Random walks on finite nilpotent groups driven by long-jump measures

Abstract

We consider a variant of simple random walk on a finite group. At each step, we choose an element, s, from a set of generators (“directions”) uniformly, and an integer, j, from a power law distribution (“speed”) associated with the chosen direction, and move from the current position, g, to $g{s}^j$. We show that if the finite group is nilpotent, the time it takes this walk to reach its uniform equilibrium is of the same order of magnitude as the diameter of a suitable pseudo-metric on the group, which is attached to the generators and speeds. Additionally, we give sharp bounds on the ${\ell }^2$-distance between the distribution of the position of the walker and the stationary distribution, and compute the relevant diameter for some examples.