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 1. Our analysis is based upon a new result on the asymptotic distribution of order statistics of certain correlated exchangeable random variables.
"Extreme Value Behavior in the Hopfield Model." Ann. Appl. Probab. 11 (1) 91 - 120, February 2001. https://doi.org/10.1214/aoap/998926988