A. Bovier, D.M. Mason
Extrme value behaviour in the Hopfield model
(185K, PS)
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.