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 $-\log {\mu }^{(2n)}(id)=o(n^{1\mathrm{/}2})$, 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 $\beta \in (0,1\mathrm{/}2)$, 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.