A well-known result of Benjamini, Lyons, Peres, and Schramm states that if $G$ is a finitely generated Cayley graph of a group $Γ$, then $Γ$ is amenable if and only if $G$ admits a $Γ$-invariant random spanning tree with at most two ends. We show that this is equivalent to the existence of a $Γ$-invariant random spanning double ray in a power of $G$.