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Disordered systems, Morita approach,
non-Gibbsian measures, mean field models, random field model
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\begin{document}
\title{How non-Gibbsianness helps a
\\ metastable Morita minimizer to provide a stable free energy}
%\title{Two wrongs can make a right.\\ Why non-Gibbsianness helps an unstable
%solution of a wrong approximation to give a correct answer}
\author{Christof K\"ulske\thanks{Research supported by the
DFG} \footnote{TU Berlin, Institut f\"ur Mathematik, Sekr. MA 7-4,
Strasse des 17. Juni 136, D-10623 Berlin, Germany, \texttt{kuelske@math.tu-berlin.de},}
\footnote{EURANDOM,
P.O.Box 513,
5600 MB Eindhoven, The Netherlands
}}
% \, and Christof
%K\"ulske\footnote{Weierstrass-Institut f\"ur
%Angewandte Analysis und Stochastik,
%Mohrenstr. 39,
%D-10117 Berlin,
%GERMANY,
%\texttt{kuelske@wias-berlin.de},
%\texttt{http://www.wias-berlin.de/people/kuelske/} }}
\maketitle
\begin{abstract}
We analyze a simple approximation scheme based
on the Morita-approach for the example of the
mean field random field Ising model
where it is claimed to be exact in some of the physics literature.
We show that the approximation scheme is flawed, but
it provides a set of equations whose metastable solutions surprisingly
yield the correct solution of the model. We explain how the same equations
appear in a different way as rigorous consistency equations. We
clarify the relation between the validity of their
solutions and the almost surely discontinuous behavior of the single-site
conditional probabilities.
\end{abstract}
\bigskip
\bigskip
\noindent{\it AMS 2000 Subject Classification.} 60G60, 82B20, 82B44.\\
\noindent{\it Keywords and phrases.} Disordered systems, Morita approach,
non-Gibbsian measures, mean field models, random field model.\\
\section{Introduction} \label{sect:intro}
The {\it Morita-approach} or {\it equilibrium ensemble approach}
to systems with quenched disorder goes back to \cite{Mor64}.
A fair and clear review from a theoretical physicist's point of view containing a quick outline of the
theory and various interesting
recent applications is given by K\"uhn in \cite{Kueh04} (see also \cite{Kueh96}).
%We will make more reference to this review in a moment.
The central idea in the Morita-approach is:
Look at the {\it joint measure} governing the
distribution of the quenched degrees of freedom and the dynamical variables,
rather than directly trying to describe
the {\it quenched measure} for the dynamical variables
for fixed realization of the disorder.
Ideally one would then like to write this joint measure as a formal equilibrium
average over the {\it joint variables} in terms of a {\it joint Hamiltonian}. This joint Hamiltonian
would then be the sum of the original one and another "disorder-Hamiltonian" depending
only on the quenched degrees of freedom. The resulting model possesses full
spatial symmetries and one might hope that it
is amenable to techniques known from systems
without disorder.
Mathematically there are problems with this idea.
In finite volume this "disorder-Hamiltonian" can in principle be chosen in such a way
that the resulting joint distribution coincides with that of the true model.
For lattice systems in the infinite volume this is however a serious problem.
In fact, for many models an absolutely summable joint Hamiltonian does not exist,
and the joint measures in the infinite volume are non-Gibbsian measures. The appearance of a non-Gibbsian
joint distribution was first discovered in the example of the
Grising model in \cite{vEMSS}
and studied in a general context in \cite{K1,K2}.
See also the discussion in \cite{EKM,KM1,KuehMaz00}.
A well-understood example for this are in
particular the joint measures of the random field Ising model \cite{BK}
in more than $3$ dimensions at low temperature and small disorder.
They provide an illuminating example of strong non-Gibbsian
pathologies. In fact, their conditional probabilities are shown to be
discontinuous functions of the conditionings, for a set of conditionings
with (joint) measure one \cite{K2}. This means
that the measure is not even "almost surely Gibbs" (in the sense of \cite{MRM}). This pathology
even causes the usual Gibbs variational principle to fail \cite{KLR02}.
Close analogies to this behavior on the lattice
can be already found in the corresponding mean-field model.
Here the corresponding functions describing the
conditional expectations can be explicitly computed \cite{K3}.
For more on the analogies between non-Gibbsian measures on the lattice
and discontinuous behavior of conditional probabilities in mean-field models
see \cite{KuH04} and \cite{K3}.
The motivation for using
the Morita-approach from the point of view of theoretical physics
is however to leave these conceptual problems aside and
take it as a source for approximation schemes \cite{Kueh04,KM1}.
Such schemes can be obtained
by taking certain simplified trial disorder-Hamiltonians that
are chosen e.g. demanding that a finite number of moments of the
distribution of the disorder variables coincide with that of the true distribution.
Then one would like to solve
the resulting Morita approximant model and hope that relevant
features of the solution are the same as that of the true model.
It might seem hopeless to justify such
approximations in general for non-trivial lattice models.
It is therefore valuable to fully understand at least simple toy models that
can be explicitly treated. This is want we want to do here.
We will give here a complete discussion
of the quick naive "solution" of the mean-field
random field Ising model, based on a very simple approximant joint measure containing only
one parameter \cite{SerPal93}. This so-called solution is fairly old and the computations
are trivial, but the justification of the resulting equations is subtle. So it is
worth to reconsider it from a rigorous point of view and straighten out
some wrong claims in the literature, answering a question of K\"uhn.
In particular we take issue with the statement of
K\"uhn \cite{Kueh04}, describing the work of \cite{SerPal93}. He writes:
"it reproduces the exact solution at the cost of introducing a single `chemical
potential' to fix the average value of the random field, which creates a term in
the modified Hamiltonian that introduces no non-locality into the model over and
above that already contained in the definition of the Curie-Weiss limit.
This is a remarkable result in the light of concerns raised about the
appearance of non-Gibbsian measures within the equilibrium ensemble approach
and the identification of the RFIM as providing a realization of a kind of
`worse-case scenario' in the non-Gibbsian world..."
We will indeed see that even for this simple model the situation is subtle and the validity
of the solution is fundamentally related to the analogue of
"non-Gibbsianness" in the mean-field context.
To see this, we will start in Section 2 by reviewing the quick "solution", following \cite{Kueh04}.
This provides us with two equations for two parameters, the magnetisation and
the Morita chemical potential.
