%PARAM.1
\magnification=\magstep1
\pageno=0
%FONTS.1
%on CARL:
%\font\huge=cmbx12
%\font\xbold=cmbx10 scaled\magstep1
%\font\sub=cmbx10
%on IBM:
\font\huge=cmbx10 scaled\magstep3
\font\xbold=cmbx10 scaled\magstep2
\font\sub=cmbx10 scaled\magstep1
\font\small=cmr7
%TITLES.2
\count5=0
\count6=1
\count7=1
\count8=1
\def\proof{\medskip\noindent{\bf Proof.\ }}
\def\qed{\hfill\smallskip
	 \line{\hfill\vrule height 1.8ex width 2ex depth +.2ex
	       \ \ \ \ \ \ }\bigskip}
\def\remark{\medskip\noindent{\bf Remark.\ }}
\def\abstract#1\par{\noindent{\sub Abstract.} #1 \par}
\def\equ(#1){\csname e#1\endcsname}
\def\clm(#1){\csname c#1\endcsname}
\def\equation(#1){\eqno\tag(#1)}
%\def\equation(#1){\eqno\tag(#1) {\rm #1}}
\def\tag(#1){(\number\count5.
	      \number\count6)
    \expandafter\xdef\csname e#1\endcsname{
    (\number\count5.\number\count6)}
    \global\advance\count6 by 1}
\def\claim #1(#2) #3\par{
    \vskip.1in\medbreak\noindent
   {\bf #1\ \number\count5.\number\count7.}
    \expandafter\xdef\csname c#2\endcsname{
    #1~\number\count5.\number\count7}
    \global\advance\count7 by 1
    {\sl #3}\par
    \ifdim\lastskip<\medskipamount
    \removelastskip\penalty55\medskip\fi}
\def\section#1\par{\vskip0pt plus.3\vsize\penalty-75
    \vskip0pt plus -.3\vsize\bigskip\bigskip
    \global\advance\count5 by 1
    \message{#1}\leftline
     {\xbold \number\count5.\ #1}
    \count6=1
    \count7=1
    \count8=1
    \nobreak\smallskip\noindent}
\def\subsection#1\par{\vskip0pt plus.2\vsize\penalty-75
    \vskip0pt plus -.2\vsize\bigskip\bigskip
    \message{#1}\leftline{\sub
    \number\count5.\number\count8.\ #1}
    \global\advance\count8 by 1
    \nobreak\smallskip\noindent}
%MACROS.7
\footline{\ifnum\pageno=0\hss\else\hss\tenrm\folio\hss\fi}
\let\cl=\centerline
\let\ov=\overline
\def\natural{{\rm I\kern-.18em N}}
\def\integer{{\rm Z\kern-.32em Z}}
\def\real{{\rm I\kern-.2em R}}
\def\complex{\kern.1em{\raise.47ex\hbox{
	    $\scriptscriptstyle |$}}\kern-.40em{\rm C}}
\def\Tanh{{\rm Tanh}}
\def\AA{{\cal A}}
\def\BB{{\cal B}}
\def\CC{{\cal C}}
\def\DD{{\cal D}}
\def\EE{{\cal E}}
\def\FF{{\cal F}}
\def\GG{{\cal G}}
\def\HH{{\cal H}}
\def\II{{\cal I}}
\def\JJ{{\cal J}}
\def\KK{{\cal K}}
\def\LL{{\cal L}}
\def\MM{{\cal M}}
\def\NN{{\cal N}}
\def\OO{{\cal O}}
\def\PP{{\cal P}}
\def\QQ{{\cal Q}}
\def\RR{{\cal R}}
\def\SS{{\cal S}}
\def\TT{{\cal T}}
\def\UU{{\cal U}}
\def\VV{{\cal V}}
\def\WW{{\cal W}}
\def\XX{{\cal X}}
\def\YY{{\cal Y}}
\def\ZZ{{\cal Z}}
\def\rA{1}
\def\rB{2}
\def\rD{4}
\def\rE{6}
\def\rF{7}
\def\rG{8}
\def\rH{10}
\def\rI{11}
\def\rJ{12}
\def\rK{3}
\def\rL{9}
\def\rM{5}
\def\rN{13}
\def\rP{14}
\def\rQ{15}
%ABSTRACT.3 Version 2
\cl{{\huge Some Rigorous Results on the}}
\cl{{\huge Hopfield Neural Network Model}}
\vskip.6in
\cl{Hans Koch
\footnote{$^*$}
{{\small Supported in Part by the
National Science Foundation under Grant No. DMS--8802590.}}}
\cl{Department of Mathematics, University of Texas at Austin}
\cl{Austin, TX 78712}
\vskip.25in
\cl{Jacques Piasko}
\cl{Institut f\"ur Theoretische Physik, ETH H\"onggerberg}
\cl{CH--8093 Z\"urich}
\vskip.8in
\abstract
We analyze the thermal equilibrium distribution of $2^p$ mean field variables
for the Hopfield model with $p$ stored patterns,
in the case where $2^p$ is small compared to the number of spins.
In particular, we give a full description
of the free energy density in the thermodynamic limit,
and of the so called ``symmetric solutions'' for the mean field equations.
\par\vfill\eject
%INTRO.6
\section Introduction and main results

We consider Hopfield's model [\rA,\rB,\rK] of an associative read--only memory
with $p$ stored patterns, in the case where $2^p$ is small
compared to the number of degrees of freedom (neurons, spins).
The time evolution of this model is the Glauber dynamics for a system of
$N$ interacting Ising spins $S_i$ with values $+1$ or $-1$,
governed by a Hamiltonian of the form
$$
H_N=-\sum_{1\le i<j\le N}J_{ij}S_iS_j\quad -\sum_{1\le i\le N}T_iS_i
\;.\equation(1)
$$
Here, the values of the coupling constants $J_{ij}$ and $T_i$
depend on the content of the memory:
If $\xi=(\xi^1,\xi^2,\ldots\,,\xi^p)$ is an arbitrary but fixed collection
of spin configurations, representing the stored patterns,
then the following constants are chosen.
$$
J_{ij}={1\over N}\sum_{\mu=1}^p\xi^\mu_i\xi^\mu_j
\,,\qquad
T_i=\eta\xi^\nu_i
\,,\equation(2)
$$
where $\nu$ and $\eta$ are external field parameters
which will be specified later.

The generalized Hopfield dynamics [\rD] describes the retrieval,
from a ``noisy'' memory,
of $p$ stored patterns by association with some input pattern.
The retrieval process can be viewed as a random walk
on the set $\SS$ of spin configurations:
If $\rho$ is the delta function on $\SS$ with peak at the input pattern,
or any other probability distribution on $\SS$,
then the spin distribution after one unit of time is given by $W\rho$,
$$
(W\rho)(S^\prime)=\sum_{S\in\SS}W(S^\prime,S)\rho(S)
\,.\equation(3)
$$
The transition probabilities $W(S^\prime,S)$
depend on the inverse temperature $\beta$ of the noise,
and are defined as follows.
If $S^\prime$ can be obtained from $S$ by flipping a single spin, then
$$
W(S^\prime,S)={1\over N}\cdot
{\omega(S^\prime)\over{\omega(S)+\omega(S^\prime)}}
\,,\equation(4)
$$
where $\omega$ denotes the normalized Gibbs distribution
$$
\omega(S)=\biggl(\sum_{S\in\SS}e^{-\beta H_N(\eta,\xi,S)}\biggr)^{-1}
\cdot e^{-\beta H_N(\eta,\xi,S)}
\,.\equation(5)
$$
All other off--diagonal entries of $W$ are zero,
and the diagonal entries are determined by requiring that
the probabilities $W(S^\prime,S)$, for any fixed $S$, add up to one.

From this definition it follows immediately that
all matrix elements of $W^N$ are positive,
and that $W$ satisfies the detailed balance condition
$\omega(S)W(S^\prime,S)=\omega(S^\prime)W(S,S^\prime)$,
for any pair $(S^\prime,S)$ of spin configurations.
Thus, by the fundamental Theorem of Monte Carlo calculus,
$W^n\rho$ converges to $\omega$ as $n\to\infty$,
for every probability distribution $\rho$ on $\SS$,
in contradistinction to the situation at zero temperature [\rB,\rM,\rN,\rP],
where every local minimum of the energy function $H_N$ corresponds to
an attractor of the Hopfield dynamics.
As long as $\beta$ is finite, the retrieval of information
is a transient process; after a sufficiently long time,
the system starts to forget its initial condition,
and approaches the thermal equilibrium state $\omega$.

The first part of our analysis deals with the equilibrium properties
of the Hopfield model with ``unbiased'' memories.
More precisely, we consider the free energy per spin,
averaged over all $2^{pN}$ possible choices of $p$ patterns,
$$
F_N(\beta,\eta)=2^{-pN}\sum_\xi{-1\over{\beta N}}
\ln\biggl(\sum_{S\in\SS}e^{-\beta H_N(\eta,\xi,S)}\biggr),
\equation(8)
$$
and we assume that $2^p\ll N$.
The averaging will be justified later by showing that,
outside a negligible set of ``biased'' patterns $\xi$,
the free energy per spin converges uniformly to the same value as $F_N$
(i.e.~it is self--averaging), for large $N$.

This model, and variations thereon, have been studied in detail
in the case of a fixed finite number of patterns [\rD,\rE,\rG,\rL,\rH].
As for the thermodynamic behavior, it is found that
a second order phase transition occurs at $\beta=1$,
from a paramagnetic high temperature phase ($\beta<1$)
to a ferromagnetic low temperature phase ($\beta>1$).
The following theorem establishes the existence of this phase transition,
in a more general situation where the number of patterns
is not necessarily finite. A proof is given in Section 2.

\claim Theorem(1) Fix $\beta\ne 1$ and $\alpha<1$. Then the
average free energy density $F_N(\beta,\eta)$ 
converges as $N\to\infty$, for any positive integer $\nu$,
and for any sequence $p=p(N)$ satisfying $\nu\le p$ and $2^p\le N^\alpha$.
The same holds for the magnetization
$m_N(\beta,\eta)={\partial\over\partial\eta}F_N(\beta,\eta)$, if $\eta\ne 0$.
The corresponding limits $F_\infty$ and $m_\infty$
only depend on $(\beta,\eta)$, and they satisfy
$$
\eqalign{
\beta F_\infty(\beta,\eta)
&=-\ln 2 -{1\over 2}\int\limits_0^\beta a_1(t)^2 dt +\OO(\eta)\,,\cr
m_\infty(\beta,\eta)&={\rm sgn}(\eta)a_1(\beta) +\OO(\eta)\,,
\quad \eta\ne 0,\cr}
\equation(9)
$$
where $a_1(\beta)$ is the largest solution of the equation
$\tanh(\beta a_1)=a_1$.

