Abstract.
We investigate the limiting fluctuations of the order parameter in the Hopfield model of spin glasses and neural networks with finitely many patterns at the critical temperature 1/β c = 1. At the critical temperature, the measure-valued random variables given by the distribution of the appropriately scaled order parameter under the Gibbs measure converge weakly towards a random measure which is non-Gaussian in the sense that it is not given by a Dirac measure concentrated in a Gaussian distribution. This remains true in the case of β = β N →β c = 1 as N→∞ provided β N converges to β c = 1 fast enough, i.e., at speed ?(1/). The limiting distribution is explicitly given by its (random) density.
Author information
Authors and Affiliations
Additional information
Received: 12 May 1998 / Revised version: 14 October 1998
Rights and permissions
About this article
Cite this article
Gentz, B., Löwe, M. The fluctuations of the overlap in the Hopfield model with finitely many patterns at the critical temperature. Probab Theory Relat Fields 115, 357–381 (1999). https://doi.org/10.1007/s004400050241
Issue Date:
DOI: https://doi.org/10.1007/s004400050241
- Mathematics Subject Classification (1991): 60F05, 60K35 (primary), 82C32 (secondary)