## Electronic Journal of Probability

### A line-breaking construction of the stable trees

#### Abstract

We give a new, simple construction of the $\alpha$-stable tree for $\alpha \in (1,2]$. We obtain it as the closure of an increasing sequence of $\mathbb{R}$-trees inductively built by gluing together line-segments one by one. The lengths of these line-segments are related to the the increments of an increasing $\mathbb{R}_+$-valued Markov chain. For $\alpha = 2$, we recover Aldous' line-breaking construction of the Brownian continuum random tree based on an inhomogeneous Poisson process.

#### Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 16, 24 pp.

Dates
Accepted: 24 February 2015
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465067122

Digital Object Identifier
doi:10.1214/EJP.v20-3690

Mathematical Reviews number (MathSciNet)
MR3317158

Zentralblatt MATH identifier
06471494

Rights

#### Citation

Goldschmidt, Christina; Haas, Bénédicte. A line-breaking construction of the stable trees. Electron. J. Probab. 20 (2015), paper no. 16, 24 pp. doi:10.1214/EJP.v20-3690. https://projecteuclid.org/euclid.ejp/1465067122

#### References

• Abraham, Romain; Delmas, Jean-FranÃ§ois. Î²-coalescents and stable Galton–Watson trees. Preprint, 2013. arXiv:1303.6882
• Addario-Berry, Louigi; Broutin, Nicolas; Goldschmidt, Christina. Critical random graphs: limiting constructions and distributional properties. Electron. J. Probab. 15 (2010), no. 25, 741–775.
• Albenque, Marie; Marckert, Jean-FranÃ§ois. Some families of increasing planar maps. Electron. J. Probab. 13 (2008), no. 56, 1624–1671.
• Aldous, David. The continuum random tree. I. Ann. Probab. 19 (1991), no. 1, 1–28.
• Aldous, David. The continuum random tree. II. An overview. Stochastic analysis (Durham, 1990), London Math. Soc. Lecture Note Ser., vol. 167, Cambridge Univ. Press, Cambridge, 1991, pp. 23–70.
• Aldous, David. The continuum random tree. III. Ann. Probab. 21 (1993), no. 1, 248–289.
• Aldous, David; Pitman, Jim. Inhomogeneous continuum random trees and the entrance boundary of the additive coalescent. Probab. Theory Related Fields 118 (2000), no. 4, 455–482.
• Amini, Omid; Devroye, Luc; Griffiths, Simon; Olver, Neil. Explosion and linear transit times in infinite trees. Preprint, 2014. arXiv:1411.4426
• Berestycki, Julien; Berestycki, NathanaÃ«l; Schweinsberg, Jason. Beta-coalescents and continuous stable random trees. Ann. Probab. 35 (2007), no. 5, 1835–1887.
• Bettinelli, Jérémie. Scaling limit of random planar quadrangulations with a boundary. Ann. Inst. Henri Poincaré Probab. Stat. (to appear), 2011+. arXiv:1111.7227
• Burago, Dmitri; Burago, Yuri; Ivanov, Sergei. A course in metric geometry. Graduate Studies in Mathematics, 33. American Mathematical Society, Providence, RI, 2001. xiv+415 pp. ISBN: 0-8218-2129-6
• Caraceni, Alessandra. The scaling limit of random outerplanar maps. Preprint, 2014. arXiv:1405.1971
• Croydon, David; Hambly, Ben. Spectral asymptotics for stable trees. Electron. J. Probab. 15 (2010), no. 57, 1772–1801.
• Curien, Nicolas; Haas, Bénédicte. The stable trees are nested. Probab. Theory Related Fields 157 (2013), no. 3-4, 847–883.
• Curien, Nicolas; Haas, Bénédicte. Random trees constructed by aggregation. Preprint, 2014. arXiv:1411.4255
• Curien, Nicolas; Haas, Bénédicte; Kortchemski, Igor. The CRT is the scaling limit of random dissections. Random Structures Algorithms (to appear), 2013+. arXiv:1305.3534
• Curien, Nicolas; Kortchemski, Igor. Random stable looptrees. Electron. J. Probab. 19 (2014), 1–35, paper no. 108 (electronic). arXiv:1304.1044
• Duquesne, Thomas. A limit theorem for the contour process of conditioned Galton-Watson trees. Ann. Probab. 31 (2003), no. 2, 996–1027.
• Duquesne, Thomas. The exact packing measure of Lévy trees. Stochastic Process. Appl. 122 (2012), no. 3, 968–1002.
• Duquesne, Thomas; Le Gall, Jean-François. Random trees, Lévy processes and spatial branching processes, Astérisque 281 (2002), vi+147.
• Duquesne, Thomas; Le Gall, Jean-François. Probabilistic and fractal aspects of Lévy trees. Probab. Theory Related Fields 131 (2005), no. 4, 553–603.
