Spectrum and ergodicity of a neutral multi-allelic Moran model

Abstract: We will present some recent results on the study of a neutral multi-allelic Moran model, which is a finite continuous-time Markov process. For this process, it is assumed that the individuals interact according to two processes: a mutation process where they mutate independently of each other according to an irreducible rate matrix, and a Moran type reproduction process, where two individuals are uniformly chosen, one dies and the other is duplicated. During this talk we will discuss some recent results for the spectrum of the generator of the neutral multi-allelic Moran process, providing explicit expressions for its eigenvalues in terms of the eigenvalues of the rate matrix that drives the mutation process. Our approach does not require that the mutation process be reversible, or even diagonalizable. Additionally, we will discuss some applications of these results to the study of the speed of convergence to stationarity of the Moran process for a process with general mutation scheme. We specially focus on the case where the mutation scheme satisfies the so called "parent independent" condition, where (and only where) the neutral Moran model becomes reversible. In this later case we can go further and prove the existence of a cutoff phenomenon for the convergence to stationarity.

This presentation is based on a recently submitted work, for which a preprint is available at arxiv.org/abs/2010.08809.

Mathematics

Audience: researchers in the topic

Pisa Online Seminar in Probability