Random walks on direct sums of discrete groups

1988 ◽  
Vol 1 (4) ◽  
pp. 341-356 ◽  
Author(s):  
Donald I. Cartwright
Keyword(s):  
Random Walks ◽  
Discrete Groups ◽  
Direct Sums

Asymptotic entropy of the ranges of random walks on discrete groups

2020 ◽  
Vol 63 (6) ◽  
pp. 1153-1168
Author(s):  
Xinxing Chen ◽  
Jiansheng Xie ◽  
Minzhi Zhao
Keyword(s):  
Random Walks ◽  
Discrete Groups ◽  
Asymptotic Entropy

scholarly journals Positive definite functions and cut-off for discrete groups

2020 ◽  
pp. 1-17
Author(s):  
Amaury Freslon
Keyword(s):  
Random Walks ◽  
Quantum Groups ◽  
Discrete Group ◽  
Coxeter Groups ◽  
Free Groups ◽  
Positive Definite Function ◽  
Positive Definite ◽  
Definite Function ◽  
Discrete Groups ◽  
Compact Quantum Groups

Abstract We consider the sequence of powers of a positive definite function on a discrete group. Taking inspiration from random walks on compact quantum groups, we give several examples of situations where a cut-off phenomenon occurs for this sequence, including free groups and infinite Coxeter groups. We also give examples of absence of cut-off using free groups again.

scholarly journals Asymptotic entropy of transformed random walks

2016 ◽  
Vol 37 (5) ◽  
pp. 1480-1491 ◽  
Author(s):  
BEHRANG FORGHANI
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Ergodic Theory ◽  
Stopping Time ◽  
Stopping Times ◽  
Discrete Groups ◽  
Differential Entropy ◽  
Random Walks On Groups ◽  
Rate Of Escape ◽  
Asymptotic Entropy

We consider general transformations of random walks on groups determined by Markov stopping times and prove that the asymptotic entropy (respectively, rate of escape) of the transformed random walks is equal to the asymptotic entropy (respectively, rate of escape) of the original random walk multiplied by the expectation of the corresponding stopping time. This is an analogue of the well-known Abramov formula from ergodic theory; its particular cases were established earlier by Kaimanovich [Differential entropy of the boundary of a random walk on a group. Uspekhi Mat. Nauk38(5(233)) (1983), 187–188] and Hartman et al [An Abramov formula for stationary spaces of discrete groups. Ergod. Th. & Dynam. Sys.34(3) (2014), 837–853].

Tail-fields of products of random variables and ergodic equivalence relations

1999 ◽  
Vol 19 (5) ◽  
pp. 1325-1341 ◽  
Author(s):  
KLAUS SCHMIDT
Keyword(s):  
Random Walks ◽  
Discrete Group ◽  
Random Variables ◽  
Stationary Sequence ◽  
Conjugacy Classes ◽  
Equivalence Relations ◽  
Discrete Groups ◽  
Ergodic Equivalence Relations ◽  
Finite Conjugacy ◽  
Tail Triviality

We prove the following result. Let $G$ be a countable discrete group with finite conjugacy classes, and let $(X_n, n\in\mathbb Z)$ be a two-sided, strictly stationary sequence of $G$-valued random variables. Then $\mathscr T_\infty =\mathscr T_\infty ^*$, where $\mathscr T_\infty$ is the two-sided tail-sigma-field $\bigcap_{M\ge1}\sigma (X_m:|m|\ge M)$ of $(X_n)$ and $T_\infty ^*$ the tail-sigma-field $\bigcap_{M\ge0}\sigma (Y_{m,n}:m,n\ge M)$ of the random variables $(Y_{m,n}, m,n\ge0)$ defined as the products $Y_{m,n}=X_n\dots X_{-m}$. This statement generalises a number of results in the literature concerning tail triviality of two-sided random walks on certain discrete groups.

Sign in / Sign up

Export Citation Format

Share Document