G.R. Guerberoff, and G.A. Raggio
ON THE FREE ENERGY OF THE HOPFIELD MODEL
(107K, LaTeX)
ABSTRACT. The general theory of inhomogeneous mean-field systems of Ref.
\cite{RaWe2} provides a variational expression for the (almost sure)
limiting free energy density of the Hopfield model
\[ H_{N,p}^{\{ \mbox{\boldmath $\xi$} \}} (S) = - \frac{1}{2N} \sum_{i,j =1}^N \sum_{\mu =1}^p \xi^{\mu}_i \xi^{\mu}_j S_i
S_j \] for Ising spins $S_i$ and $p$ random patterns $\mbox{\boldmath
$\xi$}^{\mu} = ( \xi^{\mu}_1, \xi^{\mu}_2, \cdots , \xi^{\mu}_N )$
under the assumption that
\[ \lim_{N \to \infty} N^{-1} \sum_{i=1}^N
\delta_{\mbox{\boldmath $\xi$}_i} = \lambda \; \; \; , \; \; \; \mbox{\boldmath
$\xi$}_i = ( \xi^1_i, \xi^2_i, \cdots , \xi^p_i )\] exists (almost
surely) in the space of probability measure over $p$ copies of $\{ -1
, 1 \}$. Including an ``external field'' term $ - \sum_{\mu=1}^p
h^{\mu}
\sum_{i=1}^N \xi^{\mu}_i S_i$, we give a number of general
properties of the free-energy density and compute it for
a): $p=2$ in general, and b): $p$ arbitrary when $\lambda$ is uniform and
at most the two components $ h^{\mu_1}$ and $h^{\mu_2}$ are non-zero,
obtaining the (almost sure) formula
\[ f( \beta , {\bf h}) = \frac{1}{2} f^{cw}( \beta ,
h^{\mu_1}+h^{\mu_2} ) + \frac{1}{2} f^{cw} ( \beta , h^{\mu_1}-h^{\mu_2})
\]
for the free energy, where $f^{cw}$ denotes the limiting free energy density of the
Curie-Wei{\ss} model with unit interaction constant. In both cases, we obtain explicit
formulae for the
limiting (almost sure) values of the so-called overlap parameters
\[ m_N^{\mu} ( \beta , {\bf h } ) =
N^{-1} \sum_{i=1}^N \xi^{\mu}_i < S_i > \]
in terms of the Curie-Wei{\ss} magnetizations.
For the general i.i.d. case with $Prob \{ \xi^{\mu}_i= \pm 1 \} =
(1/2) \pm \epsilon$, we obtain the lower bound $1+ 4
\epsilon^2 (p-1)$ for the temperature $T_c$ separating the trivial free
regime where the overlap vector is zero, from the non-trivial regime
where it is non-zero. This lower bound is exact for: $p=2$, or
$\epsilon =0$, or $\epsilon = \pm 1/2$. For $p=2$ we identify an
intermediate temperature region between $T_*= 1- 4 \epsilon^2$ and
$T_c = 1 + 4
\epsilon^2$ where the overlap vector is homogeneous (i.e. all its
components are equal) and non-zero. $T_*$ marks the transition to the
non-homogeneous regime where the components of the overlap vector are
distinct. We
conjecture that the homogeneous non-zero regime exists for $p \geq 3$
and that $T_* = \max \{ 1-
4 \epsilon^2 (p-1), 0 \}$.