Customizing the Mixing Time Using the p-Spin Ising Model

Abstract

We will analyze the mixing time behavior for the dynamics of the Ising model and the p-spin Ising model, the latter which is a variant where spins of various subsets of K_n will determine the dynamics. We will see that the drift function of the birth-and-death chain of the magnetization of the Ising model and p-spin Ising model dynamics on K_n plays a large role in determining the mixing time. We will show that the p-spin Ising model admits a mixing time of n*log n with cutoff, and a mixing time n^a for any 1< a<2. Particularly, if the drift function of the p-spin Ising magnetization function has highest power contact k, then the mixing time is O(n^{2-2/(k+1)}).