Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Scaling limits of random outerplanar maps with independent link-weights

Benedikt Stufler

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The scaling limit of large simple outerplanar maps was established by Caraceni using a bijection due to Bonichon, Gavoille and Hanusse. The present paper introduces a new bijection between outerplanar maps and trees decorated with ordered sequences of edge-rooted dissections of polygons. We apply this decomposition in order to provide a new, short proof of the scaling limit that also applies to the general setting of first-passage percolation. We obtain sharp tail-bounds for the diameter and recover the asymptotic enumeration formula for outerplanar maps. Our methods also enable us to treat subclasses such as bipartite outerplanar maps.


La limite d’échelle des cartes planaires extérieures simples a été établie par Caraceni via une bijection de Bonichon, Gavoille et Hanusse. Dans ce papier, nous construisons une nouvelle bijection entre les cartes planaires extérieures, et les arbres décorés par des suites ordonnées de dissections de polygones enracinées. Nous utilisons cette décomposition pour obtenir une nouvelle preuve, plus courte, du résultat de Caraceni, qui s’étend en outre au cadre général de la percolation de premier passage. Nous obtenons des bornes précises sur la queue de distribution du diamètre des cartes planaires extérieures, et retrouvons les formules d’énumération asymptotiques de ces cartes. Nos méthodes nous permettent également de traiter le cas de sous-classes comme les cartes planaires extérieures biparties.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 2 (2017), 900-915.

Received: 28 May 2015
Revised: 28 December 2015
Accepted: 13 January 2016
First available in Project Euclid: 11 April 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles 60C05: Combinatorial probability
Secondary: 05C80: Random graphs [See also 60B20]

Continuum random tree Scaling limits Random graphs Outerplanar maps


Stufler, Benedikt. Scaling limits of random outerplanar maps with independent link-weights. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 2, 900--915. doi:10.1214/16-AIHP741.

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