A dynamical phase transition in a caricature of a spin glass

Abstract

This paper studies the rate of convergence to equilibrium of Glauber dynamics (Gibbs Sampler) for a system ofN Ising spins with random energy (at inverse temperature β>0). For each of the 2^N spin configurations the energy is drawn independently from the values 0 and-logN with probabilities 1-N ⁻⁷, resp.N ^−γ (γ>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. AsN→∞ the system exhibits two dynamical phase transitions: (1) at γ=1 the coupling time changes from polynomial (γ>1) to stretched exponential (γ<1) inN; (2) if γ<1, then at β=γ the “almost coupling time” [i.e., the first time that the two dynamics are within distanceo(N)] changes from polynomial (β<γ) to stretched exponential (β>γ) inN. The techniques used to control the randomness in the coupling are static and dynamic large-deviation estimates and stochastic domination arguments.