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- ItemExponential Moments for Planar Tessellations(New York, NY [u.a.] : Springer Science + Business Media B.V., 2020) Jahnel, Benedikt; Tóbiás, AndrásIn this paper we show existence of all exponential moments for the total edge length in a unit disk for a family of planar tessellations based on stationary point processes. Apart from classical tessellations such as the Poisson–Voronoi, Poisson–Delaunay and Poisson line tessellation, we also treat the Johnson–Mehl tessellation, Manhattan grids, nested versions and Palm versions. As part of our proofs, for some planar tessellations, we also derive existence of exponential moments for the number of cells and the number of edges intersecting the unit disk.
- ItemPoisson approximation and connectivity in a scale-free random connection model([Madralin] : EMIS ELibEMS, 2021) Iyer, Srikanth K.; Jhawar, Sanjoy KrFor abstract see PDF
- ItemSharp phase transition for Cox percolation(Seattle, Wash. : Univ. of Washington, Mathematics Dep., 2022) Hirsch, Christian; Jahnel, Benedikt; Muirhead, StephenWe prove the sharpness of the percolation phase transition for a class of Cox percolation models, i.e., models of continuum percolation in a random environment. The key requirements are that the environment has a finite range of dependence, satisfies a local boundedness condition and can be constructed from a discrete iid random field, however the FKG inequality need not hold. The proof combines the OSSS inequality with a coarse-graining construction that allows us to compare different notions of influence.
- ItemCentral limit theorems for the radial spanning tree(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2014) Schulte, Matthias; Thäle, ChristophConsider 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 in- tensity 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.
- ItemStochastic homogenization on irregularly perforated domains(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Heida, Martin; Jahnel, Benedikt; Vu, Anh DucWe study stochastic homogenization of a quasilinear parabolic PDE with nonlinear microscopic Robin conditions on a perforated domain. The focus of our work lies on the underlying geometry that does not allow standard homogenization techniques to be applied directly. Instead we prove homogenization on a regularized geometry and demonstrate afterwards that the form of the homogenized equation is independent from the regularization. Then we pass to the regularization limit to obtain the anticipated limit equation. Furthermore, we show that Boolean models of Poisson point processes are covered by our approach.
- ItemSharp phase transition for Cox percolation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Hirsch, Christian; Jahnel, Benedikt; Muirhead, StephenWe prove the sharpness of the percolation phase transition for a class of Cox percolation models, i.e., models of continuum percolation in a random environment. The key requirements are that the environment has a finite range of dependence and satisfies a local boundedness condition, however the FKG inequality need not hold. The proof combines the OSSS inequality with a coarse-graining construction.
- ItemWhen do wireless network signals appear Poisson?(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Keeler, Paul; Ross, Nathan; Xia, AihuaWe consider the point process of signal strengths from transmitters in a wireless network observed from a fixed position under models with general signal path loss and random propagation effects. We show via coupling arguments that under general conditions this point process of signal strengths can be well-approximated by an inhomogeneous Poisson or a Cox point processes on the positive real line. We also provide some bounds on the total variation distance between the laws of these point processes and both Poisson and Cox point processes. Under appropriate conditions, these results support the use of a spatial Poisson point process for the underlying positioning of transmitters in models of wireless networks, even if in reality the positioning does not appear Poisson. We apply the results to a number of models with popular choices for positioning of transmitters, path loss functions, and distributions of propagation effec
- ItemSINR percolation for Cox point processes with random powers(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Jahnel, Benedikt; Tóbiás, AndrásSignal-to-interference plus noise ratio (SINR) percolation is an infinite-range dependent variant of continuum percolation modeling connections in a telecommunication network. Unlike in earlier works, in the present paper the transmitted signal powers of the devices of the network are assumed random, i.i.d. and possibly unbounded. Additionally, we assume that the devices form a stationary Cox point process, i.e., a Poisson point process with stationary random intensity measure, in two or higher dimensions. We present the following main results. First, under suitable moment conditions on the signal powers and the intensity measure, there is percolation in the SINR graph given that the device density is high and interferences are sufficiently reduced, but not vanishing. Second, if the interference cancellation factor γ and the SINR threshold τ satisfy γ ≥ 1/(2τ), then there is no percolation for any intensity parameter. Third, in the case of a Poisson point process with constant powers, for any intensity parameter that is supercritical for the underlying Gilbert graph, the SINR graph also percolates with some small but positive interference cancellation factor.