The mixing time evolution of Glauber dynamics for the mean-field Ising model

Abstract

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 $[2(1-β)]^{-1} n\log n$ 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 $n\to\infty$. 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 $1/\sqrt{n}$ around the critical temperature. In the high temperature regime, $β= 1 - δ$ for some $0 < δ< 1$ so that $δ^2 n \to\infty$ with $n$, the mixing-time has order $(n/δ)\log(δ^2 n)$, and exhibits cutoff with constant 1/2 and window size $n/δ$. In the critical window, $β= 1\pm δ$ where $δ^2 n$ is O(1), there is no cutoff, and the mixing-time has order $n^{3/2}$. At low temperature, $β= 1 + δ$ for $δ> 0$ with $δ^2 n \to\infty$ and $δ=o(1)$, there is no cutoff, and the mixing time has order $(n/δ)\exp(({3/4}+o(1))δ^2 n)$.

43 pages, 2 figures