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 .

Read more about this topic: Percolation Threshold

Famous quotes containing the words general, formulas, exact and/or results:

“... women can never do efficient and general service in hospitals until their dress is prescribed by laws inexorable as those of the Medes and Persians. Then, that dress should be entirely destitute of steel, starch, whale-bone, flounces, and ornaments of all descriptions; should rest on the shoulders, have a skirt from the waist to the ankle, and a waist which leaves room for breathing.”
—Jane Grey Swisshelm (1815–1884)

“That’s the great danger of sectarian opinions, they always accept the formulas of past events as useful for the measurement of future events and they never are, if you have high standards of accuracy.”
—John Dos Passos (1896–1970)

“For 350 years we have been taught that reading maketh a full man, conference a ready man and writing an exact man. Football’s place is to add a patina of character, a deference to the rules and a respect for authority.”
—Walter Wellesley (Red)

“We do not raise our children alone.... Our children are also raised by every peer, institution, and family with which they come in contact. Yet parents today expect to be blamed for whatever results occur with their children, and they expect to do their parenting alone.”
—Richard Louv (20th century)