Roughness of geodesics in Liouville quantum gravity

TL;DR

This paper proves that geodesics in Liouville quantum gravity have a Hausdorff dimension greater than 1, confirming a longstanding conjecture, by establishing geometric bounds using properties of the Gaussian free field.

Contribution

It confirms the conjecture that LQG geodesics have Hausdorff dimension greater than 1 for all parameters, using a new criterion and regularity properties of the GFF.

Findings

Hausdorff dimension of LQG geodesics exceeds 1

Established geometric crossing bounds for LQG geodesics

Proved analogous results for LFPP geodesics

Abstract

The metric associated with the Liouville quantum gravity (LQG) surface has been constructed through a series of recent works and several properties of its associated geodesics have been studied. In the current article we confirm the folklore conjecture that the Euclidean Hausdorff dimension of LQG geodesics is stirctly greater than 1 for all values of the so-called Liouville first passage percolation (LFPP) parameter $\xi$. We deduce this from a general criterion due to Aizenman and Burchard which in our case amounts to near-geometric bounds on the probabilities of certain crossing events for LQG geodesics in the number of crossings. We obtain such bounds using the axiomatic characterization of the LQG metric after proving a special regularity property for the Gaussian free field (GFF). We also prove an analogous result for the LFPP geodesics.