A recursive distributional equation for the stable tree

Abstract

We provide a new characterisation of Duquesne and Le Gall’s α-stable tree, $\mathit{\alpha }\in (1,2]$, as the solution of a recursive distributional equation (RDE) of the form $\mathcal{T}\stackrel{d}{=}g\left(\mathrm{\xi },{\mathcal{T}}_i,i\ge 0\right)$, where g is a concatenation operator, $\mathrm{\xi }=\left({\mathrm{\xi }}_i,i\ge 0\right)$ a sequence of scaling factors, ${\mathcal{T}}_i$, $i\ge 0$, and $\mathcal{T}$ are i.i.d. trees independent of ξ. This generalises the characterisation of the Brownian Continuum Random Tree proved by Albenque and Goldschmidt, based on self-similarity observed by Aldous. 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.