5 research outputs found
Branch merging on continuum trees with applications to regenerative tree growth
We introduce a family of branch merging operations on continuum trees and
show that Ford CRTs are distributionally invariant. This operation is new even
in the special case of the Brownian CRT, which we explore in more detail. The
operations are based on spinal decompositions and a regenerativity preserving
merging procedure of -strings of beads, that is, random
intervals equipped with a random discrete measure
arising in the limit of ordered -Chinese restaurant
processes as introduced recently by Pitman and Winkel. Indeed, we iterate the
branch merging operation recursively and give an alternative approach to the
leaf embedding problem on Ford CRTs related to -regenerative tree growth processes.Comment: 40 pages, 5 figure
A recursive distribution equation for the stable tree
We provide a new characterisation of Duquesne and Le Gall's -stable
tree, , as the solution of a recursive distribution equation
(RDE) of the form ,
where is a concatenation operator, a sequence of
scaling factors, , , and are i.i.d.
trees independent of . 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