Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Scaling limits for the peeling process on random maps

Nicolas Curien and Jean-François Le Gall

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We study the scaling limit of the volume and perimeter of the discovered regions in the Markovian explorations known as peeling processes for infinite random planar maps such as the uniform infinite planar triangulation (UIPT) or quadrangulation (UIPQ). In particular, our results apply to the metric exploration or peeling by layers algorithm, where the discovered regions are (almost) completed balls, or hulls, centered at the root vertex. The scaling limits of the perimeter and volume of hulls can be expressed in terms of the hull process of the Brownian plane studied in our previous work. Other applications include the metric exploration of the dual graph of our infinite random lattices, and first-passage percolation with exponential edge weights on the dual graph, also known as the Eden model or uniform peeling.


Nous étudions la limite d’échelle du processus des volumes et des périmètres des régions explorées par un algorithme « d’épluchage » sur les cartes infinies aléatoires telles que l’UIPT (la triangulation infinie uniforme du plan) ou son analogue quadrangulaire l’UIPQ. Nos résultats s’appliquent en particulier à l’exploration des boules (pour la distance de graphe) complétées et centrées à la racine de la carte. Dans ce cas, la limite d’échelle coïncide avec le processus du périmètre et du volume des boules complétées dans le plan brownien. Parmi les autres applications, mentionnons l’exploration des boules complétées sur la carte duale et la percolation de premier passage avec poids exponentiels sur la carte duale. Ce dernier modèle, équivalent au modèle d’Eden sur la carte initiale, correspond à l’algorithme d’épluchage uniforme.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 1 (2017), 322-357.

Received: 19 December 2014
Revised: 19 July 2015
Accepted: 1 October 2015
First available in Project Euclid: 8 February 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20] 60F17: Functional limit theorems; invariance principles
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Random planar maps Peeling process Scaling limits Lévy process


Curien, Nicolas; Le Gall, Jean-François. Scaling limits for the peeling process on random maps. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 1, 322--357. doi:10.1214/15-AIHP718.

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