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Abstract
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Let
be the planar
Gaussian free field and let
be a supercritical Liouville quantum gravity (LQG) metric associated with
. Such
metrics arise as subsequential scaling limits of supercritical Liouville first passage
percolation (Ding and Gwynne, 2020) and correspond to values of the matter central
charge
.
We show that a.s. the boundary of each complementary connected component of a
-metric
ball is a Jordan curve and is compact and finite-dimensional with respect to
.
This is in contrast to the
whole boundary of the
-metric
ball, which is noncompact and infinite-dimensional with respect to
(Pfeffer,
2021). Using our regularity results for boundaries of complementary connected components
of
-metric
balls, we extend the confluence of geodesics results of Gwynne and Miller (2019) to
the case of supercritical Liouville quantum gravity. These results show that two
-geodesics
with the same starting point and different target points coincide for a nontrivial
initial time interval.
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Keywords
Liouville quantum gravity, Gaussian free field, Liouville
quantum gravity metric, supercritical Liouville quantum
gravity, confluence of geodesics, Liouville first passage
percolation
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Mathematical Subject Classification
Primary: 60D05, 60G60
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Milestones
Received: 2 October 2022
Revised: 26 July 2023
Accepted: 2 September 2023
Published: 30 January 2024
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© 2024 MSP (Mathematical Sciences
Publishers). |
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