In this paper we characterise the distributions of the number of predecessors and of the number of successors of a given set of vertices, A, in the random mapping model, T n D̂ (see Hansen and Jaworski (2008)), with exchangeable in-degree sequence (D̂ 1,D̂ 2,…,D̂ n ). We show that the exact formulae for these distributions and their expected values can be given in terms of the distributions of simple functions of the in-degree variables D̂ 1,D̂ 2,…,D̂ n . As an application of these results, we consider two special examples of T n D̂ which correspond to random mappings with preferential and anti-preferential attachment, and determine the exact distributions for the number of predecessors and the number of successors in these cases. We also characterise, for these two special examples, the asymptotic behaviour of the expected numbers of predecessors and successors and interpret these results in terms of the threshold behaviour of epidemic processes on random mapping graphs. The families of discrete distributions obtained in this paper are also of independent interest.

We consider the set of polynomials of degree n over a finite field and put the uniform probability measure on this set. Any such polynomial factors uniquely into a product of its irreducible factors. To each polynomial we associate a step function on the interval [0,1] such that the size of each jump corresponds to the number of factors of a certain degree in the factorization of the random polynomial. We normalize these random functions and show that the resulting random process converges weakly to Brownian motion as n → ∞. This result complements earlier work by the author on the order statistics of the degree sequence of the factors of a random polynomial.

In this paper we investigate the ‘local’ properties of a random mapping model, T n D̂, which maps the set {1, 2, …, n} into itself. The random mapping T n D̂ , which was introduced in a companion paper (Hansen and Jaworski (2008)), is constructed using a collection of exchangeable random variables D̂ 1, …, D̂ n which satisfy In the random digraph, G n D̂ , which represents the mapping T n D̂ , the in-degree sequence for the vertices is given by the variables D̂ 1, D̂ 2, …, D̂ n , and, in some sense, G n D̂ can be viewed as an analogue of the general independent degree models from random graph theory. By local properties we mean the distributions of random mapping characteristics related to a given vertex v of G n D̂ - for example, the numbers of predecessors and successors of v in G n D̂ . We show that the distribution of several variables associated with the local structure of G n D̂ can be expressed in terms of expectations of simple functions of D̂ 1, D̂ 2, …, D̂ n . We also consider two special examples of T n D̂ which correspond to random mappings with preferential and anti-preferential attachment, and determine, for these examples, exact and asymptotic distributions for the local structure variables considered in this paper. These distributions are also of independent interest.

Let D be the punctured unit disk. It is easy to see that no pair x, y in D can cover D in the sense that D cannot be contained in the union of the unit disks centred at x and y. With this fact in mind, let V n = {X 1, X 2, …, X n }, where X 1, X 2, … are random points sampled independently from a uniform distribution on D. We prove that, with asymptotic probability 1, there exist two points in V n that cover all of V n .

In this paper, we introduce a compound random mapping model which can be viewed as a generalization of the basic random mapping model considered by Ross and by Jaworski. We investigate a particular example, the Poisson compound random mapping, and compare results for this model with results known for the well-studied uniform random mapping model. We show that, although the structure of the components of the random digraph associated with a Poisson compound mapping differs from the structure of the components of the random digraph associated with the uniform model, the limiting distribution of the normalized order statistics for the sizes of the components is the same as in the uniform case, i.e. the limiting distribution is the Poisson-Dirichlet (½) distribution on the simplex {{x i } : ∑ x i ≤ 1, x i ≥ x i+1 ≥ 0 for every i ≥ 1}.

Suppose that [Cscr]={cij[ratio ]i, j[ges ]1} is a collection of i.i.d. nonnegative continuous random variables and suppose T is a rooted, directed tree on vertices labelled 1,2,[ctdot ],n. Then the ‘cost’ of T is defined to be c(T)=[sum ](i,j)∈Tcij, where (i, j) denotes the directed edge from i to j in the tree T. Let Tn denote the ‘optimal’ tree, i.e. c(Tn)=min{c(T)[ratio ]T is a directed, rooted tree in with n vertices}. We establish general conditions on the asymptotic behaviour of the moments of the order statistics of the variables c11, c12, [ctdot ], cin which guarantee the existence of sequences {an}, {bn}, and {dn} such that b−1n(c(Tn)−an)→N(0, 1) in distribution, d−1nc(Tn)→1 in probability, and d−1nE(c(Tn))→1 as n→∞, and we explicitly determine these sequences. The proofs of the main results rely upon the properties of general random mappings of the set {1, 2, [ctdot ], n} into itself. Our results complement and extend those obtained by McDiarmid [9] for optimal branchings in a complete directed graph.

Every monic, degree n polynomial in Fq[x;] is the characteristic polynomial of at least one n × n matrix (with entries in the finite field Fq), but they do not appear with equal frequency. There is no a priori reason that the characteristic polynomial of a typical matrix should resemble a typical monic degree n polynomial. Nevertheless, we prove a precise version of the following heuristic statement: ‘Excepting its small factors, the characteristic polynomial of a random matrix is random.’

For each n > 0, the Ewens sampling formula from population genetics is a measure on the set of all partitions of the integer n. To determine the limiting distributions for the part sizes of a partition with respect to the measures given by this formula, we associate to each partition a step function on [0, 1]. Each jump in the function equals the number of parts in the partition of a certain size. We normalize these functions and show that the induced measures on D[0, 1] converge to Wiener measure. This result complements Kingman's frequency limit theorem [10] for the Ewens partition structure.