A new encoding of coalescent processes. Applications to the additive and multiplicative cases

摘要

We revisit the discrete additive and multiplicative coalescents, startingwith $n$ particles with unit mass. These cases are known to be related to some"combinatorial coalescent processes": a time reversal of a fragmentation ofCayley trees or a parking scheme in the additive case, and the random graphprocess $(G(n,p))_p$ in the multiplicative case. Time being fixed, encodingthese combinatorial objects in real-valued processes indexed by the line is thekey to describing the asymptotic behaviour of the masses as $n\to +\infty$. We propose to use the Prim order on the vertices instead of the classicalbreadth-first (or depth-first) traversal to encode the combinatorial coalescentprocesses. In the additive case, this yields interesting connections betweenthe different representations of the process. In the multiplicative case, itallows one to answer to a stronger version of an open question of Aldous [Ann.Probab., vol. 25, pp. 812--854, 1997]: we prove that not only the sequence of(rescaled) masses, seen as a process indexed by the time $\lambda$, convergesin distribution to the reordered sequence of lengths of the excursions abovethe current minimum of a Brownian motion with parabolic drift $(B_t+\lambda t -t^2/2, t\geq 0)$, but we also construct a version of the standard augmentedmultiplicative coalescent of Bhamidi, Budhiraja and Wang [Probab. Theory Rel.,to appear] using an additional Poisson point process.