Tesselation of a triangle by repeated barycentric subdivision.
The Apollonian decay of beer foam bubble size distribution and the lattices of Young diagrams and their correlated mixing functions.
The asymptotic behavior of the average -discrepancies and a randomized discrepancy.
The Beta(p,1) extensions of the random (uniform) Cantor sets
Starting from the random extension of the Cantor middle set in [0,1], by iteratively removing the central uniform spacing from the intervals remaining in the previous step, we define random Beta(p,1)-Cantor sets, and compute their Hausdorff dimension. Next we define a deterministic counterpart, by iteratively removing the expected value of the spacing defined by the appropriate Beta(p,1) order statistics. We investigate the reasons why the Hausdorff dimension of this deterministic fractal is greater...
The brownian cactus I. Scaling limits of discrete cactuses
The cactus of a pointed graph is a discrete tree associated with this graph. Similarly, with every pointed geodesic metric space , one can associate an -tree called the continuous cactus of . We prove under general assumptions that the cactus of random planar maps distributed according to Boltzmann weights and conditioned to have a fixed large number of vertices converges in distribution to a limiting space called the Brownian cactus, in the Gromov–Hausdorff sense. Moreover, the Brownian cactus...
The Brunn-Minkowski Inequality in Gauss Space.
The distribution of the number of nodes in the relative interior of the typical I-segment in homogeneous planar anisotropic STIT Tessellations
A result about the distribution of the number of nodes in the relative interior of the typical I-segment in homogeneous and isotropic random tessellations stable under iteration (STIT tessellations) is extended to the anisotropic case using recent findings from Schreiber/Thäle, Typical geometry, second-order properties and central limit theory for iteration stable tessellations, arXiv:1001.0990 [math.PR] (2010). Moreover a new expression for the values of this probability distribution is presented...
The distribution of unbounded random sets and the multivalued strong law of large numbers in nonreflexive Banach spaces.
The Hausdorff dimension and exact Hausdorff measure of random recursive sets with overlapping.
The Markovian hyperbolic triangulation
We construct and study the unique random tiling of the hyperbolic plane into ideal hyperbolic triangles (with the three corners located on the boundary) that is invariant (in law) with respect to Möbius transformations, and possesses a natural spatial Markov property that can be roughly described as the conditional independence of the two parts of the triangulation on the two sides of the edge of one of its triangles.
The mean of a maximum likelihood estimator associated with the Brownian bridge.
The probability of the triangle inequality in probabilistic metric squares.
The sizes of components in random circle graphs
We study random circle graphs which are generated by throwing n points (vertices) on the circle of unit circumference at random and joining them by an edge if the length of shorter arc between them is less than or equal to a given parameter d. We derive here some exact and asymptotic results on sizes (the numbers of vertices) of "typical" connected components for different ways of sampling them. By studying the joint distribution of the sizes of two components, we "go into" the structure of random...
The symmetric property () for the Gaussian measure
We give a proof, based on the Poincaré inequality, of the symmetric property () for the Gaussian measure. If is continuous, bounded from below and even, we define and we haveThis property is equivalent to a certain functional form of the Blaschke-Santaló inequality, as explained in a paper by Artstein, Klartag and Milman.
The volume fraction of a non-overlapping germ-grain model.
Thin points for brownian motion
Translative integral formulae for convex bodies.
Trees and matchings from point processes.