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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 $n\log n/f(Δ)$, where $Δ$ is the maximum degree and $f(Δ) = Θ(Δ\log^2 Δ)$. 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 $(1/4+o(1))n \log n$.