A DYNAMICAL PHASE-TRANSITION IN A CARICATURE OF A SPIN-GLASS

This paper studies the rate of convergence to equilibrium of Glauber d ynamics (Gibbs Sampler) for a system of N Ising spins with random ener gy (at inverse temperature beta > 0). For each of the 2N spin configur ations the energy is drawn independently from the values 0 and -log N with probabilities 1 - N(-gamma), resp. N(-gamma) (gamma > 0), and is kept fixed during the evolution. The main result is an estimate of the coupling time of two Glauber dynamics starting from different configu rations and coupled via the same updating noise. As N --> infinity the system exhibits two dynamical phase transitions: (1) at gamma = 1 the coupling time changes from polynomial (gamma > 1) to stretched expone ntial (gamma < 1) in N; (2) if gamma < 1, then at beta = gamma the ''a lmost coupling time'' [i.e., the first time that the two dynamics are within distance o(N)] changes from polynomial (beta < gamma) to stretc hed exponential (beta > gamma) in N. The techniques used to control th e randomness in the coupling are static and dynamic large-deviation es timates and stochastic domination arguments.