Decay of Correlations in Nearest-Neighbour Self-Avoiding Walk, Percolation, Lattice Trees and Animals

We consider nearest-neighbour self-avoiding walk, bond percolation, lattice trees, and bond lattice animals on Z. The two-point functions of these models are respectively the generating function for self-avoiding walks from the origin to x ∈ Z, 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 [6], 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 is asymptotic to const.|x|2−d as |x| → ∞.