Geodesic
The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt 2}$
Hugo Duminil-Copin1 ; Stanislav Smirnov2
1 Section de mathématiques, Université de Genève, 2-4 rue du Lièvre, case postale 64 1211, Genève 4, Switzerland
2 Section de mathématiques, Université de Genève, 2-4 rue du Lièvre, case postale 64 1211, Genève 4, Switzerland and Chebyshev Laboratory, St. Petersburg State University, Saint Petersburg 199178, Russia
Annals of mathematics, Tome 175 (2012) no. 3, pp. 1653-1665.

Voir la notice de l'article provenant de la source Annals of Mathematics website

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 nonrigorously 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($8/3$).
DOI : 10.4007/annals.2012.175.3.14

Hugo Duminil-Copin 1 ; Stanislav Smirnov 2

1 Section de mathématiques, Université de Genève, 2-4 rue du Lièvre, case postale 64 1211, Genève 4, Switzerland
2 Section de mathématiques, Université de Genève, 2-4 rue du Lièvre, case postale 64 1211, Genève 4, Switzerland and Chebyshev Laboratory, St. Petersburg State University, Saint Petersburg 199178, Russia 
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Hugo Duminil-Copin; Stanislav Smirnov. The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt 2}$. Annals of mathematics, Tome 175 (2012) no. 3, pp. 1653-1665. doi : 10.4007/annals.2012.175.3.14. http://geodesic.mathdoc.fr/articles/10.4007/annals.2012.175.3.14/

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