Mixing Time For The Ising Model: A Uniform Lower Bound For All Graphs

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 nlogn/f(Δ), where Δ is the maximum degree and f(Δ)=Θ(Δlog2Δ). 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))nlogn.