Jacobsen 2016 growth constant square lattice self avoiding

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main_estimate: States the paper's numerical estimate of the square-lattice self-avoiding walk growth constant, mu = 2.63815853032790(3), obtained by extrapolating topological transfer-matrix data, and the authors' conclusion that the value conjectured from the quartic 13t^4 - 7t^2 - 581 fails in the 12th digit; neither is a proved bound.

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Jacobsen, Jesper Lykke and Scullard, Christian R. and Guttmann, Anthony J., On the growth constant for square-lattice self-avoiding walks. J. Phys. A 49 (2016), no. 49, 494004, 18 pp. DOI 10.1088/1751-8113/49/49/494004. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1607.02984), every other right reserved. The copy read for this card is the arXiv version arXiv:1607.02984v1 (11 July 2016).

The paper compares three methods for estimating the connective constant mu of self-avoiding walks on the square lattice: finite-lattice series extrapolation for self-avoiding polygons, a method adapting the Duminil-Copin and Smirnov identity, and the recently developed Topological Transfer-Matrix method of one of the authors. They argue the transfer-matrix method is the most computationally efficient, parallelize it, and obtain mu = 2.63815853032790(3), a substantial gain on earlier estimates such as 2.63815853035(2) of Clisby and Jensen. The estimate is an extrapolation under an assumed finite-size scaling form, with an error bar rather than a proved bound. On its basis the authors conclude that Guttmann's old conjecture, that mu is the positive real root 2.6381585303417408... of 13t^4 - 7t^2 - 581 = 0, which had agreed with the increasingly precise earlier estimates, fails in the 12th digit: the conjectured radius of convergence x_c = 1/mu is too low by about 2 * 10^-12. The authors also note, as a separate ground for doubt, that the quartic has a conjugate pair of roots on the imaginary axis while numerical analysis of the walk and polygon series shows no such singularity, and they validate their numerics by repeating the extrapolation with fewer data points (n_max = 19, 20).

Source: https://arxiv.org/abs/1607.02984.

Read status. Claims checked: the principal estimate, its scaling assumptions, the conjecture and the earlier estimates it is compared with were read clause by clause on the printed pages (pp. 1--3, 7, 12, 20--23). The computations were read for their structure only and not rerun.

Bears on. #528: the paper gives a numerical estimate of C_2, with its error bar in the fourteenth decimal place, and its authors conclude that C_2 is not the positive root of 13t^4 - 7t^2 - 581. Neither is proved, so the paper does not determine or rigorously bound C_2, and it says nothing about C_k for k other than 2.

Results. The principal estimate (pp. 1, 22, unnumbered; the paper has no numbered theorems), with the conjecture it is compared with (pp. 2--3). The other estimates of the paper, by series analysis (p. 7) and by the adapted Duminil-Copin and Smirnov identity (p. 12), are recorded on that page as comparisons.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.