The genealogy of a solvable population model under selection with dynamics related to directed polymers

We consider a stochastic model describing a constant size $N$ population that may be seen as a directed polymer in random medium with $N$ sites in the transverse direction. The population dynamics is governed by a noisy traveling wave equation describing the evolution of the individual fitnesses. We show that under suitable conditions the generations are independent and the model is characterized by an extended Wright-Fisher model, in which the individual $i$ has a random fitness $\eta_i$ and the joint distribution of offspring $(\nu_1,\ldots,\nu_N)$ is given by a multinomial law with $N$ trials and probability outcomes $\eta_i$'s. We then show that the average coalescence times scales like $\log N$ and that the limit genealogical trees are governed by the Bolthausen-Sznitman coalescent, which validates the predictions by Brunet, Derrida, Mueller and Munier for this class of models. We also study the extended Wright-Fisher model, and show that, under certain conditions on $\eta_i$, the limit may be Kingman's coalescent, a coalescent with multiple collisions, or a coalescent with simultaneous multiple collisions.