# Confluence of geodesics in Liouville quantum gravity for $\gamma \in (0,2)$

## TL;DR

This paper proves a confluence property of geodesics in Liouville quantum gravity surfaces for all b3  (0,2), showing that geodesics from different points merge before reaching a fixed point, aiding in establishing the LQG metric.

## Contribution

It establishes the confluence of geodesics in LQG surfaces for b3 , a key step in proving the existence and uniqueness of the LQG metric.

## Key findings

- Geodesics from different points merge before reaching a fixed point.
- Results apply to subsequential limits of Liouville first passage percolation.
- Supports the proof of the LQG metric's existence and uniqueness.

## Abstract

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 (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 $\gamma\in (0,2)$.