On the Gibbs states of the noncritical Potts model on Z^2

TL;DR

This paper proves that all Gibbs states of the noncritical q-state Potts model on Z^2 are convex combinations of pure phases, showing they are translation invariant, with optimal error estimates based on interface fluctuations.

Contribution

It establishes that below the critical temperature, Gibbs states are convex combinations of pure phases, using finite-volume analysis and interface fluctuation scaling.

Findings

Gibbs states are convex combinations of pure phases

All Gibbs states are translation invariant below critical temperature

Finite-volume errors are optimally estimated using Brownian scaling

Abstract

We prove that all Gibbs states of the q-state nearest neighbor Potts model on Z^2 below the critical temperature are convex combinations of the q pure phases; in particular, they are all translation invariant. To achieve this goal, we consider such models in large finite boxes with arbitrary boundary condition, and prove that the center of the box lies deeply inside a pure phase with high probability. Our estimate of the finite-volume error term is of essentially optimal order, which stems from the Brownian scaling of fluctuating interfaces. The results hold at any supercritical value of the inverse temperature.