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July 2020 Confluence of geodesics in Liouville quantum gravity for $\gamma \in (0,2)$
Ewain Gwynne, Jason Miller
Ann. Probab. 48(4): 1861-1901 (July 2020). DOI: 10.1214/19-AOP1409


We prove that for any metric, which one can associate with a Liouville quantum gravity (LQG) surface for $\gamma \in (0,2)$ satisfying certain natural axioms, its geodesics exhibit the following confluence property. For any fixed point $z$, a.s. any two $\gamma $-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 (Acta Math. 205 (2010) 287–360). 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 $\gamma \in (0,2)$.


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Ewain Gwynne. Jason Miller. "Confluence of geodesics in Liouville quantum gravity for $\gamma \in (0,2)$." Ann. Probab. 48 (4) 1861 - 1901, July 2020.


Received: 1 July 2019; Revised: 1 October 2019; Published: July 2020
First available in Project Euclid: 20 July 2020

zbMATH: 07224962
MathSciNet: MR4124527
Digital Object Identifier: 10.1214/19-AOP1409

Primary: 60G52 , 60J67

Keywords: confluence of geodesics , Gaussian free field , Liouville first passage percolation , Liouville quantum gravity , LQG metric

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.48 • No. 4 • July 2020
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