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

TL;DR

This paper introduces a novel encoding method for additive and multiplicative coalescent processes using the Prim order, leading to new insights and solutions to open problems in the asymptotic analysis of these processes.

Contribution

It proposes using the Prim order for encoding coalescent processes, connecting different representations and solving an open problem about convergence in the multiplicative case.

Findings

Prim order encoding reveals new process representations.

Proves convergence of rescaled masses to Brownian excursion lengths.

Constructs a version of the augmented multiplicative coalescent with Poisson processes.

Abstract

We revisit the discrete additive and multiplicative coalescents, starting with $n$ particles with unit mass. These cases are known to be related to some "combinatorial coalescent processes": a time reversal of a fragmentation of Cayley trees or a parking scheme in the additive case, and the random graph process $(G(n,p))_p$ in the multiplicative case. Time being fixed, encoding these combinatorial objects in real-valued processes indexed by the line is the key 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 classical breadth-first (or depth-first) traversal to encode the combinatorial coalescent processes. In the additive case, this yields interesting connections between the different representations of the process. In the multiplicative case, it allows 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 $\lambda$, converges in distribution to the reordered sequence of lengths of the excursions above the 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 augmented multiplicative coalescent of Bhamidi, Budhiraja and Wang [Probab. Theory Rel., to appear] using an additional Poisson point process.