Liouville quantum gravity spheres as matings of finite-diameter trees

TL;DR

This paper presents multiple equivalent constructions of the Liouville quantum gravity sphere, including methods using Bessel excursions, correlated Brownian loops, and stable Lévy processes, linking quantum gravity with random planar maps.

Contribution

It introduces new, equivalent ways to construct the LQG sphere, especially for b3=9/3, connecting it with the Brownian map and SLE processes.

Findings

Multiple equivalent constructions of LQG spheres.

Connection between b3=9/3 LQG sphere and Brownian map.

Use of stable Lévy processes to produce quantum disks.

Abstract

We show that the unit area Liouville quantum gravity sphere can be constructed in two equivalent ways. The first, which was introduced by the authors and Duplantier, uses a Bessel excursion measure to produce a Gaussian free field variant on the cylinder. The second uses a correlated Brownian loop and a "mating of trees" to produce a Liouville quantum gravity sphere decorated by a space-filling path. In the special case that $\gamma=\sqrt{8/3}$, we present a third equivalent construction, which uses the excursion measure of a $3/2$-stable L\'evy process (with only upward jumps) to produce a pair of trees of quantum disks that can be mated to produce a sphere decorated by SLE$_6$. This construction is relevant to a program for showing that the $\gamma=\sqrt{8/3}$ Liouville quantum gravity sphere is equivalent to the Brownian map.