Related papers: Intersections of random sets

We study an inhomogeneous random connection model in the connectivity regime. The vertex set of the graph is a homogeneous Poisson point process $\mathcal{P}_s$ of intensity $s>0$ on the unit cube…

In this survey, we discuss volumetric and combinatorial results concerning (mostly finite) intersections or unions of balls (mostly of equal radii) in the $d$-dimensional real vector space, mostly equipped with the Euclidean norm. Our first…

We study the asymptotic behavior of the maximum interpoint distance of random points in a $d$-dimensional set with a unique diameter and a smooth boundary at the poles. Instead of investigating only a fixed number of $n$ points as $n$ tends…

We study the problem of coupling a stochastic block model with a planted bisection to a uniform random graph having the same average degree. Focusing on the regime where the average degree is a constant relative to the number of vertices…

We study intersection properties of two or more independent tree-like random graphs. Our setting encompasses critical, possibly long range, Bernoulli percolation clusters, incipient infinite clusters, as well as critical branching random…

Estimates are constructed for the deviation of the concentration functions of sums of independent random variables with finite variances from the folded normal distribution function without any assumptions concerning the existence of the…

We prove that, in the coupon collector's problem, the point processes given by the times of $r$-th arrivals for coupons of each type, centered and normalized in a proper way, converge toward a non-homogeneous Poisson point process. This…

Confounding matters in almost all observational studies that focus on causality. In order to eliminate bias caused by connfounders, oftentimes a substantial number of features need to be collected in the analysis. In this case, large p…

Given S_1, a finite set of points in the plane, we define a sequence of point sets S_i as follows: With S_i already determined, let L_i be the set of all the line segments connecting pairs of points of the union of S_1,...,S_i, and let…

Given $\alpha \in (0, \infty)$ and $r \in (0, \infty)$, let ${\cal D}_{r, \alpha}$ be the disc of radius $r$ in the hyperbolic plane having curvature $-\alpha^2$. Consider the Poisson point process having uniform intensity density on ${\cal…

Our interest is in the scaled joint distribution associated with $k$-increasing subsequences for random involutions with a prescribed number of fixed points. We proceed by specifying in terms of correlation functions the same distribution…

We prove that the distributional limit of the normalised number of returns to small neighbourhoods of periodic points of non-uniformly hyperbolic dynamical systems is compound Poisson. The returns to small balls around a fixed point in the…

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 present sufficient conditions for sums of dependent point processes to converge in distribution to a Poisson process. This extends the classical result of Grigelionis [Theory Probab. Appl. 8 (1963) 172--182] for sums of uniformly null…

The task of this survey is to present various results on intersection patterns of convex sets. One of main tools for studying intersection patterns is a point of view via simplicial complexes. We recall the definitions of so called…

Consider a stationary Poisson process $\eta$ in the $d$-dimensional Euclidean or hyperbolic space and construct a random graph with vertex set $\eta$ as follows. First, each point $x\in\eta$ is connected by an edge to its nearest neighbour,…

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…

When the problem of boson sampling was first proposed, it was assumed that little or no photon collisions occur. However, modern experimental realizations rely on setups where collisions are quite common, i.e. the number of photons $M$…

We investigate random graphs on the points of a Poisson process in $d$-dimensional space, which combine scale-free degree distributions and long-range effects. Every Poisson point carries an independent random mark and given marks and…

We consider random graphs with uniformly bounded edges on a Poisson point process conditioned to contain the origin. In particular we focus on the random connection model, the Boolean model and Miller-Abrahams random resistor network with…