The Annals of Applied Probability

Coagulation–fragmentation duality, Poisson–Dirichlet distributions and random recursive trees

Rui Dong, Christina Goldschmidt, and James B. Martin

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In this paper we give a new example of duality between fragmentation and coagulation operators. Consider the space of partitions of mass (i.e., decreasing sequences of nonnegative real numbers whose sum is 1) and the two-parameter family of Poisson–Dirichlet distributions PD (α,θ) that take values in this space. We introduce families of random fragmentation and coagulation operators Fragα and Coagα,θ, respectively, with the following property: if the input to Fragα has PD (α,θ) distribution, then the output has PD (α,θ+1) distribution, while the reverse is true for Coagα,θ. This result may be proved using a subordinator representation and it provides a companion set of relations to those of Pitman between PD (α,θ) and PD (αβ,θ). Repeated application of the Fragα operators gives rise to a family of fragmentation chains. We show that these Markov chains can be encoded naturally by certain random recursive trees, and use this representation to give an alternative and more concrete proof of the coagulation–fragmentation duality.

Article information

Ann. Appl. Probab. Volume 16, Number 4 (2006), 1733-1750.

First available in Project Euclid: 17 January 2007

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Zentralblatt MATH identifier

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60G51: Processes with independent increments; Lévy processes 60C05: Combinatorial probability 05C05: Trees

Coagulation fragmentation Poisson–Dirichlet distributions recursive trees time reversal


Dong, Rui; Goldschmidt, Christina; Martin, James B. Coagulation–fragmentation duality, Poisson–Dirichlet distributions and random recursive trees. Ann. Appl. Probab. 16 (2006), no. 4, 1733--1750. doi:10.1214/105051606000000655.

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