arXiv subjects

Günter Last

Publications and source records attributed to Günter Last.

At least 37 records · Page 2Linked to original sources

arXiv2020

On strongly rigid hyperfluctuating random measures

In contrast to previous belief, we provide examples of stationary ergodic random measures that are both hyperfluctuating and strongly rigid. Therefore, we study hyperplane intersection processes (HIPs) that are formed by the vertices of Poisson hyperplane tessellations. These HIPs are known to be hyperfluctuating, that is, the variance of the number of points in a bounded observation window grows faster than the size of the window. Here we show that the HIPs exhibit a particularly strong rigidity property. For any bounded Borel set $B$, an exponentially small (bounded) stopping set suffices to reconstruct the position of all points in $B$ and, in fact, all hyperplanes intersecting $B$. Therefore, also the random measures supported by the hyperplane intersections of arbitrary (but fixed) dimension, are hyperfluctuating. Our examples aid the search for relations between correlations, density fluctuations, and rigidity properties.

math.PR↗

arXiv2020

Characterizing digital microstructures by the Minkowski-based quadratic normal tensor

For material modeling of microstructured media, an accurate characterization of the underlying microstructure is indispensable. Mathematically speaking, the overall goal of microstructure characterization is to find simple functionals which describe the geometric shape as well as the composition of the microstructures under consideration, and enable distinguishing microstructures with distinct effective material behavior. For this purpose, we propose using Minkowski tensors, in general, and the quadratic normal tensor, in particular, and introduce a computational algorithm applicable to voxel-based microstructure representations. Rooted in the mathematical field of integral geometry, Minkowski tensors associate a tensor to rather general geometric shapes, which make them suitable for a wide range of microstructured material classes. Furthermore, they satisfy additivity and continuity properties, which makes them suitable and robust for large-scale applications. We present a modular algorithm for computing the quadratic normal tensor of digital microstructures. We demonstrate multigrid convergence for selected numerical examples and apply our approach to a variety of microstructures. Strikingly, the presented algorithm remains unaffected by inaccurate computation of the interface area. The quadratic normal tensor may be used for engineering purposes, such as mean-field homogenization or as target value for generating synthetic microstructures.

cs.CE↗

arXiv2020

Hyperuniform and rigid stable matchings

We study a stable partial matching $τ$ of the (possibly randomized) $d$-dimensional lattice with a stationary determinantal point process $Ψ$ on $\mathbb{R}^d$ with intensity $α>1$. For instance, $Ψ$ might be a Poisson process. The matched points from $Ψ$ form a stationary and ergodic (under lattice shifts) point process $Ψ^τ$ with intensity $1$ that very much resembles $Ψ$ for $α$ close to $1$. On the other hand $Ψ^τ$ is hyperuniform and number rigid, quite in contrast to a Poisson process. We deduce these properties by proving more general results for a stationary point process $Ψ$, whose so-called matching flower (a stopping set determining the matching partner of a lattice point) has a certain subexponential tail behaviour. For hyperuniformity, we also additionally need to assume some mixing condition on $Ψ$. Further, if $Ψ$ is a Poisson process then $Ψ^τ$ has an exponentially decreasing truncated pair correlation function.

math.PR↗

arXiv2018

Transporting random measures on the line and embedding excursions into Brownian motion

We consider two jointly stationary and ergodic random measures $ξ$ and $η$ on the real line $\mathbb{R}$ with equal intensities. An allocation is an equivariant random mapping from $\mathbb{R}$ to $\mathbb{R}$. We give sufficient and partially necessary conditions for the existence of allocations transporting $ξ$ to $η$. An important ingredient of our approach is to introduce a transport kernel balancing $ξ$ and $η$, provided these random measures are mutually singular. In the second part of the paper, we apply this result to the path decomposition of a two-sided Brownian motion into three independent pieces: a time reversed Brownian motion on $(-\infty,0]$, an excursion distributed according to a conditional Itô's law and a Brownian motion starting after this excursion. An analogous result holds for Bismut's excursion law.

