High degrees in recursive trees

Let Tn be a random recursive tree with vertex set [n] := {1, . . . , n} and edges directed towards the root. Let degn(i) denote the number of children of vertex i ∈ [n] of Tn. Devroye and Lu [1] showed that the maximum degree ∆n of Tn satisfies ∆n/ log2 n→ 1 a.s. Here we study the distribution of the maximum degree and of the number of vertices with near-maximum degree. For any d ∈ Z, let X d = |{i ∈ [n] : degn(i) = blog2 nc + d}|. Also, let P be a Poisson point process on R with rate function λ(x) = 2−x · ln 2. We show that, up to lattice effects, the vectors (X d , d ∈ Z) converge in distribution to (|P ∩ [d, d+ 1)|, d ∈ Z). This recovers and extends results of Goh and Schmutz [2].