Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Rescaled bipartite planar maps converge to the Brownian map

Céline Abraham

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Abstract

For every integer $n\geq1$, we consider a random planar map $\mathcal{M}_{n}$ which is uniformly distributed over the class of all rooted bipartite planar maps with $n$ edges. We prove that the vertex set of $\mathcal{M}_{n}$ equipped with the graph distance rescaled by the factor $(2n)^{-1/4}$ converges in distribution, in the Gromov–Hausdorff sense, to the Brownian map. This complements several recent results giving the convergence of various classes of random planar maps to the Brownian map.

Résumé

Pour tout entier $n$ strictement positif, on considère une carte planaire aléatoire $\mathcal{M}_{n}$ de loi uniforme sur l’ensemble des cartes biparties enracinées à $n$ arêtes. On montre que l’ensemble des sommets de $\mathcal{M}_{n}$ muni de la distance de graphe renormalisée par $(2n)^{-1/4}$ converge en loi au sens de Gromov–Hausdorff vers la carte brownienne. Ce travail complète une série de résultats de convergence de différents modèles de cartes aléatoires vers la carte brownienne.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 2 (2016), 575-595.

Dates
Received: 13 January 2014
Revised: 20 July 2014
Accepted: 13 October 2014
First available in Project Euclid: 4 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1462367885

Digital Object Identifier
doi:10.1214/14-AIHP657

Mathematical Reviews number (MathSciNet)
MR3498001

Zentralblatt MATH identifier
1375.60034

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

Keywords
Brownian map Planar map Graph distance Bipartite map Scaling limit Gromov–Hausdorff convergence Two-type Galton–Watson tree

Citation

Abraham, Céline. Rescaled bipartite planar maps converge to the Brownian map. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 2, 575--595. doi:10.1214/14-AIHP657. https://projecteuclid.org/euclid.aihp/1462367885


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