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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