Geodesics and metric ball boundaries in Liouville quantum gravity
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440_2022_1112_ReferencePDF.pdf
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Author(s)
Gwynne, Ewain
Pfeffer, Joshua
Sheffield, Scott
Date Issued
February 11, 2022
Publisher
Springer Berlin Heidelberg
Citation
Gwynne, Ewain, Pfeffer, Joshua and Sheffield, Scott. 2022. "Geodesics and metric ball boundaries in Liouville quantum gravity."
Version
Author's final manuscript
Abstract
Abstract
Recent works have shown that there is a canonical way to to assign a metric (distance function) to a Liouville quantum gravity (LQG) surface for any parameter
$$\gamma \in (0,2)$$
γ
∈
(
0
,
2
)
. We establish a strong confluence property for LQG geodesics, which generalizes a result proven by Angel, Kolesnik and Miermont for the Brownian map. Using this property, we also establish zero-one laws for the Hausdorff dimensions of geodesics, metric ball boundaries, and metric nets w.r.t. the Euclidean or LQG metric. In the case of a metric ball boundary, our result combined with earlier work of Gwynne (Commun Math Phys 378(1):625–689, 2020.
arXiv:1909.08588
) gives a formula for the a.s. Hausdorff dimension for the boundary of the metric ball stopped when it hits a fixed point in terms of the Hausdorff dimension of the whole LQG surface. We also show that the Hausdorff dimension of the metric ball boundary is carried by points which are not on the boundary of any complementary connected component of the ball.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1007/s00440-022-01112-5
