Mass-structure of weighted real trees

Abstract

Rooted, weighted continuum random trees are used to describe limits of sequences of random discrete trees. Formally, they are random quadruples $(\mathcal{T},d,r,p)$, where $(\mathcal{T},d)$ is a tree-like metric space, $r\in \mathcal{T}$ is a distinguished root, and $p$ is a probability measure on this space. The underlying branching structure is carried implicitly in the metric $d$. We explore various ways of describing the interaction between branching structure and mass in $(\mathcal{T},d,r,p)$ in a way that depends on $d$ only by way of this branching structure. We introduce a notion of mass-structure equivalence and show that two rooted, weighted $\mathbb{R}$-trees are equivalent in this sense if and only if the discrete hierarchies derived by i.i.d. sampling from their weights, in a manner analogous to Kingman's paintbox, have the same distribution. We introduce a family of trees, called "interval partition trees" that serve as representatives of mass-structure equivalence classes, and which naturally represent the laws of the aforementioned hierarchies.

30 pages, 4 figures

References in corpus (3)

• Spinal partitions and invariance under re-rooting of continuum random trees
• Regenerative tree growth: Binary self-similar continuum random trees and Poisson--Dirichlet compositions
• The Aldous chain on cladograms in the diffusion limit

Cited by in corpus (4)

• The Aldous chain on cladograms in the diffusion limit
• Spaces of algebraic measure trees and triangulations of the circle
• Aldous diffusion I: a projective system of continuum $k$-tree evolutions
• Exchangeable and Sampling Consistent Distributions on Rooted Binary Trees