These equation have in fact solutions for which the magnetisation-variable
takes the known value of the spontaneous magnetization.
However, we note that
this solution corresponds to a wrong (metastable) saddle-point approximation for the
approximant measure and therefore the naive derivation given above is %helplessly
flawed.
Moreover, we will prove that in the low temperature regime it is even strictly impossible
to choose a chemical potential such that the magnetic fields become symmetric. In brief,
the theory based on the Morita approximant measure with just one chemical potential fails.
%contrary to the physicists' claim.
How can we understand then that the two equations derived by
a wrong line of argument yield the correct value of the magnetization? Is this
just accidental?
We will see in Section 3 that the same two equations come up in a different way as
consistency equations for the conditional probabilities of the true joint measures
of the model without approximations. Here however the fixed Morita-chemical potential
is replaced by a random variable.
It is in this context that we will finally understand that the validity of these
equations and the almost sure discontinuity of
the conditional expectations are consequences of each other.\bigskip
\section{Invalidity of single-site Morita approximation
approach for the Curie Weiss Random Field Ising Model}
\label{sect:Morita}
We consider the mean-field random field Ising model.
It is defined in terms of the following formula for the quenched
Gibbs expectation for fixed choice of the random fields.
\bigskip
\noindent{\bf Quenched measure:}
\begin{equation}\label{quenchedmeas}
\mu_{\b,\e,h_0, N}[\eta_{[1,N]}](\s_{[1, N]})
:=\frac{\exp\Bigl(\frac{\b}{2 N}\bigl(\sum_{i=1}^N \s_i\bigr)^2 + \b
\sum_{i=1}^N (\e\eta_i+h_0)\s_i
\Bigr)}{Z_{\b,\e,h_0,N} [\eta_{[1,N]}]}
\end{equation}
Here the spins ("dynamical variables") take values $\s_i=\pm 1 $ and
the random fields take values $\eta_i=\pm 1$ with equal probability.
We denote their distribution by $\P$.
We stress that the partition function appearing in the
denominator depends on the realization of the random
fields $\eta_{[1,N]}$ describing the disorder.
We allow from the beginning also an external magnetic field $h_0$, but
we are mainly interested in the case $h_0\downarrow 0$.
What one understands by the "solution of the model" is the characterization
of the behavior of this measure on the $\s$'s for a large set of $\eta$'s,
having asymptotically $\P$-measure one. This has been done in great
detail \cite{SW85,APZ92,K97},
and so in this model
there is no need for any approximation based on the Morita-approach in order to solve the model.
Most basically, we know the phase structure in zero external field,
for any choice of the parameters $\b,\e$.
We recall that for large $\b$ and small $\e$ in zero
external magnetic field $h_0$ the
model exhibits a spontaneous magnetization whose value $m$ is a solution
of the equation
\begin{equation}\label{eos}
m=\frac{1}{2}\Bigr(\tanh \b(m+\e)+\tanh \b(m-\e)\Bigl)
\end{equation}
We also know finer properties of the quenched distribution
above, like its dependence on the volume label $N$, for fixed realisation of the
random fields. This can be asymptotically described by the in the "metastates formalism" \cite{K97}, a notion
due to Newman and Stein \cite{NS96a,NS96b}.
For general background on this notion in the theory of disordered systems see \cite{New97,Bo01}.
Knowing the correct solution,
our point in this note will be however to put the Morita approximation scheme
outlined above to the test.
Now, in the Morita-approach one looks at the joint measures on the product
space of the spin variables $\s$ and the disorder variables $\eta$.
They are simply composed from the quenched measures
and the a priori uniform distribution of the random fields by the following obvious formula.
\bigskip
\noindent{\bf True joint measure:}
\begin{equation}
K_{\b,\e,h_0, N}(\s_{[1,N]}, \eta_{[1, N]})
=\frac{1}{2^N}\mu_{\b,\e,h_0, N}[\eta_{[1,N]}](\s_{[1, N]})
\end{equation}
The approximant measure we want to consider is obtained
by putting a single-site disorder potential
$\l\sum_{i=1}^N \eta_i$ with just one free parameter $\l$ that has the meaning
of a chemical potential governing the mean value of the random fields.
\bigskip
\noindent {\bf Morita-approximant measure:}
\begin{equation}
\hat K_{\l;\b,\e,h_0, N}(\s_{[1,N]}, \eta_{[1, N]})
=\frac{\exp\Bigl(\frac{\b}{2 N}\bigl(\sum_{i=1}^N \s_i\bigr)^2 + \b
\sum_{i=1}^N (\e\eta_i+h_0)\s_i+\l\sum_{i=1}^N \eta_i
\Bigr)}{Z_{\l;\b,\e,h_0,N} }
\end{equation}
We stress that the partition function does {\it not} depend
on $\eta_{[1, N]}$, in contrast to (\ref{quenchedmeas}).
The Hamiltonian of this measure contains no non-local couplings
of the random fields.
Then the idea of the naive Morita approximation-approach is as follows:
1) For any fixed $\l$, compute
the large-$N$ limit of distribution of this model.
2) Choose $\l=\l(\b,\e,h_0)$ such that the expectation
of the random fields coincides with the true joint measures, i.e. it vanishes,
$\lim_{N}\int\hat K_{\l;\b,\e,h_0, N}(d\eta_1)\eta_1=0$.
More precisely the value of $\l$ will depend on $N$, but it will
have a well-defined limit as $N\uparrow \infty$.
3) Then, the distribution of the Morita approximant measure taken with
this value of the bias of the random fields $\l$, should be close
to the true joint measure. E.g. we should have that the distribution of the
spin average $\frac{1}{N}\sum_{i=1}^N \s_i$ has the same infinite volume
limit in the true joint measure and in the Morita
approximant measure.
Let us write down the following precise formulation in order to have
a well-defined starting point of discussion.
\noindent{\bf Single-Site approximation conjecture:} Let $\b,\e,h_0$ be fixed.
Then the conjecture is that there is a value $\l(\b,\e,h_0)$ such that
$\lim_{N}\hat K_{\l;\b,\e,h_0, N}(d\eta_1)\eta_1=0$ and that for
this value we have that
\begin{equation}
\lim_{N\uparrow\infty}K_{\b,\e,h_0, N}\Bigl(\frac{1}{N}\sum_{i=1}^N \s_i \in \,\cdot\,\Bigr)
=\lim_{N\uparrow\infty}
\hat K_{\l(\b,\e,h_0);\b,\e,h_0, N}\Bigl(\frac{1}{N}\sum_{i=1}^N \s_i \in \,\cdot\,\Bigr)\end{equation}
Is this conjecture true? How does this relate to the proved a.s. discontinuity
of the conditional expectations of the true joint measures?
