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March 2008 Decay of correlations in nearest-neighbor self-avoiding walk, percolation, lattice trees and animals
Takashi Hara
Ann. Probab. 36(2): 530-593 (March 2008). DOI: 10.1214/009117907000000231


We consider nearest-neighbor self-avoiding walk, bond percolation, lattice trees, and bond lattice animals on ℤd. The two-point functions of these models are respectively the generating function for self-avoiding walks from the origin to x∈ℤd, the probability of a connection from the origin to x, and the generating functions for lattice trees or lattice animals containing the origin and x. Using the lace expansion, we prove that the two-point function at the critical point is asymptotic to const.|x|2−d as |x|→∞, for d≥5 for self-avoiding walk, for d≥19 for percolation, and for sufficiently large d for lattice trees and animals. These results are complementary to those of [Ann. Probab. 31 (2003) 349–408], where spread-out models were considered. In the course of the proof, we also provide a sufficient (and rather sharp if d>4) condition under which the two-point function of a random walk on ℤd is asymptotic to const.|x|2−d as |x|→∞.


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Takashi Hara. "Decay of correlations in nearest-neighbor self-avoiding walk, percolation, lattice trees and animals." Ann. Probab. 36 (2) 530 - 593, March 2008.


Published: March 2008
First available in Project Euclid: 29 February 2008

zbMATH: 1142.82006
MathSciNet: MR2393990
Digital Object Identifier: 10.1214/009117907000000231

Primary: 82B27 , 82B41 , 82B43 , 82C41
Secondary: 60K35

Keywords: Critical behavior , Lace expansion , lattice trees and animals , percolation , Self-avoiding walk , two-point function

Rights: Copyright © 2008 Institute of Mathematical Statistics


Vol.36 • No. 2 • March 2008
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