Diffusion in planar Liouville quantum gravity

Abstract

We construct the natural diffusion in the random geometry of planar Liouville quantum gravity. Formally, this is the Brownian motion in a domain $D$ of the complex plane for which the Riemannian metric tensor at a point $z \in D$ is given by $\exp (γh(z) - \frac12 γ^2 \E (h(z)^2))$. Here $h$ is an instance of the Gaussian Free Field on $D$ and $γ\in (0,2)$ is a parameter. We show that the process is almost surely continuous and enjoys certain conformal invariance properties. We also estimate the Hausdorff dimension of times that the diffusion spends in the thick points of the Gaussian Free Field, and show that it spends Lebesgue-almost all its time in the set of $γ$-thick points, almost surely. The diffusion is constructed by a limiting procedure after regularisation of the Gaussian Free Field. The proof is inspired by arguments of Duplantier--Sheffield for the convergence of the Liouville quantum gravity measure, previous work on multifractal random measures, and relies also on estimates on the occupation measure of planar Brownian motion by Dembo, Peres, Rosen and Zeitouni. A similar but deeper result has been independently and simultaneously proved by Garban, Rhodes and Vargas.