%K\"uhn \cite{Kueh04} writes with reference to the latter phenomenon:
%{\it In the light of these findings, and also, because the mean-field
%version of the RFIM is technically very simple to solve, both directly and
%via the equilibrium ensemble approach, we will review these solutions in what
%follows, demonstrating their equivalence at the level of equations
%of state.}
Let us review the quick derivation of the solution of the model based on this conjecture
(we follow here \cite{Kueh04}).
\noindent {\bf Naive (problematic!) derivation: }
Look at the partition function of the Morita approximant measure, putting $h_0=0$ from the beginning,
and use a simple Gaussian identity (Hubbard-Stratonovitch transformation)
to write
\begin{equation}\label{partition-function}
Z_{\l;\b,\e,N}=2^N \int\frac{d m}{\sqrt{2\pi /(\b N)}}
\exp\Bigl(-\b N \Phi_{\l;\b,\e}(m) \Bigr)
\end{equation}
The function appearing in the exponent is $N$-independent and
is given below in (\ref{phifunction}) by putting $h_0=0$.
Using the Laplace method to compute the integral
we must have $\frac{\partial}{\partial m}\Phi_{\l;\b,\e}(m)=0$.
This is an equation for the minimizer $m$ of the form
\begin{equation}\label{mfeq1}\begin{split}
&m=\frac{\sum_{k=\pm 1}\sinh\bigl(
\b(m+\e k)
\bigr) e^{\l k}}{\sum_{k=\pm 1}\cosh\bigl(
\b(m+ \e k )
\bigr) e^{\l k}}\cr
\end{split}
\end{equation}
The parameter $\l$ is fixed such that the mean of the
magnetic field sum divided by $N$ vanishes, i.e. $\lim_{N\uparrow\infty} \frac{\partial}{\partial \l}
\log Z_{\l;\b,\e,N}=0$. This requires at the minimizer $m$
that $\frac{\partial}{\partial \l}\Phi_{\l;\b,\e}(m)=0$.
This requires that
%\begin{equation}\label{mfeq2}\begin{split}
%&\frac{\sum_{k=\pm 1}k\cosh\bigl(
%\b(m+\e k )
%\bigr) e^{\l k}}{\sum_{k=\pm 1}\cosh\bigl(
%\b(m+\e k)
%\bigr) e^{\l k}}=0\cr
%\end{split}
%\end{equation}
\begin{equation}\label{mfeq2}\begin{split}
&e^{-2 \l}=\frac{\cosh (\b(m+\e))}{\cosh(\b(m-\e))}
\end{split}
\end{equation}
The equation (\ref{mfeq2}) shows that $m$ and $\l$ are in one-to-one correspondence
to each other. From (\ref{mfeq1}) and (\ref{mfeq2}) follows
the well known (and correct) mean field equation
(\ref{eos}). So it seems that
the Morita approximation approach becomes exact in this case and we are done.
K\"uhn writes appropriately: {\it This result \cite{SerPal93} - simple and reassuring as it is -
must be regarded as remarkable in the light of concerns raised about
the appearance of non-Gibbsian measures within the equilibrium ensemble
approach \cite{vEMSS,K1,EKM} and the identification of the RFIM as providing a realization
of a system exhibiting almost surely non-Gibbsian joint measures \cite{K1,K3}.}
Indeed, we note that the "derivation" is flawed because of the following fact.
\noindent {\bf Worrisome fact why this derivation is wrong: } Suppose that $\b>1$, $\e>0$ and $\l>0$ are fixed.
Then the minimum of the function $m\mapsto\Phi_{\l;\b,\e}(m)$ is attained at a unique
positive value $m^*(\l)$ (as we will see below).
Therefore there cannot be a pair $(m^*(\l),\l)$ satisfying (\ref{mfeq2}).
So the solution $(m,\l)$ obtained by (\ref{mfeq1}), (\ref{mfeq2}) corresponds to
a wrong value for the free energy.
\noindent {\bf The remaining Morita mystery: }
Why does the wrong minimizer give the correct equation for the magnetization?
It is the purpose of this note to clarify the situation.
We will be even more general and more careful here
and allow for a possibly non-zero external magnetic field $h_0$.
This we do in order to investigate whether taking the limit $h_0\downarrow 0$
only in the end will help us to solve the problem of this approach.
We can readily solve the model for any choice of the parameters
of inverse temperature $\b$, strength of random fields $\e$, external field $h_0$
and Morita chemical potential $\l$.
As usual in mean field models there is convergence to (linear combinations)
of product measures over the sites $i$. Indeed, any limit measure must
be a mixture of product measures. This is clear by de Finetti's theorem
since the limit of exchangeable measures inherits the property of exchangeability.
Now, solving our simple model is almost trivial when we note that by summing over the
$\eta$ first we obtain a resulting effective Curie-Weiss Ising model with a new
effective homogenous magnetic field acting on the $\s$'s. The computations
are simple and will be given below for the sake of completeness. Before we do so
let us however state the most important consequence of this in the present context.
\begin{thm} {\bf (Impossibility of single-site approximation of true joint measure)}\label{thm:invalidity}
Assume that $\b>1$ and $\e> 0$ are fixed.
Then
\begin{equation}\begin{split}
&\Bigl\{h_0\in \R, \exists \l\in \R :
\lim_{N\uparrow\infty}
\hat K_{\l;\b,\e,h_0, N}(d \eta)=\P(d\eta)
\Bigr\}\cr
&=\R\backslash
\Bigl[-a(\b,\e), +a(\b,\e) \Bigr]\end{split}
\end{equation}
where $a(\b,\e)$ is strictly bigger than zero.
Here the symbol $\lim$ denotes a weak limit.
\end{thm}
In words the theorem states that
the set of external homogenous magnetic fields $h_0$
for which there exists a "compensating" Morita-field $\l$ that
reproduces the neutral i.i.d. distribution for the random fields is
bounded away from zero for any $\b>1$. This means that
the approximation scheme must necessarily fail in the relevant
low temperature regime: First of all, in zero external field
$h_0$ it is impossible to produce asymptotically symmetric i.i.d.
random fields by an appropriate choice of $\l$. This result however, might
not be too surprising.
