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In statistical physics and mathematics, percolation theory describes the behavior of a network when nodes or links are added. This is a geometric type of phase transition, since at a critical fraction of addition the network of small, disconnected clusters merge into significantly larger connected, so-called spanning clusters.
The percolation threshold is a mathematical concept in percolation theory that describes the formation ... Approximate formula for site-bond percolation on a ...
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.
There are many different effective medium approximations, [5] each of them being more or less accurate in distinct conditions. Nevertheless, they all assume that the macroscopic system is homogeneous and, typical of all mean field theories, they fail to predict the properties of a multiphase medium close to the percolation threshold due to the absence of long-range correlations or critical ...
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, .
First passage percolation is one of the most classical areas of probability theory. It was first introduced by John Hammersley and Dominic Welsh in 1965 as a model of fluid flow in a porous media. [1] It is part of percolation theory, and classical Bernoulli percolation can be viewed as a subset of first passage percolation.
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 ...
Thus the Erdős–Rényi process is the mean-field case of percolation. Some significant work was also done on percolation on random graphs. From a physicist's point of view this would still be a mean-field model, so the justification of the research is often formulated in terms of the robustness of the graph, viewed as a communication network.