scholarly journals Random walks on discrete groups of polynomial volume growth

2002 ◽  
Vol 30 (2) ◽  
pp. 723-801 ◽  
Author(s):  
Georgios K. Alexopoulos
Keyword(s):  
Random Walks ◽  
Volume Growth ◽  
Discrete Groups

scholarly journals Laws of the iterated logarithm on covering graphs with groups of polynomial volume growth

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Ryuya Namba
Keyword(s):  
Random Walks ◽  
Quadratic Forms ◽  
Volume Growth ◽  
Moderate Deviation ◽  
Point Of View ◽  
Iterated Logarithm ◽  
Large Numbers ◽  
Geometric Point ◽  
Rate Functions ◽  
The Given

AbstractModerate deviation principles (MDPs) for random walks on covering graphs with groups of polynomial volume growth are discussed in a geometric point of view. They deal with any intermediate spatial scalings between those of laws of large numbers and those of central limit theorems. The corresponding rate functions are given by quadratic forms determined by the Albanese metric associated with the given random walks. We apply MDPs to establish laws of the iterated logarithm on the covering graphs by characterizing the set of all limit points of the normalized random walks.

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

Random Walks on Graphs with Regular Volume Growth

1998 ◽  
Vol 8 (4) ◽  
pp. 656-701 ◽  
Author(s):  
T. Coulhon ◽  
A. Grigoryan
Keyword(s):  
Random Walks ◽  
Volume Growth ◽  
Random Walks On Graphs

SPECTRAL MULTIPLIERS ON DISCRETE GROUPS

2001 ◽  
Vol 33 (4) ◽  
pp. 417-424 ◽  
Author(s):  
GEORGIOS K. ALEXOPOULOS
Keyword(s):  
Volume Growth ◽  
Discrete Groups ◽  
Multiplier Theorem ◽  
Spectral Multipliers

The classical Mikhlin–Hörmander multiplier theorem is generalised to the context of discrete groups of polynomial volume growth.

scholarly journals Positive definite functions and cut-off for discrete groups

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.

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].

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