Self-avoiding walks and trees in spread-out lattices

LetGR be the graph obtained by joining all sites ofZd which are separated by a distance of at mostR. Let μ(GR) denote the connective constant for counting the self-avoiding walks in this graph. Let λ(GR) denote the coprresponding constant for counting the trees embedded inGR. Then asR→∞, μ(GR) is asymptotic to the coordination numberkR ofGR, while λ(GR) is asymptotic toekR. However, ifd is 1 or 2, then μ(GR)-kR diverges to −∞.

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