Liouville Brownian Motion and Thick Points of the Gaussian Free Field

Understand

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 $1/(2 - \frac{\gamma^2}{2})$.