Fluctuations for the number of records on subtrees of the Continuum Random Tree

TL;DR

This paper investigates the asymptotic distribution of the number of cuts needed to isolate the root in subtrees of the Continuum Random Tree, revealing new Gaussian fluctuation results beyond previous convergence findings.

Contribution

It extends existing results by establishing a distributional convergence for fluctuations of the number of cuts in CRT subtrees, using martingale and Poisson process techniques.

Findings

Convergence of scaled fluctuations to a Gaussian mixture

Extension of Abraham and Delmas's results on CRTs

Application of martingale limit theory to CRT processes

Abstract

We study the asymptotic behavior af the number of cuts $X(T_n)$ needed to isolate the root in a rooted binary random tree $T_n$ with $n$ leaves. We focus on the case of subtrees of the Continuum Random Tree generated by uniform sampling of leaves. We elaborate on a recent result by Abraham and Delmas, who showed that $X(T_n)/\sqrt{2n}$ converges a.s. towards a Rayleigh-distributed random variable $\Theta$, which gives a continuous analog to an earlier result by Janson on conditioned, finite-variance Galton-Watson trees. We prove a convergence in distribution of $n^{-1/4}(X(T_n)-\sqrt{2n}\Theta)$ towards a random mixture of Gaussian variables. The proofs use martingale limit theory for random processes defined on the CRT, related to the theory of records of Poisson point processes.