## The Annals of Probability

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

Takashi Hara

#### Abstract

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

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

Dates
First available in Project Euclid: 29 February 2008

Permanent link to this document
https://projecteuclid.org/euclid.aop/1204306960

Digital Object Identifier
doi:10.1214/009117907000000231

Mathematical Reviews number (MathSciNet)
MR2393990

Zentralblatt MATH identifier
1142.82006

#### Citation

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

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