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


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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  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].


  1. 1.

    Angel, O., Schramm, O.: Uniform infinite planar triangulation. Commun. Math. Phys. 241, 191–213 (2003)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Arista, J., Rivero, V.: Implicit renewal theory for exponential functionals of lévy processes. arXiv:1510.01809

  3. 3.

    Baur, E., Miermont, G., Ray, G.: Classification of scaling limits of uniform quadrangulations with a boundary. arXiv:1608.01129

  4. 4.

    Bernardi, O., Curien, N., Miermont, G.: A Boltzmann approach to percolation on random triangulations. arXiv:1705.04064

  5. 5.

    Bertoin, J.: Self-similar fragmentations. Ann. Inst. H. Poincaré Probab. Stat. 38, 319–340 (2002)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Bertoin, J.: Compensated fragmentation processes and limits of dilated fragmentations. Ann. Probab. 44, 1254–1284 (2016)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Bertoin, J.: Markovian growth-fragmentation processes. Bernoulli 23, 1082–1101 (2017)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Bertoin, J., Curien, N., Kortchemski, I.: Random planar maps & growth-fragmentations. Ann. Probab. (accepted)

  9. 9.

    Bertoin, J., Doney, R.A.: On conditioning a random walk to stay nonnegative. Ann. Probab. 22, 2152–2167 (1994)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Bertoin, J., Kortchemski, I.: Self-similar scaling limits of Markov chains on the positive integers. Ann. Appl. Probab. 26, 2556–2595 (2016)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Bertoin, J., Stephenson, R.: Local explosion in self-similar growth-fragmentation processes. Electron. Commun. Probab. 21, 1–12 (2016)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Bertoin, J., Watson, A.R.: Probabilistic aspects of critical growth-fragmentation equations. Adv. Appl. Probab. 48, 37–61 (2016)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Bertoin, J., Yor, M.: The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes. Potential Anal. 17, 389–400 (2002)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Bettinelli, J., Miermont, G.: Compact Brownian surfaces I: Brownian disks. Probab. Theory Relat. Fields 167, 555–614 (2017)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Biggins, J.D.: Uniform convergence of martingales in the branching random walk. Ann. Probab. 20, 137–151 (1992)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Biggins, J.D., Kyprianou, A.E.: Measure change in multitype branching. Adv. Appl. Probab. 36, 544–581 (2004)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Borot, G., Bouttier, J., Guitter, E.: Loop models on random maps via nested loops: case of domain symmetry breaking and application to the Potts model. J. Phys. A Math. Theor. 45(49), 494017 (2012)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Budd, T., The peeling process of infinite Boltzmann planar maps. Electron. J. Combin. 23, 37, Paper 1.28 (2016)

  19. 19.

    Budd, T., Curien, N.: Geometry of infinite planar maps with high degrees. Electron. J. Probab. 22, 1–37 (2017)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Budd, T., Curien, N., Marzouk, C.: Infinite random planar maps related to Cauchy processes. Preprint arXiv (2017)

  21. 21.

    Caravenna, F., Chaumont, L.: Invariance principles for random walks conditioned to stay positive. Ann. Inst. Henri Poincaré Probab. Stat. 44, 170–190 (2008)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Chaumont, L.: Conditionings and path decompositions for Lévy processes. Stoch. Processes Appl. 64, 39–54 (1996)

    Article  Google Scholar 

  23. 23.

    Chaumont, L., Pardo, J.C.: The lower envelope of positive self-similar Markov processes. Electron. J. Probab. 11, 49, 1321–1341 (2006)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Curien, N., Le Gall, J.-F.: Scaling limits for the peeling process on random maps. Ann. Inst. Henri Poincaré Probab. Stat. 53, 322–357 (2017)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Dadoun, B.: Asymptotics of self-similar growth-fragmentation processes. Electron. J. Probab. 22, 1–30 (2017)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)

    Book  Google Scholar 

  27. 27.

