The looping constant of Z^d

Details The looping constant of Z^d The looping constant of Z^d

2011-06-11

arxiv.org/abs/1106.2226v2

Mathematics

Probability

15 pages, 3 figures. Updated title and introduction to reflect the fact that Conjecture 2 of v1 is now proved (in arXiv:1106.5

Scientific paper

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 $G \uparrow Z^d$ 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.

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