On the spectrum and the ergodicity of a neutral multi-allelic Moran model

Sur le spectre et l'ergodicité d'un modèle Moran neutre multi-allélique

Josué M. Corujo (1)

1 IRMA - Institut de Recherche Mathématique Avancée

The purpose of this paper is to provide a complete description of the eigenvalues of the generator of a neutral multi-type Moran model, and the applications to the study of the speed of convergence to stationarity. The Moran model we consider is a non-reversible in general, continuous-time Markov chain with unknown stationary distribution. Specifically, we consider $N$ individuals such that each one of them is of one type among $K$ possible allelic types. The individuals interact in two ways: by an independent irreducible mutation process and by a reproduction process, where a pair of individuals is randomly chosen, one of them dies and the other reproduces. Our main result provides explicit expressions for the eigenvalues of the infinitesimal generator matrix of the Moran process, in terms of the eigenvalues of the jump rate matrix. As consequences of this result, we study the convergence in total variation of the process to stationarity. Our results include a lower bound for the mixing time of the Moran process when the mutation process allows a real eigenvalue. Furthermore, we study in detail the spectral decomposition of the neutral multi-allelic Moran model with parent independent mutation scheme, which turns to be the unique mutation scheme that makes the neutral Moran process reversible. Under the parent independent mutation, we also prove the existence of a cutoff phenomenon in the chi-square and the total variation distances when initially all the individuals are of the same type and the number of individuals tends to infinity. Additionally, in the absence of reproduction, we prove that the the total variation distance to stationarity of the parent independent mutation process, when initially all the individuals are of the same type, has a Gaussian profile.

Mots clés: neutral multi-allelic Moran process; Fleming-Viot type particle system; interacting particle system; convergence rate to stationarity; finite continuous-time Markov chains; multivariate Hahn polynomials; cutoff; multivariate Krawtchouk polynomials

Domaines: Probabilités [math.PR]

Josué M. Corujo. On the spectrum and the ergodicity of a neutral multi-allelic Moran model. ALEA : Latin American Journal of Probability and Mathematical Statistics, 2023, 20 (18), pp.505-546. ⟨10.30757/ALEA.v20-18⟩. ⟨hal-02969874v3⟩