But the theorem says more: Even choosing $h_0$ strictly positive and letting it tend
to zero afterwards won't help us.
Having said this, it is interesting to investigate the set of parameters
for which the distribution of random fields becomes
neutral i.i.d. in more detail. Let us make the following definition.
\noindent{\bf Neutral Set: } Fix the inverse temperature $\b>0$ and $\e>0$.
We call the parameter set
\begin{equation}\begin{split}
&{\cal R}(\b,\e):=\Bigl\{(h_0,\l)\in \R\times \R,
\lim_{N\uparrow\infty}
\hat K_{\l;\b,\e,h_0, N}(d \eta)=\P(d\eta)
\Bigr\}\cr
\end{split}
\end{equation}
the neutral set. Obviously ${\cal R}(\b,\e)=-{\cal R}(\b,\e)$ by the symmetry of the model.
Then we have the following theorem.
\begin{thm} {\bf (Structure of neutral set)}\label{thm:structurenl}
Assume that $\b>0$ and $\e> 0$ are fixed. Then the set $\RR(\b,\e)$ is the union
of two semi-infinite curves, related to each other by reflection
at the origin. They are connected if and only if $\b\leq 1$. \\
More precisely these curves are of the following form.
There is a continuous increasing function $h_0\mapsto l_{\b,\e}(h_0)$
that is defined an open interval of the form $(a(\b,\e),\infty)$ and takes positive values.
The left endpoint of the interval
satisfies $a(\b,\e)\begin{cases}>0 \text{ for } \b>1\cr =0 \text{ for }\b\leq1\end{cases}$.
Define
\begin{equation}\label{nlgraph}\begin{split}
&\RR^+(\b,\e):=\Bigl\{
(h_0,-l_{\b,\e}(h_0))\Bigl | h_0\in (a(\b,\e),\infty)
\Bigr\}
\end{split}
\end{equation}
Then, the neutral set has the form
\begin{equation}\label{nldecomp}\RR(\b,\e)=\begin{cases} \RR^+(\b,\e)\cup \Bigl( -\RR^+(\b,\e) \Bigr)& \text{ for }
\b>1 \cr \RR^+(\b,\e)\cup \Bigl( -\RR^+(\b,\e) \Bigr)\cup \Bigl\{(0,0)\Bigr\}& \text{ for }Ê\b\leq 1\end{cases}
\end{equation}
Hence, for $\b>1$ the neutral set is disconnected.
For $\b\leq 1$ we have moreover $l^*_{\b,\e}(0+)=0$, and hence
the neutral set is connected.
\end{thm}
This result on the neutral set is a consequence of the solution
of the Morita approximant for any choice of the parameters.
We will now describe the behavior of the Morita approximant
for general choice of the parameters. Then we will derive
as a conclusion the explicit condition for the neutral set.
We need some definitions.
Define the effective magnetic field-like parameter
\begin{equation}\begin{split}
\hat h =h_0 +\bar h_{\b,\e}(\l)
\end{split}
\end{equation}
with the function
\begin{equation} \label{eq:10}
\bar h_{\b,\e}(\l):=\frac{1}{2\b}\log\frac{\cosh(\l+\b\e)}{\cosh(\l-\b\e)}
\end{equation}
Define the joint single-site measures depending on the parameter set, and
on an additional (magnetization-like) parameter $m\in \R$.
\begin{equation}\label{limitform}\begin{split}
\pi_{\l;\b,\e,h_0}[m](\s_{i},\eta_{i})&:=\frac{\exp\Bigl( \b\bigl(
(m+\e \eta_i +h_0) \s_i
+\l\eta_i
\bigr)
\Bigr)}{2 \sum_{k=\pm 1}\cosh\bigl(
\b(m+\e k +h_0)
\bigr) e^{\l k}
}\cr
&=
\frac{\exp\bigl(\b (m +\hat h )\s_i\bigr)}{2 \cosh
\b (m +\hat h )}\frac{\exp\bigl((\b\e\s_i+\l) \eta_i \bigr)}{2 \cosh(\b\e\s_i+\l)}\cr
\end{split}
\end{equation}
Here we found it convenient to express the joint distribution on the r.h.s.
appearing under the $i$-product in the form $\text{Prob}(\s_i,\eta_i)=\text{Prob}(\s_i)\text{Prob}(\eta_i|\s_i)$.
In this way the marginal on the $\s$'s can be readily read off.
We see that the role of the parameter $\hat h$ is to
provide an "effective magnetic field" acting on the spins.
\begin{thm} {\bf (Solution of Morita approximant)}\label{thm:Morita approximant}
Assume that the parameters \linebreak $\b,\e\in(0,\infty)$ and $\l,h_0\in (-\infty,\infty)$ are fixed. \\
\noindent{\bf (i): }Assume at first that $\hat h\neq 0$.
Then we have the weak convergence
\begin{equation}\label{limitform1}\begin{split}
&\lim_{N\uparrow\infty} \hat K_{\l;\b,\e,h_0, N}(\s_{[1,N_0]},\eta_{[1,N_0]})
=\prod_{i=1}^{N_0} \pi_{\l;\b,\e,h_0}\bigl[m^{\text{CW}}(\b,\hat h)\bigr](\s_{i},\eta_{i})
\end{split}
\end{equation}
Here we have denoted by $m^{\text{CW}}(\b,h)$
the solution of $m=\tanh(\b(m+h))$ that has the sign of $h$, for $h\neq 0$. \\
\noindent{\bf (ii): }For $\hat h=0$ we have the weak convergence to the symmetric linear combination
of product measures
\begin{equation}\begin{split}
&\lim_{N\uparrow\infty} \hat K_{\l;\b,\e,h_0, N}(\s_{[1,N_0]},\eta_{[1,N_0]})\cr
&=\frac{1}{2}\prod_{i=1}^{N_0} \pi_{\l;\b,\e,h_0}\bigl[m^{\text{CW}}(\b,0+)\bigr](\s_{i},\eta_{i})
+\frac{1}{2}\prod_{i=1}^{N_0} \pi_{\l;\b,\e,h_0}\bigl[m^{\text{CW}}(\b,0-)\bigr](\s_{i},\eta_{i})
\end{split}
\end{equation}
\end{thm}
\noindent{\bf Remark: }
Of course
$m^{\text{CW}}(\b,h)$ is the magnetization of an ordinary Curie Weiss Ising model in an external field.
