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| '''Percolation threshold''' is a mathematical term related to [[percolation theory]], which is the formation of long-range connectivity in random systems. Below the threshold a giant connected component does not exist; while above it, there exists a giant component of the order of system size. In engineering and [[coffee making]], percolation represents the flow of fluids through porous media, but in the mathematics and physics worlds it generally refers to simplified [[Lattice model (physics)|lattice models]] of random systems or networks (graphs), and the nature of the connectivity in them. | | C'est vraiment une bonne ligne de conduite à penser exactement ce qui est pris dans votre sac à main bien avant de chercher la personne à commander. Un moteur accréditives offre une utilisation importante du flux d'air pour augmenter la puissance d'aspiration massively.Bottega Veneta portefeuille Plusieurs celui d'une conception de type et de conception visent maintenant et d'autres à la question en ce moment de l'oeil humain de la marchandise ne sont pas les mêmes. Un bon nombre de type d'utilisation, habillez-vous en ne vont pas invoceline solutions de nettoyage particulières sace, Jimmy Choo Sacs néanmoins vous découvrirez un petit nombre de taches gênants qui est traitée avec des produits spécifiques. Tags: robes de demoiselle d'honneur, cérémonie coûteux gownsChoosing bas prix costumes de mariage dans votre mariage par les moyens suivants: adentyler<br><br> |
| The percolation threshold is the critical value of the occupation probability
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| ''p'', or more generally a critical surface for a group of parameters ''p''<sub>1</sub>, ''p''<sub>2</sub>, ...,
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| such that infinite connectivity (''[[percolation]]'') first occurs.
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|
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| ==Percolation models==
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| The most common percolation model is to take a regular lattice, like a square lattice, and make it into a random network by randomly "occupying" sites (vertices) or bonds (edges) with a statistically independent probability ''p''. At a critical threshold ''p<sub>c</sub>'', large clusters and long-range connectivity first appears, and this is called the '''percolation threshold'''. Depending on the method for obtaining the random network, one distinguishes between the [[site percolation]] threshold and the [[bond percolation]] threshold. More general systems have several probabilities ''p''<sub>1</sub>, ''p''<sub>2</sub>, etc., and the transition is characterized by a ''critical surface'' or ''manifold''. One can also consider continuum systems, such as overlapping disks and spheres placed randomly, or the negative space ([[Swiss cheese (mathematics)|''Swiss-cheese'']] models).
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| | |
| In the systems described so far, it has been assumed that the occupation of a site or bond is completely random—this is the so-called ''[[Bernoulli process|Bernoulli]] percolation.'' For a continuum system, random occupancy corresponds to the points being placed by a [[Poisson process]]. Further variations involve correlated percolation, such as percolation clusters related to Ising and Potts models of ferromagnets, in which the bonds are put down by the [[Fortuin-Kasteleyn method]].<ref name="KasteleynFortuin69">
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| {{cite journal
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| | last = Kasteleyn
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| | first = P. W.
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| | authorlink =
| |
| | coauthors = C. M. Fortuin
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| | title = Phase transitions in lattice systems with random local properties
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| | journal = Journal of the Physical Society of Japan (Supplements)
| |
| | volume = 26
| |
| | issue =
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| | year = 1969
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| | pages = 11–14}}
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| </ref>
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| In ''bootstrap'' or ''k-sat'' percolation, sites and/or bonds are first occupied and then successively culled from a system if a site does not have at least ''k'' neighbors. Another important model of percolation, in a different [[universality class]] altogether, is [[directed percolation]], where connectivity along a bond depends upon the direction of the flow.
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| | |
| Over the last several decades, a tremendous amount of work has gone into finding exact and approximate values of the percolation thresholds for a variety of these systems. Exact thresholds are only known for certain two-dimensional lattices that can be broken up into a self-dual array, such that under a triangle-triangle transformation, the system remains the same. Studies using numerical methods have led to numerous improvements in algorithms and several theoretical discoveries.
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| | |
| The notation such as (4,8<sup>2</sup>) comes from [[Grünbaum]] and Shepard,<ref name="Grunbaum">
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| {{cite book
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| |author=Grünbaum, Branko; and Shephard, G. C.
| |
| | title=Tilings and Patterns
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| | location=New York
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| | publisher=W. H. Freeman
| |
| | year=1987
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| | isbn=0-7167-1193-1}}
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| </ref> and indicates that around a given vertex, going in the clockwise direction, one encounters first a square and then two octagons. Besides the eleven [[Archimedean lattices]] composed of regular polygons with every site equivalent, many other more complicated lattices with sites of different classes have been studied. | |
| | |
| Error bars in the last digit or digits are shown by numbers in parentheses. Thus, 0.729724(3) signifies 0.729724 ± 0.000003, and 0.74042195(80) signifies 0.74042195 ± 0.00000080. The error bars variously represent one or two standard deviations in net error (including statistical and expected systematic error), or an empirical confidence interval.
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| | |
| == Thresholds on Archimedean lattices ==
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| | |
| [[File:Archimedean-Lattice.png|600x1500px|none|Example image caption]]
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| | |
| {{clr}}
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| This is a picture of the 11 Archimedean Lattices or uniform tilings, in which all polygons are regular and each vertex is surrounded by the same sequence of polygons. The notation (<VAR >3</VAR ><sup>4</sup>, 6) for example means that every vertex is surrounded by four triangles and one hexagon. Drawings from
| |
| .<ref name="Parviainen04">{{cite book | |
| | last = Parviainen
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| | first = Robert
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| | authorlink =
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| | coauthors =
| |
| | title = Connectivity Properties of Archimedean and Laves Lattices
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| | publisher = Uppsala Dissertations in Mathematics
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| | volume = 34
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| | year = 2005
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| | pages = 37
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| | url = http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-4251
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| | isbn = 91-506-1751-6}}
| |
| </ref> See also [[List of uniform tilings|Uniform Tilings.]] | |
| | |
| <gallery> | |
| [[File:Archimedean.jpg|Archimedean lattices|250px|left]] | |
| [[File:arctes0c.jpg|Archimedean lattices|250px|left]]
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| [[File:arctes1c.jpg|Archimedean lattices|250px|left]]
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| [[File:arctes2c.jpg|Archimedean lattices|250px|left]]
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| [[File:arctes3c.jpg|Archimedean lattices|250px|left]]
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| </gallery>
| |
| {{clr}}
| |
| {| class="wikitable"
| |
| ! Lattice
| |
| ! ''z''
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| ! <math>\overline z</math>
| |
| ! Site Percolation Threshold
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| ! Bond Percolation Threshold
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| |-
| |
| | 3-12 or (3, 12<sup>2</sup> )
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| | 3
| |
| |3
| |
| | 0.807900764... = (1 - 2 sin (π/18))<sup>1/2</sup><ref name="SudingZiff99"/>
| |
| | 0.7404207988474(7), <ref name="Jacobsen14"/><ref name="JacobsenScullard13">{{cite journal
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| | coauthors = Christian R. Scullard
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| | title = Critical manifolds, graph polynomials, and exact solvability
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| | journal = StatPhys 25, Seoul, Korea July 21–26 http://www.statphys25.org/data/Statphys25%20Abstract%20Book.pdf
| |
| | volume =
| |
| | issue =
| |
| | year = 2013
| |
| | pages = }}</ref> 0.740420800(2),<ref name="ScullardJacobsen12" /> 0.74042195(80),<ref name="Parviainen07">{{cite journal
| |
| | last = Parviainen
| |
| | first = Robert
| |
| | authorlink =
| |
| | coauthors =
| |
| | title = Estimation of bond percolation thresholds on the Archimedean lattices
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| |
| </ref>
| |
| 0.74042077(2)<ref name="DingFuGuoWu10">{{cite journal
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| | last = Ding
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| | first = Chengxiang
| |
| | authorlink =
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| | coauthors = Zhe Fu. Wenan Guo, F. Y. Wu
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| | title = Critical frontier for the Potts and percolation models on triangular-type and kagome-type lattices II: Numerical analysis
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| | year = 2010
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| | pages = 061111
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| | doi =10.1103/PhysRevE.81.061111|bibcode = 2010PhRvE..81f1111D |arxiv = 1001.1488 }}</ref>
| |
| |-
| |
| | cross (4, 6, 12)
| |
| | 3
| |
| |3
| |
| | 0.7478008(2),<ref name="Jacobsen14"/> 0.747806(4)<ref name="SudingZiff99"/>
| |
| | 0.6937314(1),<ref name="Jacobsen14"/> 0.69373383(72)<ref name="Parviainen07"/>
| |
| |-
| |
| |square octagon, bathroom tile, 4-8, [[Truncated square tiling|truncated square]]
| |
| (4, 8<sup>2</sup>)
| |
| | 3
| |
| | 3
| |
| | 0.7297233(5),<ref name="Jacobsen14"/> 0.729724(3)<ref name="SudingZiff99"/>
| |
| | 0.676803124(1), <ref name="Jacobsen14"/> 0.67680232(63)<ref name="Parviainen07"/>
| |
| |-
| |
| | honeycomb (6<sup>3</sup>)
| |
| | 3
| |
| | 3
| |
| | 0.6962(6), <ref name="DjordjevicStanleyMargolina82"> {{cite journal
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| | last = Djordjevic
| |
| | first = Z. V.
| |
| | authorlink =
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| | coauthors = H. E. Stanley, Alla Margolina
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| | title =Site percolation threshold for honeycomb and square lattices
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| | volume = 15
| |
| | issue =
| |
| | year = 1982
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| | pages = L405-L412
| |
| | doi = 10.1088/0305-4470/15/8/006 }}
| |
| </ref> 0.6970413(10),<ref name="ZiffGu08"/> 0.697043(3),<ref name="SudingZiff99">{{cite journal
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| | last = Suding
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| | first = P. N.
