Annals of Probability
- Ann. Probab.
- Volume 48, Number 4 (2020), 1861-1901.
Confluence of geodesics in Liouville quantum gravity for $\gamma \in (0,2)$
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 $\gamma \in (0,2)$.
Ann. Probab., Volume 48, Number 4 (2020), 1861-1901.
Received: July 2019
Revised: October 2019
First available in Project Euclid: 20 July 2020
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60J67: Stochastic (Schramm-)Loewner evolution (SLE) 60G52: Stable processes
Gwynne, Ewain; Miller, Jason. Confluence of geodesics in Liouville quantum gravity for $\gamma \in (0,2)$. Ann. Probab. 48 (2020), no. 4, 1861--1901. doi:10.1214/19-AOP1409. https://projecteuclid.org/euclid.aop/1595232097