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Connective constant for a weighted self-avoiding walk on \(\mathbb{Z}^2\). (English) Zbl 1326.82012
Summary: We consider a self-avoiding walk on the dual \(\mathbb{Z}^2\) lattice. This walk can traverse the same square twice but cannot cross the same edge more than once. The weight of each square visited by the walk depends on the way the walk passes through it and the weight of the whole walk is calculated as a product of these weights. We consider a family of critical weights parametrized by angle \(\theta\in[\frac{\pi}{3},\frac{2\pi}{3}]\). For \(\theta=\frac{\pi}{3}\), this can be mapped to the self-avoiding walk on the honeycomb lattice. The connective constant in this case was proved to be equal to \(\sqrt{2+\sqrt{2}}\) by H. Duminil-Copin and S. Smirnov [Ann. Math. (2) 175, No. 3, 1653–1665 (2012; Zbl 1253.82012)]. We generalize their result.

82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
60J67 Stochastic (Schramm-)Loewner evolution (SLE)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82D60 Statistical mechanics of polymers
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