%\input mymag.tex \magnification=\magstephalf \baselineskip=16pt \parskip=8pt \rightskip=0.5truecm \def\a{\alpha} \def\b{\beta} \def\d{\delta} \def\e{\epsilon} \def\f{\phi} \def\g{\gamma} \def\l{\lambda} \def\k{\kappa} \def\r{\rho} \def\s{\sigma} \def\t{\tau} \def\th{\theta} \def\x{\xi} \def\z{\zeta} \def\o{\omega} \def\D{\Delta} \def\L{\Lambda} \def\G{\Gamma} \def\O{\Omega} \def\S{\Sigma} \def\T{T} \def\Th{Theta} \def\X{\Xi} \def\del #1{\frac{\partial^{#1}}{\partial\l^{#1}}} \def\tb{\tilde \b} \def\bt{\tilde \b} \def\mh{\hat \mu} \def\hm{\hat \mu} \def\Const{Const\,} \def\const{const\,} \def\Norm{Norm.\,} \def\1{1} \def\E{I\kern-.25em{E}} \def\N{I\kern-.22em{N}} \def\M{I\kern-.22em{M}} \def\R{I\kern-.22em{R}} \def\Q{I\kern-.22em{Q}} \def\Z{{Z\kern-.5em{Z}}} \def\C{C\kern-.75em{C}} \def\P{I\kern-.25em{P}} %\def\1{{1\kern-.25em\hbox{\rm I}}} %\def\P{\hskip.2em\hbox{rm P\kern-0.8em{I}\hskip.7em}} \def\der{\frac{d}{dx}} \def\del{\partial} % Spezielle Definitionen \def\pb{\Phi_\b} \def\AA{{\cal A}} \def\BB{{\cal B}} \def\CC{{\cal C}} \def\DD{{\cal D}} \def\EE{{\cal E}} \def\FF{{\cal F}} \def\GG{{\cal G}} \def\HH{{\cal H}} \def\II{{\cal I}} \def\JJ{{\cal J}} \def\KK{{\cal K}} \def\LL{{\cal L}} \def\MM{{\cal M}} \def\NN{{\cal N}} \def\OO{{\cal O}} \def\PP{{\cal P}} \def\QQ{{\cal Q}} \def\RR{{\cal R}} \def\SS{{\cal S}} \def\TT{{\cal T}} \def\UU{{\cal U}} \def\VV{{\cal V}} \def\WW{{\cal W}} \def\XX{{\cal X}} \def\YY{{\cal Y}} \def\ZZ{{\cal Z}} \def\chap #1{\line{\ch #1\hfill}} \def\sign{\,\hbox{sign}\,} \def\one{\hbox{J}\kern-.2em\hbox{I}} \def\un #1{\underline{#1}} \def\ov #1{\overline{#1}} \def\ba{{\backslash}} \def\sb{{\subset}} \def\sp{{\supset}} \def\rar{{\rightarrow}} \def\lar{{\leftarrow}} \def\em{{\emptyset}} \def\inn{{\rm int}} \def\cp{\rm \,cp\,} \def\pc{\oplus} \def\mc{\ominus} \def\lcn{\leq^{\rm cn}} \def\lwcn{\leq^{\rm wcn}} \def\pcn{\oplus^{\rm cn}} \def\pwcn{\oplus^{\rm wcn}} \def\lscn{\leq^{\rm scn}} \def\llcn{\leq^{\rm lcn}} \def\lswcn{\leq^{\rm swcn}} \def\pscn{\oplus^{\rm scn}} \def\pswcn{\oplus^{\rm swcn}} \def\Law{{\rm Law}} \def\eu{1} \def\h #1#2{h_{#1}^{({#2})}} \def\m #1{{\un m}^{({#1})}} \def\rr #1{\r^{({#1})}} \def\Y #1{Y^{({#1})}} % Non-character macros \newcount\foot \foot=1 \def\note#1{\footnote{${}^{\number\foot}$}{\ftn #1}\advance\foot by 1} \def\tag #1{\eqno{\hbox{\rm(#1)}}} \def\frac#1#2{{#1\over #2}} \def\text#1{\quad{\hbox{#1}}\quad} \def\proposition #1{\noindent{\thbf Proposition #1: }} %\def\datei #1{\headline{{}\hfill{#1.tex}\qquad\the\day.\the\month.\the\year}} \def\datei #1{\headline{\rm \the\day.\the\month.\the\year{}\hfill{#1.tex}}} \def\theo #1{\noindent{\thbf Theorem #1: }} \def\lemma #1{\noindent{\thbf Lemma #1: }} \def\definition #1{\noindent{\thbf Definition #1: }} \def\corollary {\noindent{\thbf Corollary: }} \def\proof{{\noindent\pr Proof: }} \def\proofof #1{{\noindent\pr Proof of #1: }} \def\endproof{$\diamondsuit$} \def\remark{{\bf Remark: }} \def\Korollar {\noindent{\thbf Korollar: }} \def\Zusatz {\noindent{\thbf Zusatz: }} \def\Bew{{\noindent\pr Beweis: }} \def\Bewvon #1{{\noindent\pr Beweis von #1: }} \def\endproof{$\diamondsuit$} \def\Bem{{\bf Bemerkung: }} \font\pr=cmbxsl10 scaled\magstephalf %\font\thbf=cmcsc10 scaled\magstep1 \font\thbf=cmbxsl10 scaled\magstephalf \long\def\fussnote#1#2{{\baselineskip=10pt \setbox\strutbox=\hbox{\vrule height 7pt depth 2pt width 0pt} \sevenrm \footnote{#1}{#2} }} \def\scr{\scriptstyle} % Font-Definitions \font\bfit=cmbxti10 \font\ch=cmbx12 \font\sevenit=cmti7 \font\ftn=cmr8 \font\ftit=cmti8 \font\it=cmti10 \font\bf=cmbx10 \font\srm=cmr5 % Lay-out setting %\parskip=16pt \pm 2pt %\baselineskip=12pt \overfullrule=0pt \font\tit=cmbx12 \font\aut=cmbx12 \font\aff=cmsl12 \nopagenumbers {$ $} \vskip2truecm \centerline{\tit METASTATES IN DISORDERED MEAN FIELD MODELS:} \vskip.2truecm \centerline{\tit RANDOM FIELD AND HOPFIELD MODELS\footnote{${}^*$}{\ftn Work supported by the DFG %under contract No. ... }} %\vskip.2truecm %\centerline{\tit } \vskip2truecm %\centerline{\tit } %\vskip2.5cm %\centerline{\aut Anton Bovier\footnote{${}^1$}{\ftn e-mail: %bovier@iaas-berlin.dbp.de}} %\vskip.1truecm %\centerline{\aff Institut f\"ur Angewandte Analysis und Stochastik} %\centerline{\aff Hausvogteiplatz 5-7, O-1086 Berlin, Germany} \vskip.5truecm \centerline{\aut Christof K\"ulske\footnote{${}^1$}{\ftn e-mail: kuelske@wias-berlin.de}} \vskip.1truecm \centerline{\aff WIAS} \centerline{\aff Mohrenstrasse 39} \centerline{\aff D-10117 Berlin, Germany} \vskip1.5truecm\rm \noindent {\bf Abstract:} We rigorously investigate the size dependence of disordered mean field models with finite local spin space in terms of metastates. Thereby we provide an illustration of the framework of metastates for systems of randomly competing Gibbs measures. In particular we consider the thermodynamic limit of the empirical metastate $1/N\sum_{n=1}^N \d_{\mu_n(\eta)}$ where $\mu_n(\eta)$ is the Gibbs measure in the finite volume $\{1,\dots,n\}$ and the frozen disorder variable $\eta$ is fixed. We treat explicitly the Hopfield model with finitely many patterns and the Curie Weiss Random Field Ising model. In both examples in the phase transition regime the empirical metastate is dispersed for large $N$. Moreover it does not converge for a.e. $\eta$ but rather in distribution for whose limits we give explicit expressions. We also discuss another notion of metastates, due to Aizenman and Wehr. \noindent {\bf Key Words: } Disordered Systems, Size Dependence, Random Gibbs States, Metastates, Mean Field Models, Hopfield Model, Random Field Model \vfill ${}$ \eject \chap{I. Introduction} In a recent series of papers [NS1],[NS2],[NS3], the interesting role of the volume dependence in disordered systems having more than one infinite volume Gibbs states was stressed. In a particularly interesting article [NS3] the notion of metastates, being probability measures on the states of the systems, was introduced to describe the volume dependence of system with frozen disorder. (See therein and the discussion with [P] for implications on the theory of spin glasses and the relation to the phenomena of replica symmetry breaking and non self averaging.) It is the aim of this paper to provide a rigorous step into the investigation of size dependence by metastates by our investigation of examples of random mean field systems. In the general case of disordered lattice spin systems in the presence of phase transitions, the problem of size dependence is the following. To start with a nontrivial situation, assume that the system admits more than one infinite volume Gibbs state. We consider the finite volume Gibbs measures $\mu_{\L_N}(\eta)$, for fixed realization of the disorder $\eta$, in the finite volume $\L_N$. We want to study a situation where the boundary conditions for the measures $\mu_{\L_N}(\eta)$ are such that they do not preselect one of the infinite volume Gibbs measures. (There are many natural situations, where it is (practically) impossible (or not of interest) to select Gibbs measures by boundary conditions. This is the case in spin glasses, where the Gibbs measures are not explicitly known. Note moreover that in mean field systems it is impossible to put boundary conditions at all.) To be concrete, we imagine that, for large $N$, the state of the system will be close to a mixture of random infinite volume Gibbs measures. That is, a good approximation for the finite volume Gibbs measures will often be $$ \eqalign{ &\mu_{\L_N}(\eta)\approx \sum_{m}p^{m}_N(\eta) \mu^{m}_{\infty}(\eta) \cr } \tag{1.1} $$ where $\left(\mu^{m}_{\infty}(\eta)\right)_{m\in \MM}$ are the (supposedly countably many) extremal Gibbs measures in the infinite volume. The problem of size dependence is: {\it Characterize the behavior of $\mu_{\L_N}(\eta)$ along the sequence $\L_N$.