In this talk, we consider a variant of random walks on finite groups. For this variant, at each step, we choose from a set of generators ("directions") uniformly, and an integer from a power law distribution ("speed") associated with the direction. We show that the mixing time of this walk of the same order of magnitude as the diameter of a pseudo-metric on the group that depends only on the speeds and generators. If time allows, I will talk about some of the challenges of computing this diameter. (Joint work with L. Saloff-Coste.)