  We use the lace expansion to study the long-distance decay of the two-point
function of weakly self-avoiding walk on the integer lattice $\mathbb{Z}^d$ in
dimensions $d>4$, in the vicinity of the critical point, and prove an upper
bound $|x|^{-(d-2)}\exp[-c|x|/\xi]$, where the correlation length $\xi$ has a
square root divergence at the critical point. As an application, we prove that
the two-point function for weakly self-avoiding walk on a discrete torus in
dimensions $d>4$ has a "plateau." We also discuss the significance and
consequences of the plateau for the analysis of critical behaviour on the
torus.
