Abstract: The behaviour of interfaces in lattice spin systems and their graphical representations is of multiple interest. Not only do they characterise the phases of the system, their fluctuations play an important role in the study of non-translation-invariant Gibbs measures, while more refined geometric properties and their scaling limits have received enormous attention in the last decades.

We will discuss these matters for the Potts model and FK-percolation on the square lattice. After introducing these models and briefly reviewing their relation and phase transitions, we will give an overview of results and conjectures concerning the corresponding interfaces. We will then present the rough idea of a proof of an invariance principle in the ordered regime of the Ising model, relying on the celebrated Ornstein–Zernike theory. Via couplings with the six-vertex and Ashkin–Teller models, we explain how to make use of these methods to obtain a related invariance principle and verify the so-called wetting phenomenon in the Potts model and FK-percolation at the transition point when the transition is discontinuous.

The second part of the talk is based on joint work with Alexander Glazman and Sébastien Ott.