\bigskip
\noindent {\bf Proof:}
Let us write the
Morita approximant joint measure as a marginal on the $\s$'s times
the conditional measure of the random fields given the $\s$'s, that is
\begin{equation}\begin{split}
&\hat K_{\l;\b,\e,h_0, N}(\s_{[1,N]}, \eta_{[1, N]})\cr&
=\frac{\exp\Bigl(\frac{\b}{2 N}\bigl(\sum_{i=1}^N \s_i\bigr)^2 + \b h_0
\sum_{i=1}^N \s_i + \sum_{i=1}^N \log \cosh(\b\e\s_i+\l)
\Bigr)}{\Norm. }
\prod_{i=1}^N \frac{e^{ (\b\e\s_i+\l) \eta_i } }{2 \cosh(\b\e\s_i+\l)}
\end{split}
\end{equation}
This shows us that the marginal distribution on the $\s$'s is given
by an ordinary ordered mean field Ising model of the form
$\varpropto \exp\Bigl(\frac{\b}{2 N}\bigl(\sum_{i=1}^N \s_i\bigr)^2 + \b \hat h
\sum_{i=1}^N \s_i \Bigr)/\Norm. $
with the effective field $\hat h$. From here the limit statements are
obvious by the known convergence results of the Curie Weiss Ising model
to the corresponding (linear combination of) product measures. $\Cox$
\bigskip
Now, from the explicit solution we may derive
explicit information on the neutral set.
In order to do so, note at first the elementary properties
\begin{eqnarray} \label{eq:emp-meas}
\bar h_{\b,\e}(\l)&= & \left\{
\begin{array}{ll}
\downarrow -\e, & \mbox{for } \l\downarrow -\infty \\
0, & \mbox{for } \l=0\\
\uparrow\e
& \mbox{for} \l\uparrow\infty \, ,
\end{array} \right.
\nonumber
\end{eqnarray}
and it is a monotonically increasing in $\l$ and odd.
It maps $\R$ to the interval $(-\e,\e)$.
\begin{thm} {\bf (Explicit description of neutral set)}\label{thm:explicitnl}
Assume that $\b>0$ and $\e> 0$ are fixed. The decomposition (\ref{nldecomp})
holds with
\begin{equation}\begin{split}
&\RR^+(\b,\e)=\Bigl\{
(h_0,-l)\Bigl| 0<\bar h_{\b,\e}(l)< h_0, \cr
&\qquad\qquad m^{\text{CW}}\Bigl(\b,h_0- \bar h_{\b,\e}(l)\Bigr)=\frac{\sinh( 2 l) }{\sinh(2 \b\e) }
\Bigr\}
\end{split}
\end{equation}
This set can be written as a graph in the form
(\ref{nlgraph}) with a continuous increasing $l_{\b,\e}(h_0)$ that
maps the interval $(a(\b,\e),\infty)$ onto the interval $(\bar h_{\b,\e}(a(\b,\e)), \b \e)$
where $a(\b,\e)$ is uniquely given by
$\bar h_{\b,\e}\bigl(a(\b,\e)\bigr)= \frac{1}{2}\sinh^{-1}\Bigl(
\sinh(2 \b \e)\,m^{\text{CW}}\bigl(\b,0+\bigr) \Bigr)$.
\end{thm}
\noindent{\bf Remark: }Note that the above expression for $a(\b,\e)$ implies
that $a(\b,\e)=0$ if and only if the spontaneous magnetization $m^{\text{CW}}\bigl(\b,0+\bigr)$
vanishes, i.e. $\b\leq 1$.
\bigskip
\noindent {\bf Proof:} Suppose that $\b >1$. Then, in order
to have convergence to a symmetric product measure
on the random fields we must have that the parameters $\l;\b,\e,h_0$ are such that
$\hat h \neq 0$. (Indeed, for
$\hat h=0$ the distribution on the spins
converges to a symmetric mixture of two different product measures.
But from this it is obvious that also the random field distribution
will be a mixture between two different product measures.)
This shows that $(0,0)\not\in \RR(\b,\e)$
in that case.
Suppose however $\b\leq 1$. Then $(h_0,\l)=(0,0)$
implies $\hat h=0$ which implies that the distribution
of the $\s$'s is a symmetric product measure. But this implies
that the distribution of the random fields will be a symmetric product measure
so that $(0,0)\not\in \RR(\b,\e)$
in that case. \\
So, we are left with the case $\hat h\neq 0$.
We can treat the cases $\b>1 $ and $\b\leq 1$
on a unified basis.
Now, conditional on the value of $\s$ the $\eta_i$ have
an expectation value of $\tanh(\l+\b \e \s_i)$.
We use the simple identity
\begin{equation}\begin{split}
&\tanh(\l+\b \e \s_i)=\frac{B(1-L^2)\s_i + L(1- B^2)}{1 - B^2 L^2}\text{ where}Ê\cr
&L=\tanh\l, \quad B=\tanh \b \e
\end{split}
\end{equation}
for $\s_i=\pm 1$.
So the distribution on the random fields $\eta_i$ converges
weakly to a product measure with individual expectation value
\begin{equation}\begin{split}
&\lim_{N\uparrow\infty}
\int\hat K_{\l;\b,\e,h_0, N}(d \eta_1)\eta_1=\frac{B(1-L^2) m^{\text{CW}}(\b,\hat h) + L(1- B^2)}{1 - B^2 L^2}
\end{split}
\end{equation}
Put $l=-\l$ and use $\tanh(l)/(1-\tanh^2(l))= \sinh (2l)$.
So, in order to have the desired convergence to the symmetric
product measure we must have
\begin{equation}\label{consistency}m^{\text{CW}}(\b,h_0-\bar h_{\b,\e}(l)) =\frac{\sinh(2 l)}{\sinh(2\b\e)}
\end{equation}
This equation can only hold if $h_0-\bar h_{\b,\e}(l)$ and $l$ have the same
sign. By symmetry we can assume that $l >0$. But this implies that $h_0>0$ (
since $\bar h_{\b,\e}(l)>0$.)
So, it suffices to look for all pairs $(l,h_0)$ with $l>0$ that satisfy the consistency equations
(\ref{consistency}).
The small trick we are using now is to fix the $l$ and ask for $h_0$ rather
than doing it the opposite way.
