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Author: Hirsch, Christian × Subject: continuum percolation ×

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Now showing 1 - 4 of 4
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    Sharp phase transition for Cox percolation
    (Seattle, Wash. : Univ. of Washington, Mathematics Dep., 2022) Hirsch, Christian; Jahnel, Benedikt; Muirhead, Stephen
    We 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.
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    Bounded-hop percolation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Hirsch, Christian
    Motivated by an application in wireless telecommunication networks, we consider a two-type continuum-percolation problem involving a homogeneous Poisson point process of users and a stationary and ergodic point process of base stations. Starting from a randomly chosen point of the Poisson point process,we investigate distribution of the minimum number of hops that are needed to reach some point of the second point process.In the supercritical regime of continuum percolation, we use the close relationship between Euclidean and chemical distance to identify the distributional limit of the rescaled minimum number of hops that are needed to connect a typical Poisson point to a point of the second point process as its intensity tends to infinity. In particular, we obtain an explicit expression for the asymptotic probability that a typical Poisson point connects to a point of the second point process in a given number of hops.
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    Connection intervals in multi-scale dynamic networks
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Hirsch, Christian; Jahnel, Benedikt; Cali, Elie
    We consider a hybrid spatial communication system in which mobile nodes can connect to static sinks in a bounded number of intermediate relaying hops. We describe the distribution of the connection intervals of a typical mobile node, i.e., the intervals of uninterrupted connection to the family of sinks. This is achieved in the limit of many hops, sparse sinks and growing time horizons. We identify three regimes illustrating that the limiting distribution depends sensitively on the scaling of the time horizon.
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    Sharp phase transition for Cox percolation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Hirsch, Christian; Jahnel, Benedikt; Muirhead, Stephen
    We 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.