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