The looping constant of Z^d

Understand

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 $\xi(Z^d)$.