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{1041903207,
author = {Gentz, Barbara},
title = {A central limit theorem for the overlap in the Hopfield
model},
journal = {Ann. Probab.},
volume = {24},
number = {2},
year = {1996},
pages = { 1809-1841},
language = {en},
url = {http://dml.mathdoc.fr/item/1041903207}
}
Gentz, Barbara. A central limit theorem for the overlap in the Hopfield
model. Ann. Probab., Tome 24 (1996) no. 2, pp. 1809-1841. http://gdmltest.u-ga.fr/item/1041903207/