Note that if $p$ is constant, then the $2p$ possible choices for
$\bigl(\nu,{\rm sgn}(\eta)\bigr)$ lead to $2p$ distinct phases
at low temperature (by symmetry, the magnetization in the direction
of $\xi^\mu$ is zero for all $\mu\ne\nu$).
If $2^p$ grows like $N^\alpha$, with $\alpha<1$,
then an infinite number of these low temperature phases are obtained.
The case $\alpha=1$ represents the borderline case
for the methods used in our proof of \clm(1),
and possibly also for the validity of equation \equ(9);
but the phase portrait is believed to be the same for all $\alpha$.
On the other hand, if $p$ grows at a rate proportional to $N$,
the Hopfield model is expected to exhibit a spin glass phase [\rF,\rI].

We shall now change to a reduced representation (of the Hopfield model),
in which the independent degrees of freedom
are $d=2^p$ mean field variables [\rL].
Let $\{e_1,e_2,\ldots\,,e_d\}$ be a fixed ordered set which
contains all vectors in $\real^p$ whose components are either $+1$ or $-1$.
Any choice of $p$ patterns can then be regarded as a map
$\xi:\ i\mapsto\xi_i=(\xi_i^1,\xi_i^2,\ldots,\xi_i^p)$,
which associates to every site $i$, $1\le i\le N$, one of the vectors $e_k$.
The map $\xi$ defines a partition $N=L_1+L_2+\ldots +L_d$ of $N$,
where $L_k=L_k(\xi)$ denotes the number of sites in $\xi^{-1}(e_k)$.
It also determines a partition 
of the spin configuration space $\SS$ into subsets
$$
\SS(Y)=\Bigl\{S\in\SS:
\sum_{i\in\xi^{-1}(e_k)}\!\!S_i=Y_k\,,\quad 1\le k\le d\Bigr\},
\equation(12)
$$
indexed by vectors $Y\in\integer^d$ with components 
$Y_k\in\{-L_k,-L_k+2,\ldots,L_k\}$, $1\le k\le d$.
Such vectors will be referred to as mean field configurations.

It is easy to check that the Hamiltonian $H_N$ is constant
on each of the sets $\SS(Y)$.
In addition, the Hopfield dynamics induces a random walk
on the set $\YY$ of mean field configurations,
with transition probabilities given by
$$
\ov{W}(Y^\prime,Y)={1\over|\SS(Y)|}\sum_{S\in\SS(Y)}
\sum_{S^\prime\in\SS(Y^\prime)}W(S^\prime,S).
\equation(13)
$$
To be more specific, we define a linear transformation $A$,
which maps functions on $\SS$ to functions on $\YY$,
by the equation
$$
(A\rho)(Y)=\sum_{S\in\SS(Y)}\rho(S).
\equation(14)
$$

\claim Proposition(2) The matrix $\ov{W}$ is conjugate to $W$
and satisfies detailed balance, i.e.
\item{$(i)$} $AW=\ov{W}A$,
\item{$(ii)$} $(A\omega)(Y)\ov{W}(Y^\prime,Y)
=(A\omega)(Y^\prime)\ov{W}(Y,Y^\prime)$
\quad for every $\,Y,Y^\prime\in\YY$.

From either of these two properties
(whose proof is straightforward) it follows that
every probability distribution on $\YY$ converges to $A\omega$
under the mean field dynamics defined by $\ov{W}$.
The equilibrium distribution $A\omega$ is the Gibbs distribution
for the mean field Hamiltonian $\ov{H}_N$, given by
$$
\ov{H}_N(\beta,\eta,L,Y)
\equiv H_N(\eta,\xi,S)-{1\over\beta}\ln|\SS(Y)|,\qquad S\in\SS(Y).
\equation(15)
$$

To simplify our discussion, we assume from now on that $L_k=N/d$,
for $1\le k\le N$. As far as the proof of \clm(1) is concerned,
this restriction is justified by the fact that
the average \equ(8) may be restricted to patterns
satisfying $\bigm|L_k(\xi)-N/d\bigm|<\sqrt{N/d}\ln(N)$ for all $k$,
without affecting the limit $N\to\infty$; for details see Section 2.
With all $L_k$ set to $N/d$, and for zero external field,
the mean field Hamiltonian becomes
$$
\ov{H}_N(\beta,0,L,Y)=N\beta^{-1}\ln2
+N\beta^{-1}f_\beta\bigl({d\over N}Y\bigr)+{\scriptstyle\OO}(N),
\equation(17)
$$
where
$$
f_\beta(y)={1\over d}\sum_{k=1}^d \int\limits_0^{y_k}\!dt \tanh^{-1}(t)
-{\beta\over 2d}||Py||^2.
\equation(18)
$$
Here, $||\ldotp||$ is the norm
defined by the standard inner product on $\real^d$, and
$P$ denotes the orthogonal projection in $\real^d$ onto the subspace
spanned by the $p$ vectors $e^\mu=(e_1^\mu,e_2^\mu,\ldots\,,e_d^\mu)$.
Formally, it is now clear that in the limit $N\to\infty$,
and for fixed finite $p$, the average free energy density
is determined by the minimum of $h_\beta$ on the hypercube $[-1,1]^d$.
It can be shown that $h_\beta(y)$ takes on this minimum value if and only if
$$
y=\pm a_1(\beta)e^\mu,
\qquad 1\le\mu\le p.
\equation(19)
$$

These $2p$ minimizing vectors (for $\beta>1$)
are commonly referred to as retrieval states,
since each of them is associated with exactly one of the stored patterns
($e^\mu$ is the mean field analogue of the pattern $\xi^\mu$).
Below we will discuss other local minima of $f_\beta$,
or so called spurious states,
which are are associated with several patterns
(thus corresponding to a confused memory).
In numerical experiments, both types of ``states'' behave
like attractors for the Hopfield dynamics, if $N$ is sufficiently large.
We expect that any distribution on $\YY$, whose support lies
within a distance ${\scriptstyle\OO}(N)$ of a local minimum $Y$ of $\ov{H}_N$,
will evolve first into a distribution which is essentially localized
in a ball of radius $\OO(\sqrt{N})$ around $Y$,
before spreading out significantly.
Formal calculations indicate that the time scale
for the localization process is of the order of $N$,
while the time needed to reach an approximate thermal equilibrium
grows exponentially with $N$.

In the second part of this paper we consider
the set of critical points of the function $f_\beta$,
or equivalently, the solutions of the (mean field) equation
$$
y_k=\tanh\bigl(\beta(Py)_k\bigr),\qquad 1\le k\le d.
\equation(20)
$$
Our first result describes
the so called ``symmetric solutions'' of order $n$, whose existence,
for all $\beta>1$ and $n\le p$, has been conjectured in [\rD],
based on (numerical calculations and)
expansions near $\beta=1$ and $\beta=\infty$.
A symmetric solution of order $n>0$ (the case $n=0$
corresponds to the trivial solution $y=0$) can be obtained
by making the ansatz $y=a_n(e^1+\ldots+e^n)+w$, with $Pw=0$.
As shown in Section 3,
this ansatz leads to the following equation for $a_n$.
$$
a_n=2^{-n+1}\!\sum_{0\le m<n/2}
{n\choose m}{n-2m\over n}\tanh\bigl((n-2m)\beta a_n\bigr).
\equation(23)
$$

\claim Theorem(4) Given $\beta>1$, and a positive integer $n$,
the equation \equ(23) has a unique positive solution $a_n=a_n(\beta)$.
Furthermore, if $(c_1,c_2,\ldots,c_p)$ is a vector of length $n$ in $\real^p$
whose components are either $0$ or $\pm 1$, and if $y\in\real^d$ is defined by
$$
y_k=\tanh\bigl(\beta a_n(\beta)\sum_{\mu=1}^p c_\mu e^\mu_k\bigr),
\qquad 1\le k\le d,
\equation(22)
$$
then the function $f_\beta$ has a critical point at $y$.

For $\beta\le 1$, it is easy to see
that the equation \equ(20) admits only the trivial solution,
and that $f_\beta$ takes its minimum value for $y=0$.
This minimum turns into a local maximum
as $\beta$ is increased past its critical value $\beta=1$,
and the remaining $3^p-1$ symmetric solutions bifurcate away from the origin.
A qualitative picture of what happens near $\beta=1$
can be derived from general results in bifurcation theory; see [\rH].
A more direct approach, which also allows for explicit numerical bounds,
is presented in Section 3. In particular, we prove the following result.

\claim Theorem(5)
Let $1<\beta<1+\bigl(9d+500p^8\bigr)^{-1}$ and $y\in\real^d$.
\item{$(i)$} If $f_\beta$ has a critical point at $y$, then
$y$ is a symmetric solution of some order $n\le p$.
\item{$(ii)$} If $f_\beta$ has a local minimum at $y$, then
$y$ is a symmetric solution of order $n=1$.

We note that, while some condition on $\beta$ is necessary
in order for the conclusion of \clm(5) to hold,
the bound given here is clearly too restrictive.
Numerical results [\rD] indicate that there is an increasing sequence
of inverse temperatures $\beta_m$, starting with $\beta_1\approx 2.17$,
such that if $\beta$ is larger (smaller) than $\beta_m\,$, then every
symmetric solution of order 2m+1 corresponds to a local minimum
(saddle point) of $f_\beta$.
In contrast, the symmetric solutions of even order
seem to correspond to saddle points, for all $\beta>1$.

Our last result concerns the observed qualitative difference
between solutions of even and odd order.

\claim Theorem(6) Let $m$ be a positive integer not exceeding $p/2$,
and assume that
$$
\beta\cdot 2^{-2m-1}{2m\choose m}>\ln(\beta)>1.
\equation(24)
$$
Then $f_\beta$ has a saddle point or local maximum
at every symmetric solution of order $2m$,
and if $2m<p$, $f_\beta$ has a local minimum
at every symmetric solution of order $2m+1$.

Further details, including the proofs of Theorems 1.3--5,
are given in Section 3.

