Kingman's coalescent as a random metric space
Kingman's coalescent is a Markov process with state--space the collection of partitions of the positive integers. Its initial state is the trivial partition of singletons and it evolves by successive pairwise mergers of blocks. The coalescent induces a metric on the positive integers: the distance between two integers is the time until they both belong to the same block. We investigate the completion of this (random) metric space. We show that almost surely it is a compact metric space with Hausdorff and packing dimension both $1$, and it has positive capacities in precisely the same gauges as the unit interval.