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

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 (δ, where δ is the maximum degree and f (δ = θ(δlog2 δ. Their result applies to more general spin systems, and in that generality, they showed that some dependence on δ 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. © 2011 Association des Publications de l'Institut Henri Poincaré.