In this paper we consider drift and entropy growth for symmetric finitary random walks on finitely generated groups. We construct examples of various intermediate asymptotics of the drift for such random walks. We establish general inequalities which connect drift, entropy and exponential growth rate of groups. Then we apply these inequalities to get estimates for entropy in particular examples.
"On drift and entropy growth for random walks on groups." Ann. Probab. 31 (3) 1193 - 1204, July 2003. https://doi.org/10.1214/aop/1055425775