The effect of boundary conditions on mixing of 2D Potts models at discontinuous phase transitions

TL;DR

This paper investigates how boundary conditions influence the mixing times of the Swendsen--Wang dynamics for the 2D Potts model at discontinuous phase transitions, revealing a spectrum of effects between boundary types.

Contribution

It classifies the impact of various boundary conditions on mixing times for the Potts model at discontinuous transitions, extending understanding beyond the known extreme cases.

Findings

Mixed boundary conditions induce exponential mixing times.

Dobrushin and periodic boundary conditions lead to sub-exponential mixing.

Boundary conditions critically affect the dynamics at phase transitions.

Abstract

We study Swendsen--Wang dynamics for the critical $q$-state Potts model on the square lattice. For $q=2,3,4$, where the phase transition is continuous, the mixing time $t_{\textrm{mix}}$ is expected to obey a universal power-law independent of the boundary conditions. On the other hand, for large $q$, where the phase transition is discontinuous, the authors recently showed that $t_{\textrm{mix}}$ is highly sensitive to boundary conditions: $t_{\textrm{mix}} \geq \exp(cn)$ on an $n\times n$ box with periodic boundary, yet under free or monochromatic boundary conditions, $t_{\textrm{mix}} \leq\exp(n^{o(1)})$. In this work we classify this effect under boundary conditions that interpolate between these two (torus vs. free/monochromatic). Specifically, if one of the $q$ colors is red, mixed boundary conditions such as red-free-red-free on the 4 sides of the box induce $t_{\textrm{mix}} \geq \exp(cn)$, yet Dobrushin boundary conditions such as red-red-free-free, as well as red-periodic-red-periodic, induce sub-exponential mixing.