Large deviation principles for the Hopfield model and the Kac-Hopfield model
The Hopfield model of statistical mechanics determines random interaction between spins \(\sigma_i\) by the \(N \times N\) matrix \(J\) of the form \[ J_{ij} = {1 \over N} \sum^p_{\mu = 1} \xi^\mu_i \xi^\mu_j, \quad i,j = \overline {1,N}, \] where the number \(p\) may depend on \(N\). The simplest case of \(p(N)\) bounded as \(N \to \infty\) is studied rather well. In particular, the large deviation principle for the overlap parameter \[ m_\Delta (\mu; \sigma) = |\Delta |^{-1} \sum_{i \in \Delta} \xi^\mu_i \sigma_i, \quad \Delta \subset [1,N], \] can be established by using general Gärtner-Ellis theorems. The present paper concerns much more difficult situation when \(p(N)\) is allowed to increase infinitely when \(N \to \infty\). The authors introduce the Kac version of the Hopfield model with the interaction matrix \[ J^{(\gamma)}_{ij} = {1 \over 2} \gamma \sum^M_{\mu = 1} \xi_i^\mu \xi^\mu_j I \bigl( \gamma |i - j |\bigr), \] where \(I(x) = 1\) if \(|x |\leq 1\) and \(I(x) = 0\) otherwise, and \(M\) depends on the parameter \(\gamma > 0\). For fixed \(\gamma > 0\) they define the limiting as \(N \to \infty\) rate function of the overlap parameter and show that in the limit \(\gamma \downarrow 0\), \(M (\gamma) \to \infty\), \(\gamma M (\gamma) \to 0\) the result coincides with that obtained for the Hopfield model in the limit \(N,p \to \infty\), \(pN \to 0\). This is an analogue of the Lebowitz-Penrose theorem for the distribution of the overlap parameters. The paper is clearly written and contains a series of interesting probabilistic results.
scientific article; zbMATH DE number 751348