math.PR↗

arXiv2018

Absent-minded passengers

Passengers board a fully booked airplane in order. The first passenger picks one of the seats at random. Each subsequent passenger takes his or her assigned seat if available, otherwise takes one of the remaining seats at random. It is well known that the last passenger obtains her own seat with probability $1/2$. We study the distribution of the number of incorrectly seated passengers, and we also discuss the case of several absent-minded passengers.

math.PR↗

arXiv2018

The random connection model and functions of edge-marked Poisson processes: second order properties and normal approximation

The random connection model is a random graph whose vertices are given by the points of a Poisson process and whose edges are obtained by randomly connecting pairs of Poisson points in a position dependent but independent way. We study first and second order properties of the numbers of components isomorphic to given finite connected graphs. For increasing observation windows in an Euclidean setting we prove qualitative multivariate and quantitative univariate central limit theorems for these component counts as well as a qualitative central limit theorem for the total number of finite components. To this end we first derive general results for functions of edge marked Poisson processes, which we believe to be of independent interest.

math.PR↗

arXiv2018

An integral characterization of the Dirichlet process

We give a new integral characterization of the Dirichlet process on a general phase space. To do so we first prove a characterization of the nonsymmetric Beta distribution via size-biased sampling. Two applications are a new characterization of the Dirichlet distribution and a marked version of a classical characterization of the Poisson-Dirichlet distribution via invariance under size-biased sampling.

math.PR↗

arXiv2017

On maximal hard-core thinnings of stationary particle processes

The present paper studies existence and distributional uniqueness of subclasses of stationary hard-core particle systems arising as thinnings of stationary particle processes. These subclasses are defined by natural maximality criteria. We investigate two specific criteria, one related to the intensity of the hard-core particle process, the other one being a local optimality criterion on the level of realizations. In fact, the criteria are equivalent under suitable moment conditions. We show that stationary hard-core thinnings satisfying such criteria exist and are frequently distributionally unique. More precisely, distributional uniqueness holds in subcritical and barely supercritical regimes of continuum percolation. Additionally, based on the analysis of a specific example, we argue that fluctuations in grain sizes can play an important role for establishing distributional uniqueness at high intensities. Finally, we provide a family of algorithmically constructible approximations whose volume fractions are arbitrarily close to the maximum.

math.PR↗

arXiv2017

Concentration inequalities for measures of a Boolean model

We consider a Boolean model $Z$ driven by a Poisson particle process $η$ on a metric space $\mathbb{Y}$. We study the random variable $ρ(Z)$, where $ρ$ is a (deterministic) measure on $\mathbb{Y}$. Due to the interaction of overlapping particles, the distribution of $ρ(Z)$ cannot be described explicitly. In this note we derive concentration inequalities for $ρ(Z)$. To this end we first prove two concentration inequalities for functions of a Poisson process on a general phase space.

math.PR↗

arXiv2016

On the Ornstein-Zernike equation for stationary cluster processes and the random connection model

In the first part of this paper we consider a general stationary subcritical cluster model in $\mathbb{R}^d$. The associated pair-connectedness function can be defined in terms of two-point Palm probabilities of the underlying point process. Using Palm calculus and Fourier theory we solve the Ornstein-Zernike equation (OZE) under quite general distributional assumptions. In the second part of the paper we discuss the analytic and combinatorial properties of the OZE-solution in the special case of a Poisson driven random connection model.

