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

TL;DR

This paper studies a variant of random walks on finite nilpotent groups driven by long-jump measures, establishing their mixing times and providing bounds on their convergence to stationarity.

Contribution

It introduces a new model of random walks with long jumps on finite nilpotent groups and derives sharp bounds on mixing times and $ ext{ell}^2$-distances.

Findings

Mixing time is comparable to the diameter of a specific pseudo-metric.

Sharp bounds are provided for the $ ext{ell}^2$-distance to stationarity.

D(S, a) is computed explicitly for certain examples.

Abstract

We consider a variant of random walks on finite groups. At each step, we choose an element from a set of generators ("directions") uniformly, and an integer from a power law ("speed") distribution associated with the chosen direction. We show that if the finite group is nilpotent, the mixing time of this walk is of the same order of magnitude as the diameter of a suitable pseudo-metric, D(S, a), on the group, which depends only on 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 D(S,a) for some examples.