Anton Bovier, V\'eronique Gayrard
THE RETRIEVAL PHASE OF THE HOPFIELD MODEL: A RIGOROUS ANALYSIS OF THE OVERLAP
DISTRIBUTION
(183K, PS (uuencoded, gz-compressed))
ABSTRACT. Standard large deviation estimates
or the use of the Hubbard-Stratonovich transformation reduce the
analysis of the distribution of the overlap parameters essentially to that of
an explicitly known random function $\Phi_{N,\b}$
on $\R^M$. In this article we present a
rather careful study of the structure of the minima of this random function
related to the retrieval of the stored patterns. We denote by $m^*(\b)$ the
modulus of the spontaneous magnetization in the
Curie-Weiss model and by $\a$ the ratio between the number of the
stored patterns and the system size. We show that there exist strictly positive
numbers $0<\g_a<\g_c$ such that
1) If $\sqrt\a\leq \g_a (m^*(\b))^2$, then the
absolute minima of $\Phi$ are located within small balls
around the points $\pm m^*e^\mu$, where $e^\mu$ denotes the $\mu$-th
unit vector while
2) if $\sqrt\a\leq \g_c (m^*(\b))^2$ at least a local minimum
surrounded by extensive energy barriers exists near these points.
The random location of these minima is given within precise bounds.
These are used to prove sharp estimates on the support of the Gibbs measures.