A recursive distribution equation for the stable tree

Abstract

We provide a new characterisation of Duquesne and Le Gall's $\alpha$-stable tree, $\alpha \in (1,2]$, as the solution of a recursive distribution equation (RDE) of the form $\mathcal{T}\stackrel{d}{=}g(\xi ,{\mathcal{T}}_i,i\ge 0)$, where $g$ is a concatenation operator, $\xi =({\xi }_i,i\ge 0)$ a sequence of scaling factors, ${\mathcal{T}}_i$, $i\ge 0$, and $\mathcal{T}$ are i.i.d. trees independent of $\xi$. This generalises a version of the well-known characterisation of the Brownian Continuum Random Tree due to Aldous, Albenque and Goldschmidt. By relating to previous results on a rather different class of RDE, we explore the present RDE and obtain for a large class of similar RDEs that the fixpoint is unique (up to multiplication by a constant) and attractive.

30 pages, 5 figures

References in corpus (3)

• Self-similar fragmentations derived from the stable tree II: splitting at nodes
• Spinal partitions and invariance under re-rooting of continuum random trees
• The coding of compact real trees by real valued functions