math.PR↗

arXiv2016

Cell shape analysis of random tessellations based on Minkowski tensors

To which degree are shape indices of individual cells of a tessellation characteristic for the stochastic process that generates them? Within the context of stochastic geometry and the physics of disordered materials, this corresponds to the question of relationships between different stochastic models. In the context of image analysis of synthetic and biological materials, this question is central to the problem of inferring information about formation processes from spatial measurements of resulting random structures. We address this question by a theory-based simulation study of shape indices derived from Minkowski tensors for a variety of tessellation models. We focus on the relationship between two indices: an isoperimetric ratio of the empirical averages of cell volume and area and the cell elongation quantified by eigenvalue ratios of interfacial Minkowski tensors. Simulation data for these quantities, as well as for distributions thereof and for correlations of cell shape and volume, are presented for Voronoi mosaics of the Poisson point process, determinantal and permanental point processes, and Gibbs hard-core and random sequential absorption processes as well as for Laguerre tessellations of polydisperse spheres and STIT- and Poisson hyperplane tessellations. These data are complemented by mechanically stable crystalline sphere and disordered ellipsoid packings and area-minimising foam models. We find that shape indices of individual cells are not sufficient to unambiguously identify the generating process even amongst this limited set of processes. However, we identify significant differences of the shape indices between many of these tessellation models. Given a realization of a tessellation, these shape indices can narrow the choice of possible generating processes, providing a powerful tool which can be further strengthened by density-resolved volume-shape correlations.

cond-mat.soft↗

arXiv2016

Inclusion-exclusion principles for convex hulls and the Euler relation

Consider $n$ points $X_1,\ldots,X_n$ in $\mathbb R^d$ and denote their convex hull by $Π$. We prove a number of inclusion-exclusion identities for the system of convex hulls $Π_I:=conv(X_i\colon i\in I)$, where $I$ ranges over all subsets of $\{1,\ldots,n\}$. For instance, denoting by $c_k(X)$ the number of $k$-element subcollections of $(X_1,\ldots,X_n)$ whose convex hull contains a point $X\in\mathbb R^d$, we prove that $$ c_1(X)-c_2(X)+c_3(X)-\ldots + (-1)^{n-1} c_n(X) = (-1)^{\dim Π} $$ for all $X$ in the relative interior of $Π$. This confirms a conjecture of R. Cowan [Adv. Appl. Probab., 39(3):630--644, 2007] who proved the above formula for almost all $X$. We establish similar results for the number of polytopes $Π_J$ containing a given polytope $Π_I$ as an $r$-dimensional face, thus proving another conjecture of R. Cowan [Discrete Comput. Geom., 43(2):209--220, 2010]. As a consequence, we derive inclusion-exclusion identities for the intrinsic volumes and the face numbers of the polytopes $Π_I$. The main tool in our proofs is a formula for the alternating sum of the face numbers of a convex polytope intersected by an affine subspace. This formula generalizes the classical Euler--Schläfli--Poincaré relation and is of independent interest.

math.PR↗

arXiv2016

Second-order properties and central limit theorems for geometric functionals of Boolean models

Let $Z$ be a Boolean model based on a stationary Poisson process $η$ of compact, convex particles in Euclidean space ${\mathbb{R}}^d$. Let $W$ denote a compact, convex observation window. For a large class of functionals $ψ$, formulas for mean values of $ψ(Z\cap W)$ are available in the literature. The first aim of the present work is to study the asymptotic covariances of general geometric (additive, translation invariant and locally bounded) functionals of $Z\cap W$ for increasing observation window $W$, including convergence rates. Our approach is based on the Fock space representation associated with $η$. For the important special case of intrinsic volumes, the asymptotic covariance matrix is shown to be positive definite and can be explicitly expressed in terms of suitable moments of (local) curvature measures in the isotropic case. The second aim of the paper is to prove multivariate central limit theorems including Berry-Esseen bounds. These are based on a general normal approximation result obtained by the Malliavin--Stein method.

math.PR↗

arXiv2016

Second order analysis of geometric functionals of Boolean models

This paper presents asymptotic covariance formulae and central limit theorems for geometric functionals, including volume, surface area, and all Minkowski functionals and translation invariant Minkowski tensors as prominent examples, of stationary Boolean models. Special focus is put on the anisotropic case. In the (anisotropic) example of aligned rectangles, we provide explicit analytic formulae and compare them with simulation results. We discuss which information about the grain distribution second moments add to the mean values.

