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The looping constant of Z d
Author(s) -
Levine Lionel,
Peres Yuval
Publication year - 2014
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20478
Subject(s) - mathematics , constant (computer programming) , random walk , combinatorics , spanning tree , loop erased random walk , state (computer science) , heterogeneous random walk in one dimension , statistics , computer science , algorithm , programming language
The looping constantξ ( ℤ d ) is the expected number of neighbors of the origin that lie on the infinite loop‐erased random walk inℤ d . Poghosyan, Priezzhev, and Ruelle, and independently, Kenyon and Wilson, proved recently that ξ ( ℤ 2 ) = 5 4 . We consider the infinite volume limits as G ↑ ℤ dof 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 ofℤ 2 , their respective values are 8,17 8and1 8 . © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 45, 1–13, 2014