Random walk on random walks: higher dimensions

Oriane Blondel (1, 2, 3) , Marcelo R Hilário , Renato S dos Santos , Vladas Sidoravicius (4) , Augusto Teixeira

1 ICJ - Institut Camille Jordan
2 CNRS - Centre National de la Recherche Scientifique
3 PSPM - Probabilités, statistique, physique mathématique
4 IMPA Jean-Christophe Yoccoz

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].

Mots clés: renormalization; law of large num- bers; large deviations; central limit theorem; dynamic random environment; 82C44; 82C22; Secondary 82B41; regeneration AMS MSC 2010: Primary 60F15; 60K35; Random walk

Domaines: Probabilités [math.PR]

Oriane Blondel, Marcelo R Hilário, Renato S dos Santos, Vladas Sidoravicius, Augusto Teixeira. Random walk on random walks: higher dimensions. Electronic Journal of Probability, 2019, 24, ⟨10.1214/19-EJP337⟩. ⟨hal-01616312⟩