Fluctuations Analysis of Finite Discrete Birth and Death Chains with Emphasis on Moran Models with Mutations

Abstract

The Moran model is a discrete-time birth and death Markov chain describing the evolution of the number of type 1 alleles in a haploid population with two alleles whose total size N is preserved during the course of evolution. Bias mechanisms such as mutations or selection can affect its neutral dynamics. For the ergodic Moran model with mutations, we get interested in the fixation probabilities of a mutant, the growth rate of fluctuations, the first hitting time of the equilibrium state starting from state {0}, the first return time to the equilibrium state, and the first hitting time of {N} starting from {0}, together with the time needed for the walker to reach its invariant measure, again starting from {0}. For the last point, an appeal to the notion of Siegmund duality is necessary, and a cutoff phenomenon will be made explicit. We are interested in these problems in the large population size limit N→∞. The Moran model with mutations includes the heat exchange models of Ehrenfest and Bernoulli-Laplace as particular cases; these were studied from the point of view of the controversy concerning irreversibility (H-theorem) and the recurrence of states.