Cutoff phenomenon of the Glauber dynamics for the Ising model on complete multipartite graphs in the high temperature regime

Abstract

In this paper, the Glauber dynamics for the Ising model on the complete multipartite graph $K_{np_1,\dots,np_m}$ is investigated where $0<p_i<1$ is the proportion of the vertices in the $i$th component. We show that the dynamics exhibits the cutoff phenomena at $t_n = \frac{1}{2(1-\beta/\beta_{cr})} n\ln n $ with window size $O(n)$ in the high temperature regime $\beta< \beta_{cr}$ where $\beta_{cr}$ is a constant only depending on $p_1,\dots,p_m$. Exponentially slow mixing is shown in the low temperature regime $\beta>\beta_{cr}$.