Universiteit Leiden

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Some case studies of random walks in dynamic random environments

Promotor: Promotor: W.Th.F. den Hollander, Co-promotor: V. Sidoravicius.

Renato Soares dos Santos
19 december 2012
Thesis in Leiden Repository

This thesis is dedicated to the study of random walks in dynamic random environments. These are models for the motion of a tracer particle in a disordered medium, which is called a static random environment if it stays constant in time, or dynamic otherwise. The evolution of the random walk is defined by assigning to it random jump rates which depend locally on the random environment. Such models belong to the greater area of \emph{disordered systems}, and have been studied extensively since the early seventies in the physics and mathematics literature. The goal is to understand the scaling properties, as time goes to infinity, of the path of the random walk. Several results are available in the literature for dynamic random environments which are uniformly elliptic and have uniform and fast enough mixing in space-time. However, very little is known when either of these conditions fail. In this thesis, we study examples of such situations, namely, non-elliptic cases in Chapter 2, a dynamic random environment with fast but non-uniform mixing in Chapter 4, and a dynamic random environment with both slow and non-uniform mixing in Chapters 3 and 5.