Fixing $l$ we see that the l.h.s. runs monotonically through the open interval
$(m^{\text{CW}}(\b,0+),1)$ when $h_0$ runs in the "allowed range"
$(\bar h_{\b,\e}(l),\infty)$.
So, the set of $l>0$ such that there exists a solution $h_0$ is
determined by the condition $(m^{\text{CW}}(\b,0+),1)\ni \frac{\sinh(2 l)}{\sinh(2\b\e)}$.
Equivalently, this is the open interval $l\in (a(\b,\e),\b \e)$. Moreover
the map to $h_0$ is continuous and monotone by known properties of the function
$m^{\text{CW}}(\b,h)$. So it can be inverted and this yields the claim.
$\Cox$
\bigskip
\bigskip
So what has happened in the naive (but wrong) derivation of the mean-field equations
(\ref{mfeq1}) and (\ref{mfeq2})?
In order to see this let us write down a representation of the
finite-$N$ approximant measures.
As a result of a Gaussian transformation on the level
of measures we get the following formula.
\begin{prop}
In finite volume $N$ we have the identity
\begin{equation}
\begin{split}
&\hat K_{\l;\b,\e,h_0, N}(\s_{[1,N]},\eta_{[1, N]})
=\frac{\int dm
\exp\bigl( -\b N \hat \Phi_{\b,\e,h_0}(m)\bigr)}{\int d\tilde m
\exp\bigl( -\b N \hat \Phi_{\l;\b,\e,h_0}(\tilde m) \bigr)}\prod_{i=1}^N \pi_{\l;\b,\e,h_0}[m](\s_{i},\eta_{i})
\end{split}
\end{equation}
Here
\begin{equation}\label{phifunction}\begin{split}
\hat \Phi_{\l;\b,\e,h_0}(m)
& =\frac{m^2}{2}-\frac{1}{\b}
\log \sum_{k=\pm 1}\cosh\bigl(
\b(m+\e k +h_0)
\bigr) e^{\l k}\cr
&=\frac{m^2}{2}-\frac{1}{\b}\log\cosh \bigl(
\b(m+\hat h)
\bigr)+\Const(\b,\e)
\end{split}
\end{equation}
where $\Const(\b,\e)$ does not depend on $m$.
\end{prop}
\noindent{\bf Remark: }
The second equality for $\hat\Phi$ can be seen e.g. by
reexpressing the first $\cosh$ as a sum over a spin $s=\pm 1$ and
exchanging the $s$ and $k$-sums.
\noindent{\bf Proof: }
We use a Gaussian transition kernel from the $\s$-variables
to an auxiliary real valued variable $m$ given by $T(d m|\s_{[1,N]})=
\exp\Bigl( -\frac{\b N}{2} \Bigl(m-\frac{\sum_{i=1}^n\s_{i}}{N} \Bigr)^2\Bigr)dm /\Norm.$.
We define a "big joint measure" on the spins, the random fields and also the
auxiliary magnetization-like continuous variable by the formula
\begin{equation}
\hat M_{\l;\b,\e,h_0, N}(d m,\s_{[1,N]}, \eta_{[1, N]})
:=\hat K_{\l;\b,\e,h_0, N}(\s_{[1,N]}, \eta_{[1, N]})T(d m|\s_{[1,N]})
\end{equation}
We see that $m$ concentrates very nicely around the value of
the empirical average of the true spins in this measure.
Then the non-normalized density of this "big joint measure" is given by
$\exp\Bigl( -\frac{\b N}{2} m^2
+\b\sum_{i}\bigl(
(m+\e \eta_i +h_0) \s_i
+\l\eta_i
\bigr)
\Bigr)$. Use this to express the "big joint measure" in the form of a marginal on the $m$ times a conditional
measure on the $(\s,\eta)$ given the $m$. From here it is simple to get the desired formula.
$\Cox$
\bigskip
So, conditional on a value of $m$,
the pairs $(\s_i,\eta_i)$ are independent.
We have then for their conditional mean values
\begin{equation}\label{eq:9}\begin{split}
&%\EE^1_{\b,\e,h_0}(m,\l):=
\sum_{\s_1=\pm}\pi_{\l;\b,\e,h_0}[m](\s_1)\s_1
=\frac{\sum_{k=\pm 1}\sinh\bigl(
\b(m+\e k +h_0)
\bigr) e^{\l k}}{\sum_{k=\pm 1}\cosh\bigl(
\b(m+ \e k +h_0)
\bigr) e^{\l k}}\cr
&%\EE^2_{\b,\e,h_0}(m,\l):=
\sum_{\eta_1=\pm}\pi_{\l;\b,\e,h_0}[m](\eta_1)\eta_1
=\frac{\sum_{k=\pm 1}k\cosh\bigl(
\b(m+\e k +h_0)
\bigr) e^{\l k}}{\sum_{k=\pm 1}\cosh\bigl(
\b(m+\e k +h_0)
\bigr) e^{\l k}}\cr
\end{split}
\end{equation}
We remark that, with this notation, we have that (the version for general $h_0$ of)
the saddle point equation
(\ref{mfeq1}) is equivalent to the consistency equation for the magnetization written as
\begin{equation}\label{mfeq1a}\begin{split}
&%\EE^1_{\b,\e,h_0}(m,\l):=
m=\sum_{\s_1=\pm}\pi_{\l;\b,\e,h_0}[m](\s_1)\s_1
\end{split}
\end{equation}
The (version for general $h_0$ of) the neutrality equation (\ref{mfeq2}) is written as
is equivalent to
\begin{equation}\label{mfeq2a}\begin{split}
&%\EE^1_{\b,\e,h_0}(m,\l):=
0=\sum_{\eta_1=\pm}\pi_{\l;\b,\e,h_0}[m](\eta_1)\eta_1
\end{split}
\end{equation}
Now, the large-$N$ limit of the model is obtained by looking at the absolute minimizer
of the function $m\mapsto \hat \Phi(m) $.
But note that the representation for $\hat\Phi(m)$ given in the second
line shows that is has the double-well form of the corresponding function in
an Ising model in the external field $\hat h$.
It is an elementary property of this function
that its absolute minimizer has the same sign as $\hat h$.
But this shows that the relation (\ref{mfeq2}) can not be true for
the absolute minimizer. Instead the solution of (\ref{mfeq1}), (\ref{mfeq2}) corresponds
to the second local minimum which is not the absolute minimum but the metastable minimum.
