Height of weighted recursive trees with sub-polynomially growing total weight

TL;DR

This paper investigates the height of weighted recursive trees with sub-polynomial total weight growth, revealing new behaviors and breaking universality observed in polynomial growth cases.

Contribution

It extends previous work by analyzing the height of weighted recursive trees under sub-polynomial weight growth, uncovering novel asymptotic behaviors.

Findings

Identifies new asymptotic regimes for tree height

Shows universality breaks down in sub-polynomial growth cases

Provides asymptotic formulas for different growth regimes

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 at random with probability proportional to its weight. In the case where the total weight of the tree at step $n$ grows polynomially in $n$, we obtained in (Pain-S\'enizergues 2022) an asymptotic expansion for the height of the tree, which falls into the university class of the maximum of branching random walks. In this paper, we consider the case of a total weight growing sub-polynomially in $n$ and obtain asymptotics for the height of the tree in several regimes, showing that universality is broken and exhibiting new behaviors.