The standard additive coalescent

Abstract

Regard an element of the set $$\Delta := {(x_1, x_2, \dots): x_1 \geq x_2 \geq \dots \geq 0, \Sigma_i x_i = 1}$$ as a fragmentation of unit mass into clusters of masses $x_i$. The additive coalescent of Evans and Pitman is the $\Delta$-valued Markov process in which pairs of clusters of masses ${x_i, x_j}$ merge into a cluster of mass $x_i + x_j$ at rate $x_i + x_j$. They showed that a version $(\rm X^{\infty}(t), -\infty < t < \infty)$ of this process arises as a $n \to \infty$ weak limit of the process started at time $-1/2 \log n$ with $n$ clusters of mass $1/n$. We show this standard additive coalescent may be constructed from the continuum random tree of Aldous by Poisson splitting along the skeleton of the tree. We describe the distribution of $\rm X^{\infty}(t)$ on $\Delta$ at a fixed time $t$. We show that the size of the cluster containing a given atom, as a process in $t$, has a simple representation in terms of the stable subordinator of index 1/2. As $t \to -\infty$, we establish a Gaussian limit for (centered and normalized) cluster sizes and study the size of the largest cluster.