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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Abstract

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.

Résumé

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1500624031

Digital Object Identifier
doi:10.1214/16-AIHP748

Mathematical Reviews number (MathSciNet)
MR3689961

Zentralblatt MATH identifier
1381.60029

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

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

Citation

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. https://projecteuclid.org/euclid.aihp/1500624031


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