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Percolation

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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 (see Physical Review E 96 (2017) 052119 for the method used):

triangular 1×1 2×2 3×3 4×4 5×5 6×6 7×7
square 1×1 2×2 3×3 4×4 5×5 6×6 7×7
nnsquare(a) 1×1 2×2 3×3 4×4 5×5 6×6 7×7
unionjack 2×2 4×4 6×6
hexagonal 2×2 4×4 6×6

Exact cluster densities for site percolation on 2d lattices with open boundary conditions (computed with transfer matrix method):

triangular 1×1 2×2 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
square 1×1 2×2 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
nnsquare(a) 1×1 2×2 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 1×1 2×2 3×3 4×4 5×5 6×6 7×7
square 1×1 2×2 3×3 4×4 5×5 6×6 7×7
nnsquare(a) 1×1 2×2 3×3 4×4 5×5 6×6 7×7
unionjack 2×2 4×4 6×6
hexagonal 2×2 4×4 6×6

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

triangular 1×1 2×2 3×3 4×4 5×5 6×6 7×7
square 1×1 2×2 3×3 4×4 5×5 6×6 7×7
nnsquare(a) 1×1 2×2 3×3 4×4 5×5 6×6 7×7
unionjack 2×2 4×4 6×6
hexagonal 2×2 4×4 6×6

Rv is the probability of wrapping around the vertical dimension:

square 1×1 2×2 3×3 4×4 5×5 6×6 7×7
nnsquare(a) 1×1 2×2 3×3 4×4 5×5 6×6 7×7

Exact spanning probabilities

Exact spanning probabilities in 2d lattices along the second dimension (open boundary conditions).
The data for the square lattice was computed in J. Phys. A: Math. Theor. 55 334002 (2022).
The data for the nnsquare lattice follows from the fact that a configuration with k occupied sites in the square lattice spans if and only if the configuration of the N-k empty sited spans in the nnsquere lattice.

triangular 1×1 2×2 3×3 4×4 5×5 6×6 7×7 8×8 9×9 10×10 11×11
square 1×1 2×2 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 17×17 18×18 19×19 20×20 21×21 22×22
nnsquare(a) 1×1 2×2 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 17×17 18×18 19×19 20×20 21×21 22×22

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

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updated on Friday, October 06th 2023, 16:07:44 CET;