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The percolation threshold is a mathematical concept in percolation theory that describes the formation of long-range connectivity in random systems. Below the threshold a giant connected component does not exist; while above it, there exists a giant component of the order of system size.
For site percolation on the square lattice, the value of p c is not known from analytic derivation but only via simulations of large lattices which provide the estimate p c = 0.59274621 ± 0.00000013. [7] A limit case for lattices in high dimensions is given by the Bethe lattice, whose threshold is at p c = 1 / z − 1 for a ...
Conductivity near the percolation threshold in physics, occurs in a mixture between a dielectric and a metallic component. The conductivity σ {\displaystyle \sigma } and the dielectric constant ϵ {\displaystyle \epsilon } of this mixture show a critical behavior if the fraction of the metallic component reaches the percolation threshold .
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.
Thus is a sharp threshold for the connectedness of G(n, p). Further properties of the graph can be described almost precisely as n tends to infinity. For example, there is a k ( n ) (approximately equal to 2log 2 ( n )) such that the largest clique in G ( n , 0.5) has almost surely either size k ( n ) or k ( n ) + 1.
Percolation is the study of connectivity in random systems, such as electrical conductivity in random conductor/insulator systems, fluid flow in porous media, gelation in polymer systems, etc. [1] At a critical fraction of connectivity or porosity, long-range connectivity can take place, leading to long-range flow.
Percolation clusters become self-similar precisely at the threshold density for sufficiently large length scales, entailing the following asymptotic power laws: . The fractal dimension relates how the mass of the incipient infinite cluster depends on the radius or another length measure, () at = and for large probe sizes, .
When p is above the critical percolation threshold p c, there will be a percolating cluster or pond that visits the entire system. The probability that a point belongs to the percolating or "infinite" cluster is written as P ∞ in percolation theory, and it is related to R 2 ( p ) by R 2 ( p )/ L 2 = p − P ∞ where L is the size of the square.