Average Entropy of the Ranges for Simple Random Walks on Discrete Groups

Xin-Xing Chen, Jian-Sheng Xie, Min-Zhi Zhao

Published 2016-12-03Version 1

Inspired by Benjamini et al (Ann. Inst. H. Poincar\'e Probab. Stat. 2010) and Windisch (Electron. J. Probab. 2010), we consider the entropy of the random walk range formed by a simple random walk on a discrete group. It is shown in this setting the existence of a quantity which we call the average entropy of the range. The extreme values of this quantity are also discussed. An important consequence is that, the tail $\sigma$-algebra formed by the $n$-step ranges is always trivial for a recurrent random walk.