Extreme value behavior in the Hopfield model

• 成果类型: Article
• 署名作者: Bovier, A; Mason, DM
• 署名单位: Leibniz Association; Weierstrass Institute for Applied Analysis & Stochastics; University of Delaware
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/aoap/998926988
• 发表日期: 2001
• 页码: 91-120
• 关键词: gibbs-states; field

摘要

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 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.