We prove a Poisson process approximation result for stabilising functionals of a determinantal point process. Our results use concrete couplings of determinantal processes with different Palm measures and exploit their association properties. Second, we focus on the Ginibre process and show in the asymptotic scenario of an increasing observation window that the process of points with a large nearest neighbour distance converges after a suitable scaling to a Poisson point process. As a corollary, we obtain the scaling of the maximum nearest neighbour distance in the Ginibre process, which turns out to be different from its analogue for independent points.

We consider stationary configurations of points in Euclidean space that are marked by positive random variables called scores. The scores are allowed to depend on the relative positions of other points and outside sources of randomness. Such models have been thoroughly studied in stochastic geometry, e.g. in the context of random tessellations or random geometric graphs. It turns out that in a neighborhood of a point with an extreme score it is possible to rescale positions and scores of nearby points to obtain a limiting point process, which we call the tail configuration. Under some assumptions on dependence between scores, this local limit determines the global asymptotics for extreme scores within increasing windows in $\mathbb{R}^d$. The main result establishes the convergence of rescaled positions and clusters of high scores to a Poisson cluster process, quantifying the idea of the Poisson clumping heuristic by Aldous (1989, in the point process setting). In contrast to the existing results, our framework allows for explicit calculation of essentially all extremal quantities related to the limiting behavior of extremes. We apply our results to models based on (marked) Poisson processes where the scores depend on the distance to the kth nearest neighbor and where scores are allowed to propagate through a random network of points depending on their locations.

We investigate geometric properties of invariant spatio-temporal random fields $X\colon\mathbb M^d\times \mathbb R\to \mathbb R$ defined on a compact two-point homogeneous space $\mathbb M^d$ in any dimension $d\ge 2$, and evolving over time. In particular, we focus on chi-squared-distributed random fields, and study the large-time behavior (as $T\to +\infty$) of the average on [0,T] of the volume of the excursion set on the manifold, i.e. of $\lbrace X(\cdot, t)\ge u\rbrace$ (for any $u >0$). The Fourier components of X may have short or long memory in time, i.e. integrable or non-integrable temporal covariance functions. Our argument follows the approach developed in Marinucci et al. (2021) and allows us to extend their results for invariant spatio-temporal Gaussian fields on the two-dimensional unit sphere to the case of chi-squared distributed fields on two-point homogeneous spaces in any dimension. We find that both the asymptotic variance and limiting distribution, as $T\to +\infty$, of the average empirical volume turn out to be non-universal, depending on the memory parameters of the field X.

In this paper we derive cumulant bounds for subgraph counts and power-weighted edge lengths in a class of spatial random networks known as weight-dependent random connection models. These bounds give rise to different probabilistic results, from which we mainly focus on moderate deviations of the respective statistics, but also show a concentration inequality and a normal approximation result. This involves dealing with long-range spatial correlations induced by the profile function and the weight distribution. We start by deriving the bounds for the classical case of a Poisson vertex set, and then provide extensions to α-determinantal processes.

Gaussian random polytopes have received a lot of attention, especially in the case where the dimension is fixed and the number of points goes to infinity. Our focus is on the less-studied case where the dimension goes to infinity and the number of points is proportional to the dimension d. We study several natural quantities associated with Gaussian random polytopes in this setting. First, we show that the expected number of facets is equal to $C(\alpha)^{d+o(d)}$, where $C(\alpha)$ is some constant which depends on the constant of proportionality $\alpha$. We also extend this result to the expected number of k-facets. We then consider the more difficult problem of the asymptotics of the expected number of pairs of estranged facets of a Gaussian random polytope. When the number of points is 2d, we determine the constant C such that the expected number of pairs of estranged facets is equal to $C^{d+o(d)}$.

We provide two constructions of Gaussian random holomorphic sections of a Hermitian holomorphic line bundle $(L,h_{L})$ on a Hermitian complex manifold $(X,\Theta )$, that are particularly interesting in the case where the space of $\mathcal {L}^2$-holomorphic sections $H^{0}_{(2)}(X,L)$ is infinite dimensional. We first provide a general construction of Gaussian random holomorphic sections of L, which, if $H^{0}_{(2)}(X,L)$ is infinite dimensional, are almost never $\mathcal {L}^2$-integrable on X. The second construction combines the abstract Wiener space theory with the Berezin–Toeplitz quantization and yields a Gaussian ensemble of random $\mathcal {L}^2$-holomorphic sections. Furthermore, we study their random zeros in the context of semiclassical limits, including their distributions, large deviation estimates, local fluctuations and hole probabilities.

