Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

The scaling limit of random outerplanar maps

Alessandra Caraceni

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A planar map is outerplanar if all its vertices belong to the same face. We show that random uniform outerplanar maps with $n$ vertices suitably rescaled by a factor $1/\sqrt{n}$ converge in the Gromov–Hausdorff sense to ${7\sqrt{2}}/{9}$ times Aldous’ Brownian tree. The proof uses the bijection of Bonichon, Gavoille and Hanusse (J. Graph Algorithms Appl. 9 (2005) 185–204).


Une carte planaire est dite outerplanaire si tous ses sommets appartiennent à la même face. Nous montrons que les cartes outerplanaires aléatoires uniformes à $n$ sommets, multipliées par le facteur d’échelle $1/\sqrt{n}$, convergent au sens de Gromov–Hausdorff vers ${7\sqrt{2}}/{9}$ fois l’arbre Brownien d’Aldous. La preuve utilise la bijection de Bonichon, Gavoille et Hanusse (J. Graph Algorithms Appl. 9 (2005) 185–204).

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 4 (2016), 1667-1686.

Received: 17 May 2014
Revised: 24 February 2015
Accepted: 16 June 2015
First available in Project Euclid: 17 November 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability 60F05: Central limit and other weak theorems

Random outerplanar maps Scaling limits Galton–Watson trees Brownian continuum random tree Gromov–Hausdorff topology


Caraceni, Alessandra. The scaling limit of random outerplanar maps. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 4, 1667--1686. doi:10.1214/15-AIHP694.

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