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October 1996 A central limit theorem for the overlap in the Hopfield model
Barbara Gentz
Ann. Probab. 24(4): 1809-1841 (October 1996). DOI: 10.1214/aop/1041903207


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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Barbara Gentz. "A central limit theorem for the overlap in the Hopfield model." Ann. Probab. 24 (4) 1809 - 1841, October 1996.


Published: October 1996
First available in Project Euclid: 6 January 2003

zbMATH: 0885.60016
MathSciNet: MR1415230
Digital Object Identifier: 10.1214/aop/1041903207

Primary: 60F05
Secondary: 60K35 , 82B44 , 82C32

Keywords: Fluctuations , Hopfield model , Laplace's method , neural networks , Overlap

Rights: Copyright © 1996 Institute of Mathematical Statistics

Vol.24 • No. 4 • October 1996
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