Moderate deviations for the overlap parameter in the Hopfield model

• 成果类型: Article
• 署名作者: Eichelsbacher, P; Löwe, M
• 署名单位: Ruhr University Bochum; University of Munster
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-004-0349-8
• 发表日期: 2004
• 页码: 441-472
• 关键词: CENTRAL-LIMIT-THEOREM; capacity; field

摘要

We derive moderate deviation principles for the overlap parameter in the Hopfield model of spin glasses and neural networks. If the inverse temperature beta is different from the critical inverse temperature beta(c)=1 and the number of patterns M(N) satisfies M(N)/N --> 0, the overlap parameter multiplied by N-gamma, 1/2 < gamma < 1, obeys a moderate deviation principle with speed N1-2gamma and a quadratic rate function (i.e. the Gaussian limit for gamma = 1/2 remains visible on the moderate deviation scale). At the critical temperature we need to multiply the overlap parameter by N-gamma stop, 1/4 < gamma < 1. If then M(N) satisfies (M(N)(6) log N boolean AND M(N)N-2(4gamma) log N)/N --> 0, the rescaled overlap parameter obeys a moderate deviation principle with speed N1-4gamma and a rate function that is basically a fourth power. The random term occurring in the Central Limit theorem for the overlap at beta(c) = 1 is no longer present on a moderate deviation scale. If the scaling is even closer to N-1/4, e.g. if we multiply the overlap parameter by N-1/4 log log N the moderate deviation principle breaks down. The case of variable temperature converging to one is also considered. If beta(N) converges to beta(c) fast enough, i.e. faster than O(N-2gamma), the non-Gaussian rate function persists, whereas for beta(N) converging to one slower than O(N-2gamma) the moderate deviations principle is given by the Gaussian rate. At the borderline the moderate deviation rate function is the one at criticality plus an additional Gaussian term.