The looping constant of \(\mathbb Z^d\) (Q2877765)

The paper provides explicit formulas expressing limits of three statistics of \(\mathbb Z^d\) in terms of the looping constant. More precisely, for a finite subgraph \(G\) of \(\mathbb Z^d\), sample a uniform spanning subgraph of \(G\) having exactly one cycle (called a spanning unicycle), and let \(\lambda(G)\) be the expected length of that cycle. Let \(\tau(G)\) be the number of spanning unicycles of \(G\) divided by the number of rooted spanning trees of \(G\). Finally, let \(\zeta(G)\) be the number of particles at the origin in a stationary sandpile on \(G\) as defined in [\textit{D. Dhar}, Phys. Rev. Lett. 64, No.14, 1613--1616 (1990; Zbl 0943.82553)]. It is proved that, for every \(d\geq 2\) and every sequence \((G_n)\) of finite subgraphs of \(\mathbb Z^d\) converging to \(\mathbb Z^d\), all three \(\lambda(G_n)\), \(\tau(G_n)\) and \(\zeta(G_n)\) converge, and the limits can be expressed as NEWLINE\[NEWLINE\lambda = \frac{2d-2}{\xi -1},\quad \tau= \frac{\xi -1}{2},\quad \zeta= d+ \frac{\xi -1}{2},NEWLINE\]NEWLINE where \(\xi\) denotes the looping constant of \(\mathbb Z^d\), i.e., the expected number of neighbors of the origin that lie on the infinite loop-erased random walk in \(\mathbb Z^d\) as defined in [\textit{G. F. Lawler}, Duke Math. J. 47, 655--693 (1980; Zbl 0445.60058)].NEWLINENEWLINENEWLINENEWLINE For the square grid \(\mathbb Z^2\), the value of \(\xi\) is known to be \(5/4\), yielding \(\tau= 1/8\) and \(\lambda = 8\), which is interesting as it is not clear apriori why these quantities have to be rational. The authors point out that the above results are ``readily assembled from known ingredients in the literature'' (which in the reviewer's opinion does not reduce the value of the paper, which still contributes original proofs). In particular, the following ideas are used in order to relate \(\lambda\), \(\tau\) and \(\zeta\) to each other: ``(I) the burning bijection of \textit{S. N. Majumdar} and \textit{D. Dhar} [``Equivalence between the Abelian sandpile model and the \(q\to 0\) limit of the Potts model'', Physica A 185 129--145 (1992)] between spanning trees and recurrent sandpiles; (II) the theorem of \textit{C. M. López} [Ann. Comb. 1, No. 3, 253--259 (1997; Zbl 0901.05004)] relating recurrent sandpiles to a specialization of the Tutte polynomial (which is related to \(\tau\)); and (III) a result of \textit{S. R. Athreya} and \textit{A. A. Járai} [Commun. Math. Phys. 249, No. 1, 197--213 (2004; Zbl 1085.82005)] which shows that in the infinite volume limit, the bulk average height of a recurrent sandpile coincides with the expected height at the origin.''