Explosion and linear transit times in infinite trees

TL;DR

This paper characterizes when infinite randomly weighted trees exhibit explosive behavior or linear growth in transit times, providing new insights into their probabilistic structure and applications to real trees.

Contribution

It precisely characterizes linear growth in a class of infinite weighted trees and applies this to explosion phenomena and real tree height analysis.

Findings

Identifies conditions for linear growth in Poisson-weighted infinite trees.

Provides criteria for explosion in spherically-symmetric trees.

Determines finite height conditions for trees built from decreasing-length sticks.

Abstract

Let $T$ be an infinite rooted tree with weights $w_e$ assigned to its edges. Denote by $m_n(T)$ the minimum weight of a path from the root to a node of the $n$th generation. We consider the possible behaviour of $m_n(T)$ with focus on the two following cases: we say $T$ is explosive if \[ \lim_{n\to \infty}m_n(T) < \infty, \] and say that $T$ exhibits linear growth if \[ \liminf_{n\to \infty} \frac{m_n(T)}{n} > 0. \] We consider a class of infinite randomly weighted trees related to the Poisson-weighted infinite tree, and determine precisely which trees in this class have linear growth almost surely. We then apply this characterization to obtain new results concerning the event of explosion in infinite randomly weighted spherically-symmetric trees, answering a question of Pemantle and Peres. As a further application, we consider the random real tree generated by attaching sticks of deterministic decreasing lengths, and determine for which sequences of lengths the tree has finite height almost surely.