Extreme value behavior in the Hopfield model

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)IN tends to zero. In this model the unbiased Gibbs state in volume N can essentially be decompo sed into M(N) pairs of disjoint measures. We investigate the distributions of the corresponding weights, and show, in particular, that these weights c oncentrate for any given N very closely to one of the pairs, with probabili ty tending to 1. Our analysis is based upon a new result on the asymptotic distribution of order statistics of certain correlated exchangeable random variables.