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