Percolation Threshold - General Formulas For Exact Results

General Formulas For Exact Results

Inhomogeneous triangular lattice bond percolation

$1 - p_1 - p_2 - p_3 + p_1 p_2 p_3 = 0$

Inhomogeneous honeycomb lattice bond percolation = kagome lattice site percolation

$1 - p_1 p_2 - p_1 p_3 - p_2 p_3+ p_1 p_2 p_3 = 0$

Inhomogeneous (3,12^2) lattice, site percolation

$1 - 3(s_1s_2)^2 + (s_1s_2)^3 = 0,$ or $s_1 s_2 = 1 - 2 \sin(\pi/18)$

Inhomogeneous martini lattice, bond percolation $1 - (p_1 p_2 r_3 + p_2 p_3 r_1 + p_1 p_3 r_2) - (p_1 p_2 r_1 r_2 + p_1 p_3 r_1 r_3 + p_2 p_3 r_2 r_3) + p_1 p_2 p_3 ( r_1 r_2 + r_1 r_3 + r_2 r_3) +$ $r_1 r_2 r_3 (p_1 p_2 + p_1 p_3 + p_2 p_3) - 2 p_1 p_2 p_3 r_1 r_2 r_3 = 0$

Inhomogeneous martini lattice, site percolation). r = site in the star

$1 - r (p_1 p_2 + p_1 p_3 + p_2 p_3 - p_1 p_2 p_3) = 0$

Inhomogeneous martini-A (3–7) lattice, bond percolation. Left side (top of "A" to bottom): . Right side: . Cross bond: .

$1 - p_1 r_2 - p_2 r_1 - p_1 p_2 r_3 - p_1 r_1 r_3 - p_2 r_2 r_3 + p_1 p_2 r_1 r_3 + p_1 p_2 r_2 r_3 + p_1 r_1 r_2 r_3+ p_2 r_1 r_2 r_3 - p_1 p_2 r_1 r_2 r_3 = 0$

Inhomogeneous martini-B (3–5) lattice, bond percolation

Inhomogeneous checkerboard lattice, bond percolation

$1 - (p_1 p_2 + p_1 p_3 + p_1 p_4 + p_2 p_3 + p_2 p_4 + p_3 p_4) + p_1 p_2 p_3 + p_1 p_2 p_4 + p_1 p_3 p_4 + p_2 p_3 p_4 = 0$

Inhomogeneous bow-tie lattice, bond percolation

$1 - (p_1 p_2 + p_1 p_3 + p_1 p_4 + p_2 p_3 + p_2 p_4 + p_3 p_4) + p_1 p_2 p_3 + p_1 p_2 p_4 + p_1 p_3 p_4 + p_2 p_3 p_4 +$ $u(1 - p_1 p_2 - p_3 p_4 + p_1 p_2 p_3 p_4) = 0$

where are the four bonds around the square and is the diagonal bond connecting the vertex between bonds and .

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