The Annals of Applied Probability

Mutations on a random binary tree with measured boundary

Jean-Jil Duchamps and Amaury Lambert

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Consider a random real tree whose leaf set, or boundary, is endowed with a finite mass measure. Each element of the tree is further given a type, or allele, inherited from the most recent atom of a random point measure (infinitely-many-allele model) on the skeleton of the tree. The partition of the boundary into distinct alleles is the so-called allelic partition.

In this paper, we are interested in the infinite trees generated by supercritical, possibly time-inhomogeneous, binary branching processes, and in their boundary, which is the set of particles “coexisting at infinity”. We prove that any such tree can be mapped to a random, compact ultrametric tree called the coalescent point process, endowed with a “uniform” measure on its boundary which is the limit as $t\to\infty$ of the properly rescaled counting measure of the population at time $t$.

We prove that the clonal (i.e., carrying the same allele as the root) part of the boundary is a regenerative set that we characterize. We then study the allelic partition of the boundary through the measures of its blocks. We also study the dynamics of the clonal subtree, which is a Markovian increasing tree process as mutations are removed.

Article information

Ann. Appl. Probab., Volume 28, Number 4 (2018), 2141-2187.

Received: January 2017
Revised: September 2017
First available in Project Euclid: 9 August 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C05: Trees 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 54E45: Compact (locally compact) metric spaces 60G51: Processes with independent increments; Lévy processes 60G55: Point processes 60G57: Random measures 60K15: Markov renewal processes, semi-Markov processes 92D10: Genetics {For genetic algebras, see 17D92}

Coalescent point process branching process random point measure allelic partition regenerative set tree-valued process


Duchamps, Jean-Jil; Lambert, Amaury. Mutations on a random binary tree with measured boundary. Ann. Appl. Probab. 28 (2018), no. 4, 2141--2187. doi:10.1214/17-AAP1353.

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