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

This paper studies the rate of convergence to equilibrium of Glauber dynamics (Gibbs Sampler) for a system of N Ising spins with random energy (at inverse temperature beta > 0). For each of the 2N spin configurations 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 configurations 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 exponential (gamma < 1) in N; (2) if gamma < 1, then at beta = gamma the ''almost coupling time'' [i.e., the first time that the two dynamics are within distance o(N)] changes from polynomial (beta < gamma) to stretched exponential (beta > gamma) in N. The techniques used to control the randomness in the coupling are static and dynamic large-deviation estimates and stochastic domination arguments.