Relaxation of Disordered Magnets in the Griffiths' Regime

We study the relaxation to equilibrium of discrete spin systems with random finite range (not necessarily ferromagnetic) interactions in the Griffiths' regime. We prove that the speed of convergence to the unique reversible Gibbs measure is almost surely faster than any stretched exponential, at least if the probability distribution of the interaction decays faster than exponential (e.g. Gaussian). Furthermore, if the interaction is uniformly bounded, the average over the disorder of the time-autocorrelation function, goes to equilibrium as exp[-k(log t)(d/(d-1))] (in d > 1), in agreement with previous results obtained for the dilute Ising model.