Electronic Journal of Probability

Scaling limits for some random trees constructed inhomogeneously

Nathan Ross and Yuting Wen

Full-text: Open access

Abstract

We define some new sequences of recursively constructed random combinatorial trees, and show that, after properly rescaling graph distance and equipping the trees with the uniform measure on vertices, each sequence converges almost surely to a real tree in the Gromov-Hausdorff-Prokhorov sense. The limiting real trees are constructed via line-breaking the half real-line with a Poisson process having rate $(\ell +1)t^\ell dt$, for each positive integer $\ell $, and the growth of the combinatorial trees may be viewed as an inhomogeneous generalization of Rémy’s algorithm.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 5, 35 pp.

Dates
Received: 17 December 2016
Accepted: 31 August 2017
First available in Project Euclid: 3 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1517626965

Digital Object Identifier
doi:10.1214/17-EJP101

Mathematical Reviews number (MathSciNet)
MR3761565

Zentralblatt MATH identifier
1390.60046

Subjects
Primary: 60C05: Combinatorial probability 05C10: Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25]

Keywords
Scaling limit continuum random tree Gromov-Hausdorff-Prokhorov topology generalized Pólya urn line-breaking

Rights
Creative Commons Attribution 4.0 International License.

Citation

Ross, Nathan; Wen, Yuting. Scaling limits for some random trees constructed inhomogeneously. Electron. J. Probab. 23 (2018), paper no. 5, 35 pp. doi:10.1214/17-EJP101. https://projecteuclid.org/euclid.ejp/1517626965


Export citation

References

  • [1] Abraham, R., Delmas, J.-F., & Hoscheit, P. (2013). A note on the Gromov-Hausdorff-Prokhorov distance between (locally) compact metric measure spaces. Electron. J. Probab. 18, 21 pp.
  • [2] Addario-Berry, L., Broutin, N., Goldschmidt, C., & Miermont, G. (2013). The scaling limit of the minimum spanning tree of the complete graph. arXiv:1301.1664.
  • [3] Addario-Berry, L., & Wen, Y. (2015). Joint convergence of random quadrangulations and their cores. arXiv:1503.06738. To appear in Ann. Inst. H. Poincaré Probab. Statist. (B).
  • [4] Aldous, D. (1991). The continuum random tree. I. Ann. Probab. 19, 1-28.
  • [5] Aldous, D. (1991). The continuum random tree. II. An overview. London Math. Soc. Lecture Note Ser. 167, 23-70.
  • [6] Aldous, D. (1993). The continuum random tree III. Ann. Probab. 21, 248-289.
  • [7] Aldous, D. (1983). Exchangeability and related topics. In École d’été de probabilités de Saint-Flour, XIII 1983 (pp. 1-198). Springer Berlin Heidelberg.
  • [8] Aldous, D., & Pitman, J. (1999). A family of random trees with random edge lengths. Random Struct. Alg. 15, 176-195.
  • [9] Aldous, D., & Pitman, J. (1999). Inhomogeneous continuum random trees and the entrance boundary of the additive coalescent. Probab. Theory Related Fields 118, 455-482.
  • [10] Aldous, D., Miermont, G., & Pitman, J. (2004). The exploration process of inhomogeneous continuum random trees, and an extension of Jeulin’s local time identity. Probab. Theory Related Fields 129, 182-218.
  • [11] Amini, O., Devroye, L., Griffiths, S., & Olver, N. (2017). Explosion and linear transit times in infinite trees. Probab. Theory Related Fields 167, 325-347.
  • [12] Chen, B., Ford, D., & Winkel, M. (2009). A new family of Markov branching trees: the alpha-gamma model. Electron. J. Probab. 14, 400-430.
  • [13] Curien, N., & Haas, B. (2014). Random trees constructed by aggregation. arXiv:1411.4255.
  • [14] Curien, N., & Haas, B. (2013). The stable trees are nested. Probab. Theory Related Fields 157, 847-883.
  • [15] Duquesne, T., & Le Gall, J.-F. (2002). Random trees, Lévy processes and spatial branching processes. Astérisque 281, vi+147.
  • [16] Evans, S. (2008). Probability and real trees. Lecture Notes in Mathematics, 1920. Springer Berlin Heidelberg.
  • [17] Goldschmidt, C., & Haas, B. (2015). A line-breaking construction of the stable trees. Electron. J. Probab. 20, 24 pp.
  • [18] Haas, B. (2016). Asymptotics of heights in random trees constructed by aggregation. Electron. J. Probab. 22, 25 pp.
  • [19] Haas, B. (2016). Scaling limits of Markov-Branching trees and applications.arXiv:1605.07873.
  • [20] Haas, B., & Miermont, G. (2012). Scaling limits of Markov branching trees with applications to Galton-Watson and random unordered trees. Ann. Probab. 40, 2589-2666.
  • [21] Haas, B., & Stephenson, R. (2015). Scaling limits of $ k $-ary growing trees. Ann. Inst. H. Poincaré Probab. Statist 51, 1314-1341.
  • [22] Kortchemski, I. Invariance principles for Galton-Watson trees conditioned on the number of leaves. Stochastic Process. Appl. 122, 3126-3172.
  • [23] Marchal, P. (2008). A note on the fragmentation of a stable tree. Fifth Colloquium on Mathematics and Computer Science. Discrete Math. Theor. Comput. Sci. Proc., AI, 489-499.
  • [24] Marckert, J. F., & Miermont, G. (2011). The CRT is the scaling limit of unordered binary trees. Random Struct. & Alg. 38, 467-501.
  • [25] McDiarmid, C. (1998). Concentration. Probabilistic methods for algorithmic discrete mathematics (pp. 195-248). Springer Berlin Heidelberg.
  • [26] Miermont, G. (2009). Tessellations of random maps of arbitrary genus. Ann. Scientifiques de l’École Normale Supérieuree 42, 725-781.
  • [27] Peköz, E., Röllin, A., & Ross, N. (2016). Generalized gamma approximation with rates for urns, walks and trees. Ann. Probab. 44, 1776-1816.
  • [28] Peköz, E., Röllin, A., & Ross, N. (2017). Joint degree distributions of preferential attachment random graphs. Adv. in Appl. Probab. 49, 368-387.
  • [29] Pitman, J., & Winkel, M. (2009). Regenerative tree growth: binary self-similar continuum random trees and Poisson-Dirichlet compositions. Ann. Probab. 37, 1999-2041.
  • [30] Pitman, J., & Winkel, M. (2015). Regenerative tree growth: Markovian embedding of fragmenters, bifurcators, and bead splitting processes. Ann. Probab. 43, 2611-2646.
  • [31] Pitman, J., Rizzolo, D., & Winkel, M. (2014). Regenerative tree growth: structural results and convergence. Electron. J. Probab. 19, no. 70, 27 pp.
  • [32] Rémy, J. L. (1985). Un procédé itératif de dénombrement d’arbres binaires et son application à leur génération aléatoire. RAIRO, Informatique théorique 19, 179-195.
  • [33] Rembart, F., & Winkel, M. (2016). Recursive construction of continuum random trees. arXiv:1607.05323.
  • [34] Rizzolo, D. (2015). Scaling limits of Markov branching trees and Galton-Watson trees conditioned on the number of vertices with out-degree in a given set. Ann. Inst. H. Poincaré Probab. Statist. (B) 51, 512-532.
  • [35] Strassen, V. (1965). The existence of probability measures with given marginals. Ann. Math. Statist., 423-439.
  • [36] Wen, Y. (2017). The Brownian plane with minimal neck baby universe. Random Struct. Alg. doi:10.1002/rsa.20722