The distribution of the quasispecies for a Galton--Watson process on the sharp peak landscape

We study a classical multitype Galton--Watson process with mutation and selection. The individuals are sequences of fixed length over a finite alphabet. On the sharp peak fitness landscape together with independent mutations per locus, we show that, as the length of the sequences goes to $\infty$ and the mutation probability goes to 0, the asymptotic relative frequency of the sequences differing on $k$ digits from the master sequence approaches $$(\sigma e^{-a}-1)\frac{a^k}{k!}\sum_{i\geq1}\frac{i^k}{\sigma^i}\,,$$ where $\sigma$ is the selective advantage of the master sequence and $a$ is the product of the length of the chains with the mutation probability. The probability distribution $\mathcal{Q}(\sigma,a)$ on the non negative integers given by the above formula is the quasispecies distribution with parameters $\sigma$ and $a$.