalsize\it De Vonderkampen 116} \\[-1.5mm] {\normalsize\it 9411RG, Beilen} \\[-1.5mm] {\normalsize\it THE NETHERLANDS} \\[-1mm] %{\normalsize\it Institute of Applied Mathematics and Mechanics} \\[-1.5mm] %{\normalsize\it Warsaw University} \\[-1.5mm] %{\normalsize\it ul. Banacha 2} \\[-1.5mm] %{\normalsize\it 02-097 Warsaw} \\[-1.5mm] %{\normalsize\it POLAND} \\[-1mm] %{\normalsize\tt miekisz@mimuw.edu.pl} \\[-1mm] {\normalsize\tt ellisdegroote@hotmail.com} \\[-1mm]} {\protect\makebox[5in]{\quad}} \pagenumbering{arabic} %\begin{document} \maketitle \baselineskip=14pt \noindent {\bf Abstract.} We compute the Parisi overlap distribution for paperfolding sequences. %as well as some tiling systems on cubic (square?) lattices. It turns out to be discrete, and to live on the dyadic rationals. Hence it is a pure point measure whose support is the full interval $[-1,+1]$. The space of paperfolding sequences has an ultrametric structure. Our example provides an illustration of some properties which were suggested to occur for pure states in spin glass models. \newtheorem{theorem}{Theorem} % 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})} \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}} \bibliographystyle{plain} \section{Introduction} \bigskip In \cite{EHM} the study of the Parisi overlap distribution for various classes of non-periodically ordered sequences was undertaken. \newline The considered sequences were members of $\{-,+\}^{\Z}$ and their orbit closure forms typically a uniquely ergodic system, with a unique shift-invariant measure $\mu$. This measure then can be a ground state for a translation-invariant interaction which one can construct \cite{Aub1,rad,Rad1}, and the individual sequences will be the pure (extremal) ground states. \newline Their overlap distribution gives the behaviour under the product measure of $\mu$ with itself, describing a two-replica system, written as $\mu \times \mu'$ of the overlap between two randomly (from this product measure) chosen, bi-infinite sequences $\sigma$ and $\sigma'$: \begin{equation} q_{\sigma \sigma'}=lim \frac{1}{N} \sum_{i=1,...N} \sigma_i \sigma'_{i}. \end{equation} Note that the overlap $q_{\sigma \sigma'}$ between two sequences is directly related to their Hamming distance $d_H(\sigma, \sigma')= \frac{1-q_{\sigma \sigma'}}{2}$. \smallskip If we take two random sequences, each chosen according to the same shift-invariant measure, with probability one the above limit will exist, as follows from the ergodic theorem. The product measure of two ergodic measures, however, although shift-invariant, in general will not be ergodic, thus one may obtain different values for the overlap with positive probability. A simple example illustrating this is the symmetric measure which give equal weights to the two alternating sequences: \begin{equation} \mu_{alt}(\sigma) = \frac{1}{2}(\delta(+-..., \sigma) + \delta(-+..., \sigma)) \end{equation} This measure is ergodic under the shift transformation, but the product measure of $\mu_{alt}$ with itself is not. For the overlaps it holds that with probability $\frac{1}{2}$ the same sequence is chosen (overlap one) and with probability $\frac{1}{2}$ two different sequences are chosen (overlap minus one). Thus in this simple example \begin{equation} p(q) = \frac{1}{2}(\delta(q,1) + \delta(q,-1)) \end{equation} In more general situations the overlap can take any value in the interval $[-1,+1]$, and the induced measure, which is the overlap distribution, describable by a probability distribution function on this interval, can be discrete ( that is, it is a pure point measure) or nondiscrete, and thus