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  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 .