Editing Percolation theory (section)

==Introduction==<!-- [[Bond percolation]] and [[Site percolation]] redirect to here -->
[[File:perc-wiki.png|thumb|left|A three-dimensional site percolation graph]]
[[File:Transition de percolation 2.gif|thumb|left|Bond percolation in a square lattice from p=0.3 to p=0.52]]

A representative question (and the [[etymology|source]] of the name) is as follows. Assume that some liquid is poured on top of some [[porosity|porous]] material. Will the liquid be able to make its way from hole to hole and reach the bottom? This physical question is [[mathematical model|modelled]] mathematically as a [[Grid graph|three-dimensional network]] of {{math|''n'' × ''n'' × ''n''}} [[graph (discrete mathematics)|vertices]], usually called "sites", in which the [[graph (discrete mathematics)|edge]] or "bonds" between each two neighbors may be open (allowing the liquid through) with probability {{math|''p''}}, or closed with probability {{math|1 – ''p''}}, and they are assumed to be independent. Therefore, for a given {{math|''p''}}, what is the probability that an open path (meaning a path, each of whose links is an "open" bond) exists from the top to the bottom? The behavior for large&nbsp;{{math|''n''}} is of primary interest. This problem, called now '''bond percolation''', was introduced in the mathematics literature by {{harvtxt|Broadbent|Hammersley|1957}},<ref name="BroadbentHammersley1957">{{cite journal |last1=Broadbent |first1=Simon |last2=Hammersley |first2=John |author-link2=John Hammersley |title=Percolation processes I. Crystals and mazes |journal=Mathematical Proceedings of the Cambridge Philosophical Society |volume=53 |issue=3 |year=1957 |pages=629–641 |issn=0305-0041 |doi=10.1017/S0305004100032680 |bibcode=1957PCPS...53..629B|s2cid=84176793 }}</ref> and has been studied intensively by mathematicians and physicists since then.

In a slightly different mathematical model for obtaining a random graph, a site is "occupied" with probability {{math|''p''}} or "empty" (in which case its edges are removed) with probability {{math|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? Similarly, one can ask, given a connected graph at what fraction {{math|1 – ''p''}} of failures the graph will become disconnected (no large component).
[[File:Tube Network Percolation.gif|thumb|A 3D tube network percolation determination]]
The same questions can be asked for any lattice dimension. As is quite typical, it is actually easier to examine [[Infinite graph|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 {{math|''p''}}, the probability that an infinite cluster exists is either zero or one. Since this probability is an increasing function of {{math|''p''}} (proof via [[Coupling (probability)|coupling]] argument), there must be a '''critical''' {{math|''p''}} (denoted by&nbsp;{{math|''p''<sub>c</sub>}}) below which the probability is always 0 and above which the probability is always&nbsp;1. In practice, this criticality is very easy to observe. Even for {{math|''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&nbsp;{{math|''p''}}.

[[Image:Bond percolation p 51.png|thumb|Detail of a bond percolation on the square lattice in two dimensions with percolation probability {{math|''p'' {{=}} 0.51}}]]