%LIMIT.7
\section The Thermodynamic Limit

In this section we will prove Theorem 1.1.
We start by deriving an explicit expression
for the mean field Hamiltonian $\ov{H}_N$, as defined in \equ(15).
Note that if $S\in\SS(Y)$, then
$$
\sum_{i=1}^N\xi^\mu_iS_i
=\sum_{k=1}^d \sum_{i\in\xi^{-1}(e_k)}\xi^\mu_iS_i
=\sum_{k=1}^d e^\mu_k\sum_{i\in\xi^{-1}(e_k)}S_i
=\sum_{k=1}^d e^\mu_kY_k\equiv\langle e^\mu,Y\rangle .
\equation(101)
$$
Using this identity, the mean field Hamiltonian can be written as follows.
$$
\eqalign{
\ov{H}_N(\beta,\eta,L,Y)
&\equiv -{1\over 2N}\sum_{\mu=1}^p\Bigl(\sum_{i=1}^N\xi^\mu_iS_i\Bigr)^2
-\eta\sum_{i=1}^N\xi^\nu_iS_i +{p\over 2}
-{1\over\beta}\ln|\SS(Y)|\cr
&=-{1\over 2N}\sum_{\mu=1}^p\langle e^\mu,Y\rangle^2
-\eta\langle e^\nu,Y\rangle +{p\over 2}
-{1\over\beta}\sum_{k=1}^d\ln\biggl({L_k\choose {1\over 2}(L_k+Y_k)}\biggr) .
\cr}
\equation(102)
$$
The entropy term may be represented more conveniently
by using Stirling's formula:
There is a function $g$, satisfying the bound
$|g(L_k,Y_k)|\le\ln(L_k)+1$, such that
$$
\ln\biggl({L_k\choose {1\over 2}(L_k+Y_k)}\biggr)
=L_k\ln2 -L_k\!\!\int\limits_0^{Y_k/L_k}\!\!dt \tanh^{-1}(t) +g(L_k,Y_k).
\equation(103)
$$

In order to discuss the $N$--dependence of $\ov{H}_N$,
let us now change to normalized variables by writing
$$
L_k=(1+\lambda_k){N\over d},\qquad Y_k=L_ky_k.
\equation(104)
$$
The range of values for $(\lambda,y)$
is determined from that of the original variables $(L,Y)$.
In particular, $y$ takes on values in the set
$\XX=\{y\in [-1,1]^d: L_k(1+y_k)/2\in\natural, 1\le k\le d\}$.
Denoting by $\Lambda$ the diagonal $d\times d$ matrix
with entries $\Lambda_{kk}=\lambda_k$,
we arrive at the following expression for $\ov{H}_N$.
$$
\beta\ov{H}_N(\beta,\eta,L,Y)=-N\ln2 +{N\over d}f(\beta,\eta,\lambda,y)
+{p\over 2} +\sum_{k=1}^d g(L_k,Y_k),
\equation(105)
$$
where
$$
f(\beta,\eta,\lambda,y)=-{\beta\over 2}||P(1+\Lambda)y||^2
-\beta\eta\langle e^\nu,(1+\Lambda)y\rangle
+\sum_{k=1}^d(1+\lambda_k)\int\limits_0^{y_k}\!dt \tanh^{-1}(t).
\equation(106)
$$

The quantities of interest in this section
are the free energy density $F_{N,d}$, and the magnetization $m_{N,d}$,
$$
\eqalign{
F_{N,d}(\beta,\eta,L)
&=-{1\over\beta N}\ln\Bigl(\sum_{y\in\XX}
e^{-\beta\ov{H}_N(\beta,\eta,L,Y)}\Bigr),\cr
m_{N,d}(\beta,\eta,L)
&={d\over d\eta}F_{N,d}(\beta,\eta,L).\cr}
\equation(107)
$$
More precisely, if $\phi_{N,d}(L)$ denotes one of the functions
defined in \equ(107), we would like to compute the average of
$\phi_{N,d}(L(\xi))$ over all possible choices of the patterns $\xi$,
$$
2^{-pN}\sum_\xi\phi_{N,d}(L(\xi))
=d^{-N}\!\sum_{L\in\PP_{N,d}}{N!\over L_1!L_2!\cdots L_d!}\ \phi_{N,d}(L).
\equation(108)
$$
Here, $\PP_{N,d}$ denotes the set of vectors in $\real^d$,
whose components are nonnegative integers which add up to $N$.
The following Proposition (a simple large deviations estimate)
will be used to approximate the sum \equ(108)
by a sum over ``unbiased'' patterns, represented by the set
$\UU_{N,d}=\{L\in\PP_{N,d}: |\lambda_k|<\delta, 1\le k\le d\}$,
for some fixed $\delta>0$.

\claim Proposition(1l) There exists $\delta_o>0$, such that
if ${d\over N}\le\delta\le\delta_o$, then
$$
d^{-N}\!\!\sum_{L\in\PP_{N,d}\setminus\UU_{N,d}}
{N!\over L_1! L_2!\cdots L_d!}\quad<dN \exp\bigl(-{N\delta^2\over 2d}\bigr).
\equation(1l)
$$

\proof
Assume that $d\le\delta N$, and denote by $\BB$
the set of all nonnegative integers $n\le N$, which satisfy
$\bigm|n-{N\over d}\bigm|\ge{N\over d}\delta$.
Since every vector in $\PP_{N,d}\setminus\UU_{N,d}$
has at least one of its $d$ components in $\BB$, 
the left hand side of \equ(1l) is bounded by
$$
\sigma\equiv d\cdot d^{-N}\!\!\!\!\sum_{L\in\PP_{N,d} :\ L_1\in\BB}
{N!\over L_1! L_2!\cdots L_d!}\quad
=\;d\cdot d^{-N}\sum_{L_1\in\BB}\biggl[{N\choose L_1}(d-1)^{N-L_1}\biggr].
\equation(2l)
$$
It is easy to check that the expression in square brackets,
when considered as a function of $L_1$,
is increasing for $L_1<(N-d+1)/d$, and decreasing for $L_1>(N+1)/d$.
Thus, if $\varepsilon$ is chosen such that
$L_1=(1+\varepsilon)N/d$ maximizes $[\ldots]$ on $\BB$, then
$\delta\le |\varepsilon|\le 2\delta$, and
$$
\sigma\le dN\cdot d^{-N}{N\choose {N\over d}(1+\varepsilon)}
(d-1)^{N-(1+\varepsilon)N/d}.
\equation(4l)
$$
By applying Stirling's formula to the combinatorial factor
on the right hand side of inequality \equ(4l),
and then simplifying the result, we obtain
$$
\ln(\sigma)\le\ln(dN)-Ng(\varepsilon)
-{1\over 2}\ln\Bigl({N\over d}(1+\varepsilon)\Bigr) + const,
\equation(5l)
$$
where
$$
g(\varepsilon)={1\over d}(1+\varepsilon)\ln(1+\varepsilon)
+\bigl(1-{1\over d}(1+\varepsilon)\bigr)
\ln\bigl(1-{\varepsilon\over d-1}\bigr).
\equation(6l)
$$
An explicit calculation shows that $g(0)=g^\prime(0)=0$, and that
$g^{\prime\prime}(\varepsilon)>1/d$,
for $\varepsilon$ sufficiently close to zero.
Since $\delta\le |\varepsilon|\le 2\delta$,
we can now bound $\sigma$ as follows.
$$
\sigma\le dN e^{-Ng(\varepsilon)}\le dN\exp\bigl(-{N\delta^2\over 2d}\bigr),
\equation(8l)
$$
provided that $\delta$ is sufficiently small, and $d\le\delta N$.
This proves the assertion of \clm(1l).
\qed 

In what follows, the number $p$ of patterns is assumed to be
sufficiently small, such that $d\equiv 2^p\le N^\alpha$,
for some fixed positive constant $\alpha<1$.
We also choose, once and for all,
$$
\delta=\sqrt{{d\over N}}\ln(N).
\equation(109)
$$

\claim Corollary(100) {\rm (self--averaging) }
Let $(N,d)\mapsto(\phi_{N,d}:\PP_{N,d}\to\real)$ be a two--parameter
sequence of functions, and assume that there are constants $\phi_\infty$
and $\kappa,\lambda,M>0$, such that for $N>M$, and for $d\le N^\alpha$,
the following holds.
\hfill\break
$(a)$\qquad $|\phi_{N,d}(L)|\le N^\lambda$ for all $L\in\PP_{N,d}$,
\hfill\break
$(b)$\qquad $|\phi_{N,d}(L)-\phi_\infty|\le N^{-\kappa}$ for $L\in\UU_{N,d}$.
\hfill\break
Then
$$
\Bigm| d^{-N}\sum_{L\in\PP_{N,d}}{N!\over L_1!L_2!\cdots L_d!}\ 
\phi_{N,d}(L) -\phi_\infty\Bigm| \le 2N^{-\kappa},
\equation(110)
$$
provided that $N$ is sufficiently large, and $d\le N^\alpha$.

\proof
Using \clm(1l), and assuming properties $(a)$ and $(b)$,
we can bound the left hand side of \equ(110) as follows.
$$
d^{-N}\!\!\!\sum_{L\in\PP_{N,d}\setminus\UU_{N,d}}
{N!\over L_1!\cdots L_d!}\bigm|\phi_{N,d}(L)-\phi_\infty\bigm|
+d^{-N}\!\sum_{L\in\UU_{N,d}}
{N!\over L_1!\cdots L_d!}\bigm|\phi_{N,d}(L)-\phi_\infty\bigm|
$$
$$
\le (N^\lambda+|\phi_\infty|)N^2 e^{-{1\over 2}(\ln N)^2} +N^{-\kappa}.
\equation(111)
$$
For sufficiently large $N$, the last expression is bounded by $2N^{-\kappa}$.
\qed

Our aim is to apply this Corollary to the free energy and to the
magnetization, as defined in \equ(107).
The hypothesis $(a)$ is easy to check in these two cases:
$F_{N,d}$ and $m_{N,d}$ are bounded in absolute value by
${p\over 2}+const=\OO(\ln N)$ and 1, respectively.
We shall now work towards the proof of property $(b)$.

