Let $G$ be a group generated by a finite set $S$. Let $P$ be a Markov operator defined by the uniform measure on $S$. A function is superharmonic if $Pf\leq f$.
Assume that there is a set of non-negative superharmonic functions $X$, which satisfies
(1) $X$ is convex and invariant under $P$
(2) There exists a sequence $(x_n)$ in $G$ such that $f(g.x_n)/f(x_n)\rightarrow 1$ for all g in $G$.
What are the conditions on this set one can impose in order to conclude amenability of the group? If the set of all non-negative superharmonic functions satisfies (2) then the paper "Amenability and superharmonic functions" by Northshield  implies amenability. It seems one should be able to read more information provided extra conditions on this set. I would be interested to have any citation on related articles.