Metastability in the generalized Hopfield model with finitely many patterns

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 ξ1, . . . , ξP where the patterns have i.i.d. components and the components of patterns ξ1, . . . ξp have absolutely continuous distributions on [−1, 1] whereas the components of patterns ξp+1, . . . , ξP have discrete distributions on [−1, 1] with no atom at 0. We show that metastable behaviour occurs if there is at least one pattern of each type and 2p+7 < P . In this case we provide sharp asymptotics on metastable exit times and the corresponding capacities. Our main result can be applied to generalized Hopfield models with any values of 0 ≤ p ≤ P by adding auxiliary patterns and choosing a potential which depends only on the original patterns and coincides with the original potential. We employ the potential theoretic approach developed by Bovier et al. and use an analysis of the discrete Laplacian in the space of spin configurations 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.