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

TL;DR

This paper analyzes the cutoff phenomenon of Glauber dynamics for the Ising model on complete multipartite graphs, revealing a sharp transition at high temperature and slow mixing at low temperature.

Contribution

It establishes the cutoff time and window for Glauber dynamics on multipartite graphs in the high temperature regime, extending understanding of mixing times in these models.

Findings

Cutoff occurs at t_n = (1/2(1 - β/β_cr)) n log n in high temperature.

Window size for cutoff is O(n).

Exponential slow mixing in low 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 $\beta>\beta_{cr}$.