Amenability and superharmonic functions

Let $G$ be a countable group and $\mu$ a symmetric and aperiodic probability measure on $G$. We show that $G$ is amenable if and only if every positive superharmonic function is nearly constant on certain arbitrarily large subsets of $G$. We use this to show that if $G$ is amenable, then the Martin boundary of $G$ contains a fixed point. More generally, we show that $G$ is amenable if and only if each member of a certain family of $G$-spaces contains a fixed point.