Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

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).

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5


  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)

  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)

  6. 6.

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

  7. 7.

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

  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)

  10. 10.

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

  11. 11.

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

  12. 12.

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

  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)

  14. 14.

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

  15. 15.

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

  16. 16.

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

  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)

  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)

  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)

  22. 22.

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

  23. 23.

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

  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)

  25. 25.

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

  26. 26.

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

  27. 27.

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

  28. 28.

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

  29. 29.

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

  30. 30.

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

  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)

  32. 32.

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

  33. 33.

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

  34. 34.

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

  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)

  37. 37.

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

  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)

  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)

  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)

  43. 43.

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

  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). https://doi.org/10.1007/s10959-016-0707-3

  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)

  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

Download references


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.

Author information

Correspondence to Igor Kortchemski.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

Mathematics Subject Classification

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