## Annals of Applied Probability

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

### Extreme Value Behavior in the Hopfield Model

Anton Bovier and David M. Mason

#### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 27 August 2001

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/998926988

**Digital Object Identifier**

doi:10.1214/aoap/998926988

**Mathematical Reviews number (MathSciNet)**

MR1825461

**Zentralblatt MATH identifier**

1024.82015

**Subjects**

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

**Keywords**

Hopfield model extreme values order statistics metastates chaotic size dependence

#### Citation

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. https://projecteuclid.org/euclid.aoap/998926988