Percolation transitions in two dimensions

Xiaomei Feng; Youjin Deng; Henk W. J. Blote

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arXiv · 0901.1370

Percolation transitions in two dimensions

Xiaomei Feng·Youjin Deng·Henk W. J. Blote

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Abstract

We investigate bond- and site-percolation models on several two-dimensional lattices numerically, by means of transfer-matrix calculations and Monte Carlo simulations. The lattices include the square, triangular, honeycomb kagome and diced lattices with nearest-neighbor bonds, and the square lattice with nearest- and next-nearest-neighbor bonds. Results are presented for the bond-percolation thresholds of the kagome and diced lattices, and the site-percolation thresholds of the square, honeycomb and diced lattices. We also include the bond- and site-percolation thresholds for the square lattice with nearest- and next-nearest-neighbor bonds. We find that corrections to scaling behave according to the second temperature dimension $X_{t2}=4$ predicted by the Coulomb gas theory and the theory of conformal invariance. In several cases there is evidence for an additional term with the same exponent, but modified by a logarithmic factor. Only for the site-percolation problem on the triangular lattice such a logarithmic term appears to be small or absent. The amplitude of the power-law correction associated with $X_{t2}=4$ is found to be dependent on the orientation of the lattice with respect to the cylindrical geometry of the finite systems.

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Xiaomei Feng, Youjin Deng, Henk W. J. Blote. 2009-01-10. Percolation transitions in two dimensions. https://arxiv.org/abs/0901.1370

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