The looping constant of ℤd

The looping constanti¾?i¾?d is the expected number of neighbors of the origin that lie on the infinite loop-erased random walk in i¾?d. Poghosyan, Priezzhev, and Ruelle, and independently, Kenyon and Wilson, proved recently that i¾?i¾?2=54. We consider the infinite volume limits as Gi¾?i¾?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 i¾?i¾?d. In the case of i¾?2, their respective values are 8, 178 and 18. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 45, 1-13, 2014

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