  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.
  KEYWORDS: Hopfield model, neural networks, storage capacity, Gibbs measures,
self-averaging, random matrices
