Moderate deviations for the overlap parameter in the Hopfield model (Q1765113)

The authors consider the following model. Given \(N, M\in \mathbb{N}\), the model Hamiltonian is set to be \[ H_N (\sigma) = - \frac{1}{2N}\sum_{i, j = 1}^N J_{ij}( M) \sigma_i \sigma_j, \quad J_{ij} (M) = \sum_{\mu=1}^M \xi_i^{\mu} \xi_j^\mu , \] where all \(\sigma_i\), \(\xi^\mu_i\), \(i = 1, \dots , N\), \(\mu = 1, \dots , M\), are random variables taking values \(\pm 1\). The \(\xi\)'s are symmetric i.i.d.; the a priori distribution of the \(\sigma\)'s is the same as the one for the \(\xi\)'s. The Hopfield model at temperature \(\beta^{-1} >0\) is identified with the Gibbs measure \[ \varrho_{N, \beta} (\sigma) = 2^{-N} \exp ( - \beta H_N (\sigma)] /Z_{N, \beta}. \] The vectors \(m_N (\sigma) = ( m_N^\mu (\sigma))_{\mu = 1}^N\), with \(m_N^\mu (\sigma) = \frac{1}{N} \sum_{i = 1}^N \xi_i^\mu \sigma_i, \) are called overlaps. Then a number of statements describing thermodynamic limits, as \(N \rightarrow + \infty\), \(\alpha (N) = M/N \rightarrow 0\), of the distributions of the re-scaled overlaps \(N^\gamma m_N (\sigma)\), \(\gamma >0\), obtained from the measures \(\varrho_{N, \beta}\), are proven. The paper may be viewed as a continuation of the research of the same authors [Markov Process. Relat. Fields 10, No. 2, 345--366 (2004; Zbl 1059.60031)].