  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 an 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 and show a
lower bound for the mixing time of the Moran process. Furthermore, we study in
detail the spectral decomposition of the neutral multi-allelic Moran model with
parent independent mutation scheme, which is 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 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.
