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

In this paper, the Glauber dynamics for the Ising model on the complete multipartite graph K-np1, ...,(npm) is investigated where 0 < p(i) < 1 is the proportion of the vertices in the ith component and m is an element of N is the fixed number of components. We show that the dynamics exhibits the cutoff phenomena at t(n) := 1/2(1 - beta/beta(cr))n In 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), ...,p(m). Exponentially slow mixing is shown in the low temperature regime beta > beta(cr).