Random walks and the additive coagulation equation

Numerical simulations of hierarchical gravitational clustering in an expanding universe show that, as time evolves, small clusters merge with each other to form larger clusters, whereas fragmentation of clusters is relatively uncommon. Stochastic models of the hierarchical clustering process can provide insight into, as well as useful approximations to, the evolution measured in these simulations. The Poisson random walk excursion model, the Poisson Galton–Watson branching process, and the monodisperse additive coagulation equation are three examples of such stochastic models. When initially identical particles cluster from an initially Poisson spatial distibution, all three approaches give essentially the same description of how clusters grow.  This paper shows that clustering from an initially Poisson distribution in which the initial particles do not all have the same mass can be described by simple generalizations of the models above. Such an initial distribution is said to be ‘compound Poisson’. Therefore, excursions of random walks associated with compound Poisson distributions are studied here. In such an excursion set model, clusters grow in essentially the same way as they do in the polydisperse additive coagulation model. Thus, the interrelations between excursion set, branching process and coagulation models of clustering, associated with the Poisson distribution, also apply to compound Poisson distributions. This means that, within the context of these models, when the initial conditions are compound Poisson then merger and accretion rates, and the entire merger history tree, can all be written analytically, just as for clustering from Poisson initial conditions.