Abstract
AbstractIt is well known that the height profile of a critical conditioned Galton–Watson tree with finite offspring variance converges, after a suitable normalisation, 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 Hölder 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 it 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.
Highlights
Consider a random generated tree. (For definitions of this and other concepts in the introduction, see Sections 2–3.) Under some technical conditions, amounting to the tree being equivalent to a critical conditioned Galton–Watson tree with finite offspring variance, the profile of the tree converges in distribution, as a random function in C[0, ∞)
The distance profile is naturally defined 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 generated trees
The limiting random function can be identified with the local time of a standard Brownian excursion; this was conjectured by Aldous [3] and proved by Drmota and Gittenberger [17], see Drmota [16, Section 4.2], and in general by Kersting [33] in a paper that remains unpublished
Summary
Consider a random generated tree. (For definitions of this and other concepts in the introduction, see Sections 2–3.) Under some technical conditions, amounting to the tree being equivalent to a critical conditioned Galton–Watson tree with finite offspring variance, the (height) profile of the tree converges in distribution, as a random function in C[0, ∞). The distance profile is defined for unrooted trees, and we will find it convenient to use unrooted trees in parts of the proof This leads us to consider random unrooted generated trees. As a preparation for the unrooted case, we give (Section 4) some results (partly from Kortchemski and Marzouk [37]) on modified rooted generated trees (Galton–Watson trees), where the root has different weights (offspring distribution) than all other vertices. The central parts of the proof of Theorem 1.4 are given in Sections 10–12, where we use both rooted and unrooted trees.
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