Improved lower bounds on the connective constants for two-dimensional self-avoiding walks

TL;DR

This paper improves lower bounds on the connective constants for self-avoiding walks on various 2D lattices using transfer-matrix enumeration of irreducible bridges, providing tighter bounds compared to previous methods.

Contribution

It introduces enhanced lower bounds for 2D self-avoiding walk connective constants via transfer-matrix enumeration of irreducible bridges, advancing prior bounds.

Findings

New lower bounds for connective constants on multiple lattices

Transfer-matrix techniques enable enumeration of bridges of many steps

Comparison shows improved bounds over previous estimates

Abstract

We calculate improved lower bounds for the connective constants for self-avoiding walks on the square, hexagonal, triangular, $(4.8^2)$, and $(3.12^2)$ lattices. The bound is found by Kesten's method of irreducible bridges. This involves using transfer-matrix techniques to exactly enumerate the number of bridges of a given span to very many steps. Upper bounds are obtained from recent exact enumeration data for the number of self-avoiding walks and compared to current best available upper bounds from other methods.