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

The near-critical two-point function and the torus plateau for weakly self-avoiding walk in high dimensions