have continuous components. In particular, it was found in \cite{EHM} that whenever the atomic ("diffraction") spectrum of the unique translation-invariant measure on the sequences does not contain a pure point component, the overlap distribution is trivial; this applies in particular to all weakly mixing systems (in which case their product measures are ergodic), as well as to the (Prouhet-(Thue-))Morse system. On the other hand for the Fibonacci sequences the overlap distribution was found to contain a(n absolutely) continuous part. In fact, in explicit form it is \cite{EHM} \begin{equation} p(q)dq = (2 \gamma -1) \delta(q, 1-4(1-\gamma))+ \frac{1}{2}1_{[1-4(1- \gamma),1]}(q)dq \end{equation} where $\gamma= \frac{2}{1+\sqrt 5}$. We notice here that the same argument which was used in \cite{EHM} applies to more general Sturmian sequences, which thus also have a continuous part in their overlap distribution. The reason is that these are also rotation sequences, see e.g. \cite{LW}. They also have recently been studied as ground states of some explicit interaction functions on lattice systems in \cite{ADJR}). Moreover, it was found that the Toeplitz (or period-doubling) sequences give rise to an overlap distribution which is concentrated on a countable number of values, with $1$ as its only limit point. Also, the set of these Toeplitz sequences has a tree (ultrametric) structure, and the overlap distribution could be obtained via this structure. Here we show that the paperfolding sequences (see \cite{BaMoRicSi,DekMenPoo} and references mentioned there), which also display such an ultrametric structure, give rise to an overlap distribution living on a {\em dense} countable number of points. As such the paperfolding system, which, in contrast to the Toeplitz system, is symmetric under the spin-flip (plus-minus) symmetry, is even closer to what is expected in Parisi's replica symmetry breaking theory of the Sherrington-Kirkpatrick model \cite{MPV, PT}. In Parisi's theory for the SK model, this countable overlap distribution is supposed to show up for a fixed disorder, while averaging over the disorder distribution is supposed to result in a continuous piece in the overlap distribution. Often the countable overlap distribution is interpreted as being related to or implying a countable number of pure states; however there seems not to be much justification for this \cite{Bol}. \smallskip %\newline Conceptually, our example illustrates a number of points: \smallskip \noindent Newman and Stein \cite{NS1,NS2,NS3,NS4,NS5, N} have shown that self-averaging arguments, based on the use of the ergodic theorem, apply to many finite-dimensional. disordered (spin-glass) models (although not to the equivalent-neighbour Sherrington-Kirkpatrick model). In particular they find that for systems with a spatial structure, in contrast to the Parisi theory's predictions for the Sherrington-Kirkpatrick model \cite{MPV,PT}, overlap quantities cannot depend on the realization of the disorder parameter, that is, they are "self-averaging". For example, for Edwards-Anderson-type spin-glass models there is no difference between a typical (Parisi SK-prediction: supported on countably many values, dense) and an averaged (Parisi SK-prediction: continuous) overlap distribution. \bigskip As mentioned before, for many nonperiodic systems, the individual (nonperiodic) sequences %-- or tilings -- are ground states for some translation invariant deterministic interaction. Thus they are often used as models for non-periodic (quasi-)crystals, \cite{Aub1,rad,Rad1, GERM,Mi1,Rue1,SBGC,Sen}. It therefore becomes natural to consider