On the Hopfield model at the critical temperature |
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| Authors: | Michel Talagrand |
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| Institution: | (1) Equipe d'Analyse, Tour 46, ESA au CNRS no. 7064, Université de Paris VI, 4, Place Jussieu, 75230 Paris Cedex 05, France (e-mail: mit@ccr.jussieu.fr), FR;(2) Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, OH 43210-1174, USA (e-mail: talagran@math.ohio-state.edu), US |
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| Abstract: | We study the Hopfield model at temperature 1, when thenumber M(N) of patterns grows a bit slower than N. We reach a goodunderstanding of the model whenever M(N)≤N/(log N)11. For example, we show that if M(N)→∞, for two typical configurations σ
1, σ
2, (∑
i
≤
N
σ1
i
σ2
i
)2 is close to NM(N).
Received: 15 December 1999 / Revised version: 8 December 2000 / Published online: 23 August 2001 |
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