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Liouville quantum gravity spheres as matings of finite-diameter trees

Author(s)
Miller, Jason E.; Sheffield, Scott Roger
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Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/
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Abstract
We show that the unit area Liouville quantum gravity sphere can be constructed in two equivalent ways. The first, which was introduced by the authors and Duplantier in (Liouville quantum gravity as a mating of trees (2014) Preprint), uses a Bessel excursion measure to produce a Gaussian free field variant on the cylinder. The second uses a correlated Brownian loop and a "mating of trees" to produce a Liouville quantum gravity sphere decorated by a space-filling path. In the special case that γ = √8/3, we present a third equivalent construction, which uses the excursion measure of a 3/2-stable Levy process (with only upward jumps) to produce a pair of trees of quantum disks that can be mated to produce a sphere decorated by SLE6. This construction is relevant to a program for showing that the γ =√8/3 Liouville quantum gravity sphere is equivalent to the Brownian map.
Date issued
2019-08
URI
https://hdl.handle.net/1721.1/126480
Department
Massachusetts Institute of Technology. Department of Mathematics
Journal
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
Publisher
Institute of Mathematical Statistics
Citation
Miller, Jason and Scott Sheffield. "Liouville quantum gravity spheres as matings of finite-diameter trees." Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 55, 3 (August 2019): 1712-1750 © 2019 Association des Publications de l'Institut Henri Poincaré
Version: Author's final manuscript
ISSN
0246-0203

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