In a previous work,we proved the validity ofthe replica-symmetric solution for the Hop field model in a nontrivial domain of parameters. This was done at the accuracy ofthe LLN In a somewhat larger domain, we obtain a description at the level ofthe CLT, or, in the terminology of physics, we calculate the fluctuations around the mean field. This is obtained by refining the tools we developed for a rigorous use of the cavity method and proving new a priori estimates about the “localization”of the Gibbs measure.
"Exponential inequalities and convergence of moments in the replica-symmetric regime of the Hopfield model." Ann. Probab. 28 (4) 1393 - 1469, October 2000. https://doi.org/10.1214/aop/1019160495