[Paper Review] Record process on the Continuum Random Tree

Romain Abraham, Jean‐François Delmas|arXiv (Cornell University)|Jul 19, 2011

Stochastic processes and statistical mechanics39 references20 citations

TL;DR

This paper constructs a continuous pruning procedure on Aldous's continuum random tree (CRT) that generates a random variable Θ, which, conditionally on the tree, follows the same distribution as the limit of the number of cuts needed to isolate the root in critical Galton-Watson trees—previously known only via weak convergence. The key result establishes that Θ equals the integral of the mass of the remaining tree over time, and that the rescaled variable Z = √(2α/σ)Θ converges almost surely to Janson’s limit law Z_T under the CRT.

ABSTRACT

By considering a continuous pruning procedure on Aldous's Brownian tree, we construct a random variable $Θ$ which is distributed, conditionally given the tree, according to the probability law introduced by Janson as the limit distribution of the number of cuts needed to isolate the root in a critical Galton-Watson tree. We also prove that this random variable can be obtained as the a.s. limit of the number of cuts needed to cut down the subtree of the continuum tree spanned by $n$ leaves.

Motivation & Objective

• To provide a constructive description of the conditional limit law Z_T, previously known only through weak convergence in Janson's work.
• To define a continuous pruning procedure on the CRT using a Poisson point process to model random cuts.
• To show that the total time until root isolation, represented as an integral over the tree's mass process, matches the conditional distribution of Z_T.
• To establish almost sure convergence of discrete cut-down processes on n-leaf subtrees to the continuous limit under the CRT.

Proposed method

• Model the CRT as a real tree coded by a Brownian excursion under excursion measure N, with length measure ℓ(dx) and mass measure m^T(dx) on leaves.
• Introduce a Poisson point process on T × [0, ∞) with intensity αℓ(dx)dθ to model random cut points in space and time.
• Define θ(x) as the first time a cut occurs on the path from the root to x, representing when x is severed from the root.
• Define Θ = ∫_T θ(x) m^T(dx), and rescale it as Z = √(2α/σ)Θ to match Janson’s unconditional Rayleigh distribution.
• Use the identity Θ = ∫₀^∞ σ_q dq, where σ_q is the mass of the tree remaining at time q, to express Θ as a time integral of the remaining mass.
• Leverage Laplace transforms and excursion theory to compute moments and establish convergence under the excursion measure.

Experimental results

Research questions

• RQ1 Can a continuous pruning procedure on the CRT be constructed such that the number of cuts to isolate the root matches Janson’s conditional limit law Z_T?
• RQ2 Is there a pathwise or almost sure convergence of discrete cut-down processes on finite subtrees to the continuous limit on the CRT?
• RQ3 Can the total time Θ required to isolate the root be represented as an integral over the remaining mass process σ_q?
• RQ4 Does the rescaled variable Z = √(2α/σ)Θ have the same distribution as Z_T conditionally on the CRT?
• RQ5 What is the first moment of Θ under the excursion measure, and how does it behave asymptotically?

Key findings

• The random variable Θ, defined as the integral of the cut-time function over the mass measure, satisfies Z = √(2α/σ)Θ ≡ Z_T in law conditionally on the CRT.
• The identity Θ = ∫₀^∞ σ_q dq holds almost surely, where σ_q is the mass of the tree remaining after time q.
• The first moment of Θ under the excursion measure satisfies H_q(r) = E[Θ | σ = r] with the bound 0 ≤ qr − H_q(r) ≤ (1/2)√(πα) q² r^{3/2}.
• The Laplace transform of the total cut time Θ is derived via excursion theory and the generating function F(q), leading to an explicit integral representation.
• The asymptotic expansion of F(q) in λ shows that the first derivative at λ = 0 is (1/(2α)) log(1 + z), with z = q√(α/μ), confirming the moment structure.
• The paper establishes almost sure convergence of the discrete cut-down process on n leaves to the continuous limit, extending Janson’s distributional result to pathwise convergence.