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 \(nlogn/f(\Delta )\), 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 \((1/4+o(1))nlogn\).