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.
Publié le : 2001-02-14
Classification:
Hopfield model,
extreme values,
order statistics,
metastates,
chaotic size dependence,
82B44,
60G70,
60K356
@article{998926988,
author = {Bovier, Anton and Mason, David M.},
title = {Extreme Value Behavior in the Hopfield Model},
journal = {Ann. Appl. Probab.},
volume = {11},
number = {2},
year = {2001},
pages = { 91-120},
language = {en},
url = {http://dml.mathdoc.fr/item/998926988}
}
Bovier, Anton; Mason, David M. Extreme Value Behavior in the Hopfield Model. Ann. Appl. Probab., Tome 11 (2001) no. 2, pp. 91-120. http://gdmltest.u-ga.fr/item/998926988/