The critical fugacity for surface adsorption of self-avoiding walks on the honeycomb lattice is 1+√2

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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 μ=√2+√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∈ [-2,2] (the case n=0 corresponding to SAWs). 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 c=1+2/√2-n. This value plays a crucial role in our generalized identity, just as the value of growth constant did in Smirnov's identity. For the case n=0, corresponding to 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.