  The looping constant $\xi(Z^d)$ is the expected number of neighbors of the
origin that lie on the infinite loop-erased random walk in $Z^d$. Poghosyan,
Priezzhev and Ruelle, and independently, Kenyon and Wilson, proved recently
that $\xi(Z^2)=5/4$.
  We consider the infinite volume limits as $G \uparrow Z^d$ of three different
statistics: (1) The expected length of the cycle in a uniform spanning unicycle
of G; (2) The expected density of a uniform recurrent state of the abelian
sandpile model on G; and (3) The ratio of the number of spanning unicycles of G
to the number of rooted spanning trees of G. We show that all three limits are
rational functions of the looping constant $\xi(Z^d)$. In the case of $Z^2$
their respective values are 8, 17/8 and 1/8.
