On trees invariant under edge contraction

Abstract

We study random trees which are invariant in law under the operation of contracting each edge independently with probability $p\in (0,1)$. We show that all such trees can be constructed through Poissonian sampling from a certain class of random measured $\mathbb{R}$-trees satisfying a natural scale invariance property. This has connections to exchangeable partially ordered sets, real-valued self-similar increasing processes and quasi-stationary distributions of Galton--Watson processes.

33 pages, 2 figures, journal version

References in corpus (2)

• A note on Gromov-Hausdorff-Prokhorov distance between (locally) compact measure spaces
• The gap between Gromov-vague and Gromov-Hausdorff-vague topology

Cited by in corpus (1)

• The $\lambda$-invariant measures of subcritical Bienaymé--Galton--Watson processes