Metastability in the generalized Hopfield model with finitely many patterns

TL;DR

This paper investigates metastable behavior in a generalized Hopfield model with finitely many patterns, providing precise estimates on exit times and capacities, especially in complex energy landscapes with multiple saddle points.

Contribution

It extends metastability analysis to a generalized Hopfield model with discrete pattern distributions, including degenerate energy surfaces and multiple saddle points.

Findings

Metastable behavior occurs in the model.

Sharp asymptotics for exit times and capacities are derived.

Analysis includes complex energy landscapes with degeneracies.

Abstract

This paper continues the study of metastable behaviour in disordered mean field models initiated in [2], [3]. We consider the generalized Hopfield model with finitely many independent patterns $\xi_1,...,\xi_p$ where the patterns have i.i.d. components and follow discrete distributions on $[-1,1]$. We show that metastable behaviour occurs and provide sharp asymptotics on metastable exit times and the corresponding capacities. We apply the potential theoretic approach developed by Bovier et al. in the space of appropriate order parameters and use an analysis of the discrete Laplacian to obtain lower bounds on capacities. Moreover, we include the possibility of multiple saddle points with the same value of the rate function and the case that the energy surface is degenerate around critical points.