Fluctuations of the Horton-Strahler number of stable Galton-Watson trees

TL;DR

This paper investigates the fluctuations of the Horton-Strahler number in stable Galton-Watson trees, introducing a real-valued variant to analyze its distributional limits and connection to stable Lévy trees.

Contribution

It introduces a real-valued variant of the Horton-Strahler number and proves its rescaled exponential converges to a limit related to stable Lévy trees.

Findings

Rescaled exponential of the new variant converges in distribution

The limit, called Strahler dilation, relates to the stable Lévy tree

Fluctuations are characterized without deterministic oscillations

Abstract

The Horton-Strahler number -- also called the register function -- is a combinatorial tool that quantifies the branching complexity of a rooted tree. We study the law of the Horton-Strahler number of stable Galton-Watson trees conditioned to have size $n$ (including the Catalan trees), which are the finite-dimensional marginals of stable L\'evy trees. While these random variables are known to grow as a multiple of $\ln n$ in probability, their fluctuations are not well understood because they are coupled with deterministic oscillations. To rule out the latter, we introduce a real-valued variant of the Horton-Strahler number. We show that a rescaled exponential of this quantity jointly converges in distribution to a measurable function of the scaling limit of the trees, i.e. the stable L\'evy tree. We call this limit the Strahler dilation and we discuss its similarities with the Horton-Strahler number.