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Brownian half‐plane excursion and critical Liouville quantum gravity

Aru, Juhan; Holden, Nina; Powell, Ellen; Sun, Xin

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Juhan Aru

Nina Holden

Xin Sun


In a groundbreaking work, Duplantier, Miller and Sheffield showed that subcritical Liouville quantum gravity (LQG) coupled with Schramm–Loewner evolutions (SLE) can be obtained by gluing together a pair of Brownian motions. In this paper, we study the counterpart of their result in the critical case via a limiting argument. In particular, we prove that as one sends in the subcritical setting, the space-filling SLE in a disk degenerates to the CLE (where CLE is conformal loop ensembles) exploration introduced by Werner and Wu, along with a collection of independent and identically distributed coin tosses indexed by the branch points of the exploration. Furthermore, in the same limit, we observe that although the pair of initial Brownian motions collapses to a single one, one can still extract two different independent Brownian motions from this pair, such that the Brownian motion encodes the LQG distance from the CLE loops to the boundary of the disk and the Brownian motion encodes the boundary lengths of the CLE loops. In contrast to the subcritical setting, the pair does not determine the CLE-decorated LQG surface. Our paper also contains a discussion of relationships to random planar maps, the conformally invariant CLE metric and growth fragmentations.


Aru, J., Holden, N., Powell, E., & Sun, X. (2023). Brownian half‐plane excursion and critical Liouville quantum gravity. Journal of the London Mathematical Society, 107(1), 441-509.

Journal Article Type Article
Acceptance Date Aug 22, 2022
Online Publication Date Dec 14, 2022
Publication Date 2023-01
Deposit Date Jan 4, 2023
Publicly Available Date Mar 29, 2023
Journal Journal of the London Mathematical Society
Print ISSN 0024-6107
Electronic ISSN 1469-7750
Publisher Wiley
Peer Reviewed Peer Reviewed
Volume 107
Issue 1
Pages 441-509
Public URL


Published Journal Article (961 Kb)

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Copyright Statement
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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