High degrees in random recursive trees

For n≥1, let Tn be a random recursive tree (RRT) on the vertex set [n]={1,…,n}. Let deg⁡Tn(v) be the degree of vertex v in Tn, that is, the number of children of v in Tn. Devroye and Lu showed that the maximum degree Δn of Tn satisfies Δn/⌊log⁡2n⌋→1 almost surely; Goh and Schmutz showed distributional convergence of Δn−⌊log⁡2n⌋ along suitable subsequences. In this work we show how a version of Kingman's coalescent can be used to access much finer properties of the degree distribution in Tn. For any i∈ℤ, let Xi(n)=|{v∈[n]:deg⁡Tn(v)=⌊log⁡n⌋+i}|. 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 (Xi(n), i∈ℤ) converge weakly in distribution to (P[i,i+1), i∈ℤ). We also prove asymptotic normality of Xi(n) when i=i(n)→−∞ slowly, and obtain precise asymptotics for P(Δn−log⁡2n>i) when i(n)→∞ and i(n)/log⁡n is not too large. Our results recover and extends the previous distributional convergence results on maximal and near‐maximal degrees in RRT.