Extreme Value Behavior in the Hopfield Model
- Bovier, Anton
- Mason, David M.
2010 Mathematics Subject Classification
- 82B44 60G70 60K35
- Hopfield model, extreme values, order statistics, metastates, chaotic size-dependence
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 one. Our analysis is based upon a new result on the asymptotic distribution of order statistics of certain correlated exchangeable random variables.
- Ann. Appl. Probab. 11 (2001), pp. 91-120