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| |
| </ref>
| |
| | 0.652703645... = 1-2 sin (π/18), 1+ ''p''<sup>3</sup>-3''p''<sup>2</sup>=0<ref name="SykesEssam">{{cite journal
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| | last = Sykes
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| | first = M. F.
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| | title = Exact critical percolation probabilities for site and bond problems in two dimensions
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| | year = 1964
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| | pages = 1117–1127
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| | doi = 10.1063/1.1704215|bibcode = 1964JMP.....5.1117S }}
| |
| </ref>
| |
| |-
| |
| |[[Kagome lattice|kagome]] (3, 6, 3, 6)
| |
| | 4
| |
| |4
| |
| | 0.652703645... = 1 - 2 sin(π/18)<ref name="SykesEssam"/>
| |
| | 0.524404978(5),<ref name="DingFuGuoWu10"/> 0.52440499(2),<ref name="FengDengBlote08">{{cite journal
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| | last = Feng
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| | first = Xiaomei
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| | title = Percolation transitions in two dimensions
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| 0.52440572...,<ref name="Scullard12">{{cite journal
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| | doi = 10.1088/0305-4470/30/15/021|arxiv = cond-mat/9707110 |bibcode = 1997JPhA...30.5351Z }}</ref>
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| 0.524404999134(2) <ref name="Jacobsen14"/><ref name="JacobsenScullard13"/>
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| |-
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| |ruby<ref name="LinMa83">{{cite journal
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| | last = Lin
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| |bibcode = 1983JPhA...16.3895L }}
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| </ref> (3, 4, 6, 4)
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| | 4
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| |4
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| | 0.621819(3)<ref name="SudingZiff99"/>
| |
| | 0.5248311(1),<ref name="Jacobsen14"/> 0.52483258(53)<ref name="Parviainen07"/>
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| |-
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| | square (4<sup>4</sup>)
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| </ref> 0.59274605(3)<ref name="FengDengBlote08"/>
| |
| | 1/2
| |
| |-
| |
| |[[snub hexagonal tiling|snub hexagonal]], maple leaf <ref name="Betts95">{{cite journal
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| |
| | doi =}}</ref> (3<sup>4</sup>,6 )
| |
| | 5
| |
| | 5
| |
| | 0.579498(3)<ref name="SudingZiff99"/>
| |
| | 0.43432764(3), <ref name="Jacobsen14"/> 0.43430621(50)<ref name="Parviainen07"/>
| |
| |-
| |
| |[[snub square tiling|snub square]], puzzle (3<sup>2</sup>, 4, 3, 4 )
| |
| | 5
| |
| | 5
| |
| | 0.550806(3)<ref name="SudingZiff99"/>
| |
| | 0.4141378476 (7),<ref name="Jacobsen14"/> 0.41413743(46)<ref name="Parviainen07"/>
| |
| |-
| |
| |(3<sup>3</sup>, 4<sup>2</sup>)
| |
| | 5
| |
| |5
| |
| | 0.550213(3)<ref name="SudingZiff99"/>
| |
| | 0.41964044(1),<ref name="Jacobsen14"/> 0.41964191(43)<ref name="Parviainen07"/>
| |
| |-
| |
| | triangular (3<sup>6</sup>)
| |
| | 6
| |
| |6
| |
| | 1/2
| |
| | 0.347296355... = 2 sin (π/18), 1+ ''p''<sup>3</sup>-3''p''=0<ref name="SykesEssam"/>
| |
| |-
| |
| |}
| |
| | |
| Note: sometimes "hexagonal" is used in place of honeycomb, although
| |
| in some fields, a triangular
| |
| lattice is also called a [[hexagonal lattice]]). ''z'' = bulk [[coordination number]].
| |
| | |
| == Square lattice with complex neighborhoods ==
| |
| | |
| {| class="wikitable"
| |
| ! Lattice
| |
| ! z
| |
| ! Site Percolation Threshold
| |
| ! Bond Percolation Threshold
| |
| |-
| |
| |square: 3N, 4N, 6N
| |
| | 4
| |
| | 0.592...<ref name="Malarz2005">{{cite journal
| |
| | last = Malarz
| |
| | first = K.
| |
| | authorlink =
| |
| | coauthors = S. Galam
| |
| | title = Square-lattice site percolation at increasing ranges of neighbor bonds
| |
| | journal = Physical Review E
| |
| | volume = 71
| |
| | issue = 1
| |
| | year = 2005
| |
| | pages = 016125
| |
| | doi = 10.1103/PhysRevE.71.016125
| |
| | arxiv=cond-mat/0408338
| |
| |bibcode = 2005PhRvE..71a6125M }}
| |
| </ref><ref name="Majewski2007">{{cite journal
| |
| | last = Majewski
| |
| | first = M.
| |
| | authorlink =
| |
| | coauthors = K. Malarz
| |
| | title = Square lattice site percolation thresholds for complex neighbourhoods
| |
| | journal = Acta Phys. Pol. B
| |
| | volume = 38
| |
| | issue = 38
| |
| | year = 2007
| |
| | pages = 2191
| |
| | arxiv=cond-mat/0609635
| |
| |bibcode = 2007AcPPB..38.2191M }}
| |
| </ref>
| |
| |
| |
| |-
| |
| |square: 3N+2N, 4N+3N, 6N+4N
| |
| | 8
| |
| | 0.407...<ref name="Malarz2005"/><ref name="Majewski2007"/><ref>{{cite web|last=Collier|first=Andrew|title=Percolation Threshold: Including Next-Nearest Neighbours|url=http://www.exegetic.biz/blog/2013/11/percolation-threshold-including-next-nearest-neighbours/}}</ref>
| |
| |
| |
| |-
| |
| |square: 4N+2N
| |
| | 8
| |
| | 0.337...<ref name="Malarz2005"/><ref name="Majewski2007"/>
| |
| |
| |
| |-
| |
| |square: 6N+3N
| |
| | 8
| |
| | 0.337...<ref name="Majewski2007"/>
| |
| |
| |
| |-
| |
| |square: 5N
| |
| | 8
| |
| | 0.270...<ref name="Majewski2007"/>
| |
| |
| |
| |-
| |
| |square: 6N+2N
| |
| | 8
| |
| | 0.277...<ref name="Majewski2007"/>
| |
| |
| |
| |-
| |
| |square: 4N+3N+2N
| |
| | 12
| |
| | 0.288...<ref name="Malarz2005"/><ref name="Majewski2007"/>
| |
| |
| |
| |-
| |
| |square: 6N+4N+3N
| |
| | 12
| |
| | 0.288...<ref name="Majewski2007"/>
| |
| |
| |
| |-
| |
| |square: 5N+2N
| |
| | 12
| |
| | 0.236...<ref name="Majewski2007"/>
| |
| |
| |
| |-
| |
| |square: 5N+3N
| |
| | 12
| |
| | 0.225...<ref name="Majewski2007"/>
| |
| |
| |
| |-
| |
| |square: 5N+4N
| |
| | 12
| |
| | 0.221...<ref name="Majewski2007"/>
| |
| |
| |
| |-
| |
| |square: 6N+3N+2N
| |
| | 12
| |
| | 0.240...<ref name="Majewski2007"/>
| |
| |
| |
| |-
| |
| |square: 6N+4N+2N
| |
| | 12
| |
| | 0.233...<ref name="Majewski2007"/>
| |
| |
| |
| |-
| |
| |square: 6N+5N
| |
| | 12
| |
| | 0.199...<ref name="Majewski2007"/>
| |
| |
| |
| |-
| |
| |square: 5N+3N+2N
| |
| | 16
| |
| | 0.219...<ref name="Majewski2007"/>
| |
| |
| |
| |-
| |
| |square: 5N+4N+2N
| |
| | 16
| |
| | 0.208...<ref name="Majewski2007"/>
| |
| |
| |
| |-
| |
| |square: 5N+4N+3N
| |
| | 16
| |
| | 0.202...<ref name="Majewski2007"/>
| |
| |
| |
| |-
| |
| |square: 6N+5N+2N
| |
| | 16
| |
| | 0.187...<ref name="Majewski2007"/>
| |
| |
| |
| |-
| |
| |square: 6N+5N+3N
| |
| | 16
| |
| | 0.182...<ref name="Majewski2007"/>
| |
| |
| |
| |-
| |
| |square: 6N+5N+4N
| |
| | 16
| |
| | 0.179...<ref name="Majewski2007"/>
| |
| |
| |
| |-
| |
| |square: 6N+4N+3N+2N
| |
| | 16
| |
| | 0.208...<ref name="Majewski2007"/>
| |
| |
| |
| |-
| |
| |square: 5N+4N+3N+2N
| |
| | 20
| |
| | 0.196...<ref name="Majewski2007"/>
| |
| |
| |
| |-
| |
| |square: 6N+5N+3N+2N
| |
| | 20
| |
| | 0.177...<ref name="Majewski2007"/>
| |
| |
| |
| |-
| |
| |square: 6N+5N+4N+2N
| |
| | 20
| |
| | 0.172...<ref name="Majewski2007"/>
| |
| |
| |
| |-
| |
| |square: 6N+5N+4N+3N
| |
| | 20
| |
| | 0.167...<ref name="Majewski2007"/>
| |
| |
| |
| |-
| |
| |square: 6N+5N+4N+3N+2N
| |
| | 24
| |
| | 0.164...<ref name="Majewski2007"/>
| |
| |
| |
| |-
| |
| |}
| |
| 2N = nearest neighbours, 3N = next-nearest neighbours, 4N = next-next-nearest neighbours, 5N = next-next-next-nearest neighbours, etc.
