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The basic idea of percolation is the existence of a sharp transition at which the long-range connectivity of the system disappears(or, going the other way, appears). This transition occurs abruptly when some generalized density in this system reaches a critical value(percolation threshold).
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You will investigate how the probability of a site matrix having a spanning cluster depends on the site occupation probability p. Site percolation is an example of a critical phenomenon. There is a special value of p, called the critical site occupation probability pc, such that for p < pc spanning clusters never occur. While for p > pc they always occur. The case p=pc is called a critical point.
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Critical Points
At critical points a qualitative transition in the behaviour of a system occurs: typically between ordered and disordered states. In our case the transition is from no spanning clusters to always spanning clusters.
Such qualitative changes are known as phase transitions. Examples are the freezing of water and demagnetisation of a ferromagnet at the Curie temperature. Strictly, critical points only exist for infinite systems. For finite systems, like we will investigate using the computer, the transition is not sharp but smeared out over a parameter range.
Percolation models are good physical descriptions of many phenomena, especially in condensed matter physics. Examples are discussed by R. Zallen in his book "The Physics of Amorphous Solids".
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