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:

“In former times and in less complex societies, children could find their way into the adult world by watching workers and perhaps giving them a hand; by lingering at the general store long enough to chat with, and overhear conversations of, adults...; by sharing and participating in the tasks of family and community that were necessary to survival. They were in, and of, the adult world while yet sensing themselves apart as children.”
—Dorothy H. Cohen (20th century)

“You treat world history as a mathematician does mathematics, in which nothing but laws and formulas exist, no reality, no good and evil, no time, no yesterday, no tomorrow, nothing but an eternal, shallow, mathematical present.”
—Hermann Hesse (1877–1962)

“‘Tis very certain that each man carries in his eye the exact indication of his rank in the immense scale of men, and we are always learning to read it. A complete man should need no auxiliaries to his personal presence.”
—Ralph Waldo Emerson (1803–1882)

“Life and language are alike sacred. Homicide and verbicide—that is, violent treatment of a word with fatal results to its legitimate meaning, which is its life—are alike forbidden.”
—Oliver Wendell Holmes, Sr. (1809–1894)