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arXiv · 2010.11260

Geodesic networks in Liouville quantum gravity surfaces

Ewain Gwynne
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Abstract

Recent work has shown that for $γ\in (0,2)$, 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 $γ$-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.

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Ewain Gwynne. 2021-07-29. Geodesic networks in Liouville quantum gravity surfaces. https://doi.org/10.2140/pmp.2021.2.643

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