various questions naturally arising in statistical mechanics about such systems, whether about their spectral properties \cite{ BaHo,EMw, LeeMo, LeeMoSo} or, as was done in \cite{EHM,E} and also here, about their overlap distributions. Typically, there are uncountably many such ground states, and as mentioned before, they can have nontrivial overlap distributions; somewhat surprisingly even without the presence of any disorder. %In the period doubling and paperfolding sequences, %the number of possible overlap values is countable, and the set of sequences %has an ultrametric structure. It turns out that: \begin{itemize} \item There is no general connection between the number of pure states -uncountable- and the number of possible overlap values --which can be one, countably or uncountably many--. \item Triviality, non-triviality, and ultrametricity of the Parisi overlap distribution all can be realized without any role for the disorder. \item A dense discrete overlap distribution, with a hierarchical structure, can be realised (by the paperfolding system). \end{itemize} \section{Main Result} For background on the paperfolding system we refer to \cite{BaMoRicSi, DekMenPoo, AS}, and references mentioned there. \smallskip \noindent A paperfolding sequence can be constructed as follows: \newline In step 1, we choose $k_1$ to be a number from the set $\{1,2,3,4 \}$. \newline Now we fill all sites of the form $k_1+4 \Z$ with pluses, and all sites of the form $k_1+2+4 \Z$ with minuses. \newline Thus we have the period-4 structure covering half the sites: \newline $ .....\ +.-.+.-.+.-.+.-. \ ...$ \newline In step 2, we again choose $k_2$ to be a number from the set $\{1,2,3,4 \}$. \newline Now we occupy all sites of the form $k_1+1+ 2k_2 + 8\Z$ with pluses and all sites $k_1+5+ 2k_2 + 8 \Z$ with minuses. We repeat this step and in the $m$th step we again occupy one quarter of the remaining empty sites with pluses, periodically with period $2 \times 2^m$, and the quarter of the sites at distance $2^m$ from the pluses we occupy with minuses, periodically with the same period. Thus in each step one half of the remaining sites are filled. \smallskip After making a choice for $k_m$, for all $m \in \N$, all sites will be filled, and we have a paperfolding sequence. Notice that in every step the uncountable set of sequences is divided into four times as many subsets as in the previous step, which generates the hierarchical (ultrametric) structure of the uncountable set of sequences. %which is obtained by having ones on a sublattice of period 4, and minus ones %on the sublattice at distance 2 from it, and then step by step %filling up the empty parts %with ones on and minus ones at sublattices of periods $2^{(m+2)} Z$, With probability $\frac{1}{2}$ we find the overlap value $q =0$, which happens if in step 1 one ``odd'' and one ``even'' sequence was chosen, that is, $k_1 - k_1'$ is odd. With probability $\frac{1}{8}$ one finds overlap values $q$ equal to plus or minus $\frac{1}{2}$ respectively, which happens if in the first step either the same ($k_1=k_1'$), or the opposite ($|k_1- k_1'|=2$) choice was made, and in step 2 a different even-odd choice for $k_2$ and $k_2'$. We repeat this argument, and we find that with probability $\frac{1}{2^{2n+1}}$ one finds overlap values $\frac{(2p+1)}{2^{n}}$, with $p= -2^{n-1}, ......, 2^{n-1}-1$ \cite{Grothe}. Thus in the end, the overlap distribution becomes concentrated on the dyadic rationals. \smallskip \newline Summarizing we have proven: \begin{theorem}\label{theorempaperfolding} The overlap distribution of paperfolding sequences is given by: \begin{equation} p(q) = \sum_{n=0}^{\infty} \sum^{'}_{m} \frac{1}{2} \left(\frac{1}{4} \right) ^{n} \delta\left(q, \frac{m}{2^{n}} \right) \end{equation} where the integers $n,m$ must satisfy the following conditions: \begin{itemize} \item if $n=0$ then $m=0$, \item if $n>0$ then $m$ is odd and $|m|<2^{n}$. \end{itemize} \end{theorem} {\bf Comments:} Properties which are common between the period-doubling and the paperfolding system are: 1) Limit periodicity. The sequences are a countable union of periodic subsequences of pluses and minuses. This implies a countable number of possible values for the overlap. 2) There is a tree (hierarchical, ultrametric) structure in the set of sequences-- the pure states--. As the overlap between two sequences depends on the level of the tree at which a different even-odd choice is made, the overlap distribution reflects this. 3) Both paperfolding and period-doubling sequences form model sets, obtained by a cut and project scheme, with an internal space of 2-adic numbers. \cite{BaMoSc,BaMoRicSi, Sc}. This agrees with the existence of a connection between Parisi's replica-symmetry-breaking ideas and p-adic numbers as was suggested in \cite{ParSou}. \smallskip However, in contrast to the period-doubling case, the set of paperfolding sequences is spin-flip symmetric, and the set on which the overlap distribution is concentrated now lies dense in the interval $[-1, +1]$. %can be obtained by a kind of cut- and project approach with an %internal space of p-adic numbers. {\bf As p-adic numbers have an %ultrametric structure, this should be related to 2). More comments here %already???} %Remark1: diffraction spectrum is what in \cite{EMw} we called atomic spectrum, %dynamical spectrum in the terms of that paper is the union of the %atomic and molecular spectrum, compare \cite{BaHo, BaLe, BaMo, LeeMoSo}. %Remark 2: Tilings. %We want to comment especially on Wang tilings, that is tilings which %live on some periodic lattice , se e.g. \cite{Mi1} ({\bf and further refs???)}. %Such tiling models, including their overlaps, have been a.o. studied %by Parisi-Leuzzi \cite{LeuPa}, as models for glasses. Although it is not %expected that 2-dimensional non-periodic order can occur in such models at %finite T, one can either study the ground state properties of such systems or %otherwise stabilize the non-periodic structure by adding an extra direction %\cite{EMZ, Gaehler} . As remarked in \cite{LeuPa}, with identical %fixed boundary conditions one has overlap 1, however, in contrast to %that we note that with periodic %boundary conditions (or two copies of a translation-invariant %infinite-system measure) one obtains an ultrametric overlap distribution. %It is known that various of these Wang tilings also can be obtained as %model sets --cut and project schemes-- starting from p-adic originals. %\cite{BaMoSc, LeeMo, Sc} {\bf (you mentioned Robinson tiles before, further %comments???). %Questions: %1.Are there "Sturmian or rotation tilings", with continuous overlap %distributions or is this impossible? %The tiles should live on a regular lattice (Wang tiles?). %2. If the number of tiles (spin values) is larger than one, do we get a %distribution on overlap matrices? Is this a useful object? %In the paperfolding example, where one starts with 4 values abcd %a or b =1 , c or d =-1 %the ac overlap distribution is trivial (1 with prob. one quarter, zero %otherwise, the ab overlap is nontrivial, probably the rest is %expressible in those). %3) Do you know something we can say about Ammann-Beenker, Robinson %Kari-Culik, Parisi-Leuzzi etc tilings?