Electronic Communications in Probability

Connective constant for a weighted self-avoiding walk on $\mathbb{Z}^2$

Alexander Glazman

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We consider a self-avoiding walk on the dual $\mathbb{Z}^2$ lattice.This walk can traverse the same square twice but cannot cross the same edge more than once. The weight of each square visited by the walk depends on the way the walk passes through it and the weight of the whole walk is calculated as a product of these weights. We consider a family of critical weights parametrized by angle $\theta\in[\frac{\pi}{3},\frac{2\pi}{3}]$. For $\theta=\frac{\pi}{3}$, this can be mapped to the self-avoiding walk on the honeycomb lattice. The connective constant in this case was proved to be equal to $\sqrt{2+\sqrt{2}}$ by Duminil-Copin and Smirnov. We generalize their result.

Article information

Electron. Commun. Probab., Volume 20 (2015), paper no. 86, 13 pp.

Accepted: 13 November 2015
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]
Secondary: 60J67 60K35 82D60: Polymers

weighted self-avoiding walks connective constant integrable weights Yang-Baxter equation parafermionic observable

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Glazman, Alexander. Connective constant for a weighted self-avoiding walk on $\mathbb{Z}^2$. Electron. Commun. Probab. 20 (2015), paper no. 86, 13 pp. doi:10.1214/ECP.v20-3844. https://projecteuclid.org/euclid.ecp/1465321013

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  • Alam, I. T.; Batchelor, M. T. Integrability as a consequence of discrete holomorphicity: loop models. J. Phys. A 47 (2014), no. 21, 215201, 17 pp.
  • M. T. Batchelor. Personal communications.
  • Bauerschmidt, Roland; Duminil-Copin, Hugo; Goodman, Jesse; Slade, Gordon. Lectures on self-avoiding walks. Probability and statistical physics in two and more dimensions, 395–467, Clay Math. Proc., 15, Amer. Math. Soc., Providence, RI, 2012.
  • Beaton, Nicholas R.; Bousquet-Mélou, Mireille; de Gier, Jan; Duminil-Copin, Hugo; Guttmann, Anthony J. The critical fugacity for surface adsorption of self-avoiding walks on the honeycomb lattice is $1+\sqrt{2}$. Comm. Math. Phys. 326 (2014), no. 3, 727–754.
  • Chelkak, Dmitry; Smirnov, Stanislav. Universality in the 2D Ising model and conformal invariance of fermionic observables. Invent. Math. 189 (2012), no. 3, 515–580.
  • de Gier, Jan; Lee, Alexander; Rasmussen, Jørgen. Discrete holomorphicity and integrability in loop models with open boundaries. J. Stat. Mech. Theory Exp. 2013, no. 2, P02029, 27 pp.
  • Duminil-Copin, Hugo; Smirnov, Stanislav. Conformal invariance of lattice models. Probability and statistical physics in two and more dimensions, 213–276, Clay Math. Proc., 15, Amer. Math. Soc., Providence, RI, 2012.
  • Duminil-Copin, Hugo; Smirnov, Stanislav. The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt{2}}$. Ann. of Math. (2) 175 (2012), no. 3, 1653–1665.
  • P. Flory. Principles of Polymer Chemistry. Cornell University Press, 1953.
  • Guttmann, A. J.; Enting, I. G. The size and number of rings on the square lattice. J. Phys. A 21 (1988), no. 3, L165–L172.
  • Hammersley, J. M.; Welsh, D. J. A. Further results on the rate of convergence to the connective constant of the hypercubical lattice. Quart. J. Math. Oxford Ser. (2) 13 1962 108–110.
  • Ikhlef, Yacine; Cardy, John. Discretely holomorphic parafermions and integrable loop models. J. Phys. A 42 (2009), no. 10, 102001, 11 pp.
  • Ikhlef, Y.; Weston, R.; Wheeler, M.; Zinn-Justin, P. Discrete holomorphicity and quantized affine algebras. J. Phys. A 46 (2013), no. 26, 265205, 34 pp.
  • Lawler, Gregory F.; Schramm, Oded; Werner, Wendelin. On the scaling limit of planar self-avoiding walk. Fractal geometry and applications: a jubilee of Benoît Mandelbrot, Part 2, 339–364, Proc. Sympos. Pure Math., 72, Part 2, Amer. Math. Soc., Providence, RI, 2004.
  • Madras, Neal; Slade, Gordon. The self-avoiding walk. Probability and its Applications. Birkhäuser Boston, Inc., Boston, MA, 1993. xiv+425 pp. ISBN: 0-8176-3589-0
  • Nienhuis, Bernard. Exact critical point and critical exponents of ${\rm O}(n)$ models in two dimensions. Phys. Rev. Lett. 49 (1982), no. 15, 1062–1065.
  • Nienhuis, Bernard. Critical and multicritical ${\rm O}(n)$ models. Statistical physics (Rio de Janeiro, 1989). Phys. A 163 (1990), no. 1, 152–157.
  • W.J.C. Orr. Statistical treatment of polymer solutions at infinite dilution. Transactions of the Faraday Society, 43:12–27, 1947.
  • Smirnov, Stanislav. Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. of Math. (2) 172 (2010), no. 2, 1435–1467.
  • Smirnov, Stanislav. Discrete complex analysis and probability. Proceedings of the International Congress of Mathematicians. Volume I, 595–621, Hindustan Book Agency, New Delhi, 2010.