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

43 pages, 2 figures