The Annals of Probability
- Ann. Probab.
- Volume 36, Number 2 (2008), 530-593.
Decay of correlations in nearest-neighbor self-avoiding walk, percolation, lattice trees and animals
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|→∞.
Ann. Probab., Volume 36, Number 2 (2008), 530-593.
First available in Project Euclid: 29 February 2008
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
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]
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. https://projecteuclid.org/euclid.aop/1204306960