Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Convergence of the free Boltzmann quadrangulation with simple boundary to the Brownian disk

Ewain Gwynne and Jason Miller

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Abstract

We prove that the free Boltzmann quadrangulation with simple boundary and fixed perimeter, equipped with its graph metric, natural area measure, and the path which traces its boundary converges in the scaling limit to the free Boltzmann Brownian disk. The topology of convergence is the so-called Gromov–Hausdorff–Prokhorov-uniform (GHPU) topology, the natural analog of the Gromov–Hausdorff topology for curve-decorated metric measure spaces. From this we deduce that a random quadrangulation of the sphere decorated by a $2l$-step self-avoiding loop converges in law in the GHPU topology to the random curve-decorated metric measure space obtained by gluing together two independent Brownian disks along their boundaries.

Résumé

Nous démontrons que la quadrangulation de Boltzmann libre avec un bord simple de périmètre fixé, munie de sa métrique de graphe, de sa mesure d’aire naturelle, et du chemin qui décrit sa frontière, converge dans la limite d’échelle vers le disque brownien libre de Boltzmann. La topologie de cette convergence est celle de Gromov–Hausdorff–Prokhorov-uniforme (GHPU), qui est l’analogue naturel de la topologie de Gromov-Hausdroff pour des espaces métriques mesurés décorés par une courbe. Nous déduisons de cela qu’une quadrangulation aléatoire de la sphère, décorée par une marche aléatoire auto-évitante de longueur $2l$, converge en loi pour la topologie GHPU vers l’espace métrique mesuré et décoré par une courbe que l’on obtient en recollant ensemble deux disques browniens indépendants le long de leurs bords.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 1 (2019), 551-589.

Dates
Received: 31 January 2017
Revised: 23 December 2017
Accepted: 8 February 2018
First available in Project Euclid: 18 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1547802409

Digital Object Identifier
doi:10.1214/18-AIHP891

Mathematical Reviews number (MathSciNet)
MR3901655

Zentralblatt MATH identifier
07039779

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60F17: Functional limit theorems; invariance principles 05C80: Random graphs [See also 60B20]

Keywords
Random planar maps Brownian map Brownian disk Quadrangulation with simple boundary Self-avoiding walk Gromov–Hausdorff–Prokhorov-uniform topology

Citation

Gwynne, Ewain; Miller, Jason. Convergence of the free Boltzmann quadrangulation with simple boundary to the Brownian disk. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 1, 551--589. doi:10.1214/18-AIHP891. https://projecteuclid.org/euclid.aihp/1547802409


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