We consider the count of subgraphs with an arbitrary configuration of endpoints in the random-connection model based on a Poisson point process on ${\mathord{\mathbb R}}^d$. We present combinatorial expressions for the computation of the cumulants and moments of all orders of such subgraph counts, which allow us to estimate the growth of cumulants as the intensity of the underlying Poisson point process goes to infinity. As a consequence, we obtain a central limit theorem with explicit convergence rates under the Kolmogorov distance and connectivity bounds. Numerical examples are presented using a computer code in SageMath for the closed-form computation of cumulants of any order, for any type of connected subgraph, and for any configuration of endpoints in any dimension $d{\geq} 1$. In particular, graph connectivity estimates, Gram–Charlier expansions for density estimation, and correlation estimates for joint subgraph counting are obtained.

Let $\mathbb{P}_\kappa(n)$ be the probability that n points $z_1,\ldots,z_n$ picked uniformly and independently in $\mathfrak{C}_\kappa$, a regular $\kappa$-gon with area 1, are in convex position, that is, form the vertex set of a convex polygon. In this paper, we compute $\mathbb{P}_\kappa(n)$ up to asymptotic equivalence, as $n\to+\infty$, for all $\kappa\geq 3$, which improves on a famous result of Bárány (Ann. Prob. 27, 1999). The second purpose of this paper is to establish a limit theorem which describes the fluctuations around the limit shape of an n-tuple of points in convex position when $n\to+\infty$. Finally, we give an asymptotically exact algorithm for the random generation of $z_1,\ldots,z_n$, conditioned to be in convex position in $\mathfrak{C}_\kappa$.

Persistent Betti numbers are a major tool in persistent homology, a subfield of topological data analysis. Many tools in persistent homology rely on the properties of persistent Betti numbers considered as a two-dimensional stochastic process $ (r,s) \mapsto n^{-1/2} (\beta^{r,s}_q ( \mathcal{K}(n^{1/d} \mathcal{X}_n))-\mathbb{E}[\beta^{r,s}_q ( \mathcal{K}( n^{1/d} \mathcal{X}_n))])$. So far, pointwise limit theorems have been established in various settings. In particular, the pointwise asymptotic normality of (persistent) Betti numbers has been established for stationary Poisson processes and binomial processes with constant intensity function in the so-called critical (or thermodynamic) regime; see Yogeshwaran et al. (Prob. Theory Relat. Fields 167, 2017) and Hiraoka et al. (Ann. Appl. Prob. 28, 2018).

In this contribution, we derive a strong stabilization property (in the spirit of Penrose and Yukich, Ann. Appl. Prob. 11, 2001) of persistent Betti numbers, and we generalize the existing results on their asymptotic normality to the multivariate case and to a broader class of underlying Poisson and binomial processes. Most importantly, we show that multivariate asymptotic normality holds for all pairs (r, s), $0\le r\le s<\infty$, and that it is not affected by percolation effects in the underlying random geometric graph.

We investigate the hyperuniformity of marked Gibbs point processes that have weak dependencies among distant points whilst the interactions of close points are kept arbitrary. Various stability and range assumptions are imposed 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 the Widom–Rowlinson model with random radii and Gibbs point processes with interactions based on Voronoi tessellations and nearest-neighbour graphs.

Given a sequence of independent random vectors taking values in ${\mathbb R}^d$ and having common continuous distribution function F, say that the $n^{\rm \scriptsize}$th observation sets a (Pareto) record if it is not dominated (in every coordinate) by any preceding observation. Let $p_n(F) \equiv p_{n, d}(F)$ denote the probability that the $n^{\rm \scriptsize}$th observation sets a record. There are many interesting questions to address concerning pn and multivariate records more generally, but this short paper focuses on how pn varies with F, particularly if, under F, the coordinates exhibit negative dependence or positive dependence (rather than independence, a more-studied case). We introduce new notions of negative and positive dependence ideally suited for such a study, called negative record-setting probability dependence (NRPD) and positive record-setting probability dependence (PRPD), relate these notions to existing notions of dependence, and for fixed $d \geq 2$ and $n \geq 1$ prove that the image of the mapping pn on the domain of NRPD (respectively, PRPD) distributions is $[p^*_n, 1]$ (resp., $[n^{-1}, p^*_n]$), where $p^*_n$ is the record-setting probability for any continuous F governing independent coordinates.

As a generalization of random recursive trees and preferential attachment trees, we consider random recursive metric spaces. These spaces are constructed from random blocks, each a metric space equipped with a probability measure, containing a labelled point called a hook, and assigned a weight. Random recursive metric spaces are equipped with a probability measure made up of a weighted sum of the probability measures assigned to its constituent blocks. At each step in the growth of a random recursive metric space, a point called a latch is chosen at random according to the equipped probability measure, and a new block is chosen at random and attached to the space by joining together the latch and the hook of the block. We use martingale theory to prove a law of large numbers and a central limit theorem for the insertion depth, the distance from the master hook to the latch chosen. We also apply our results to further generalizations of random trees, hooking networks, and continuous spaces constructed from line segments.

