# A recursive distribution equation for the stable tree

## TL;DR

This paper characterizes the alpha-stable tree as a solution to a recursive distribution equation, generalizing known results for the Brownian continuum random tree and establishing uniqueness and attractiveness of the fixpoint.

## Contribution

It introduces a new recursive distribution equation characterization for alpha-stable trees, extending previous work on Brownian trees and analyzing fixpoint properties.

## Key findings

- Unique fixpoint solutions for a class of RDEs
- The alpha-stable tree is characterized as a recursive distribution fixpoint
- The fixpoint is attractive and unique up to scaling

## 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 $\mathcal{T}\overset{d}{=}g(\xi,\mathcal{T}_i, i\geq0)$, where $g$ is a concatenation operator, $\xi = (\xi_i, i\geq 0)$ a sequence of scaling factors, $\mathcal{T}_i$, $i \geq 0$, 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.