On groups, slow heat kernel decay yields Liouville property and sharp entropy bounds

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.