| |
| | |
| == Approximate formulas for thresholds of Archimedean lattices ==
| |
| {| class="wikitable"
| |
| ! Lattice
| |
| ! z
| |
| ! Site Percolation Threshold
| |
| ! Bond Percolation Threshold
| |
| |-
| |
| |(3, 12<sup>2</sup> )
| |
| | 3
| |
| |
| |
| |
| |
| |-
| |
| |(4, 6, 12)
| |
| | 3
| |
| |
| |
| |
| |
| |-
| |
| |(4, 8<sup>2</sup>)
| |
| | 3
| |
| |
| |
| | 0.676835..., 4''p''<sup>3</sup> + 3''p''<sup>4</sup> - 6 ''p''<sup>5</sup>- 2 ''p''<sup>6</sup> = 1 <ref name="ScullardZiff10">{{cite journal
| |
| | last = Scullard
| |
| | first = C. R.
| |
| | authorlink =
| |
| | coauthors = R. M. Ziff
| |
| | title = Critical surfaces for general inhomogeneous bond percolation problems
| |
| | journal = J. Stat. Mech: Th. Exp
| |
| | volume = 2010
| |
| | issue = 3
| |
| | year = 2010
| |
| | pages = P03021
| |
| | doi = 10.1088/1742-5468/2010/03/P03021 |bibcode = 2010JSMTE..03..021S |arxiv = 0911.2686 }}
| |
| </ref>
| |
| |-
| |
| | honeycomb (6<sup>3</sup>)
| |
| | 3
| |
| |
| |
| |
| |
| |-
| |
| |[[kagome lattice|kagome]] (3, 6, 3, 6)
| |
| | 4
| |
| |
| |
| | 0.524430..., 3''p''<sup>2</sup> + 6''p''<sup>3</sup> - 12 ''p''<sup>4</sup>+ 6 ''p''<sup>5</sup> - ''p''<sup>6</sup> = 1 <ref name="Wu79"/>
| |
| |-
| |
| |(3, 4, 6, 4)
| |
| | 4
| |
| |
| |
| |
| |
| |-
| |
| | square (4<sup>4</sup>)
| |
| | 4
| |
| |
| |
| | 1/2 (exact)
| |
| |-
| |
| |(3<sup>4</sup>,6 )
| |
| | 5
| |
| |
| |
| | 0.434371..., 12''p''<sup>3</sup> + 36''p''<sup>4</sup> -21 ''p''<sup>5</sup>- 327 ''p''<sup>6</sup> + 69''p''<sup>7</sup> + 2532''p''<sup>8</sup> - 6533 ''p''<sup>9</sup>
| |
| + 8256 ''p''<sup>10</sup> - 6255''p''<sup>11</sup> + 2951''p''<sup>12</sup> - 837 ''p''<sup>13</sup>+ 126 ''p''<sup>14</sup> - 7''p''<sup>15</sup>= 1 <ref name="Scullard10">
| |
| {{cite journal
| |
| | last = Scullard
| |
| | first = C. R.
| |
| | authorlink =
| |
| | coauthors =
| |
| | title =
| |
| | journal = To be published
| |
| | volume =
| |
| | issue =
| |
| | year = 2010
| |
| | pages =
| |
| | doi = }}
| |
| </ref>
| |
| |-
| |
| | snub square, puzzle (3<sup>2</sup>, 4, 3, 4 )
| |
| | 5
| |
| |
| |
| |
| |
| |-
| |
| |(3<sup>3</sup>, 4<sup>2</sup>)
| |
| | 5
| |
| |
| |
| |
| |
| |-
| |
| | triangular (3<sup>6</sup>)
| |
| | 6
| |
| | 1/2 (exact)
| |
| |
| |
| |-
| |
| |}
| |
| | |
| == Formulas for site-bond percolation ==
| |
| {| class="wikitable"
| |
| ! Lattice
| |
| ! z
| |
| ! <math> \overline z </math>
| |
| ! Threshold
| |
| ! Notes
| |
| |-
| |
| |(6<sup>3</sup>) honeycomb
| |
| | 3
| |
| | 3
| |
| | <math> b s [1 - (\sqrt{t}/(3-t))(\sqrt{b} - \sqrt{t})] = t </math>,
| |
| when equal: b = s = 0.82199
| |
| | approximate formula, s = site prob., b = bond prob., t = 1 - 2 sin (π/18) <ref name = "ZiffGu08"/>
| |
| |-
| |
| |}
| |
| | |
| == Archimedean Duals (Laves Lattices) ==
| |
| [[File:Dual Archimedean.png|600x1500px|none|Example image caption]]
| |
| Laves lattices are the duals to the Archimedean lattices. Drawings from.<ref name="Parviainen04"/> See also [[List of uniform tilings|Uniform Tilings]].
| |
| | |
| {|class="wikitable"
| |
| ! Lattice
| |
| ! z
| |
| ! <math>\overline z</math>
| |
| ! Site Percolation Threshold
| |
| ! Bond Percolation Threshold
| |
| |-
| |
| | Cairo pentagonal
| |
| D(3<sup>2</sup>,4,3,4)=(2/3)(5<sup>3</sup>)+(1/3)(5<sup>4</sup>)
| |
| |3,4
| |
| |3⅓
| |
| |0.650184<ref name="Parviainen04"/>
| |
| |p<sub>c</sub><sup>bond</sup>=1-p<sub>c</sub><sup>bond</sup>(3<sup>2</sup>,4,3,4)
| |
| 0.58586256(54)<ref name="Parviainen07"/>
| |
| |-
| |
| |D(3<sup>3</sup>,4<sup>2</sup>)=(1/3)(5<sup>4</sup>)+(2/3)(5<sup>3</sup>)
| |
| |3,4
| |
| |3⅓
| |
| |0.647084<ref name="Parviainen04"/>
| |
| |p<sub>c</sub><sup>bond</sup>=1-P<sub>c</sub><sup>bond</sup>(3<sup>3</sup>,4<sup>2</sup>)
| |
| 0.58035808(57)<ref name="Parviainen07"/>
| |
| |-
| |
| |D(3<sup>4</sup>,6)=(1/5)(4<sup>6</sup>)+(4/5)(4<sup>3</sup>)
| |
| |3,6
| |
| |3 3/5
| |
| |0.639447<ref name="Parviainen04"/>
| |
| |p<sub>c</sub><sup>bond</sup>=1-p<sub>c</sub><sup>bond</sup>(3<sup>4</sup>,6 )
| |
| 0.56569378(50)<ref name="Parviainen07"/>
| |
| |-
| |
| |[[Rhombille tiling|dice, rhombille tiling]]
| |
| D(3,6,3,6)=(1/3)(4<sup>6</sup>)+(2/3)(4<sup>3</sup>)
| |
| |3,6
| |
| |4
| |
| |0.5851(4),<ref name="SakamotoYonezawaHori89">{{cite journal
| |
| | last = Sakamoto
| |
| | first = S.
