{
  "id": "2209.01855",
  "version": "v1",
  "published": "2022-09-05T09:26:31.000Z",
  "updated": "2022-09-05T09:26:31.000Z",
  "title": "Multiple Partition Structures and Harmonic Functions on Branching Graphs",
  "authors": [
    "Eugene Strahov"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.CO",
    "math.MP",
    "math.RT"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-05T09:26:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "harmonic functions",
      "branching graph",
      "study multiple partition structures",
      "young diagrams",
      "population genetics"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}