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 does such a logarithmic term appear 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.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 does such a logarithmic term appear 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.

arXiv:0901.1370Physical Review ESep 30, 2008142 citations

Site Percolation in a Square Lattice with Second- and Dilute First-Neighbor Bonds

P.M.C. de Oliveira

A two-parameter real-space renormalization-group approach to the site-percolation problem in a square lattice is proposed. The two parameters are the concentration $p$ of active first-neighbor bonds, and the concentration $q$ of occupied sites. From these two parameters one can calculate two independent critical exponents, and the others can be obtained through scaling relations, with no need of external applied fields.A two-parameter real-space renormalization-group approach to the site-percolation problem in a square lattice is proposed. The two parameters are the concentration $p$ of active first-neighbor bonds, and the concentration $q$ of occupied sites. From these two parameters one can calculate two independent critical exponents, and the others can be obtained through scaling relations, with no need of external applied fields.

Physical Review LettersNov 16, 198111 citations

Universal scaling functions for site and bond percolations on planar lattices

Chin‐Kun Hu, Chai‐Yu Lin, Jau-Ann Chen

Physica A Statistical Mechanics and its ApplicationsNov 1, 199537 citations

Renormalization Group Approach for Critical Percolation Behavior in a Two Dimensional Bond Percolating Square Lattice

Chiaki Murase

A renormalization group approach in real space is applied to the bond percolating square lattice in two dimension using two scaling transformations proposed by Reynolds et al. (Reynolds's method) and by Yuge (Yuge's method). The renormalization transformation R b ( p ), the eigenvalue λ( b ) and the correlation length critical exponent v ( b ) are calculated exactly by changing the scale of length b =4 for Reynolds's method and $b=5/\sqrt{2}$ and $6/\sqrt{2}$ for Yuge's method. The true critical exponent v is estimated v =1.341 and 1.337 by extrapolation, respectively. Discussions are made on the results.A renormalization group approach in real space is applied to the bond percolating square lattice in two dimension using two scaling transformations proposed by Reynolds et al. (Reynolds's method) and by Yuge (Yuge's method). The renormalization transformation R b ( p ), the eigenvalue λ( b ) and the correlation length critical exponent v ( b ) are calculated exactly by changing the scale of length b =4 for Reynolds's method and $b=5/\sqrt{2}$ and $6/\sqrt{2}$ for Yuge's method. The true critical exponent v is estimated v =1.341 and 1.337 by extrapolation, respectively. Discussions are made on the results.

Journal of the Physical Society of JapanApr 1, 19810 citations

A two-parameter real-space renormalization-group approach to the site-percolation problem in a square lattice is proposed. The two parameters are the concentration

p

of active first-neighbor bonds, and the concentration

q

of occupied sites. From these two parameters one can calculate two independent critical exponents, and the others can be obtained through scaling relations, with no need of external applied fields.

b=5/\sqrt{2}

and

6/\sqrt{2}

for Yuge's method. The true critical exponent v is estimated v =1.341 and 1.337 by extrapolation, respectively. Discussions are made on the results.