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

TL;DR

This paper analyzes the decay of the two-point function for weakly self-avoiding walk in high dimensions near criticality, revealing a plateau phenomenon on the torus and its implications for critical behavior.

Contribution

It provides a rigorous upper bound on the two-point function near criticality and demonstrates the existence of a plateau on the torus, advancing understanding of high-dimensional self-avoiding walks.

Findings

Two-point function decays as |x|^{-(d-2)} exp[-c|x|/ξ] near criticality

Correlation length ξ diverges as a square root at the critical point

Existence of a plateau in the two-point function on the torus 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.