Related papers: Limit theory for planar Gilbert tessellations
Shape is one of the important characteristics for the structures observed in living organisms. Whereas biologists have proposed models where the shape is controlled on a molecular level [1], physicists, following Turing [2] and d'Arcy…
We investigate the existence and first percolation properties of general stopped germ-grain models. They are defined via a random set of germs generated by a homogeneous planar Poisson point process in $\mathbf{R}^{2}$. From each germ, a…
In Euclidean space, the asymptotic shape of large cells in various types of Poisson driven random tessellations has been the subject of a famous conjecture due to David Kendall. Since shape is a geometric concept and large cells are…
A branching random tessellation (BRT) is a stochastic process that transforms a coarse initial tessellation of $\mathbb{R}^d$ into a finer tessellation by means of random cell divisions in continuous time. This concept generalises the…
We show that the random point measures induced by vertices in the convex hull of a Poisson sample on the unit ball, when properly scaled and centered, converge to those of a mean zero Gaussian field. We establish limiting variance and…
We pursue two goals in this article. As our first goal, we construct a family $\mathcal{M}_G$ of Gibbs like measures on the set of piecewise linear convex functions $g:\mathbb{R}^2\to\mathbb{R}$. It turns out that there is a one-to-one…
It is well known that, under broad assumptions, the time-scaled point process of exceedances of a high level by a stationary sequence converges to a compound Poisson process as the level grows. The purpose of this note is to demonstrate…
The experimental actualisation of organoids modelling organs from brains to pancreases has revealed that much of the diverse morphologies of organs are emergent properties of simple intercellular "rules" and not the result of top-down…
We study point processes that consist of certain centers of point tuples of an underlying Poisson process. Such processes arise in stochastic geometry in the study of exceedances of various functionals describing geometric properties of the…
We study a class of discrete-time random walks in $\mathbb{R}^d$ whose conditional drift decays polynomially in time and grows polynomially with the distance from the origin to the current position. This class is related to several models…
The objective of this study is to investigate the limiting behavior of a subgraph counting process. The subgraph counting process we consider counts the number of subgraphs having a specific shape that exist outside an expanding ball as the…
We consider Gaussian approximation in three particular models of Poisson-Laguerre tessellations, namely, the $\beta$-, $\beta'$- and Gaussian-Voronoi tessellations. The tessellations are constructed based on inhomogeneous Poisson point…
In this article we establish two fundamental results for the sublevel set persistent homology for stationary processes indexed by the positive integers. The first is a strong law of large numbers for the persistence diagram (treated as a…
This continues work started in part I on a general branching-within-branching model for host-parasite co-evolution. Here we focus on asymptotic results for relevant processes in the case when parasites survive. In particular, limit theorems…
We consider self-loops and multiple edges in the configuration model as the size of the graph tends to infinity. The interest in these random variables is due to the fact that the configuration model, conditioned on being simple, is a…
We study light propagation in a medium with uniform torsion, modeled as a continuum of screw dislocations within the geometric theory of defects. By solving Maxwell's equations in covariant form, we show that torsion induces intrinsic…
A central limit theorem is established for a sum of random variables belonging to a sequence of random fields. The fields are assumed to have zero mean conditional on the past history and to satisfy certain conditional $\alpha$-mixing…
The patterns arising from the differential swelling of gels are investigated experimentally and theoretically as a model for the differential growth of living tissues. Two geometries are considered: a thin strip of soft gel clamped to a…
In this paper, we consider a regular tessellation of the Euclidean plane and the sequence of its geometric scalings by negative powers of a fixed integer. We generate iteratively random sets as the union of adjacent tiles from these…
In this paper, we are interested in the behavior of the typical Poisson-Voronoi cell in the plane when the radius of the largest disk centered at the nucleus and contained in the cell goes to infinity. We prove a law of large numbers for…