Some old and basic facts about random walks on groups

TL;DR

This paper reviews fundamental facts about random walks on groups, including periodicity classes and ratio limits, serving as a supplement to the author's previous monograph.

Contribution

It provides a concise overview of classical results on random walks on groups, emphasizing periodicity and ratio limits, with historical context.

Findings

Exhibits a basic exercise on periodicity classes

Discusses foundational aspects of ratio limits in random walks

Serves as a supplement to existing monograph

Abstract

This note contains old instead of new results about random walks on groups, which may serve as a small supplement to the author's monograph ``Random Walks on Infinite Graphs and Groups'' (Cambridge Univ. Press 2000/2009). First, we exhibit a basic exercise on the periodicity classes of random walk. The second topic concerns some basics on ratio limits for random walks, which had been published ``only'' in German in the 1970ies.