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Communications in Mathematical Physics

, Volume 326, Issue 3, pp 727–754 | Cite as

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

  • Nicholas R. Beaton
  • Mireille Bousquet-Mélou
  • Jan de Gier
  • Hugo Duminil-Copin
  • Anthony J. Guttmann
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.

Keywords

Surface Adsorption Growth Constant Loop Model Honeycomb Lattice Weighted Vertex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Nicholas R. Beaton
    • 2
  • Mireille Bousquet-Mélou
    • 1
  • Jan de Gier
    • 2
  • Hugo Duminil-Copin
    • 3
  • Anthony J. Guttmann
    • 2
  1. 1.CNRS, LaBRI, UMR 5800Université de BordeauxTalence CedexFrance
  2. 2.Department of Mathematics and StatisticsThe University of MelbourneMelbourneAustralia
  3. 3.Section de MathématiquesUniversité de GenèveGenevaSwitzerland

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