Conformal Invariance of FK Ising loop ensemble

Stochastic Analysis Seminar Series

Abstract: In this talk I’ll describe some recent progress in the study of random geometry of interfaces in statistical physics models. The conformal invariance of an interface in spin Ising and FK Ising models in 2D was established in a sequence of results by Smirnov, Chelkak&Smirnov and Chelkak&al. I will present a result showing the full conformal invariance (i.e., of multiple interfaces) of Fortuin-Kasteleyn representation of Ising model (FK Ising model) at criticality. The loop ensemble of all the interfaces, which in a planar model are closed loops, in the FK Ising model at criticality defined on a lattice approximation of a planar domain is shown to converge to a conformally invariant scaling limit as the mesh size is decreased. More specifically, the scaling limit can be described using a branching SLE(kappa,kappa-6) with kappa=16/3, a variant of Oded Schramm's SLE curves. We consider the exploration tree of the loop collection and the main step of the proof is to find a discrete holomorphic observable which is a martingale for the branch of the exploration tree.

Location:
Speaker(s):

Antti Kemppainen (Helsinki University)

Date:
Monday, April 27, 2015 - 16:15
to 17:15