  We consider spread-out models of self-avoiding walk, bond percolation,
lattice trees and bond lattice animals on the d-dimensional hyper cubic lattice
having long finite-range connections, above their upper critical dimensions d=4
(self-avoiding walk), d=6 (percolation) and d=8 (trees and animals). The
two-point functions for these models are respectively the generating function
for self-avoiding walks from the origin to x, the probability of a connection
from 0 to x, and the generating function for lattice trees or lattice animals
containing 0 and x. We use the lace expansion to prove that for sufficiently
spread-out models above the upper critical dimension, the two-point function of
each model decays, at the critical point, as a multiple of $|x|^{2-d}$ as x
goes to infinity. We use a new unified method to prove convergence of the lace
expansion. The method is based on x-space methods rather than the Fourier
transform. Our results also yield unified and simplified proofs of the bubble
condition for self-avoiding walk, the triangle condition for percolation, and
the square condition for lattice trees and lattice animals, for sufficiently
spread-out models above the upper critical dimension.