} %\cite{SBGC}, %\cite{Mi1,BQ,Rad1,Sen}. %\cite{EM2,Rue1,Aub1,rad}. %{GERM,Aub1,rad}. %\cite{EM1,EZ}. \bigskip \noindent {\em Acknowledgements}: %for a financial support under the grant KBN 2 P03A 015 11. A.C.D. v.E became interested in these issues during an earlier collaboration with Bert Hof and Jacek Mi\c{e}kisz. He also acknowledges very useful discussions with Michael Baake, Jacek Mi\c{e}kisz and Dimitri Petritis. We also thank Michael Baake for some helpful advice on the manuscript. \addcontentsline{toc}{section}{\bf References} \begin{thebibliography}{10} \bibitem{ADJR} V. Anagnostopoulou, K. D\'iaz-Ordaz, O. Jenkinson and C. Richard. newblock Sturmian maximizing measures for the piecewise-linear cosine family. \newblock Dresden preprint, 2010. \bibitem{AS} J.-P. Allouche and J. Shallit \newblock Automatic sequences. \newblock Cambridge University Press, Cambridge, 2003. \bibitem{Aub1} S. Aubry. \newblock Weakly periodic structures and example. \newblock {\em J.Phys. (Paris) Coll.}, C3-50:97--106, 1989. \bibitem{BaHo} M.~Baake and M.~H\"offe. \newblock Diffraction of random tilings:Some rigorous results. \newblock {\em J. Stat. Phys.}99:219--261, 2000. %\bibitem{BaLe} %M.~Baake and D.~Lenz. %\newblock Dynamical systems on translation bounded measures: Pure point %dynamical and diffraction spectra. %\newblock Ergodic Theory and Dynamical Systems 24, p1867--1893, arXiv:math.DS{/}0302061, 2004. \bibitem{BaMoRicSi} M.~Baake, R.~V.~Moody, C.~Richard, B~Sing. \newblock Which distributions of matter diffract?-- Some answers. \newblock Quasicrystals:Structure and Physical Properties: Ed. H.-R. Trebin, Wiley-VCH p.188--207, arXiv:math-ph{/}0301019, 2003. \bibitem{BaMoSc} M.~Baake, R.~V.~Moody and M.~Schlottmann. \newblock Limit-(quasi)periodic point sets as quasicrystals with p-adic internal spaces. \newblock {\em J. Phys. A, Math.Gen.} 31:5755--5765, 1998. \bibitem{BaMo} M.~Baake, R.~V.~Moody. \newblock Weighted Dirac combs with pure point diffraction. \newblock {\em J. Reine und Angewandte Mathematik} 573, p61--94, arXiv:math.MG{/}0203030, 2004. \bibitem{Bol} E. Bolthausen. In {\em Spin glasses}, Eds.E. Bolthausen and A. Bovier. Springer LNM 1900. 2007. See in particular p16. %\bibitem{BQ} %{\em Beyond Quasicrystals} (les Houches 1994), Eds. F. Axel and D.Gratias. %\newblock Springer, Berlin etc., 1994. %\bibitem{brisla} %J.~Bricmont and J.~Slawny. %\newblock Phase transitions in systems with a finite number of dominant %ground states. %\newblock {\em J. Stat. Phys.}, 54:89--161, 1989. %\bibitem{sin1} %E. I. Dinaburg and Ya.~G.~Sinai. %\newblock An analysis of the ANNNI model by the Peierls contour method. %\newblock {\em Commun. Math. Phys.} 98:119-144, 1985. %\bibitem{sin2} %E. I. Dinaburg, A. E. Mazel, and Ya. G. Sinai. %\newblock The ANNNI model and contour models with interaction. %\newblock {\em Sov. Sci. Rev. C Math/Phys,} 6:113-168, 1987. \bibitem{DekMenPoo} M.~Dekking, M.~Mendes-France, A.~van der Poorten \newblock FOLDS! (I,II and III) \newblock {\em The Mathematical Intelligencer}, 4, p. 130--138, p. 173--181, p. 190--195, 1982. %\bibitem{Dob1} %R.~L.~Dobrushin. %\newblock Gibbs states describing coexistence of phases for a %three-dimensional Ising model. %\newblock {\em Th. Prob. Appl.}, 17:582--600, 1972. %\bibitem{DobShl1} %R.~L.~Dobrushin and S.~B.~Shlosman. %\newblock The problem of translation invariance of Gibbs states %at low temperatures. %\newblock {\em Sov. Sci. Rev.}, 5:53--195, 1985. \bibitem{E} A.~C.~D. van Enter. \newblock On the set of pure states for some systems with non-periodic long-range order. \newblock {\em Physica A} 232:600--607, 1996. \bibitem{EHM} A.~C.~D. van Enter, A.Hof and J.