On the growth constant for square-lattice self-avoiding walks

TL;DR

This paper refines the estimate of the growth constant for square-lattice self-avoiding walks, introduces an efficient computational method, and provides a more precise value that challenges previous conjectures.

Contribution

It presents a highly efficient topological transfer-matrix method for estimating the growth constant, achieving unprecedented precision and refuting a longstanding conjecture.

Findings

New estimate of the growth constant: 2.63815853032790(3)

The topological transfer-matrix method is the most efficient among tested methods

The conjecture fails in the 12th digit of the estimate.

Abstract

The growth constant for two-dimensional self-avoiding walks on the honeycomb lattice was conjectured by Nienhuis in 1982, and since that time the corresponding results for the square and triangular lattices have been sought. For the square lattice, a possible conjecture was advanced by one of us (AJG) more than 20 years ago, based on the six significant digit estimate available at the time. This estimate has improved by a further six digits over the intervening decades, and the conjectured value continued to agree with the increasingly precise estimates. We discuss the three most successful methods for estimating the growth constant, including the most recently developed Topological Transfer-Matrix method, due to another of us (JLJ). We show this to be the most computationally efficient of the three methods, and by parallelising the algorithm we have estimated the growth constant significantly more precisely, incidentally ruling out the conjecture, which fails in the 12th digit. Our new estimate of the growth constant is $$\mu(\mathrm{square}) = 2.63815853032790\, (3).$$