The rate of escape for random walks on polycyclic and metabelian 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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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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Rate of escape of random walks on wreath products and related groups

The Annals of Probability ◽

10.1214/aop/1068646371 ◽

2003 ◽

Vol 31 (4) ◽

pp. 1917-1934 ◽

Cited By ~ 13

Author(s):

David Revelle

Keyword(s):

Random Walks ◽

Wreath Products ◽

Rate Of Escape

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Rate of Escape of Random Walks on Regular Languages and Free Products by Amalgamation of Finite Groups

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3580 ◽

2008 ◽

Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽

Author(s):

Lorenz A. Gilch

Keyword(s):

Random Walks ◽

Finite Groups ◽

Transition Probabilities ◽

Finite Alphabet ◽

Free Products ◽

Rate Of Escape ◽

International Audience ◽

Special Case ◽

Current Word ◽

Natural Way

International audience We consider random walks on the set of all words over a finite alphabet such that in each step only the last two letters of the current word may be modified and only one letter may be adjoined or deleted. We assume that the transition probabilities depend only on the last two letters of the current word. Furthermore, we consider also the special case of random walks on free products by amalgamation of finite groups which arise in a natural way from random walks on the single factors. The aim of this paper is to compute several equivalent formulas for the rate of escape with respect to natural length functions for these random walks using different techniques.

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Rate of escape of random walks on free products

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700036375 ◽

2007 ◽

Vol 83 (1) ◽

pp. 31-54 ◽

Cited By ~ 6

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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