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2015 Connective constant for a weighted self-avoiding walk on $\mathbb{Z}^2$
Alexander Glazman
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Electron. Commun. Probab. 20: 1-13 (2015). DOI: 10.1214/ECP.v20-3844


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 Duminil-Copin and Smirnov. We generalize their result.


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Alexander Glazman. "Connective constant for a weighted self-avoiding walk on $\mathbb{Z}^2$." Electron. Commun. Probab. 20 1 - 13, 2015.


Accepted: 13 November 2015; Published: 2015
First available in Project Euclid: 7 June 2016

zbMATH: 1326.82012
MathSciNet: MR3434203
Digital Object Identifier: 10.1214/ECP.v20-3844

Primary: 82B41
Secondary: 60J67 , 60K35 , 82D60

Keywords: Connective constant , integrable weights , parafermionic observable , weighted self-avoiding walks , Yang-Baxter equation

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