## The Annals of Probability

### Recursive construction of continuum random trees

#### Abstract

We introduce a general recursive method to construct continuum random trees (CRTs) from independent copies of a random string of beads, that is, any random interval equipped with a random discrete probability measure, and from related structures. We prove the existence of these CRTs as a new application of the fixpoint method for recursive distribution equations formalised in high generality by Aldous and Bandyopadhyay.

We apply this recursive method to show the convergence to CRTs of various tree growth processes. We note alternative constructions of existing self-similar CRTs in the sense of Haas, Miermont and Stephenson, and we give for the first time constructions of random compact $\mathbb{R}$-trees that describe the genealogies of Bertoin’s self-similar growth fragmentations. In forthcoming work, we develop further applications to embedding problems for CRTs, providing a binary embedding of the stable line-breaking construction that solves an open problem of Goldschmidt and Haas.

#### Article information

Source
Ann. Probab., Volume 46, Number 5 (2018), 2715-2748.

Dates
Revised: October 2017
First available in Project Euclid: 24 August 2018

https://projecteuclid.org/euclid.aop/1535097638

Digital Object Identifier
doi:10.1214/17-AOP1237

Mathematical Reviews number (MathSciNet)
MR3846837

Zentralblatt MATH identifier
06964347

#### Citation

Rembart, Franz; Winkel, Matthias. Recursive construction of continuum random trees. Ann. Probab. 46 (2018), no. 5, 2715--2748. doi:10.1214/17-AOP1237. https://projecteuclid.org/euclid.aop/1535097638

