The Annals of Probability

Random planar maps and growth-fragmentations

Jean Bertoin, Nicolas Curien, and Igor Kortchemski

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We are interested in the cycles obtained by slicing at all heights random Boltzmann triangulations with a simple boundary. We establish a functional invariance principle for the lengths of these cycles, appropriately rescaled, as the size of the boundary grows. The limiting process is described using a self-similar growth-fragmentation process with explicit parameters. To this end, we introduce a branching peeling exploration of Boltzmann triangulations, which allows us to identify a crucial martingale involving the perimeters of cycles at given heights. We also use a recent result concerning self-similar scaling limits of Markov chains on the nonnegative integers. A motivation for this work is to give a new construction of the Brownian map from a growth-fragmentation process.

Article information

Ann. Probab., Volume 46, Number 1 (2018), 207-260.

Received: February 2016
Revised: January 2017
First available in Project Euclid: 5 February 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60G51: Processes with independent increments; Lévy processes 60G18: Self-similar processes

Planar maps growth-fragmentations self-similar Markov processes scaling limits


Bertoin, Jean; Curien, Nicolas; Kortchemski, Igor. Random planar maps and growth-fragmentations. Ann. Probab. 46 (2018), no. 1, 207--260. doi:10.1214/17-AOP1183.

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