A general approach to massive upper bound for two-point function with application to self-avoiding walk torus plateau

TL;DR

This paper establishes a general condition ensuring the two-point function remains uniformly bounded near criticality for models on high-dimensional lattices, and applies it to self-avoiding walk, revealing a plateau phenomenon on the torus.

Contribution

It introduces a convolution-based criterion for bounding the two-point function and demonstrates its application to self-avoiding walk in high dimensions, including on the torus.

Findings

Proves a uniform bound for the two-point function near criticality.

Verifies the condition for self-avoiding walk in dimensions > 4.

Identifies a plateau in the two-point function on the torus for high-dimensional models.

Abstract

We prove a sufficient condition for the two-point function of a statistical mechanical model on $\mathbb{Z}^d$, $d > 2$, to be bounded uniformly near a critical point by $|x|^{-(d-2)} \exp [ -c|x| / \xi ]$, where $\xi$ is the correlation length. The condition is given in terms of a convolution equation satisfied by the two-point function, and we verify the condition for strictly self-avoiding walk in dimensions $d > 4$ using the lace expansion. As an example application, we use the uniform bound to study the self-avoiding walk on a $d$-dimensional discrete torus with $d > 4$, proving a ``plateau'' of the torus two-point function, a result previously obtained for weakly self-avoiding walk in dimensions $d > 4$ by Slade. Our method has the potential to be applied to other statistical mechanical models on $\mathbb{Z}^d$ or on the torus.