High degrees in random recursive trees

For n >= 1, let T-n 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 T-n. Devroye and Lu showed that the maximum degree Delta(n) of T-n satisfies Delta(n)/left perpendicularlog(2 )nright perpendicular -> 1 almost surely; Goh and Schmutz showed distributional convergence of Delta(n) - left perpendicularlog(2 )nright perpendicular 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 T-n. For any i is an element of Z, let X-i((n)) = vertical bar{v is an element of [n] : deg(Tn) (v) = left perpendicularlog nright perpendicular + i}vertical bar. Also, let P be a Poisson point process on R with rate function lambda(x) = 2(-x). In 2. We show that, up to lattice effects, the vectors (X-i((n)), i is an element of Z) converge weakly in distribution to (P[i, i + 1), i is an element of Z). We also prove asymptotic normality of X-i((n)) when i = i(n) -> -infinity slowly, and obtain precise asymptotics for P(Delta(n) - log(2) n > i) when i(n) -> infinity and i(n)/log n is not too large. Our results recover and extends the previous results on maximal and near-maximal degrees in RRT.