\claim Proposition(101) For $L\in\UU_{N,d}$,
$$
\Bigm|\beta\ov{H}_N(\beta,\eta,L,Y) +N\ln 2
-{N\over d}f(\beta,\eta,0,y)\Bigm|
\le (1+\beta)(3+|\eta|)\delta N.
\equation(112)
$$

\proof By using equation \equ(105), and the fact
that $|y_k|\le 1$ and $|\lambda_k|\le\delta$,
we have
$$
\eqalign{&\phantom{=}
\Bigm|\beta\ov{H}_N(\beta,\eta,L,Y) +N\ln 2
-{N\over d}f(\beta,\eta,0,y)\Bigm|\cr
&=\Bigm|{N\over d}f(\beta,\eta,\lambda,y)-{N\over d}f(\beta,\eta,0,y)
+{p\over 2} -\sum_{k=1}^d g(L_k,Y_k)\Bigm|\cr
&\le{N\over d}\Bigm|{\beta\over 2}\langle P(2+\Lambda)y,\Lambda y\rangle
+\beta\eta\langle e^\nu,\Lambda y\rangle
-\sum_{k=1}^d\lambda_k\int\limits_0^{y_k}dt\tanh^{-1}(t)\Bigm|
+{p\over 2} +\sum_{k=1}^d |g(L_k,Y_k)|\cr
&\le{N\over d}\Bigl[
{\beta\over 2}(2+\delta)\delta d +\beta|\eta|\delta d +\delta d\ln 2
\Bigr] +2d\ln(N) \le(1+\beta)(3+|\eta|)\delta N.\cr}
\equation(113)
$$
\qed

Since the dominant contributions to the free energy density
and the magnetization come from mean field configurations $Y$
which minimize $\ov{H}_N$, we continue by estimating
$f(\beta,\eta,0,y)$ near its minimum. To do so, let us write
$$
f(\beta,\eta,0,y)=-\sum_{k=1}^d h_k(y_k) +{\beta\over 2}||(1-P)y||^2,
\equation(114)
$$
where, for $|s|\le 1$,
$$
h_k(s)=\int\limits_0^s dt
\bigl[\beta\eta e^\nu_k +\beta t -\tanh^{-1}(t)\bigr].
\equation(115)
$$

At this point, it is necessary to distinguish between the
high temperature phase $(\beta<1)$ and the low temperature phase $(\beta>1)$,
and between large and small external fields $\eta$.
To avoid undue repetition, we will limit our discussion to $\beta>1$,
and to small values of $\eta$; the other cases can be treated similarly.

If $\eta$ is sufficiently small (depending on $\beta$, but not on $d$),
then each of the functions $h_k$ has a unique positive maximum
on each of the intervals $(-1,0)$ and $(0,+1)$.
Denote by $v_k(-)$ and $v_k(+)$ the location of these maxima;
then $v_k(\pm)=\pm a_1(\beta)+\OO(\eta)$,
where $a_1(\beta)$ is the positive solution of the equation
$a_1=\tanh(\beta a_1)$.

\claim Definition(102) Given a vector $y\in\real^d$,
define $v(y)$ to be the vector in $\real^d$
whose $k^{th}$ component is $v_k\bigl({\rm sgn}(y_k)\bigr)$, for $1\le k\le d$.
Here, we use e.g.~the convention that ${\rm sgn(0)}$ is positive.
Furthermore, define $u=v(\eta e^\nu)$.

\claim Proposition(103) Let $\beta>1$. Then there are two positive constants
$c_l<c_u$, such that for any vector $y$ in $[-1,1]^d$,
and for $|\eta|$ sufficiently small,
$$
\eqalign{
f(\beta,\eta,0,y)&=f(\beta,\eta,0,u)\cr
&+\beta|\langle u-v(y),\ell e^\nu\rangle|
+c(y)||y-v(y)||^2 +{\beta\over 2}||(1-P)y||^2,\cr}
\equation(118)
$$
where $c(y)$ and $\ell$ are constants (depending on $\eta$)
which satisfy the bounds
$c_l\le c(y)\le c_u$ and $|\ell-\eta|=\OO(\eta^3)$, respectively.

\proof  $\ell$ is defined by writing the local minima of $h_k$ in the form
$h_k\bigl(v_k(\pm)\bigr)=E_k+\beta\ell e^\nu_k v_k(\pm)$.
By computing the difference of these two values, we obtain
$$
\ell e^\nu_k=\eta e^\nu_k +{\beta^{-1}\over |v_k(+)|+|v_k(-)|}
\int\limits_{|v_k(-)|}^{|v_k(+)|}\!\!dt\bigl[\beta t-\tanh^{-1}(t)\bigr]
=\bigl(\eta+\OO(\eta^3)\bigr)e^\nu_k.
\equation(120)
$$
Given $y\in [-1,1]^d$, define $x_k={\rm sgn}(y_k)$.
Then we can write $f(\beta,\eta,0,y)$ as follows.
$$
f(\beta,\eta,0,y)=-\beta\langle v(y),\ell e^\nu\rangle
+\sum_{k=1}^d\bigl[h_k\bigl(v_k(x_k)\bigr)-h_k(y_k)-E_k\bigr]
+{\beta\over 2}||(1-P)y||^2.
\equation(121)
$$
Since, for the values of $\beta$ and $\eta$ considered,
$h_k$ has a quadratic maximum at $v_k(x_k)$, which is unique
in the interval bounded by 0 and $x_k$, we have
$$
c_l\le{h_k\bigl(v_k(x_k)\bigr)-h_k(y_k)\over |v_k(x_k)-y_k|^2}\le c_u,
\equation(122)
$$
with bounds $c_l, c_u$ that are independent of $k$ and $y$.
As a consequence,
$$
f(\beta,\eta,0,y)=-\beta\langle v(y),\ell e^\nu\rangle
+c(y)||y-v(y)||^2 -\sum_{k=1}^d E_k +{\beta\over 2}||(1-P)y||^2,
\equation(123)
$$
for some constant $c(y)$ contained in the interval $[c_l,c_u]$.

The equation \equ(118) is now obtained by using
that the two norms in \equ(123) vanish for $y=u$,
and that $\langle v(y),\ell e^\nu\rangle$ is maximized by $y=u$
(note also that $v(u)=u$).
Both of these properties follow from the fact that for $1\le k\le d$,
$$
{\rm sgn}(u_k)={\rm sgn}(\eta e^\nu_k)={\rm sgn}(\ell e^\nu_k),\qquad
|u_k|=\max\{|v_k(-)|, |v_k(+)|\}=t,
\equation(124)
$$
where $t$ is the largest solution
of the equation $\beta|\eta| +\beta t -\tanh^{-1}(t)=0$.
\qed

The following two Propositions, together with \clm(100),
prove the assertions of \clm(1).

\claim Proposition(104)
There is a function $F_\infty(\beta,\eta)$ with the following properties.
If $\beta>1$, and if $|\eta|$ is sufficiently small, then
$$
|F_{N,d}(\beta,\eta,L)-F_\infty(\beta,\eta)|\le\bigl(d/N\bigr)^{1/3},
\equation(125)
$$
for all $L\in\UU_{N,d}$, and for $N$ sufficiently large. Furthermore,
$$
F_\infty(\beta,\eta)=-{\ln 2\over\beta}
-{1\over 2\beta}\int\limits_1^\beta\!dt\;a_1(t)^2 +\OO(\eta).
\equation(125a)
$$

\proof  From the definition \equ(107), and from \clm(101), it follows that
$$
F_{N,d}(\beta,\eta,L)
=-{\ln 2\over\beta}+{1\over\beta d}f(\beta,\eta,0,u)+\OO(\delta)
-{1\over\beta N}\ln\Bigl(\sum_{y\in\XX}
e^{-[f(\beta,\eta,0,y)-f(\beta,\eta,0,u)]N/d}\Bigr),
\equation(126)
$$
for all $L\in\UU_{N,d}$. To get an upper bound on the sum over $y$,
we will use that, for large $N$, the number of elements in $\XX$ is bounded by
$$
|\XX|=\prod_{k=1}^d(L_k+1)\le\bigl(2N/d\bigr)^d \le e^{\delta N}.
\equation(130)
$$
To get the corresponding lower bound, we note that 
there is a vector $y^{\prime}\in\XX$, which is sufficiently close to $u$,
such that ${\rm sgn}(y_k^\prime)={\rm sgn}(u_k)$,
and $|y^{\prime}_k-u_k|\le 3d/N$, for $1\le k\le d$. By \clm(103),
$$
0\le|f(\beta,\eta,0,y^{\prime})-f(\beta,\eta,0,u)|
\le (c_u+{\beta\over 2})||y^{\prime}-u||^2
\le 9(c_u+{\beta\over 2}){d^3\over N^2}\le\delta d,
\equation(127)
$$
and therefore
$$
|\XX|\ge\sum_{y\in\XX}e^{-[f(\beta,\eta,0,y)-f(\beta,\eta,0,u)]N/d}
\ge e^{-\delta N}.
\equation(128)
$$
This inequality, together with \equ(130), shows
that the last term in \equ(126) is bounded by $\pm\delta$.
The bound \equ(125) follows, if we define
$F_\infty(\beta,\eta)\equiv\bigl(-\ln2+f(\beta,\eta,0,u)/d\bigr)/\beta$.

In order to prove \equ(125a), we need only consider
the case $\eta=0$, since
$$
|f(\beta,\eta,0,u)-f(\beta,0,0,u)|
=|\beta\eta\langle e^\nu,u\rangle|\le\beta d|\eta|.
\equation(132)
$$
From equation \equ(106), using the fact that
$y=u(\beta)={\rm sgn}(\eta)a_1(\beta)e^\nu$ minimizes $f(\beta,0,0,y)$,
we obtain
$$
\eqalign{
{d\over d\beta}f(\beta,0,0,u(\beta))
&=\bigl({\partial\over\partial\beta}f\bigr)(\beta,0,0,u(\beta))
+\langle\bigl({\delta\over\delta y}f\bigr)(\beta,0,0,u(\beta)),
{d\over d\beta}u(\beta)\rangle \cr
&=-{1\over 2}||Pu(\beta)||^2 +0 =-{1\over 2}a_1(\beta)^2 d.\cr}
\equation(131)
$$
The assertion now follows since $f(1,0,0,u(1))=f(1,0,0,0)=0$.
\qed

\claim Proposition(105)
There is a function $m_\infty(\beta,\eta)$ with the following properties.
If $\beta>1$, and if $|\eta|>0$ is sufficiently small, then
$$
|m_{N,d}(\beta,\eta,L)-m_\infty(\beta,\eta)|\le 3\bigl({d\over N}\bigr)^{1/5},
\equation(134)
$$
for all $L\in\UU_{N,d}$, and for $N$ sufficiently large. Furthermore,
$$
m_\infty(\beta,\eta)={\rm sgn}(\eta)a_1(\beta) +\OO(\eta).
\equation(134a)
$$