~Mi\c{e}kisz. \newblock Overlap distributions for deterministic systems with many pure states. \newblock {\em J. of Phys. A, Math. Gen.} 25:L1133--1137, 1992. %\bibitem{EM1} %A.~C.~D. van Enter and J.~Mi\c{e}kisz. %\newblock Breaking of periodicity at positive temperatures. %\newblock {\em Comm. Math. Phys.} 134:647--651, 1990. \bibitem{EMw} A.~C.~D. van Enter and J.~Mi\c{e}kisz. \newblock How should one define a (weak) crystal? \newblock {\em J. Stat. Phys.} 66:1147--1153, 1992. %\bibitem{EMZ} %A.~C.~D. van Enter, J.~Mi\c{e}kisz and M.~Zahradn\'ik. %\newblock Nonperiodic long-range order for fast-decaying interactions at %positive temperatures. %\newblock {\em J. Stat. Phys.} 90:1441--1447, 1998. %\bibitem{EZ} %A.~C.~D. van Enter and B.~Zegarlinski. %\newblock Non-periodic long-range order for one-dimensional pair interactions. %\newblock To appear in {\em J. of Phys. A, Math. Gen.} %\bibitem{sel} %M. E. Fisher and W. Selke. %\newblock Low twemperature analysis of the axial next-nearest %neighbour Ising model near its multiphase point. %\newblock {\em Phil. Trans. Royal Soc.} 302:1, 1981. %\bibitem{fish} %M. E. Fisher and A. M. Szpilka. %\newblock Domain-wall interactions. I. General features %and phase diagrams for spatially modulated phases. %\newblock {\em Phys. Rev.} B36:644-666, 1987. %\bibitem{Gaehler} %F.~G\"ahler and M.~Reichert. %\newblock Finite-Temperature Order-Disorder phase transition in a Culster %Model of Decagonal Tilings. %\newblock arXiv:cond-mat{/}0302074, 2003. \bibitem{GERM} C.~Gardner, J.~Mi\c{e}kisz, C.~Radin and A.~C.~D.~van Enter. \newblock Fractal symmetry in an Ising model. \newblock {\em J. Phys. A, Math. Gen.}, 22:L1019--1023, 1989. \bibitem{Grothe} E.~de Groote. \newblock The overlap distribution of paperfolding sequences. \newblock Groningen bachelor thesis, 2010. %\bibitem{HoZa} %P.~Holick\'y and M.~Zahradn\'ik. %\newblock Stratified low temperature phases of stratified spin models. A %general Pirogov-Sinai approach. %\newblock ESI preprint, 1996. %\bibitem{kak} S. Kakutani. %\newblock Ergodic theory of shift transformations. %\newblock {\em Proc. Fifth Berkeley Sympos. Math. Statist. Probability II,} %405:4414, 1967. %\bibitem{kea} M. Keane. %\newblock Generalized Morse sequences. %\newblock {\em Zeit. Wahr.} 10:335-353, 1968. %\bibitem{laan} %L. Laanait and N. Moussa. %\newblock Rigorous study of the spin-1/2 Ising model in a layered %magnetic field at low temperatures. %\newblock {\em J. Phys. A: Math. Gen.} 30:1059-1068, 1997. \bibitem{LeeMo} J.-Y.~Lee, and R.~V.~Moody. \newblock Lattice substitution systems and model sets. \newblock {\em Discrete and Computational Geometry} 25:173--201, 2001. \bibitem{LeeMoSo} J.-Y.~Lee, R.~V.~Moody and B.~Solomyak. \newblock Pure point and dynamical diffraction spectra. \newblock {\em Ann. Inst. Poincar\'e, Theor. Math. Phys.} 3:1003--1018, 2002. %\bibitem{LeuPa} %L.~Leuzzi and G.~Parisi. %\newblock Thermodynamics of a tiling model. %\newblock {\em J. Phys. A: Math. Gen.} 33:4215--4225, 2000. \bibitem{LW} K.~L\"u and J.~Wang \newblock Construction of Sturmian sequences. \newblock {\em J. Phys. A: Math. Gen.} 38:2891--2897, 2005. \bibitem{MPV} M.~M\'ezard, G.~Parisi and M.~A.~Virasoro. \newblock {\em Spin Glass Theory and Beyond.} World Scientific, Singapore, 1987. \bibitem{Mi1} J.~Mi\c{e}kisz. \newblock {\em Quasicrystals. Microscopic models of non-periodic structures} \newblock Leuven Lecture Notes in Mathematical and Theoretical Physics, Vol. 5, 1993. %\bibitem{mier2} J. Mi\c{e}kisz and C. Radin. %\newblock The third law of thermodynamics. %\newblock {\em Mod. Phys. Lett.