#### References

• [1] Abraham, R., Delmas, J.-F. and He, H. (2015). Pruning of CRT-sub-trees. Stochastic Process. Appl. 125 1569–1604.
• [2] Albenque, M. and Goldschmidt, C. (2015). The Brownian continuum random tree as the unique solution to a fixed point equation. Electron. Commun. Probab. 20 no. 61, 14.
• [3] Aldous, D. (1991). The continuum random tree. I. Ann. Probab. 19 1–28.
• [4] Aldous, D. (1991). The continuum random tree. II. An overview. In Stochastic Analysis (Durham, 1990). 167 23–70. Cambridge Univ. Press, Cambridge.
• [5] Aldous, D. (1993). The continuum random tree. III. Ann. Probab. 21 248–289.
• [6] Aldous, D. and Bandyopadhyay, A. (2005). A survey of max-type recursive distributional equations. Ann. Appl. Probab. 15 1047–1110.
• [7] Aldous, D., Miermont, G. and Pitman, J. (2004). Brownian bridge asymptotics for random p-mappings. Electron. J. Probab. 9 37–56.
• [8] Bertoin, J. (2002). Self-similar fragmentations. Ann. Inst. Henri Poincaré Probab. Stat. 38 319–340.
• [9] Bertoin, J. (2006). Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics 102. Cambridge Univ. Press, Cambridge.
• [10] Bertoin, J. (2017). Markovian growth-fragmentation processes. Bernoulli 23 1082–1101.
• [11] Bertoin, J., Budd, T., Curien, N. and Kortchemski, I. (2016). Martingales in self-similar growth-fragmentations and their connections with random planar maps. Preprint, arXiv:1605.00581.
• [12] Bertoin, J., Curien, N. and Kortchemski, I. (2018). Random planar maps and growth-fragmentations. Ann. Probab. 46 207–260.
• [13] Bertoin, J. and Pitman, J. (1994). Path transformations connecting Brownian bridge, excursion and meander. Bull. Sci. Math. 118 147–166.
• [14] Bertoin, J. and Stephenson, R. (2016). Local explosion in self-similar growth-fragmentation processes. Electron. Commun. Probab. 21 12 pp.
• [15] Bolley, F. (2008). Separability and completeness for the Wasserstein distance. In Séminaire de Probabilités XLI. Lecture Notes in Math. 1934 371–377.
• [16] Broutin, N. and Sulzbach, H. (2016). Self-similar real trees defined as fixed-points and their geometric properties. Preprint, arXiv:1610.05331.
• [17] Burago, D., Burago, Y. and Ivanov, S. (2001). A Course in Metric Geometry. Amer. Math. Soc., Providence, RI.
• [18] Carmona, P., Petit, F. and Yor, M. (1997). On the distribution and asymptotic results for exponential functionals of Lévy processes. In Exponential Functionals and Principal Values Related to Brownian Motion 73–130.
• [19] Curien, N., Le Gall, J.-F. and Miermont, G. (2013). The Brownian cactus I. Scaling limits of discrete cactuses. Ann. Inst. Henri Poincaré Probab. Stat. 49 340–373.
• [20] Duquesne, T. (2003). A limit theorem for the contour process of conditioned Galton–Watson trees. Ann. Probab. 31 996–1027.
• [21] Duquesne, T. (2006). The coding of compact real trees by real valued functions. arXiv:math.PR/0604106.
• [22] Duquesne, T. and Le Gall, J.-F. (2002). Random trees, Lévy processes and spatial branching processes. Astérisque 281.
• [23] Duquesne, T. and Le Gall, J.-F. (2005). Probabilistic and fractal aspects of Lévy trees. Probab. Theory Related Fields 131 553–603.
• [24] Duquesne, T. and Winkel, M. (2007). Growth of Lévy trees. Probab. Theory Related Fields 139 313–371.
• [25] Duquesne, T. and Winkel, M. (2012). Hereditary tree growth and Lévy forests. arXiv:1211.2179.
• [26] Evans, S. (2008). Probability and Real Trees: École D’été de Probabilités de Saint-Flour XXXV-2005. Springer, Berlin.
• [27] Evans, S. N., Pitman, J. and Winter, A. (2006). Rayleigh processes, real trees, and root growth with re-grafting. Probab. Theory Related Fields 134 81–126.
• [28] Gnedin, A. and Pitman, J. (2005). Regenerative composition structures. Ann. Probab. 33 445–479.
• [29] Goldschmidt, C. and Haas, B. (2015). A line-breaking construction of the stable trees. Electron. J. Probab. 20 24 pp.
• [30] Haas, B. and Miermont, G. (2004). The genealogy of self-similar fragmentations with negative index as a continuum random tree. Electron. J. Probab. 9 57–97.
• [31] Haas, B. and Miermont, G. (2012). Scaling limits of Markov branching trees with applications to Galton–Watson and random unordered trees. Ann. Probab. 40 2589–2666.
• [32] Haas, B., Pitman, J. and Winkel, M. (2009). Spinal partitions and invariance under re-rooting of continuum random trees. Ann. Probab. 37 1381–1411.
• [33] Le Gall, J.-F. (1991). Brownian excursions, trees and measure-valued branching processes. Ann. Probab. 19 1399–1439.
• [34] Le Gall, J.-F. (2006). Random real trees. Ann. Fac. Sci. Toulouse Math. (6) 41 35–62.
• [35] Le Gall, J.-F. and Miermont, G. (2011). Scaling limits of random planar maps with large faces. Ann. Probab. 39 1–69.
• [36] Marchal, P. (2008). A note on the fragmentation of a stable tree. In Fifth Colloquium on Mathematics and Computer Science. Discrete Math. Theor. Comput. Sci. Proc., AI 489–499. Assoc. Discrete Math. Theor. Comput. Sci., Nancy.
• [37] Miermont, G. (2003). Self-similar fragmentations derived from the stable tree I: Splitting at heights. Probab. Theory Related Fields 127 423–454.
• [38] Miermont, G. (2005). Self-similar fragmentations derived from the stable tree II: Splitting at nodes. Probab. Theory Related Fields 131 341–375.
• [39] Miermont, G. (2009). Tessellations of random maps of arbitrary genus. Ann. Sci. Éc. Norm. Supér. (4) 42 725–781.
• [40] Pitman, J. (1999). Coalescents with multiple collisions. Ann. Probab. 27 1870–1902.
• [41] Pitman, J. (2006). Combinatorial stochastic processes. In Lectures from the 32nd Summer School on Probability Theory Held in Saint-Flour, July 724, 2002. Lecture Notes in Math. 1875 7–24. Springer, Berlin.
• [42] Pitman, J. and Winkel, M. (2009). Regenerative tree growth: Binary self-similar continuum random trees and Poisson–Dirichlet compositions. Ann. Probab. 37 1999–2042.
• [43] Pitman, J. and Winkel, M. (2015). Regenerative tree growth: Markovian embedding of fragmenters, bifurcators, and bead splitting processes. Ann. Probab. 43 2611–2646.
• [44] Rembart, F. and Winkel, M. (2016). A binary embedding of the stable line-breaking construction. Preprint, arXiv:1611.02333.
• [45] Stephenson, R. (2013). General fragmentation trees. Electron. J. Probab. 18 1–45.