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 $\mathrm{\ell }$ 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 $w^{\ast }$). 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 $\mathrm{\ell }$ goes to infinity, and both the population size and the inverse of the mutation rate are of order $\mathrm{\ell }$. 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.