Liouville quantum gravity spheres as matings of finite-diameter trees
Name
1506.03804.pdf
Description
Accepted version
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958.74 KB
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Author(s)
Miller, Jason E.
Sheffield, Scott Roger
Date Issued
August 2019
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
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.
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/10.1214/18-aihp932
