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