The resolvent for simple random walks on the free product of two discrete groups

1986 ◽  
Vol 192 (1) ◽  
pp. 109-116 ◽  
Author(s):  
Paolo M. Soardi
Keyword(s):  
Random Walks ◽  
Free Product ◽  
Discrete Groups

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 C*-algebras associated with amalgamated products of groups

1987 ◽  
Vol 29 (2) ◽  
pp. 143-148
Author(s):  
Bola O. Balogun
Keyword(s):  
Free Product ◽  
Discrete Groups ◽  
Verbal Subgroup ◽  
C Algebras ◽  
Products Of Groups ◽  
Amalgamated Products

Let V denote the class of discrete groups G which satisfy the following conditions (a), (b) and (c):(a) G = (A * B; K = φ(H)) is the free product of two groups A and B with the subgroup H amalgamated.(b) H does not contain the verbal subgroup A(X2) of A and K does not contain the verbal subgroup B(X2)of B.

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].

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

scholarly journals Rate of escape of random walks on free products

2007 ◽  
Vol 83 (1) ◽  
pp. 31-54 ◽  
Author(s):  
Lorenz A. Gilch
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Free Product ◽  
Finite Family ◽  
Block Length ◽  
Probability Measures ◽  
Free Products ◽  
Rate Of Escape ◽  
Transient Random Walk

AbstractSuppose we are given the free product V of a finite family of finite or countable sets (Vi)i∈∮ and probability measures on each Vi, which govern random walks on it. We consider a transient random walk on the free product arising naturally from the random walks on the Vi. We prove the existence of the rate of escape with respect to the block length, that is, the speed at which the random walk escapes to infinity, and furthermore we compute formulae for it. For this purpose, we present three different techniques providing three different, equivalent formulae.

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