\proof
The magnetization $m_{N,d}$ is given by the following expression.
$$
m_{N,d}(\beta,\eta,L)
=\biggl(\sum_{y\in\XX}e^{-\beta\ov{H}_N(\beta,\eta,L,Y)}\biggr)^{-1}
\sum_{y\in\XX}{1\over d}\langle e^\nu,(1+\Lambda)y\rangle
e^{-\beta\ov{H}_N(\beta,\eta,L,Y)}.
\equation(135)
$$
We may assume that $\eta$ is nonzero,
since by symmetry, $m_{N,d}(\beta,0,L)=0$.
By writing the inner product in equation \equ(135) as the sum
$$
{1\over d}\langle e^\nu,(1+\Lambda)y\rangle
={1\over d}\langle e^\nu,u\rangle
+{1\over d}\langle e^\nu,(1+\Lambda)y-u\rangle,
\equation(136)
$$
we split the magnetization into a leading term
$m_\infty(\beta,\eta)\equiv{1\over d}\langle e^\nu,u\rangle$, and a remainder.
The sum over $y$ in the remainder is now estimated separately
on the set $R=\{y\in\XX: |\langle y-u,e^\nu\rangle|\le\varepsilon d\}$,
and on its complement, where $\varepsilon=(d/N)^{1/5}$.
For $y\in R$, we have
$$
{1\over d}|\langle e^\nu,(1+\Lambda)y-u\rangle|
\le {1\over d}|\langle e^\nu,\Lambda y\rangle|
+{1\over d}|\langle e^\nu,y-u\rangle|
\le\delta +\varepsilon .
\equation(137)
$$
Using \clm(101), and \clm(104), we arrive at the bound
$$
\eqalign{
\bigm|m_{N,d}&(\beta,\eta,L)-{1\over d}\langle e^\nu,u\rangle\bigm|\cr
&\le\delta +\varepsilon+
\biggl(\sum_{y\in\XX}e^{-\beta\ov{H}_N(\beta,\eta,L,Y)}\biggr)^{-1}
\sum_{y\in\XX\setminus R}{1\over d}|\langle e^\nu,(1+\Lambda)y-u\rangle|
e^{-\beta\ov{H}_N(\beta,\eta,L,Y)}\cr
&\le 2\varepsilon+e^{\OO(\delta N)}\sum_{y\in\XX\setminus R}
e^{-[f(\beta,\eta,0,y)-f(\beta,\eta,0,u)]N/d}.\cr}
\equation(138)
$$
In order to estimate the last term in \equ(138),
we use the fact that for $y\in\XX\setminus R$,
$f(\beta,\eta,0,y)$ cannot be very close to its minimum value.
More precisely, if $y$ lies in $\XX\setminus R$,
then either $|\langle e^\nu,u-v(y)\rangle|>\varepsilon d/2$ holds,
or $|\langle e^\nu,y-v(y)\rangle|>\varepsilon d/2$.
In the first case, it follows from \clm(103) that, for small $|\eta|$,
$$
f(\beta,\eta,0,y)-f(\beta,\eta,0,u)
\ge{\beta\over 3}|\eta|\varepsilon d.
\equation(139)
$$
In the second case, we combine \clm(103) with the inequality
$$
||y-v(y)||^2\ge|\langle e^\nu,y-v(y)\rangle|^2 ||e^\nu||^{-2}
\ge{1\over 4}\varepsilon^2 d,
\equation(140)
$$
to obtain a similar result:
$$
f(\beta,\eta,0,y)-f(\beta,\eta,0,u)
\ge{c_l\over 4}\varepsilon^2 d.
\equation(141)
$$
By substituting these two bounds into \equ(138),
we find that, for $\varepsilon<|\eta|$,
$$
\bigm|m_{N,d}(\beta,\eta,L)-{1\over d}\langle e^\nu,u\rangle\bigr|
\le 2\varepsilon +e^{\OO(\delta N)}
\sum_{y\in\XX\setminus R}e^{-\kappa\varepsilon^2 N},
\equation(142)
$$
where $\kappa$ is some positive constant (depending on $\beta$ and $\eta$). 
The number of terms in the sum over $y$ is bounded by $\exp(\OO(\delta N))$,
as in \equ(130). The assertion \equ(134) now follows
since $\delta/\varepsilon^2\to 0$, as $N$ tends to infinity,
while \equ(134a) follows from the fact that
$u={\rm sgn}(\eta)a_1(\beta)e^\nu +\OO(\eta)$.
\qed

%SYMM.7
\section The Symmetric Solutions

In this section we describe in more detail the set of critical points
of the function $f_\beta$,
$$
f_\beta(y)\equiv{1\over d}f(\beta,0,0,y)
={1\over d}\sum_{k=1}^d \int\limits_0^{y_k}\!dt \tanh^{-1}(t)
-{\beta\over 2d}||Py||^2,
\equation(201)
$$
defined for $y\in(-1,1)^d$. The number of patterns $p$
is assumed to be fixed, but arbitrary, and $d=2^p$.
As mentioned in the introduction,
the local minima of $f_\beta$ are expected to play an important role
for the dynamics of the Hopfield model.

A well known procedure for finding (e.g.~numerically)
the local minima of a function $g$ is the method of steepest descent,
which (in its simplest form) consists in iterating a map
$\Omega: y\mapsto y-\lambda\nabla g(y)$.
If $\lambda>0$ is chosen sufficiently small
(such that the Hessian of $\lambda g$ has only eigenvalues smaller than 2)
then the stable fixed points of $\Omega$
are precisely the local minima of $g$.
In the following, a map of this type will be used in order to distinguish
local minima of $f_\beta$ from other critical points;
this map is also closely related to the one used in [\rG,\rH],
and somewhat similar to the learning algorithm of [\rQ].

Before applying the method of steepest descent,
we may of course perform a change of variables $z\mapsto y$,
such as the one defined by the equation
$$
y=\Tanh(\beta z)\equiv
\bigl(\tanh(\beta z_1),\tanh(\beta z_2),\ldots,\tanh(\beta z_d)\bigr).
\equation(901)
$$

\claim Proposition(901) If $y$ is a local minimum of $f_\beta$
in the hypercube $(-1,1)^d$, then $z=Py$ is a stable fixed point of the map
$$
\Omega_\beta: z\mapsto P\Tanh(\beta Pz),\quad z\in\real^d.
\equation(902)
$$
Conversely, if $z$ is a stable fixed point of $\Omega_\beta$,
then $y=\Tanh(\beta z)$ is a local minimum of $f_\beta$.

\proof
The derivative of the function $g_\beta=f_\beta\bigl(\Tanh(\beta\,.)\bigr)$
can be written as follows.
$$
Dg_\beta(z;u)={\beta^2\over d}\langle
z-P\Tanh(\beta z),\Tanh^\prime(\beta z)\bullet u\rangle ,
\equation(903)
$$
where ``$v\,\bullet$'' denotes the diagonal matrix
associated with a vector $v$, i.e. $(v\bullet u)_k=v_ku_k$.
Since $\tanh^\prime(\beta z)>0$, we see that the critical points of $g_\beta$
coincide with the fixed points of $\Omega_\beta$.
Assume now that $\Omega_\beta(z)=z$.
Then the second derivative of $g_\beta$ at $z$ is given by
$$
D^2g_\beta(z;u,v)={\beta^2\over d}\bigl\langle
\bigl[{\rm Id}-\beta P\bigl(\Tanh^\prime(\beta z)\bullet\bigr)\bigr]v,
\Tanh^\prime(\beta z)\bullet u\bigr\rangle .
\equation(904)
$$
Note that the matrix $[\cdots]$ in \equ(904) is self--adjoint
with respect to the inner product
$(v,u)=\langle v,\Tanh^\prime(\beta z)\bullet u\rangle$. Thus, we have
$$
\inf_{(v,v)=1}D^2g(z;v,v)={\beta^2\over d}(1-\lambda),
\equation(905)
$$
where $\lambda$ is the largest eigenvalue of
$\beta P\bigl(\Tanh(\beta z)\bullet\bigr)$, or equivalently
(if $\lambda\ne 0$), the largest eigenvalue of the tangent map
$D\Omega_\beta(z)=\beta P\bigl(\Tanh^\prime(\beta Pz)\bullet\bigr)P$
of $\Omega_\beta$ at the fixed point $z$.
\qed

At high temperatures ($\beta<1$), the map $\Omega_\beta$
is easily seen to be a contraction, with fixed point $z=0$.
For $\beta\ge 1$ the situation is more complicated;
but fortunately, the Hopfield model with orthogonal patterns
(i.e.~when $L_k=N/d$ for all $k$) has many symmetries.
In order to describe these symmetries,
let us denote by $C_p$ the set of corners of the hypercube $[-1,1]^p$,
and by $E$ the map $k\mapsto e_k$ which was introduced earlier
for the purpose of (arbitrarily) enumerating the elements of $C_p$.
If $\psi$ is a permutation of the set $C_p$, we associate with $\psi$
a linear transformation $\Psi$ on $\real^d$, by defining
$$
(\Psi y)_k=y_j,\qquad j=E^{-1}\bigl(\psi^{-1}\bigl(E(k)\bigr)\bigr),
\equation(209)
$$
for all $y\in\real^d$, and for $1\le k\le d$. Note that $\Psi$ is orthogonal
with respect to the standard inner product in $\real^d$.
The following permutations are of particular interest; see also [\rJ].
For $1\le\nu,\kappa,\lambda\le p$
we define $\psi_\nu$ and $\psi_{\kappa\lambda}$ by setting
$$
\eqalign{
\bigl(\psi_\nu(c)\bigr)^\mu&=\cases{
-c^\mu, &if $\mu=\nu$,\cr +c^\mu, &if $\mu\ne\nu$,\cr}\cr
\bigl(\psi_{\kappa\lambda}(c)\bigr)^\mu&=\cases{
c^\kappa, &if $\mu=\lambda$,\cr c^\lambda, &if $\mu=\kappa$,\cr
c^\mu, &if $\mu\not\in\{\kappa,\lambda\}$,\cr}\cr}
\equation(211)
$$
for all $c$ in $C_p$.
If $\psi$ is any of these permutations,
then $\Psi^2={\rm Id}$, and thus $\Psi$ is symmetric.
Furthermore, the identity $(\Psi e^\mu)_k=\bigl(\psi^{-1}(e_k)\bigr)^\mu$,
which follows directly from \equ(209), shows that
$\Psi_\nu$ acts on the set $S=\{e^1,e^2,\ldots,e^p\}$
by multiplying the vector $e^\nu$ by $-1$,
and $\Psi_{\kappa\lambda}$ acts on $S$
by exchanging $e^\kappa$ with $e^\lambda$.
As a consequence, all of these transformations commute with $P$,
the orthogonal projection onto the span of $S$.
This proves the following Proposition.

\claim Proposition(202) Let $\psi$ be one of the permutations
defined in \equ(211). Then $\Omega_\beta\circ\Psi=\Psi\circ\Omega_\beta$
for all $\beta$.

