Spinal partitions and invariance under re-rooting of continuum random trees

TL;DR

This paper introduces a theory of spinal decompositions for fragmentation trees, showing how fine partitions derive from coarse ones and proving stable trees are uniquely invariant under re-rooting.

Contribution

It develops spinal decomposition theory for fragmentation trees and establishes the invariance of stable trees under re-rooting among continuous fragmentation trees.

Findings

Fine partitions are obtained by shattering coarse parts independently.

Stable trees are the only continuous fragmentation trees invariant under re-rooting.

The theory applies to the two-parameter Poisson-Dirichlet family, including stable trees.

Abstract

We develop some theory of spinal decompositions of discrete and continuous fragmentation trees. Specifically, we consider a coarse and a fine spinal integer partition derived from spinal tree decompositions. We prove that for a two-parameter Poisson--Dirichlet family of continuous fragmentation trees, including the stable trees of Duquesne and Le Gall, the fine partition is obtained from the coarse one by shattering each of its parts independently, according to the same law. As a second application of spinal decompositions, we prove that among the continuous fragmentation trees, stable trees are the only ones whose distribution is invariant under uniform re-rooting.