Non-Gibbsian Multivariate Ewens Probability Distributions on Regular Trees

TL;DR

This paper investigates whether Ewens' sampling formula distributions are Gibbsian on regular trees, concluding they are non-Gibbsian, and establishes a sufficient condition for their consistency, opening new research directions.

Contribution

It demonstrates that Ewens distributions are non-Gibbsian on regular trees and provides a sufficient condition for their consistency, advancing the theory of non-Gibbsian distributions on trees.

Findings

Ewens distributions are non-Gibbsian on regular trees

A sufficient condition for the consistency of these distributions is derived

Lays groundwork for future research in non-Gibbsian probability distributions

Abstract

Ewens' sampling formula (ESF) provides the probability distribution governing the number of distinct genetic types and their respective frequencies at a selectively neutral locus under the infinitely-many-alleles model of mutation. A natural and significant question arises: ``Is the Ewens probability distribution on regular trees Gibbsian?" In this paper, we demonstrate that Ewens probability distributions can be regarded as non-Gibbsian distributions on regular trees and derive a sufficient condition for the consistency condition. This study lays the groundwork for a new direction in the theory of non-Gibbsian probability distributions on trees.