The Annals of Probability

A central limit theorem for the overlap in the Hopfield model

Barbara Gentz

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We consider the Hopfield model with n neurons and an increasing number $p = p(n)$ of randomly chosen patterns. Under the condition $(p^3 \log p)/n \to 0$, we prove for every fixed choice of overlap parameters a central limit theorem as $n \to \infty$, which holds for almost all realizations of the random patterns. In the special case where the temperature is above the critical one and there is no external magnetic field, the condition $(p^2 \log p)/n \to 0$ suffices. As in the case of a finite number of patterns, the central limit theorem requires a centering which depends on the random patterns.

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Ann. Probab., Volume 24, Number 4 (1996), 1809-1841.

First available in Project Euclid: 6 January 2003

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.) 82C32: Neural nets [See also 68T05, 91E40, 92B20]

Fluctuations Hopfield model overlap neural networks Laplace's method


Gentz, Barbara. A central limit theorem for the overlap in the Hopfield model. Ann. Probab. 24 (1996), no. 4, 1809--1841. doi:10.1214/aop/1041903207.

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