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Percolation

Publications

• Exact Site Percolation Probability on the Square Lattice
  J. Phys. A: Math. Theor. 55 334002 (2022)
• Percolation is Odd (with C. Moore)
  Physical Review Letters 123 230605 (2019)
• Percolation thresholds and Fisher exponents in hypercubic lattices (with C. Moore)
  Physical Review E 98 (2018) 022120
• Series expansion of the percolation threshold on hypercubic lattices (with C. Moore)
  Journal of Physics A 51 (2018) 475001
• Universal features of cluster numbers in percolation (with I. Jensen and R.M. Ziff)
  Physical Review E 96 (2017) 052119
• Percolation Thresholds in Hyperbolic Lattices (with Cris Moore)
  Physical Review E 96 (2017) 042116
• Percolation in Finite Matching Lattices (with Robert M. Ziff)
  Physical Review E 94 (2016) 062152
• Continuum Percolation Thresholds in Two Dimensions (with Cris Moore)
  Physical Review E 86 (2012) 061109

Data

Exact cluster densities

The cluster density is n_c,L(p) = N^-1 ∑_k a_k p^k (1-p)^N-k with N=L² (for site percolation) or N=2 L² (for bond percolation) and the a_k 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 R^x(p) = ∑_k a_k p^k (1-p)^N-k with N=L² and the a_k taken from the data files and x indicates the kind of wrapping event.

R^b 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

R^e 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

R^v 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).