Asymptotic statistics of the n-sided planar Voronoi cell: II. Heuristics

We develop a set of heuristic arguments to explain several results on planar Poisson-Voronoi tessellations that were derived earlier at the cost of considerable mathematical effort. The results concern Voronoi cells having a large number n of sides. The arguments start from an entropy balance applied to the arrangement of n neighbors around a central cell. It is followed by a simplified evaluation of the phase space integral for the probability p_n that an arbitrary cell be n-sided. The limitations of the arguments are indicated. As a new application we calculate the expected number of Gabriel (or full) neighbors of an n-sided cell in the large-n limit.Comment: 22 pages, 10 figure

New Monte Carlo method for planar Poisson-Voronoi cells

By a new Monte Carlo algorithm we evaluate the sidedness probability p_n of a planar Poisson-Voronoi cell in the range 3 \leq n \leq 1600. The algorithm is developed on the basis of earlier theoretical work; it exploits, in particular, the known asymptotic behavior of p_n as n\to\infty. Our p_n values all have between four and six significant digits. Accurate n dependent averages, second moments, and variances are obtained for the cell area and the cell perimeter. The numerical large n behavior of these quantities is analyzed in terms of asymptotic power series in 1/n. Snapshots are shown of typical occurrences of extremely rare events implicating cells of up to n=1600 sides embedded in an ordinary Poisson-Voronoi diagram. We reveal and discuss the characteristic features of such many-sided cells and their immediate environment. Their relevance for observable properties is stressed.Comment: 35 pages including 10 figures and 4 table

Note on a q-modified central limit theorem

A q-modified version of the central limit theorem due to Umarov et al. affirms that q-Gaussians are attractors under addition and rescaling of certain classes of strongly correlated random variables. The proof of this theorem rests on a nonlinear q-modified Fourier transform. By exhibiting an invariance property we show that this Fourier transform does not have an inverse. As a consequence, the theorem falls short of achieving its stated goal.Comment: 10 pages, no figure

Continuous and first-order jamming transition in crossing pedestrian traffic flows

After reviewing the main results obtained within a model for the intersection of two perpendicular flows of pedestrians, we present a new finding: the changeover of the jamming transition from continuous to first order when the size of the intersection area increases.Comment: 14 pages, 9 figure

Exact domain wall theory for deterministic TASEP with parallel update

Domain wall theory (DWT) has proved to be a powerful tool for the analysis of one-dimensional transport processes. A simple version of it was found very accurate for the Totally Asymmetric Simple Exclusion Process (TASEP) with random sequential update. However, a general implementation of DWT is still missing in the case of updates with less fluctuations, which are often more relevant for applications. Here we develop an exact DWT for TASEP with parallel update and deterministic (p=1) bulk motion. Remarkably, the dynamics of this system can be described by the motion of a domain wall not only on the coarse-grained level but also exactly on the microscopic scale for arbitrary system size. All properties of this TASEP, time-dependent and stationary, are shown to follow from the solution of a bivariate master equation whose variables are not only the position but also the velocity of the domain wall. In the continuum limit this exactly soluble model then allows us to perform a first principle derivation of a Fokker-Planck equation for the position of the wall. The diffusion constant appearing in this equation differs from the one obtained with the traditional `simple' DWT.Comment: 5 pages, 4 figure

Crossing pedestrian traffic flows,diagonal stripe pattern, and chevron effect

We study two perpendicular intersecting flows of pedestrians. The latter are represented either by moving hard core particles of two types, eastbound (\symbp) and northbound (\symbm), or by two density fields, \rhop_t(\brr) and \rhom_t(\brr). Each flow takes place on a lattice strip of width $M$ so that the intersection is an $M\times M$ square. We investigate the spontaneous formation, observed experimentally and in simulations, of a diagonal pattern of stripes in which alternatingly one of the two particle types dominates. By a linear stability analysis of the field equations we show how this pattern formation comes about. We focus on the observation, reported recently, that the striped pattern actually consists of chevrons rather than straight lines. We demonstrate that this `chevron effect' occurs both in particle simulations with various different update schemes and in field simulations. We quantify the effect in terms of the chevron angle $\Delta\theta_0$ and determine its dependency on the parameters governing the boundary conditions.Comment: 36 pages, 22 figure

Asymptotic statistics of the n-sided planar Poisson-Voronoi cell. I. Exact results

We achieve a detailed understanding of the $n$-sided planar Poisson-Voronoi cell in the limit of large $n$. Let ${p}\_n$ be the probability for a cell to have $n$ sides. We construct the asymptotic expansion of $\log {p}\_n$ up to terms that vanish as $n\to\infty$. We obtain the statistics of the lengths of the perimeter segments and of the angles between adjoining segments: to leading order as $n\to\infty$, and after appropriate scaling, these become independent random variables whose laws we determine; and to next order in $1/n$ they have nontrivial long range correlations whose expressions we provide. The $n$-sided cell tends towards a circle of radius (n/4\pi\lambda)^{\half}, where $\lambda$ is the cell density; hence Lewis' law for the average area $A\_n$ of the $n$-sided cell behaves as $A\_n \simeq cn/\lambda$ with $c=1/4$. For $n\to\infty$ the cell perimeter, expressed as a function $R(\phi)$ of the polar angle $\phi$, satisfies $d^2 R/d\phi^2 = F(\phi)$, where $F$ is known Gaussian noise; we deduce from it the probability law for the perimeter's long wavelength deviations from circularity. Many other quantities related to the asymptotic cell shape become accessible to calculation.Comment: 54 pages, 3 figure