1106.2226 (Lionel Levine et al.)

The looping constant of Z^d    [PDF]

Lionel Levine, Yuval Peres

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.

View original: http://arxiv.org/abs/1106.2226