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The standard additive coalescent


Regard an element of the set Δ := {(x1, x2, . . .): x1 ≥ x2 ≥ ⋯ ≥ 0, ∑i xi = 1} as a fragmentation of unit mass into clusters of masses xi. The additive coalescent of Evans and Pitman is the Δ-valued Markov process in which pairs of clusters of masses {xi, x j} merge into a cluster of mass xi + x j at rate xi + x j. They showed that a version (X∞(t), -∞ < t < ∞) of this process arises as a n → ∞ 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 X∞(t) on Δ 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.

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