The Looping Constant of Z^d

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 \(G↑Z_d\) 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.