We derive explicit formulas for the two first moments of he site frequency spectrum (SF S n,b ) 1≤b≤n−1 of the Bolthausen-Sznitman coalescent along with some precise and efficient approximations, even for small sample sizes n. These results provide new L 2 -asymptotics for some values of b = o(n). We also study the length of internal branches carrying b > n/2 individuals. In this case we obtain the distribution function and a convergence in law. Our results rely on the random recursive tree construction of the Bolthausen-Sznitman coalescent.

We derive explicit formulas for the two first moments of the site frequency spectrum (SF S n,b ) 1≤b≤n−1 of the Bolthausen-Sznitman coalescent along with some precise and efficient approximations, even for small sample sizes n. These results provide new L 2 -asymptotics for some values of b = o(n). We also study the length of internal branches carrying b > n/2 individuals, we provide their joint distribution function as well as a convergence in law for their marginal distribution. Our results rely on the random recursive tree construction of the Bolthausen-Sznitman coalescent.

Since the foundation of population genetics, it is believed that directional selection should reduce genetic diversity. We present an exactly solvable population model which contradicts this intuition. The population is modelled as a cloud of particles evolving in a 1-dimensional fitness space (fitness wave). We show the existence of a phase transition which separates the parameter space into a weak and a strong selection regimes. We find that genetic diversity is highly non-monotone in the selection strength and, in contrast with the common intuition, our model predicts that genetic diversity is typically higher in the strong selection regime. This apparent paradox is resolved by observing that a higher selection strength increases the absolute fitness of the wave, but typically generate lower relative fitness between individuals within the wave. These findings entail that inferring the magnitude of natural selection from genetic data may raise some serious conceptual issues.Along the way, we uncover a new phase transition in front propagation. Namely, we show that the transition from weak to strong selection can be reformulated in terms of a transition from fully-pulled to semi-pulled waves. This transition is the pulled analog to the semi-pushed to fully-pushed regimes observed in noisy-FKPP travelling waves in the presence of Allee effect.

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