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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.
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.
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 ...
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.
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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.
Percolation threshold; R. Random cluster model; W. Water retention on random surfaces This page was last edited on 20 May 2018, at 05:31 (UTC). Text is available ...
Localized percolation refers to removing a node its neighbors, next nearest neighbors etc. until a fraction of of nodes from the network is removed. It was shown that for random graph with Poisson distribution of degrees p c = 1 k {\displaystyle p_{c}={\tfrac {1}{\langle k\rangle }}} exactly as for random removal.