Abstract
LetG R be the graph obtained by joining all sites ofZ d which are separated by a distance of at mostR. Let μ(G R ) denote the connective constant for counting the self-avoiding walks in this graph. Let λ(G R ) denote the coprresponding constant for counting the trees embedded inG R . Then asR→∞, μ(G R ) is asymptotic to the coordination numberk R ofG R , while λ(G R ) is asymptotic toek R. However, ifd is 1 or 2, then μ(G R )-k R diverges to −∞.
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Dedicated to Oliver Penrose on this occasion of his 65th birthday.
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Penrose, M.D. Self-avoiding walks and trees in spread-out lattices. J Stat Phys 77, 3–15 (1994). https://doi.org/10.1007/BF02186829
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DOI: https://doi.org/10.1007/BF02186829