WIAS Preprint No. 2435, (2017)

Random walk on random walks: Higher dimensions


  • Blondel, Oriane
  • Hilário, Marcelo R.
  • Soares dos Santos, Renato
  • Sidoravicius, Vladas
  • Teixeira, Augusto

2010 Mathematics Subject Classification

  • 60F15 60K35 82B41 82C22 82C44


  • Random walk, dynamic random environment, law of large numbers, central limit theorem, large deviations, renormalization, regeneration




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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