Behaviors of entropy on finitely generated groups
Jérémie Brieussel · The Annals of Probability · 2013
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 \[\liminf\frac{\log{\mathbb{E}}\|Y_{n}\|}{\log n}=\frac{1}{2},\qquad\limsup\frac{\log{\mathbb{E}}\|Y_{n}\|}{\log n}=1.\]