Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Brownian disks and the Brownian snake

Jean-François Le Gall

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Abstract

We provide a new construction of the Brownian disks, which have been defined by Bettinelli and Miermont as scaling limits of quadrangulations with a boundary when the boundary size tends to infinity. Our method is very similar to the construction of the Brownian map, but it makes use of the positive excursion measure of the Brownian snake which has been introduced recently. This excursion measure involves a continuous random tree whose vertices are assigned nonnegative labels, which correspond to distances from the boundary in our approach to the Brownian disk. We provide several applications of our construction. In particular, we prove that the uniform measure on the boundary can be obtained as the limit of the suitably normalized volume measure on a small tubular neighborhood of the boundary. We also prove that connected components of the complement of the Brownian net are Brownian disks, as it was suggested in the recent work of Miller and Sheffield. Finally, we show that connected components of the complement of balls centered at the distinguished point of the Brownian map are independent Brownian disks, conditionally on their volumes and perimeters.

Résumé

Nous donnons une nouvelle construction des disques browniens, qui ont été définis par Bettinelli et Miermont comme limites d’échelle de quadrangulations avec frontière quand la taille de la frontière tend vers l’infini. Notre méthode est semblable à la construction de la carte brownienne, mais elle utilise la mesure d’excursion positive du serpent brownien introduite récemment. Cette mesure d’excursion implique un arbre aléatoire continu dont les sommets reçoivent des labels positifs, qui correspondent aux distances depuis la frontière dans notre approche du disque brownien. Nous donnons plusisurs applications de cette construction. En particulier, nous montrons que la mesure uniforme sur la frontière peut être obtenue comme limite de la mesure de volume (convenablement normalisée) sur un petit voisinage tubulaire de la frontière. Nous montrons aussi que les composantes connexes du complémentaire du filet brownien sont des disques browniens, comme cela est suggéré dans le travail récent de Miller et Sheffield. Finalement, nous montrons que les composantes connexes du complémentaire d’une boule centrée au point distingué de la carte brownienne sont, conditionnellement à leurs volumes et leurs périmètres, des disques browniens indépendants.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 1 (2019), 237-313.

Dates
Received: 5 May 2017
Revised: 20 October 2017
Accepted: 8 January 2018
First available in Project Euclid: 18 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1547802401

Digital Object Identifier
doi:10.1214/18-AIHP882

Mathematical Reviews number (MathSciNet)
MR3901647

Zentralblatt MATH identifier
07039771

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60C05: Combinatorial probability

Keywords
Brownian disk Brownian map Excursion measure Brownian snake Uniform measure on boundary Continuous random tree Connected components of complement of balls

Citation

Le Gall, Jean-François. Brownian disks and the Brownian snake. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 1, 237--313. doi:10.1214/18-AIHP882. https://projecteuclid.org/euclid.aihp/1547802401


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