On support sets of the critical Liouville Quantum Gravity

TL;DR

This paper characterizes the fractal support sets of critical Liouville Quantum Gravity, showing they have zero Hausdorff dimension, and provides a complete description of gauge functions for finite measure support.

Contribution

It fully characterizes gauge functions for support sets of cLQG and confirms the conjecture that cLQG is supported on a set of zero Hausdorff dimension.

Findings

cLQG support sets have zero Hausdorff dimension

Complete characterization of gauge functions for finite measure support

Supports are related to near-critical level sets of Discrete Gaussian Free Field

Abstract

We study fractal properties of support sets of the critical Liouville Quantum Gravity (cLQG) associated with the Gaussian Free Field in planar domains. Specifically, we completely characterize the gauge functions $\phi$ (subject to mild monotonicity conditions) for which the cLQG admits a support set of finite $\phi$-Hausdorff measure. As a corollary, we settle the conjecture that the cLQG is supported on a set of vanishing Hausdorff dimension. Our proofs are based on the fact that the cLQG describes the near-critical level sets of Discrete Gaussian Free Field.