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$β$-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 Galton-Watson tree with $n$ labeled leaves, we give a representation of the discrete $β(1+α,1-α)$-coalescent, with $α\in [1/2,1)$ starting from the trivial partition of the $n$ first integers. The construction can also be made directly on the stable continuum L{é}vy 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.