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 Z(d). The two-point functions of these models are respectively the generating function for self-avoiding walks from the origin to x epsilon 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.vertical bar x vertical bar(2-d) as vertical bar x vertical bar -> infinity, 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.vertical bar x vertical bar(2-d) as vertical bar x vertical bar -> infinity.