The distance profile of rooted and unrooted simply generated trees

TL;DR

This paper investigates the asymptotic behavior of the distance profile in critical simply generated trees, showing convergence to a continuous limit related to the Brownian continuum random tree and exploring properties like Hölder continuity and differentiability.

Contribution

It introduces the convergence of the distance profile for both rooted and unrooted simply generated trees to a continuous limit, extending known results from height profiles.

Findings

Distance profile converges to a density function in the Brownian CRT

The limiting function is Hölder continuous of any order < 1

The function is almost everywhere differentiable

Abstract

It is well-known that the height profile of a critical conditioned Galton-Watson tree with finite offspring variance converges, after a suitable normalization, to the local time of a standard Brownian excursion. In this work, we study the distance profile, defined as the profile of all distances between pairs of vertices. We show that after a proper rescaling the distance profile converges to a continuous random function that can be described as the density of distances between random points in the Brownian continuum random tree. We show that this limiting function a.s. is H\"older continuous of any order $\alpha<1$, and that it is a.e. differentiable. We note that it cannot be differentiable at $0$, but leave as open questions whether it is Lipschitz, and whether is continuously differentiable on the half-line $(0,\infty)$. The distance profile is naturally defined also for unrooted trees contrary to the height profile that is designed for rooted trees. This is used in our proof, and we prove the corresponding convergence result for the distance profile of random unrooted simply generated trees. As a minor purpose of the present work, we also formalize the notion of unrooted simply generated trees and include some simple results relating them to rooted simply generated trees, which might be of independent interest.