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Martingales in self-similar growth-fragmentations and their connections with random planar maps

  • Jean Bertoin
  • Timothy Budd
  • Nicolas Curien
  • Igor Kortchemski
Article
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Abstract

The purpose of the present work is twofold. First, we develop the theory of general self-similar growth-fragmentation processes by focusing on martingales which appear naturally in this setting and by recasting classical results for branching random walks in this framework. In particular, we establish many-to-one formulas for growth-fragmentations and define the notion of intrinsic area of a growth-fragmentation. Second, we identify a distinguished family of growth-fragmentations closely related to stable Lévy processes, which are then shown to arise as the scaling limit of the perimeter process in Markovian explorations of certain random planar maps with large degrees (which are, roughly speaking, the dual maps of the stable maps of Le Gall and Miermont in Ann Probab 39:1–69, 2011). As a consequence of this result, we are able to identify the law of the intrinsic area of these distinguished growth-fragmentations. This generalizes a geometric connection between large Boltzmann triangulations and a certain growth-fragmentation process, which was established in Bertoin et al. (Ann Probab, accepted).

Mathematics Subject Classification

60F17 60C05 05C80 60G51 60J80 

Notes

Acknowledgements

NC and IK acknowledge partial support from Agence Nationale de la Recherche, Grant Number ANR-14-CE25-0014 (ANR GRAAL), ANR-15-CE40-0013 (ANR Liouville) and from the City of Paris, Grant “Emergences Paris 2013, Combinatoire à Paris”. TB acknowledges support from the ERC-Advance Grant 291092, “Exploring the Quantum Universe” (EQU). Finally, we would like to thank two anonymous referees for useful comments.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  • Jean Bertoin
    • 1
  • Timothy Budd
    • 2
    • 3
  • Nicolas Curien
    • 4
  • Igor Kortchemski
    • 5
  1. 1.Universität ZürichZurichSwitzerland
  2. 2.Niels Bohr InstituteUniversity of CopenhagenCopenhagenDenmark
  3. 3.IPhT, CEAUniversité Paris-SaclayParisFrance
  4. 4.Université Paris-SudOrsayFrance
  5. 5.CNRS, CMAPÉcole polytechniquePalaiseauFrance

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