2020/10/31 by Ewain Gwynne
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Combinatorics #Geodesic #Geometry #Geometry and complex manifolds #Linear-quadratic-Gaussian control #Mathematical analysis #Mathematical optimization #Mathematical physics #Mathematics #Metric (unit) #Optimal control #Physics #Point (geometry) #Pure mathematics #Quantum #Quantum gravity #Quantum mechanics #Stochastic processes and statistical mechanics #Surface (topology) #Topology (electrical circuits) #math-ph #math.MP #math.PR
paper · pdf · doi:10.2140/pmp.2021.2.643
published as Prob. Math. Phys. 2 (2021) 643-684 · 40 pages, 10 figures; accepted to PMP
arxiv created 2021/07/29 · openalex publication_date 2021/10/15 · arxiv updated 2021/11/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Recent work has shown that for γ∈ (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∈\mathbb N such that almost surely any two points are joined by at most m distinct LQG geodesics.