Near optimal bounds for weak and strong spatial mixing for the anti-ferromagnetic Potts model on trees

TL;DR

This paper establishes near-optimal bounds for weak and strong spatial mixing in the anti-ferromagnetic Potts model on trees, extending previous results to a broader parameter range.

Contribution

It provides new bounds for spatial mixing in the anti-ferromagnetic Potts model on trees, expanding understanding beyond the infinite temperature case.

Findings

Strong spatial mixing holds for a near-optimal parameter range.

Weak spatial mixing results are also established.

Extends previous results from uniform proper colorings to broader settings.

Abstract

We show that the anti-ferromagnetic Potts model on trees exhibits strong spatial mixing for a near-optimal range of parameters. Our work complements recent results of Chen, Liu, Mani, and Moitra [arXiv.2304.01954] who showed this to be true in the infinite temperature setting, corresponding to uniform proper colorings. We furthermore prove weak spatial mixing results complementing results in [arXiv.2304.01954].