Censored Glauber Dynamics for the mean field Ising Model

Abstract

We study Glauber dynamics for the Ising model on the complete graph on $n$ vertices, known as the Curie-Weiss Model. It is well known that at high temperature ($β< 1$) the mixing time is $Θ(n\log n)$, whereas at low temperature ($β> 1$) it is $\exp(Θ(n))$. Recently, Levin, Luczak and Peres considered a censored version of this dynamics, which is restricted to non-negative magnetization. They proved that for fixed $β> 1$, the mixing-time of this model is $Θ(n\log n)$, analogous to the high-temperature regime of the original dynamics. Furthermore, they showed \emph{cutoff} for the original dynamics for fixed $β<1$. The question whether the censored dynamics also exhibits cutoff remained unsettled. In a companion paper, we extended the results of Levin et al. into a complete characterization of the mixing-time for the Currie-Weiss model. Namely, we found a scaling window of order $1/\sqrt{n}$ around the critical temperature $β_c=1$, beyond which there is cutoff at high temperature. However, determining the behavior of the censored dynamics outside this critical window seemed significantly more challenging. In this work we answer the above question in the affirmative, and establish the cutoff point and its window for the censored dynamics beyond the critical window, thus completing its analogy to the original dynamics at high temperature. Namely, if $β= 1 + δ$ for some $δ> 0$ with $δ^2 n \to \infty$, then the mixing-time has order $(n / δ)\log(δ^2 n)$. The cutoff constant is $(1/2+[2(ζ^2 β/ δ- 1)]^{-1})$, where $ζ$ is the unique positive root of $g(x)=\tanh(βx)-x$, and the cutoff window has order $n / δ$.