Scaling limits for some random trees constructed inhomogeneously

TL;DR

This paper introduces new inhomogeneous recursive random trees, proves their convergence to real trees in the Gromov-Hausdorff-Prokhorov sense, and describes their limits via Poisson line-breaking processes.

Contribution

It presents a novel class of inhomogeneous recursive trees and establishes their almost sure convergence to continuum limits using advanced probabilistic techniques.

Findings

Convergence of rescaled trees to real trees in Gromov-Hausdorff-Prokhorov sense

Construction of limiting trees via Poisson line-breaking processes

Generalization of Rémys algorithm to inhomogeneous settings

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.