Glauber dynamics for the mean-field Ising model: cut-off, critical power law, and metastability

TL;DR

This paper analyzes the Glauber dynamics for the mean-field Ising model, revealing a sharp cut-off for subcritical temperatures, polynomial mixing at criticality, and metastability phenomena in the supercritical regime.

Contribution

It provides rigorous results on the mixing times and phase transition behaviors of Glauber dynamics in the Curie-Weiss model, including cut-off, critical power law, and metastability.

Findings

Cut-off at beta<1 with window of order n log n

Order n^{3/2} mixing time at critical temperature beta=1

Metastability with O(n log n) mixing time for beta>1

Abstract

We study the Glauber dynamics for the Ising model on the complete graph, also known as the Curie-Weiss Model. For beta < 1, we prove that the dynamics exhibits a cut-off: the distance to stationarity drops from near 1 to near 0 in a window of order n centered at [2(1-beta)]^{-1} n log n. For beta = 1, we prove that the mixing time is of order n^{3/2}. For beta > 1, we study metastability. In particular, we show that the Glauber dynamics restricted to states of non-negative magnetization has mixing time O(n log n).