Percolation transitions in two dimensions - Condensed Matter > Statistical MechanicsReport as inadecuate

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Abstract: We investigate bond- and site-percolation models on several two-dimensionallattices numerically, by means of transfer-matrix calculations and Monte Carlosimulations. The lattices include the square, triangular, honeycomb kagome anddiced lattices with nearest-neighbor bonds, and the square lattice withnearest- and next-nearest-neighbor bonds. Results are presented for thebond-percolation thresholds of the kagome and diced lattices, and thesite-percolation thresholds of the square, honeycomb and diced lattices. Wealso include the bond- and site-percolation thresholds for the square latticewith nearest- and next-nearest-neighbor bonds.We find that corrections to scaling behave according to the secondtemperature dimension $X {t2}=4$ predicted by the Coulomb gas theory and thetheory of conformal invariance. In several cases there is evidence for anadditional term with the same exponent, but modified by a logarithmic factor.Only for the site-percolation problem on the triangular lattice such alogarithmic term appears to be small or absent. The amplitude of the power-lawcorrection associated with $X {t2}=4$ is found to be dependent on theorientation of the lattice with respect to the cylindrical geometry of thefinite systems.

Author: Xiaomei Feng, Youjin Deng, Henk W.J. Blote


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