Speaker: Dr. Minjae Park (University of Chicago). 

​Title: Geometry of uniform meandric systems 

 

 

Abstract: I will discuss how a random geometry perspective can provide new insights into classical combinatorial objects, using meandric systems as an example. A meandric system of size n consists of loops formed by two arc diagrams—non-crossing perfect matchings on {1,…,2n}—with one drawn above and the other below the real line. Equivalently, it is a coupled collection of meanders with a total size of n. I will present a conjecture describing the large-scale geometry of a uniformly sampled meandric system of size n in terms of Liouville quantum gravity (LQG) surfaces decorated by Schramm-Loewner evolution (SLE)-type curves. I will then outline several rigorous results supporting this conjecture. In particular, a uniform meandric system exhibits macroscopic loops; and its half-plane version has no infinite paths.

 

The results are based on joint work with Jacopo Borga (MIT) and Ewain Gwynne (UChicago). ​

 

​If time permits, I will also discuss connections between this approach and other probabilistic structures, such as permutons (limits of permutations), the directed landscape, and other conformal field theory models. ​