Compact metric measure spaces and Λ-coalescents coming down from infinity

We study topological properties of random metric spaces which arise by Λ-coalescents. These are stochastic processes, which start with an infinite number of lines and evolve through multiple mergers in an exchangeable setting. We show that the resulting Λ-coalescent measure tree is compact iff the Λ-coalescent comes down from infinity, i.e. only consists of finitely many lines at any positive time. If the Λ-coalescent stays infinite, the resulting metric measure space is not even locally compact. Our results are based on general notions of compact and locally compact (isometry classes of) metric measure spaces. In particular, we give characterizations for general (random) metric measure spaces to be (locally) compact using the Gromov-weak topology.