Spectral dimension of Liouville quantum gravity

TL;DR

This paper calculates the spectral dimension of 2D-Liouville quantum gravity, showing it is 2, and also finds it is 1 for boundary cases, using advanced probabilistic methods and heat kernel analysis.

Contribution

It provides the first rigorous computation of the spectral dimension for 2D-Liouville quantum gravity, confirming physicists' predictions with new mathematical techniques.

Findings

Spectral dimension is 1 for boundary Liouville quantum gravity.

Spectral dimension is 2 for 2D-Liouville quantum gravity.

Uses Gaussian multiplicative chaos and heat kernel analysis.

Abstract

This paper is concerned with computing the spectral dimension of 2d-Liouville quantum gravity. As a warm-up, we first treat the simple case of boundary Liouville quantum gravity. We prove that the spectral dimension is 1 via an exact expression for the boundary Liouville Brownian motion and heat kernel. Then we treat the 2d-case via a decomposition of time integral transforms of the Liouville heat kernel into Gaussian multiplicative chaos of Brownian bridges. We show that the spectral dimension is 2 in this case, as announced by physicists (see Ambj\orn and al. in \cite{amb}) fifteen years ago.