Random trees constructed by aggregation

Nicolas Curien, Bénédicte Haas

Published 2014-11-16Version 1

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.