Optimal Mixing Time for the Ising Model in the Uniqueness Regime

Abstract

We prove an optimal $O(n\log n)$ mixing time of the Glauber dynamics for the Ising models with edge activity $\beta \in \left(\frac{\Delta -2}{\Delta },\frac{\Delta }{\Delta -2}\right)$. This mixing time bound holds even if the maximum degree $\Delta$ is unbounded. We refine the boosting technique developed in [CFYZ21], and prove a new boosting theorem by utilizing the entropic independence defined in [AJK+21]. The theorem relates the modified log-Sobolev (MLS) constant of the Glauber dynamics for a near-critical Ising model to that for an Ising model in a sub-critical regime.

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• The computational hardness of counting in two-spin models on d-regular graphs