In this talk I first review at an elementary level a selection of central limit theorems, including some lesser known cases, for sums and maxima of uncorrelated and correlated random variables. I recall why several of them appear in physics. Next, I show that there is room for new versions of central limit theorems applicable to specific classes of… (More)
I present a concise review of advances realized over the past three years on planar Poisson-Voronoi tessellations. These encompass new analytic results, a new Monte Carlo method, and application to experimental data.
Let p n be the probability for a planar Poisson-Voronoi cell to have exactly n sides. We construct the asymptotic expansion of log p n up to terms that vanish as n → ∞. Along with it comes a nearly complete understanding of the structure of the large cell. We show that two independent biased random walks executed by the polar angle determine the trajectory… (More)
We consider two stochastic processes, the Gribov process and the general epidemic process, that describe the spreading of an infectious disease. In contrast to the usually assumed case of short-range infections that lead, at the critical point, to directed and isotropic percolation respectively, we consider long-range infections with a probability… (More)
In planar cellular systems m n denotes the average sidedness of a cell neighboring an n-sided cell. Aboav's empirical law states that nm n is linear in n. A downward curvature is nevertheless observed in the numerical nm n data of the Random Voronoi Froth. The exact large-n expansion of m n obtained in the present work, viz. m n = 4+3(π/n) 1 2 +. . .,… (More)
By a new Monte Carlo algorithm we evaluate the sidedness probability p n of a planar Poisson-Voronoi cell in the range 3 ≤ n ≤ 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 → ∞. Our p n values all have between four and six significant digits. Accurate n… (More)
We study the support (i.e. the set of visited sites) of a t step random walk on a two-dimensional square lattice in the large t limit. A broad class of global properties M (t) of the support is considered, including, e.g., the number S(t) of its sites; the length of its boundary; the number of islands of unvisited sites that it encloses; the number of such… (More)
We consider the d-dimensional Poisson-Voronoi tessellation and investigate the applicability of heuristic methods developed recently for two dimensions. Let p n (d) be the probability that a cell have n neighbors (be 'n-faced') and m n (d) the average facedness of a cell adjacent to an n-faced cell. We obtain the leading order terms of the asymptotic… (More)