Abstract
AbstractThe looping constant is the expected number of neighbors of the origin that lie on the infinite loop‐erased random walk in . Poghosyan, Priezzhev, and Ruelle, and independently, Kenyon and Wilson, proved recently that . We consider the infinite volume limits as 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 . In the case of , their respective values are 8, and . © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 45, 1–13, 2014
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