Geodesic distances in Liouville quantum gravity

TL;DR

This paper introduces a new way to measure distances in Liouville quantum gravity, uses it to numerically estimate the Hausdorff dimension, and compares results with theoretical predictions and random triangulations.

Contribution

It proposes a novel definition of geodesic distance in Liouville quantum gravity and applies it to numerically analyze the quantum geometry of random surfaces.

Findings

Agreement with Watabiki's conjectured formula for Hausdorff dimension

Numerical validation of geodesic distances in quantum geometry

Comparison of cycle length distributions with random triangulations

Abstract

In order to study the quantum geometry of random surfaces in Liouville gravity, we propose a definition of geodesic distance associated to a Gaussian free field on a regular lattice. This geodesic distance is used to numerically determine the Hausdorff dimension associated to shortest cycles of 2d quantum gravity on the torus coupled to conformal matter fields, showing agreement with a conjectured formula by Y. Watabiki. Finally, the numerical tools are put to test by quantitatively comparing the distribution of lengths of shortest cycles to the corresponding distribution in large random triangulations.