Proceedings of the American Mathematical Society
Let G be a countable group and u 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.
Northshield, S. (1993). Amenability and superharmonic functions. Proceedings of the American Mathematical Society, 119 (2), 561-566.