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

Abstract

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.