The looping constant of Z^d

2011/06/11 by Lionel Levine, Yuval Peres, Levine, Lionel +1
Computer Science · Mathematics · Physics and Astronomy · #60G50 #82B20 #FOS: Mathematics #FOS: Physical sciences #Probability (math.PR) #Statistical Mechanics (cond-mat.stat-mech) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Topological and Geometric Data Analysis #cond-mat.stat-mech #math.PR #msc:60G50 #msc:82B20

paper · pdf · doi:10.48550/arxiv.1106.2226

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

openalex publication_date 2011/06/11 · arxiv created 2012/07/17 · arxiv updated 2012/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The looping constant ξ(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 ξ(Z²)=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 ξ(Z^d). In the case of Z² their respective values are 8, 17/8 and 1/8.

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