Correction terms for the height of weighted recursive trees

TL;DR

This paper refines the understanding of the height of weighted recursive trees by deriving second and third order asymptotic expansions, using methods inspired by branching random walks, and extends results to related preferential attachment models.

Contribution

It provides the second and third order asymptotic expansions for the height of weighted recursive trees, advancing previous first-order results.

Findings

Derived second order asymptotics for tree height

Derived third order asymptotics for tree height

Results apply to preferential attachment trees with additive fitnesses

Abstract

Weighted recursive trees are built by adding successively vertices with predetermined weights to a tree: each new vertex is attached to a parent chosen randomly proportionally to its weight. Under some assumptions on the sequence of weights, the first order for the height of such trees has been recently established in [Electron. J. Probab. 26 (2021), Paper No. 80] by one of the authors. In this paper, we obtain the second and third orders in the asymptotic expansion of the height of weighted recursive trees, under similar assumptions. Our methods are inspired from those used to prove similar results for branching random walks. Our results also apply to a related model of growing trees, called the preferential attachment tree with additive fitnesses.