Asymptotic entropy-constrained performance of tessellating and universal randomized lattice quantization

T. Linder; K. Zeger

doi:10.1109/18.312189

Asymptotic entropy-constrained performance of tessellating and universal randomized lattice quantization

IEEE Transactions on Information Theory ◽

10.1109/18.312189 ◽

1994 ◽

Vol 40 (2) ◽

pp. 575-579 ◽

Cited By ~ 32

Author(s):

T. Linder ◽

K. Zeger

Keyword(s):

Asymptotic Entropy

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Asymptotic entropy of the ranges of random walks on discrete groups

Science China Mathematics ◽

10.1007/s11425-017-9369-6 ◽

2020 ◽

Vol 63 (6) ◽

pp. 1153-1168

Author(s):

Xinxing Chen ◽

Jiansheng Xie ◽

Minzhi Zhao

Keyword(s):

Random Walks ◽

Discrete Groups ◽

Asymptotic Entropy

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"MÜNCHHAUSEN TRICK" AND AMENABILITY OF SELF-SIMILAR GROUPS

International Journal of Algebra and Computation ◽

10.1142/s0218196705002694 ◽

2005 ◽

Vol 15 (05n06) ◽

pp. 907-937 ◽

Cited By ~ 7

Author(s):

VADIM A. KAIMANOVICH

Keyword(s):

Random Walk ◽

Probability Measure ◽

Group Identity ◽

Convex Combination ◽

Similar Group ◽

Grigorchuk Group ◽

Intermediate Growth ◽

Group A ◽

Self Similar ◽

Asymptotic Entropy

The structure of a self-similar group G naturally gives rise to a transformation which assigns to any probability measure μ on G and any vertex w in the action tree of the group a new probability measure μw. If the measure μ is self-similar in the sense that μw is a convex combination of μ and the δ-measure at the group identity, then the asymptotic entropy of the random walk (G, μ) vanishes; therefore, the random walk is Liouville and the group G is amenable. We construct self-similar measures on several classes of self-similar groups, including the Grigorchuk group of intermediate growth.

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Asymptotic entropy of transformed random walks

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2015.90 ◽

2016 ◽

Vol 37 (5) ◽

pp. 1480-1491 ◽

Cited By ~ 1

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].

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Assessment of the Effectiveness of Random and Real-Networks Based on the Asymptotic Entropy

Complex Networks XI - Springer Proceedings in Complexity ◽

10.1007/978-3-030-40943-2_4 ◽

2020 ◽

pp. 39-50

Author(s):

Raihana Mokhlissi ◽

Dounia Lotfi ◽

Joyati Debnath ◽

Mohamed El Marraki

Keyword(s):

Asymptotic Entropy

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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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