An almost sure central limit theorem for the overlap parameters in the Hopfield model
The Hopfield model is considered with the Hamiltonian \[ H_n(\sigma ,\xi ) = -{1\over 2n} \sum _{\mu =1}^p \sum _{i,j=1}^n \xi _i^\mu \xi _j^\mu \sigma _i \sigma _j, \] where \((\sigma _i)_{i=1,\ldots ,n}\) and \((\xi _i^\mu )_{i=1,\ldots ,n}\) with values in \(\{-1,1\}^n\) are the spin configuration and the codification of the \(\mu \)-th pattern, \(\mu =1,\ldots ,p\), respectively. The asymptotic behaviour of the random vector \( {1\over n} S_n(\sigma ,\xi ) = {1\over n} \sum _{i=1}^n \xi _i\sigma _i, \) where \(\xi _i=(\xi _i^1,\ldots ,\xi _i^p)\), is studied under the (random) Gibbs measure \( d P_{n,\beta ,\xi } = e^{-\beta H_n(\sigma ,\xi )}/Z_{n,\beta ,\xi }. \) With a proper centering and conditioning the Gaussian fluctuations are proved for non-critical \(\beta \neq \beta _c\) and almost every \(\xi \) with respect to \( P_\xi = \Big (\big (\tfrac 12 \delta _{-1}+\tfrac 12 \delta _1\big )^{\otimes p}\Big )^{\otimes \mathbb N} \) which corresponds to the uniform i.i.d. assumption.
scientific article; zbMATH DE number 1001248