Random walk on random walks: higher dimensions

• Autor(a) principal: Oriane Blondel
• Data de Publicação: 2019
• Outros Autores: Marcelo Richard Hilário, Renato Soares dos Santos, Vladas Sidoravicius, Augusto Quadros Teixeira
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Institucional da UFMG
• Texto Completo: https://doi.org/10.1214/19-EJP337; http://hdl.handle.net/1843/56442; https://orcid.org/0000-0001-9864-5533; https://orcid.org/0000-0002-8681-5176
• topic: Random walk; Dynamical random environment; Strong law of large numbers; Functional central limit theorem; Large deviation bound; Renormalization regeneration times; Probabilidades; Matemática; Passeio aleatório (Matemática); Lei dos grandes números; Teorema central do limite

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].