 93139 Anton Bovier , V'eronique Gayrard , Pierre Picco
 Gibbs states of the Hopfield model in the regime of perfect memory
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May 18, 93

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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.
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