Abstract
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($8/3$).
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@misc{BGG,
author={Beaton, N. and de Gier, J. and Guttmann, A.},
TITLE={The critical fugacity for surface adsorption of {SAW} on the honeycomb lattice is $1+\sqrt{2}$},
YEAR={2011},
NOTE={13 pages},
ARXIV={1109.0358},
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AUTHOR = {Ikhlef, Yacine and Cardy, John},
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JOURNAL = {J. Phys. A},
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NOTE={{\it Invent. Math.},
to appear},
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@incollection {Smirnov, MRKEY = {2275653},
AUTHOR = {Smirnov, Stanislav},
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@inproceedings {smirnov-icm2010, MRKEY = {2827906},
AUTHOR = {Smirnov, Stanislav},
TITLE = {Discrete complex analysis and probability},
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MRNUMBER = {2827906},
ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
ZBLNUMBER = {05971075},
}