Skip to main content

The Critical Fugacity for Surface Adsorption of Self-Avoiding Walks on the Honeycomb Lattice is \({1+\sqrt{2}}\)

Communications in Mathematical Physics volume 326, pages 727–754 (2014)Cite this article

Abstract

In 2010, Duminil-Copin and Smirnov proved a long-standing conjecture of Nienhuis, made in 1982, that the growth constant of self-avoiding walks on the hexagonal (a.k.a. honeycomb) lattice is \({\mu=\sqrt{2+\sqrt{2}}}\). A key identity used in that proof was later generalised by Smirnov so as to apply to a general O(n) loop model with \({n\in [-2,2]}\) (the case n = 0 corresponding to self-avoiding walks).

We modify this model by restricting to a half-plane and introducing a surface fugacity y associated with boundary sites (also called surface sites), and obtain a generalisation of Smirnov’s identity. The critical value of the surface fugacity was conjectured by Batchelor and Yung in 1995 to be \({y_{\rm c}=1+2/\sqrt{2-n}}\). This value plays a crucial role in our generalized identity, just as the value of the growth constant did in Smirnov’s identity.

For the case n = 0, corresponding to self-avoiding walks interacting with a surface, we prove the conjectured value of the critical surface fugacity. A crucial part of the proof involves demonstrating that the generating function of self-avoiding bridges of height T, taken at its critical point 1/μ, tends to 0 as T increases, as predicted from SLE theory.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

References

  1. Alm S.E., Janson S.: Random self-avoiding walks on one-dimensional lattices. Comm. Stat. Stoch. Models 6(2), 169–212 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  2. Balian R., Toulouse G.: Critical exponents for transitions with n = −2 components of the order parameter. Phys. Rev. Lett. 20, 544–546 (1973)

    ADS  Article  Google Scholar 

  3. Batchelor M.T., Bennet-Wood D., Owczarek A.L.: Two-dimensional polymer networks at a mixed boundary: surface and wedge exponents. Eur. Phys. J. B 5(1), 139–142 (1998)

    ADS  Article  Google Scholar 

  4. Batchelor M.T., Yung C.M.: Exact results for the adsorption of a flexible self-avoiding polymer chain in two dimensions. Phys. Rev. Lett. 74, 2026–2029 (1995)

    ADS  Article  Google Scholar 

  5. Beaton, N.R.: The critical surface fugacity of self-avoiding walks on a rotated honeycomb lattice. Journal of Physics A: Mathematical and Theoretical 47(2014), 075003+

    Google Scholar 

  6. Beaton, N.R., Guttmann, A.J., Jensen, I.: A numerical adaptation of SAW identities from the honeycomb to other 2D lattices. J. Phys. A, 45(3), 035201 (18pp) (2012). Arxiv:1110.1141

  7. Beaton, N.R., Guttmann, A.J., Jensen, I.: Two-dimensional self-avoiding walks and polymer adsorption: critical fugacity estimates. J. Phys. A 45(5), 055208 (12pp) (2012). Arxiv:1110.6695

    Google Scholar 

  8. Binder, K.: Critical behaviour at surfaces. In: Phase transitions and critical phenomena, Vol. 8. London: Academic Press, 1983, pp. 1–144

  9. Gennes P.-G.: Exponents for the excluded-volume problem as derived by the Wilson method. Phys. Lett. A 38, 339–340 (1972)

    ADS  Article  Google Scholar 

  10. Domany E., Mukamel D., Nienhuis B., Schwimmer A.: Duality relations and equivalences for models with O(N) and cubic symmetry. Nucl. Phys. B 190, 279–287 (1981)

    ADS  Article  Google Scholar 

  11. Duminil-Copin, H., Hammond, A.: Self-avoiding walk is sub-ballistic. Comm. Math. Phys. 324(2), 401–423 (2013). Arxiv:1205.0401

    Google Scholar 

  12. Duminil-Copin, H., Smirnov, S.: The connective constant of the honeycomb lattice equals \({\sqrt{2+\sqrt 2}}\). Ann. Math. 175(3), 1653–1665 (2012). Arxiv:1007.0575

