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Item type:Publication, Access status: without files , First passage percolation on Erdős–Rényi graphs with general weightsWe consider first passage percolation on the Erdős–Rényi graph with n vertices in which each pair of distinct vertices is connected independently by an edge with probability λ/n for some λ>1. The edges of the graph are given nonnegative i.i.d. weights with a nondegenerate distribution such that the probability of zero is not too large. We consider the paths with small total weight between two distinct typical vertices and analyse the joint behaviour of the numbers of edges on such paths, the so-called hopcounts, and the total weights of these paths. For n→∞, we show that, after a suitable transformation, the pairs of hopcounts and total weights of these paths converge in distribution to a Cox process, that is, a Poisson process with a random intensity measure. The random intensity measure is controlled by two independent random variables, whose distribution is the solution of a distributional fixed point equation and is related to branching processes. For nonarithmetic and arithmetic edge-weight distributions we observe different behaviour. In particular, we derive the limiting distribution for the minimal total weight and the corresponding hopcount(s). Our results generalise earlier work of Bhamidi, van der Hofstad and Hooghiemstra, who assume that edge weights have an absolutely continuous distribution. The main tool we employ is the method of moments.Publicationtype: Journal Article16 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , The radial spanning tree in hyperbolic space: degree and edge-length*(Institute of Mathematical Statistics, 2025-08-07); ; ; Consider a stationary Poisson process η in a d-dimensional hyperbolic space of constant curvature −κ and let the points of η together with a fixed origin o be the vertices of a graph. Connect each point x ∈ η with its radial nearest neighbour, which is the hyperbolically nearest vertex to x that is closer to o than x. This construction gives rise to the hyperbolic radial spanning tree, whose geometric properties are in the focus of this paper. In particular, the degree of the origin is studied. For increasing balls around o as observation windows, expectation and variance asymptotics as well as a quantitative central limit theorem for a class of edge-length functionals are derived. The results are contrasted with those for the Euclidean radial spanning tree.Publicationtype: Journal Article17 24 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , Normal approximation of Kabanov–Skorohod Integrals on poisson spaces(2024-06); ; We consider the normal approximation of Kabanov–Skorohod integrals on a general Poisson space. Our bounds are for the Wasserstein and the Kolmogorov distance and involve only difference operators of the integrand of the Kabanov–Skorohod integral. The proofs rely on the Malliavin–Stein method and, in particular, on multiple applications of integration by parts formulae. As examples, we study some linear statistics of point processes that can be constructed by Poisson embeddings and functionals related to Pareto optimal points of a Poisson process.Publicationtype: Journal ArticleScopus© Citations 1 27 63 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , The random connection model and functions of edge-marked poisson processes: Second order properties and normal approximation(2021-02); ; 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.Publicationtype: Journal Article48