Random trees constructed by aggregation

TL;DR

This paper investigates how the asymptotic behavior of branch lengths in recursively constructed random trees affects their geometric properties, revealing phase transitions in compactness and Hausdorff dimension based on branch length decay rates.

Contribution

It characterizes the geometric properties of limiting trees constructed by recursive aggregation, linking branch length decay to Hausdorff dimension and compactness.

Findings

For branch lengths ~ n^{-eta}, the limiting tree is compact with Hausdorff dimension 1/β for β in (0,1].

When β > 1, the limiting tree is non-compact with Hausdorff dimension 1.

The results include the classical Brownian tree as a special case.

Abstract

We study a general procedure that builds random $\mathbb R$-trees by gluing recursively a new branch on a uniform point of the pre-existing tree. The aim of this paper is to see how the asymptotic behavior of the sequence of lengths of branches influences some geometric properties of the limiting tree, such as compactness and Hausdorff dimension. In particular, when the sequence of lengths of branches behaves roughly like $n^{-\alpha}$ for some $\alpha \in (0,1]$, we show that the limiting tree is a compact random tree of Hausdorff dimension $\alpha^{-1}$. This encompasses the famous construction of the Brownian tree of Aldous. When $\alpha >1$, the limiting tree is thinner and its Hausdorff dimension is always 1. In that case, we show that $ \alpha^{-1}$ corresponds to the dimension of the set of leaves of the tree.