  This article is concerned with the long time behavior of neutral genetic
population models, with fixed population size. We design an explicit, finite,
exact, genealogical tree based representation of stationary populations that
holds both for finite and infinite types (or alleles) models. We then analyze
the decays to the equilibrium of finite populations in terms of the convergence
to stationarity of their first common ancestor. We estimate the Lyapunov
exponent of the distribution flows with respect to the total variation norm. We
give bounds on these exponents only depending on the stability with respect to
mutation of a single individual; they are inversely proportional to the
population size parameter.
