Anton Bovier, Veronique Gayrard
METASTATES IN THE HOPFIELD MODEL IN THE REPLICA SYMMETRIC REGIME
(220K, PS)
ABSTRACT. 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 $\a>0$. 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 in some length. As a byproduct, in a certain
regime of the parameters $\a$ and $\b$ (the inverse temperature),
we also give a
simple proof of Talagrand's [T1] recent result that the
replica symmetric solution found by Amit, Gutfreund, and Sompolinsky [AGS]
can be rigorously justified.