scholarly journals Sharp lower bounds for the asymptotic entropy of symmetric random walks

2015 ◽  
Vol 9 (3) ◽  
pp. 711-735 ◽  
Author(s):  
Sébastien Gouëzel ◽  
Frédéric Mathéus ◽  
François Maucourant
Keyword(s):  
Random Walks ◽  
Lower Bounds ◽  
Asymptotic Entropy

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 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].

Hausdorff dimension of the harmonic measure on trees

1998 ◽  
Vol 18 (3) ◽  
pp. 631-660 ◽  
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.

scholarly journals Asymptotic entropy of random walks on Fuchsian buildings and Kac–Moody groups

2016 ◽  
Vol 285 (3-4) ◽  
pp. 707-738
Author(s):  
Lorenz Gilch ◽  
Sebastian Müller ◽  
James Parkinson
Keyword(s):  
Random Walks ◽  
Asymptotic Entropy

scholarly journals Asymptotic entropy and Green speed for random walks on countable groups

2008 ◽  
Vol 36 (3) ◽  
pp. 1134-1152 ◽  
Author(s):  
Sébastien Blachère ◽  
Peter Haïssinsky ◽  
Pierre Mathieu
Keyword(s):  
Random Walks ◽  
Asymptotic Entropy

Spherical Mean and the Fundamental Group

1991 ◽  
Vol 34 (1) ◽  
pp. 3-11 ◽  
Author(s):  
Toshiaki Adachi
Keyword(s):  
Riemannian Manifold ◽  
Fundamental Group ◽  
Random Walks ◽  
Isoperimetric Inequality ◽  
Lower Bounds ◽  
Compact Riemannian Manifold ◽  
Universal Covering ◽  
Spherical Means ◽  
Universal Covering Space ◽  
Spherical Mean

AbstractWe investigate some properties of spherical means on the universal covering space of a compact Riemannian manifold. If the fundamental group is amenable then the greatest lower bounds of the spectrum of spherical Laplacians are equal to zero. If the fundamental group is nontransient so are geodesic random walks. We also give an isoperimetric inequality for spherical means.

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