The looping constant of Z d

The looping constantξ ( ℤ d ) is the expected number of neighbors of the origin that lie on the infinite loop‐erased random walk inℤ d . Poghosyan, Priezzhev, and Ruelle, and independently, Kenyon and Wilson, proved recently that ξ ( ℤ 2 ) = 5 4 . We consider the infinite volume limits as G ↑ ℤ dof 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 ( ℤ d ) . In the case ofℤ 2 , their respective values are 8,17 8and1 8 . © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 45, 1–13, 2014