The fluctuations of the overlap in the Hopfield model with finitely many patterns at the critical temperature

• 成果类型: Article
• 署名作者: Gentz, B; Löwe, M
• 署名单位: Swiss Federal Institutes of Technology Domain; ETH Zurich; University of Bielefeld
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s004400050241
• 发表日期: 1999
• 页码: 357-381
• 关键词: CENTRAL-LIMIT-THEOREM; associative memory; gibbs-states; Lower bounds; capacity

摘要

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).