A recursive distribution equation for the stable tree

Abstract

We provide a new characterisation of Duquesne and Le Gall's α-stable tree, α ∈ (1, 2], as the solution of a recursive distribution equation (RDE) of the formwhere g is a concatenation operator, ξ = (ξ i , i ≥ 0) a sequence of scaling factors, T i , i ≥ 0, and T are i.i.d. trees independent of ξ. 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.

Self-similar random structures defined as fixed points of distributional equations

“…One may recognize the same scaling paramaters (R i ) i as in [14] on the βstable tree, already mentioned in Section 3.3. Indeed these parameters are directly computed from the β-stable process coding both the β-stable tree and looptree.…”

“…Another work to mention is the article from Chee, Rembart and Winkel [14], concerning the β-stable trees. It is inspired by Marchal's random growth algorithm introduced in [33]: the stable trees are the unique fixed point of a recursive equation that consists in gluing together infinitely many rescaled copies of a random compact metric space at a single branchpoint.…”

“…When we rescale looptrees, we multiply measures by c and distances by c 1/β . The authors of [14] were not interested in the size of the loop nor in the repartition of the substructures around it since it does not appear when we look at the stable tree. From [35,Equation (1)] and the linked remark of [15], one can recover the width of the hubs of the stable tree with the formula…”

“…When we come back to the β-stable looptrees, each jump of height ∆ s of the excursion is sent to a circle with lenght ∆ s (again from [15], see the discussion at the end of Section 2). It remains to determine the law of the local time, or the width, of the internal node around which rescaled stable trees are glued in the construction of [14] which will be the length of the circle for our case of the β-stable looptree L β . From [36] we know that it will be a rescaled mutiple of the 1/β-diversty of the sequence (R i ) i :…”

“…This result has been generalized to construct a larger class of random real trees by Broutin and Sulzbach in [10], from which we borrow the notations. Note also that the study of the β-stable trees by Rembart and Winkel in [38] and also in [14] with Chee fall in this general setting. The fact that these classical objects verify a distributional fixed-point equation pushes us to wonder about general properties of such an equation or their fixed-points.…”