Editing Percolation theory (section)

===Subcritical and supercritical===

The main fact in the subcritical phase is "exponential decay". That is, when {{math|''p'' < ''p''<sub>c</sub>}}, the probability that a specific point (for example, the origin) is contained in an open cluster (meaning a maximal connected set of "open" edges of the graph) of size {{math|''r''}} decays to zero [[Big O notation#Orders of common functions|exponentially]] in&nbsp;{{math|''r''}}. This was proved for percolation in three and more dimensions by {{harvtxt|Menshikov|1986}} and independently by {{harvtxt|Aizenman|Barsky|1987}}. In two dimensions, it formed part of Kesten's proof that {{math|''p''<sub>c</sub> {{=}} {{sfrac|1|2}}}}.<ref name="Kesten1982">{{Cite book |last1=Kesten |first1=Harry |author-link1=Harry Kesten |title=Percolation Theory for Mathematicians |publisher=Birkhauser |year=1982 |doi=10.1007/978-1-4899-2730-9 |isbn=978-0-8176-3107-9}}</ref>

The [[dual graph]] of the square lattice {{math|'''ℤ'''<sup>2</sup>}} is also the square lattice. It follows that, in two dimensions, the supercritical phase is dual to a subcritical percolation process. This provides essentially full information about the supercritical model with {{math|''d'' {{=}} 2}}. The main result for the supercritical phase in three and more dimensions is that,  for sufficiently large&nbsp;{{math|''N''}}, there is almost certainly an infinite open cluster in the two-dimensional slab {{math|'''ℤ'''<sup>2</sup> × [0, ''N'']<sup>''d'' − 2</sup>}}. This was proved by {{harvtxt|Grimmett|Marstrand|1990}}.<ref name="GrimmettMarstrand1990">{{cite journal |last1=Grimmett |first1=Geoffrey |author-link1=Geoffrey Grimmett |last2=Marstrand |first2=John |title=The Supercritical Phase of Percolation is Well Behaved |journal=Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |volume=430 |issue=1879 |year=1990 |pages=439–457 |issn=1364-5021 |doi=10.1098/rspa.1990.0100 |bibcode=1990RSPSA.430..439G |s2cid=122534964}}</ref>

In two dimensions with {{math|''p'' < {{sfrac|1|2}}}}, there is with probability one a unique infinite closed cluster (a closed cluster is a maximal connected set of "closed" edges of the graph). Thus the subcritical phase may be described as finite open islands in an infinite closed ocean. When {{math|''p'' > {{sfrac|1|2}}}} just the opposite occurs, with finite closed islands in an infinite open ocean. The picture is more complicated when {{math|''d'' ≥ 3}} since {{math|''p''<sub>c</sub> < {{sfrac|1|2}}}}, and there is coexistence of infinite open and closed clusters for {{math|''p''}} between {{math|''p''<sub>c</sub>}} and&nbsp;{{math|1 − ''p''<sub>c</sub>}}.