On the geometry of uniform meandric systems

A meandric system of size $n$ is the set of loops formed from two arc diagrams (non-crossing perfect matchings) on $\{1,\dots,2n\}$, one drawn above the real line and the other below the real line. Equivalently, a meandric system is a coupled collection of meanders of total size $n$. I will discuss a conjecture which describes the large-scale geometry of a uniformly sampled meandric system of size $n$ in terms of Liouville quantum gravity (LQG) decorated by certain Schramm-Loewner evolution (SLE) type curves. I will then present several rigorous results which are consistent with this conjecture. In particular, a uniform meandric system admits macroscopic loops; and the half-plane version of the meandric system has no infinite paths. No prior knowledge about meanders, SLE, or LQG is needed to understand the talk. Based on joint work with Jacopo Borga and Minjae Park.