The connective constant of the honeycomb lattice equals \(\sqrt{2+\sqrt 2}\). (English) Zbl 1253.82012

Let \(c_n\) be the number of \(n\)-step self-avoiding walks on the hexagonal lattice started from some fixed vertex. It is known that there exists \(\mu\in (0,+\infty)\) such that \(\mu=\lim_{n\to \infty}c_n^{1/n}\). The positive real number \(\mu\) is called the connective constant of the hexagonal lattice. In 1982, using Coulomb gas formalism, B. Nienhuis proposed physical arguments for \(\mu\) to have the value \(\sqrt{2+\sqrt{2}}\). In the paper under review, the authors give a rigorous prove of this result, using a parafermionic observable for the self-avoiding walk, which satisfies a half of the discrete Cauchy-Riemann relations.


82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
Full Text: DOI arXiv


[1] N. Beaton, J. de Gier, and A. Guttmann, The critical fugacity for surface adsorption of SAW on the honeycomb lattice is \(1+\sqrt2\), 2011. · Zbl 1288.82006 · doi:10.1007/s00220-014-1896-1
[2] Y. Ikhlef and J. Cardy, ”Discretely holomorphic parafermions and integrable loop models,” J. Phys. A, vol. 42, iss. 10, p. 102001, 2009. · Zbl 1159.81041 · doi:10.1088/1751-8113/42/10/102001
[3] D. Chelkak and S. Smirnov, Universality in the 2D Ising model and conformal invariance of fermionic observables, 2009. · Zbl 1257.82020 · doi:10.1007/s00222-011-0371-2
[4] P. A. Flory, Principles of Polymer Chemistry, Cornell University Press, 1953.
[5] J. M. Hammersley and D. J. A. Welsh, ”Further results on the rate of convergence to the connective constant of the hypercubical lattice,” Quart. J. Math. Oxford Ser., vol. 13, pp. 108-110, 1962. · Zbl 0123.00304 · doi:10.1093/qmath/13.1.108
[6] W. Kager and B. Nienhuis, ”A guide to stochastic Löwner evolution and its applications,” J. Statist. Phys., vol. 115, iss. 5-6, pp. 1149-1229, 2004. · Zbl 1157.82327 · doi:10.1023/B:JOSS.0000028058.87266.be
[7] G. F. Lawler, O. Schramm, and W. Werner, ”On the scaling limit of planar self-avoiding walk,” in Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 2, Providence, RI: Amer. Math. Soc., 2004, vol. 72, pp. 339-364. · Zbl 1069.60089
[8] G. F. Lawler, O. Schramm, and W. Werner, ”Conformal invariance of planar loop-erased random walks and uniform spanning trees,” Ann. Probab., vol. 32, iss. 1B, pp. 939-995, 2004. · Zbl 1126.82011 · doi:10.1214/aop/1079021469
[9] N. Madras and G. Slade, The Self-Avoiding Walk, Boston, MA: Birkhäuser, 1993. · Zbl 0780.60103
[10] B. Nienhuis, ”Exact critical point and critical exponents of \({ O}(n)\) models in two dimensions,” Phys. Rev. Lett., vol. 49, iss. 15, pp. 1062-1065, 1982. · doi:10.1103/PhysRevLett.49.1062
[11] B. Nienhuis, ”Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas,” J. Statist. Phys., vol. 34, iss. 5-6, pp. 731-761, 1984. · Zbl 0595.76071 · doi:10.1007/BF01009437
[12] S. Smirnov, ”Towards conformal invariance of 2D lattice models,” in International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1421-1451. · Zbl 1112.82014
[13] S. Smirnov, ”Discrete complex analysis and probability,” in Proceedings of the International Congress of Mathematicians. Volume I, New Delhi, 2010, pp. 595-621. · Zbl 1251.30049
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.