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.