The Looping Constant of Z Lionel Levine and Yuval Peres

The looping constant ξ(Z) is the expected number of neighbors of the origin that lie on the infinite loop-erased random walk in Z. Poghosyan, Priezzhev and Ruelle, and independently, Kenyon and Wilson, proved recently that ξ(Z) = 54 . We consider the infinite volume limits as G ↑ Z 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 ξ(Z). In the case of Z their respective values are 8, 17 8 and 1 8 . Fix an integer d ≥ 2, and let ξ = ξ(Z) be the expected number of neighbors of the origin on the infinite loop-erased random walk in Z (defined in section 1). We call this number ξ the looping constant of Z. We explore several statistics of Z that can be expressed as rational functions of ξ. Recently Poghosyan, Priezzhev, and Ruelle [PPR11] and Kenyon and Wilson [KW11] independently proved ξ(Z) = 5 4 , which implies that all of these statistics have rational values for Z. A unicycle is a connected graph with the same number of edges as vertices. Such a graph has exactly one cycle (Figure 1). If G is a finite (multi)graph, a spanning subgraph of G is a graph containing all of the vertices of G and a subset of the edges. A uniform spanning unicycle (USU) of G is a spanning subgraph of G which is a unicycle, selected uniformly at random. We regard the d-dimensional integer lattice Z as a graph with the usual nearest-neighbor adjacencies: x ∼ y if and only if x − y ∈ {±e1, . . . ,±ed}, where the ei are the standard basis vectors. An exhaustion of Z is a sequence V1 ⊂ V2 ⊂ · · · of finite subsets such that ⋃ n≥1 Vn = Z. Let Gn be the multigraph obtained from Z by collapsing V c n to a single vertex sn, and removing self-loops at sn. We do not collapse edges, so Gn may have edges Date: July 16, 2012. 2010 Mathematics Subject Classification. 60G50, 82B20.