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. Intuitively, these trees have a combinatorial “underlying branching structure” implied by their topology but otherwise independent of the metric $d$. We explore various ways of making this rigorous, using the weight $p$ to do so without losing the fractal complexity possible in continuum trees. We introduce a notion of mass-structural 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.
Highlights
This paper explores three related ideas: a notion of “mass-structural equivalence” between rooted, weighted real trees; a family of such trees, called “interval partition (IP) trees,” in which the metric is, in a sense, specified by the weight and underlying branching structure; and continuum random tree representations of exchangeable random hierarchies on N
Aldous introduced the Brownian continuum random trees (CRTs) (BCRT), which arises as a scaling limit of various families of random discrete trees, including critical Galton-Watson trees conditioned on total progeny
The BCRT is a random fractal in the sense that, if we decompose it around a suitably chosen random branch point, the components are each distributed as scaled copies of a BCRT and are conditionally independent given their sizes
Summary
This paper explores three related ideas: a notion of “mass-structural equivalence” between rooted, weighted real trees; a family of such trees, called “interval partition (IP) trees,” in which the metric is, in a sense, specified by the weight and underlying branching structure; and continuum random tree representations of exchangeable random hierarchies on N. (ii) The map Θ is a bijection from the set of isomorphism classes of IP trees to the set of e.i.g. laws of hierarchies on N This theorem is a hierarchies analogue to Kingman’s paintbox theorem [22], which describes exchangeable partitions of N, or to de Finetti’s theorem for exchangeable sequences of random variables [21]. In [14], the authors pose the “Naïve conjecture” that exchangeable hierarchies are characterized by a mixture of the three behaviors exhibited in Example 1.9: macroscopic splitting, broom-like explosion, and comb-like erosion This is formalized in Conjecture 2 of that paper, which is verified by Theorem 1.8 above and the following. Explosions, erosion, and macroscopic splitting correspond to pa, ps, and branch points, respectively, with pl corresponding to the singletons that are eventually isolated by repeated splitting
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