As another immediate consequence we have the following orthogonality property.
Let $I=\{1,2,\ldots,p\}$, and for every subset $J\subset I$ let
$$
e^{(J)}_k=\prod_{\mu\in J}e^\mu_k\,,\qquad 1\le k\le d,
\equation(212)
$$
where the value of an empty product is defined to be 1.
It is easy to see that $e^{(J)}$ is an eigenvector of $\Psi_\nu$,
for every $J\subset I$ and every $\nu\in I$;
the corresponding eigenvalue is $-1$ if $\nu\in J$, and 1 if $\nu\not\in J$.
Since the operators $\Psi_\nu$ are symmetric and commute with each other,
the set $\{e^{(J)}: J\subset I\}$ is an orthogonal basis for $\real^d$.

The next two propositions establish, for $\beta>1$,
the existence of $3^p$ ``symmetric'' fixed points for $\Omega_\beta$.
Each of these fixed points is associated with a nonnegative integer $n\le p$,
and with one of the following $2^n{p\choose n}$ vectors $v\in\real^d$.
$$
v=\sum_{\mu=1}^pc_\mu e^\mu,
\qquad c_\mu\in\{-1,0,1\},\qquad\sum_{\mu=1}^pc_\mu^2 =n.
\equation(213)
$$

\claim Proposition(204) Let $1\le n\le p$. If $v$ satisfies \equ(213),
and if $a$ is any real number, then $\Omega_\beta(av)=\gamma_n(\beta a)v$,
where $\gamma_n$ is the function defined by the following equation.
$$
\gamma_n(x)=2^{-n+1}\!\sum_{0\le m<n/2}
{n\choose m}{n-2m\over n}\tanh\bigl((n-2m)x\bigr).
\equation(214)
$$

\proof
Given $n>0$ and a vector $v$ as in \equ(213), denote by $S$ the set
of linear transformations $\Psi$, which contains:
$\;\Psi_\nu$ if and only if $c_\nu=0$,
$\Psi_{\kappa\lambda}$ if and only if $c_\kappa=c_\lambda\ne 0$,
$\Psi_\lambda\Psi_{\kappa\lambda}\Psi_\lambda$
if and only if $-c_\kappa=c_\lambda\ne 0$,
and no other elements. It is easy to check that the only
vectors $z\in P\real^d$ which satisfy $\Psi z=z$, for all $\Psi$ in $S$,
are the multiples of $v$. Since by \clm(202) we have
$\Psi\Omega_\beta(av)=\Omega_\beta(a\Psi v)=\Omega_\beta(av)$,
for all $\Psi$ in $S$, it follows that $\Omega_\beta(av)=a^\prime v$
for some real number $a^\prime$. 

In order to see that $a^\prime=\gamma_n(\beta a)$,
it is useful to write the components of $v$ in the form
$$
v_k=\sum_{\mu=1}^p c_\mu e^\mu_k=m\cdot 1+(n-m)\cdot(-1)+(p-n)\cdot 0,
\equation(217)
$$
where $m=m(k)$ is the number of elements $\mu$ in $\{1,2,\ldots,p\}$
for which $c_\mu e^\mu_k$ is equal to 1.
A moment's reflection shows that, given $j$, there are exactly
$$
\sum_{m=0}^n 2^{p-n}{n\choose m}\delta(2m-n-j)
\equation(217)
$$
values of $k$ for which $v_k$ is equal to $j$. Thus we have
$$
\eqalign{
a^\prime&=||v||^{-2}\langle v,\Omega_\beta(av)\rangle
={1\over nd}\sum_{k=1}^d v_k\tanh(\beta av_k)\cr
&={1\over nd}2^{p-n}\!\!\sum_{j=0}^n\sum_{m=0}^n
{n\choose m}\delta(2m-n-j)j\tanh(\beta aj)\cr
&=2^{-n}\!\sum_{m=0}^n
{n\choose m}{2m-n\over n}\tanh\bigl((2m-n)\beta a\bigr)\cr
&=2^{-n+1}\!\sum_{0\le m<n/2}
{n\choose m}{n-2m\over n}\tanh\bigl((n-2m)\beta a\bigr).\cr}
\equation(218)
$$
\qed

\claim Proposition(205) For $\beta>1$, the equation $\gamma_n(\beta a)=a$
has a unique positive solution $a=a_n(\beta)$. Moreover,
\hfill\break
$(a)$\quad $a_n(\beta)$ is an increasing function of $\beta$,
\hfill\break
$(b)$\quad $a_n(\beta)\to
2^{-n+1}{n-1\choose\lfloor(n-1)/2\rfloor}$ as $\beta\to\infty$,\quad
where $\lfloor r\rfloor$ denotes the integer part of $r$,
\hfill\break
$(c)$\quad $a_n(\beta)^2={3\over 3n-2}(\beta-1)+\OO\bigl((\beta-1)^2\bigr)$
as $\beta\downarrow 0$.

\noindent
The proof of these statements is straightforward,
given the following properties of $\gamma_n$.

\claim Proposition(206) The functions $\gamma_n$ are odd and satisfy
\hfill\break
$(a)$\quad $\gamma_n(x)>0$,\quad $\gamma_n^\prime(x)>0$,
     \quad $\gamma_n^{\prime\prime}(x)<0$,\quad for all $x>0$,
\hfill\break
$(b)$\quad $\gamma_n(x)\to
2^{-n+1}{n-1\choose\lfloor(n-1)/2\rfloor}$ as $x\to\infty$,
\hfill\break
$(c)$\quad $\gamma_n^\prime(0)=1$,
     \quad $\gamma_n^{\prime\prime\prime}(0)=-2(3n-2)$.

\proof
The inequalities (a) are obtained from the corresponding inequalities
for the function $\tanh$. Property (b) follows from \equ(214) :
If we define ${n-1\choose m}=0$ for $m<0$, then
$$
\eqalign{
\lim_{x\to\infty}\gamma_n(x)&=
2^{-n+1}\!\sum_{0\le m<n/2}{n\choose m}{n-2m\over n}\cr
&=2^{-n+1}\!\sum_{m<n/2}\biggl({n-1\choose m}-{n-1\choose m-1}\biggr)
=2^{-n+1}{n-1\choose\lfloor(n-1)/2\rfloor}.\cr}
\equation(219)
$$

To prove (c), we use the representation
$\gamma_n(x)={1\over nd}\langle v,\Tanh(xv)\rangle$,
with $v=\sum_{\mu=1}^n e^\mu$. The $m^{th}$ derivative of $\gamma_n$
at the origin is then given by the following equation.
$$
\eqalign{
\gamma_n^{(m)}(0)&={1\over nd}\tanh^{(m)}(0)\sum_{k=1}^d v_k^{m+1}\cr
&=\tanh^{(m)}(0)\cdot{1\over n}\sum_M\biggl[{1\over d}\sum_{k=1}^d
e_k^{\mu_1}e_k^{\mu_2}\cdots e_k^{\mu_{m+1}}\biggr],\cr}
\equation(220)
$$
where $\sum_M$ denotes the sum over all ordered sets
$M=(\mu_1,\mu_2,\ldots,\mu_{m+1})$ with $1\le\mu_j\le n$.
Because of the orthogonality of the vectors \equ(212),
the expression $[\ldots]$ in \equ(220) vanishes,
unless every element of $M$ occurs an even number of times in $M$.
If it does not vanish, then $[\ldots]=1$. For $m=1$ there are $n$
sets left which contribute to $\sum_M$, and thus $\gamma_n^\prime(0)=1$.
If $m=3$ then there are $n(3n-2)$ such sets, and
$\gamma_n^{\prime\prime\prime}(0)=-2(3n-2)$ follows since
$\tanh^{\prime\prime\prime}(0)=-2$.
\qed

In the remaining part of this section,
we discuss the stability of the symmetric fixed points,
for $\beta$ in the interval
$$
1<\beta<1+\bigl(9d+500p^8\bigr)^{-1},
\equation(905a)
$$
as well as for large values of $\beta$. In addition, we show
that the symmetric fixed points are unique if $\beta$ satisfies \equ(905a);
the following estimate is the first step of the proof.

\claim Proposition(902) For $0<\beta<1+{1\over 9d}$,
every nonzero fixed point $z$ of $\Omega_\beta$ satisfies
$$
||z||^2<23d\cdot(\beta-1).
\equation(906)
$$

\proof
Assume that $\Omega_\beta(z)=z$, and let $y=\Tanh(\beta z)$.
By using that $Py=z$, and that $z_ky_k=z_k\tanh(\beta z_k)\ge 0$ for all $k$,
we obtain the following identities.
$$
\eqalign{
(\beta-1)||z||^2&=(\beta-1)||z||^2
+\sum_{k=1}^dy_k\bigl(y_k-\tanh(\beta z_k)\bigr)\cr
&=||y-z||^2+\sum_{k=1}^dy_k\bigl(\beta z_k-\tanh(\beta z_k)\bigr)\cr
&=||y-z||^2+\sum_{k=1}^d|y_k|\bigl(\beta|z_k|-\tanh(\beta|z_k|)\bigr).\cr}
\equation(907)
$$
Since none of the terms in the last sum is negative,
it follows that either $z=0$, or $\beta>1$.

Let us now assume that $1<\beta<1+{1\over 9d}$.
Then \equ(907) implies that $||y-z||^2$ has to be small;
in particular, it follows that
$$
\eqalign{
|\beta z_k|&=\beta\bigl(|y_k|+|z_k-y_k|\bigr)
\le\beta(1+\sqrt{\beta-1}\cdot||z||)\cr
&<\Bigl(1+{1\over 9d}\Bigr)\Bigl(1+{1\over 3\sqrt{d}}\cdot\sqrt{d}\Bigr)
<\sqrt{2}.\cr}
\equation(908)
$$
The last term in \equ(907) can now be bounded from below, by using
\equ(908) and the inequality $\tanh^{\prime\prime\prime}(x)\le -2+8x^2$.
$$
\eqalign{
\sum_{k=1}^d|&y_k|\bigl(\beta|z_k|-\tanh(\beta|z_k|)\bigr)\;
\ge\sum_{k=1}^d|y_k|\Bigl({1\over 3}(\beta|z_k|)^3
-{2\over 15}(\beta|z_k|)^5\Bigr)\cr
&\ge\sum_{k=1}^d|y_k|\cdot{1\over 15}(\beta|z_k|)^3\;
\ge{\beta^3\over 15}\Bigl[\sum_{k=1}^dz_k^4-
\sum_{k=1}^d|y_k-z_k|\cdot|z_k|^3\Bigr].\cr}
\equation(909)
$$
The two sums in the square bracket can be compared
by using equation \equ(907) again, together with the Schwartz inequality:
$$
\eqalign{
\sum_{k=1}^d|y_k-z_k|\cdot|z_k|^3
&\le||y-z||\cdot||z||\Bigl(\sum_{k=1}^dz_k^4\Bigr)^{1/2}\cr
&\le\sqrt{\beta-1}\cdot||z||^2\Bigl(\sum_{k=1}^dz_k^4\Bigr)^{1/2}
\;\le{1\over 3}\sum_{k=1}^3z_k^4.\cr}
\equation(910)
$$
As a consequence of \equ(907),\equ(909), and \equ(910), we have
$$
(\beta-1)||z||^2\;\ge{\beta^3\over 15}\cdot{2\over 3}\sum_{k=1}^dz_k^4
\;\ge{2\beta^3\over 45d}||z||^4,
\equation(911)
$$
and the bound \equ(906) follows.
\qed

After having localized the fixed points in a small ball around $z=0$,
we can now use perturbation theory to rule out the existence
of non--symmetric fixed points of $\Omega_\beta$,
for $\beta$ in the interval \equ(905a).

