The long time asymptotics for random walks on infinite graphs is a principal topic in both geometry and probability theory. A covering graph of a finite graph with a nilpotent covering transformation group is called a nilpotent covering graph, regarded as a generalization of a crystal lattice or the Cayley graph of a finite generated group of polynomial growth.

In this talk, we discuss non-symmetric random walks on nilpotent covering graphs from a view point of the theory of discrete geometric analysis developed by Kotani and Sunada, and give central limit theorems for them. We also mention a relationship between the limiting diffusion and (the Lyons lift of) distorted Brownian rough path.

This talk is based on joint work with Satoshi Ishiwata (Yamagata University) and Ryuya Namba (Kyoto Sangyo University).