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 (β < 1) has order n log n, whereas the mixing-time in the case β > 1 is exponential in n. Recently, Levin, Luczak and Peres proved that for any fixed β < 1 there is cutoff at time $${\frac{1}{2(1-\beta)}n\log n}$$ with a window of order n, whereas the mixing-time at the critical temperature β = 1 is Θ(n3/2). It is natural to ask how the mixing-time transitions from Θ(n log n) to Θ(n3/2) 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 βc = 1. In particular, we find a scaling window of order $${1/\sqrt{n}}$$ around the critical temperature. In the high temperature regime, β = 1 − δ for some 0 < δ < 1 so that δ2n → ∞ with n, the mixing-time has order (n/δ) log(δ2n), and exhibits cutoff with constant $${\frac{1}{2}}$$ and window size n/δ. In the critical window, β = 1± δ, where δ2n is O(1), there is no cutoff, and the mixing-time has order n3/2. At low temperature, β = 1 + δ for δ > 0 with δ2n → ∞ and δ = o(1), there is no cutoff, and the mixing time has order $${\frac{n}{\delta}{\rm exp}\left((\frac{3}{4}+o(1))\delta^2n\right)}$$.