The looping constant of Z^d

Abstract

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.

15 pages, 3 figures, to appear in Random Structures & Algorithms

References in corpus (3)

• Spanning forests and the vector bundle Laplacian
• Spanning trees of graphs on surfaces and the intensity of loop-erased random walk on planar graphs
• The approach to criticality in sandpiles

Cited by in corpus (1)

• Random two-component spanning forests