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The connective constant of the honeycomb lattice equals \(\sqrt{2+\sqrt 2}\). (English) Zbl 1253.82012

Let \(c_n\) be the number of \(n\)-step self-avoiding walks on the hexagonal lattice started from some fixed vertex. It is known that there exists \(\mu\in (0,+\infty)\) such that \(\mu=\lim_{n\to \infty}c_n^{1/n}\). The positive real number \(\mu\) is called the connective constant of the hexagonal lattice. In 1982, using Coulomb gas formalism, B. Nienhuis proposed physical arguments for \(\mu\) to have the value \(\sqrt{2+\sqrt{2}}\). In the paper under review, the authors give a rigorous prove of this result, using a parafermionic observable for the self-avoiding walk, which satisfies a half of the discrete Cauchy-Riemann relations.

MSC:

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
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