} This has some analogy with studying the orbit of a dynamical system with `time' $N$ (see [NS3]). Possible `extremes' that could occur here, are e.g. 1) convergence to one infite volume Gibbs measure or 2) an `erratic' sequence of states, a behavior that was named chaotic size dependence in [NS3]. A first question one may ask is: What Gibbs measures can be constructed through any subsequences $\L_{N_k}$ at all? More interesting even, lead by the dynamical system analogy, the following object was introduced in [NS3] to describe the `trajectory' $N\mapsto \mu_{\L_N}(\eta)$ in more detail:. $$ \eqalign{ &\k_N(\eta):=\frac{1}{N}\sum_{n=1}^N \d_{\mu_{\L_n}(\eta)} } \tag{1.2} $$ We will refer to $\k_N(\eta)$ as the `empirical metastate' and it will the main object of our study. Note that $\k_N$ is a random measure (through its $\eta$-dependence) on the states of the system. For large $N$ it will effectively be centered on the infinite volume Gibbs measures. There are various scenarios for the large $N$-behavior of $\k_N(\eta)$. If the system admits just one infinite volume Gibbs measure $\mu_\infty(\eta)$, the situation is easy: $\k_N(\eta)$ will converge to $\d_{\mu_\infty(\eta)}$. But note that also in the presence of phase transitions $\k_N$ can converge to a $\d$-measure. (Take as an example the ordinary ferromagnetic $2d$ Ising model without disorder, at low temperatures and put periodic boundary conditions. Then $\mu_N\rightarrow\frac{1}{2}(\mu_\infty^+ + \mu_\infty^-)$ with $N\uparrow\infty$. Consequently $\k_N\rightarrow \d_{\frac{1}{2}(\mu_\infty^+ + \mu_\infty^-)}$.) Nondegeneracy for the metastate can arise for random systems because, for fixed realization of the disorder, the finite volume fluctuations of the underlying random quantities could favor one of different phases even when they are equivalent in the average. While the structure of the phase diagram is nonrandom, the degeneracy between the phases in the finite volume would then be lifted in a random fashion. The information about how this is done lies in the $p^{m}_N(\eta)$. A variety of large-$N$ behavior is then possible: $\k_N$ can be the dirac measure on a mixture of states, it can be a mixture of dirac measures on pure states, it can be a mixture of dirac measures on mixtures. The second aspect is that $\k_N$ itself is a random object: In what way will the behavior of $\k_N$ depend on the realization? One could be tempted to expect that, as a generic behavior, $\k_N(\eta)$ will converge at (almost) all fixed $\eta$ (see [NS3] for a conjecture in that direction for certain systems). This were the case if the random objects $\mu_{\L_N}(\eta)$ lost memory very rapidly along the path $N\rightarrow \mu_N(\eta)$. In this paper we provide examples where this is {\it not } the case. Nevertheless, the limiting behavior of the empirical metastate can be described in our examples in two ways: First, it is possible to give {\it pathwise} approximation results, that hold for all typical realizations. Second, we suggest to study the behavior of the empirical metastate in {\it law}. This idea extends [APZ] where convergence of the Gibbs measures themselves was considered in law. We will see that, in our examples, infinite volume limits exist in law and give interesting information about the asymptotic behavior of the system along the path. In order to make sense out of this, one has to speak about notions of convergence of $\k_N$ with $N\uparrow\infty$. As it is common practice, we will choose the weak topologies that are inherited on the space of states and on the space of metastates when we choose as a starting point the product topology on the spin space (see Chapter 2). It makes the two spaces polish. The physical content of this notion of convergence is that convergence is checked locally on all levels. In the first part of this paper we will outline the general treatment of random mean field models with quadratic interaction and finite state space. Then we will consider two representatives of this class in detail. The advantage our mean field models is that they allow for explicit expressions for the weights $p^{m}_N(\eta)$ and good enough approximations (1). Our two examples are: \item{(i)} The Curie Weiss Random Field Ising Model (CWRFIM): Denote $\O:=\{1,-1\}^{\N}$ the space of Ising spin configurations $\s=(\s_i)_{i\in\N}$. We will denote the set of states (which is the set of probability measures on $\O$) by $\PP(\O)$. Let $\eta=(\eta_i)_{i\in\N}$, denote a sequence of i.i.d. Bernoulli variables taking the values $\e,-\e$ with probability $\frac{1}{2}$. For the inverse temperature $\b$ define the Gibbs measures $$ \eqalign{ &\mu_N(\eta)[(\s_i)_{i=1,\dots,N}]:=\frac{1}{\Norm}\exp\left( \frac{\b}{2N}\sum_{1\leq i,j\leq N}\s_i\s_j +\b\sum_{1\leq i\leq N}\eta_i\s_i\right) } \tag{1.3} $$ in the finite volume\footnote{$^1$}{ As usual they can also be viewed as measures on $\O$ by tensoring with arbitrary product measures for the spins at sites $i>N$.} $\{1,\dots,N\}$. The phase diagram of the system is well known (see [SW],[APZ]). At low temperatures and small $\e$ the model is ferromagnetic, i.e. there exist two `pure' phases, a ferromagnetic $+$ phase $\mu_\infty^+(\eta)$ and a $-$ phase $\mu_\infty^-(\eta)$. We restrict our interest to this region of the phase diagram. \item{(ii)} The Hopfield model with finite number of patterns: Let $\O$ be the space of Ising spins as above. Let $\xi=(\xi^\mu_i)_{i\in\N,\mu=1,\dots,M}$ denote i.i.d. Bernoulli variables taking the values $1,-1$ with probability $\frac{1}{2}$. $\xi^\mu=(\xi_i^\mu)_{i\in\N}$ are the {\it patterns} the model is supposed to learn ([Ho]). For $\b>0$ define the finite volume Gibbs measures $$ \eqalign{ &\mu_N(\xi)[(\s_i)_{i=1,\dots,N}]:=\frac{1}{\Norm}\exp\left( \frac{\b}{2N}\sum_{1\leq i,j\leq N}\sum_{1\leq \nu\leq M} \x^\nu_i\x^\nu_j \s_i\s_j\right) } \tag{1.4} $$ Due to our restriction on the number of patterns to remain fixed when $N\uparrow\infty$, we are deep inside the `region of perfect memory' if $\b>1$. This means that, for large $N$, the system is approximately in a mixture of the $M$ `Mattis states' $\mu^\nu_{\infty}(\xi)$. The latter is a state, associated the $\nu$-th pattern, s.t. the overlap vector $\left(\frac{1}{N}\sum_{i=1}^N\x_i^\r \s_i \right)_{\r=1,\dots,M}$ is centered around $\pm m^*(\b)a^\nu$, where $a^\nu$ is the $\nu$-th unity vector in $\R^M$. ($m^*(\b)$ is the solution of the ordinary Curie Weiss Mean Field equation.) For precise statements, see e.g. [BGP],[BG1]. For the state of the art in the Hopfield model with $\lim_{N\uparrow\infty}\frac{M(N)}{N}=\a$ small and positive we refer to [BG2] where a beautiful proof of the validity of the replica symmetric solution is given. One reason for treating the Hopfield model here is of course, that it can be viewed as an interpolation between a ferromagnet and a spin. For the limiting distribution