Mixing time for the Ising model: a uniform lower bound for all graphs

TL;DR

This paper establishes a universal lower bound of approximately (1/4) n log n on the mixing time of Glauber dynamics for the ferromagnetic Ising model across all graphs, improving understanding of convergence rates.

Contribution

It proves a uniform lower bound on the mixing time for Glauber dynamics in the ferromagnetic Ising model applicable to all graphs, regardless of maximum degree.

Findings

Lower bound of (1/4 + o(1)) n log n for mixing time

Universal bound applies to all n-vertex graphs

Improves understanding of convergence rates for Glauber dynamics

Abstract

Consider Glauber dynamics for the Ising model on a graph of $n$ vertices. Hayes and Sinclair showed that the mixing time for this dynamics is at least $n\log n/f(\Delta)$, where $\Delta$ is the maximum degree and $f(\Delta) = \Theta(\Delta \log^2 \Delta)$. Their result applies to more general spin systems, and in that generality, they showed that some dependence on $\Delta$ is necessary. In this paper, we focus on the ferromagnetic Ising model and prove that the mixing time of Glauber dynamics on any $n$-vertex graph is at least $(1/4+o(1))n \log n$.