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\u2191Z_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.<\/p>\n","protected":false},"excerpt":{"rendered":"
The 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 . 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