of the empirical metastates in these two models we will prove the following theorems. (For the pathwise approximation results and related information, see Theorems 1' and 2'). These show that even in these simple models there is some richness in the empirical metastate. \theo{1}{\it For all bounded continuous functions $F:\PP(\O)\mapsto \R$ we have the limit in law $$ \eqalign{ &\lim_{N\uparrow \infty}\frac{1}{N}\sum_{n=1}^N F(\mu_n(\eta)) =^{\hbox{law}}n_\infty F\left(\mu_\infty^+(\eta)\right) +(1-n_\infty) F\left(\mu_\infty^-(\eta)\right) } \tag{1.5} $$ where $n_\infty$ is a random variable, {\it independent of $\eta$} on the r.h.s, distributed according to $$ \eqalign{ &\P\left[ n_\infty\r \right] } \tag{3.21} $$ with a standard normal variable $G$. >From that we have, for $\eta\in\un\HH$, $$ \eqalign{ &\lim_{N\uparrow\infty} \left(\tilde\mu_N(\eta)\left[B_{\r}(m)\right] -p^m_N(\eta) \right)\leq \lim_{N\uparrow\infty} \left(\bar\mu_N(\eta)\left[B_{2\r}(m)\right] -p^m_N(\eta) \right) } \tag{3.22} $$ Similarly we can obtain the lower bound $$ \eqalign{ &\lim_{N\uparrow\infty} \left(\tilde\mu_N(\eta)\left[B_{\r}(m)\right] -p^m_N(\eta) \right)\geq \lim_{N\uparrow\infty} \left(\bar\mu_N(\eta)\left[B_{\frac{\r}{2}}(m)\right] -p^m_N(\eta) \right) } \tag{3.23} $$ which proves the claim. \endproof We come to the \proofof{Lemma 2} Take $\eta\in\un\HH$. We only have to check convergence on a local event of the form $A:=\{\s_{\L}=\o_{\L}\}$ with fixed $\o_{\L}$. Then, using the factorization formula (3.8), we have $$ \eqalign{ &\left|\mu_N(\eta)[A] -\sum_{m\in \MM}p^m_N(\eta)\mu^0_\infty(m,\eta)[A]\right| \leq \tilde\mu_N(\eta)\left[ \left(\cup_{m\in\MM}B_{\r_N}(m)\right)^c\right]\cr &+\quad\sum_{m\in\MM} \left|\int_{B_{\r_N}(m)}\tilde\mu_N(\eta)(d\tilde m) \mu^0_\infty(\tilde m, \eta)[A] -p^m_N(\eta)\mu^0_\infty(m,\eta)[A]\right|\cr } \tag{3.24} $$ where the first term on the r.h.s. vanishes under the $N$-limit (see first remark). We pick one $m$ in the sum and write $$ \eqalign{ &\left|\int_{B_{\r_N}(m)}\tilde\mu_N(\eta)(d\tilde m) \mu^0_\infty(\tilde m, \eta)[A] -p^m_N(\eta)\mu^0_\infty(m,\eta)[A]\right|\cr &\leq \int_{B_{\r_N}(m)}\tilde\mu_N(\eta)(d\tilde m) \left|\mu^0_\infty(\tilde m, \eta)[A]-\mu^0_\infty(m, \eta)[A]\right|\cr &\quad+\left| \tilde\mu_N(\eta)\left[B_{\r_N}(m)\right] -p^m_N(\eta)\right|\mu^0_\infty(m,\eta)[A]\cr } \tag{3.25} $$ The first term goes to zero with $\r_N\downarrow 0$, due to the continuity of the function $$ \eqalign{ &\tilde m\mapsto\mu^0_\infty(\tilde m, \eta)[A] } \tag{3.26} $$ (In fact, it is $\CC^\infty$ everywhere; all derivatives exist for all $\tilde m\in\R^M$, due to the assumed boundedness of the order parameter.) The second term goes to zero according to the assumption (3.19). \endproof Putting the pieces from the Lemmata 1 and 2 together, we immediately obtain the following approximation result that we fix as \proposition{1}{\it Suppose that we are given a quadratic random mean field model of the above type %with infinite volume Gibbs measures %$\left(\mu^0_\infty(m,\eta)\right)_{m\in \MM}$ whose Hubbard-Stratonovich measures $\tilde \mu_N(\eta)$ obey the approximation property CR$(\r_N)$ with probability vector $\left(p^m_N(\eta)\right)_{m\in \MM}$. Then \item{(i)} For all $\eta\in\un\HH$ we have for the set of weak cluster points in $\PP(\O)$ $$ \eqalign{ &\CC\PP\left(\mu_N(\eta)\,\,, N=1,2,\dots\right) =\CC\PP\left(\sum_{m\in \MM}p^m_N(\eta)\mu^0_\infty(m,\eta)\,\,, N=1,2,\dots \right) } \tag{3.27} $$ \item{(ii)} Define the metastate $$ \eqalign{ &\tilde \k_N(\eta):=\frac{1}{N}\sum_{n=1}^N \d_{\sum_{m\in \MM}p^m_N(\eta)\mu^0_\infty(m,\eta)} } \tag{3.28} $$ Then, for all $\eta\in \HH'=\left\{\eta, \lim_{N\uparrow\infty} \frac{1}{N} \sum_{n=1}^N 1_{\eta\in {\HH(n)}^c} =0 \right\}$ we have $$ \eqalign{ &\lim_{N\uparrow\infty}\left(\int\k_N(\eta)(d\mu)F(\mu) -\int\tilde\k_N(\eta)(d\mu)F(\mu)\right)=0 } \tag{3.29} $$ for all bounded continuous $F$ on $\PP(\O)$. \item{(iii)} Assume that $\lim_{N\uparrow\infty}\P[\HH(N)]=1$. Then, for any bounded continuous function $G:\PP(\O)\times\HH\mapsto \R$ $$ \eqalign{ &\lim_{N\uparrow \infty}\left(\E\left[ G\bigl(\mu_N(\eta),\eta\bigr)\right] -\E\left[ G\left(\sum_{m\in \MM}p^m_N(\eta)\mu^0_\infty(m,\eta),\eta\right)\right] \right)=0 } \tag{3.30} $$ } \remark Again a word of care about the difference of $\un\HH$ and $\HH'$: The CWRFIM will give an example where, due to this difference, the set of cluster points becomes a.s. larger than the set of measures the metastate will be asymptotically supported in (See Chapter 4, Theorem 1'). Let us exploit another piece of information that we expect to hold in these models. Due to the permutation symmetry in mean field models the weights should behave asymptotically in the same way if the random variables in a {\it finite} volume are changed. This will be easy to verify in our examples, but we refrain from a general investigation here. So, we take this as an assumption and look for the consequence on the distribution of $\k_N(\eta)$. Due to the fact that we check convergence of $\k_N(\eta)(F)$ with {\it local} $F's$, the weights will then become {\it asymptotically independent} from the random variables the function $F$ feels. Let us use the notation $\Vert p-p' \Vert$ for two weights $p,p'$, viewed as elements in $\R^M$, for any norm on $\R^M$.\footnote{$^1$}{ Due to the finiteness of $M$, the choice of the norm doesn't matter; if we allowed $M$ to increase with $N$, this would become an important point.} The precise consequence of this phenomenon for the distribution of the empirical and for the conditioned metastate is \proposition{2}{\it Suppose, in addition to the assumption of proposition 1, that for all $\eta\in\un\HH$, for all finite $V\sb \N$, $$ \eqalign{ &\lim_{N\uparrow\infty}\sup_{\tilde\eta_V} \Vert p_N(\eta)-p_N(\eta+\tilde\eta_V) \Vert=0 } \tag{3.31} $$ where $\tilde\eta_V$ is a local perturbation in the finite volume $V$ s.t. $\eta_V+\tilde \eta_V$ lies in the support of the distribution $\P$. Let $\eta'$ denote a copy of disorder variables, independent of $\eta$. \item{(i)}If $\P[\HH']=1$, we have for the empirical metastate $$ \eqalign{ &\lim_{N\uparrow\infty}\int\k_N(\eta)(d\mu)F(\mu) =^{\hbox{law}}\lim_{N\uparrow\infty} \frac{1}{N}\sum_{n=1}^N F\left(\sum_{m\in \MM}p^m_N(\eta')\mu^0_\infty(m,\eta)\right) } \tag{3.32} $$ for all bounded continuous $F$ on $\PP(\O)$, whenever the limit on the r.h.s. exists. \item{(ii)} If $\lim_{N\uparrow\infty}\P[\HH(N)]=1$, we have for the conditioned metastate $$ \eqalign{ &\int\bar\k(\eta)(d\mu)F(\mu) =\lim_{N\uparrow\infty} \int \P(d\eta') F\left(\sum_{m\in \MM}p^m_N(\eta')\mu^0_\infty(m,\eta)\right) } \tag{3.33} $$ for all bounded continuous $F$ on $\PP(\O)$, whenever the limit on the r.h.s. exists. } \proof We may restrict to local functions $F$ of the form (2.2). To prove (i) it suffices to show that, given $F$, there exist versions $\eta_1,\eta_2$, of disorder variables, mutually independent, s.t. for all $\eta\in \un\HH$ we have the pointwise limit $$ \eqalign{ &\lim_{n\uparrow\infty}\left( F\left(\sum_{m\in \MM}p^m_n(\eta)\mu^0_\infty(m,\eta)\right) -F\left(\sum_{m\in \MM}p^m_n(\eta_1)\mu^0_\infty(m,\eta_2)\right)\right)=0 } \tag{3.34} $$ But note that such a function can be written in the form $$ \eqalign{ & F\left(\sum_{m\in \MM}p^m_n(\eta)\mu^0_\infty(m,\eta)\right) =\hat