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February 2001 Extreme Value Behavior in the Hopfield Model
Anton Bovier, David M. Mason
Ann. Appl. Probab. 11(1): 91-120 (February 2001). DOI: 10.1214/aoap/998926988


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.


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Anton Bovier. David M. Mason. "Extreme Value Behavior in the Hopfield Model." Ann. Appl. Probab. 11 (1) 91 - 120, February 2001.


Published: February 2001
First available in Project Euclid: 27 August 2001

zbMATH: 1024.82015
MathSciNet: MR1825461
Digital Object Identifier: 10.1214/aoap/998926988

Primary: 60G70 , 60K356 , 82B44

Keywords: chaotic size dependence , Extreme values , Hopfield model , metastates , order statistics

Rights: Copyright © 2001 Institute of Mathematical Statistics


Vol.11 • No. 1 • February 2001
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