• Duquesne, Thomas; Le Gall, Jean-François. The Hausdorff measure of stable trees. ALEA Lat. Am. J. Probab. Math. Stat. 1 (2006), 393–415.
• Duquesne, Thomas; Le Gall, Jean-François. On the re-rooting invariance property of Lévy trees. Electron. Commun. Probab. 14 (2009), 317–326.
• Duquesne, Thomas; Winkel, Matthias. Growth of Lévy trees. Probab. Theory Related Fields 139 (2007), no. 3-4, 313–371.
• Evans, Steven N. Probability and real trees. Lectures from the 35th Summer School on Probability Theory held in Saint-Flour, July 6-23, 2005. Lecture Notes in Mathematics, 1920. Springer, Berlin. 2008. xii+193 pp. ISBN: 978-3-540-74797-0
• Haas, Bénédicte; Miermont, Grégory. The genealogy of self-similar fragmentations with negative index as a continuum random tree. Electron. J. Probab. 9 (2004), no. 4, 57–97 (electronic).
• Haas, Bénédicte; Miermont, Grégory. Scaling limits of Markov branching trees with applications to Galton-Watson and random unordered trees. Ann. Probab. 40 (2012), no. 6, 2589–2666.
• Haas, Bénédicte; Miermont, Grégory; Pitman, Jim; Winkel, Matthias. Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models. Ann. Probab. 36 (2008), no. 5, 1790–1837.
• Haas, Bénédicte; Pitman, Jim; Winkel, Matthias. Spinal partitions and invariance under re-rooting of continuum random trees. Ann. Probab. 37 (2009), no. 4, 1381–1411.
• Jamison, Benton; Orey, Steven; Pruitt, William. Convergence of weighted averages of independent random variables. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 4 1965 40–44.
• Janson, Svante. Limit theorems for triangular urn schemes. Probab. Theory Related Fields 134 (2006), no. 3, 417–452.
• Janson, Svante; Stefánsson, Sigurdur Orn. Scaling limits of random planar maps with a unique large face. Ann. Probab. (to appear), 2012+. arXiv:1212.5072
• Le Gall, Jean-François. Random real trees. Ann. Fac. Sci. Toulouse Math. (6) 15 (2006), no. 1, 35–62.
• Le Gall, Jean-François. Uniqueness and universality of the Brownian map. Ann. Probab. 41 (2013), no. 4, 2880–2960.
• Le Gall, Jean-François; Le Jan, Yves. Branching processes in Lévy processes: the exploration process. Ann. Probab. 26 (1998), no. 1, 213–252.
• Le Gall, Jean-François; Miermont, Grégory. Scaling limits of random planar maps with large faces. Ann. Probab. 39 (2011), no. 1, 1–69.
• Le Jan, Yves. Superprocesses and projective limits of branching Markov process. Ann. Inst. H. Poincaré Probab. Statist. 27 (1991), no. 1, 91–106.
• Marchal, Philippe. A note on the fragmentation of a stable tree. Fifth Colloquium on Mathematics and Computer Science, 489–499, Discrete Math. Theor. Comput. Sci. Proc., AI, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2008.
• Marckert, Jean-François; Miermont, Grégory. The CRT is the scaling limit of unordered binary trees. Random Structures Algorithms 38 (2011), no. 4, 467–501.
• Miermont, Grégory. Self-similar fragmentations derived from the stable tree. I. Splitting at heights. Probab. Theory Related Fields 127 (2003), no. 3, 423–454.
• Miermont, Grégory. Self-similar fragmentations derived from the stable tree. II. Splitting at nodes. Probab. Theory Related Fields 131 (2005), no. 3, 341–375.
• Miermont, Grégory. Invariance principles for spatial multitype Galton-Watson trees. Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008), no. 6, 1128–1161.
• Miermont, Grégory. The Brownian map is the scaling limit of uniform random plane quadrangulations. Acta Math. 210 (2013), no. 2, 319–401.
• Pillai, R. N. On Mittag-Leffler functions and related distributions. Ann. Inst. Statist. Math. 42 (1990), no. 1, 157–161.
• Pitman, Jim. Combinatorial stochastic processes. Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7-24, 2002. With a foreword by Jean Picard. Lecture Notes in Mathematics, 1875. Springer-Verlag, Berlin, 2006. x+256 pp. ISBN: 978-3-540-30990-1; 3-540-30990-X
• Rémy, Jean-Luc. Un procédé itératif de dénombrement d'arbres binaires et son application á leur génération aléatoire. (French) [An iterative procedure for enumerating binary trees and its application to their random generation] RAIRO Inform. Théor. 19 (1985), no. 2, 179–195.