R. W. R. Darling, Robin Pemantle
Continuous time branching models are used to create random fractals in a Euclidean space, whose Hausdorff dimension is controlled by an input parameter. Finite realizations are applied in modelling the set of sites visited in models of human and animal mobility.
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This paper studies, by employing analytical tools, the small time behavior of the heat content for the Poisson kernel over the unit ball in $\Rd$, $d\geq 2$ by working with its related set covariance function. As a result, we obtain a representation for the third term in the expansion of the heat content over the unit ball and provide the explicit form of the this term in the particular cases $d=2$ and $d=3$.
Let us consider Euclidean first-passage percolation on the Poisson-Delaunay triangulation. We prove almost sure coalescence of any two semi-infinite geodesics with the same asymptotic direction. The proof is based on an adapted Burton-Keane argument and makes use of the concentration property for shortest-path lengths in the considered graphs. Moreover, by considering the specific example of the relative neighborhood graph, we illustrate that our approach extends to further well-known graphs in computational geometry. As an application, we show that the expected number of semi-infinite geodesi…
We consider the Poisson cylinder model in ${\mathbb R}^d$, $d\ge 3$. We show that given any two cylinders ${\mathfrak c}_1$ and ${\mathfrak c}_2$ in the process, there is a sequence of at most $d-2$ other cylinders creating a connection between ${\mathfrak c}_1$ and ${\mathfrak c}_2$. In particular, this shows that the union of the cylinders is a connected set, answering a question appearing in a previous paper. We also show that there are cylinders in the process that are not connected by a sequence of at most $d-3$ other cylinders. Thus, the diameter of the cluster of cylinders equals $d-2$.
We consider the extension of the Euclidean stochastic geometry Poisson Hail model to the case where the service speed is zero in some subset of the Euclidean space and infinity in the complement. We use and develop tools pertaining to sub-additive ergodic theory in order to establish shape theorems for the growth of the ice-heap under light tail assumptions on the hailstone characteristics. The asymptotic shape depends on the statistics of the hailstones, the intensity of the underlying Poisson point process and on the geometrical properties of the zero speed set.
We consider a queue where the server is the Euclidean space, and the customers are random closed sets (RACS) of the Euclidean space. These RACS arrive according to a Poisson rain and each of them has a random service time (in the case of hail falling on the Euclidean plane, this is the height of the hailstone, whereas the RACS is its footprint). The Euclidean space serves customers at speed 1. The service discipline is a hard exclusion rule: no two intersecting RACS can be served simultaneously and service is in the First In First Out order: only the hailstones in contact with the ground melt …
We consider the Poisson cylinder model in $d$-dimensional hyperbolic space. We show that in contrast to the Euclidean case, there is a phase transition in the connectivity of the collection of cylinders as the intensity parameter varies. We also show that for any non-trivial intensity, the diameter of the collection of cylinders is infinite.
For a measure mu supported on a compact connected subset of a Euclidean space which satisfies a uniform d-dimensional decay of the volume of balls we show that the maximal edge in the minimum spanning tree of n indepndent samples from mu is, with high probability (log n/n)^(1/d).
We consider a stationary Poisson hyperplane process with given directional distribution and intensity in $d$-dimensional Euclidean space. Generalizing the zero cell of such a process, we fix a convex body $K$ and consider the intersection of all closed halfspaces bounded by hyperplanes of the process and containing $K$. We study how well these random polytopes approximate $K$ (measured by the Hausdorff distance) if the intensity increases, and how this approximation depends on the directional distribution in relation to properties of $K$.
