Behaviors of entropy on finitely generated groups

TL;DR

This paper explores diverse entropy behaviors of random walks on finitely generated groups, constructing examples with specific asymptotic properties for entropy, return probability, and drift, using groups of automorphisms of extended rooted trees.

Contribution

It demonstrates the existence of finitely generated groups with prescribed entropy growth rates and analyzes their random walk properties, extending understanding of entropy dynamics in group theory.

Findings

Constructed groups with entropy liminf and limsup between 1/2 and 1.

Provided examples of groups with specific return probability decay rates.

Analyzed drift behavior showing liminf and limsup between 1/2 and 1.

Abstract

A variety of behaviors of entropy functions of random walks on finitely generated groups is presented, showing that for any $\frac{1}{2}\leq \alpha\leq\beta\leq1$, there is a group $\Gamma$ with measure $\mu$ equidistributed on a finite generating set such that \[\liminf\frac{\log H_{\Gamma ,\mu}(n)}{\log n}=\alpha ,\qquad \limsup \frac{\log H_{\Gamma ,\mu}(n)}{\log n}=\beta .\] 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 \[\liminf\frac{{\log}|{\log P}(Y_n=_{\Gamma}1)|}{\log n}=\frac{1}{3},\qquad \limsup\frac{{\log}|{\log P}(Y_n=_{\Gamma}1)|}{\log n}=1\] and drift satisfying \[\liminf\frac{\log {\mathbb{E}}\|Y_n\|}{\log n}=\frac{1}{2},\qquad \limsup\frac{\log {\mathbb{E}}\|Y_n\|}{\log n}=1.\]