Strong confluence of geodesics in Liouville quantum gravity

TL;DR

This paper proves a strong confluence property of geodesics in Liouville quantum gravity, showing that geodesics targeted to a fixed point almost surely merge, extending previous results to all subcritical gamma values.

Contribution

It establishes a strong confluence result for geodesics in Liouville quantum gravity for all gamma in (0,2), ruling out the existence of non-overlapping limit geodesics.

Findings

Geodesics targeted to a fixed point almost surely merge.

No sequence of geodesics converging to an exceptional geodesic can remain disjoint.

Extends previous confluence results from gamma=√8/3 to all gamma in (0,2).

Abstract

$\gamma$-Liouville quantum gravity ($\gamma$-LQG) constitutes a family of planar random geometries whose geodesics exhibit intricate fractal behaviour. As is observed in various planar models of random geometry as part of the phenomenon of geodesic confluence, geodesics in $\gamma$-LQG tend to merge with each other. In particular, in Gwynne-Miller '19, it was established that in $\gamma$-LQG, geodesics targeted to a fixed point do coalesce in the sense that any two such geodesics almost surely merge before reaching their common target. However, in view of the randomness inherent to the geometry, it is a priori possible that while geodesics targeted to a fixed point do coalesce, there exists a sequence of geodesics $P_n$ converging to an exceptional geodesic $P$ as $n\rightarrow \infty$ such that $P_n$ does not overlap with $P$ for any $n$. In this paper, we prove that this is not possible, thereby establishing a strong confluence statement for $\gamma$-LQG for all $\gamma\in (0,2)$. This extends the results obtained in Miller-Qian '20 for $\gamma=\sqrt{8/3}$ to all subcritical values of $\gamma$. We discuss applications to the study of geodesic stars and geodesic networks and include a list of open questions.