Geodesic networks in Liouville quantum gravity surfaces

TL;DR

This paper studies geodesic networks in Liouville quantum gravity surfaces, classifying their structures and showing that any two points are connected by a bounded number of geodesics, extending known results from Brownian maps.

Contribution

It provides a complete classification of geodesic networks in LQG surfaces and establishes a uniform bound on the number of geodesics between points, advancing the understanding of LQG geometry.

Findings

Classified all possible geodesic networks from a typical point to any point.

Identified dense sets of point pairs with specific geodesic network types.

Proved a uniform bound on the number of geodesics connecting any two points.

Abstract

Recent work has shown that for $\gamma \in (0,2)$, a Liouville quantum gravity (LQG) surface can be endowed with a canonical metric. We prove several results concerning geodesics for this metric. In particular, we completely classify the possible networks of geodesics from a typical point on the surface to an arbitrary point on the surface, as well as the types of networks of geodesics joining two points which occur for a dense set of pairs of points on the surface. This latter result is the $\gamma$-LQG analog of the classification of geodesic networks in the Brownian map due to Angel, Kolesnik, and Miermont (2017). We also show that there is a deterministic $m\in\mathbb N$ such that almost surely any two points are joined by at most $m$ distinct LQG geodesics.