Spectral dimension of Liouville quantum gravity

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.

18 pages

References in corpus (5)

• Uniqueness and universality of the Brownian map
• The topological structure of scaling limits of large planar maps
• KPZ in one dimensional random geometry of multiplicative cascades
• Gaussian multiplicative chaos and KPZ duality
• Another derivation of the geometrical KPZ relations

Cited by in corpus (6)

• The fractal dimension of Liouville quantum gravity: universality, monotonicity, and bounds
• KPZ formula derived from Liouville heat kernel
• Heat kernel for Liouville Brownian motion and Liouville graph distance
• Liouville heat kernel: regularity and bounds
• Precision measurements of Hausdorff dimensions in two-dimensional quantum gravity
• Return probability and recurrence for the random walk driven by two-dimensional Gaussian free field