Asymptotic entropy of the ranges of random walks on discrete groups  

作　　者：Xinxing Chen Jiansheng Xie Minzhi Zhao 

机构地区：[1]School of Mathematical Sciences,Shanghai Jiaotong University,Shanghai 200240,China [2]Shanghai Center of Mathematics,Fudan University,Shanghai 200433,China [3]School of Mathematical Sciences,Fudan University,Shanghai 200433,China [4]School of Mathematical Sciences,Zhejiang University,Hangzhou 310027,China

出　　处：《Science China Mathematics》2020年第6期1153-1168,共16页中国科学：数学（英文版）

基　　金：National Natural Science Foundation of China (Grant Nos. 11790273, 11771286, 11531001, 11371317 and 11271077);the Laboratory of Mathematics for Nonlinear Science, Fudan University;supported by the Zhejiang Provincial Natural Science Foundation of China (Grant No. LQ18A010007)。

摘　　要：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.

关 键 词：random walk ENTROPY RANGE RECURRENT 

分 类 号：O211[理学—概率论与数理统计]