\bigskip
\bigskip
\section{Validity of consistency equations and almost sure discontinuity of conditional expectations}
\label{sect:true Morita}
So how can we understand the fact that the correct
solution of the model is obtained by
solving equations (\ref{mfeq1a}) and (\ref{mfeq2a}) although the solution
corresponds to the wrong saddle point? The solution to this
puzzle is due to the fact that the naive equations have rigorous counterparts in the following sense.
The equations we are going to state now appear as consistency equations for
the conditional probabilities of the {\it true} joint measures.
\begin{prop}{\bf (Consistency equations for true joint measure)}
There is a function $\l_N(\eta_{[2,N]})$, depending on the parameters
$\b,\e,h_0$, which is invariant under permutation of $(\eta_i)_{i=2,\dots,N}$
such that we have
\begin{equation}\begin{split}\label{true1}
&\sum_{\s_1}
K_{\b,\e,h_0, N}(\s_1)\s_1\cr
&= \sum_{\s_{[2,N]}, \eta_{[2, N]}}K_{\b,\e,h_0, N}(\s_{[2,N]}, \eta_{[2, N]})
\,\,\Biggl(\sum_{\s_1=\pm}\pi_{\l_N(\eta_{[2,N]});\b,\e,h_0}\Bigl[\frac{1}{N}\sum_{i=2}^N \s_i \Bigr](\s_1)\s_1\Biggr)
\end{split}
\end{equation}
\begin{equation}\begin{split}\label{true2}
&0=
\sum_{\s_{[2,N]}, \eta_{[2, N]}}
K_{\b,\e,h_0, N}(\s_{[2,N]}, \eta_{[2, N]})\,\,\Biggl(\sum_{\eta_1=\pm 1}
\pi_{\l_N(\eta_{[2,N]});\b,\e,h_0}\Bigl[\frac{1}{N}\sum_{i=2}^N \s_i \Bigr](\eta_1)\eta_1\Biggr)\cr
\end{split}
\end{equation}
\end{prop}
\bigskip
\noindent{\bf Proof of the proposition: }
The proof is based on the following lemma.
\begin{lem}{\bf (Representation of conditional probability of true joint measure)}
The single-site conditional probabilities can be written in the form
\begin{equation}\begin{split}
&K_{\b,\e,h_0, N}(\s_1,\eta_1|\s_{[2,N]}, \eta_{[2, N]})
=\pi_{\l_N(\eta_{[2,N]});\b,\e,h_0}\Bigl[\frac{1}{N}\sum_{i=2}^N \s_i \Bigr](\s_1,\eta_1)
\end{split}
\end{equation}
where
\begin{equation}\begin{split}
\l_{N}(\eta_{[2,N]})%\equiv\l_{\b,\e,h_0,N}\Bigl( \frac{1}{N}
%\sum_{i=2}^N \eta_i
%\Bigr)
&=\frac{1}{2}\log\frac{Z_{\b,\e,h_0,N} [\eta_1=-,
\eta_{[2,N]}]}{Z_{\b,\e,h_0,N} [\eta_1=+,
\eta_{[2,N]}]}\cr
%&=\frac{1}{2}\log \mu_{\b,\e,h_0,N} [\eta_1=+,
%\eta_{[2,N]}]\Bigl(\exp\bigl( - 2 \b \e \s_1 \bigr) \Bigr)
\end{split}
\end{equation}
% the quenched average
%we see that this function is decreasing in $\eta_{[2,N]}$.
\end{lem}
\noindent{\bf Proof of the lemma: }
By a simple computation we have for the single-site distribution
\begin{equation}\begin{split}
&K_{\b,\e,h_0, N}(\s_1,\eta_1|\s_{[2,N]}, \eta_{[2, N]})\cr
&= \frac{1}{\Norm}
\exp \Bigl(\b\bigl(\frac{1}{N}\sum_{i=2}^N \s_i+\e\eta_1+h_0 \bigr)\s_1
+\frac{1}{2}\log\frac{Z_{\b,\e,h_0,N} [\eta_1=-,
\eta_{[2,N]}]}{Z_{\b,\e,h_0,N} [\eta_1=+,
\eta_{[2,N]}]}\times\eta_1
\Bigr)
\end{split}
\end{equation}
and this shows the claim.
$\Cox$
Continuing with the proof of the proposition we use the formula for the
conditional probabilities writing
\begin{equation}\begin{split}
&K_{\b,\e,h_0, N}(\s_1,\eta_1)\cr
&= \sum_{\s_{[2,N]}, \eta_{[2, N]}}
K_{\b,\e,h_0, N}(\s_{[2,N]}, \eta_{[2, N]})\,\,\,\,
\pi_{\l_N(\eta_{[2,N]});\b,\e,h_0}\Bigl[\frac{1}{N}\sum_{i=2}^N \s_i \Bigr](\s_1,\eta_1)
\cr
\end{split}
\end{equation}
But this equation gives the equation for the magnetization (\ref{true1}) by summing over $\s_1$.
Using the symmetry of the distribution of $\eta_1$ we get (\ref{true2}).
$\Cox$
\bigskip
\bigskip
Let us now summarize what we know by the rigorous
solution of the random field model about
the limiting distribution of the pair of random quantities entering the single-site kernel $\pi$.
In words, the distribution becomes sharp in the case of non-zero external field.
It becomes sharp but double valued in the case of vanishing external field.
In view of the last lemma this statement is a different way of saying that there
is a jump of the conditional probabilities when the empirical
random field sum of the conditioning is infinitesimally perturbed around
its typical value $0$. Now, the rigorous statement is as follows.
\bigskip
\bigskip
\begin{thm}{\bf (Convergence of true joint measures)}
\noindent{\bf (i)} Suppose that $h_0>0$. Then we have the weak limit
$$\lim_{N\uparrow\infty}K_{\b,\e,h_0, N}\Biggl (\frac{1}{N}\sum_{i=2}^N \s_i \in\,\cdot\,\,\quad ,\,
\l_N(\eta_{[2, N]})\in \,\cdot \,\Biggr)
\rightarrow \d_{m^*(h_0)}\times\d_{\l^*(h_0)}
$$
Here $(m^*(h_0),\l^*(h_0))$ is a
solution of the consistency equations
(\ref{mfeq1a}) and (\ref{mfeq2a}). % with $m^*>0$.
