2019/05/01 by Ewain Gwynne, Gwynne, Ewain, Jason Miller +1
Computer Science · Mathematics · Physics and Astronomy · #Bayesian Methods and Mixture Models #Complex Variables (math.CV) #Computer science #FOS: Mathematics #FOS: Physical sciences #Fixed point #Geodesic #Linear-quadratic-Gaussian control #Mathematical Physics (math-ph) #Mathematical analysis #Mathematical physics #Mathematics #Merge (version control) #Optimal control #Physics #Probability (math.PR) #Pure mathematics #Quantum #Quantum gravity #Quantum mechanics #Random Matrices and Applications #Stochastic processes and statistical mechanics #Uniqueness #math-ph #math.CV #math.MP #math.PR
paper · pdf · doi:10.48550/arxiv.1905.00381
44 pages, 9 figures; to appear in Annals of Probability
openalex publication_date 2019/05/01 · arxiv created 2020/02/01 · arxiv updated 2020/02/04 · openalex created_date 2022/07/29 · openalex updated_date 2026/08/05
We prove that for any metric which one can associate with a Liouville quantum gravity (LQG) surface for γ∈ (0,2) satisfying certain natural axioms, its geodesics exhibit the following confluence property. For any fixed point z, a.s. any two γ-LQG geodesics started from distinct points other than z must merge into each other and subsequently coincide until they reach z. This is analogous to the confluence of geodesics property for the Brownian map proven by Le Gall (2010). Our results apply for the subsequential limits of Liouville first passage percolation and are an important input in the proof of the existence and uniqueness of the LQG metric for all γ∈ (0,2).