Dudley Stark

The characteristic polynomial of a random permutation matrix

We establish a central limit theorem for the logarithm of the characteristic polynomial of a random permutation matrix. We relate this result to a central limit theorem of Wieand for the counting function for the eigenvalues lying in some interval on the unit circle.

The vertex degree distribution of passive random intersection graph models

In a random passive intersection graph model the edges of the graph are decided by taking the union of a fixed number of cliques of random size. We give conditions for a random passive intersection graph model to have a limiting vertex degree distribution, in particular to have a Poisson limiting vertex degree distribution. We give related conditions which, in addition to implying a limiting vertex degree distribution, imply convergence of expectation.

Explicit Limits of Total Variation Distance in Approximations of Random Logarithmic Assemblies by Related Poisson Processes

Assemblies are labelled combinatorial objects that can be decomposed into components. Examples of assemblies include set partitions, permutations and random mappings. In addition, a distribution from population genetics called the Ewens sampling formula may be treated as an assembly. Each assembly has a size n, and the sum of the sizes of the components sums to n. When the uniform distribution is put on all assemblies of size n, the process of component counts is equal in distribution to a process of independent Poisson variables Zi conditioned on the event that a weighted sum of the independent variables is equal to n. Logarithmic assemblies are assemblies characterized by some θ > 0 for which iEZi → θ. Permutations and random mappings are logarithmic assemblies; set partitions are not a logarithmic assembly. Suppose b = b(n) is a sequence of positive integers for which b/n → β e (0, 1]. For logarithmic assemblies, the total variation distance db(n) between the laws of the first b coordinates of the component counting process and of the first b coordinates of the independent processes converges to a constant H(β). An explicit formula for H(β) is given for β e (0, 1] in terms of a limit process which depends only on the parameter θ. Also, it is shown that db(n) → 0 if and only if b/n → 0, generalizing results of Arratia, Barbour and Tavare for the Ewens sampling formula. Local limit theorems for weighted sums of the Zi are used to prove these results.

A Darboux-Type Theorem for Slowly Varying Functions

For functionsg(z) satisfying a slowly varying condition in the complex plane, we find asymptotics for the Taylor coefficients of the functionformula]when?>0. As applications we find asymptotics for the number of permutations with cycle lengths all lying in a given setS, and for the number having unique cycle lengths.

Compound Poisson approximations of subgraph counts in random graphs

Poisson approximation, random graphs, Steins method Poisson approximations for the counts of a given subgraph in large random graphs were accomplished using Steins method by Barbour and others. Compound Poisson approximation results, on the other hand, have not appeared, at least partly because of the lack of a suitable coupling. We address that problem by introducing the concept of cluster determining pairs, leading to a useful coupling for a large class of subgraphs we call local. We find bounds on the compound Poisson approximation of counts of local subgraphs in large random graphs.

Poissonian Behaviour of Ising Spin Systems in an External Field

We apply the Stein–Chen method for Poisson approximation to spin-half Ising-type models in positive external field which satisfy the FKG inequality. In particular, we show that, provided the density of minus spins is low and can be expanded as a convergent power series in the activity (fugacity) variable, the distribution of minus spins is approximately Poisson. The error of the approximation is inversely proportional to the number of lattice sites (we obtain upper and lower bounds on the total variation distance between the exact distribution and its Poisson approximation). We illustrate these results by application to specific models, including the mean-field and nearest neighbor ferromagnetic Ising models.

Asymptotic enumeration and logical limit laws for expansive multisets and selections

Given a sequence of integers

Asymptotic enumeration of incidence matrices

a_j, j\ge 1,

Logical limit laws for logarithmic structures

a multiset is a combinatorial object composed of unordered components, such that there are exactly

Information Loss in Riffle Shuffling

a_j