Anton Bovier , V'eronique Gayrard , Pierre Picco
Gibbs states of the Hopfield model in the regime of perfect memory
(344K, PS)
ABSTRACT. We study the thermodynamic properties of the
Hopfield model of an autoassociative memory. If $N$ denotes the number of
neurons and $M(N)$ the number of stored patterns, we prove the following
results: If $\frac MN\downarrow 0$ as $N\uparrow \infty$, then there exists an
infinite number of infinite volume Gibbs measures for all temperatures
$T<1$ concentrated on spin configurations that have overlap with
exactly one specific pattern. Moreover, the measures
induced on the overlap parameters are Dirac measures concentrated
on a single point. If $\frac MN\rightarrow \a$,
as $N\uparrow \infty$ for $\a$ small enough, we show that for
temperatures $T$ smaller than some $T(\a)<1$, the induced measures
can have support only on a disjoint union of
balls around the previous points, but we cannot construct the infinite volume
measures through convergent sequences of measures.