Random walks in Dirichlet environments on Z with bounded jumps

We examine a class of random walks in random environments on Z with bounded jumps, a generalization of the classic one-dimensional model. The environments we study have i.i.d. transition probability vectors drawn from Dirichlet distributions. For the transient case of this model, we characterize ballisticity-nonzero limiting velocity. We do this in terms of two parameters, kappa(0) and kappa(1). The parameter kappa(0) governs finite trapping effects. The parameter kappa(1), which already is known to characterize directional transience, also governs repeated traversals of arbitrarily large regions of the graph. We show that the walk is ballistic if and only if min(kappa(0), |kappa(1)|) > 1. We prove some stronger results regarding moments of the quenched Green function and other functions that the quenched Green function dominates. These results help us to better understand the phenomena and parameters affecting ballisticity.