Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Liouville quantum gravity spheres as matings of finite-diameter trees

Jason Miller and Scott Sheffield

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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 $\gamma=\sqrt{8/3}$, 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 $\mathrm{SLE}_{6}$. This construction is relevant to a program for showing that the $\gamma=\sqrt{8/3}$ Liouville quantum gravity sphere is equivalent to the Brownian map.

Résumé

Nous montrons que la sphère de gravité quantique de Liouville d’aire unité peut être construite de deux façons équivalentes. La première, introduite par les auteurs et Duplantier dans (Liouville quantum gravity as a mating of trees (2014) Preprint), utilise la mesure d’excursion d’un processus de Bessel pour définir une variante du champ libre gaussien sur le cylindre. La seconde utilise une boucle d’un mouvement brownien corrélé et un « accouplement d’arbres » pour produire une sphère de gravité quantique de Liouville décorée par un chemin remplissant l’espace.

Dans le cas particulier où $\gamma=\sqrt{8/3}$, nous présentons une troisième construction équivalente, utilisant la mesure d’excursion d’un processus de Lévy stable d’exposant $3/2$ (sans sauts négatifs) pour produire une paire d’arbres de disques quantiques que l’on peut accoupler pour obtenir une sphère décorée par un $\mathrm{SLE}_{6}$. Cette construction intervient dans un programme ayant pour but de montrer que la sphère de gravité quantique de Liouville pour $\gamma=\sqrt{8/3}$ est équivalente à la carte brownienne.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 3 (2019), 1712-1750.

Dates
Received: 10 April 2017
Revised: 6 September 2018
Accepted: 17 September 2018
First available in Project Euclid: 25 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1569398883

Digital Object Identifier
doi:10.1214/18-AIHP932

Mathematical Reviews number (MathSciNet)
MR4010949

Subjects
Primary: 60J67: Stochastic (Schramm-)Loewner evolution (SLE) 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11]

Keywords
Gaussian free field Liouville quantum gravity Schramm-Loewner evolution Continuum random tree Conformal welding

Citation

Miller, Jason; Sheffield, Scott. Liouville quantum gravity spheres as matings of finite-diameter trees. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 3, 1712--1750. doi:10.1214/18-AIHP932. https://projecteuclid.org/euclid.aihp/1569398883


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