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

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 $$\mu (\mathrm{square})=2.63815853032790(3).$$

arXiv comment24 pages, 3 figures, 4 tables

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• Square lattice self-avoiding walks and biased differential approximants
• Two-dimensional interacting self-avoiding walks: new estimates for critical temperatures and exponents
• Self-avoiding walks and polygons crossing a domain on the square and hexagonal lattices
• Bootstrap approach to geometrical four-point functions in the two-dimensional critical $Q$-state Potts model: A study of the $s$-channel spectra
• Robustness of flat bands on the perturbed Kagome and the perturbed Super-Kagome lattice
• On the existence of critical exponents for self-avoiding walks
• Numerical estimates of square lattice star vertex exponents
• Self-avoiding walk on ${\mathbb{Z}}^2$ with Yang-Baxter weights: universality of critical fugacity and 2-point function
• The escape transition in a self-avoiding walk model of linear polymers
• Entropic exponents of grafted lattices stars