Inspired by Benjamini et al.(2010) and Windisch(2010),we consider the entropy of the random walk ranges Rn formed by the first n steps of a random walk S on a discrete group.In this setting,we show the existence of hR...Inspired by Benjamini et al.(2010) and Windisch(2010),we consider the entropy of the random walk ranges Rn formed by the first n steps of a random walk S on a discrete group.In this setting,we show the existence of hR:=limn→∞H(Rn)/n called the asymptotic entropy of the ranges.A sample version of the above statement in the sense of Shannon(1948) is also proved.This answers a question raised by Windisch(2010).We also present a systematic characterization of the vanishing asymptotic entropy of the ranges.Particularly,we show that hR=0 if and only if the random walk either is recurrent or escapes to negative infinity without left jump.By introducing the weighted digraphs Γn formed by the underlying random walk,we can characterize the recurrence property of S as the vanishing property of the quantity limn→∞H(Γn)/n,which is an analogue of hR.展开更多
基金National Natural Science Foundation of China (Grant Nos. 11790273, 11771286, 11531001, 11371317 and 11271077)the Laboratory of Mathematics for Nonlinear Science, Fudan Universitysupported by the Zhejiang Provincial Natural Science Foundation of China (Grant No. LQ18A010007)。
文摘Inspired by Benjamini et al.(2010) and Windisch(2010),we consider the entropy of the random walk ranges Rn formed by the first n steps of a random walk S on a discrete group.In this setting,we show the existence of hR:=limn→∞H(Rn)/n called the asymptotic entropy of the ranges.A sample version of the above statement in the sense of Shannon(1948) is also proved.This answers a question raised by Windisch(2010).We also present a systematic characterization of the vanishing asymptotic entropy of the ranges.Particularly,we show that hR=0 if and only if the random walk either is recurrent or escapes to negative infinity without left jump.By introducing the weighted digraphs Γn formed by the underlying random walk,we can characterize the recurrence property of S as the vanishing property of the quantity limn→∞H(Γn)/n,which is an analogue of hR.
