Range-renewal structure of transient simple random walk

Abstract

Dvoretzky and Erdös (1950) proved limn→+∞Rnn=γ almost surely for the simple symmetric random walk on Zd (d≥3) with Rn being the number of distinct cites visited in the time interval [0,n) and γ=γd being the escape rate. Following Derriennic’s approach, we prove further in this note that limn→+∞Rn,kRn=γ(1−γ)k−1,limn→+∞Rn,kRn,k+=γ,k=1,2,⋯ almost surely with Rn,k (resp. Rn,k+) being the number of distinct cites visited exactly (resp. at least) k times in the time interval [0,n). Such results are generalized to transient simple random walks on discrete groups.