Extreme value behavior in the Hopfield model.(English)Zbl 1024.82015

Given $$N \in {\mathbb N}$$, let $$\sigma_i$$, $$i = 1, \dots , N$$, be spins, i.e., the variables taking values $$\pm 1$$. The set of all possible spin configurations is denoted by $${\mathcal S}_N$$. Let $$(\Omega , {\mathcal F}, P)$$ be a probability space on which one defines independent identically distributed random variable $$\xi_i^\mu$$, where $$i = 1, \dots , N$$ and $$\mu = 1, \dots , M$$, $$M\in {\mathbb N}$$, taking values $$\pm 1$$ with equal probabilities. They define random maps $$m_N^\mu: {\mathcal S}_N \rightarrow [-1, 1]$$, being $m_N^\mu(\sigma) = {1 \over N}\sum_{i =1}^N \xi_i^\mu \sigma_i .$ These maps give vectors $$m_N (\sigma) = (m_N^1(\sigma), \dots m_N^M(\sigma)) \in {\mathbb R}^M$$. Let $$|m_N (\sigma)|_2$$ denote the Euclidean norms of such vectors. The Hamiltonian $H_N (\sigma) = - {N \over 2} |m_N (\sigma)|_2^2,$ determines interaction between the spins. By means of $$H_N$$ one introduces the corresponding local Gibbs measure $d \mu_{N, \beta} (\sigma) = {e^{-\beta H_N (\sigma)} \over Z_{N, \beta}} d P_\sigma,$ where $$P_\sigma = ((1/2)\delta_{-1} + (1/2)\delta_{+1})^{\otimes N}$$. The measure $$\mu_{N, \beta}$$ defines a probability distribution on $${\mathbb R}^M$$ by $Q_{N , \beta} = \mu_{N, \beta } \circ m_N^{-1}.$ In [A. Bovier and V. Gayrard, Probab. Theory Related Fields 107, 61-98 (1997; Zbl 0866.60085)] it was proved that this distribution can be asymptotically ($$N \rightarrow +\infty$$) decomposed into the sum of pairs of disjoint measures. In this article, it is shown that, if $$M,N \rightarrow +\infty$$ in such a way that $$M^2 N^{-1} \log M \rightarrow 0$$, then the weight of one pair in this decomposition becomes close to one with probability tending to 1.

MSC:

 82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics 60G70 Extreme value theory; extremal stochastic processes 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82C32 Neural nets applied to problems in time-dependent statistical mechanics 82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)

Zbl 0866.60085
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