PUB - Publikationen an der Universität Bielefeld

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/beta(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 beta = beta(N) --> beta(c) = 1 as N --> infinity provided beta(N) converges to beta(c) = 1 fast enough, i.e., at speed partial derivative(1/root N). The Limiting distribution is explicitly given by its (random) density. Mathematics Subject Classification (1991): 60F05, 60K35 (primary), 82C32 (secondary).