Depth of vertices with high degree in random recursive trees

Let Tₙ be a random recursive tree with n nodes. List vertices of Tₙ in decreasing order of degree as v¹, , vⁿ, and write dⁱ and hⁱ for the degree of vⁱ and the distance of vⁱ from the root, respectively. We prove that, as n along suitable subsequences, \ (dⁱ - ₂ n, hⁱ - n² n) ( (Pᵢ, i 1), (Nᵢ, i 1) ) \, , \ where =1- (₂ e) /2, ²=1- (₂ e) /4, (Pᵢ, i 1) is a Poisson point process on Z and (Nᵢ, i 1) is a vector of independent standard Gaussians. We additionally establish joint normality for the depths of uniformly random vertices in Tₙ, which extends results for the case of a single random vertex. The joint limit holds even if the random vertices are conditioned to have large degree, provided the normalizing constants are adjusted accordingly; however, both the mean and variance of the conditinal depths remain of orden n. Our results are based on a simple relationship between random recursive trees and Kingman's n-coalescent; a utility that seems to have been largely overlooked.