Liouville quantum gravity spheres as matings of finite-diameter trees
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Authors
Miller, Jason
Sheffield, Scott
Publication Date
2019-08-01Journal Title
Annales de l'institut Henri Poincare (B) Probability and Statistics
ISSN
0246-0203
Publisher
Institute of Mathematical Statistics
Volume
55
Issue
3
Pages
1712-1750
Type
Article
This Version
AM
Metadata
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Miller, J., & Sheffield, S. (2019). Liouville quantum gravity spheres as matings of finite-diameter trees. Annales de l'institut Henri Poincare (B) Probability and Statistics, 55 (3), 1712-1750. https://doi.org/10.1214/18-AIHP932
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, 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 $γ=\sqrt{8/3}$, we present a third equivalent construction, which uses the excursion measure of a $3/2$-stable Lévy process (with only upward jumps) to produce a pair of trees of quantum disks that can be mated to produce a sphere decorated by SLE$_6$. This construction is relevant to a program for showing that the $γ=\sqrt{8/3}$ Liouville quantum gravity sphere is equivalent to the Brownian map.
Sponsorship
Engineering and Physical Sciences Research Council (EP/I03372X/1)
Engineering and Physical Sciences Research Council (EP/L018896/1)
Identifiers
External DOI: https://doi.org/10.1214/18-AIHP932
This record's URL: https://www.repository.cam.ac.uk/handle/1810/287994
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