| |
| | authorlink =
| |
| | coauthors = F. Yonezawa and M. Hori
| |
| | title = A proposal for the estimation of percolation thresholds in two-dimensional lattices
| |
| | journal = J. Phys. A
| |
| | volume = 22
| |
| | issue = 14
| |
| | pages = L699–L704
| |
| | doi = 10.1088/0305-4470/22/14/009
| |
| | year = 1989|bibcode = 1989JPhA...22L.699S }}
| |
| </ref> 0.585040<ref name="Parviainen04"/>
| |
| |p<sub>c</sub><sup>bond</sup>=1-p<sub>c</sub><sup>bond</sup></sub>(3,6,3,6 )
| |
| 0.475595021(5),<ref name="DingFuGuoWu10"/> 0.47559500(8),<ref name="FengDengBlote08"/> 0.47559483(90),<ref name="ZiffGu08"/> 0.475594(7)<ref name="ZiffSuding97"/>
| |
| |-
| |
| | ruby dual
| |
| D(3,4,6,4)=(1/6)(4<sup>6</sup>)+(2/6)(4<sup>3</sup>)+(3/6)(4<sup>4</sup>)
| |
| |3,4,6
| |
| |4
| |
| |0.582410<ref name="Parviainen04"/>
| |
| |p<sub>c</sub><sup>bond</sup>=1-p<sub>c</sub><sup>bond</sup>(3,4,6,4 )
| |
| 0.47516741(47)<ref name="Parviainen07"/>
| |
| |-
| |
| | bisected hexagon,<ref name="DengEtAl11">{{cite journal
| |
| | last = Deng
| |
| | first = Y.
| |
| | authorlink =
| |
| | coauthors = Y. Huang, J. L. Jacobsen, J. Salas, and A. D. Sokal,
| |
| | title = Finite-temperature phase transition in a class of four-state Potts
| |
| antiferromagnets
| |
| | journal = Physical Review Letters
| |
| | volume = 107
| |
| | issue =
| |
| | year = 2006
| |
| | pages = 150601
| |
| | doi = 10.1103/PhysRevLett.107.150601|bibcode = 2011PhRvL.107o0601D |arxiv = 1108.1743 }}
| |
| </ref> cross dual
| |
| D(4,6,12)= (1/6)(3<sup>12</sup>)+(2/6)(3<sup>6</sup>)+(1/2)(3<sup>4</sup>)
| |
| |4,6,12
| |
| |6
| |
| |1/2
| |
| |p<sub>c</sub><sup>bond</sup>=1-p<sub>c</sub><sup>bond</sup>(4,6,12)
| |
| 0.30626616(28)<ref name="Parviainen07"/>
| |
| |-
| |
| |asanoha (hemp leaf)<ref name="Syozi82">{{Cite book
| |
| | first = I
| |
| | last = Syozi
| |
| | author-link =
| |
| | first2 =
| |
| | last2 =
| |
| | author2-link =
| |
| | editor-last = Domb
| |
| | editor-first = C
| |
| | editor2-last = Green
| |
| | editor2-first = M. S.
| |
| | contribution =
| |
| | contribution-url =
| |
| | series = Phase Transitions in Critical Phenomena
| |
| | year = 1972
| |
| | pages = 270–329
| |
| | place =
| |
| | publisher = Academic Press, London
| |
| | url =
| |
| | doi =
| |
| | id =
| |
| | postscript = <!--None--> }}
| |
| </ref>
| |
| D(3, 12<sup>2</sup>)=(2/3)(3<sup>3</sup>)+(1/3)(3<sup>12</sup>)
| |
| | 3,12
| |
| | 6
| |
| |1/2
| |
| | p<sub>c</sub><sup>bond</sup>=1-p<sub>c</sub><sup>bond</sup>(3, 12<sup>2</sup>) = 0.25957804(20),<ref name="Parviainen07"/> 0.25957918(90),<ref name="ZiffGu08"/> 0.25957922(8)<ref name="DingFuGuoWu10"/>
| |
| |-
| |
| |[[Tetrakis square tiling|union jack, tetrakis square tiling]]
| |
| D(4,8<sup>2</sup> )=(1/2)(3<sup>4</sup>)+(1/2)(3<sup>8</sup>)
| |
| |4,8
| |
| |6
| |
| |1/2
| |
| |p<sub>c</sub><sup>bond</sup>=1-p<sub>c</sub><sup>bond</sup>(4,8<sup>2</sup> )
| |
| 0.23219767(37)<ref name="Parviainen07"/>
| |
| |}
| |
| | |
| Site bond percolation (both thresholds apply simultaneously to one system).
| |
| | |
| {|class="wikitable"
| |
| ! Lattice
| |
| ! z
| |
| ! <math>\overline z</math>
| |
| ! Site Percolation Threshold
| |
| ! Bond Percolation Threshold
| |
| |-
| |
| | square
| |
| | 4
| |
| | 4
| |
| | 0.615185(15)<ref name="HoviAharony96">{{cite journal
| |
| | last = Hovi
| |
| | first = J.-P.
| |
| | authorlink =
| |
| | coauthors = A. Aharony
| |
| | title = Scaling and universality in the spanning probability for percolation
| |
| | journal = Physical Review E
| |
| | volume = 53
| |
| | issue = 1
| |
| | pages = 235–253
| |
| | doi = 10.1103/PhysRevE.53.235
| |
| | year = 1996|bibcode = 1996PhRvE..53..235H }}
| |
| </ref>
| |
| | 0.95
| |
| |-
| |
| |
| |
| |
| |
| |
| |
| | 0.667280(15)<ref name="HoviAharony96"/>
| |
| | 0.85
| |
| |-
| |
| |
| |
| |
| |
| |
| |
| |0.732100(15)<ref name="HoviAharony96"/>
| |
| | 0.75
| |
| |-
| |
| |
| |
| |
| |
| |
| |
| |0.75
| |
| | 0.726195(15)<ref name="HoviAharony96"/>
| |
| |-
| |
| |
| |
| |
| |
| |
| |
| | 0.815560(15)<ref name="HoviAharony96"/>
| |
| | 0.65
| |
| |-
| |
| |
| |
| |
| |
| |
| |
| | 0.85
| |
| | 0.615810(30)<ref name="HoviAharony96"/>
| |
| |-
| |
| |
| |
| |
| |
| |
| |
| |0.95
| |
| |0.533620(15)<ref name="HoviAharony96"/>
| |
| |}
| |
| <nowiki>*</nowiki> For more values, see [http://arxiv.org/pdf/cond-mat/9906078v1.pdf An Investigation of site-bond percolation]
| |
| | |
| == 2-Uniform Lattices ==
| |
| Top 3 Lattices: #13 #12 #36
| |
| <br />
| |
| Bottom 3 Lattices: #34 #37 #11
| |
| {{clr}}
| |
| [[File:2uni4m1.gif|20 2 uniform lattices|400px|left]]
| |
| <ref name="Grunbaum"/>
| |
| {{clr}}
| |
| Top 2 Lattices: #35 #30
| |
| <br />
| |
| Bottom 2 Lattices: #41 #42
| |
| {{clr}}
| |
| [[File:2uni4m2.gif|20 2 uniform lattices|400px|left]]
| |
| <ref name="Grunbaum"/>
| |
| {{clr}}
| |
| Top 4 Lattices: #22 #23 #21 #20
| |
| <br />
| |
| Bottom 3 Lattices: #16 #17 #15
| |
| [[File:2uni4m3.gif|20 2 uniform lattices|400px|left]]
| |
| <ref name="Grunbaum"/>
| |
| {{clr}}
| |
| Top 2 Lattices: #31 #32
| |
| <br />
| |
| Bottom Lattice: #33
| |
| [[File:2uni4m4.gif|20 2 uniform lattices|400px|left]]
| |
| <ref name="Grunbaum"/>
| |
| {{clr}}
| |
| | |
| {| class="wikitable"
| |
| ! #
| |
| ! Lattice
| |
| ! z
| |
| ! <math>\overline z</math>
| |
| ! Site Percolation Threshold
| |
| ! Bond Percolation Threshold
| |
| |-
| |
| | 41
| |
| |(1/2)(3,4,3,12) + (1/2)(3, 12<sup>2</sup>)
| |
| | 4,3
| |
| |
| |
| | 0.7680(2)<ref name="NeherEtAl08">{{cite journal
| |
| | last = Neher
| |
| | first = Richard
| |
| | coauthors = Mecke, Klaus and Wagner, Herbert
| |
| | title = Topological estimation of percolation thresholds
| |
| | journal = Journal of Statistical Mechanics: Theory and Experiment
| |
| | pages = P01011
| |
| | doi = 10.1088/1742-5468/2008/01/P01011
| |
| | year = 2008
| |
| | volume = 2008|bibcode = 2008JSMTE..01..011N |arxiv = 0708.3250 }}
| |
| </ref>
| |
| | 0.67493252(36)<ref name="Sirius"/>
