Extreme value behavior in the Hopfield model (Q1872473)

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 [\textit{A. Bovier} and \textit{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.