Random planar maps and growth-fragmentations

• Jean Bertoin ^[1] ; Nicolas Curien ^[2] ; Igor Kortchemski ^[3]
  1. [1] University of Zurich

     University of Zurich

     Zürich, Suiza

     
  2. [2] University of Paris-Sud

     University of Paris-Sud

     Arrondissement de Palaiseau, Francia

     
  3. [3] École Polytechnique (France)

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• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 46, Nº. 1, 2018, págs. 207-260
• Idioma: inglés
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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.