The looping constant of Z^d

TL;DR

This paper explores the looping constant in Z^d and links it to three statistical measures in graph theory and sandpile models, providing explicit rational function relationships and specific values in two dimensions.

Contribution

It establishes that three key statistics' limits are rational functions of the looping constant and computes their explicit values in Z^2.

Findings

The limits are rational functions of the looping constant $\xi(Z^d)$.

Explicit values for the three statistics in Z^2 are 8, 17/8, and 1/8.

The looping constant $\xi(Z^2)$ equals 5/4.

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.