Abstract
We define some new sequences of recursively constructed random combinatorial trees, and show that, after properly rescaling graph distance and equipping the trees with the uniform measure on vertices, each sequence converges almost surely to a real tree in the Gromov-Hausdorff-Prokhorov sense. The limiting real trees are constructed via line-breaking the real half-line with a Poisson process having rate $(\ell+1)t^\ell dt$, for each positive integer $\ell$, and the growth of the combinatorial trees may be viewed as an inhomogeneous generalization of R\'emy's algorithm.
Highlights
Understanding the structure of large random trees and graphs is an important topic of much recent interest in mathematics, statistics, and science
One important approach to studying a large random discrete structure is to determine limiting behavior as its size tends to infinity, in particular the structure may converge in a suitable sense to a limit object
In this paper we are interested in a third setting that has been an important and active research area for the last 25 years: continuum tree limits of combinatorial trees; here trees are viewed as measured metric spaces and convergence is in the Gromov-Hausdorff-Prokhorov (GHP) topology
Summary
Understanding the structure of large random trees and graphs is an important topic of much recent interest in mathematics, statistics, and science. One direction of extension is showing convergence of other families of rescaled combinatorial trees to the BCRT; see, e.g., Haas & Miermont [20], Kortchemski [22], Marckert & Miermont [24], Rizzolo [34] Another type of extension, and that considered in this paper, is constructing and studying other continuum random trees (CRTs) via some analog of part or all of Items (1-4) above. We extend these ideas by defining a new family of sequences of growing recursively constructed combinatorial trees in the spirit of Rémy’s algorithm and show these sequences of trees can be embedded into appropriate Poisson line-breaking constructions. See the recent works of Amini, Devroye, Griffiths & Olver [11] and Haas [18] for related constructions and discussions
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