Percolation

During the last decades, percolation theory, the mathematical study of percolation, has brought new understanding and techniques to a broad range of topics in physics, materials science, complex networks, epidemiology, and other fields. For example, in geology, percolation refers to filtration of water through soil and permeable rocks. The water flows to recharge the groundwater in the water table and aquifers. In places where infiltration basins or septic drain fields are planned to dispose of substantial amounts of water, a percolation test is needed beforehand to determine whether the intended structure is likely to succeed or fail. In two dimensional square lattice percolation is defined as follows. A site is "occupied" with probability p or "empty" (in which case its edges are removed) with probability 1 – p; the corresponding problem is called site percolation, see Fig. 2.

Percolation typically exhibits universality. Statistical physics concepts such as scaling theory, renormalization, phase transition, critical phenomena and fractals are used to characterize percolation properties. Combinatorics is commonly employed to study percolation thresholds.

Due to the complexity involved in obtaining exact results from analytical models of percolation, computer simulations are typically used. The current fastest algorithm for percolation was published in 2000 by Mark Newman and Robert Ziff.[1]

• Coffee percolation (see Fig. 1), where the solvent is water, the permeable substance is the coffee grounds, and the soluble constituents are the chemical compounds that give coffee its color, taste, and aroma.
• Movement of weathered material down on a slope under the earth's surface.
• Cracking of trees with the presence of two conditions, sunlight and under the influence of pressure.
• Collapse and robustness of biological virus shells to random subunit removal (experimentally-verified fragmentation of viruses).[2][3]
• Robustness of networks to random and targeted attacks.[4]
• Transport in porous media.
• Epidemic spreading.[5][6][7]
• Surface roughening.
• Dental percolation, increase rate of decay under crowns because of a conducive environment for strep mutants and lactobacillus
• Potential sites for septic systems are tested by the "perk test". Example/theory: A hole (usually 6–10 inches in diameter) is dug in the ground surface (usually 12–24" deep). Water is filled in to the hole, and the time is measured for a drop of one inch in the water surface. If the water surface quickly drops, as usually seen in poorly-graded sands, then it is a potentially good place for a septic "leach field". If the hydraulic conductivity of the site is low (usually in clayey and loamy soils), then the site is undesirable.
• Traffic percolation.[8][9]
• Percolation in the presence of reinforced (decentralization of the network) has been studied by Yuan et al.[10]

• Kesten, Harry; "What is percolation?", in Notices of the AMS, May 2006.
• Sahimi, Muhammad; Applications of Percolation Theory, Taylor & Francis, 1994. ISBN 0-7484-0075-3 (cloth), ISBN 0-7484-0076-1 (paper).
• Grimmett, Geoffrey; Percolation (2. ed). Springer Verlag, 1999.
• Stauffer, Dietrich ; and Aharony, Ammon; Introduction to Percolation Theory, Taylor & Francis, 1994, revised second edition, ISBN 9780748402533.
• Bunde, Armin; Havlin, Shlomo (editors); Fractals and Disordered Systems, Springer, 1996.
• Kirkpatrick, Scott; "Percolation and Conduction", in Reviews of Modern Physics, 45, 574, 1973.
• Ben-Avraham, Daniel; Havlin, Shlomo; Diffusion and Reactions in Fractals and Disordered Systems, Cambridge University Press, 2000.
• Rodrigues, Edouard; Remarkable properties of pawns on a hexboard
• Cohen, Reuven; and Havlin, Shlomo; Complex Networks: Structure, Robustness and Function, Cambridge University Press, 2010, ISBN 978-0521841566.
• Bollobás, Béla; Riordan, Oliver; Percolation, Cambridge University Press, 2006, ISBN 0521872324.
• Grimmett, Geoffrey; Percolation, Springer, 1999

• Introduction to Percolation Theory: short course by Shlomo Havlin