Open Access
January 2018 Random planar maps and growth-fragmentations
Jean Bertoin, Nicolas Curien, Igor Kortchemski
Ann. Probab. 46(1): 207-260 (January 2018). DOI: 10.1214/17-AOP1183


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.


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Jean Bertoin. Nicolas Curien. Igor Kortchemski. "Random planar maps and growth-fragmentations." Ann. Probab. 46 (1) 207 - 260, January 2018.


Received: 1 February 2016; Revised: 1 January 2017; Published: January 2018
First available in Project Euclid: 5 February 2018

zbMATH: 06865122
MathSciNet: MR3758730
Digital Object Identifier: 10.1214/17-AOP1183

Primary: 60D05 , 60F17
Secondary: 60G18 , 60G51

Keywords: growth-fragmentations , planar maps , scaling limits , Self-similar Markov processes

Rights: Copyright © 2018 Institute of Mathematical Statistics

Vol.46 • No. 1 • January 2018
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