## The Annals of Probability

### A central limit theorem for the overlap in the Hopfield model

Barbara Gentz

#### Abstract

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.

#### Article information

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

Dates
First available in Project Euclid: 6 January 2003

https://projecteuclid.org/euclid.aop/1041903207

Digital Object Identifier
doi:10.1214/aop/1041903207

Mathematical Reviews number (MathSciNet)
MR1415230

Zentralblatt MATH identifier
0885.60016

#### Citation

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. https://projecteuclid.org/euclid.aop/1041903207

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