On Dirichlet eigenvectors for neutral two-dimensional Markov chains

TL;DR

This paper characterizes the eigenvectors of transition matrices for neutral two-type population Markov chains, revealing their structure as products of universal polynomials and functions of total population, with applications to quasistationary analysis.

Contribution

It provides a novel explicit description of eigenvectors for a broad class of neutral two-dimensional Markov chains, linking them to Dirichlet eigenvectors and ordering eigenvalues.

Findings

Eigenvectors are products of universal polynomials and functions of total size.

Eigenvalues are ordered based on Dirichlet eigenvector properties.

Application to quasistationary behavior of nearly neutral populations.

Abstract

We consider a general class of discrete, two-dimensional Markov chains modeling the dynamics of a population with two types, without mutation or immigration, and neutral in the sense that type has no influence on each individual's birth or death parameters. We prove that all the eigenvectors of the corresponding transition matrix or infinitesimal generator \Pi\ can be expressed as the product of "universal" polynomials of two variables, depending on each type's size but not on the specific transitions of the dynamics, and functions depending only on the total population size. These eigenvectors appear to be Dirichlet eigenvectors for \Pi\ on the complement of triangular subdomains, and as a consequence the corresponding eigenvalues are ordered in a specific way. As an application, we study the quasistationary behavior of finite, nearly neutral, two-dimensional Markov chains, absorbed in the sense that 0 is an absorbing state for each component of the process.