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

By H. Duminil-Copin, Stanislav Smirnov. 2010

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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).

Subfields: Atomic and Molecular Physics, and Optics, Condensed Matter Physics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6472278852bd2cfa9a0232036ddd9057766c2410