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.

v4: 32 pages. An error in Lemma 4.2 has been corrected (does not affect conclusions). v3: New title. Major revisions to Section 1, editorial changes throughout, new references to recent literature on torus plateau [21], [28]