The Looping Constant of Z^d

The looping constant ξ(Zd) is the expected number of neighbors of the origin that lie on the infinite loop-erased random walk in Zd. Poghosyan, Priezzhev and Ruelle, and independently, Kenyon and Wilson, proved recently that ξ(Z2)=5/4.
We consider the infinite volume limits as G↑Zd 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 ξ(Zd). In the case of Z2 their respective values are 8, 17/8 and 1/8.