Non-Gibbsian Multivariate Ewens Probability Distributions on Regular Trees
F. H. Haydarov, Z. E. Mustafoyeva, U. A. Rozikov
TL;DR
The paper addresses whether Ewens' sampling formula distributions on regular trees admit a Gibbsian representation under standard Gibbs specifications. It translates the ESF into a tree-based spin system, defines a potential $U_\Lambda$, a Hamiltonian $H_\Lambda$, and finite-volume measures with normalizing constants $Z_{|\Lambda|}( heta)$, and then demonstrates that the ESF-induced potential fails the usual summability condition, rendering the measures non-Gibbsian. It develops a consistency framework for tree extensions, analyzing two vertex-addition scenarios and providing explicit expressions for the partition-function ratio $Z_\Delta/Z_\Lambda$. The work establishes a foundational link between population-genetics distributions and non-Gibbsian probability theory on trees, enabling RG-like analyses in combinatorial settings and motivating further study of non-Gibbsian measures on graphical structures.
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.