Abstract
We introduce regenerative tree growth processes as consistent families of random trees with n labelled leaves, n>=1, with a regenerative property at branch points. This framework includes growth processes for exchangeably labelled Markov branching trees, as well as non-exchangeable models such as the alpha-theta model, the alpha-gamma model and all restricted exchangeable models previously studied. Our main structural result is a representation of the growth rule by a sigma-finite dislocation measure kappa on the set of partitions of the natural numbers extending Bertoin's notion of exchangeable dislocation measures from the setting of homogeneous fragmentations. We use this representation to establish necessary and sufficient conditions on the growth rule under which we can apply results by Haas and Miermont for unlabelled and not necessarily consistent trees to establish self-similar random trees and residual mass processes as scaling limits. While previous studies exploited some form of exchangeability, our scaling limit results here only require a regularity condition on the convergence of asymptotic frequencies under kappa, in addition to a regular variation condition.
Highlights
Introduction to regenerative tree growth processesFor each n ≥ 1, denote by Tn the set of rooted leaf-labelled combinatorial trees with no degree2 vertices and n + 1 degree-1 vertices, one of which is called the root, the others leaves
Taking our cues from the exchangeable case, cf. [4], one thing we want from our dislocation measures is to be able to use them to embed regenerative tree growth processes in continuous time, making the trees the genealogical trees of continuous-time fragmentation processes
Proposition 11 If (Tn, n ≥ 1) is a regenerative tree growth process with associated dislocation measure κ and (Tn◦, n ≥ 1) is the sequence of trees such that Tn◦ is obtained from Tn by delabelling the leaves, (Tn◦, n ≥ 1) is a Markov branching model based on the functions p◦n(n1
Summary
For each n ≥ 1, denote by Tn the set of rooted leaf-labelled combinatorial trees with no degree vertices and n + 1 degree-1 vertices, one of which is called the root, the others leaves. Theorem 4 Let (Tn, n ≥ 1) be a regenerative tree growth process associated with a dislocation measure κ. We remark that when considering Tn◦ purely as a tree we treat it as an element of the set T◦n of rooted unlabelled trees with n leaves and no degree-2 vertices This theorem provides conditions for the existence of a scaling limit of Tn◦, where the label structure of Tn has been forgotten. Theorem 5 Let (Tn, n ≥ 1) be a regenerative tree growth process with dislocation measure κ and X(n) the residual mass process of {1} in Tn. Assume that the first block Γ1 of κ-a.e. Γ ∈ P has an asymptotic frequency |Γ1| ∈ (0, 1) and define Λ as push-forward of κ under Γ → − log(|Γ1|).
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