Random Walks in Dirichlet Environments with Bounded Jumps

This thesis studies non-nearest-neighbor random walks in random environments (RWRE) on the integers and on the d-dimensional integer lattic that are drawn in an i.i.d. way according to a Dirichlet distribution. We complete a characterization of recurrence and transience in a given direction for random walks in Dirichlet environments (RWDE) by proving directional recurrence in the case where the Dirichlet parameters are balanced and the annealed drift is zero. As a step toward this, we prove a 0-1 law for directional transience of i.i.d. RWRE on the 2-dimensional integer lattice with bounded jumps. Such a 0-1 law was proven by Zerner and Merkl for nearest-neighbor RWRE in 2001, and Zerner gave a simpler proof in 2007. We modify the latter argument to allow for bounded jumps. We then characterize ballisticity, or nonzero liiting velocity, of transienct RWDE on the integers. It turns out that ballisticity is controlled by two parameters, kappa0 and kappa1. The parameter kappa0, which controls finite traps, is known to characterize ballisticity for nearest-neighbor RWDE on the d-dimensional integer lattice for dimension d at least 3, where transient walks are ballistic if and only if kappa0 is greater than 1. The parameter kappa1, which controls large-scale backtracking, is known to characterize ballisticity for nearest-neighbor RWDE on the one-dimensional integer lattice, where transient walks are ballistic if and only if the absolute value of kappa1 is greater than 1. We show that in our model, transient walks are ballistic if and only if both parameters are greater than 1. Our characterization is thus a mixture of known characterizations of ballisticity for nearest-neighbor one-dimensional and higher-dimensional cases. We also prove more detailed theorems that help us better understand the phenomena affecting ballisticity.