The looping constant of ℤd. Issue 1 (13th February 2013) - Levine, Lionel
Abstract The looping constantξ(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)=54. 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, 178 and 18. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 45, 1–13, 2014