## EVENT DETAILS AND ABSTRACT

**Analysis Seminar**
**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.

**Date:** Monday, October 12, 2015

**Time:** 4:00pm

**Where:** Lunt 103

**Contact Person:** Prof. Jared Wunsch

**Contact email:** jwunsch@math.northwestern.edu

**Contact Phone:** 847-491-5580

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