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A functional limit theorem is proved for multitype continuous time Markov branching processes. As consequences, we obtain limit theorems for the branching process stopped by some stopping rule, for example when the total number of particles reaches a given level. Using the

. The asymptotic distribution of the number of Hamilton cycles in a random regular graph is determined. The limit distribution is of an unusual type; it is the distribution of a variable whose logarithm can be written as an infinite linear combination of independent Poisson variables, and thus the logarithm has an infinitely divisible distribution with a certain discrete L'evy measure. Similar results are found for some related problems. These limit results imply that some different models of random regular graphs are contiguous, which means that they are qualitatively asymptotically equivalent. For example, if r 3, then the usual (uniformly distributed) random r-regular graph is contiguous to the one constructed by taking the union of r perfect matchings on the same vertex set (assumed to be of even cardinality), conditioned on there being no multiple edges. Some consequences of contiguity for asymptotic distributions are discussed. 0. Introduction In two remarkable papers, Robinso...

. The minimal weight of a spanning tree in a complete graph Kn with independent, uniformly distributed random weights on the edges, is shown to have an asymptotic normal distribution. The proof uses a functional limit extension of results by Barbour and Pittel on the distribution of the number of tree components of given sizes in a random graph. 1. Introduction and results Assign random weights T ij , 1 i ! j n, to the edges of the complete graph K n with vertex set f1; : : : ; ng, and let W n be the minimum weight of a spanning tree of K n . We assume that the weights are independent and identically distributed, with a uniform distribution on [0; 1]. It was proved by Frieze [5] that W n ! i(3) = 1 X k=1 k \Gamma3 = 1:202 : : : in probability as n ! 1, see also Bollobs [3]. The main purpose of the present paper is to show that W n has an asymptotic normal distribution. Theorem 1. Let W n be the weight of the minimal spanning tree. Then n 1=2 \Gamma W n \Gamma i(3) \Delta ...

�������� � Fix a small graph H and let YH denote the number of copies of H in the random graph G(n, p). We investigate the degree of concentration of YH around its mean, motivated by the following questions • For what λ does YH have sub-Gaussian behaviour, namely P r(|YH − �(YH) | ≥ (λvar(YH)) 1/2) ≤ e −cλ. where c is a positive constant. • What is P r(YH ≥ (1 + ɛ)�(YH)), for a constant ɛ, i.e, when the upper tail is of order Ω(�(YH))? • Fixing λ = ω(1) in advance, find a reasonably small tail T = T (λ) such that P r(|YH − �(YH) | ≥ T) ≤ e −λ. We prove a general concentration result which contains a partial answer to each of these questions. Given a graph H with k vertices {v1,..., vk} and m edges. Consider a bigger graph G and a subgraph H ′ of G. The density of H is the ratio m/k; H is balanced

The numbers of spanning trees, Hamilton cycles and perfect matchings in a random graph Gnm are shown to be asymptotically normal if m is neither too large nor too small. At the lower limit m n 3=2 , these numbers are asymptotically log-normal. For Gnp , the numbers are asymptotically log-normal for a wide range of p, including p constant. The same results are obtained for random directed graphs and bipartite graphs. The results are proved using decomposition and projection methods.

We study the profile Xn,k of random search trees including binary search trees and m-ary search trees. Our main result is a functional limit theorem of the normalized profile Xn,k/EXn,k for k =⌊α log n ⌋ in a certain range of α. A central feature of the proof is the use of the contraction method to prove convergence in distribution of certain random analytic functions in a complex domain. This is based on a general theorem concerning the contraction method for random variables in an infinite-dimensional Hilbert space. As part of the proof, we show that the Zolotarev metric is complete for a Hilbert space. 1. Introduction. Search

We study the k-core of a random (multi)graph on n vertices with a given degree sequence. In our previous paper [Random Structures Algorithms 30 (2007) 50–62] we used properties of empirical distributions of independent random variables to give a simple proof of the fact that the size of the giant k-core obeys a law of large numbers as n →∞. Here we develop the method further and show that the fluctuations around the deterministic limit converge to a Gaussian law above and near the threshold, and to a non-normal law at the threshold. Further, we determine precisely the location of the phase transition window for the emergence of a giant k-core. Hence, we deduce corresponding results for the k-core in G(n, p) and G(n, m).

Abstract. We study the evolution of the susceptibility in the subcritical random graph G(n, p) as n tends to infinity. We obtain precise asymptotics of its expectation and variance, and show it obeys a law of large numbers. We also prove that the scaled fluctuations of the susceptibility around its deterministic limit converge to a Gaussian law. We further extend our results to higher moments of the component size of a random vertex, and prove that they are jointly asymptotically normal. 1.

Abstract. We study the space requirements of a sorting algorithm where only items that at the end will be adjacent are kept together. This is equivalent to the following combinatorial problem: Consider a string of fixed length n that starts as a string of 0’s, and then evolves by changing each 0 to 1, with the n changes done in random order. What is the maximal number of runs of 1’s? We give asymptotic results for the distribution and mean. It turns out that, as in many problems involving a maximum, the maximum is asymptotically normal, with fluctuations of order n 1/2, and to the first order well approximated by the number of runs at the instance when the expectation is maximized, in this case when half the elements have changed to 1; there is also a second order term of order n 1/3. We also treat some variations, including priority queues and socksorting. 1.