New Lower Bounds on the Self-Avoiding-Walk Connective Constant

TL;DR

This paper introduces a simple, elementary method using loop erasure and restoration to rigorously establish lower bounds on the self-avoiding walk connective constant on hypercubic lattices, especially effective in high dimensions.

Contribution

The paper presents a novel, elementary approach for lower bounding the connective constant without requiring exact enumeration, improving bounds for dimensions three and higher.

Findings

Bounds match the first four terms of the 1/d expansion

Bounds are the best to date for dimensions d ≥ 3

Lower bounds are within a few percent of estimated values for d=3 to 6

Abstract

We give an elementary new method for obtaining rigorous lower bounds on the connective constant for self-avoiding walks on the hypercubic lattice $Z^d$. The method is based on loop erasure and restoration, and does not require exact enumeration data. Our bounds are best for high $d$, and in fact agree with the first four terms of the $1/d$ expansion for the connective constant. The bounds are the best to date for dimensions $d \geq 3$, but do not produce good results in two dimensions. For $d=3,4,5,6$, respectively, our lower bound is within 2.4\%, 0.43\%, 0.12\%, 0.044\% of the value estimated by series extrapolation.