text: Let @math be a symmetric probability measure of finite entropy on a group @math . We show that if @math , then the pair @math has the Liouville property (all bounded @math -harmonic functions on @math are constant). Furthermore, if @math where @math , then the entropy of the @math -fold convolution power @math satisfies @math . This improves earlier results of Gournay and of Saloff-Coste and the second author. We extend the bounds to transitive graphs and illustrate their sharpness on a family of groups.
  @cite_12: Let @math be a finitely generated group equipped with a finite symmetric generating set and the associated word length function @math . We study the behavior of the probability of return for random walks driven by symmetric measures @math that are such that @math . For this purpose we develop new relations between the isoperimetric profiles associated with different symmetric probability measures. These techniques allow us to obtain a sharp @math -version of Erschler's inequality concerning the F lner functions of wreath products. Examples and assorted applications are included.
summary: In the work of the second author with Saloff-Coste @cite_12 , an upper bound similar to Theorem was proved. More precisely, suppose @math is a symmetric probability measure of finite second moment on @math such that @math for some @math , then by [Theorem 1.7] Saloff-Coste2014 [ H_ (n) (n ^ 1+ n )^ 1- for any >0. ] In particular, this bound implies the Liouville property of @math for @math . Our results in this paper imply a sharper upper bound @math under the same assumption, and are applicable to a larger class of decay lower bounds.