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Author: Jahnel, Benedikt × Subject: Boolean model ×

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Now showing 1 - 5 of 5
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    Phase transitions for chase-escape models on Poisson–Gilbert graphs
    ([Madralin] : EMIS ELibEMS, 2020) Hinsen, Alexander; Jahnel, Benedikt; Cali, Elie; Wary, Jean-Philippe
    We present results on phase transitions of local and global survival in a two-species model on Poisson–Gilbert graphs. Initially, there is an infection at the origin that propagates on the graph according to a continuous-time nearest-neighbor interacting particle system. The graph consists of susceptible nodes and nodes of a second type, which we call white knights. The infection can spread on susceptible nodes without restriction. If the infection reaches a white knight, this white knight starts to spread on the set of infected nodes according to the same mechanism, with a potentially different rate, giving rise to a competition of chase and escape. We show well-definedness of the model, isolate regimes of global survival and extinction of the infection and present estimates on local survival. The proofs rest on comparisons to the process on trees, percolation arguments and finite-degree approximations of the underlying random graphs.
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    Continuum percolation in a nonstabilizing environment
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Jahnel, Benedikt; Jhawar, Sanjoy Kumar; Vu, Anh Duc
    We prove nontrivial phase transitions for continuum percolation in a Boolean model based on a Cox point process with nonstabilizing directing measure. The directing measure, which can be seen as a stationary random environment for the classical Poisson--Boolean model, is given by a planar rectangular Poisson line process. This Manhattan grid type construction features long-range dependencies in the environment, leading to absence of a sharp phase transition for the associated Cox--Boolean model. Our proofs rest on discretization arguments and a comparison to percolation on randomly stretched lattices established in [MR2116736].
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    Phase transitions for the Boolean model of continuum percolation for Cox point processes
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Jahnel, Benedikt; Tóbiás, András; Cali, Eli
    We consider the Boolean model with random radii based on Cox point processes. Under a condition of stabilization for the random environment, we establish existence and non-existence of subcritical regimes for the size of the cluster at the origin in terms of volume, diameter and number of points. Further, we prove uniqueness of the infinite cluster for sufficiently connected environments.
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    Phase transitions for chase-escape models on Gilbert graphs
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Hinsen, Alexander; Jahnel, Benedikt; Cali, Eli; Wary, Jean-Philippe
    We present results on phase transitions of local and global survival in a two-species model on Gilbert graphs. At initial time there is an infection at the origin that propagates on the Gilbert graph according to a continuous-time nearest-neighbor interacting particle system. The Gilbert graph consists of susceptible nodes and nodes of a second type, which we call white knights. The infection can spread on susceptible nodes without restriction. If the infection reaches a white knight, this white knight starts to spread on the set of infected nodes according to the same mechanism, with a potentially different rate, giving rise to a competition of chase and escape. We show well-definedness of the model, isolate regimes of global survival and extinction of the infection and present estimates on local survival. The proofs rest on comparisons to the process on trees, percolation arguments and finite-degree approximations of the underlying random graphs.
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    SINR percolation for Cox point processes with random powers
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Jahnel, Benedikt; Tóbiás, András
    Signal-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.