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End date: 2019 × Start date: 2010 × Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik × Subject: central limit theorem × Subject: large deviations ×

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    Random walk on random walks: Higher dimensions
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Blondel, Oriane; Hilário, Marcelo R.; Santos, Renato Soares dos; Sidoravicius, Vladas; Teixeira, Augusto
    We study the evolution of a random walker on a conservative dynamic random environment composed of independent particles performing simple symmetric random walks, generalizing results of [16] to higher dimensions and more general transition kernels without the assumption of uniform ellipticity or nearest-neighbour jumps. Specifically, we obtain a strong law of large numbers, a functional central limit theorem and large deviation estimates for the position of the random walker under the annealed law in a high density regime. The main obstacle is the intrinsic lack of monotonicity in higher-dimensional, non-nearest neighbour settings. Here we develop more general renormalization and renewal schemes that allow us to overcome this issue. As a second application of our methods, we provide an alternative proof of the ballistic behaviour of the front of (the discrete-time version of) the infection model introduced in [23].
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    Random walk on random walks: Low densities
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Blondel, Oriane; Hilário, Marcelo R.; Santos, Renato dos; Sidoravicius, Vladas; Teixeira, Augusto
    We consider a random walker in a dynamic random environment given by a system of independent simple symmetric random walks. We obtain ballisticity results under two types of perturbations: low particle density, and strong local drift on particles. Surprisingly, the random walker may behave very differently depending on whether the underlying environment particles perform lazy or non-lazy random walks, which is related to a notion of permeability of the system. We also provide a strong law of large numbers, a functional central limit theorem and large deviation bounds under an ellipticity condition.