The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt2}$

Details The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt2}$ The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt2}$

2010-07-04

arxiv.org/abs/1007.0575v2

Physics

Mathematical Physics

11 pages, 4 figures

Scientific paper

We provide the first mathematical proof that the connective constant of the hexagonal lattice is equal to $\sqrt{2+\sqrt 2}$. This value has been derived non rigorously by B. Nienhuis in 1982, using Coulomb gas approach from theoretical physics. Our proof uses a parafermionic observable for the self avoiding walk, which satisfies a half of the discrete Cauchy-Riemann relations. Establishing the other half of the relations (which conjecturally holds in the scaling limit) would also imply convergence of the self-avoiding walk to SLE(8/3).

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