Editing Percolation theory (section)

== Phases ==

===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>}}.

===Criticality===
[[File:Percolation zoom.gif|right|thumb|Zoom in a critical percolation cluster (Click to animate)]]
Percolation has a [[mathematical singularity|singularity]] at the critical point {{math|''p'' {{=}} ''p''<sub>c</sub>}} and many properties behave as of a power-law with <math>p-p_c</math>, near <math>p_c</math>. [[Critical scaling|Scaling theory]] predicts the existence of [[critical exponents]], depending on the number ''d'' of dimensions, that determine the class of the singularity. When {{math|''d'' {{=}} 2}} these predictions are backed up by arguments from [[conformal field theory]] and [[Schramm–Loewner evolution]], and include predicted numerical values for the exponents. Most of these predictions are conjectural except when the number {{math|''d''}} of dimensions satisfies either {{math|''d'' {{=}} 2}} or {{math|''d'' ≥ 6}}. They include:

* There are no infinite clusters (open or closed)
* The probability that there is an open path from some fixed point (say the origin) to a distance of {{math|''r''}} decreases ''polynomially'', i.e. is [[big O notation|on the order of]] {{math|''r''<sup>''α''</sup>}} for some&nbsp;{{math|''α''}}
** {{math|''α''}} does not depend on the particular lattice chosen, or on other local parameters. It depends only on the dimension {{math|''d''}} (this is an instance of the [[Universality (dynamical systems)|universality]] principle).
** {{math|''α<sub>d</sub>''}} decreases from {{math|''d'' {{=}} 2}} until {{math|''d'' {{=}} 6}} and then stays fixed.
** {{math|''α''<sub>2</sub> {{=}} −{{sfrac|5|48}}}}
** {{math|''α''<sub>6</sub> {{=}} −1}}.
* The shape of a large cluster in two dimensions is [[conformal map|conformally invariant]].

See {{harvtxt|Grimmett|1999}}.<ref name="Grimmett1999">{{Cite book |last1=Grimmett |first1=Geoffrey |author-link1=Geoffrey Grimmett |title=Percolation |volume=321 |year=1999 |issn=0072-7830 |doi=10.1007/978-3-662-03981-6 |series=Grundlehren der mathematischen Wissenschaften |place=Berlin |publisher=Springer |isbn=978-3-642-08442-3 |url=http://www.statslab.cam.ac.uk/~grg/papers/perc/perc.html |access-date=2009-04-18 |archive-date=2020-02-23 |archive-url=https://web.archive.org/web/20200223024219/http://www.statslab.cam.ac.uk/~grg/papers/perc/perc.html |url-status=live }}</ref> In 11 or more dimensions, these facts are largely proved using a technique known as the [[lace expansion]]. It is believed that a version of the lace expansion should be valid for 7 or more dimensions, perhaps with implications also for the threshold case of 6 dimensions.  The connection of percolation to the lace expansion is found in {{harvtxt|Hara|Slade|1990}}.<ref name="HaraSlade1990">{{cite journal |last1=Hara |first1=Takashi |last2=Slade |first2=Gordon |title=Mean-field critical behaviour for percolation in high dimensions |journal=Communications in Mathematical Physics |volume=128 |issue=2 |year=1990 |pages=333–391 |issn=0010-3616 |doi=10.1007/BF02108785 |bibcode=1990CMaPh.128..333H |s2cid=119875060 |url=http://projecteuclid.org/euclid.cmp/1104180434 |access-date=2022-10-30 |archive-date=2021-02-24 |archive-url=https://web.archive.org/web/20210224085300/https://projecteuclid.org/euclid.cmp/1104180434 |url-status=live }}</ref>

In two dimensions, the first fact ("no percolation in the critical phase") is proved for many lattices, using duality. Substantial progress has been made on two-dimensional percolation through the conjecture of [[Oded Schramm]] that the [[scaling limit]] of a large cluster may be described in terms of a [[Schramm&ndash;Loewner evolution]]. This conjecture was proved by {{harvtxt|Smirnov|2001}}<ref name="Smirnov2001">{{cite journal |last1=Smirnov |first1=Stanislav |author-link1=Stanislav Smirnov |title=Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits |journal=Comptes Rendus de l'Académie des Sciences |series=I |volume=333 |issue=3 |year=2001 |pages=239–244 |issn=0764-4442 |doi=10.1016/S0764-4442(01)01991-7 |bibcode=2001CRASM.333..239S |arxiv=0909.4499 |citeseerx=10.1.1.246.2739}}</ref> in the special case of site percolation on the triangular lattice.