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

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 (beta < 1) has order n log n, whereas the mixing-time in the case beta > 1 is exponential in n. Recently, Levin, Luczak and Peres proved that for any fixed beta < 1 there is cutoff at time 1 /2(1-beta) n log n with a window of order n, whereas the mixing-time at the critical temperature beta = 1 is Theta (n(3/2)). It is natural to ask how the mixing-time transitions from Theta(n log n) to Theta (n(3/2)) and finally to exp (Theta(n)). That is, how does the mixing-time behave when beta = beta( n) is allowed to tend to 1 as n -> infinity. 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 beta(c) = 1. In particular, we find a scaling window of order 1/root n around the critical temperature. In the high temperature regime, beta = 1 - delta for some 0 < delta < 1 so that delta(2)n -> infinity with n, the mixing-time has order (n/delta) log( d2n), and exhibits cutoff with constant 1 2 and window size n/delta. In the critical window, beta = 1 +/- delta, where delta(2)n is O(1), there is no cutoff, and the mixing-time has order n(3/2). At low temperature, beta = 1+delta ford > 0 with delta(2)n -> infinity and delta = o(1), there is no cutoff, and the mixing time has order n/delta exp ((3/4 + o(1))delta(2)n).