arXiv subjects

Daniela Flimmel

Publications and source records attributed to Daniela Flimmel.

5 recordsLinked to original sources

arXiv2025

Fitting regular point patterns with a hyperuniform perturbed lattice

We introduce a flexible methodology for modelling regular spatial point patterns using hyperuniform perturbed lattices. We show that, under suitable mixing conditions on the displacement field, lattices perturbed by stationary random fields are hyperuniform in arbitrary dimension. In particular, Gaussian perturbations with absolutely summable covariances yield class-I hyperuniform point processes. We further derive an explicit formula for the $K$-function of Gaussian models, which enables efficient parameter estimation via the minimum contrast method. The proposed framework provides a computationally tractable alternative to classical Gibbs models for repulsive data. The methodology is illustrated on three-dimensional data describing grain centers in a polycrystalline nickel-titanium alloy.

math.PR↗

arXiv2024

(Non)-hyperuniformity of perturbed lattices

We ask whether a stationary lattice in dimension $d$ whose points are shifted by identically distributed but possibly dependent perturbations remains hyperuniform. When $d = 1$ or $2$, we show that it is the case when the perturbations have a finite $d$-moment, and that this condition is sharp. When $d \geq 3$, we construct arbitrarily small perturbations such that the resulting point process is not hyperuniform. As a side remark of independent interest, we exhibit hyperuniform processes with arbitrarily slow decay of their number variance.

math.PR↗

arXiv2023

Non-hyperuniformity of Gibbs point processes with short range interaction

We investigate the hyperuniformity of marked Gibbs point processes with weak dependencies among distant points whilst the interactions of close points are kept arbitrary. Some variants of stability and range assumptions are posed on the Papangelou intensity in order to prove that the resulting point process is not hyperuniform. The scope of our results covers many frequently used models including Gibbs point processes with a superstable, lower-regular, integrable pair potential as well as Widom--Rowlinson model with random radii or Gibbs point processes with interactions based on Voronoi tessellation and nearest neighbour graph.

math.PR↗

arXiv2021

On the Variance of the Area of Planar Cylinder Processes Driven by Brillinger-Mixing Point Processes

We study some asymptotic properties of cylinder processes in the plane defined as union sets of dilated straight lines (appearing as mutually overlapping infinitely long strips) derived from a stationary independently marked point process on the real line, where the marks describe thickness and orientation of individual cylinders. Such cylinder processes form an important class of (in general non-stationary) planar random sets. We observe the cylinder process in an unboundedly growing domain $\rho K$ when $\rho \to \infty\,$, where the set $K$ is compact and star-shaped w.r.t. the origin ${\bf o}$ being an inner point of $K$. Provided the unmarked point process satisfies a Brillinger-type mixing condition and the thickness of the typical cylinder has a finite second moment we prove a (weak) law of large numbers as well as a formula of the asymptotic variance for the area of the cylinder process in $\rho K$. Due to the long-range dependencies of the cylinder process, this variance increases proportionally to $\rho^3$.

math.PR↗

arXiv2019

Limit theory for unbiased and consistent estimators of statistics of random tessellations

We observe a realization of a stationary generalized weighted Voronoi tessellation of the d-dimensional Euclidean space within a bounded observation window. Given a geometric characteristic of the typical cell, we use the minus-sampling technique to construct an unbiased estimator of the average value of this geometric characteristic. Under mild conditions on the weights of the cells, we establish variance asymptotics and the asymptotic normality of the unbiased estimator as the observation window tends to the whole space. Moreover, the weak consistency is shown for this estimator.

math.PR↗