Liouville Brownian Motion and Thick Points of the Gaussian Free Field

TL;DR

This paper establishes bounds on the Hausdorff dimension of the set of points where Liouville Brownian motion spends time within thick points of the Gaussian Free Field, revealing multifractal properties and diffusivity behavior.

Contribution

It completes a conjecture by providing a lower bound and estimates on diffusivity exponents, deepening understanding of Liouville Brownian motion's multifractal structure.

Findings

Lower bound for Hausdorff dimension in thick points

Diffusivity exponent depends on starting point

Path is differentiable almost everywhere for certain parameters

Abstract

We find a lower bound for the Hausdorff dimension that a Liouville Brownian motion spends in $\alpha$-thick points of the Gaussian Free Field, where $\alpha$ is not necessarily equal to the parameter used in the construction of the geometry. This completes a conjecture in \cite{berestycki2013diffusion}, where the corresponding upper bound was shown. In the course of the proof, we obtain estimates on the (Euclidean) diffusivity exponent, which depends strongly on the nature of the starting point. For a Liouville typical point, it is $1/(2 - \frac{\gamma^2}{2})$. In particular, for $\gamma > \sqrt{2}$, the path is Lebesgue-almost everywhere differentiable, almost surely. This provides a detailed description of the multifractal nature of Liouville Brownian motion.