Geodesics and metric ball boundaries in Liouville quantum gravity

TL;DR

This paper proves a confluence property for geodesics in Liouville quantum gravity, establishes zero-one laws for Hausdorff dimensions of key structures, and provides formulas for the boundary dimensions of metric balls in LQG surfaces.

Contribution

It generalizes geodesic confluence results to LQG, derives Hausdorff dimension laws, and connects boundary dimensions to the overall LQG surface.

Findings

Strong confluence property for LQG geodesics.

Zero-one laws for Hausdorff dimensions of geodesics and boundaries.

Dimension formulas for metric ball boundaries in LQG.

Abstract

Recent works have shown that there is a canonical way to to assign a metric (distance function) to a Liouville quantum gravity (LQG) surface for any parameter $\gamma \in (0,2)$. We establish a strong confluence property for LQG geodesics, which generalizes a result proven by Angel, Kolesnik and Miermont for the Brownian map. Using this property, we also establish zero-one laws for the Hausdorff dimensions of geodesics, metric ball boundaries, and metric nets w.r.t. the Euclidean or LQG metric. In the case of a metric ball boundary, our result combined with earlier work of Gwynne (2020) gives a formula for the a.s. Hausdorff dimension for the boundary of the metric ball stopped when it hits a fixed point in terms of the Hausdorff dimension of the whole LQG surface. We also show that the Hausdorff dimension of the metric ball boundary is carried by points which are not on the boundary of any complementary connected component of the ball.