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{$ $}
\vskip2truecm
\centerline{\tit METASTATES IN DISORDERED MEAN FIELD MODELS:}
\vskip.2truecm
\centerline{\tit RANDOM FIELD AND HOPFIELD MODELS\footnote{${}^*$}{\ftn Work
supported by the DFG %under contract No. ...
}}
%\vskip.2truecm
%\centerline{\tit }
\vskip2truecm
%\centerline{\tit }
%\vskip2.5cm
%\centerline{\aut Anton Bovier\footnote{${}^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.
But note that for the solution of (5.33) with $\l_\mu\equiv 0$
and $b>0$, we have
$\l_1=b>0$ for $M$ odd (resp. $\l_1=2b>0$ for $M$ even),
$\l_\mu\geq 0$ for $2\leq \mu\leq M$.
Thus, by adding a sufficiently large $b>0$
to the fixed $l_\mu$'s one can always force the corresponding
$\l_1$ to become positive without destroying the positivity of the other
$\l_\mu$'s.
This proves the claim.\endproof
\endproof\endproof
\bigskip
\chap{Acknowledgments:}
The author thanks the WIAS, Berlin for its kind hospitality;
he thanks A.Bovier and
%, during the finishing of this paper,
Charles Newman for interesting discussions.
This work was supported by the DFG.
\bigskip
\bigskip%\input sosdef
%\datei{metaref}
%\pageno=500
\chap {References}
\item{[APZ]} J.M.G. Amaro de Matos, A.E.Patrick, V.A.Zagrebnov,
Random Infinite-Volume Gibbs States for the Curie-Weiss Random Field
Ising Model., J.Stat.Phys {\bf 66}, 139-164 (1992)
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\end