Behaviors of entropy on finitely generated groups

Abstract

A variety of behaviors of entropy functions of random walks on finitely generated groups is presented, showing that for any $\frac{1}{2}\le \alpha \le \beta \le 1$, there is a group $\Gamma$ with measure $\mu$ equidistributed on a finite generating set such that \[\liminf\frac{\log H_{Γ,μ}(n)}{\log n}=α,\qquad \limsup \frac{\log H_{Γ,μ}(n)}{\log n}=β.\] The groups involved are finitely generated subgroups of the group of automorphisms of an extended rooted tree. The return probability and the drift of a simple random walk $Y_n$ on such groups are also evaluated, providing an example of group with return probability satisfying \[\liminf\frac{{\log}|{\log P}(Y_n=_Γ1)|}{\log n}=\frac{1}{3},\qquad \limsup\frac{{\log}|{\log P}(Y_n=_Γ1)|}{\log n}=1\] and drift satisfying \[\liminf\frac{\log {\mathbb{E}}\|Y_n\|}{\log n}=\frac{1}{2},\qquad \limsup\frac{\log {\mathbb{E}}\|Y_n\|}{\log n}=1.\]

Published in at http://dx.doi.org/10.1214/12-AOP761 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Groups of given intermediate word growth
• Groups of Oscillating Intermediate Growth

Cited by in corpus (5)

• The Liouville property for groups acting on rooted trees
• Ordering the space of finitely generated groups
• Speed exponents of random walks on groups
• Non-Liouville groups with return probability exponent at most 1/2
• Amenability of quadratic automaton groups