The critical fugacity for surface adsorption of SAW on the honeycomb lattice is 1 + √ 2

Recently Duminil-Copin and Smirnov proved a long-standing conjecture of Nienhuis, made in 1982, that the connective constant of self-avoiding walks on the honeycomb lattice is √ 2 + √ 2. A key identity used in that proof was later generalised by Smirnov so as to apply to a general O(n) model with n ∈ [−2,2]. We modify this model by restricting to a half-plane and introducing a fugacity associated with surface sites, and obtain a further generalisation of the Smirnov identity. Our identity depends naturally on the conjectured value of the critical surface fugacity and thus provides an independent prediction for this value. For the case n = 0, characterising the surface adsorption transition of self-avoiding walks, we provide a proof for the value of the critical surface fugacity.