The critical fugacity for surface adsorption of SAW on the honeycomb lattice is $1+\sqrt{2}$

Details The critical fugacity for surface adsorption of SAW on the honeycomb lattice is $1+\sqrt{2}$ The critical fugacity for surface adsorption of SAW on the honeycomb lattice is $1+\sqrt{2}$

2011-09-02

arxiv.org/abs/1109.0358v2

Physics

Mathematical Physics

13 pages, 6 figures

Scientific paper

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 $\sqrt{2+\sqrt{2}}.$ A key identity used in that proof was later generalised by Smirnov so as to apply to a general O(n) model with $n\in [-2,2]$. 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 \emph{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.

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