F\left( \left(\sum_{m\in \MM} p^m_n(\eta)\mu^0_\infty(m,\eta)(f_j)\right)_{j=1,\dots,l}\right) =:\tilde F\left(p_n(\eta),\eta_J \right) } \tag{3.35} $$ Due to the $\mu^0_\infty(m,\eta)$ being product measures with local dependence on the randomness, the $\eta$-dependence of $F$ other than through $p_n(\eta)$ itself remains {\it local}; the finite support $J$ of $\eta_J$ depends of course on the special choice of the functions $f_j$. Now, define the variable $\eta_1$ to coincide with $\eta$ on $J^c$ and to coincide with an independent copy on $J$. Define $\eta_2$ to coincide with $\eta$ on $J$ and to coincide with an independent copy on $J^c$. Since $\tilde F$ is a uniformly continuous function on the compact space of probability vectors (3.34) follows from the assumption (3.31). The same type of argument proves (ii). \endproof \bigskip Let us comment on the relations between the various objects we have obtained and the picture that arises from the above propositions, assuming the approximation properties (3.19) and (3.31). The full information on the level of metastates is contained in the object $\tilde\k_N(\eta)$ (3.28). It is centered on the infinite volume Gibbs states and contains the asymptotic form of the weights in the extremal decomposition. The weights will depend on the overall information of the random variables; therefore they will be asymptotically independent from the variables in a fixed finite volume. But, a local observable feels the underlying randomness only locally. Thus, for the limit {\it of the distribution} of the empirical metastate, the weights can be replaced with an independent copy, giving rise to an `additional randomness'. The limiting distribution of $\tilde\k_N(\eta)$ contains information about the asymptotic behavior along a path $N\mapsto \mu_N(\eta)$. On the other hand, the conditioned metastate contains no path properties at all: The weights, replaced with independent copies with the same distribution are integrated out. In that case, the whole size dependence is averaged `over infinity'. Its interpretation, suggested by the asymptotic independence, is then: Having no particular knowledge of the given realization of the disorder globally, the conditioned metastate gives the weights with one expects to find a specific mixture of states. This same metastate could be constructed by `thinning out' the sequence of volumes which occur in the empirical metastate in a nonrandom way, as has been shown for lattice systems in [N]. \bigskip \bigskip %\input sosdef %\datei{meta4} \chap{4. The Curie Weiss Random Field Ising Model in the 2 phase region} In this chapter we prove Theorem 1 for our first example, the CWRFIM, and the fixed realization results of Theorem 1'. By this we provide an easy example of the mean field picture of the last chapter. We will also see in this example that the set of fixed realization cluster can be strictly larger, almost surely, than the support of all the metastates. In the CWRFIM the logarithmic moment generating function of the order parameter (3.11) becomes $$ \eqalign{ &L(t,\eta_i)=\frac{1}{\b}\log\cosh(\b(t+\eta_i)) } \tag{4.1} $$ Due to our assumption that $\eta_i=\pm\e$ takes only two values it can be written in the form $$ \eqalign{ &L(t,\eta_i)=L_{+}(t)+L_{-}(t)\frac{\eta_i}{\e} } \tag{4.2} $$ where $$ \eqalign{ &L_{+}(t):=\frac{1}{2\b}\left( \log\cosh(\b(t+\e))+\log\cosh(\b(t-\e)) \right)\cr &L_{-}(t):=\frac{1}{2\b}\left( \log\cosh(\b(t+\e))-\log\cosh(\b(t-\e)) \right)\cr } \tag{4.3} $$ Thus the function $\Phi_N(m,\eta)$ becomes $$ \eqalign{ &\Phi_N(m,\eta)=\frac{m^2}{2}-L_{+}(m)-L_{-}(m)\frac{W_N}{N} } \tag{4.4} $$ where the dependence on the randomness on only through the random walk $$ \eqalign{ &W_N:=\sum_{1\leq i\leq N}\frac{\eta_i}{\e} } \tag{4.5} $$ This will make the analysis particularly easy, in that it reduces questions on the metastates to questions about the walk $W_N$. As said before in Chapter 3, the structure of the phase diagram is determined by the averaged function $\Phi^0_N(m)=\frac{m^2}{2} -L_{+}(m)$ which has been analysed in detail (see [SW],[APZ]): For `large magnetic fields' $\e>\frac{1}{2}$, it has only one global quadratic minimum at $m=0$. For $0\leq \e\leq \frac{1}{2}$ there exists a critical inverse temperature $\b_c(\e)$ s.t. for $\b>\b_c(\e)$ the system has two symmetric global quadratic minima at positions $\pm m^*\equiv \pm m^*(\b,\e)$; for $\b<\b_c(\e)$ the system has one global quadratic minimum at $m=0$. We assume for the rest of this chapter that we are in this two phase region.\footnote{$^1$} {At the phase transition line itself there exist two regions: For small $\e$ there is a unique global quartic minimum at $m=0$, as for the usual CW ferromagnet; for large $\e$ there are three global quadratic minima. These two line segments are separated by a tricritical point, where there is one global sixth order minimum.} The results about the metastate are now easy to understand heuristically: Define $\mu^\pm_{\infty}(\eta):= \mu_{\infty}^0(\pm m^*,\eta)$. Let us just replace the integral over $m$ in the definition of $\tilde\mu_N(\eta)$ by two delta functions at $\pm m^*$ with weights determined by the values of $\Phi_N(\pm m^*,\eta)$. Let us thus introduce the weights $$ \eqalign{ &p_N(W_N):=\frac{e^{c(\b)W_N}}{e^{c(\b)W_N}+e^{-c(\b)W_N}} } \tag{4.6} $$ with $c(\b)=\b L_{-}(m^*)$. Heuristically we have then $$ \eqalign{ &\mu_N(\eta)\approx p(W_N)\mu_{\infty}^+(\eta)+ (1-p(W_N))\mu_{\infty}^-(\eta) \cr } \tag{4.7} $$ Now, the argument in the exponent of $p(W_N)$, $W_N\sim N^\frac{1}{2}$, moves on a scale increasing with $N$. Thus, for the empirical metastate, we might even use the approximation $p(W_N)\approx 1_{W_N>0}$. Let us thus define $$ \eqalign{ &n_N(\eta):=\frac{1}{N}\#\{1\leq n\leq N|W_n>0\} } \tag{4.8} $$ Then, for a continuous function $F$ on $\PP(\O)$ we would have $$ \eqalign{ &\frac{1}{N}\sum_{1\leq n\leq N} F\left(\mu_n(\eta)\right) \approx n_N(\eta) F(\mu_{\infty}^+(\eta)) + (1-n_N(\eta))F(\mu_{\infty}^-(\eta)) } \tag{4.9} $$ which explains the results for the empirical metatate. Denote, following the notation of the last chapter, $$ \eqalign{ &\tilde\k_{N}(\eta):=n_N(\eta) \d_{\mu^+(\eta)} +(1-n_N(\eta))\d_{\mu^-(\eta)} } \tag{4.10} $$ Then the precise results are given by Theorem 1 and \theo{1'}{\it \item{(i)} For all $\eta$ in a full measure set, the set of weak cluster points equals $$ \eqalign{ &\CC\PP\{\mu_N(\eta),\,N=1,2,\dots\}\cr &\quad=\left\{q \mu^+(\eta)+(1-q)\mu^+(\eta),\,\, \frac{1}{q}=1+\exp(-2 c(\b)z) ,\, z\in\Z\cup\{+\infty\}\cup\{-\infty\}\right\} } \tag{4.11} $$ \item{(ii)} For all $\eta$ in a full measure set, for any continuous function $F:\PP(\O)\mapsto \R$ the empirical metastate is approximated by $$ \eqalign{ &\lim_{N\uparrow \infty}\left(\int\k_{N}(\eta)(d\mu)F(\mu) -\int\tilde\k_{N}(\eta)(d\mu)F(\mu)\right)=0 } \tag{4.12} $$ \item{(iii)} For all $\eta$ in a full measure set the conditioned metastate exists and equals $$ \eqalign{ &\bar\k(\eta) =\frac{1}{2} \d_{\mu^+(\eta)} +\frac{1}{2}\d_{\mu^-(\eta)} } \tag{4.13} $$ } \remark Note explicitly, that the conditioned metastate contains only the equal weight distribution on $\{-\frac{1}{2},\frac{1}{2}\}$, which is obtained by averaging over the variable $n_\infty$ of Theorem 1. The information it contains at all, it thus that, for