In this article, we introduce fractional Poisson felds of order k in n-dimensional Euclidean space $R_n^+$. We also work on time-fractional Poisson process of order k, space-fractional Poisson process of order k and tempered version of time-space fractional Poisson process of order k in one dimensional Euclidean space $R_1^+$. These processes are defined in terms of fractional compound Poisson processes. Time-fractional Poisson process of order k naturally generalizes the Poisson process and Poisson process of order k to a heavy tailed waiting times counting process. The space-fractional Poiss…
We explore a generalisation of the Lévy fractional Brownian field on the Euclidean space based on replacing the Euclidean norm with another norm. A characterisation result for admissible norms yields a complete description of all self-similar Gaussian random fields with stationary increments. Several integral representations of the introduced random fields are derived. In a similar vein, several non-Euclidean variants of the fractional Poisson field are introduced and it is shown that they share the covariance structure with the fractional Brownian field and converge to it. The shape parameter…
Concentration inequalities are obtained on Poisson space, for random functionals with finite or infinite variance. In particular, dimension free tail estimates and exponential integrability results are given for the Euclidean norm of vectors of independent functionals. In the finite variance case these results are applied to infinitely divisible random variables such as quadratic Wiener functionals, including Lévy's stochastic area and the square norm of Brownian paths. In the infinite variance case, various tail estimates such as stable ones are also presented.
We establish inequalities for assessing the distance between the distribution of a (possibly multidimensional) functional of a Poisson random measure and that of a Gaussian element. Our bounds only involve add-one cost operators at the order one - that we evaluate and compare at two different scales - and are specifically tailored for studying the Gaussian fluctuations of sequences of geometric functionals displaying a form of weak stabilization - see Penrose and Yukich (2001) and Penrose (2005). Our main bounds extend the estimates recently exploited by Chatterjee and Sen (2017) in the proof …
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 compone…
The Euclidean first-passage percolation model of Howard and Newman is a rotationally invariant percolation model built on a Poisson point process. It is known that the passage time between 0 and $ne_1$ obeys a diffusive upper bound: $\mbox{Var}\, T(0,ne_1) \leq Cn$, and in this paper we improve this inequality to $Cn/\log n$. The methods follow the strategy used for sublinear variance proofs on the lattice, using the Falik-Samorodnitsky inequality and a Bernoulli encoding, but with substantial technical difficulties. To deal with the different setup of the Euclidean model, we represent the pas…
The main goal of this paper is to determine the Poisson boundary of lamplighter random walks over a general class of discrete groups $Γ$ endowed with a rich boundary. The starting point is the Strip Criterion of identification of the Poisson boundary for random walks on discrete groups due to Kaimanovich. A geometrical method for constructing the strip as a subset of the lamplighter group starting with a smaller strip in the base group $Γ$ is developed. Then, this method is applied to several classes of base groups $Γ$: groups with infinitely many ends, hyperbolic groups in the sense of Gromov…
In the spherical Poisson Boolean model, one takes the union of random balls centred on the points of a Poisson process in Euclidean $d$-space with $d \geq 2$. We prove that whenever the radius distribution has a finite $d$-th moment, there exists a strictly positive value for the intensity such that the vacant region percolates.
We investigate asymptotics for the minimal spanning acycles of the (Alpha)-Delaunay complex on a stationary Poisson process on $\mathbb{R}^d, d \geq 2$. Minimal spanning acycles are topological (or higher-dimensional) generalization of minimal spanning trees. We establish a central limit theorem for total weight of the minimal spanning acycle on a Poisson-Delaunay complex. Our approach also allows us to establish central limit theorems for sum of birth times and lifetimes in the persistent diagram of the Delaunay complex. The key to our proof is in showing the so-called weak stabilization of m…
This paper considers the time evolution of a queue that is embedded in a Poisson point process of moving wireless interferers. The queue is driven by an external arrival process and is subject to a time-varying service process that is a function of the SINR that it sees. Static configurations of interferers result in an infinite queue workload with positive probability. In contrast, a generic stability condition is established for the queue in the case where interferers possess any non-zero mobility that results in displacements that are both independent across interferers and oblivious to int…
We consider two independent symmetric Markov random flights $\bold Z_1(t)$ and $\bold Z_2(t)$ performed by the particles that simultaneously start from the origin of the Euclidean plane $\Bbb R^2$ in random directions distributed uniformly on the unit circumference $S_1$ and move with constant finite velocities $c_1>0, \; c_2>0$, respectively. The new random directions are taking uniformly on $S_1$ at random time instants that form independent homogeneous Poisson flows of rates $λ_1>0, \; λ_2>0$. The probability distribution function of the Euclidean distance $$ρ(t)=\Vert \bold Z_1(t) - \bold …
Consider a homogeneous Poisson point process in a compact convex set in $d$-dimensional Euclidean space which has interior points and contains the origin. The radial spanning tree is constructed by connecting each point of the Poisson point process with its nearest neighbour that is closer to the origin. For increasing intensity of the underlying Poisson point process the paper provides expectation and variance asymptotics as well as central limit theorems with rates of convergence for a class of edge functionals including the total edge length.
