Systematic scanning Glauber dynamics for the mean-field Ising model

TL;DR

This paper analyzes the mixing times of systematic scan Glauber dynamics for the mean-field Ising model on complete graphs, revealing cutoff phenomena and precise mixing time orders across different temperature regimes.

Contribution

It provides the first detailed analysis of systematic scan Glauber dynamics on the mean-field Ising model, including cutoff results and mixing time estimates for various temperature regimes.

Findings

Cutoff occurs for $k = o(n^{1/3})$ in high temperature regime.

Mixing time is of order $n^{3/2}k^{-1}$ at critical temperature $eta=1$.

Mixing time is of order $nk^{-1} ext{log} n$ for $eta > 1$.

Abstract

We study the mixing time of systematic scan Glauber dynamics Ising model on the complete graph. On the complete graph $K_n$, at each time, $k \leq n$ vertices are chosen uniformly random and are updated one by one according to the uniformly randomly chosen permutations over the $k$ vertices. We show that if $k = o(n^{1/3})$, the high temperature regime $\beta < 1$ exhibits cutoff phenomena. For critical temperature regime $\beta = 1$, We prove that the mixing time is of order $n^{3/2}k^{-1}$. For $\beta > 1$, we prove the mixing time is of order $nk^{-1}\log n$ under the restricted dynamics.