large $N$, the system will be in one of the pure phases. The set of cluster points has also been observed by [APZ]. We would like to point our here that, a.s., it is strictly bigger than the support of the metastates. The special structure is of course due to the discrete nature of the distribution of the random fields; if their distribution were continuous, we would expect to get in fact all mixtures. The proof is of course an application of the general propositions 1 and 2 plus the model dependent estimates of the laplace asymptotics for the measure $\tilde\mu_N(\eta)$. To this end we will now introduce two sorts of `regular sets' of realizations of the disorder. One is $$ \eqalign{ &\HH_1(N):=\left\{\eta: |W_N(\eta)|\leq N^{\frac{1+\d}{2}} \right\} } \tag{4.14} $$ with some $0<\d<\frac{1}{2}$. We consider balls around the minima $\pm m^*$ with radii $$ \eqalign{ &\r_N:=N^{-\frac{1}{4}+\frac{\d}{2}} } \tag{4.15} $$ Then an estimation of the occurring integrals gives \proposition{3}{\it There exists a nonrandom $N_0=N_0(\b,\e)$ s.t. for all $N\geq N_0$ for all $\eta\in\HH_1(N)$ $$ \eqalign{ &\tilde\mu_N \left[B_{\r_N}(m^*)\cup B_{\r_N}(-m^*)\right] \geq 1-\exp\left(-\const(\b,\e) N^{\frac{1}{2}+\d}\right) } \tag{4.16} $$ and $$ \eqalign{ &\left|\log\frac{\tilde\mu_N \left[B_{\r_N}(m^*)\right]}{\tilde \mu_N\left[B_{\r_N}(-m^*)\right]} -2 c(\b) W_N \right|\leq \Const(\b,\e) N^{-\frac{1}{4}+\frac{\d}{2}} } \tag{4.17} $$ } \remark The proposition shows that outside exceptional sets one has a fairly explicit control about the cluster properties of $\tilde\mu_N(\eta)$, including the relative weights. We only remark that it is easy to see same bounds hold for the measure $\bar\mu_N$ (with a possible degradation of $\const(\b,\e)$ and $N_0$). We will postpone the proof to the end of this chapter. Let us also introduce the smaller regular sets $\HH_2(N)$ by imposing as a second condition that the $|W_N(\eta)|$ be not too small: $$ \eqalign{ &\HH_2(N):=\left\{\eta: |W_N(\eta)|\leq N^{\frac{1+\d}{2}} \text{and} |W_N(\eta)|\geq N^{\tilde\d} \right\} } \tag{4.18} $$ for $0<\tilde\d<\frac{1}{2}$. Denote, following our usual notation, $$ \eqalign{ &\HH'_{1,2}:=\left\{\lim_{N\uparrow\infty} \frac{1}{N} \sum_{n=1}^N 1_{\eta\in {\HH_{1,2}(n)}^c} =0 \right\}\cr &\un\HH_{1,2}:=\liminf_{N\uparrow\infty}\HH_{1,2}(N) } \tag{4.19} $$ Then we have \lemma{3}{\it \item{(i)} $\P[\HH_2']=1$, $\P[\un\HH_2]=0$ \item{(ii)} $\P[\HH_1']=\P[\un\HH_1]=1$ } \proof To prove the first claim in (i) we must show that $$ \eqalign{ &S_N:=\frac{1}{N} \sum_{1\leq n\leq N} 1_{W_n\in B_n}\rightarrow 0 } \tag{4.20} $$ a.s. where $B_n=\left\{x\in\R: |x|\geq N^{\frac{1+\d}{2}} \text{or} |x|\leq N^{\tilde\d}\right\}$. $S_N$ is nothing but the mean time of the walk spent in the `bad regions' $B_n$. Note that $S_{n}\leq 2 S_{2^{k+1}}$ for $2^k\leq n\leq 2^{k+1}$. Therefor it suffices to show that $S_{2^{k}}\rightarrow 0$ a.s. with $k\uparrow\infty$. By Borel-Cantelli it suffices to show that, for any (rational) $\e$, $$ \eqalign{ &\sum_{k=1}^\infty \P\left[ S_{2^k}>\e \right]<\infty } \tag{4.21} $$ But this follows simply from the Chebycheff inequality since $$ \eqalign{ &\P\left[ S_N>\e \right]\leq \frac{\E S_N}{\e} = \frac{1}{\e N} \sum_{1\leq n\leq N}\P\left[ W_n\in B_n \right]\leq \frac{\Const}{\e} N^{-\frac{1}{2}+\tilde\d} } \tag{4.22} $$ where we have used that, by standard estimates, $\P\left[W_n\in B_n \right]\leq \Const\left( N^{-\frac{1}{2}+\tilde\d}+ e^{-\const N^\d}\right)$ The second claim in (i) follows from the recurrence of the random walk. (ii) follows from the law of iterated logarithm. \endproof (i) shows that we really need to distinguish between the sets $\HH'_2$ and $\un\HH_2$. With these preparations we come to the \proofof{Theorem 1 and 1'} It is easy to check that from the estimates in proposition 3 follows property CR$(\r_N)$ along the sets $\HH_1(N)$ with the weights defined by (4.6). To show Theorem 1'(i), we note that it follows from proposition 1 (ii) that the cluster points are described in terms of the cluster points of the weights (4.6), for all $\eta$ in the full measure set $\un \HH_1$. But, due to the recurrence of the walk, these are of the form as in written in (4.11), a.s. To prove the rest of the statements, we use the different, `trivial' weights $$ \eqalign{ &p^{m^*}_N(\eta)=1_{W_N>0}\cr &p^{-m^*}_N(\eta)=1_{W_N\leq 0}\cr } \tag{4.23} $$ For Theorem 1'(ii), note that from proposition 3 also follows property CR$(\r_N)$ along the smaller sets $\HH_2(N)$ for the weights (4.23). This is a simple consequence of the imposed minimum size of $|W_N|$. Thus, Theorem 1'(ii) follows from proposition 1(ii) with the full measure set $\HH'_2$. To prove Theorem 1'(iii) and Theorem 1 note that we have for $\eta\in\un\HH_2$, because of the minimum size of $|W_N|$, $$ \eqalign{ &\lim_{N\uparrow\infty}\sup_{\tilde\eta_V} (1_{\sum_{i=1}^N\eta_i>0}-1_{\sum_{i=1}^N \eta_i+\sum_{i\in V}\tilde\eta_i)>0})=0 } \tag{4.24} $$ Note further, that $\lim_{N\uparrow\infty}\P[\HH_2(N)]=1$ (as has been seen in the proof of Lemma 3). Thus, Theorem 1' (iii) follows from proposition 2 (ii). To obtain Theorem 1, remark that, according to proposition 2 (i), the distributional limit is given by the expression $$ \eqalign{ &\lim_{N\uparrow \infty}\frac{1}{N}\sum_{n=1}^N F(\mu_n(\eta)) =^{\hbox{\it law}} \lim_{N\uparrow \infty}\left( n_N F\left(\mu_\infty^+(\eta)\right) +(1-n_N) F\left(\mu_\infty^-(\eta)\right)\right) } \tag{4.25} $$ where now $n_N$ are random variables with distribution as in (4.8), but independent of $\eta$. Now, it is a well known result from elementary fluctuation theory (see e.g. [Fe]) that the $n_N$ converge in distribution to a variable $n_\infty$ which is distributed according to the $\arcsin$-law (1.6). (And {\it not} to the equidistribution on $\{\frac{1}{2},-\frac{1}{2}\}$!) This concludes the proof.\endproof \bigskip Let us finally give the proof of proposition (3). The type of estimates used here are standard; we apply parts of what was used in [BG1] in a far more complicated situation. However, we include these computations here since they are prototypical for random mean field models. Thus, let $m^*>0$ is the largest solution of the mean field equation $m=L'_{+}(m)$. We define $R_\r:=\left(B_\r(m^*) \cup B_\r(-m^*)\right)^c$. We will have to estimate the corresponding integrals $$ \eqalign{ &I_\r^\pm:=\int_{B_\r(\pm m^*)}dm\exp\left( -\b N \left(\Phi_N(m)-\Phi^0(m^*)\right) \right)\cr &J_\r:=\int_{R_\r}dm\exp\left( -\b N \left(\Phi_N(m)-\Phi^0(m^*)\right) \right)\cr } \tag{4.26} $$ where we have dropped the $\eta$ in our notation. To prove the proposition we show that there exist $N_0=N_0(\b,\e)$ and $\const(\b,\e)>0$ s.t. for all $N\geq N_0$ and for all $\eta\in \HH_1(N)$ $$ \eqalign{ &\frac{J_{\r_N}}{I_{\r_N}^\pm}\leq \exp\left( -\const(\b,\e)N^{\frac{1}{2}+\d} \right) } \tag{4.27} $$ and $$ \eqalign{ &\frac{I_{\r_N}^\pm}{I_{\r_N}^\mp}\exp\left( \mp 2\b L_{-}(m^*) W_N \right) \geq 1-\const(\b,\e) N^{-\frac{1}{4}+\frac{\d}{2}} } \tag{4.28} $$ Before we start, we remark for later use that the higher derivatives of $L_\pm$ vanish at infinity: $$ \eqalign{ &\lim_{|m|\uparrow\infty} \left|\left(\frac{\del}{\del m}\right)^k L_+(m)\right|=0\quad,k\geq 2\cr &\lim_{|m|\uparrow\infty} \left|\left(\frac{\del}{\del m}\right)^k L_-(m)\right|=0\quad,k\geq 1\cr } \tag{4.29} $$ and are therefore uniformly bounded. We write $m=\pm m^*+v$ and treat the two cases $\pm$ at the same time. Then we have for $|v|\leq \r$, using the symmetry properties of the functions and of their derivatives, $$ \eqalign{ &\Phi_N(\pm m^*+v)-\Phi^0(m^*) \pm\frac{W_N}{N}L_{-}(m^*)\cr &=\frac{{\Phi^0}^{''}( m^*+\th v)}{2}v^2 -\frac{W_N}{N}L'_{-}(m^*)v -\frac{W_N}{N}\frac{L^{''}_{-}(\pm m^*+\th' v)}{2}v^2 } \tag{4.30} $$ with some $0\leq \th,\th'\leq 1$. Thus, on $|v|\leq \r$, $$ \eqalign{ &\Phi_N(\pm m^*+v)-\Phi^0(m^*)\pm\frac{W_N}{N}L_{-}(m^*) \leq\frac{b_+}{2}v^2- z v } \tag{4.31} $$ with $$ \eqalign{ &z:=\frac{W_N}{N}L'_{-}(m^*) } \tag{4.32} $$ and $$ \eqalign{ &b_+:= b_+(\r):=\sup_{v,|v|\leq\r} {\Phi^0}^{''}(m^*+v) +\left|\frac{W_N}{N}\right|\sup_{v,|v|\leq\r} \left|L^{''}_{-}(m^*+v)\right| } \tag{4.33} $$ Similarly we have $$ \eqalign{ &\Phi_N(\pm m^*+v)-\Phi^0(m^*) \pm\frac{W_N}{N}L_{-}(m^*) \geq\frac{b_-}{2}v^2- z v } \tag{4.34} $$ with $$ \eqalign{ &b_-:=b_-(\r):=\inf_{v,|v|\leq\r} {\Phi^0}^{''}(m^*+v) -\left|\frac{W_N}{N}\right|\sup_{v,|v|\leq\r} \left|L^{''}_{-}(m^*+v)\right| } \tag{4.35} $$ \lemma{4}{Denote $P(x)=\P[G\geq x]$ for a standard Normal $G$. If $a\in [-\g,\g]$, $\g>0$ $$ \eqalign{ &\int_{|x|\geq \g}e^{-\frac{x^2}{2}+ax}\frac{dx}{\sqrt {2\pi}} =e^{\frac{a^2}{2}}\left(P(\g-a)+P(-\g-a) \right) \leq e^{-\frac{\g^2}{2}+a\g} +e^{-\frac{\g^2}{2}-a\g} } \tag{4.36} $$ \proof From the well known estimate $P(x)\leq \exp(-x^2/2)$.\endproof } This gives, for $\r\geq 4 |z|/b_+$, $$ \eqalign{ &\int_{|v|\geq\r}e^{-\b N\left(\frac{b_+}{2}v^2- z v\right)}dv \leq 2\sqrt{\frac{2\pi}{\b N b_+}} e^{-\b N\left(\frac{b_+}{2}\r^2- |z| \r\right)} \leq\sqrt{\frac{8\pi}{\b N b_+}} \exp\left(-\frac{\b N b_+ \r^2}{4} \right)\cr } \tag{4.37} $$ With $$ \eqalign{ &\int_{\R}e^{-\b N\left(\frac{b_+}{2}v^2- z v\right)}dv =\sqrt{\frac{2\pi}{\b N b_+}}\exp\left( \frac{z^2\b N}{2b_+} \right) } \tag{4.38} $$ we obtain from this $$ \eqalign{ &I_\r^\pm\geq \exp\left( \pm\b L_{-}(m^*) W_N \right) \sqrt{\frac{2\pi}{\b N b_+}} \left(\exp\left( \frac{z^2\b N}{2b_+} \right)- 2\exp\left(-\frac{\b N b_+ \r^2}{4} \right)\right)\cr } \tag{4.39} $$ For the upper bound we simply write $$ \eqalign{ &I_\r^\pm\leq \exp\left( \pm\b L_{-}(m^*) W_N \right) \int_{\R}e^{-\b N\left(\frac{b_-}{2}v^2- z v\right)}dv\cr &=\exp\left( \pm\b L_{-}(m^*) W_N \right) \sqrt{\frac{2\pi}{\b N b_-}} \exp\left( \frac{z^2\b N}{2b_-} \right) \cr } \tag{4.40} $$ Next we estimate the integral over the outer region. We use the following rough estimate. \lemma{5}{\it For each $\e,\b$ in the two phase region there exists a constant $\hat c(\b,\e)$ s.t. for all $v\geq -m^*$ $$ \eqalign{ &\Phi^0(m^*+v)-\Phi^0(m^*)\geq \hat c(\b,\e)v^2\cr &\sup_{m\in\R} |L_-(m)|=:c_2(\b,\e)<\infty\cr } \tag{4.41} $$ } \proof The first claim states that $\Phi^0$ is bounded below by a parabol on $\R_\geq$. It can be chosen to coincide with $\Phi$ at the points $m=0$ and $m^*$ (where the absolute minimum is attained.) The proof is elementary. To prove the second claim it suffices to verify that $\lim_{m\rightarrow\pm\infty} |L_-(m)|<\infty$ which is again elementary. \endproof >From this we have $$ \eqalign{ &J_\r:=\int_{R_\r}dm\exp\left( -\b N \left(\Phi_N(m)-\Phi^0(m^*)\right) \right)\cr &\leq 2 \exp\left(c_2(\b,\e)|W_N| \right) \int_{|v|\geq \r}dv\exp\left( -\b N \hat c(\b,\e)v^2 \right)\cr &\leq 2 \exp\left(c_2(\b,\e)|W_N|-\b N \hat c(\b,\e)\r^2 \right) } \tag{4.42} $$ Thus, on $\HH_1(N)$, $$ \eqalign{ &J_\r \leq \Const \exp\left(\Const(\b,\e) N^{\frac{1+\d}{2}} -\const(\b,\e)N^{\frac{1}{2}+\d} \right) } \tag{4.43} $$ The choice of $\r_N$ was made to make the last estimate hold. Since $\Phi^0$ has bounded third derivatives we have further $$ \eqalign{ &|\sup_{v,|v|\leq\r}{\Phi^0}^{''}(\pm m^*+v)-{\Phi^0}^{''}(m^*)| \leq \Const(\b,\e)\r } \tag{4.44} $$ Thus, on $\HH_1(N)$, $$ \eqalign{ &|b_+(\r_N) -{\Phi^0}^{''}(m^*)| \leq \Const(\b,\e)N^{{-\frac{1}{4}+\frac{\d}{2}}} } \tag{4.45} $$ We have from these estimates $$ \eqalign{ &\frac{J_{\r_N}}{I_{\r_N}^\pm}\leq \Const' \exp\left(\Const'(\b,\e) N^{\frac{1+\d}{2}} -\const(\b,\e)N^{\frac{1}{2}+\d} \right) } \tag{4.46} $$ and $$ \eqalign{ &\frac{I_{\r_N}^\pm}{I_{\r_N}^\mp}\exp\left( \mp 2\b L_{-}(m^*) W_N \right) \geq \sqrt\frac{b_+(\r_N)}{b_-(\r_N)}\left( 1-2 e^{-\frac{\b N b_+(\r_N)\r_N^2}{4}} \right)\cr &\geq 1-\const(\b,\e)\r_N = 1-\const(\b,\e) N^{{-\frac{1}{4}+\frac{\d}{2}}} } \tag{4.47} $$ from which the claim follows for large enough $N$.\endproof \bigskip\vfill\eject \bigskip%\input sosdef %\datei{meta5} \chap{5. The Hopfield model below the critical temperature} The logarithmic moment generating function of the order parameter is $$ \eqalign{ &L(t,\x_i)=\frac{1}{\b}\log\cosh(\b t\cdot\x_i)) } \tag{5.1} $$ The structure of the phase diagram is determined by the averaged function $\Phi^0_N(m)=\frac{m^2}{2}-\E L(m,\x_1)$. For $\b>1$ there exist precisely $2 M$ global minima at positions $s m^* a^\nu$, $s=\pm 1$, $a^\nu$ being the $\nu$th unity vector of $\R^M$. These are solutions of the averaged mean field equation $$ \eqalign{ \E\left[ \x_1 \tanh(m\cdot \x_1) \right]=m \cr } \tag{5.2} $$ $m^*$ is the largest solution of the ordinary Curie Weiss equation $m=\tanh \b m$. The $M$ symmetric mixtures of the above product measures $$ \eqalign{ &\mu^\nu_\infty(\xi):= \frac{1}{2}\left( \mu_{\infty}^0(m^*a^\nu,\xi) +\mu_{\infty}^0(-m^*a^\nu,\xi) \right) } \tag{5.3} $$ are called `Mattis states'. They always come in pairs due to the $\pm$ symmetry of the model. For more precise information on the Hopfield model, also in the case where the number of patterns is allowed to go to infinity, see [BGP],[BG1],[BG2]. An important role will be played now by the $M\times M$ matrix $b_N(\xi)$, defined by $$ \eqalign{ &b^{\mu \nu}_N(\xi) :=\sum_{i=1}^N \left(\xi_i^\mu \xi_i^\nu -\d^{\mu \nu} \right) } \tag{5.4} $$ $b_N$ is symmetric and has vanishing diagonal; note that different elements are uncorrelated (unless prescribed by symmetry) but {\it not} independent. $b_N$ will describe the random symmetry breaking between the Mattis states in finite volume. Thus, the role that has been played by the random walk $N\mapsto W_N$ in the CWRFIM will now be played by the multidimensional random walk $N\rightarrow b_N$. The asymptotic form of the weights in the extremal decomposition is then given as follows. Let us denote by $\AA$ the $\frac{M(M-1)}{2}$ dimensional vector space of $M\times M$ symmetric matrices with vanishing diagonal. Let us denote by $\SS=\{(p^\mu)_{\mu=1,\dots,M}\}$ the simplex of $M$-dimensional probability vectors. Let us now define the map $p:\AA\rightarrow \SS$ given by $$ \eqalign{ &p^\nu(V):=\frac{\tilde p^\nu(V)}{\sum_{\mu=1}^M\tilde p^\mu(V)}\text{where} \tilde p^\nu(V):=\exp\left( c(\b) (V^2)^{\nu \nu} \right)\cr } \tag{5.5} $$ with $$ \eqalign{ & c(\b)=\frac{\b m^{*}}{2\left( 1-\b(1-m^{*})^2 \right)} } \tag{5.6} $$ To obtain the weights in Theorem 2(1.7) from (5.5) take $M(M-1)/2$ independent onedimensional Brownian motions $W_t^{\mu\nu}$ for $\mu<\nu$; we set $W_t^{\nu\mu}:=W_t^{\mu\nu}$ and $W_t^{\mu\mu}:=0$ to obtain a Brownian motion $W_t=(W_t^{\mu\nu})_{1\leq \mu,\nu\leq M}$ with values in $\AA$. With this definition we have the approximate formula $$ \eqalign{ &\mu_N(\xi)\approx \sum_{1\leq\nu\leq M}p^{\nu}(N^{-\frac{1}{2}} b_N(\xi)) \mu^\nu_\infty(\xi)\cr } \tag{5.7} $$ Note that (not only $M=1$ but also) $M=2$ is a trivial case: For $M=2$ we have $p^{(1)}(V)\equiv p^{(2)}(V)\equiv \frac{1}{2}$, for all $V\in\AA$. Nontrivial size dependence in the Hopfield model occurs only if $M\geq 3$. We remark that the occurrence of the matrix $N^{-\frac{1}{2}} b_N(\xi)$ in the weights can be easily understood: In fact, its diagonal elements describe the energy difference between the $M$ pairs of groundstates $\s=\pm\xi^\mu$, since $E_N(\s=\xi^\mu,\xi)=\frac{1}{2 N}\left(b_N^2(\xi)\right)^{\mu\mu} +\frac{N}{2}$. For finite temperature the formula (5.7) can then be understood if one performs a perturbational calculation for the depth of the minima of the random function $m\mapsto\Phi_N(m,\xi)$, thereby considering the deviation from its mean value as a perturbation. Precise estimates (analogues of proposition 3 for the CWRFIM) that allow for the application of proposition 1 and 2 have in fact been done in a different context, so that we need not repeat their proofs here; they can be readily read off from [Gen], where central limit behavior for the measures $\bar\mu_N$ around the randomly shifted minima of the function $\Phi_N(m,\xi)$ was proved. It is important to note that, while in the CWRFIM the arguments in the exponents of the weights were moving on a scale $\sim N^\frac{1}{2}$, now the normalization of the central limit theorem is taken. This was the reason for favoring the extremal states in the first case. In the Hopfield model, the weights will remain spread over all mixtures when $N\uparrow\infty$. To state the results precisely we introduce the following objects. Following old notations we set $$ \eqalign{ &\tilde\k_N(\xi)=\frac{1}{N}\sum_{n=1}^N \d_{\sum _{\nu=1}^M p^\nu\left(\frac{b_n(\xi)}{\sqrt n}\right)\mu^\nu_{\infty}(\xi)} } \tag{5.8} $$ It is possible to get an even nicer form: We find it instructive to introduce also a metastate that differs from the above by strong approximation of $b_N(\xi)$ by a Gaussian process of particularly simple form. To do so, we apply the powerful strong invariance principle for partial sum processes for $\R^k$-valued independent random variables, whose proof can be found in a general context in [Rio]. It states that a sequence of Gaussian random variables can be constructed on a {\it common} probability space having the same $k\times k$ covariance matrix that approximates the partial sum process for a.e. realization. In our case, from [Rio], page 1712, Cor. 4 follows that there exist onedimensional random variables $\g_n^{\mu \nu}=\g_n^{\nu \mu}$ for $\nu\neq \mu$, $\g_n^{\mu \mu}\equiv 0$, on a {\it common} probability space with $\xi$ s.t.: \item{(i)} $\g=\left(\g_n^{\mu \nu}\right)_{1\leq \mu\neq\nu\leq M;n=1,2,\dots}$ are i.i.d. Normal Gaussians (for different $\{\mu,\nu\}$ and $n$) \item{(ii)} $$ \eqalign{ &\sup_{N=1,2,\dots}\Vert b_N^{\mu \nu}-g_N^{\mu \nu}\Vert=\OO(\log N) } \tag{5.9} $$ a.s., where $$ \eqalign{ &g_N^{\mu \nu}= \sum_{n=1}^N \g_n^{\mu \nu} } \tag{5.10} $$ Then we put $$ \eqalign{ &\hat\k_N(\g,\xi):=\frac{1}{N}\sum_{n=1}^N \d_{\sum _{\nu} p^\nu\left(\frac{g_N}{\sqrt n}\right)\mu^\nu_{\infty}(\xi)} } \tag{5.11} $$ \remark Note that the matrix elements of $g_N$ have the advantage not only of being Gaussian but also {\it independent} (unless prescribed by the symmetry of the matrix) which was not true for the matrices $b_N$. Thus, they form a $M(M-1)/2$ dimensional random walk with Standard Gaussian increments. With these definitions, the analogue of Theorems 1,1' are Theorem 2 and \theo{2'}{\it \item{(i)} For all $\xi$ in a full measure set, the set of weak cluster points equals $$ \eqalign{ &\CC\PP\{\mu_N(\xi),\,N=1,2,\dots\} =\left\{\sum_{1\leq\nu\leq M}q^{\nu} \mu^\nu_\infty(\xi),\,\,(q^\nu)_{\nu=1,\dots,M}\in\SS'\right\} } \tag{5.12} $$ where $\SS'=\{(\frac{1}{2},\frac{1}{2} )\}$ for $M=2$ and $\SS'=\SS$ for $M\geq 3$. \item{(ii)} For all $\xi$ in a full measure set, for any continuous function $F:\PP(\O)\mapsto \R$ the empirical metastate is approximated by $$ \eqalign{ &\lim_{N\uparrow \infty}\left(\int\k_{N}(\xi)(d\mu)F(\mu) -\int\tilde\k_{N}(\xi)(d\mu)F(\mu)\right)=0 } \tag{5.13} $$ \item{(iii)} A.s., for any continuous function $F:\PP(\O)\mapsto \R$ the empirical metastate is approximated by $$ \eqalign{ &\lim_{N\uparrow \infty}\left(\int\k_{N}(\xi)(d\mu)F(\mu) -\int\hat\k_{N}(\g,\xi)(d\mu)F(\mu)\right)=0 } \tag{5.14} $$ \item{(iv)} For all $\xi$ in a full measure set the conditioned metastate exists and equals $$ \eqalign{ &\bar\k(\xi)(F) =\E_g F\left(\sum _{\nu=1}^M p^\nu\left(g\right) \mu^\nu_{\infty}(\xi)\right) } \tag{5.15} $$ where $g$ is a Normal Gaussian in $\AA$. } In the course of the proof we will have to compare the map $p(V)$ at different arguments in the noncompact space $\AA$. To be able to do so, we need some information about the continuity of $V\mapsto p(V)$. We have \lemma{6}{\it Define the norm $$ \eqalign{ &\Vert V\Vert_{ss}^2:= \sup_{\mu}\sum_{\nu}\left(V^{\nu\mu}\right)^2 } \tag{5.16} $$ Then $$ \eqalign{ &\Vert p(V)- p(V') \Vert_1 \leq 4 c(\b) \left(\Vert V\Vert_{ss} +\Vert V-V'\Vert_{ss}\right) \Vert V-V'\Vert_{ss} } \tag{5.17} $$ } \proof Writing $V_{\a\b}=V_{\b\a}$ we view $p(V)$ as a function of the $M(M-1)/2$ variables $V^{\a\b}$ for $\a<\b$. Then the Taylor formula gives $$ \eqalign{ &p^\nu(V')-p^\nu(V) =\sum_{\a<\b}\frac{\del p^\nu}{\del V^{\a \b}} (\tilde V)(V'-V)^{\a\b} } \tag{5.18} $$ where $\tilde V=V+\th (V'-V)$. It is easy to compute that $$ \eqalign{ &\frac{\del p^\nu}{\del V^{\a \b}} =p^\nu( 1-p^\nu)\frac{\del\log \tilde p^\nu}{\del V^{\a \b}} -\left( p^\nu \right)^2 \frac{1}{\tilde p^\nu}\sum_{\r,\r\neq \nu}\tilde p^\r \frac{\del\log \tilde p^\r}{\del V^{\a \b}} } \tag{5.19} $$ Now $$ \eqalign{ &\frac{\del\log \tilde p^\r}{\del V^{\a \b}} =2 c V^{\a \b}\left(\d_{\a\r}+\d_{\b\r}\right) } \tag{5.20} $$ where we write $c\equiv c(\b)$. Therefore $$ \eqalign{ &\sum_{\a<\b} \frac{\del\log \tilde p^\r}{\del V^{\a \b}}(\tilde V) (V'-V)^{\a\b} =2 c \left( \tilde V(V-V') \right)^{\r\r} } \tag{5.21} $$ Then $$ \eqalign{ &|p^\nu(V')-p^\nu(V)| =2 c \left|p^\nu( 1-p^\nu) \left(\tilde V(V-V') \right)^{\nu\nu} -\left( p^\nu \right)^2 \frac{1}{\tilde p^\nu}\sum_{\r,\r\neq \nu} \tilde p^\r \left( \tilde V(V-V') \right)^{\r\r} \right|\cr &\leq 2 c \left(p^\nu( 1-p^\nu) +\left( p^\nu \right)^2 \frac{1}{\tilde p^\nu}\sum_{\r,\r\neq \nu} \tilde p^\r \right) \sup_{\l}\left|\left( \tilde V(V-V') \right)^{\l\l}\right|\cr &= 4 c p^\nu( 1-p^\nu) \sup_{\l}\left|\left( \tilde V(V-V') \right)^{\l\l}\right|\cr } \tag{5.22} $$ where all $p,\tilde p$'s are taken at the argument $\tilde V$. Note that $$ \eqalign{ &\sup_{\l}\left|\left( \tilde V(V-V') \right)^{\l\l}\right|\leq \Vert \tilde V\Vert_{ss} \Vert V-V'\Vert_{ss} \leq \left(\Vert V\Vert_{ss} +\Vert V-V'\Vert_{ss}\right) \Vert V-V'\Vert_{ss} } \tag{5.23} $$ Summing over $\nu$ gives the lemma.\endproof Finally we come to the \proofof{Theorem 2 and 2'} >From [Gen], proposition 1.3. immediately follows that for any $0<\d<\frac{1}{2}$, $\r<\frac{m^*}{2}$, $s=\pm 1$, $$ \eqalign{ &\bar\mu_N(\xi)\left[ B_\r(s m^* a^\nu) \right] =\frac{\tilde p^\nu\left(\frac{b_N(\xi)}{\sqrt N}\right)(1+\OO(N^{-\d}))} {\sum_{\mu=1}^M \tilde p^\mu\left(\frac{b_N(\xi)}{\sqrt N}\right)(1+\OO(N^{-\d}))} } \tag{5.24} $$ $\OO(N^{-\d})$ is here {\it nonuniform} in $\x$.