Large-scale behaviour and hydrodynamic limit of beta coalescents

Abstract

We quantify the behaviour at large scales of the beta coalescent Π = {Π(t), t ≥ 0} with parameters a, b > 0. Specifically, we study the rescaled block size spectrum of Π(t) and of its restriction Πn(t) to {1, . . . , n}. Our main result is a Law of Large Numbers type of result if Π comes down from infinity. In the case of Kingman’s coalescent the derivation of this so-called hydrodynamic limit has been known since the work of Smoluchowski [30]. We extend Smoluchowski’s result to beta coalescents and show that if Π comes down from infinity both rescaled spectra