  Let $\mu$ be a symmetric probability measure of finite entropy on a group
$G$. We show that if $-\log \mu^{(2n)}(id)=o(n^{1/2})$, then the pair $(G,\mu)$
has the Liouville property (all bounded $\mu$-harmonic functions on $G$ are
constant). Furthermore, if $-\log \mu^{(2n)}(id)=O(n^{\beta})$ where
$\beta\in(0,1/2)$, then the entropy of the $n$-fold convolution power
$\mu^{(n)}$ satisfies $H(\mu^{(n)})=O\left(n^{\frac{\beta}{1-\beta}}\right)$.
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.
