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 ∈ D is given by exp(γ h(z)), appropriately normalised. Here h is an instance of the Gaussian free field on D and γ ∈ (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. Similar but deeper results have been independently and simultaneously proved by Garban, Rhodes and Vargas.