Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Joint convergence of random quadrangulations and their cores

Louigi Addario-Berry and Yuting Wen

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Abstract

We show that a uniform quadrangulation, its largest $2$-connected block, and its largest simple block jointly converge to the same Brownian map in distribution for the Gromov–Hausdorff–Prokhorov topology. We start by deriving a local limit theorem for the asymptotics of maximal block sizes, extending the result in (Random Structures Algorithms 19 (2001) 194–246). The resulting diameter bounds for pendant submaps of random quadrangulations straightforwardly lead to Gromov–Hausdorff convergence. To extend the convergence to the Gromov–Hausdorff–Prokhorov topology, we show that exchangeable “uniformly asymptotically negligible” attachments of mass simply yield, in the limit, a deterministic scaling of the mass measure.

Résumé

Nous montrons qu’une quadrangulation uniformément aléatoire, sa plus grande composante 2-connexe, et sa plus grande composante simple convergent conjointement en loi vers la même carte brownienne dans le sens Gromov–Hausdorff–Prokhorov. En premier, nous étendons l’analyse de (Random Structures Algorithms 19 (2001) 194–246) afin de démontrer un théorème limite local pour les tailles des plus grandes composantes. Les bornes sur les diamètres ainsi obtenues impliquent directement la convergence dans le sens Gromov–Hausdorff. Pour obtenir la convergence pour la topologie Gromov–Hausdorff–Prokhorov, nous prouvons que l’effet de l’attachement des masses sur l’objet limite est déterministe, si les masses sont attachées de manière échangeable et les masses sont uniformément asymptotiquement négligeables.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 4 (2017), 1890-1920.

Dates
Received: 30 March 2015
Revised: 27 April 2016
Accepted: 22 June 2016
First available in Project Euclid: 27 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1511773730

Digital Object Identifier
doi:10.1214/16-AIHP775

Mathematical Reviews number (MathSciNet)
MR3729639

Zentralblatt MATH identifier
1382.60017

Subjects
Primary: 60C05: Combinatorial probability
Secondary: 68P10: Searching and sorting 68W40: Analysis of algorithms [See also 68Q25]

Keywords
Brownian map Gromov–Hausdorff–Prokhorov convergence Singularity analysis Connectivity Random quadrangulations

Citation

Addario-Berry, Louigi; Wen, Yuting. Joint convergence of random quadrangulations and their cores. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 4, 1890--1920. doi:10.1214/16-AIHP775. https://projecteuclid.org/euclid.aihp/1511773730


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