    Google Scholar 

  13. Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge: Cambridge University Press, 2009

  14. Flory, P.: Principles of Polymer Chemistry. New York: Cornell University Press, 1953

  15. Hammersley J.M., Torrie G.M., Whittington S.G.: Self-avoiding walks interacting with a surface. J. Phys. A 15(2), 539–571 (1982)

    ADS  Article  MathSciNet  Google Scholar 

  16. Janse van Rensburg E.J., Orlandini E., Whittingon S.G.: Self-avoiding walks in a slab: rigorous results. J. Phys. A 39(45), 13869–13902 (2006)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  17. Kesten H.: On the number of self-avoiding walks. J. Math. Phys. 4, 960–969 (1963)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  18. Klazar, M.: On the theorem of Duminil-Copin and Smirnov about the number of self-avoiding walks in the hexagonal lattice. Arxiv:1102.5733

  19. Lawler, G.F., Schramm, O., Werner, W.: On the scaling limit of planar self-avoiding walk. In: Fractal geometry and applications: a jubilee of Benoî t Mandelbrot, Part 2. Proc. Sympos. Pure Math. Vol. 72, Providence, RI: Amer. Math. Soc., 2004, pp. 339–364

  20. Madras, N., Slade, G.: The self-avoiding walk. In: Probability and its Applications. Boston, MA: Birkhäuser Boston Inc., 1993

  21. Nienhuis B.: Exact critical point and critical exponents of O(n) models in two dimensions. Phys. Rev. Lett. 49, 1062–1065 (1982)

    ADS  Article  MathSciNet  Google Scholar 

  22. Orr W.J.C.: Statistical treatment of polymer solutions at infinite dilution. Trans. Faraday Soc. 43, 12–27 (1947)

    Article  Google Scholar 

  23. Rychlewski G., Whittington S.G.: Self-avoiding walks and polymer adsorption: low temperature behaviour. J. Stat. Phys. 145(3), 661–668 (2011)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  24. Smirnov, S.: Discrete complex analysis and probability. In: Proceedings of the International Congress of Mathematicians. Vol. I, New Delhi: Hindustan Book Agency, 2010, pp. 595–621

  25. Stanley H.E.: Dependence of critical properties on dimensionality of spins. Phys. Rev. Lett. 20, 589–593 (1968)

    ADS  Article  Google Scholar 

  26. Janse van Rensburg, E.J.: The statistical mechanics of interacting walks, polygons, animals and vesicles. In: Oxford Lecture Series in Mathematics and its Applications, Vol. 18. Oxford: Oxford University Press, 2000

  27. Whittington S.G.: Self-avoiding walks terminally attached to an interface. J. Chem. Phys. 63, 779–785 (1975)

    ADS  Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CNRS, LaBRI, UMR 5800, Université de Bordeaux, 351 cours de la Libération, 33405, Talence Cedex, France

    Mireille Bousquet-Mélou

  2. Department of Mathematics and Statistics, The University of Melbourne, 3010, Melbourne, VIC, Australia

    Nicholas R. Beaton, Jan de Gier & Anthony J. Guttmann

  3. Section de Mathématiques, Université de Genève, Geneva, Switzerland

    Hugo Duminil-Copin

Authors
  1. Nicholas R. Beaton
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mireille Bousquet-Mélou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jan de Gier
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Hugo Duminil-Copin
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Anthony J. Guttmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jan de Gier.

Additional information

Communicated by S. Smirnov

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Beaton, N.R., Bousquet-Mélou, M., de Gier, J. et al. The Critical Fugacity for Surface Adsorption of Self-Avoiding Walks on the Honeycomb Lattice is \({1+\sqrt{2}}\) . Commun. Math. Phys. 326, 727–754 (2014). https://doi.org/10.1007/s00220-014-1896-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-014-1896-1

Keywords

  • Surface Adsorption
  • Growth Constant
  • Loop Model
  • Honeycomb Lattice
  • Weighted Vertex