| |
| |-
| |
| | 42
| |
| |(1/3)(3,4,6,4) + (2/3)(4,6,12)
| |
| | 4,3
| |
| |
| |
| | 0.7157(2)<ref name="NeherEtAl08"/>
| |
| | 0.64536587(40)<ref name="Sirius"/>
| |
| |-
| |
| | 36
| |
| |(1/7)(3<sup>6</sup>) + (6/7)(3<sup>2</sup>,4,12)
| |
| | 6,4
| |
| |
| |
| | 0.6808(2)<ref name="NeherEtAl08"/>
| |
| | 0.55778329(40)<ref name="Sirius"/>
| |
| |-
| |
| | 15
| |
| |(2/3)(3<sup>2</sup>,6<sup>2</sup>) + (1/3)(3,6,3,6)
| |
| | 4,4
| |
| |
| |
| | 0.6499(2)<ref name="NeherEtAl08"/>
| |
| | 0.53632487(40)<ref name="Sirius"/>
| |
| |-
| |
| | 34
| |
| |(1/7)(3<sup>6</sup>) + (6/7)(3<sup>2</sup>,6<sup>2</sup>)
| |
| | 6,3
| |
| |
| |
| | 0.6329(2)<ref name="NeherEtAl08"/>
| |
| | 0.51707873(70)<ref name="Sirius"/>
| |
| |-
| |
| | 16
| |
| |(4/5)(3,4<sup>2</sup>,6) + (1/5)(3,6,3,6)
| |
| | 4,4
| |
| |
| |
| | 0.6286(2)<ref name="NeherEtAl08"/>
| |
| | 0.51891529(35)<ref name="Sirius"/>
| |
| |-
| |
| | 17
| |
| |(4/5)(3,4<sup>2</sup>,6) + (1/5)(3,6,3,6)*
| |
| | 4,4
| |
| |
| |
| | 0.6279(2)<ref name="NeherEtAl08"/>
| |
| | 0.51769462(35)<ref name="Sirius"/>
| |
| |-
| |
| | 35
| |
| |(2/3)(3,4<sup>2</sup>,6) + (1/3)(3,4,6,4)
| |
| | 4,4
| |
| |
| |
| | 0.6221(2)<ref name="NeherEtAl08"/>
| |
| | 0.51973831(40)<ref name="Sirius"/>
| |
| |-
| |
| | 11
| |
| |(1/2)(3<sup>4</sup>,6) + (1/2)(3<sup>2</sup>,6<sup>2</sup>)
| |
| | 5,4
| |
| |
| |
| | 0.6171(2)<ref name="NeherEtAl08"/>
| |
| | 0.48921280(37)<ref name="Sirius"/>
| |
| |-
| |
| | 37
| |
| |(1/2)(3<sup>3</sup>,4<sup>2</sup>) + (1/2)(3,4,6,4)
| |
| | 5,4
| |
| |
| |
| | 0.5885(2)<ref name="NeherEtAl08"/>
| |
| | 0.47229486(38)<ref name="Sirius"/>
| |
| |-
| |
| | 30
| |
| |(1/2)(3<sup>2</sup>,4,3,4) + (1/2)(3,4,6,4)
| |
| | 5,4
| |
| |
| |
| | 0.5883(2)<ref name="NeherEtAl08"/>
| |
| | 0.46573078(72)<ref name="Sirius"/>
| |
| |-
| |
| | 23
| |
| |(1/2)(3<sup>3</sup>,4<sup>2</sup>) + (1/2)(4<sup>4</sup>)
| |
| | 5,4
| |
| |
| |
| | 0.5720(2)<ref name="NeherEtAl08"/>
| |
| | 0.45844622(40)<ref name="Sirius">{{cite journal
| |
| | last = Gu
| |
| | first = Hang
| |
| | coauthors = R. M. Ziff
| |
| | title = Percolation thresholds of 2-uniform lattice
| |
| | journal = To be published
| |
| | date =
| |
| | year = 2007}}
| |
| </ref>
| |
| |-
| |
| | 22
| |
| |(2/3)(3<sup>3</sup>,4<sup>2</sup>) + (1/3)(4<sup>4</sup>)
| |
| | 5,4
| |
| |
| |
| | 0.5648(2)<ref name="NeherEtAl08"/>
| |
| | 0.44528611(40)<ref name="Sirius"/>
| |
| |-
| |
| | 12
| |
| |(1/4)(3<sup>6</sup>) + (3/4)(3<sup>4</sup>,6)
| |
| | 6,5
| |
| |
| |
| | 0.5607(2) <ref name="NeherEtAl08"/>
| |
| | 0.41109890(37) <ref name="Sirius"/>
| |
| |-
| |
| | 33
| |
| |(1/2)(3<sup>3</sup>,4<sup>2</sup>) + (1/2)(3<sup>2</sup>,4,3,4)
| |
| | 5,5
| |
| |
| |
| | 0.5505(2) <ref name="NeherEtAl08"/>
| |
| | 0.41628021(35) <ref name="Sirius"/>
| |
| |-
| |
| | 32
| |
| |(1/3)(3<sup>3</sup>,4<sup>2</sup>) + (2/3)(3<sup>2</sup>,4,3,4)
| |
| | 5,5
| |
| |
| |
| | 0.5504(2) <ref name="NeherEtAl08"/>
| |
| | 0.41549285(36) <ref name="Sirius"/>
| |
| |-
| |
| | 31
| |
| |(1/7)(3<sup>6</sup>) + (6/7)(3<sup>2</sup>,4,3,4)
| |
| | 6,5
| |
| |
| |
| | 0.5440(2) <ref name="NeherEtAl08"/>
| |
| | 0.40379585(40) <ref name="Sirius"/>
| |
| |-
| |
| | 13
| |
| |(1/2)(3<sup>6</sup>) + (1/2)(3<sup>4</sup>,6)
| |
| | 6,5
| |
| |
| |
| | 0.5407(2) <ref name="NeherEtAl08"/>
| |
| | 0.38914898(35) <ref name="Sirius"/>
| |
| |-
| |
| | 21
| |
| |(1/3)(3<sup>6</sup>) + (2/3)(3<sup>3</sup>,4<sup>2</sup>)
| |
| | 6,5
| |
| |
| |
| | 0.5342(2) <ref name="NeherEtAl08"/>
| |
| | 0.39491996(40) <ref name="Sirius"/>
| |
| |-
| |
| | 20
| |
| |(1/2)(3<sup>6</sup>) + (1/2)(3<sup>3</sup>,4<sup>2</sup>)
| |
| | 6,5
| |
| |
| |
| | 0.5258(2) <ref name="NeherEtAl08"/>
| |
| | 0.38285085(38) <ref name="Sirius"/>
| |
| |-
| |
| |}
| |
| | |
| == Inhomogeneous 2-Uniform Lattice ==
| |
| | |
| [[File:2uniformLattice37.pdf|2uniformLattice37]]
| |
| | |
| This figure shows the 2-uniform lattice #37 in the isoradial representation in which each polygon is inscribed in a circle of unit radius.
| |
| The squares in the 2-uniform lattice must now be represented as rectangles in order to satisfy the isoradial condition.
| |
| The lattice is shown by black edges, and the dual lattice by red dashed lines. The green circles show the isoradial constraint on both the
| |
| original and dual lattices. The yellow polygons highlight the three types of polygons on the lattice, and the pink polygons highlight the two
| |
| types of polygons on the dual lattice. The lattice has vertex types (1/2)(3<sup>3</sup>,4<sup>2</sup>) + (1/2)(3,4,6,4), while
| |
| the dual lattice has vertex types (1/15)(4<sup>6</sup>)+(6/15)(4<sup>2</sup>,5<sup>2</sup>)+(2/15)(5<sup>3</sup>)+(6/15)(5<sup>2</sup>,4). The critical point is where the longer
| |
| bonds (on both the lattice and dual lattice) have occupation probability p = 2 sin (π/18) = 0.347296... which is the bond percolation threshold on a triangular lattice, and the shorter bonds have
| |
| occupation probability 1 - 2 sin(π/18) = 0.652703..., which is the bond percolation on a hexagonal lattice. These results follow from the isoradial condition
| |
| <ref name="GrimmettManolescu12">{{cite journal
| |
| | last = Grimmett
| |
| | first = G.
| |
| | authorlink =
| |
| | coauthors = Manolescu, I
| |
| | title = Bond percolation on isoradial graphs
| |
| | journal =
| |
| | volume =
| |
| | issue =
| |
| | year =
| |
| | pages =
| |
| | arxiv = 1204.0505 |bibcode = 2012arXiv1204.0505G }}</ref> but also follow from applying the star-triangle transformation to certain stars on the honeycomb lattice.
| |
| Finally, it can be generalized to having three different probabilities in the three different directions, p<sub>1</sub>,
| |
| p<sub>2</sub> and p<sub>3</sub> for the long bonds, and 1 - p<sub>1</sub>, 1 - p<sub>2</sub>, and 1 - p<sub>3</sub> for
| |
| the short bonds, where p<sub>1</sub>,
| |
| p<sub>2</sub> and p<sub>3</sub> satisfy the critical surface for the inhomogenous triangular lattice.
