99-309 A. Bovier, D.M. Mason

Extrme value behaviour in the Hopfield model (185K, PS) Aug 25, 99

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Abstract. We study a Hopfield model whose number of patterns $M$ grows to infinity with the system size $N$, in such a way that $M(N)^{2}\log M(N)/N$ tends to zero. In this model the unbiased Gibbs state in volume $N$ can essentially be decomposed into $M(N)$ pairs of disjoint measures. We investigate the distributions of the corresponding weights, and show, in particular, that these weights concentrate for any given $N$ very closely to one of the pairs, with probability tending to one. Our analysis is based upon a new result on the asymptotic distribution of order statistics of certain correlated exchangeable random variables.

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