    Haas, B.: Loss of mass in deterministic and random fragmentations. Stoch. Process. Appl. 106, 245–277 (2003)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Jagers, P.: General branching processes as Markov fields. Stoch. Process. Appl. 32, 183–212 (1989)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Jelenković, P.R., Olvera-Cravioto, M.: Implicit renewal theory and power tails on trees. Adv. Appl. Probab. 44, 528–561 (2012)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Kuznetsov, A.: Wiener–Hopf factorization and distribution of extrema for a family of Lévy processes. Ann. Appl. Probab. 20, 1801–1830 (2010)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Kuznetsov, A., Pardo, J.C.: Fluctuations of stable processes and exponential functionals of hypergeometric Lévy processes. Acta Appl. Math. 123, 113–139 (2013)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Kyprianou, A.E.: Martingale convergence and the stopped branching random walk. Probab. Theory Relat. Fields 116, 405–419 (2000)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Kyprianou, A.E.: Fluctuations of Lévy Processes with Applications, Introductory lectures, Universitext, 2nd edn. Springer, Heidelberg (2014)

    Book  Google Scholar 

  34. 34.

    Lamperti, J.: Semi-stable Markov processes I. Z. Wahrscheinlichkeitstheor. verwandte Gebi. 22, 205–225 (1972)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Le Gall, J.-F.: Brownian disks and the Brownian snake. arXiv:1704.08987

  36. 36.

    Le Gall, J.-F., Miermont, G.: Scaling limits of random planar maps with large faces. Ann. Probab. 39, 1–69 (2011)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Liu, Q.: On generalized multiplicative cascades. Stoch. Processes Appl. 86, 263–286 (2000)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Lyons, R., Pemantle, R., Peres, Y.: Conceptual proofs of \(L\log L\) criteria for mean behavior of branching processes. Ann. Probab. 23, 1125–1138 (1995)

    MathSciNet  Article  Google Scholar 

  39. 39.

    Miller, J., Sheffield, S.: An axiomatic characterization of the Brownian map. arXiv:1506.03806

  40. 40.

    Nerman, O.: On the convergence of supercritical general (C–M–J) branching processes. Z. Wahrscheinlichkeitstheor. verwandte Gebi. 57, 365–395 (1981)

    MathSciNet  Article  Google Scholar 

  41. 41.

    Rembardt, F., Winkel, M.: Recursive construction of continuum random trees. arXiv:1607.05323

  42. 42.

    Rivero, V.: Tail asymptotics for exponential functionals of lévy processes: the convolution equivalent case. Ann. Inst. H. Poincaré Probab. Stat. 48, 1081–1102 (2012)

    Article  Google Scholar 

  43. 43.

    Shi, Q.: Growth-fragmentation processes and bifurcators. Electron. J. Probab. 22, 1–25 (2017)

    MathSciNet  Article  Google Scholar 

  44. 44.

    Shi, Z.: Branching random walks, vol. 2151 of Lecture Notes in Mathematics. Springer, Cham (2015). Lecture notes from the 42nd Probability Summer School held in Saint Flour, 2012, École d’Été de Probabilités de Saint-Flour [Saint-Flour Probability Summer School]

  45. 45.

    Stephenson, R.: Local convergence of large critical multi-type Galton–Watson trees and applications to random maps. J. Theor. Probab. (2016).

    MathSciNet  Article  Google Scholar 

  46. 46.

    Uribe Bravo, G.: The falling apart of the tagged fragment and the asymptotic disintegration of the Brownian height fragmentation. Ann. Inst. Henri Poincaré Probab. Stat. 45, 1130–1149 (2009)

    MathSciNet  Article  Google Scholar 

  47. 47.

    Whittaker, E.T., Watson, G.N.: A course of modern analysis. Cambridge Mathematical Library, Cambridge University Press, Cambridge (1996). An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions, Reprint of the fourth (1927) edition

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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).

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Mathematics Subject Classification

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