Β-coalescents and Stable Galton-watson Trees

Representation of coalescent process using pruning of trees has been used by Goldschmidt and Martin for the Bolthausen-Sznitman coalescent and by Abraham and Delmas for the β(3/2, 1/2)-coalescent. By considering a pruning procedure on stable GaltonWatson tree with n labeled leaves, we give a representation of the discrete β(1 + α, 1− α)coalescent, with α ∈ [1/2, 1) starting from the trivial partition of the n first integers. The construction can also be made directly on the stable continuum Lévy tree, with parameter 1/α, simultaneously for all n. This representation allows to use results on the asymptotic number of coalescence events to get the asymptotic number of cuts in stable Galton-Watson tree (with infinite variance for the reproduction law) needed to isolate the root. Using convergence of the stable Galton-Watson tree conditioned to have infinitely many leaves, one can get the asymptotic distribution of blocks in the last coalescence event in the β(1+α, 1−α)coalescent.