Percolation

In physics, chemistry and materials science, percolation (from Lat. percolare, to filter or trickle through) concerns the movement and filtering of fluids through porous materials (for more details see percolation theory). During the last five decades, percolation theory, an extensive mathematical model of percolation, has brought new understanding and techniques to a broad range of topics in physics, materials science, complex networks, epidemiology as well as in geography. In Geography percolation is filtration of water through soil and permeable rocks.the water flows to groundwater storage (aquifers) Percolation typically exhibits universality. Statistical physics concepts such as scaling theory, renormalization, phase transition, critical phenomena and fractals are useful to characterize percolation properties. Combinatorics is commonly employed to study percolation thresholds. Applications / specific examples include: coffee percolation, 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 the act of 'upwards' claiming; whereby a claimed subject who is claimed by another entity, is funneled to their claimer cracking of trees with the presence of two conditions, sunlight and under the influence of pressure Robustness of networks to random and targeted attacks Transport in porous media Epidemic spreading Surface roughening By analytical studies, only few exact results can be obtained for percolation. Hence, many results have been obtained from computer simulations. The current fastest algorithm for percolation was published in 2000 by Mark Newman and Robert Ziff. A representative question (and the source of the name) is as follows. Assume that some liquid is poured on top of some porous material. Will the liquid be able to make its way from hole to hole and reach the bottom? This physical question is modelled mathematically as a three-dimensional network of n ? n ? n vertices, usually called "sites", in which the edge or "bonds" between each two neighbors may be open (allowing the liquid through) with probability p, or closed with probability 1 Ц p, and they are

ssumed to be independent. Therefore, for a given p, what is the probability that an open path exists from the top to the bottom? The behavior for large n is of primary interest. This problem, called now bond percolation, was introduced in the mathematics literature by Broadbent & Hammersley (1957), and has been studied intensively by mathematicians and physicists since. In a slightly different mathematical model for obtaining a random graph, 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. The question is the same: for a given p, what is the probability that a path exists between top and bottom? Of course the same questions can be asked for any lattice dimension. As is quite typical, it is actually easier to examine infinite networks than just large ones. In this case the corresponding question is: does an infinite open cluster exist? That is, is there a path of connected points of infinite length "through" the network? By Kolmogorov's zero-one law, for any given p, the probability that an infinite cluster exists is either zero or one. Since this probability is an increasing function of p (proof via coupling argument), there must be a critical p (denoted by pc) below which the probability is always 0 and above which the probability is always 1. In practice, this criticality is very easy to observe. Even for n as small as 100, the probability of an open path from the top to the bottom increases sharply from very close to zero to very close to one in a short span of values of p. Detail of a bond percolation on the square lattice in two dimensions with percolation probability p = 0.51 In some cases pc may be calculated explicitly. For example, for the square lattice Z2 in two dimensions, pc = 1/2 for bond percolation, a fact which was an open question for more than 20 years and was finally resolved by Harry Kesten in the early 1980s, see Kesten (1982). A limit case for lattices in many dimensions is given by the Bethe lattice, whose threshold is at pc = 1/(z ? 1) for a coordination number z. For most infinite lattice graphs, pc cannot be calculated exactly. For example, pc is not known for bond percolation in the hypercubic lattice in two dimensions.