Site Frequency Spectrum of the Bolthausen-Sznitman Coalescent

We derive explicit formulas for the two first moments of he site frequency spectrum ( SFS 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.