The Mixing Time Evolution of Glauber Dynamics for the Mean-Field Ising Model

摘要

We consider Glauber dynamics for the Ising model on the complete graph on n vertices, known as the Curie-Weiss model. It is well-known that the mixing-time in the high temperature regime (β 1 is exponential in n. Recently, Levin, Luczak and Peres proved that for any fixed β < 1 there is cutoff at time with a window of order n, whereas the mixing-time at the critical temperature β = 1 is Θ(n 3/2). It is natural to ask how the mixing-time transitions from Θ(n log n) to Θ(n 3/2) and finally to exp (Θ(n)). That is, how does the mixing-time behave when β = β(n) is allowed to tend to 1 as n → ∞. In this work, we obtain a complete characterization of the mixing-time of the dynamics as a function of the temperature, as it approaches its critical point β _c = 1. In particular, we find a scaling window of order around the critical temperature. In the high temperature regime, β = 1 − δ for some 0 < δ 0 with δ 2 n → ∞ and δ = o(1), there is no cutoff, and the mixing time has order . Communicated by H. Spohn