Heat kernels on 2d Liouville quantum gravity: a numerical study

TL;DR

This paper numerically investigates the heat kernel on 2D Liouville quantum gravity, confirming spectral dimension 2 and revealing superdiffusive behavior that suggests Euclidean metrics may not accurately describe LQG geometry.

Contribution

It provides the first numerical analysis of heat kernels on LQG, demonstrating spectral dimension and superdiffusive scaling, challenging Euclidean geometric assumptions.

Findings

Spectral dimension of LQG is 2

Superdiffusive space-time scaling observed

Euclidean metric may not be suitable for LQG geometry

Abstract

We numerically compute the heat kernel on a square lattice torus equipped with the measure corresponding to Liouville quantum gravity (LQG). From the on-diagonal heat kernel we verify that the spectral dimension of LQG is 2. Furthermore, when diffusion is started from a high point of the underlying Gaussian free field, our numerics indicates superdiffusive space-time scaling with respect to the Euclidean metric in the small space-to-time regime. The implications of this result require further investigation, but seem to coincide with the notion that the Euclidean metric is not the right geodesic for characterizing the geometry of LQG.