   Stephan Mertens - Research Home | Research | Publications | Teaching | Smorgasbord ## Percolation   ### Data

#### Exact cluster densities

The cluster density is nc,L(p) = N-1k ak pk (1-p)N-k with N=L2 (for site percolation) or N=2 L2 (for bond percolation) and the ak taken from the data files.

Exact cluster densities for site percolation on 2d lattices with periodic boundary conditions:

 triangular square nnsquare(a) unionjack hexagonal 3×3 4×4 5×5 6×6 7×7 3×3 4×4 5×5 6×6 7×7 3×3 4×4 5×5 6×6 7×7 4×4 6×6 4×4 6×6

Exact cluster densities for site percolation on 2d lattices with open boundary conditions:

 triangular square nnsquare(a) 3×3 4×4 5×5 6×6 7×7 8×8 9×9 10×10 11×11 12×12 13×13 14×14 15×15 16×16 3×3 4×4 5×5 6×6 7×7 8×8 9×9 10×10 11×11 12×12 13×13 14×14 15×15 16×16 3×3 4×4 5×5 6×6 7×7 8×8 9×9 10×10 11×11 12×12 13×13 14×14 15×15 16×16

#### Exact wrapping probabilities

Exact wrapping probabilities for site percolation on 2d lattices with periodic boundary conditions. The probabilities are Rx(p) = ∑k ak pk (1-p)N-k with N=L2 and the ak taken from the data files and x indicates the kind of wrapping event.

Rb is the probability that it wraps in both dimensions.

 triangular square nnsquare(a) unionjack hexagonal 3×3 4×4 5×5 6×6 7×7 3×3 4×4 5×5 6×6 7×7 3×3 4×4 5×5 6×6 7×7 4×4 6×6 4×4 6×6

Re is the probability that a configuration wraps in either dimension.

 triangular square nnsquare(a) unionjack hexagonal 3×3 4×4 5×5 6×6 7×7 3×3 4×4 5×5 6×6 7×7 3×3 4×4 5×5 6×6 7×7 4×4 6×6 4×4 6×6

Rv is the probability of wrapping around the vertical dimension:

 square 3×3 4×4 5×5 6×6 7×7 8×8 9×9 10×10 11×11 nnsquare(a) 3×3 4×4 5×5 6×6 7×7 8×8 9×9 10×10 11×11

#### Exact spanning probabilities

Exact spanning probabilities in 2d lattices along the second dimension (open boundary conditions).

 triangular square nnsquare(a) 3×3 4×4 5×5 6×6 7×7 8×8 9×9 10×10 11×11 3×3 4×4 5×5 6×6 7×7 8×8 9×9 10×10 11×11 3×3 4×4 5×5 6×6 7×7 8×8 9×9 10×10 11×11

(a) The nnsquare lattice is the square lattice with additional next nearest neighbor links (Moore neighborhood).