Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Horton self-similarity of Kingman’s coalescent tree

Yevgeniy Kovchegov and Ilya Zaliapin

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The paper establishes Horton self-similarity for a tree representation of Kingman’s coalescent process. The proof is based on a Smoluchowski-type system of ordinary differential equations that describes evolution of the number of branches of a given Horton–Strahler order in a tree that represents Kingman’s $N$-coalescent, in a hydrodynamic limit. We also demonstrate a close connection between the combinatorial Kingman’s tree and the combinatorial level set tree of a white noise, which implies Horton self-similarity for the latter.


Cet article prouve l’auto-similarité à la Horton pour la représentation par arbres du processus de coalescence de Kingman. La preuve est basée sur un système d’équations différentielles ordinaires de type Smoluchowski décrivant, dans la limite hydrodynamique, l’évolution du nombre de branches d’un ordre de Horton–Strahler donné dans un arbre représentant le $N$-coalescent de Kingman. Nous prouvons aussi un lien étroit entre l’arbre de Kingman combinatoire et l’arbre combinatoire des ensembles de niveaux d’un bruit blanc, ce qui implique l’auto-similarité à la Horton de ce dernier.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 3 (2017), 1069-1107.

Received: 3 January 2015
Revised: 20 January 2016
Accepted: 15 February 2016
First available in Project Euclid: 21 July 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability
Secondary: 82B99: None of the above, but in this section

Coalescent Kingman’s coalescent Horton–Strahler order Horton self-similarity


Kovchegov, Yevgeniy; Zaliapin, Ilya. Horton self-similarity of Kingman’s coalescent tree. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 3, 1069--1107. doi:10.1214/16-AIHP748.

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  • [1] D. J. Aldous. Deterministic and stochastic models for coalescence (aggregation and coagulation): A review of the mean-field theory for probabilists. Bernoulli 5 (1999) 3–48.
  • [2] N. Berestycki. Recent progress in coalescent theory. Ensaios Mat. 16 (2009) 1–193.
  • [3] J. Bertoin. Random Fragmentation and Coagulation Processes. Cambridge University Press, Cambridge, 2006.
  • [4] G. A. Burd, E. C. Waymire and R. D. Winn. A self-similar invariance of critical binary Galton–Watson trees. Bernoulli 6 (2000) 1–21.
  • [5] R. Darling and J. Norris. Differential equation approximations for Markov chains. Probab. Surv. 5 (2008) 37–79.
  • [6] L. Devroye and P. Kruszewski. A note on the Horton–Strahler number for random trees. Inform. Process. Lett. 56 (1995) 95–99.
  • [7] P. S. Dodds and D. H. Rothman. Scaling, universality, and geomorphology. Annu. Rev. Earth Planet. Sci. 28 (2000) 571–610.
  • [8] M. Drmota. The height of increasing trees. Ann. Comb. 12 (2009) 373–402.
  • [9] R. E. Horton. Erosional development of streams and their drainage basins: Hydrophysical approach to quantitative morphology. Geol. Soc. Am. Bull. 56 (1945) 275–370.
  • [10] W. I. Newman, D. L. Turcotte and A. M. Gabrielov. Fractal trees with side branching. Fractals 5 (1997) 603–614.
  • [11] J. R. Norris. Smoluchowski’s coagulation equation: Uniqueness, nonuniqueness and a hydrodynamic limit for the stochastic coalescent. Ann. Appl. Probab. 9 (1) (1999) 78–109.
  • [12] S. D. Peckham. New results for self-similar trees with applications to river networks. Water Resour. Res. 31 (1995) 1023–1029.
  • [13] J. Pitman. Combinatorial Stochastic Processes. Lecture Notes in Mathematics 1875. Springer-Verlag, Berlin, 2006.
  • [14] R. L. Shreve. Statistical law of stream numbers. J. Geol. 74 (1966) 17–37.
  • [15] R. L. Shreve. Infinite topologically random channel networks. J. Geol. 75 (1967) 178–186.
  • [16] A. N. Strahler. Quantitative analysis of watershed geomorphology. Trans. – Am. Geophys. Union 38 (1957) 913–920.
  • [17] X. G. Viennot. Trees everywhere. In CAAP’90 18–41. Springer, Berlin, 1990.
  • [18] I. Zaliapin and Y. Kovchegov. Tokunaga and Horton self-similarity for level set trees of Markov chains. Chaos Solitons Fractals 45 (3) (2012) 358–372.
  • [19] S. Zanardo, I. Zaliapin and E. Foufoula-Georgiou. Are American rivers Tokunaga self-similar? New results on fluvial network topology and its climatic dependence. J. Geophys. Res. 118 (2013) 166–183.