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

Abstract

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 Frag α and Coag α,θ , respectively, with the following property: if the input to Frag α has PD(α, θ) distribution, then the output has PD(α, θ + 1) distribution, while the reverse is true for Coag α,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 Frag α 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.

Regenerative tree growth: Binary self-similar continuum random trees and Poisson–Dirichlet compositions

“…In the (α, 1 − α) case, growth by crushing beads is closely connected to growth rules for random recursive trees studied by Dong, Goldschmidt and Martin [6]. Specifically, we can associate with R k a tree V k with k vertices labeled by [k] and infinitely many unlabeled vertices, all marked by weights; let V 1 consist of a root labeled 1 and infinitely many unlabeled children marked by the sequence m 1 of masses of the string of beads on R 1 ; to construct V k+1 from V k , identify the unlabeled leaf in V k marked by the size of the chosen bead s k , label it by k + 1 and add infinitely many children of vertex k + 1, marked by the sizes m k+1 of the crushed bead.…”

Regenerative tree growth: Markovian embedding of fragmenters, bifurcators, and bead splitting processes

“…In the terminology of [14], part (i) is a particular case of their result that crushing a size-biased pick from a ranked list with distribution PD(α, θ) into PD(α, 1 − α)-distributed proportions yields a PD(α, θ + 1) ranked vector. This can also be read from Aldous's sequential description of the growth of (R n , n ≥ 1).…”

Asymptotic Properties of a Random Graph with Duplications

“…Coagulation-fragmentation models are frequently used in several areas; see, e.g. [9]. These models have been applied to random graph models [15], namely, for the Erdős-Rényi model, which is completely different from ours.…”