| |
| | |
| {{clr}}
| |
| | |
| == Thresholds on 2D bowtie and martini lattices ==
| |
| | |
| To the left, center, and right are: the martini lattice, the martini-A lattice, the martini-B lattice. Below: the martini covering/medial lattice, same as the 2x2, 1x1 subnet for kagome-type lattices (removed).
| |
| | |
| [[File:Martini.png|400x1000px|none|Example image caption]]
| |
| | |
| <!-- Deleted image removed: [[File:martinicov.png|200x500px|none|Example image caption]] -->
| |
| | |
| Some other examples of generalized bow-tie lattices (a-d) and the duals of the lattices (e-h)
| |
| | |
| [[File:Bow-tie.png|600x1500px|none|Example image caption]]
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| ! Lattice
| |
| ! z
| |
| ! <math>\overline z</math>
| |
| ! Site Percolation Threshold
| |
| ! Bond Percolation Threshold
| |
| |-
| |
| | martini (3/4)(3,9<sup>2</sup>)+(1/4)(9<sup>3</sup>)
| |
| | 3
| |
| | 3
| |
| | 0.764826..., 1 +p<sup>4</sup> - 3p<sup>3</sup>=0<ref name="Scullard06">{{cite journal
| |
| | last = Scullard
| |
| | first = C. R.
| |
| | authorlink =
| |
| | coauthors =
| |
| | title =Exact site percolation thresholds using a site-to-bond transformation and the star-triangle transformation
| |
| | journal = Physical Review E
| |
| | volume = 73
| |
| | issue =
| |
| | year = 2006
| |
| | pages = 016107
| |
| | doi =10.1103/PhysRevE.73.016107 |arxiv = cond-mat/0507392 |bibcode = 2006PhRvE..73a6107S }}
| |
| </ref>
| |
| | 0.707107... = 1/√2 <ref name="Ziff06">{{cite journal
| |
| | last = Ziff
| |
| | first = R. M.
| |
| | authorlink =
| |
| | coauthors =
| |
| | title = Generalized cell–dual-cell transformation and exact thresholds for percolation
| |
| | journal = Physical Review E
| |
| | volume = 73
| |
| | issue =
| |
| | year = 2006
| |
| | pages = 016134
| |
| | doi = 10.1103/PhysRevE.73.016134 |bibcode = 2006PhRvE..73a6134Z }}
| |
| </ref>
| |
| |-
| |
| |bow-tie (c)
| |
| | 3,4
| |
| |3 1/7
| |
| |
| |
| | 0.672929..., 1-2p<sup>3</sup>-2p<sup>4</sup>-2p<sup>5</sup>-7p<sup>6</sup>+18p<sup>7</sup>+11p<sup>8</sup>-35p<sup>9</sup>+21p<sup>10</sup>-4p<sup>11</sup>=0 <ref name="ScullardZiff06">{{cite journal
| |
| | last = Scullard
| |
| | first = C. R.
| |
| | authorlink =
| |
| | coauthors = Robert M Ziff
| |
| | title =Exact bond percolation thresholds in two dimensions
| |
| | journal = Journal of Physics A
| |
| | volume = 39
| |
| | issue = 49
| |
| | year = 2006
| |
| | pages = 15089
| |
| | doi =10.1088/0305-4470/39/49/003 |arxiv = cond-mat/0610813 |bibcode = 2006JPhA...3915083Z }}
| |
| </ref>
| |
| |-
| |
| |bow-tie (d)
| |
| | 3,4
| |
| |3⅓
| |
| |
| |
| | 0.625457..., 1-2p<sup>2</sup>-3p<sup>3</sup>+4p<sup>4</sup>-p<sup>5</sup>=0 <ref name="ScullardZiff06"/>
| |
| |-
| |
| |martini-A (2/3)(3,7<sup>2</sup>)+(1/3)(3,7<sup>3</sup>)
| |
| |3,4
| |
| |3⅓
| |
| |1/√2<ref name="ScullardZiff06"/>
| |
| |0.625457..., 1-2p<sup>2</sup>-3p<sup>3</sup>+4p<sup>4</sup>-p<sup>5</sup>=0 <ref name="ScullardZiff06"/>
| |
| |-
| |
| |bow-tie dual lattice (e)
| |
| | 3,4
| |
| |3⅔
| |
| |
| |
| | 0.595482..., 1-p<sub>c</sub><sup>bond</sup> (bow-tie (a)) <ref name="ScullardZiff06"/>
| |
| |-
| |
| | bow-tie (b)
| |
| | 3,4,6
| |
| |3⅔
| |
| |
| |
| | 0.533213..., 1-p- 2p<sup>3</sup> -4p<sup>4</sup>-4p<sup>5</sup>+15<sup>6</sup>+ 13p<sup>7</sup>-36p<sup>8</sup>+19p<sup>9</sup>+ p<sup>10</sup> + p<sup>11</sup>=0 <ref name="ScullardZiff06"/>
| |
| |-
| |
| | martini covering/medial (1/2)(3<sup>3</sup>,9)+(1/2)(3,9,3,9)
| |
| | 4
| |
| | 4
| |
| | | 0.707107... = 1/√2 <ref name="Ziff06"/>
| |
| | 0.57086651(33) <ref name="GuBeckerZiff12">{{cite journal
| |
| | last = Gu
| |
| | first = Hang
| |
| | authorlink =
| |
| | coauthors = A. Becker, R. M. Ziff
| |
| | title = Percolation on the Voronoi covering/medial diagram and several other four-coordinated lattices
| |
| | journal = To be published
| |
| | volume =
| |
| | issue =
| |
| | year = 2009
| |
| | pages =
| |
| | doi = }}
| |
| </ref>
| |
| |-
| |
| | martini-B (1/2)(3,5,3,5<sup>2</sup>)+(1/2)(3,5<sup>2</sup>)
| |
| | 3, 5
| |
| | 4
| |
| | 0.618034... = 2/(1 +√5)..., 1- p<sup>2</sup>-p=0<ref name="Scullard06"/><ref name="ScullardZiff06"/>
| |
| | 1/2 <ref name="Ziff06"/><ref name="ScullardZiff06"/>
| |
| |-
| |
| |bow-tie dual lattice (f)
| |
| | 3,4,8
| |
| |4 2/5
| |
| |
| |
| | 0.466787..., 1-p<sub>c</sub><sup>bond</sup> (bow-tie (b))<ref name="ScullardZiff06"/>
| |
| |-
| |
| | bow-tie (a) (1/2)(3<sup>2</sup>,4,3<sup>2</sup>,4)+(1/2)(3,4,3)
| |
| | 4,6
| |
| |5
| |
| |0.5472(2) <ref name="vanderMarck97">{{cite journal
| |
| | last = van der Marck
| |
| | first =S. C.
| |
| | authorlink =
| |
| | coauthors =
| |
| | title = Percolation thresholds and universal formulas
| |
| | journal = Physical Review E
| |
| | volume = 55
| |
| | issue = 2
| |
| | year = 1997
| |
| | pages = 1514–1517
| |
| | doi = 10.1103/PhysRevE.55.1514|bibcode = 1997PhRvE..55.1514V }}
| |
| </ref>
| |
| | 0.404518..., 1 - p - 6p<sup>2</sup> +6p<sup>3</sup>-p<sup>5</sup>=0 <ref name="ScullardZiff06"/>
| |
| |-
| |
| |bow-tie dual lattice (h)
| |
| | 3,6,8
| |
| |5
| |
| |
| |
| | 0.374543..., 1-p<sub>c</sub><sup>bond</sup>(bow-tie (d))<ref name="ScullardZiff06"/>
| |
| |-
| |
| |bow-tie dual lattice (g)
| |
| | 3,6,10
| |
| |5½
| |
| |
| |
| | 0.327071..., 1-p<sub>c</sub><sup>bond</sup>(bow-tie (c))<ref name="ScullardZiff06"/>
| |
| |-
| |
| |}
| |
| | |
| == Thresholds on other 2D lattices ==
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| ! Lattice
| |
| ! z
| |
| ! <math>\overline z</math>
| |
| ! Site Percolation Threshold
| |
| ! Bond Percolation Threshold
| |
| |-
| |
| | (4, 6, 12) covering/medial
| |
| | 4
| |
| | 4
| |
| | p<sub>c</sub><sup>bond</sup>(4, 6, 12) 0.693731...
| |
| | 0.5593140(2),<ref name="Jacobsen14"/> 0.559315(1) <ref name="Ziff13">{{cite journal
| |
| | last = Ziff
| |
| | first = R. M.
| |
| | coauthors =
| |
| | title =
| |
| | journal = To be published
| |
| | date =
| |
| | year = 2013}}
| |
| </ref>
| |
| |-
| |
| | (4, 8<sup>2</sup>) covering/medial, square kagome
| |
| | 4
| |
| | 4
| |
| | p<sub>c</sub><sup>bond</sup>(4,8<sup>2</sup>) = 0.676803...
