Liouville Brownian motion and quantum cones in dimension $d > 2$

TL;DR

This paper studies Liouville Brownian motion in higher dimensions, computes its spectral dimension, relates Gaussian fields to Markov processes, constructs quantum cones, and proves invariance properties under Brownian motion shifts.

Contribution

It extends Liouville quantum gravity concepts to dimensions greater than two, introducing higher-dimensional quantum cones and analyzing their properties.

Findings

Spectral dimension depends on both 3 and starting point thickness.

Spherical average process identified with a Gaussian Markov process.

Law of the b1-quantum cone is shift-invariant under Liouville Brownian motion.

Abstract

For $d > 2$ and $\gamma \in (0, \sqrt{2d})$, we study the Liouville Brownian motion associated with the whole-space log-correlated Gaussian field in $\mathbb{R}^d$. We compute its spectral dimension, i.e., the short-time asymptotics of the heat kernel along the diagonal, which, in contrast to the two-dimensional case, depends on both $\gamma$ and on the thickness of the starting point. Furthermore, for even dimensions $d > 2$, we show that the spherical average process of the whole-space log-correlated Gaussian field in $\mathbb{R}^d$ can be identified with the integral of a stationary Gaussian Markov process of order $(d-2)/2$. Exploiting this representation, we construct the higher-dimensional analogue of the $\beta$-quantum cone for $\beta \in (-\infty, Q)$, with $Q = d/\gamma + \gamma/2$. Lastly, for $\alpha = Q - \sqrt{Q^2-4}$, we prove that the law of the $d$-dimensional $\alpha$-quantum cone is invariant under shifts along the trajectories of the associated Liouville Brownian motion.