Annals of Applied Probability

Extreme Value Behavior in the Hopfield Model

Anton Bovier and David M. Mason

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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.

Article information

Ann. Appl. Probab., Volume 11, Number 1 (2001), 91-120.

First available in Project Euclid: 27 August 2001

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.) 60G70: Extreme value theory; extremal processes 60K356

Hopfield model extreme values order statistics metastates chaotic size dependence


Bovier, Anton; Mason, David M. Extreme Value Behavior in the Hopfield Model. Ann. Appl. Probab. 11 (2001), no. 1, 91--120. doi:10.1214/aoap/998926988.

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