TY - JOUR

T1 - Decay of correlations in nearest-neighbor self-avoiding walk, percolation, lattice trees and animals

AU - Hara, Takashi

PY - 2008/3/1

Y1 - 2008/3/1

N2 - 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| → ∞.

AB - 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| → ∞.

UR - http://www.scopus.com/inward/record.url?scp=50249118414&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=50249118414&partnerID=8YFLogxK

U2 - 10.1214/009117907000000231

DO - 10.1214/009117907000000231

M3 - Article

AN - SCOPUS:50249118414

VL - 36

SP - 530

EP - 593

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 2

ER -