Cutoff Phenomenon of the Glauber Dynamics for the Ising Model on Complete Multipartite Graphs in the High Temperature Regime

#### Heejune Kim

2022, v.28, Issue 1, 113-148

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 and $m\in \mathbb N$ is the fixed number of components.

We show that the dynamics exhibits the cutoff phenomena at $t_n \colonequals \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 $\beta>\beta_{cr}$.

Keywords: Markov chains, Ising model, mixing time, cutoff, coupling, Glauber dynamics, Heat-bath dynamics, mean-ﬁeld model

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