Multiple Partition Structures and Harmonic Functions on Branching Graphs

TL;DR

This paper introduces multiple partition structures related to harmonic functions on branching graphs, generalizing Kingman's structures, with applications to population genetics and a new multiple Poisson-Dirichlet distribution.

Contribution

It establishes a bijective correspondence between harmonic functions and probability measures on a generalized Thoma set, extending the theory of partition structures.

Findings

Established a representation theorem for multiple partition structures.

Constructed a probability measure on wreath products related to population genetics.

Defined a multiple Poisson-Dirichlet distribution as a generalization of the classical distribution.

Abstract

We introduce and study multiple partition structures which are sequences of probability measures on families of Young diagrams subjected to a consistency condition. The multiple partition structures are generalizations of Kingman's partition structures, and are motivated by a problem of population genetics. They are related to harmonic functions and coherent systems of probability measures on a certain branching graph. The vertices of this graph are multiple Young diagrams (or multiple partitions), and the edges depend on the Jack parameter. Our main result establishes a bijective correspondence between the set of harmonic functions on the graph and probability measures on the generalized Thoma set. The correspondence is determined by a canonical integral representation of harmonic functions. As a consequence we obtain a representation theorem for multiple partition structures. We give an example of a multiple partition structure which is expected to be relevant for a model of population genetics for the genetic variation of a sample of gametes from a large population. Namely, we construct a probability measure on the wreath product of a finite group with the symmetric group. The constructed probability measure defines a multiple partition structure which is a generalization of the Ewens partition structure studied by Kingman. We show that this multiple partition structure can be represented in terms of a multiple analogue of the Poisson-Dirichlet distribution called the multiple Poisson-Dirichlet distribution in the paper.