Discontinuous transition in 2D Potts: II. Order-Order Interface convergence

TL;DR

This paper rigorously analyzes the geometric behavior of order-order interfaces in the 2D Potts model at the discontinuous transition point, revealing Brownian motion convergence and entropic repulsion phenomena.

Contribution

It establishes the first geometric proof of wetting and interface convergence for the Potts model at q>4, extending previous surface tension results.

Findings

Disordered layer boundaries converge to non-intersecting Brownian motions.

Wetting phenomenon is precisely characterized at the critical temperature.

Interfaces are coupled to non-intersecting random walks with entropic repulsion.

Abstract

The $q$-state Potts model is an archetypical model for various types of phase transitions. We consider it on the square grid and focus on the regime where it undergoes a discontinuous transition, that is $q>4$. At the transition point $T_c(q)$, there are exactly $q+1$ extremal Gibbs measures (pure phases): $q$ ordered (monochromatic) and one disordered (free). This work establishes for the first time the wetting phenomenon in a precise geometric form and in the entire regime of discontinuity $q>4$: at $T_c(q)$, between two ordered phases a disordered layer emerges and, in the diffusive scaling, its boundaries converge to a pair of Brownian motions conditioned not to intersect. This is starkly different from the subcritical ($T<T_c(q)$) behaviour. At $T_c(q)$, previous results (Bricmont--Lebowitz '87, Messager--Miracle-Sole--Ruiz--Shlosman '91) were limited to the construction and properties of the surface tension for large enough $q$. In a companion work, arXiv:2502.04129, we provide a detailed study of the Potts model under order-disorder Dobrushin conditions. That work also develops a ``renewal picture'' \`a la Ornstein-Zernike for a suitable percolation model, which plays a central part in our study of the Potts interfaces. The latter is the random-cluster representation of an Ashkin--Teller model (ATRC), and is related to the Potts model via a chain of couplings going through the six-vertex model. In the current work, we extend the analysis to a pair of interacting order-disorder interfaces forming the separation between the two ordered phases, and couple them to a pair of well-behaved random walks conditioned not to intersect. The construction of the coupling is based on rigorously deriving entropic repulsion between the two interfaces. We also prove convergence of interfaces in the FK-percolation model at $p_c(q)$ when $q>4$.