| |
| | 0.544798005(8),<ref name="Jacobsen14"/> 0.54479793(34) <ref name="Ziff13"/>
| |
| |-
| |
| | (3<sup>3</sup>, 4<sup>2</sup>) medial
| |
| | 4
| |
| | 4
| |
| |
| |
| | 0.51252459859(2) <ref name="Jacobsen14"/>
| |
| |-
| |
| | square covering (non-planar)
| |
| | 6
| |
| |6
| |
| | 1/2
| |
| | 0.3371(1) <ref name="Ziff07">{{cite journal
| |
| | last = Ziff
| |
| | first = R. M.
| |
| | authorlink =
| |
| | coauthors = Scullard, C. R.
| |
| | title = Critical surfaces for general inhomogeneous bond percolation problems
| |
| | journal = J. Stat. Mech
| |
| | volume = 2010
| |
| | issue = 3
| |
| | year = 2010
| |
| | pages = P03021
| |
| | doi = 10.1088/1742-5468/2010/03/P03021 |bibcode = 2010JSMTE..03..021S |arxiv = 0911.2686 }}
| |
| </ref>
| |
| |-
| |
| | square matching lattice (non-planar)
| |
| | 8
| |
| |8
| |
| | 1 - p<sub>c</sub><sup>site</sup>(square) = 0.407253...
| |
| | 0.25036834(6) <ref name="FengDengBlote08"/>
| |
| |-
| |
| |}
| |
| | |
| [[File:4,6,12covering.pdf|200x500px|none|4, 6, 12, Covering/medial lattice]]
| |
| (4, 6, 12) covering/medial lattice
| |
| | |
| [[File:4,82coveringlattice.pdf|200x500px|none|(4, 8^2) Covering/medial lattice]]
| |
| (4, 8<sup>2</sup>) covering/medial lattice
| |
| | |
| [[File:312coveringdual.pdf|200x500px|none|(3,12^2) Covering/medial lattice]]
| |
| (3,12<sup>2</sup>) covering/medial lattice (in light grey), equivalent to the kagome (2 x 2) subnet, and in black, the dual of these lattices.
| |
| | |
| == Thresholds on subnet lattices ==
| |
| | |
| [[File:Kagomesubnets.png|500x1200px|none|Example image caption]]
| |
| The 2 × 2 subnet is known as the "triangular kagome" lattice
| |
| <ref name="Okubo98">{{cite journal
| |
| | last = Okubo
| |
| | first = S.
| |
| | authorlink =
| |
| | coauthors = M. Hayashi, S. Kimura, H. Ohta, M. Motokawa, H. Kikuchi and H. Nagasawa
| |
| | title = Submillimeter wave ESR of triangular-kagome antiferromagnet Cu9X2(cpa)6 (X=Cl, Br)
| |
| | journal = Physica B
| |
| | volume = 246--247
| |
| | issue = 2
| |
| | year = 1998
| |
| | doi = 10.1016/S0921-4526(97)00985-X
| |
| | pages = 553–556|bibcode = 1998PhyB..246..553O }}
| |
| </ref>
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| ! Lattice
| |
| ! z
| |
| ! <math>\overline z</math>
| |
| ! Site Percolation Threshold
| |
| ! Bond Percolation Threshold
| |
| |-
| |
| | checkerboard – 2 × 2 subnet
| |
| | 4,3
| |
| |
| |
| |
| |
| | 0.596303(1) <ref name="HajiAkbariZiff08">{{cite journal
| |
| | last = Haji Akbari
| |
| | first = Amir
| |
| | authorlink =
| |
| | coauthors = R. M. Ziff
| |
| | title = Percolation in networks with voids and bottlenecks
| |
| | journal = Physical Review E
| |
| | volume = 79
| |
| | issue = 2
| |
| | year = 2009
| |
| | pages = 021118
| |
| | doi = 10.1103/PhysRevE.79.021118|bibcode = 2009PhRvE..79b1118H |arxiv = 0811.4575 }}
| |
| </ref>
| |
| |-
| |
| | checkerboard – 4 × 4 subnet
| |
| | 4,3
| |
| |
| |
| |
| |
| | 0.633685(9) <ref name="HajiAkbariZiff08"/>
| |
| |-
| |
| | checkerboard – 8 × 8 subnet
| |
| | 4,3
| |
| |
| |
| |
| |
| | 0.642318(5) <ref name="HajiAkbariZiff08"/>
| |
| |-
| |
| | checkerboard – 16 × 16 subnet
| |
| | 4,3
| |
| |
| |
| |
| |
| | 0.64237(1) <ref name="HajiAkbariZiff08"/>
| |
| |-
| |
| | checkerboard- 32 × 32 subnet
| |
| | 4,3
| |
| |
| |
| |
| |
| | 0.64219(2) <ref name="HajiAkbariZiff08"/>
| |
| |-
| |
| | checkerboard – <math>\infty</math> subnet
| |
| | 4,3
| |
| |
| |
| |
| |
| | 0.642216(10) <ref name="HajiAkbariZiff08"/>
| |
| |-
| |
| | kagome – 2 × 2 subnet
| |
| | 4
| |
| |
| |
| | 0.74042077... (same as (3, 12<sup>2</sup>) bond)
| |
| | 0.6008624(10),<ref name="ZiffGu08"/> 0.60086193(3) <ref name="DingFuGuoWu10"/>
| |
| |-
| |
| | kagome – 3 × 3 subnet
| |
| | 4
| |
| |
| |
| |
| |
| | 0.6193296(10),<ref name="ZiffGu08"/> 0.61933176(5),<ref name="DingFuGuoWu10"/> 0.61933044(32)<ref name="ZiffGu10"/>
| |
| |-
| |
| | kagome – 4 × 4 subnet
| |
| | 4
| |
| |
| |
| |
| |
| | 0.625365(3),<ref name="ZiffGu08"/> 0.62536424(7)<ref name="DingFuGuoWu10"/>
| |
| |-
| |
| | kagome – <math>\infty</math> subnet
| |
| | 4
| |
| |
| |
| |
| |
| | 0.628961(2) <ref name="ZiffGu08"/>
| |
| |-
| |
| | kagome – (1 × 1):(2 × 2) subnet
| |
| | 4
| |
| |
| |
| | 0.707107... = 1/√2 <ref name="Ziff06"/> (martini bond)
| |
| | 0.57086648(36) <ref name="GuBeckerZiff12"/>
| |
| |-
| |
| | kagome – (1 × 1):(3 × 3) subnet
| |
| | 4,3
| |
| |
| |
| | 0.728355596425196...<ref name="DingFuGuoWu10"/>
| |
| | 0.58609776(37) <ref name="ZiffGu10">{{cite journal
| |
| | last = Gu
| |
| | first = Hang
| |
| | authorlink =
| |
| | coauthors = R. M. Ziff
| |
| | title =
| |
| | journal = To be published
| |
| | volume =
| |
| | issue =
| |
| | year = 2010
| |
| | doi =
| |
| | pages =}}
| |
| </ref>
| |
| |-
| |
| | kagome – (1 × 1):(4 × 4) subnet
| |
| |
| |
| |
| |
| | 0.738348473943256...<ref name="DingFuGuoWu10"/>
| |
| |
| |
| |-
| |
| | kagome – (1 × 1):(5 × 5) subnet
| |
| |
| |
| |
| |
| | 0.743548682503071...<ref name="DingFuGuoWu10"/>
| |
| |
| |
| |-
| |
| | kagome – (1 × 1):(6 × 6) subnet
| |
| |
| |
| |
| |
| | 0.746418147634282...<ref name="DingFuGuoWu10"/>
| |
| |
| |
| |-
| |
| | kagome – (2 × 2):(3 × 3) subnet
| |
| |
| |
| |
| |
| |
| |
| | 0.61091770(30) <ref name="ZiffGu10"/>
| |
| |-
| |
| | triangular – 2 × 2 subnet
| |
| | 6,4
| |
| |
| |
| |
| |
| | 0.471628788 <ref name="HajiAkbariZiff08"/>
| |
| |-
| |
| | triangular – 3 × 3 subnet
| |
| | 6,4
| |
| |
| |
| |
| |
| | 0.509077793 <ref name="HajiAkbariZiff08"/>
| |
| |-
| |
| | triangular – 4 × 4 subnet
| |
| | 6,4
| |
| |
| |
| |
| |
| | 0.524364822 <ref name="HajiAkbariZiff08"/>
| |
| |-
| |
| | triangular – 5 × 5 subnet
| |
| | 6,4
| |
| |
| |
| |
| |
| | 0.5315976(10) <ref name="HajiAkbariZiff08"/>
| |
| |-
| |
| | triangular – <math>\infty</math> subnet
| |
| | 6,4
| |
| |
| |
| |
| |
| | 0.53993(1) <ref name="HajiAkbariZiff08"/>
| |
| |}
| |
| | |
| == Thresholds of dimers a square lattice ==
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| ! system
| |
| ! z
| |
| ! Site Threshold
| |
| |-
| |
| | unoriented dimers
| |
| | 4
| |
| | 0.5617 <ref name="Cherkasova 09">{{cite journal
| |
| | last = Cherkasova
| |
| | first = V. A.
