The Wright--Fisher model for class--dependent fitness landscapes

We consider a population evolving under mutation and selection. The genotype of an individual is a word of length $\ell$ over a finite alphabet. Mutations occur during reproduction, independently on each locus; the fitness depends on the Hamming class (the distance to a reference sequence $w^*$). Evolution is driven according to the classical Wright--Fisher process. We focus on the proportion of the different classes under the invariant measure of the process. We consider the regime where the length of the genotypes $\ell$ goes to infinity, and both the population size and the inverse of the mutation rate are of order $\ell$. We prove the existence of a critical curve, which depends both on the population size and the mutation rate. Below the critical curve, the proportion of any fixed class converges to $0$, whereas above the curve, it converges to a positive quantity, for which we give an explicit formula.