Probability and Mathematical Physics Seminar

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Permutons, meanders, and SLE-decorated Liouville quantum gravity

Speaker: Ewain Gwynne, University of Chicago

Location: Warren Weaver Hall 1302

Date: Friday, February 17, 2023, 11:10 a.m.

Synopsis:

A permuton is a probability measure on $[0,1]^2$ whose two coordinate marginals are Lebesgue measure. Permutons describe the large-scale geometry of random permutations. I will discuss a geometric construction of a certain class of random permutons using a pair of random space-filling curves called Schramm-Loewner evolution (SLE) and a random measure arising from Liouville quantum gravity (LQG). This class includes the limits of various types of random pattern-avoiding permutations as well as the conjectural limit of meandric permutations (permutations arising from a simple loop which crosses a line a specified number of times). I will then discuss some results about random permutations which can be proven using SLE and LQG, concerning, e.g., the length of the longest increasing subsequence and the fractal dimension of the support.