Scale-invariant random geometry from mating of trees: a numerical study

TL;DR

This paper numerically investigates higher-dimensional generalizations of Liouville Quantum Gravity via mating of trees, aiming to identify new universality classes of scale-invariant random geometries beyond known models.

Contribution

It develops a numerical method to simulate non-planar random graphs from higher-dimensional correlated Brownian motions, exploring their scaling limits and potential new universality classes.

Findings

Accurately reproduces known 2D Hausdorff dimensions

Provides first estimates of 3D scale-invariant geometries

Suggests existence of a three-parameter family of new geometries

Abstract

The search for scale-invariant random geometries is central to the Asymptotic Safety hypothesis for the Euclidean path integral in quantum gravity. In an attempt to uncover new universality classes of scale-invariant random geometries that go beyond surface topology, we explore a generalization of the mating of trees approach introduced by Duplantier, Miller and Sheffield. The latter provides an encoding of Liouville Quantum Gravity on the 2-sphere decorated by a certain random space-filling curve in terms of a two-dimensional correlated Brownian motion, that can be viewed as describing a pair of random trees. The random geometry of Liouville Quantum Gravity can be conveniently studied and simulated numerically by discretizing the mating of trees using the Mated-CRT maps of Gwynne, Miller and Sheffield. Considering higher-dimensional correlated Brownian motions, one is naturally led to a sequence of non-planar random graphs generalizing the Mated-CRT maps that may belong to new universality classes of scale-invariant random geometries. We develop a numerical method to efficiently simulate these random graphs and explore their possible scaling limits through distance measurements, allowing us in particular to estimate Hausdorff dimensions in the two- and three-dimensional setting. In the two-dimensional case these estimates accurately reproduce previous known analytic and numerical results, while in the three-dimensional case they provide a first window on a potential three-parameter family of new scale-invariant random geometries.