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Confluence of geodesics in Liouville quantum gravity for \(\gamma \in (0,2)\). (English) Zbl 1453.60141
Summary: 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 J.-F. Le Gall [Acta Math. 205, No. 2, 287–360 (2010; Zbl 1214.53036)]. 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 \(\gamma \in (0,2)\).

MSC:
60J67 Stochastic (Schramm-)Loewner evolution (SLE)
60G52 Stable stochastic processes
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