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

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).

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Notes

  1. 1.

    The assumption \(\int _{y>1}\hbox {e}^{y}\Lambda (\hbox {d}y)<\infty \) may be replaced by the weaker assumption that \(\int _{y>1}\hbox {e}^{ qy}\Lambda (\hbox {d}y)<\infty \) for a certain \(q>0\), but then a cutoff should be added to \(q(1-\hbox {e}^y)\), such as e.g. \(q(1-\hbox {e}^{y}) {{\mathbb {1}}}_{{y \le 1}}\) in (1).

  2. 2.

    The condition \(\int _{y>1}\hbox {e}^{y}\Lambda (\hbox {d}y)<\infty \) may be replaced by the weaker condition that there exists \(q>0\) such that \(\int _{y>1}\hbox {e}^{qy}\Lambda (\hbox {d}y)<\infty \), and by considering an additional cutoff in (22) but we shall not enter such considerations.

  3. 3.

    More precisely, keeping the notation of [39, Sect. 4], Theorem 4.6 in [39] indicates that under \({\widetilde{\mu }}^{1,L}_{\text {DISK}}\), the process describing the boundary lengths of increasing balls from the root is a self-similar growth-fragmentation under the tilted probability measure \({\mathbb {P}}^{-}_{L}\). The process \((L_{r})\) is \(Y^{-}\) (the size of the tagged fragment under \( \widehat{\mathcal {{P}}}^{-}_{L}\)), the process \((M^{1}_{r})\) is the size of the Eve cell when one uses the locally largest cell process for the Eve cell under the tilted probability measure \( \widehat{\mathcal {{P}}}^{-}_{L}\) and the process \((M_{r})\) is the size of the Eve cell when one uses the locally largest cell process for the Eve cell under the non-tilted probability measure \( \widehat{\mathcal {{P}}}_{L}\). Theorem 4.6 in [39] indicates that the process \((L_{r})\) evolves as a time-reversed \(\theta \)-stable continuous state branching process.

  4. 4.

    Contrary to [8], the cycles cannot be seen as self-avoiding loops on the original map \(\mathfrak {m}\) since they are closed paths which may visit twice the same edge; they are called frontiers in [18].

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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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Correspondence to Igor Kortchemski.

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Bertoin, J., Budd, T., Curien, N. et al. Martingales in self-similar growth-fragmentations and their connections with random planar maps. Probab. Theory Relat. Fields 172, 663–724 (2018). https://doi.org/10.1007/s00440-017-0818-5

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Mathematics Subject Classification

  • 60F17
  • 60C05
  • 05C80
  • 60G51
  • 60J80