Abstract : Let $X$ be a set with a group $G$ acting on it. We will consider the question of when there exists a $G$-homomorphism between two i.i.d. processes which are indexed by $X$. In the case where $X=G$, we will see that amenable and nonamenable groups are characterized by very different behavior with respect to this question. We will also consider the case where $X$ is a graph. The proofs involve applications of interesting ideas from percolation theory.