In this paper we study the covariance structure of the number of nodes k and l steps away from the root in random recursive trees. We give an analytic expression valid for all k, l and tree sizes N . The fraction of nodes k steps away from the root is a random probability distribution in k. The expression for the covariances allows us to show that the total variation distance between this (rand...

We introduce the problem of isolating several nodes in random recursive trees by successively removing random edges, and study the number of random cuts that are necessary for the isolation. In particular, we analyze the number of random cuts required to isolate ` selected nodes in a size-n random recursive tree for three different selection rules, namely (i ) isolating all of the nodes labelle...

Cutting random recursive trees, and the Bolthausen–Sznitman coalescent Christina Goldschmidt, University of Warwick (joint work with James Martin) The Bolthausen–Sznitman coalescent was introduced in the context of spin glasses in [1]. These days, it is usually thought of as a special case of a more general class of coalescent processes introduced by Pitman [5] and Sagitov [7] and usually refer...

We study the limiting distribution of the degree of a given node in a scaled attachment random recursive tree, a generalized random recursive tree, which is introduced by Devroye et. al (2011). In a scaled attachment random recursive tree, every node i is attached to the node labeled biXic where X0, . . . , Xn is a sequence of i.i.d. random variables, with support in [0, 1) and distribution fun...

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, ( d − blog2 nc, hi − μ lnn √ σ2 lnn ) → ((Pi, i ≥ 1), (Ni, i ≥ 1)) , where μ = 1 − (log2 e)/2, σ2 = 1 − (log2 e)/4, (Pi, i ≥ 1) is a Poi...

We describe a representation of the Bolthausen-Sznitman coalescent in terms of the cutting of random recursive trees. Using this representation, we prove results concerning the final collision of the coalescent restricted to [n]: we show that the distribution of the number of blocks involved in the final collision converges as n → ∞, and obtain a scaling law for the sizes of these blocks. We al...

In this extended abstract, we outline how to derive limit theorems for the number of subtrees of size k on the fringe of random plane-oriented recursive trees. Our proofs are based on the method of moments, where a complex-analytic approach is used for constant k and an elementary approach for k which varies with n. Our approach is of some generality and can be applied to other simple classes o...

In this paper we give a new example of duality between fragmentation and coagulation operators. Consider the space of partitions of mass (i.e., decreasing sequences of nonnegative real numbers whose sum is 1) and the twoparameter family of Poisson–Dirichlet distributions PD(α, θ) that take values in this space. We introduce families of random fragmentation and coagulation operators Fragα and Co...

We present a new technique for proving logarithmic upper bounds for diameters of evolving random graph models, which is based on defining a coupling between random graphs and variants of random recursive trees. The advantage of the technique is three-fold: it is quite simple and provides short proofs, it is applicable to a broad variety of models including those incorporating preferential attac...