math.PR↗

arXiv2014

Two lilypond systems of finite line-segments

The paper discusses two models for non-overlapping finite line-segments constructed via the lilypond protocol, operating here on a given array of points in the plane with which are associated directions. At time 0, each line-segment starts growing at unit rate around its center in the given direction; each line-segment, under Model 1, ceases growth when one of its ends hits another line, while under Model 2, its growth ceases either when one of its ends hits another line, or when it is hit by the growing end of some other line. The paper shows that these procedures are well-defined and gives constructive algorithms to compute the lengths of the segments. Moreover it specifies assumptions under which stochastic versions, i.e. models based on point processes, exist. Afterwards it deals with the question as to whether there is percolation in Model 1. The paper concludes with a section containing several conjectures and final remarks.

math.PR↗

arXiv2014

Stochastic analysis for Poisson processes

This survey is a preliminary version of a chapter of the forthcoming book "Stochastic Analysis for Poisson Point Processes: Malliavin Calculus, Wiener-Itô Chaos Expansions and Stochastic Geometry" edited by Giovanni Peccati and Matthias Reitzner. The paper develops some basic theory for the stochastic analysis of Poisson process on a general $σ$-finite measure space. After giving some fundamental definitions and properties (as the multivariate Mecke equation) the paper presents the Fock space representation of square-integrable functions of a Poisson process in terms of iterated difference operators. This is followed by the introduction of multivariate stochastic Wiener-Itô integrals and the discussion of their basic properties. The paper then proceeds with proving the chaos expansion of square-integrable Poisson functionals, and defining and discussing Malliavin operators. Further topics are products of Wiener-Itô integrals and Mehler's formula for the inverse of the Ornstein-Uhlenbeck generator based on a dynamic thinning procedure. The survey concludes with covariance identities, the Poincaré inequality and the FKG-inequality.

math.PR↗

arXiv2014

Perturbation analysis of Poisson processes

We consider a Poisson process $Φ$ on a general phase space. The expectation of a function of $Φ$ can be considered as a functional of the intensity measure $λ$ of $Φ$. Extending earlier results of Molchanov and Zuyev [Math. Oper. Res. 25 (2010) 485-508] on finite Poisson processes, we study the behaviour of this functional under signed (possibly infinite) perturbations of $λ$. In particular, we obtain general Margulis-Russo type formulas for the derivative with respect to non-linear transformations of the intensity measure depending on some parameter. As an application, we study the behaviour of expectations of functions of multivariate Lévy processes under perturbations of the Lévy measure. A key ingredient of our approach is the explicit Fock space representation obtained in Last and Penrose [Probab. Theory Related Fields 150 (2011) 663-690].

math.PR↗

arXiv2014

Unbiased shifts of Brownian motion

Let $B=(B_t)_{t\in {\mathbb{R}}}$ be a two-sided standard Brownian motion. An unbiased shift of $B$ is a random time $T$, which is a measurable function of $B$, such that $(B_{T+t}-B_T)_{t\in {\mathbb{R}}}$ is a Brownian motion independent of $B_T$. We characterise unbiased shifts in terms of allocation rules balancing mixtures of local times of $B$. For any probability distribution $ν$ on ${\mathbb{R}}$ we construct a stopping time $T\ge0$ with the above properties such that $B_T$ has distribution $ν$. We also study moment and minimality properties of unbiased shifts. A crucial ingredient of our approach is a new theorem on the existence of allocation rules balancing stationary diffuse random measures on ${\mathbb{R}}$. Another new result is an analogue for diffuse random measures on ${\mathbb{R}}$ of the cycle-stationarity characterisation of Palm versions of stationary simple point processes.

math.PR↗