\medskip\noindent
{\bf Proof of Theorem 1.4.\ }
By \clm(901), we are lead to consider the system
$$
{1\over d}\langle e^\nu,\Tanh(\beta\sum_{\mu=1}^p b_\mu e^\mu)\rangle
=b_\nu\,,\qquad 1\le\nu\le p,
\equation(222)
$$
which is equivalent to the fixed point equation $\Omega_\beta(z)=z$,
if we set $b_\nu={1\over d}\langle e^\nu,z\rangle$. Note that,
since the map $\Tanh$ commutes with all transformations $\Psi_\mu$,
the inner product in \equ(222) is an odd function of $b_\nu$,
and even in all the other coefficients $b_\mu$. Thus, we may define
$$
g^\nu(b_1^2,b_2^2,\ldots,b_p^2)\equiv{1\over db_\nu}\sum_{k=1}^d
e^\nu_k\tanh\bigl(\beta\sum_{\mu=1}^p b_\mu e^\mu_k\bigr),
\quad 1\le\nu\le p.
\equation(223)
$$
Furthermore, since the hyperbolic tangent is analytic
on the disc $|\zeta|<\pi/2$, the functions $t\mapsto g^\nu(t_1,\ldots,t_p)$
can be continued analytically to the polydisk
$$
|t_\mu|<\Bigl({\pi\over 2p}\Bigr)^2,\qquad 1\le\mu\le p.
\equation(912)
$$
The same holds for the functions $g_5^\nu$, which we define as in \equ(223),
but with $\tanh$ replaced by $\tanh_5$, where
$$
\tanh_5(\zeta)=\tanh(\zeta)-\zeta-{1\over 3}\zeta^3,\qquad \zeta\in\real.
\equation(224)
$$
In addition, $g_5^\nu$ vanishes at the origin, together with
all its first partial derivatives, for $1\le\nu\le p$.
The second derivatives can be bounded by using
the maximum principle for analytic functions, and
Cauchy's formula with circular integration contours of radius $2p^{-2}$.
Since $|\tanh_5(\zeta)|<4$, for $|\zeta|<\pi/2$, we have
$$
\Bigm|{\partial^2\over\partial t_\lambda\partial t_\mu}
g_5^\nu(t_1,\ldots,t_p)\Bigm|
<{2!\over\bigl(2p^{-2}\bigr)^2}\cdot{4\over{3\over 2p}}
<{4\over 3}p^5,
\equation(913)
$$
if $|t_\mu|\le\bigl({1\over 2p}\bigr)^2$ for all $\mu$.
This bound will be used below, together with Taylor's formula,
in order to estimate $g_5$ near the origin.

The difference $(g^\nu-g_5^\nu)$ can be computed explicitely,
by using the orthogonality of the vectors defined in \equ(212);
see also the discussion of \equ(220). We obtain
$$
\eqalign{
\bigl(g^\nu-g_5^\nu\bigr)&(b_1^2,\ldots,b_p^2)
=\beta+{1\over db_\nu}\sum_{k=1}^d e^\nu_k
\bigl(-{1\over 3}\bigr)\Bigl(\beta\sum_{\mu=1}^p b_\mu e^\mu\Bigr)^3\cr
&=\beta-{\beta^3\over 3b_\nu}\!\sum_{\mu,\kappa,\lambda=1}^p
\Bigl[b_\mu b_\kappa b_\lambda\cdot{1\over d}\sum_{k=1}^d
e^\nu_k e^\mu_k e^\kappa_k e^\lambda_k\Bigr]
=\beta-\beta^3\sum_{\mu=1}^p b_\mu^2 +{2\over 3}\beta^3 b_\nu^2.\cr}
\equation(225)
$$
This allows us to rewrite equation \equ(222) as follows.
$$
b_\nu\Bigl[(\beta-1)-\beta^3\sum_{\mu=1}^p b_\mu^2
+{2\over 3}\beta^3b_\nu^2 +g_5^\nu(b_1^2,\ldots,b_p^2)\Bigr]=0,
\quad 1\le\nu\le p.
\equation(226)
$$
Given a solution of these equations, let us define
$V=\{\nu: 1\le\nu\le p, b_\nu\ne 0\}$ and $n=|V|$.
By summing the expression $[\cdots]$ in \equ(226) over all $\nu\in V$,
it is possible to write the sum $b_1^2+\ldots +b_p^2$ in terms of
$\beta$, $n$, and $g_5$. After substituting the result back into \equ(226),
we obtain the following equation for the variables $t_\nu=b_\nu^2$.
$$
t_\nu={3\over(3n-2)\beta^3}(\beta-1)
-{3\over 2\beta^3}g_5^\nu(t_1,\ldots,t_p)
+{9\over(6n-4)\beta^3}\sum_{\mu\in V}g_5^\mu(t_1,\ldots,t_p)
\equation(227)
$$
if $\nu\in V$, and $t_\nu=0$ otherwise.
In order to establish the existence of a unique solution to \equ(227),
we will use the contraction mapping principle in a ball
$\BB(\varrho)=\{t\in\real^V:\,|t_\nu-\theta_\nu|\le\varrho,\;\nu\in V\}$,
centered at the vector $\theta$,
$$
\theta_\nu={3\over (3n-2)\beta^3}(\beta-1),\qquad \nu\in V.
\equation(914)
$$
For $\varrho$ we choose the value $26(\beta-1)$,
so that if $z=\sum_{\nu\in V}b_\nu e^\nu$ satisfies the bound \equ(906),
then the vector $t=\bigl(b_\nu^2\bigr)_{\nu\in V}$ lies in $\BB(\varrho)$.
This guarantees that all (real) fixed points of $\Omega_\beta$
will be obtained. Furthermore, as is easy to check,
the restriction \equ(905a) on $\beta$ ensures that
$|t_\nu|\le 29(\beta-1)\le\bigl({1\over 2p}\bigr)^2$
for all $t\in\BB(\varrho)$, so that the bound \equ(913) may be used.

Denote by $M_\nu(t)$ the right hand side of \equ(227).
Since $|\theta_\nu|\le 3(\beta-1)$, we obtain
$$
|M_\nu(\theta)-\theta_\nu|
\le\Bigl({3\over 2\beta^3}+{9n\over(6n-4)\beta^3}\Bigr)
\max_\mu|g_5^\mu(\theta)|
\le 6\cdot{4\over 3}p^7\cdot{1\over 2}\theta_\nu^2
\le{1\over 40p}\theta_\nu\,,
\equation(915)
$$
if $\beta$ satisfies \equ(905a). This shows that the transformation $M$,
defined by $\bigl(M(t)\bigr)_\nu=M_\nu(t)$,
maps the center of $\BB(\varrho)$ into $\BB(\varrho/2)$.
Thus, in order for $M$ to have a unique fixed point in $\BB(\varrho)$,
it is sufficient that $M$ contracts distances
(in the norm $\max_\mu|t_\mu|$) by a factor of 2 or more.
The following bound shows that this is indeed the case.
By \equ(227), \equ(913), and \equ(905a), we have
$$
\eqalign{
\max_{\nu\in V}&\sum_{\lambda\in V}
\Bigm|{\partial\over\partial t_\lambda}M_\nu(t)\Bigm|
\le n\cdot\max_{\lambda,\nu\in V}
\Bigm|{\partial\over\partial t_\lambda}M_\nu(t)\Bigm|\cr
&\le p\cdot\Bigl({3\over 2\beta^3}+{9n\over(6n-4)\beta^3}\Bigr)\cdot
\Bigm|{\partial\over\partial t_\lambda}g_5^\mu(t)\Bigm|\cr
&\le p\cdot 6\cdot{4\over 3}p^6\cdot 29(\beta-1)\;
<{1\over 2}.\cr}
\equation(916)
$$

As a consequence, the only solution of equation \equ(227)
is the symmetric solution, i.e. $t_\nu=a_n(\beta)^2$ for all $\nu\in V$.
The assertion now follows from Proposition 3.7. below.
\qed

For the following discussion of stability, let $n\ge 2$, and assume
that $z$ is a symmetric fixed point of order $n$ for $\Omega_\beta$, i.e.
that $z=a_n(\beta)v$, with $v$ satisfying equation \equ(213).
Denote by $P_1$ and $P_3$ the orthogonal projections in $\real^d$
onto the subspaces ${\rm span}\{v\}$ and
$\{w\in P\real^d: \langle w,c_\mu e^\mu\rangle=0, 1\le\mu\le p\}$,
respectively, and let $P_2=P-P_1-P_3$.
By using \clm(202), it can be shown that the linearization of
$\Omega_\beta$ at the point $z$ has the following spectral representation.
$$
D\Omega_\beta(z)=sP_3 +(s-r)P_2 +\bigl(s+(n-1)r\bigr)P_1\,,
\equation(229)
$$
where $s$ and $r$ are given by the equations
$$
\eqalign{
s&=\beta -{\beta\over d}\sum_{k=1}^d\tanh^2(\beta z_k),\cr
r&=-{\beta\over d}c_\mu c_\nu\sum_{k=1}^d\tanh^2(\beta z_k)e^\mu_k e^\nu_k,
\qquad\mu\ne\nu,\quad c_\mu c_\nu\ne 0.\cr}
\equation(230)
$$

We note that the eigenvalue $s$ of $D\Omega_\beta(z)$ is always bounded
on one side by $(s-r)$, and on the other side by $(s+(n-1)r)$.
Furthermore, as a consequence of \clm(205), the eigenvalue
$(s+(n-1)r)$ lies between 0 and 1 for all $\beta>1$;
it is the slope of the function $a\mapsto\gamma_n(\beta a)$
at the fixed point $a=a_n(\beta)$.
Thus, the eigenvalue $(s-r)$ alone determines whether or not
the fixed point $z$ is stable.