} 1:61-66, 1987. %\bibitem{mier} J. Mi\c{e}kisz and C. Radin. %\newblock The unstable chemical structure of the quasicrystalline alloys. %\newblock {\em Phys. Lett.} 119A:133-134, 1986. %\bibitem{mie2} J. Mi\c{e}kisz. %\newblock A microscopic model with quasicrystalline properties. %\newblock {\em J. Stat. Phys.} 58:1137, 1990. %\bibitem{mie3} J. Mi\c{e}kisz. %\newblock Stable quasicrystalline ground states. %\newblock To appear in {\em J. Stat. Phys}. \bibitem{N} C.~M.~Newman. \newblock Topics in Disordered Systems. \newblock ETH Lectures in mathematics, Birkh\"auser, Basel, Boston, Berlin 1997. \bibitem{NS1} C.~M.~Newman and D.~L.~Stein. \newblock The metastate approach to thermodynamic chaos. \newblock {\em Phys. Rev. E} 55: 594--5211, 1997. \bibitem{NS2} C.~M.~Newman and D.~L.~Stein. \newblock Thermodynamic chaos and the structure of short-range spin glasses. \newblock {\em Mathematical Aspects of Spin Glasses and Neural Networks} 243--287, eds A.~Bovier and P.~Picco, Birkh\"auser, Boston-Basel-Berlin, 1998. \bibitem{NS3} C.~M.~Newman and D.~L.~Stein. \newblock Ordering and Broken Symmetry in Short-ranged Spin Glasses. \newblock arXiv:cond-mat{/}0301403 {\em J. Phys., Cond. Matt.}, 15, R1319--R1364, Topical Review, 2003. \bibitem{NS4} C.~M.~Newman and D.~L.~Stein. \newblock The state(s) of replica symmetry breaking: Mean field theories versus short-ranged spin glasses. (Formerly known as ``Replica Symmetry Breaking's New Clothes'') \newblock {\em J. Stat.Phys.} 106: 213--244, 2002. \bibitem{NS5} C.~M.~Newman and D.~L.~Stein. \newblock Distribution of Pure States in Short-Range Spin Glasses \newblock {\em Int. J. Mod. Phys.},B24, 2091--2106, 2010. \bibitem{ParSou} G.~Parisi and N.~Sourlas. \newblock P-adic numbers and replica symmetry breaking. \newblock {\em Eur. J. Phys.}, B14, 235--242, 2000. \bibitem{PT} G.~Parisi and M.~Talagrand. On the distribution of the overlaps at given disorder. \newblock {\em C.R Acad. Sci. Paris, Ser I}, 339, 303--3006, 2004. \bibitem{rad} C.~Radin. \newblock Disordered ground states of classical lattice models. \newblock {\em Rev. Math. Phys.}, 3:125--135, 1991. \bibitem{Rad1} C.~Radin. \newblock Low temperature and the origin of crystalline symmetry. \newblock {\em Int. J. Mod. Phys. B}, 1:1157--1191, 1987. %\bibitem{RadSchu} %C.~Radin and L.~S.~Schulman. %\newblock Periodicity and classical ground states. %\newblock {\em Phys. Rev. Lett.}, 51:621--624, 1983. %\bibitem{rad2} C. Radin. %\newblock Crystals and quasicrystals: a lattice gas model. %\newblock {\em Phys. Lett.} 114A:381 1986. \bibitem{Rue1} D.~Ruelle. \newblock Do turbulent crystals exist? \newblock {\em Physica}, 113A:619--623, 1982. \bibitem{SBGC} D.~Schechtman, I.~Blech, D.~Gratias and J.~W.~Cahn. \newblock Metallic phase with long-range orientational order and no translation symmetry. \newblock {\em Phys. Rev. Lett.}, 53:1951--1953, 1984. \bibitem{Sc} M.~Schlottmann. \newblock Generalized model sets and dynamical systems. \newblock: In {\em Directions in Quasicrystals}, 143--159, eds M.~Baake and R.~V.~Moody, CRM Monograph Series, vol 13, AMS, providence, RI, 2000. \bibitem{Sen} M.S\'en\'echal. \newblock {\em Quasicrystals and geometry}. \newblock Cambridge University Press, 1995. %\bibitem{Sin} %Ya.~G.~Sinai. %\newblock {\em Theory of phase transitions, rigorous results.} %\newblock Pergamon Press, Oxford, 1982. %\bibitem{Sla} %J.~Slawny %\newblock Low-temperature properties of classical lattice systems: Phase %transitions and phase diagrams. %\newblock {\em Phase Transitions and Critical Phenomena}, vol. 11, %127--205. Eds. C.~Domb and J.~L.~Lebowitz. %\newblock Academic Press, 1987. \end{thebibliography} \end{document} ---------------1010120433707--