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