Probability Theory and Related Fields

, Volume 172, Issue 3–4, pp 663–724 | Cite as

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

  • Jean Bertoin
  • Timothy Budd
  • Nicolas Curien
  • Igor KortchemskiEmail author


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 



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.


  1. 1.
    Angel, O., Schramm, O.: Uniform infinite planar triangulation. Commun. Math. Phys. 241, 191–213 (2003)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bertoin, J.: Compensated fragmentation processes and limits of dilated fragmentations. Ann. Probab. 44, 1254–1284 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bertoin, J.: Markovian growth-fragmentation processes. Bernoulli 23, 1082–1101 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bertoin, J., Curien, N., Kortchemski, I.: Random planar maps & growth-fragmentations. Ann. Probab. (accepted) Google Scholar
  9. 9.
    Bertoin, J., Doney, R.A.: On conditioning a random walk to stay nonnegative. Ann. Probab. 22, 2152–2167 (1994)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bertoin, J., Stephenson, R.: Local explosion in self-similar growth-fragmentation processes. Electron. Commun. Probab. 21, 1–12 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bertoin, J., Watson, A.R.: Probabilistic aspects of critical growth-fragmentation equations. Adv. Appl. Probab. 48, 37–61 (2016)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Bettinelli, J., Miermont, G.: Compact Brownian surfaces I: Brownian disks. Probab. Theory Relat. Fields 167, 555–614 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Biggins, J.D.: Uniform convergence of martingales in the branching random walk. Ann. Probab. 20, 137–151 (1992)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Biggins, J.D., Kyprianou, A.E.: Measure change in multitype branching. Adv. Appl. Probab. 36, 544–581 (2004)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Budd, T., The peeling process of infinite Boltzmann planar maps. Electron. J. Combin. 23, 37, Paper 1.28 (2016)Google Scholar
  19. 19.
    Budd, T., Curien, N.: Geometry of infinite planar maps with high degrees. Electron. J. Probab. 22, 1–37 (2017)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Budd, T., Curien, N., Marzouk, C.: Infinite random planar maps related to Cauchy processes. Preprint arXiv (2017)Google Scholar
  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)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Chaumont, L.: Conditionings and path decompositions for Lévy processes. Stoch. Processes Appl. 64, 39–54 (1996)CrossRefGoogle 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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Dadoun, B.: Asymptotics of self-similar growth-fragmentation processes. Electron. J. Probab. 22, 1–30 (2017)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  27. 27.
    Haas, B.: Loss of mass in deterministic and random fragmentations. Stoch. Process. Appl. 106, 245–277 (2003)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Jagers, P.: General branching processes as Markov fields. Stoch. Process. Appl. 32, 183–212 (1989)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Jelenković, P.R., Olvera-Cravioto, M.: Implicit renewal theory and power tails on trees. Adv. Appl. Probab. 44, 528–561 (2012)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Kyprianou, A.E.: Martingale convergence and the stopped branching random walk. Probab. Theory Relat. Fields 116, 405–419 (2000)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Kyprianou, A.E.: Fluctuations of Lévy Processes with Applications, Introductory lectures, Universitext, 2nd edn. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  34. 34.
    Lamperti, J.: Semi-stable Markov processes I. Z. Wahrscheinlichkeitstheor. verwandte Gebi. 22, 205–225 (1972)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Liu, Q.: On generalized multiplicative cascades. Stoch. Processes Appl. 86, 263–286 (2000)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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)CrossRefGoogle Scholar
  43. 43.
    Shi, Q.: Growth-fragmentation processes and bifurcators. Electron. J. Probab. 22, 1–25 (2017)MathSciNetCrossRefGoogle 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]Google Scholar
  45. 45.
    Stephenson, R.: Local convergence of large critical multi-type Galton–Watson trees and applications to random maps. J. Theor. Probab. (2016). MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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) editionGoogle Scholar

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
    Email author
  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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