Electronic Communications in Probability

Connective constant for a weighted self-avoiding walk on $\mathbb{Z}^2$

Alexander Glazman

Full-text: Open access

Abstract

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.

Article information

Source
Electron. Commun. Probab., Volume 20 (2015), paper no. 86, 13 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465321013

Digital Object Identifier
doi:10.1214/ECP.v20-3844

Mathematical Reviews number (MathSciNet)
MR3434203

Zentralblatt MATH identifier
1326.82012

Subjects
Primary: 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]
Secondary: 60J67 60K35 82D60: Polymers

Keywords
weighted self-avoiding walks connective constant integrable weights Yang-Baxter equation parafermionic observable

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Glazman, Alexander. Connective constant for a weighted self-avoiding walk on $\mathbb{Z}^2$. Electron. Commun. Probab. 20 (2015), paper no. 86, 13 pp. doi:10.1214/ECP.v20-3844. https://projecteuclid.org/euclid.ecp/1465321013


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