Metastates in the Hopfield Model in the Replica Symmetric Regime

We study the finite dimensional marginals of the Gibbs measure in the Hopfield model at low temperature when the number of patterns, M, is proportional to the volume with a sufficiently small proportionality constant α > 0. It is shown that even when a single pattern is selected (by a magnetic field or by conditioning), the marginals do not converge almost surely, but only in law. The corresponding limiting law is constructed explicitly. We fit our result in the recently proposed language of 'metastates' which we discuss some length. As a byproduct, in a certain regime of the parameters α and β (the inverse temperature), we also give a simple proof of Talagrand's recent result that the replica symmetric solution found by Amit, Gutfreund, and Sompolinsky can be rigorously justified