High Degrees of Random Recursive Trees

For n ≥ 1, let Tn be a random recursive tree (RRT) on the vertex set [n] = {1, . . . , n}. Let degTn (v) be the degree of vertex v in Tn, that is, the number of children of v in Tn. Devroye and Lu [6] showed that the maximum degree ∆n of Tn satisfies ∆n/⌊log2 n⌋ → 1 almost surely; Goh and Schmutz [7] showed distributional convergence of ∆n − ⌊log2 n⌋ 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 X i = |{v ∈ [n] : degTn (v) = ⌊logn⌋+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 (X (n) i , i ∈ Z) converge weakly in distribution to (P[i, i + 1), i ∈ Z). We also prove asymptotic normality of X (n) i when i = i(n) → −∞ slowly, and obtain precise asymptotics for P (∆n − log2 n > i) when i(n) → ∞ and i(n)/ logn is not too large. Our results recover and extends the previous results on maximal and near-maximal degrees in random recursive trees. 1. Statement of results The process of random recursive trees (Tn, n ≥ 1) is defined as follows. T1 has a single node with label 1, which its root. The tree Tn+1 is obtained from Tn by directing an edge from a new vertex n + 1 to v ∈ [n]; the choice of v is uniformly random and independent for each n ∈ N. We call Tn a random recursive tree (RRT) of size n. As a consequence of the construction, vertex-labels in Tn increase along root-to-leaf paths. Rooted labelled trees with such property are called increasing trees. It is not difficult to see that, in fact, Tn is uniformly chosen among the set In of increasing trees with vertex set [n]. We write degTn(v) to denote the number of children of v in Tn. The degree distribution of Tn is encoded by the variables Z (n) i = |{v ∈ [n] : degTn(v) = i}|, for i ≥ 0. In fact, the study of RRT’s started with a paper by Na and Rapoport [12] in which they obtained, for any fixed i ≥ 0, the convergence E(Z i )/n → 2−i−1 as n → ∞. Mahmoud and Smythe [11] derived the asymptotic joint normality of Z (n) i for i ∈ {0, 1, 2}; Janson [8] extended the joint normality to Z (n) i for i ≥ 0 and gave explicit formulae for the covariance matrix (this is not an exhaustive account of the results concerning the random variables Z (n) i ). The above results concern typical degrees; the focus in this work is large degree vertices, and in particular the maximum degree in Tn, which we denote ∆n = maxv∈[n] degTn(v). For the rest of the paper we write log to denote logarithms with base 2, and ln to denote natural logarithms. For n ∈ N let εn = log n− ⌊log n⌋. A heuristic to find the order of ∆n is that, if E(Z i ) ≈ n2−i−1 were to hold for all i, as it does when i is fixed, then we would have E(Z ⌊logn⌋) ≈ n2 −⌊logn⌋−1 = 2−1+εn . Date: July 21, 2015. 2010 Mathematics Subject Classification. 60C05, 05C80. 1 2 LOUIGI ADDARIO-BERRY AND LAURA ESLAVA This heuristic suggests that ∆n is of order log n. This is indeed the case: Szymanski [14] proved that E [∆n] / log n → 1 as n → ∞, and Devroye and Lu [6] later established that ∆n/ log n → 1 a.s.. Finally, Goh and Schmutz [7] showed that ∆n − ⌊log n⌋ converges in distribution along suitable subsequences, and identified the possible limiting laws. Since we focus on maximal degrees, it is useful to let X (n) i = Z (n) i+⌊logn⌋ = |{v ∈ [n] : degTn(v) = ⌊log n⌋+ i}|, for n ∈ N and i ≥ −⌊log n⌋. The following is a simplified version of one of our main results. Theorem 1.1. Fix ε ∈ [0, 1]. Let (nl)l≥1 be an increasing sequence of integers satisfying εnl → ε as l → ∞. Then, as l → ∞ (X (nl) i , i ∈ Z) d −→ (P ε i , i ∈ Z) jointly for all i ∈ Z where the P ε i are independent Poisson r.v.’s with mean 2−i−1+ε. The random variables X (n) i do not converge in distribution as n → ∞ without taking subsequences; this is essentially a lattice effect caused by the floor ⌊log n⌋ in the definition of X (n) i . Theorem 1.1 can be stated in terms of weak convergence of point processes (which is equivalent to convergence of finite dimensional distributions (FDD’s); see Theorem 11.1.VII in [4]). In fact, we will also prove convergence (along subsequences) of