TY - JOUR

T1 - Brownian half-plane excursion and critical Liouville quantum gravity

AU - Aru, Juhan

AU - Holden, Nina

AU - Powell, Ellen

AU - Sun, Xin

N1 - Funding Information:
J. Aru was supported by Eccellenza grant 194648 of the Swiss National Science Foundation. N. Holden was supported by grant 175505 of the Swiss National Science Foundation, along with Dr. Max Rössler, the Walter Haefner Foundation and the ETH Zürich Foundation. E. Powell was supported by grant 175505 of the Swiss National Science Foundation. X. Sun was supported by the NSF grant DMS-2027986 and the NSF Career grant DMS-2046514. J. Aru and N. Holden were both part of SwissMAP. We all thank Wendelin Werner and ETH for their hospitality. We also thank Elie Aïdékon, Nicolas Curien, William Da Silva, Ewain Gwynne, Laurent Ménard, Avelio Sepúlveda and Samuel Watson for useful discussions. Finally, we thank the anonymous referee for their careful reading of this paper, and helping to improve the exposition in numerous places.
Funding Information:
J. Aru was supported by Eccellenza grant 194648 of the Swiss National Science Foundation. N. Holden was supported by grant 175505 of the Swiss National Science Foundation, along with Dr. Max Rössler, the Walter Haefner Foundation and the ETH Zürich Foundation. E. Powell was supported by grant 175505 of the Swiss National Science Foundation. X. Sun was supported by the NSF grant DMS‐2027986 and the NSF Career grant DMS‐2046514. J. Aru and N. Holden were both part of SwissMAP. We all thank Wendelin Werner and ETH for their hospitality. We also thank Elie Aïdékon, Nicolas Curien, William Da Silva, Ewain Gwynne, Laurent Ménard, Avelio Sepúlveda and Samuel Watson for useful discussions. Finally, we thank the anonymous referee for their careful reading of this paper, and helping to improve the exposition in numerous places.
Publisher Copyright:
© 2022 The Authors. Journal of the London Mathematical Society is copyright © London Mathematical Society.

PY - 2023/1

Y1 - 2023/1

N2 - 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 (Formula presented.) in the subcritical setting, the space-filling SLE (Formula presented.) in a disk degenerates to the CLE (Formula presented.) (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 (Formula presented.) from this pair, such that the Brownian motion (Formula presented.) encodes the LQG distance from the CLE loops to the boundary of the disk and the Brownian motion (Formula presented.) encodes the boundary lengths of the CLE (Formula presented.) loops. In contrast to the subcritical setting, the pair (Formula presented.) does not determine the CLE-decorated LQG surface. Our paper also contains a discussion of relationships to random planar maps, the conformally invariant CLE (Formula presented.) metric and growth fragmentations.

AB - 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 (Formula presented.) in the subcritical setting, the space-filling SLE (Formula presented.) in a disk degenerates to the CLE (Formula presented.) (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 (Formula presented.) from this pair, such that the Brownian motion (Formula presented.) encodes the LQG distance from the CLE loops to the boundary of the disk and the Brownian motion (Formula presented.) encodes the boundary lengths of the CLE (Formula presented.) loops. In contrast to the subcritical setting, the pair (Formula presented.) does not determine the CLE-decorated LQG surface. Our paper also contains a discussion of relationships to random planar maps, the conformally invariant CLE (Formula presented.) metric and growth fragmentations.

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UR - http://www.scopus.com/inward/citedby.url?scp=85144107290&partnerID=8YFLogxK

U2 - 10.1112/jlms.12689

DO - 10.1112/jlms.12689

M3 - Article

AN - SCOPUS:85144107290

SN - 0024-6107

VL - 107

SP - 441

EP - 509

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

IS - 1

ER -