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