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

Gentz, B (1996). A central limit theorem for the overlap in the Hopfield model. The Annals of Probability, 24(4):1809-1841.

## 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.

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.

## Citations

7 citations in Web of Science®
8 citations in Scopus®

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics Fluctuations; Hopfield model; overlap; neural networks; Laplace's method English 1996 29 Nov 2010 16:28 05 Apr 2016 13:27 Institute of Mathematical Statistics 0091-1798 https://doi.org/10.1214/aop/1041903207 http://www.ams.org/mathscinet-getitem?mr=1415230http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0872.60015
Permanent URL: https://doi.org/10.5167/uzh-22548