The Annals of Probability

Decay of correlations in nearest-neighbor self-avoiding walk, percolation, lattice trees and animals

Takashi Hara

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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|→∞.

Article information

Ann. Probab., Volume 36, Number 2 (2008), 530-593.

First available in Project Euclid: 29 February 2008

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Zentralblatt MATH identifier

Primary: 82B27: Critical phenomena 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41] 82B43: Percolation [See also 60K35] 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Critical behavior two-point function self-avoiding walk percolation lattice trees and animals lace expansion


Hara, Takashi. Decay of correlations in nearest-neighbor self-avoiding walk, percolation, lattice trees and animals. Ann. Probab. 36 (2008), no. 2, 530--593. doi:10.1214/009117907000000231.

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