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

Abstract

We consider nearest-neighbor self-avoiding walk, bond percolation, lattice trees, and bond lattice animals on ℤ^d. The two-point functions of these models are respectively the generating function for self-avoiding walks from the origin to x∈ℤ^d, 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 ℤ^d is asymptotic to const.|x|^2−d as |x|→∞.