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

We study a $q$-state Potts model on the square grid when $q>4$ at the point $T_c(q)$ of its (discontinous) transition. This model exhibits exactly $q+1$ extremal Gibbs measures: $q$ ordered (monochromatic) and one disordered (free). The current work deals with the Dobrushin order--disorder boundary conditions on a finite $N\times N$ box. 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'' \`a 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.