Field dynamics: a new tool to boost mixing results

We study the mixing time of the Glauber dynamics for anti-ferromagnetic two-spin systems with n vertices in the tree uniqueness regime. Specifically, we prove the following results.

1. An optimal Ω(1/n)lower bound for the spectral gap, which implies the O(n^3) mixing time.
2. An optimal Ω(1/n)lower bound for the modified log-Sobolev constant if the spin system further satisfies a marginal ratio bound, which implies the optimal O(n log n) mixing time.

For applications, we prove the first optimal O(n log n)  mixing time for the hardcore model and Ising model in the tree uniqueness regime, even if the maximum degree of the underlying graph is unbounded.

The key ingredient of our proof is a novel Markov chain called the field dynamics, which can be viewed as an adaptive variant of the block dynamics. The field dynamics makes a new connection between distributions in sub-critical regimes and distributions in near-critical regimes. With this new Markov chain, we prove two boosting theorems for the Glauber dynamics: one boosts the spectral gap and the other boosts the modified log-Sobolev constant.

This talk is based on three joint works (arXiv:2105.15005, arXiv:2111.03034 and arXiv:2203.07771) with Xiaoyu Chen, Yitong Yin and Xinyuan Zhang. The first work appeared in FOCS 2021.