Consider two independent Goldstein-Kac telegraph processes $X_1(t)$ and $X_2(t)$ on the real line $\Bbb R$. The processes $X_k(t), \; k=1,2,$ are performed by stochastic motions at finite constant velocities $c_1>0, \; c_2>0,$ that start at the initial time instant $t=0$ from the origin of the real line $\Bbb R$ and are controlled by two independent homogeneous Poisson processes of rates $λ_1>0, \; λ_2>0$, respectively. Closed-form expression for the probability distribution function of the Euclidean distance $$ρ(t) = |X_1(t) - X_2(t) |, \qquad t>0,$$ between these processes at arbitrary time …
Some Spatial Point Processes of Poisson Family
本文提出了一种允许在有限区域内观察多个点的广义泊松随机场，并分析了其薄化、分数变体及空间Skellam型点过程。
Efron type identities for stopping sets and Poisson hulls
研究一般空间上泊松过程的停止集，利用其空间马尔可夫性质推导分布恒等式，结果可推广到有限泊松过程凸包及更一般随机集的相关恒等式。
Ideal Poisson-Voronoi tessellations on hyperbolic spaces
本文研究双曲空间中低强度 Poisson–Voronoi 镶嵌的极限，证明其收敛为非平凡的理想镶嵌，并分析该等距不变分解中无界单元的几何性质。
Sharp asymptotics for the maximal distance from the boundary to the nucleus of a typical Poisson-Voronoi cell
本文研究d维欧氏空间中典型Poisson-Voronoi胞腔顶点到核的最大距离D，推导其尾分布精确渐近，证实相关极值指数猜想并确定尾概率估计中的显式常数。
For a hyperbolic Brownian motion on the Poincaré half-plane $\mathbb{H}^2$, starting from a point of hyperbolic coordinates $z=(η, α)$ inside a hyperbolic disc $U$ of radius $\barη$, we obtain the probability of hitting the boundary $\partial U$ at the point $(\bar η,\bar α)$. For $\barη \to \infty$ we derive the asymptotic Cauchy hitting distribution on $\partial \mathbb{H}^2$ and for small values of $η$ and $\bar η$ we obtain the classical Euclidean Poisson kernel. The exit probabilities $\mathbb{P}_z\{T_{η_1}<T_{η_2}\}$ from a hyperbolic annulus in $\mathbb{H}^2$ of radii $η_1$ and $η_2$ ar…
Using the geodesic distance on the $n$-dimensional sphere, we study the expected radius function of the Delaunay mosaic of a random set of points. Specifically, we consider the partition of the mosaic into intervals of the radius function and determine the expected number of intervals whose radii are less than or equal to a given threshold. Assuming the points are not contained in a hemisphere, the Delaunay mosaic is isomorphic to the boundary complex of the convex hull in $\mathbb{R}^{n+1}$, so we also get the expected number of faces of a random inscribed polytope. We find that the expectati…
Aldous has introduced a notion of scale-invariant random spatial network (SIRSN) as a mathematical formalization of road networks. Intuitively, those are random processes that assign a route between each pair of points in Euclidean space, while being invariant under rotation, translation, and change of scale, and such that the routes are not too long and mainly on "main roads". The only known example was somewhat artificial since invariance had to be added at the end of the construction. We prove that the network of geodesics in the random metric space generated by a Poisson line process marke…
Central limit theorems for the nearest neighbour embracing graph in Euclidean and hyperbolic space
本文针对欧氏与双曲空间中由平稳泊松过程构造的最近邻包围图，推导其相关几何泛函（总边长、长度幂次泛函、给定出度顶点数）的高斯涨落定量描述，建立了相应的中心极限定理。
The Poisson stick model in hyperbolic space
本文研究了双曲空间中泊松棒模型的临界行为，发现渗流相变临界点随长度L的平方反比变化，而唯一性相变临界点随L反比变化。
On a Class of Dynamical Poisson-Voronoi Tessellations
