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Liouville quantum gravity spheres as matings of finite-diameter trees

Accepted version
Peer-reviewed

Type

Article

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Authors

Miller, JP 
Sheffield, Scott 

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=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 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.

Description

Keywords

Journal Title

Annales de l'institut Henri Poincare (B) Probability and Statistics

Conference Name

Journal ISSN

0246-0203

Volume Title

55

Publisher

Institute of Mathematical Statistics
Sponsorship
Engineering and Physical Sciences Research Council (EP/I03372X/1)
Engineering and Physical Sciences Research Council (EP/L018896/1)