Asymptotics of the visibility function in the Boolean model

香农熵的布尔类比与单调性问题

A Boolean analogue of Shannon's entropy and monotonicity problem

该研究通过分析两类随机矩阵模型的渐近概率，定义了布尔独立性的熵，证明其速率函数的性质、由Rademacher分布最大化且沿布尔中心极限定理单调。

Asymptotics of visibility in the hyperbolic plane

At each point of a Poisson point process of intensity $λ$ in the hyperbolic place, center a ball of bounded random radius. Consider the probability $P_r$ that from a fixed point, there is some direction in which one can reach distance $r$ without hitting any ball. It is known \cite{BJST} that if $λ$ is strictly smaller than a critical intensity $λ_{gv}$ then $P_r$ does not go to $0$ as $r\to \infty$. The main result in this note shows that in the case $λ=λ_{gv}$, the probability of reaching distance larger than $r$ decays essentially polynomial, while if $λ>λ_{gv}$, the decay is exponential. W…

Visibility to infinity in the hyperbolic plane, despite obstacles

Suppose that $Z$ is a random closed subset of the hyperbolic plane $\H^2$, whose law is invariant under isometries of $\H^2$. We prove that if the probability that $Z$ contains a fixed ball of radius 1 is larger than some universal constant $p<1$, then there is positive probability that $Z$ contains (bi-infinite) lines. We then consider a family of random sets in $\H^2$ that satisfy some additional natural assumptions. An example of such a set is the covered region in the Poisson Boolean model. Let $f(r)$ be the probability that a line segment of length $r$ is contained in such a set $Z$. We s…

From heavy-tailed Boolean models to scale-free Gilbert graphs

Define the scale-free Gilbert graph based on a Boolean model with heavy-tailed radius distribution on the $d$-dimensional torus by connecting two centers of balls by an edge if at least one of the balls contains the center of the other. We investigate two asymptotic properties of this graph as the size of the torus tends to infinity. First, we determine the tail index associated with the asymptotic distribution of the sum of all power-weighted incoming and outgoing edge lengths at a randomly chosen vertex. Second, we study the behavior of chemical distances on scale-free Gilbert graphs and sho…

Asymptotics of a time bounded cylinder model

One way to model telecommunication networks are static Boolean models. However, dynamics such as node mobility have a significant impact on the performance evaluation of such networks. Consider a Boolean model in $\mathbb{R}^d$ and a random direction movement scheme. Given a fixed time horizon $T>0$, we model these movements via cylinders in $\mathbb{R}^d \times [0,T]$. In this work, we derive central limit theorems for functionals of the union of these cylinders. The volume and the number of isolated cylinders and the Euler characteristic of the random set are considered and give an answer to…

Variance asymptotics and central limit theory for geometric functionals of Poisson cylinder processes

This paper deals with the union set of a stationary Poisson process of cylinders in $\mathbb{R}^n$ having an $(n-m)$-dimensional base and an $m$-dimensional direction space, where $m\in\{0,1,\ldots,n-1\}$ and $n\geq 2$. The concept simultaneously generalises those of a Boolean model and a Poisson hyperplane or $m$-flat process. Under very general conditions on the typical cylinder base a Berry-Esseen bound for the volume of the union set within a sequence of growing test sets is derived. Assuming convexity of the cylinder bases and of the window a similar result is shown for a broad class of g…

The Boolean Model in the Shannon Regime: Three Thresholds and Related Asymptotics

Consider a family of Boolean models, indexed by integers $n \ge 1$, where the $n$-th model features a Poisson point process in ${\mathbb{R}}^n$ of intensity $e^{n ρ_n}$ with $ρ_n \to ρ$ as $n \to \infty$, and balls of independent and identically distributed radii distributed like $\bar X_n \sqrt{n}$, with $\bar X_n$ satisfying a large deviations principle. It is shown that there exist three deterministic thresholds: $τ_d$ the degree threshold; $τ_p$ the percolation threshold; and $τ_v$ the volume fraction threshold; such that asymptotically as $n$ tends to infinity, in a sense made precise in …

Comparisons and asymptotics for empty space hazard functions of germ-grain models

We study stochastic properties of the empty space for stationary germ-grain models in $\R^d$, in particular we deal with the inner radius of the empty space with respect to a general structuring element which is allowed to be lower-dimensional. We consider Poisson cluster germ-grain models and Boolean models with grains that are clusters of convex bodies and show that more variable size of clusters results in stochastically greater empty space in terms of the empty space hazard function. We also study impact of clusters being more spread in the space on the value of the empty space hazard. Fur…

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.

布尔函数的分布复杂性度量研究

A study of distributional complexity measures for Boolean functions

本文研究了布尔函数在独立输入下的分布复杂性度量，探讨了各度量在期望意义下的“渐近分离”与“多项式相关”性，并引入了局部见证复杂性与涉及部分信息的新版算法复杂性，还通过渗流例子建立了与理论计算机科学的联系。

Statistics for Poisson models of overlapping spheres

The paper considers the stationary Poisson Boolean model with spherical grains and proposes a family of nonparametric estimators for the radius distribution. These estimators are based on observed distances and radii, weighted in an appropriate way. They are ratio-unbiased and asymptotically consistent for growing observation window. It is shown that the asymptotic variance exists and is given by a fairly explicit integral expression. Asymptotic normality is established under a suitable integrability assumption on the weight function. The paper also provides a short discussion of related estim…

对Courtade-Kumar猜想的进展：最优高噪声熵界与广义坐标-wise互信息

Progress on the Courtade-Kumar Conjecture: Optimal High-Noise Entropy Bounds and Generalized Coordinate-wise Mutual Information

本研究解决了Courtade-Kumar猜想中的关键问题，通过证明高噪声下的最优熵界，扩展了该猜想的适用范围。