本文研究动态泊松-沃洛诺伊分割中用户切换事件序列的性质，分析其平稳性及强度，探讨多速度场景下的切换 Palm 分布。
Crossings and diffusion in Poisson driven marked random connection models
该研究分析了标记泊松点过程构建的随机连接模型的穿越统计，得到超临界强度下的指数尾界与板状渗流结果，进而证明相关扩散、输运过程的有效参数非退化，为Mott定律推导提供基础。
We consider the stable matching of two independent Poisson processes in $\mathbb{R}^d$ under an asymmetric color restriction. Blue points can only match to red points, while red points can match to points of either color. It is unknown whether there exists a choice of intensities of the red and blue processes under which all points are matched. We prove that for any fixed intensities, there are unmatched blue points in sufficiently high dimension. Indeed, if the ratio of red to blue intensities is $ρ$ then the intensity of unmatched blue points converges to $e^{-ρ}/(1+ρ)$ as $d\to\infty$. We a…
We construct solutions to the randomly-forced Navier--Stokes--Poisson system in periodic three-dimensional domains or in the whole three-dimensional Euclidean space. These solutions are weak in the sense of PDEs and also weak in the sense of probability. As such, they satisfy the system in the sense of distributions and the underlying probability space and the stochastic driving force are also unknowns of the problem. Additionally, these solutions dissipate energy, satisfies a relative energy inequality in the sense of [4] and satisfy a renormalized form of the continuity equation in the sense…
Poisson boundaries of building lattices and rigidity with hyperbolic-like targets
本文证明正则厚欧氏建筑及其格的泊松边界为无穷远房室空间并带调和测度类，进而推广了从建筑格到负曲率群的态射与上闭链的刚性结果。
Minkowski's First Theorem and Dirichlet's Approximation Theorem provide upper bounds on certain minima taken over lattice points contained in domains of Euclidean spaces. We study the distribution of such minima and show, under some technical conditions, that they exhibit Weibull asymptotics with respect to different natural measures on the space of unimodular lattices in $\bR^d$. This follows from very general Poisson approximation results for shrinking targets which should be of independent interest. Furthermore, we show in the appendix that the logarithm laws of Kleinbock-Margulis, Khinchin…
Hyperuniformity of random measures on Euclidean and hyperbolic spaces
该研究探究欧氏与实双曲空间上不变局部平方可积随机测度的数方差渐近下界，推广了欧氏情形的经典结果，证明双曲空间随机测度无几何超均匀性并定义了谱超均匀性与隐身性。
Boolean models in hyperbolic space
本文将欧氏空间布尔模型推广到双曲空间，研究其与凸观测窗口交集体积等几何泛函，给出期望、方差、协方差的精确与渐近公式，并建立单变量和多变量中心极限定理，揭示了双曲框架下的新现象。
Conditional Independence under Infinite Measures and Poisson Point Processes
本文研究了在刺穿乘积空间上的无限测度下的条件独立性概念，通过泊松点过程的坐标投影建立了与经典条件独立性的等价关系，并扩展了其应用范围。
The metric $D_α(q,q')$ on the set $Q$ of particle locations of a homogeneous Poisson process on $R^d$, defined as the infimum of $(\sum_i |q_i - q_{i+1}|^α)^{1/α}$ over sequences in $Q$ starting with $q$ and ending with $q'$ (where $| . |$ denotes Euclidean distance) has nontrivial geodesics when $α> 1$. The cases $1 <α< \infty$ are the Euclidean first-passage percolation (FPP) models introduced earlier by the authors while the geodesics in the case $α= \infty$ are exactly the paths from the Euclidean minimal spanning trees/forests of Aldous and Steele. We compare and contrast results and conj…
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