Capacity of the range of random walks on groups

Abstract

In this paper, we discuss asymptotic behavior of the capacity of the range of symmetric simple random walks on finitely generated groups. We show the corresponding strong law of large numbers and central limit theorem. Here, τ +A denotes the first return time of {S n } n≥0 to the set A, i.e. τ + A = inf{n ≥ 1 : S n ∈ A}. We are interested in the asymptotic behavior of the process {C n } n≥0 defined asObserve that when {S n } n≥0 is recurrent then C n ≡ 0.