The Near-Critical Two-Point Function and the Torus Plateau for Weakly Self-avoiding Walk in High Dimensions

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.