We establish power-series expansions for the asymptotic expectations of the vertex number and missed area of random disc-polygons in planar convex bodies with $C^{k+1}_+$-smooth boundaries. These results extend asymptotic formulas proved in Fodor et al. (2014).

A version of the classical Buffon problem in the plane naturally extends to the setting of any Riemannian surface with constant Gaussian curvature. The Buffon probability determines a Buffon deficit. The relationship between Gaussian curvature and the Buffon deficit is similar to the relationship that the Bertrand–Diguet–Puiseux theorem establishes between Gaussian curvature and both circumference and area deficits.

We consider Gaussian approximation in a variant of the classical Johnson–Mehl birth–growth model with random growth speed. Seeds appear randomly in $\mathbb{R}^d$ at random times and start growing instantaneously in all directions with a random speed. The locations, birth times, and growth speeds of the seeds are given by a Poisson process. Under suitable conditions on the random growth speed, the time distribution, and a weight function $h\;:\;\mathbb{R}^d \times [0,\infty) \to [0,\infty)$, we prove a Gaussian convergence of the sum of the weights at the exposed points, which are those seeds in the model that are not covered at the time of their birth. Such models have previously been considered, albeit with fixed growth speed. Moreover, using recent results on stabilization regions, we provide non-asymptotic bounds on the distance between the normalized sum of weights and a standard Gaussian random variable in the Wasserstein and Kolmogorov metrics.

We consider the minimum spanning tree problem on a weighted complete bipartite graph $K_{n_R, n_B}$ whose $n=n_R+n_B$ vertices are random, i.i.d. uniformly distributed points in the unit cube in $d$ dimensions and edge weights are the $p$-th power of their Euclidean distance, with $p\gt 0$. In the large $n$ limit with $n_R/n \to \alpha _R$ and $0\lt \alpha _R\lt 1$, we show that the maximum vertex degree of the tree grows logarithmically, in contrast with the classical, non-bipartite, case, where a uniform bound holds depending on $d$ only. Despite this difference, for $p\lt d$, we are able to prove that the total edge costs normalized by the rate $n^{1-p/d}$ converge to a limiting constant that can be represented as a series of integrals, thus extending a classical result of Avram and Bertsimas to the bipartite case and confirming a conjecture of Riva, Caracciolo and Malatesta.

We determine almost sure limits of rescaled intrinsic volumes of the construction steps of fractal percolation in ${\mathbb R}^d$ for any dimension $d\geq 1$. We observe a factorization of these limit variables which allows one, in particular, to determine their expectations and covariance structure. We also show the convergence of the rescaled expectations and variances of the intrinsic volumes of the construction steps to the expectations and variances of the limit variables, and we give rates for this convergence in some cases. These results significantly extend our previous work, which addressed only limits of expectations of intrinsic volumes.

Processes of random tessellations of the Euclidean space $\mathbb{R}^d$, $d\geq 1$, are considered that are generated by subsequent division of their cells. Such processes are characterized by the laws of the life times of the cells until their division and by the laws for the random hyperplanes that divide the cells at the end of their life times. The STIT (STable with respect to ITerations) tessellation processes are a reference model. In the present paper a generalization concerning the life time distributions is introduced, a sufficient condition for the existence of such cell division tessellation processes is provided, and a construction is described. In particular, for the case that the random dividing hyperplanes have a Mondrian distribution—which means that all cells of the tessellations are cuboids—it is shown that the intrinsic volumes, except the Euler characteristic, can be used as the parameter for the exponential life time distribution of the cells.

Stationary Poisson processes of lines in the plane are studied, whose directional distributions are concentrated on $k\geq 3$ equally spread directions. The random lines of such processes decompose the plane into a collection of random polygons, which form a so-called Poisson line tessellation. The focus of this paper is to determine the proportion of triangles in such tessellations, or equivalently, the probability that the typical cell is a triangle. As a by-product, a new deviation of Miles’s classical result for the isotropic case is obtained by an approximation argument.

The Minkowski functionals, including the Euler characteristic statistics, are standard tools for morphological analysis in cosmology. Motivated by cosmic research, we examine the Minkowski functional of the excursion set for an isotropic central limit random field, whose k-point correlation functions (kth-order cumulants) have the same structure as that assumed in cosmic research. Using 3- and 4-point correlation functions, we derive the asymptotic expansions of the Euler characteristic density, which is the building block of the Minkowski functional. The resulting formula reveals the types of non-Gaussianity that cannot be captured by the Minkowski functionals. As an example, we consider an isotropic chi-squared random field and confirm that the asymptotic expansion accurately approximates the true Euler characteristic density.