Geodesic networks in Liouville quantum gravity surfaces

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.

arXiv comment40 pages, 10 figures; accepted to PMP

References in corpus (8)

• Uniqueness and universality of the Brownian map
• Liouville quantum gravity and the Brownian map I: The QLE(8/3,0) metric
• The fractal dimension of Liouville quantum gravity: universality, monotonicity, and bounds
• Mating of trees for random planar maps and Liouville quantum gravity: a survey
• Heat kernel for Liouville Brownian motion and Liouville graph distance
• Stability of geodesics in the Brownian map
• The dimension of the boundary of a Liouville quantum gravity metric ball
• Geodesics in the Brownian map: Strong confluence and geometric structure

Cited by in corpus (2)

• Uniqueness of the critical and supercritical Liouville quantum gravity metrics
• Introduction to the Liouville quantum gravity metric