93-139 Anton Bovier , V'eronique Gayrard , Pierre Picco

Gibbs states of the Hopfield model in the regime of perfect memory (344K, PS) 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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