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

TL;DR

This work proves that the order-disorder interface in the planar $q$-state Potts model with $q>4$ converges to a Brownian bridge under diffusive scaling, in the Dobrushin boundary setup. The authors engineer a novel chain of graphical couplings linking FK-percolation, the six-vertex model, and the Ashkin–Teller random-cluster representation, and develop an Ornstein–Zernike renewal framework for long ATRC clusters. Exponential mixing and renewal structure enable a precise OZ description, which is then transferred to the FK-percolation and Potts interfaces, establishing the scaling limit and fluctuations. A companion paper addresses the order-order Dobrushin case and the emergence of a wetting layer, completing the scaling-branch picture for discontinuous two-dimensional phase transitions and their interface geometry.

Abstract

It is known that the planar q-state Potts model undergoes a discontinuous phase transition when q > 4 and there are exactly q + 1 extremal Gibbs measures at the transition point: q ordered (monochromatic) measures and one disordered (free). We focus on the Potts model under the Dobrushin order-disorder boundary conditions on a finite $N\times N$ part of the square grid. Our main result is that this interface is a well defined object, has $\sqrt{N}$ fluctuations, and converges to a Brownian bridge under diffusive scaling. The same holds also for the corresponding FK-percolation model for all q > 4. Our proofs rely on a coupling between FK-percolation, the six-vertex model, and the random-cluster representation of an Ashkin-Teller model (ATRC), and on a detailed study of the latter. The coupling relates the interface in FK-percolation to a long subcritical cluster in the ATRC model. For this cluster, we develop a ``renewal picture'' à la Ornstein-Zernike. This is based on fine mixing properties of the ATRC model that we establish using the link to the six-vertex model and its height function. Along the way, we derive various properties of the Ashkin-Teller model, such as Ornstein-Zernike asymptotics for its two-point function. In a companion work, we provide a detailed study of the Potts model under order-order Dobrushin conditions. We show emergence of a free layer of width $\sqrt{N}$ between the two ordered phases (wetting) and establish convergence of its boundaries to two Brownian bridges conditioned not to intersect.

Figures (29)

• Figure 1: Sample of a 600x600 Potts model with 25 colours at $T_c(25)$ with Dobrushin boundary conditions. Left: order-disorder interface; upper part has blue b.c., bottom has white b.c. (no colour favoured), colours are: blue for the first and interpolate between yellow and red for colours 2 to 25. Right: colours are: blue for the first, red for the second, and interpolate between yellow and orange for colours 3 to 25.
• Figure 2: Left: wired-free Dobrushin boundary condition. Right: the graph $G_{2}$.
• Figure 3: Left: Tile associated to a mid-edge. Right: Tile centred at the middle of a horizontal primal edge (solid black) or its associated vertical dual edge (dashed black), with its two possible local loop configurations.
• Figure 4: Dobrushin boundary condition for the six-vertex height function.
• Figure 5: Left: the sets $\Lambda_{2,2}$ (solid) and $\Lambda'_{2,2}$ (hollow). Right: the inner tiles $A^\mathrm{i}_{2,2}$ (white) and the boundary tiles $\partial A_{2,2}$ (grey).

Theorems & Definitions (108)

• Theorem 1
• Theorem 2
• Proposition 1.1
• Theorem 3
• Lemma 3.1: PfiVel97
• Proposition 3.2
• Theorem 4
• Corollary 3.3
• Theorem 5
• proof