\claim Proposition(208) The symmetric fixed points of order $n=1$
are stable for all $\beta>1$. For $\beta$ in the interval \equ(905a),
all other fixed points of $\Omega_\beta$ are unstable.

\proof
Let $z=a_n(\beta)v$ be a symmetric fixed point of order $n$.
If $n=0$, then $z$ is clearly unstable for all $\beta>1$,
since $z=0$ and $D\Omega_\beta(0)=\beta P$.
If $n=1$, we obtain $D\Omega_\beta(z)=sP$,
with $s$ given by equation \equ(230), and it is easy to check
that $s<1$ for all $\beta>1$. 

Consider now $n\ge 2$, and $\beta$ in the interval \equ(905a).
By repeating the discussion of equation \equ(227),
but this time for the ball $\BB\bigl({1\over 4p}\theta_\nu\bigr)$,
we get the following bound on $a_n(\beta)$.
$$
a_n(\beta)^2<\bigl(1+{1\over 4p}\bigr)\theta_\nu<4(\beta-1),
\equation(917)
$$
where $\theta_\nu$ is given by \equ(914). In particular, since
$|\tanh^{\prime\prime\prime}(x)|\le 2$ and $|v_k|\le p$, we have
$$
\tanh\bigl(|\beta z_k|\bigr)
\le\bigl(1+{\textstyle{1\over 3}}|\beta z_k|^2\bigr)|\beta z_k|
\le\Bigl(1+{1\over 4p}\Bigr)|\beta z_k|.
\equation(918)
$$
The eigenvalue $(s-r)$ of $D\Omega_\beta(z)$ can now be estimated as follows.
$$
\eqalign{
s-r&=\beta-\sum_{k=1}^d\tanh^2(\beta z_k)
\bigl(1-c_\mu c_\nu e^\mu_k e^\nu_k\bigr)\cr
&\ge\beta-{\beta\over d}\Bigl(1+{1\over 4p}\Bigr)^2
\sum_{k=1}^d(\beta z_k)^2\bigl(1-c_\mu c_\nu e^\mu_k e^\nu_k\bigr)\cr
&\ge\beta-\beta^3\Bigl(1+{1\over 4p}\Bigr)^3\theta_\nu
\Bigl[{1\over d}\sum_{k=1}^d v_k^2
\bigl(1-c_\mu c_\nu e^\mu_k e^\nu_k\bigr)\Bigr]\cr
&\ge\beta-\beta^3\Bigl(1+{1\over p}\Bigr)\theta_\nu(n-2)
=1+{4-3{n-2\over p}\over 3n-2}(\beta-1)\;>1.\cr}
\equation(919)
$$
This shows that the fixed point $z$ is unstable.
For details on how to evaluate the expression $[\cdots]$ in \equ(919)
we refer to the disussion following equation \equ(220).
\qed

\remark
For $\beta>1$, the symmetric fixed points of order $n=1$ are
not only stable, but they also minimize $f_\beta$.
This follows from \clm(103).

\claim Proposition(209) Let $n$ be an even positive integer,
and assume that $\beta$ satisfies
$$
\beta\cdot 2^{-n}{n\choose n/2} >1.
\equation(232)
$$
Then all symmetric fixed points of order $n$ are unstable.

\proof
Let $z=a_n(\beta)v$ be a symmetric fixed point of even order $n>0$,
and define $Z=\{k: 1\le k\le d, v_k=0\}$. It is easy to see that
$Z$ contains exactly $2^{p-n}{n\choose n/2}$ elements. 
Thus, if $\beta$ satisfies \equ(232), the eigenvalue $s$
of $D\Omega_\beta(z)$ can be bounded as follows.
$$
s={\beta\over d}\sum_{k=1}^d\bigl(1-\tanh^2(\beta z_k)\bigr)
\ge{\beta\over d}2^{p-n}{n\choose n/2}
=\beta\cdot 2^{-n}{n\choose n/2} >1.
\equation(233)
$$
\qed

\claim Proposition(210) Let $n$ be an odd integer larger than 1,
and assume that $\beta$ satisfies
$$
\beta\cdot 2^{-n}{n-1\choose (n-1)/2} >\ln(\beta) >1.
\equation(234)
$$
Then all symmetric fixed points of order $n$ are stable.

\proof
Let $z=a_n(\beta)v$ be a symmetric fixed point of order $n>1$.
If we assume that $n$ is odd, then $|v_k|\ge 1$ for all $k$,
and the eigenvalue $(s-r)$ of $D\Omega_\beta(z)$ can be bounded
as follows: If $\mu\ne\nu$ and $c_\mu c_\nu\ne 0$, then
$$
\eqalign{
s-r&={\beta\over d}\sum_{k=1}^d
\Bigl(1-\tanh^2\bigl(\beta a_n(\beta)v_k\bigr)\Bigr)
\bigl[1-c_\mu c_\nu e^\mu_k e^\nu_k\bigr]\cr
&\le\beta\Bigl(1-\tanh^2\bigl(\beta a_n(\beta)\bigr)\Bigr)
<4\beta e^{-2\beta a_n(\beta)}.\cr}
\equation(236)
$$
In the first inequality we have used that $[\cdots]=2$
for half of the values of $k$, and $[\cdots]=0$ for the other half.
Since $a_n(\beta)$ converges to a positive value as $\beta\to\infty$,
it is clear that $z$ is stable for large $\beta$.

In order to estimate $\beta a_n(\beta)$, we bound the factors
$\tanh\bigl((n-2m)x\bigr)$ in \equ(214)  from below by $\tanh(x)$,
and then sum as in \equ(219):
$$
\eqalign{
a_n(\beta)
&=\gamma_n\bigl(\beta a_n(\beta)\bigr)\ge\sum_{0\le m\le n/2}
{n\choose m}{n-2m\over n}\tanh\bigl(\beta a_n(\beta)\bigr)\cr
&=2^{-n+1}{n-1\choose (n-1)/2}\tanh\bigl(\beta a_n(\beta)\bigr).\cr}
\equation(237)
$$
By using \equ(234),
and the fact that $\tanh^{\prime\prime}(x)>-2x$, for $x>0$, we obtain
$$
\beta a_n(\beta)>2\tanh\bigl(\beta a_n(\beta)\bigr)
\ge 2\beta a_n(\beta) -{2\over 3}\bigl(\beta a_n(\beta)\bigr)^3,
\equation(238)
$$
which implies that $\beta a_n(\beta)>1$.
This bound can be improved by applying \equ(237) and \equ(234) again:
Since $\tanh(1)>1/2$, we have
$$
\beta a_n(\beta) >\beta\cdot 2^{-n}{n-1\choose (n-1)/2} >\ln(\beta).
\equation(239)
$$
Substituting this into \equ(236), and using the fact that $\beta>4$
by assumption \equ(234), we see that the eigenvalue $(s-r)$
is smaller than 1. This completes the proof of of \clm(210).
\qed

%REF.3
\bigskip\bigskip\noindent
{\bf Acknowledgements}\hfill\break
\smallskip\noindent
H.K.~would like to thank Prof.~K.~Hepp for his hospitality
at the ETH in Z\"urich, where part of this work has been carried out.
\bigskip\bigskip\noindent
\baselineskip=13pt plus 1pt minus 0pt
{\bf References}\hfill\break

\item{[\rA]} W.A.~Little, {\it
The Existence of Persistent States in the Brain},
Math. Biosci. {\bf 19}, 101 (1974).

\item{[\rB]} J.J.~Hopfield, {\it
Neural Networks and Physical Systems with Emergent Collective
Computational Abilities},
Proc. Natl. Acad. Sci. USA {\bf 79}, 2554 (1982).

\item{[\rK]} J.P.~Provost, G.~Vall\'ee, {\it
Ergodicity of the Coupling Constants and the Symmetric n-Replicas Trick
for a Class of Mean--Field Spin--Glass Models}
Phys. Rev. Lett. {\bf 50}, 598 (1983).

\item{[\rD]} D.J.~Amit, H.~Gutfreund, H.~Sompolinsky, {\it
Spin--Glass Models of Neural Networks},
Phys. Rev. A {\bf 32}, 1007 (1985).

\item{[\rM]} R.J.~McEliece, E.C.~Posner, E.R.~Rodemich, S.S.~Venkatesh, {\it
The Capacity of the Hopfield Associative Memory},
IEEE Transactions on Information Theory {\bf 33}, 461 (1986).

\item{[\rE]} D.J.~Amit, {\it
The Properties of Models of Simple Neural Networks},
in ``Heidelberg Colloquium on Glassy Dynamics (1986)'',
J.L.~van~Hemmen and I.~Morgenstern (eds.),
Lecture Notes in Physics {\bf 275}, 430 (1987).

\item{[\rF]} D.J.~Amit, H.~Gutfreund, H.~Sompolinsky, {\it
Statistical Mechanics of Neural Networks near Saturation},
Ann. Phys. {\bf 173}, 30 (1987).

\item{[\rG]} J.L.~van~Hemmen, {\it
Spin--Glass Model of a Neural Network},
Phys. Rev. A {\bf 34}, 3435 (1986).

\item{[\rL]} D.~Grensing, R.~K\"uhn, {\it
Random--Site Spin--Glass Models},
J. Phys. A {\bf 19}, L1153 (1986).

\item{[\rH]} D.~Grensing, J.L.~van~Hemmen, A.~Huber, R.~K\"uhn, {\it
Nonlinear Neural Networks:}
\hfill\break
{\it I. General Theory},
J. Stat. Phys. {\bf 50}, 231 (1987).
\hfill\break
{\it II. Information Processing},
J. Stat. Phys. {\bf 50}, 259 (1987).

\item{[\rI]} J.L.~van~Hemmen, V.A.~Zagrebnov, {\it
Storing Extensively Many Weighted Patterns in a Saturated Neural Network},
J. Phys. A {\bf 20}, 3989 (1987).

\item{[\rJ]} P.~Baldi, {\it
Symmetries and Learning in Neural Network Models},
Phys. Rev. Lett. {\bf 59}, 1976 (1987).

\item{[\rN]} C.M.~Newman, {\it
Memory Capacity in Neural Network Models: Rigorous Lower Bounds},
Neural Networks {\bf 1} (1988).

\item{[\rP]} J.~Koml\'os, R.~Paturi, {\it
Convergence Results in the Hopfield Model},
Preprint, UC San Diego (1987).

\item{[\rQ]} C.~Peterson, J.R.~Anderson, {\it
A Mean Field Theory Learning Algorithm for Neural Networks},
Preprint MCC--EI--259--87, MCC Austin (1987).

\bye

\bye