\footnote{$^1$}{ It means precisely that for a.e. $\xi$ there exist $N_0(\xi)$ and $\Const(\xi)$, s.t. for all $N\geq N_0(\xi)$ the term is bounded by $\Const(\xi)N^{-\d}$.} We have to use information on the minimum and maximum size of $\frac{b_N(\xi)}{\sqrt N}$. In fact, from the Law of Iterated Logarithm for partial sums of $\R^k$-valued random variables (see for this statement, which is true more generally in Banach spaces, e.g. [LT], Theorem 8.2) we have $$ \eqalign{ &\left\Vert\frac{b_N(\xi)}{\sqrt N}\right\Vert\leq \Const \sqrt{\ln\ln N} } \tag{5.25} $$ a.s. for $N\geq N_0(\xi)$ sufficiently large (with some arbitrary matrix norm.) This gives $$ \eqalign{ &\tilde p\left(\frac{b_N(\xi)}{\sqrt N}\right) \leq \left(\ln N\right)^{K}\cr } \tag{5.26} $$ with some constants $K=K(\b)$, for $N$ sufficiently large. It is easy to see with this information that from (5.24) follows that $$ \eqalign{ &\lim_{N\uparrow\infty}\left( \bar\mu_N(\xi)\left[B_\r(s m^* a^\nu)\right] -\frac{\tilde p^\nu\left(\frac{b_N(\xi)}{\sqrt N}\right)} {\sum_{\mu=1}^M \tilde p^\mu\left(\frac{b_N(\xi)}{\sqrt N}\right)} \right)=0 } \tag{5.27} $$ This, in the language of Chapter 3, is property CR$(\r)$ along a sequence of $N$-independent exceptional sets $\HH(N)\equiv \HH'$ for the fixed full measure set $\HH'$ where the assumptions necessary for the above estimates hold. Now we apply our general reasoning. >From the third remark after Lemma 2 in Chapter 3 follows that this implies CR$(\r)$ for $\tilde\mu_N$. (In fact, technically, it is typically proven before!) Due to the second remark after Lemma 2 we have then CR$(\r_N)$ which suffices for all our needs. Note further, that because of the $N$-independence of $\HH(N)=\HH'$ we don't have to worry about exceptional sets any more when applying any of the propositions 1 or 2. Thus, Theorem 2'(ii) follows from proposition 1(ii). Theorem 2'(iii) follows from proposition 1(ii) and the following fact: Property CR$(\r)$ with the probability vector $p\left(\frac{b_N}{\sqrt N}\right))$ implies the property CR$(\r)$ with the probability vector $p(\frac{g_N}{\sqrt N})$. To show the latter it suffices to show that, a.s. $$ \eqalign{ &\lim_{N\uparrow\infty}\left\Vert p\left(\frac{b_N}{\sqrt N}\right)- p\left(\frac{g_N}{\sqrt N}\right) \right\Vert_1=0 } \tag{5.28} $$ But Lemma 6 implies $$ \eqalign{ &\left\Vert p\left(\frac{b_N}{\sqrt N}\right)- p\left(\frac{g_N}{\sqrt N}\right) \right\Vert_1 \leq \frac{4 c(\b)}{N} \left(\Vert b_N\Vert_{ss} +\Vert b_N-g_N\Vert_{ss}\right) \Vert b_N-g_N\Vert_{ss} } \tag{5.29} $$ Using now the law of iterated logarithm (5.25) and the strong approximation property (5.9) for $\Vert b_N-g_N\Vert_{ss}$ the desired estimate (5.28) follows. To prove Theorem 2'(iv) and Theorem 2, let us first note the finite volume perturbation property, necessary for proposition 2: It is clear that, for fixed finite volume $V$, $\sup_{\xi_V}\Vert b_N(\xi)-b_N(\xi+\xi_V)\Vert\leq \Const(V)$. Then, we have from Lemma 6 $$ \eqalign{ &\lim_{N\uparrow\infty}\sup_{\xi_V} \left\Vert p\left(\frac{b_N(\xi)}{\sqrt N}\right)- p\left(\frac{b_N(\xi+\xi_V)}{\sqrt N}\right) \right\Vert_1 \leq \frac{4 c(\b)}{N} \left(\Vert b_N(\xi)\Vert_{ss} +\Const(V)\right) \Const(V) } \tag{5.30} $$ Using (5.25) the r.h.s. goes to zero for almost all $\eta$. Let us now denote by $\xi'$ an independent copy of $\xi$. Note that we have the two approximation properties given by proposition 2(i) and (ii). Then we construct, as above, a strongly approximating process $g'$, but this time for $\xi'$, such that it is independent of $\xi$. It follows that $$ \eqalign{ &F\left(\sum_{\nu=1}^M p^\nu\left(\frac{b_N(\xi')}{\sqrt N}\right)\mu^\nu_\infty(\xi)\right) -F\left(\sum_{\nu=1}^M p^\nu\left(\frac{g'_N}{\sqrt N}\right)\mu^\nu_\infty(\xi)\right) \rightarrow 0 } \tag{5.31} $$ a.s., for bounded continuous $F$, with $N\uparrow\infty$. Putting this together with proposition 2(ii), we obtain directly Theorem 2'(iv). For Theorem 2 we get from proposition 2(i) $$ \eqalign{ &\lim_{N\uparrow\infty}\int\k_N(\xi)(d\mu)F(\mu) =^{\hbox{\it law}}\lim_{N\uparrow\infty} \frac{1}{N}\sum_{n=1}^N F\left( p^\nu\left(\frac{g'_n}{\sqrt n}\right)\mu^\nu_\infty(\xi) \right) } \tag{5.32} $$ Since we are only interested in distributions, we replace $\frac{g'_n}{\sqrt n}$ by $\frac{W_{t_n}}{\sqrt {t_n}}$ with $t_n=\frac{n}{N}$ where $W_t$ is a Brownian motion. But then (5.32) is nothing but a Riemann sum for the continuous function $t\mapsto F\left( p^\nu\left(\frac{W_t}{\sqrt {t}}\right)\mu^\nu_\infty(\xi) \right)$. Thus it converges for almost all realizations of $W_t$ to the corresponding integral with $N\uparrow\infty$. But, from this follows that the distribution of (5.32) is the same as that of (1.7) which proves Theorem 2. To prove the result about the cluster points, Theorem 1'(i), it suffices to consider the cluster points of the weights $p\left(\frac{b_N}{\sqrt N}\right)$, $N=1,2,\dots$. Now we use the following \lemma{7}{\it Let $X_i$, $i=1,2,\dots$ be a sequence of i.i.d. $k$-dimensional Normal Gaussians. Then, a.s., the set of the cluster points of the sequence $\frac{1}{\sqrt{N}}\sum_{i=1}^N X_i$, $N=1,2,\dots$ equals all of $\R^k$. } The proof is not difficult: Given a neighborhood of a rational point in $\R^k$ it is easy to construct a sparse subsequence that hits it infinitely often with probability one. We don't give the details here. But from that we have in particular $\CC\PP\left(\frac{b_N}{\sqrt N}, N=1,2,\dots\right)=\AA$, a.s. This implies Theorem 1'(i) by continuity of $p$ and \lemma{8}{\it $\ov {p(\AA)}$ equals all of $\SS$ for $M\geq 3$.} \proof It suffices to show that, given any vector $l=(l_\mu)_{\mu=1,\dots, M}\in\R^M$, there exist a real number $b$ and a matrix $V\in\AA$, s.t. $$ \eqalign{ &l_\mu+b= (V^2)^{\mu \mu},\quad \mu=1,\dots,M } \tag{5.33} $$ The difficulty about this linear system of equations for the $M(M-1)/2$ quantities $\left(V^{\mu\nu}\right)^2$ is that it fails to give {\it nonnegative} solutions for arbitrary choices of $l$ and $b$. Thus the freedom in the choice of $b$ is really necessary. As an ansatz we consider a matrix of the type $$ \eqalign{ &V^{12}=V^{21}=\sqrt{\frac{\l_1}{2}},\,\,\,\, V^{13}=V^{31}=\sqrt{\frac{\l_2}{2}},\,\,\,\, V^{23}=V^{32}=\sqrt{\frac{\l_3}{2}},\cr &V^{\mu-1,\mu}=V^{\mu,\mu-1}=\sqrt{\l_\mu},\quad \mu=4,\dots,M,\cr &V^{\mu\nu}=V^{\nu\mu}=0\text{otherwise} } \tag{5.34} $$ with $\l_\mu\geq 0$, where the condition in the second line is empty for $M=3$. It turns out then that the solution of (5.33) with $b=0$ has the general form $$ \eqalign{ &\l_1=l_1+l_2-l_3\,\,+\,\,\left(l_4-l_5+l_6-l_7\pm\dots+(-1)^{M}l_M\right) \cr &\l_2=l_1-l_2+l_3\,\,-\,\,\left(l_4-l_5+l_6-l_7\pm\dots+(-1)^{M}l_M\right)\cr &\l_3=-l_1+l_2+l_3\,\,-\,\,\left(l_4-l_5+l_6-l_7\pm\dots+(-1)^{M}l_M\right) \cr } \tag{5.35} $$ and $$ \eqalign{ &\l_4=l_4-l_5+l_6-l_7\pm\dots+(-1)^{M}l_M \cr &\l_5=l_5-l_6+l_7\pm\dots+(-1)^{M+1}l_M \cr &\l_6=l_6-l_7+l_8\pm\dots+(-1)^{M}l_M \cr &\dots\cr &\l_M=l_M\cr } \tag{5.36} $$ It suffices to prove the statement for $l$'s in the special form $l_3\geq l_1\geq l_2$ and $(l_2\geq) \,\, l_4\geq l_5\geq \dots\geq l_M\geq 0$. But, using this order relation, it follows for the solution of (5.33) with $b=0$ that $\l_\mu\geq 0$ for all $2\leq \mu\leq M$, whereas $\l_1$ can be possibly negative. 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