Asymptotic entropy of the ranges of random walks on discrete groups

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].

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

Mathematische Zeitschrift ◽

10.1007/bf01162024 ◽

1986 ◽

Vol 192 (1) ◽

pp. 109-116 ◽

Cited By ~ 3

Author(s):

Paolo M. Soardi

Keyword(s):

Random Walks ◽

Free Product ◽

Discrete Groups

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Positive definite functions and cut-off for discrete groups

Canadian Mathematical Bulletin ◽

10.4153/s0008439520000466 ◽

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.

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Hausdorff dimension of the harmonic measure on trees

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385798108180 ◽

1998 ◽

Vol 18 (3) ◽

pp. 631-660 ◽

Cited By ~ 15

Author(s):

VADIM A. KAIMANOVICH

Keyword(s):

Hausdorff Dimension ◽

Random Walks ◽

Large Class ◽

Harmonic Measure ◽

Random Environment ◽

Free Groups ◽

Equivalence Relations ◽

Markov Operators ◽

Rate Of Escape ◽

Asymptotic Entropy

For a large class of Markov operators on trees we prove the formula ${\bf HD}\,\nu=h/l$ connecting the Hausdorff dimension of the harmonic measure $\nu$ on the tree boundary, the rate of escape $l$ and the asymptotic entropy $h$. Applications of this formula include random walks on free groups, conditional random walks, random walks in random environment and random walks on treed equivalence relations.

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