Some old and basic facts about random walks on groups
Wolfgang Woess
TL;DR
The paper revisits foundational results on random walks on groups, focusing on (i) the aperiodicity structure via the period $d(\mu)$ and the normal subgroup $\Gamma_0$ of index $d$, which allows reducing to the aperiodic case; and (ii) ratio-limit behavior and spectral radius, establishing that for irreducible walks the asymptotic growth rate is governed by $\rho(\mu)$ and that, in the aperiodic setting, the ratios of consecutive $n$-step probabilities converge to $\rho(\mu)$. A key tool is the $h$-transform (the $h$-process) built from a positive $\rho$-harmonic function, enabling a decomposition that yields precise ratio limits. When $S_{\mu}$ is finite, normalized convolution powers converge to a $\rho$-harmonic measure $\nu$ solving $\mu*\nu=\rho\nu$, with uniqueness in Abelian and certain isotropic settings leading to local-limit-type results. The results connect classical ratio-limit theorems to the landscape of random walks on groups and provide a compact, accessible account of older German-language results reinterpreted for modern references.
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.