Moderate Deviations for the overlap parameter in the Hopfield model

Abstract

We derive moderate deviation principles for the overlap parameter in the Hopfield model of spin glasses and neural networks. If the inverse temperature β is different from the critical inverse temperature β c =1 and the number of patterns M(N) satisfies M(N)/N → 0, the overlap parameter multiplied by N γ , 1/2 < γ < 1, obeys a moderate deviation principle with speed N 1−2γ and a quadratic rate function (i.e. the Gaussian limit for γ = 1/2 remains visible on the moderate deviation scale). At the critical temperature we need to multiply the overlap parameter by N γ , 1/4 < γ < 1. If then M(N) satisfies (M(N)6 log N ∧ M(N)2N4γ log N)/N → 0, the rescaled overlap parameter obeys a moderate deviation principle with speed N1−4γ and a rate function that is basically a fourth power. The random term occurring in the Central Limit theorem for the overlap at β c = 1 is no longer present on a moderate deviation scale. If the scaling is even closer to N1/4, e.g. if we multiply the overlap parameter by N1/4 log log N the moderate deviation principle breaks down. The case of variable temperature converging to one is also considered. If β N converges to β c fast enough, i.e. faster than the non-Gaussian rate function persists, whereas for β N converging to one slower than 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.