On the geometry of uniform meandric systems

TL;DR

This paper investigates the geometric properties of uniform meandric systems, conjectures their scaling limits relate to Liouville quantum gravity, SLE, and CLE, and proves several results consistent with these conjectures.

Contribution

It provides rigorous results on the geometry of uniform meandric systems and supports conjectures linking them to LQG, SLE, and CLE in the scaling limit.

Findings

Loops of nearly macroscopic diameter exist with high probability.

The infinite meandric system with boundary has no infinite path.

Boundary-modified system has a unique infinite path conjectured to be SLE_6.

Abstract

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$_6$.