[15] On the geometry of uniform meandric systems (with Ewain Gwynne and Minjae Park).

Preprints / By

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. A uniform random meandric system can be viewed as a random planar map decorated by a Hamiltonian path (corresponding to the real line) and a collection of loops (formed by the arcs).

Based on physics heuristics and numerical evidence, we conjecture that the scaling limit of this decorated random planar map is given by an independent triple consisting of a Liouville quantum gravity (LQG) sphere with parameter \(\gamma=\sqrt{2}\), a Schramm-Loewner evolution (SLE) curve with parameter \(\kappa=8\), and a conformal loop ensemble (CLE) with parameter \(\kappa=6\). 

We prove several rigorous results which are consistent with this conjecture. In particular, a uniform meandric system admits loops of nearly macroscopic graph-distance diameter with high probability. Furthermore, a.s., the uniform infinite meandric system with boundary has no infinite path. But, a.s., its boundary-modified version has a unique infinite path whose scaling limit is conjectured to be chordal SLE\(_6\).

arXiv

Simulation of a uniform meandric system with boundary of size 1000000 (see Section 7.4 of the papervfor a precise definition and for the details of simulations). The left picture shows the corresponding arc diagrams. The right picture shows the associated planar map, embedded in the disk via the Tutte embedding, together with some of the loops of the meandric system. The largest 300 loops (in terms of number of vertices) are each shown in color, as indicated by the color bar. Smaller loops and edges between consecutive vertices of the real line are shown in gray. Note that the distribution of colors in the arc diagram picture is rather chaotic – this is consistent with the fact that the meandric system loops are a complicated functional of the arc diagrams. According to our conjectures, the embedded planar map together with the path induced by the real line, and the loops, should converge to √ 2-LQG decorated by SLE8 and CLE6.