Metastability in Glauber dynamics in the low-temperature limit: Beyond exponential asymptotics

Abstract

We consider Glauber dynamics of classical spin systems of Ising type in the limit when the temperature tends to zero in finite volume. We show that information on the structure of the most profound minima and the connecting saddle points of the Hamiltonian can be translated into sharp estimates on the distribution of the times of metastable transitions between such minima as well as the low lying spectrum of the generator. In contrast with earlier results on such problems, where only the asymptotics of the exponential rates is obtained, we compute the precise pre-factors up to multiplicative errors that tend to 1 as $T\downarrow 0$. As an example we treat the nearest neighbor Ising model on the 2 and 3 dimensional square lattice. Our results improve considerably earlier estimates obtained by Neves-Schonmann and Ben Arous-Cerf. Our results employ the methods introduced by Bovier, Eckhoff, Gayrard, and Klein.