%Furthermore the limit $\lim_{h_0\downarrow 0}
%(m^*(h_0),\l^*(h_0))=(m^*,\l^*)$ exists and is a solution of the
%naive equations for $h_0=0$.
\medskip
\noindent{\bf (ii)} Suppose that $h_0=0$. Then
$$\lim_{N\uparrow\infty}
K_{\b,\e,h_0=0, N}\Biggl (
\frac{1}{N}\sum_{i=2}^N \s_i \in\,\cdot\,\,\quad ,\,
\l_N(\eta_{[2, N]})\in \,\cdot \,
\Biggl| \sum_{i=1}^N\eta_i>0\Biggr)
\rightarrow \d_{m^*}\times\d_{\l^*}
$$
where $(m^*,\l^*)$ is the
unique solution of the consistency equations
(\ref{mfeq1a}) and (\ref{mfeq2a}) with $m^*>0$ (and, as a consequence $\l^*<0$).
As a consequence we have
$$
\lim_{N\uparrow\infty}K_{\b,\e,h_0=0, N}\Biggl (
\frac{1}{N}\sum_{i=2}^N \s_i \in\,\cdot\,\,\quad ,\,
\l_N(\eta_{[2, N]})\in \,\cdot \,\Biggr)
\rightarrow \frac{1}{2}\d_{m^*}\times\d_{\l^*}+\frac{1}{2}\d_{-m^*}\times\d_{-\l^*}
$$
\end{thm}
\noindent{\bf Remark: }
We see that the system chooses the particular value of $\l_N(\eta_{[2, N]})$ (that has the opposite
sign of the magnetisation) itself!
\bigskip
\noindent{\bf Proof: } We only sketch the proof.
We rewrite the quotient of partition functions appearing in the definition of $\l_N(\eta_{[2,N]})$ in the form
\begin{equation}\begin{split}\label{lam}
\l_{N}(\eta_{[2,N]})=\frac{1}{2}\log \mu_{\b,\e,h_0,N} [\eta_1=+,
\eta_{[2,N]}]\Bigl(\exp\bigl( - 2 \b \e \s_1 \bigr) \Bigr)
\end{split}
\end{equation}
From here Theorem 3.3 follows from the work done for the quenched
model in \cite{K97,K3}. Let us focus here only on the interesting case of vanishing
external magnetic field $h_0=0$.
In this case it was shown that, under the condition of positive sum of the random fields
the empirical average of the spins concentrates sharply around
the positive magnetisation $m^*$ (positive solution of (\ref{eos}))
w.r.t. to the quenched Gibbs probability.
(This is true for "typical values" of the random field sum,
that is for $N^{\frac{1}{2}-\d}\leq
\sum_{i=1}^N\eta_i \leq N^{\frac{1}{2}+\d}$, and these values get all mass w.r.t. $\P$ in the large-$N$ limit).
At the same time the
quenched Gibbs probability $\mu_{\b,\e,h_0,N} [\eta_1=+,
\eta_{[2,N]}](\s_1=+)$ aquires a sharp value that is related in a simple way to $m^*$.
From (\ref{lam}) this gives the value of $\l^*$.
$\Cox$
\bigskip
\bigskip
Not assuming the knowledge of the solution of the quenched model
we can reverse the argument in the following way in order to solve the model.
Look at the consistency equations for the true joint measure (\ref{true1}),(\ref{true2}).
Take $h_0>0$. Then it is very plausible without much a priori knowledge that the
distribution of the
pair $\Bigl( \frac{1}{N}\sum_{i=2}^N \s_i,\l_{N}(\eta_{[2,N]})
\Bigr)$ under the true joint measure should converge to a Dirac measure $\d_{m,\l}$.
(This is in particular clear, if we assume that $\l_N$ has the form (\ref{lam}) and assume
that the quenched magnetization becomes sharp for typical realization of the random fields in a
positive homogeneous external field.)
But this means that the outer integrals in the
rigorous consistency equations become sharp.
So, the limiting value $(m,\l)$ must necessarily satisfy the
naive consistency equations
(\ref{mfeq1a}) and (\ref{mfeq2a}).
These equations can then be solved and afterwards we let the external magnetic field
$h_0$ tend to zero from above to discover
the known solutions for the model.
Let us finally see that, in the case of $h_0=0$ the
validity of the naive equations implies that
{\it there must be} discontinuous behavior of the conditional expectations as a function
of the average of the random fields appearing in the conditioning.
Indeed, suppose that $\l_N(\eta_{[2,N]})$ were a continuous function
of $\frac{1}{N}\sum_{i=2}^N \eta_i$. Then, by the law
of large numbers it would have to be constant in the large-$N$ limit.
But by reasons of symmetry this constant would have
to be zero in the case of $h_0=0$.
But this is in contradiction to the non-trivial solution of the
naive equations (\ref{mfeq1}) and (\ref{mfeq2}).
To summarize the last line of
argument in catchy terms: Non-Gibbsianness is necessary
to help the metastable solution to provide the right answer.
\bigskip
We remark that the purpose of this note is not
to attack the Morita approach in general as a valuable heuristic method
in theoretical physics to predict the behavior of disordered systems
when a rigorous analysis is not available or not yet available.
As pointed out to us by Reimer K\"uhn, one could also argue that the second of
the naive equations (\ref{mfeq2}), which
demands that $m$ and $\lambda$ at the physical fixed point
must have opposite sign, renders the region of integration for the partition function
(\ref{partition-function})
which includes the other fixed point as {\it unphysical} and so
to be excluded from the domain of integration.
This line of reasoning would render the "naive" argument
correct and this is not the first occasion in physics where such things
happen. %, (both in the narrow and in the wide sense).
While there seems no direct rigorous justification for this procedure
we have shown that one is able to understand the validity of the naive equations
by viewing the parameter $\l$ properly as a stochastic quantity.
This might give hope that results
obtained by approximation schemes based on the Morita approach
provide correct answers also in more complicated situations where
a rigorous analysis is lacking. A better understanding of this
would pose a fascinating challenge.
\bigskip
\bigskip
{\bf Acknowledgments:}
The author thanks Aernout van Enter and Reimer K\"uhn for discussions at the conference
"Gibbs vs. non-Gibbs" (EURANDOM, december 2003)
which led to this paper and valuable comments on an earlier draft.
\bigskip
\bigskip
\begin{thebibliography}{19}
\small
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