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Conformal invariance of boundary touching loops of FK Ising model

Date: 
Tuesday 27th January 2015 - 15:00 to 16:00
Venue: 
INI Seminar Room 1
Abstract: 
Co-author: Stanislav Smirnov (University of Geneva and St. Petersburg State University)

I will present a result showing the full conformal invariance of Fortuin-Kasteleyn representation of Ising model (FK Ising model) at criticality. The collection 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(?,?-6) with ?=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.

This is a joint work with Stanislav Smirnov (University of Geneva and St. Petersburg State University)

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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons