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

Details Coagulation--fragmentation duality, Poisson--Dirichlet distributions and random recursive trees Coagulation--fragmentation duality, Poisson--Dirichlet distributions and random recursive trees

2005-07-28

arxiv.org/abs/math/0507591v3

Annals of Applied Probability 2006, Vol. 16, No. 4, 1733-1750

Mathematics

Probability

Published at http://dx.doi.org/10.1214/105051606000000655 in the Annals of Applied Probability (http://www.imstat.org/aap/) by

Scientific paper

10.1214/105051606000000655

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 $\operatorname {PD}(\alpha,\theta)$ that take values in this space. We introduce families of random fragmentation and coagulation operators $\mathrm {Frag}_{\alpha}$ and $\mathrm {Coag}_{\alpha,\theta}$, respectively, with the following property: if the input to $\mathrm {Frag}_{\alpha}$ has $\operatorname {PD}(\alpha,\theta)$ distribution, then the output has $\operatorname {PD}(\alpha,\theta+1)$ distribution, while the reverse is true for $\mathrm {Coag}_{\alpha,\theta}$. This result may be proved using a subordinator representation and it provides a companion set of relations to those of Pitman between $\operatorname {PD}(\alpha,\theta)$ and $\operatorname {PD}(\alpha\beta,\theta)$. Repeated application of the $\mathrm {Frag}_{\alpha}$ 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.

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