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.
Categories: math.PR math-ph math.MP