The Standard Additive Coalescent

as a fragmentation of unit mass into clusters of masses xi. The additive coalescent of Evans and Pitman is the -valued Markov process in which pairs of clusters of masses xi xj merge into a cluster of mass xi + xj at rate xi + xj. They showed that a version X∞ t −∞ < t < ∞ of this process arises as a n → ∞ weak limit of the process started at time − 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 X∞ t on 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 → −∞, we establish a Gaussian limit for (centered and normalized) cluster sizes and study the size of the largest cluster.