From conformal invariance of parafermionic observables to conformal invariance of interfaces of planar random-cluster models

Hugo Duminil - Copin , University of Geneva
Jadwin Hall A06

In this talk we will explain how the determination of the scaling limit of parafermionic observables can be used to deduce the conformal invariance of interfaces in planar random-cluster models with cluster-weight 1 ≤ Q ≤ 4 (1\leq Q \leq 4). The strategy was introduced in the context of the loop-erased random walk by Lawler-Schramm-Werner, and was implemented for the FK Ising model (a.k.a. the random-cluster model with cluster-weight equal to 2) by Chelkak, Duminil-Copin, Hongler, Kemppainen and Smirnov, based on a main contribution of Smirnov.