Scaling limits for the Lego discrepancy

For the Lego discrepancy with M bins, which is equivalent with a chi^2-statistic with M bins, we present a procedure to calculate the moment generating function of the probability distribution perturbatively if M and N, the number of uniformly and randomly distributed data points, become large. Furthermore, we present a phase diagram for various limits of the probability distribution in terms of the standardized variable if M and N become infinite.Comment: 16 page

A fast algorithm for generating a uniform distribution inside a high-dimensional polytope

We describe a uniformly fast algorithm for generating points \vec{x} uniformly in a hypercube with the restriction that the difference between each pair of coordinates is bounded. We discuss the quality of the algorithm in the sense of its usage of pseudo-random source numbers, and present an interesting result on the correlation between the coordinates.Comment: 7 pages, cpu-time table added to illustrate efficienc

Computer-aided analysis of Riemann sheet structures

We report on experience with an investigation of the analytic structure of the solution of certain algebraic complex equations. In particular the behavior of their series expansions around the origin is discussed. The investigation imposes the need for an analysis of the singularities and the Riemann sheets of the solution, in which numerical methods are used.Comment: 11 pages, 3 figures, uses a4wide.sty, amsmath.sty and axodraw.st

SARGE: an algorithm for generating QCD-antennas

We present an algorithm to generate any number of random massless momenta in phase space, with a distribution that contains the kinematical pole structure that is typically found in multi-parton QCD-processes. As an application, we calculate the cross-section of some \eplus\eminus \to partons processes, and compare SARGE's performance with that of the uniform-phase space generator RAMBO.Comment: 9 pages, affiliation correcte

Quantum field theory for discrepancies

The concept of discrepancy plays an important role in the study of uniformity properties of point sets. For sets of random points, the discrepancy is a random variable. We apply techniques from quantum field theory to translate the problem of calculating the probability density of (quadratic) discrepancies into that of evaluating certain path integrals. Both their perturbative and non-perturbative properties are discussed.Comment: 26 page

Gaussian limits for discrepancies. I: Asymptotic results

We consider the problem of finding, for a given quadratic measure of non-uniformity of a set of $N$ points (such as $L_2$ star-discrepancy or diaphony), the asymptotic distribution of this discrepancy for truly random points in the limit $N\to\infty$. We then examine the circumstances under which this distribution approaches a normal distribution. For large classes of non-uniformity measures, a Law of Many Modes in the spirit of the Central Limit Theorem can be derived.Comment: 25 pages, Latex, uses fleqn.sty, a4wide.sty, amsmath.st