EVENT DETAILS AND ABSTRACT

Title: Scaling limit of the topological structure of critical Fortuin-Kasteleyn planar maps
Speaker: Ewain Gwynne
Speaker Info: MIT
Brief Description:
Special Note:
Abstract:

A critical Fortuin-Kasteleyn (FK) planar map of size n with parameter q > 0 is a random a planar map with n edges decorated by a collection of loops, sampled from the uniform measure on such objects weighted by q^{K/2}, where K is the number of loops. It is conjectured that the scaling limit of the critical FK planar map is a Liouville quantum gravity (LQG) surface decorated by an independent conformal loop ensemble (CLE_\kappa). In this talk, we give an overview of the proof that this scaling limit occurs in a certain topology, assuming only basic background in probability theory. In particular, we introduce a structure called a lamination which encodes all of the topological information about a collection of loops and a measure on the plane and show that the lamination of a critical FK planar map converges in distribution to the lamination of a CLE on an LQG surface. Our proof uses a bijective encoding of critical FK planar maps due to Sheffield (2011) and an analogous encoding of a CLE on an LQG surface due to Duplantier, Miller, and Sheffield (2014). As an application, we obtain that the law of whole-plane CLE_\kappa for \kappa \in (4,8) is invariant under inversion. This talk is based on joint works with various subsets of Cheng Mao, Jason Miller, and Xin Sun.