On the spectrum of a neutral multi-allelic Moran model

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

Josué M. Corujo (1, 2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

The purpose of this paper is to provide a complete description of the eigenvalues of the generator of a neutral multi-type Moran model, which is non-reversible in general, continuous-time Markov chain with unknown stationary distribution. Specifically, we consider N individuals, where 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. We also 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 assumption we provide a complete description of the left and right eigenfunctions of the infinitesimal generator of the Markov process. We use spectral techniques to provide non-asymptotic bounds for the rate of convergence of the neutral multi-allelic Moran process with parent independent mutation to stationarity in the total variation and chi-square distances. Moreover, we prove the existence of a strongly optimal cutoff phenomenon in the chi-square distance 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 mutation process presents a strongly optimal total variation cutoff with Gaussian profile when initially all the individuals are of the same type.

Domaines: Probabilités [math.PR]

Josué M. Corujo. On the spectrum of a neutral multi-allelic Moran model. 2020. ⟨hal-02969874v1⟩