| |
| | authorlink =
| |
| | coauthors = Yu. Yu. Tarasevich, N. I. Lebovka, and N.V. Vygornitskii
| |
| | title = Percolation of the aligned dimers on a square lattice
| |
| | journal = Eur. Phys. J. B
| |
| | volume = 74
| |
| | issue = 2
| |
| | year = 2010
| |
| | doi = 10.1140/epjb/e2010-00089-2
| |
| | url =
| |
| | pages = 205–209 |bibcode = 2010EPJB...74..205C |arxiv = 0912.0778 }}
| |
| </ref>
| |
| |-
| |
| | parallel dimers
| |
| | 4
| |
| | 0.5683<ref name="Cherkasova 09"/>
| |
| |-
| |
| |}
| |
| | |
| == Thresholds of polymers (random walks) on a square lattice ==
| |
| | |
| System is composed of ordinary (non-avoiding) random walks of length l on the square lattice.
| |
| <ref name="Zia09">{{cite journal
| |
| | last = Zia
| |
| | first = R. K. P.
| |
| | authorlink =
| |
| | coauthors = W. Yong, B. Schmittmann
| |
| | title = Percolation of a collection of finite random walks: a model for gas permeation through thin polymeric membranes
| |
| | journal = Journal of Mathematical Chemistry
| |
| | volume = 45
| |
| | issue =
| |
| | year = 2009
| |
| | pages = 58–64
| |
| | doi = 10.1007/s10910-008-9367-6 }}
| |
| </ref>
| |
| {| class="wikitable"
| |
| |-
| |
| ! ''l'' (polymer length)
| |
| ! z
| |
| ! Bond Percolation
| |
| |-
| |
| | 1
| |
| | 4
| |
| | 0.5(exact) <ref name="Yong07">{{cite journal
| |
| | last = Wu
| |
| | first = Yong
| |
| | authorlink =
| |
| | coauthors = B. Schmittmann, R. K. P. Zia
| |
| | title = Two-dimensional polymer networks near percolation
| |
| | journal = Journal of Physics A
| |
| | volume = 41
| |
| | issue = 2
| |
| | year = 2008
| |
| | pages = 025008
| |
| | doi = 10.1088/1751-8113/41/2/025004 |bibcode = 2008JPhA...41b5004W }}
| |
| </ref>
| |
| |-
| |
| | 2
| |
| | 4
| |
| | 0.47697(4)<ref name="Yong07"/>
| |
| |-
| |
| | 4
| |
| | 4
| |
| | 0.44892(6) <ref name="Yong07"/>
| |
| |-
| |
| | 8
| |
| | 4
| |
| | 0.41880(4)<ref name="Yong07"/>
| |
| |-
| |
| |}
| |
| | |
| == Thresholds of self-avoiding walks of length k added by random sequential adsorption ==
| |
| | |
| {| class="wikitable" border="1"
| |
| |-
| |
| ! k
| |
| ! z
| |
| ! Site Thresholds
| |
| ! Bond Thresholds
| |
| |-
| |
| | 1
| |
| | 4
| |
| | 0.593(2) <ref name="Corn03">{{cite journal
| |
| | last = Cornette
| |
| | first = V.
| |
| | authorlink =
| |
| | coauthors = A.J. Ramirez-Pastor, F. Nieto
| |
| | title = Two-dimensional polymer networks near percolation
| |
| | journal = European Physical Journal B
| |
| | volume = 36
| |
| | issue = 3
| |
| | year = 2003
| |
| | pages = 397
| |
| | doi = 10.1140/epjb/e2003-00358-1 |bibcode = 2003EPJB...36..391C }}
| |
| </ref>
| |
| | 0.5009(2) <ref name="Corn03"/>
| |
| |-
| |
| | 2
| |
| | 4
| |
| | 0.564(2) <ref name="Corn03"/>
| |
| | 0.4859(2) <ref name="Corn03"/>
| |
| |-
| |
| | 3
| |
| | 4
| |
| | 0.552(2) <ref name="Corn03"/>
| |
| | 0.4732(2) <ref name="Corn03"/>
| |
| |-
| |
| | 4
| |
| | 4
| |
| | 0.542(2) <ref name="Corn03"/>
| |
| | 0.4630(2) <ref name="Corn03"/>
| |
| |-
| |
| | 5
| |
| | 4
| |
| | 0.531(2) <ref name="Corn03"/>
| |
| | 0.4565(2) <ref name="Corn03"/>
| |
| |-
| |
| | 6
| |
| | 4
| |
| | 0.522(2) <ref name="Corn03"/>
| |
| | 0.4497(2) <ref name="Corn03"/>
| |
| |-
| |
| | 7
| |
| | 4
| |
| | 0.511(2) <ref name="Corn03"/>
| |
| | 0.4423(2) <ref name="Corn03"/>
| |
| |-
| |
| | 8
| |
| | 4
| |
| | 0.502(2) <ref name="Corn03"/>
| |
| | 0.4348(2) <ref name="Corn03"/>
| |
| |-
| |
| | 9
| |
| | 4
| |
| | 0.493(2) <ref name="Corn03"/>
| |
| | 0.4291(2) <ref name="Corn03"/>
| |
| |-
| |
| | 10
| |
| | 4
| |
| | 0.488(2) <ref name="Corn03"/>
| |
| | 0.4232(2) <ref name="Corn03"/>
| |
| |-
| |
| | 11
| |
| | 4
| |
| | 0.482(2) <ref name="Corn03"/>
| |
| | 0.4159(2) <ref name="Corn03"/>
| |
| |-
| |
| | 12
| |
| | 4
| |
| | 0.476(2) <ref name="Corn03"/>
| |
| | 0.4114(2) <ref name="Corn03"/>
| |
| |-
| |
| | 13
| |
| | 4
| |
| | 0.471(2) <ref name="Corn03"/>
| |
| | 0.4061(2) <ref name="Corn03"/>
| |
| |-
| |
| | 14
| |
| | 4
| |
| | 0.467(2) <ref name="Corn03"/>
| |
| | 0.4011(2) <ref name="Corn03"/>
| |
| |-
| |
| | 15
| |
| | 4
| |
| | 0.4011(2) <ref name="Corn03"/>
| |
| | 0.3979(2) <ref name="Corn03"/>
| |
| |-
| |
| |}
| |
| | |
| == Thresholds on 2D inhomogeneous lattices ==
| |
| {| class="wikitable"
| |
| |-
| |
| ! Lattice
| |
| ! z
| |
| ! Site Percolation Threshold
| |
| ! Bond Percolation Threshold
| |
| |-
| |
| | bowtie with p = 1/2 on one non-diagonal bond
| |
| | 3
| |
| |
| |
| | 0.3819654(5),<ref name="ZiffEtAl12"/> <math>(3 - \sqrt{5})/2 </math><ref name="ScullardZiff10"/>
| |
| |-
| |
| |}
| |
| | |
| == Thresholds for 2D continuum models ==
| |
| {| class="wikitable"
| |
| |-
| |
| ! System
| |
| ! Φ<sub>c</sub>
| |
| ! η<sub>c</sub>
| |
| ! n<sub>c</sub>
| |
| |-
| |
| | Disks of radius r
| |
| | 0.67634831(2),<ref name="MertensMoore12"/> 0.6763475(6),<ref name="QuintanillaZiff">{{cite journal
| |
| | last = Quintanilla
| |
| | first = John A.
| |
| | authorlink =
| |
| | coauthors = R. M. Ziff
| |
| | title = Near symmetry of percolation thresholds of fully penetrable disks with two different radii
| |
| | journal = Physical Review E
| |
| | volume = 76
| |
| | issue = 5
| |
| | year = 2007
| |
| | pages = 051115 [6 pages]
| |
| | doi = 10.1103/PhysRevE.76.051115|bibcode = 2007PhRvE..76e1115Q }}
| |
| </ref> 0.676339(4) <ref name="QuintanillaTorquatoZiff">{{cite journal
| |
| | last = Quintanilla
| |
| | first = J
| |
| | authorlink =
| |
| | coauthors = S. Torquato; R. M. Ziff
| |
| | title = Efficient measurement of the percolation threshold for fully penetrable discs
| |
| | journal = J. Phys. A: Math. Gen.
| |
| | volume = 33
| |
| | issue = 42
| |
| | year = 2000
| |
| | pages = L399–L407.
| |
| | doi = 10.1088/0305-4470/33/42/104 |bibcode = 2000JPhA...33L.399Q }}</ref>
| |
| | 1.12808737(6),<ref name="MertensMoore12"/> 1.128085(2),<ref name="QuintanillaZiff"/> 1.128059(12) <ref name="QuintanillaTorquatoZiff"/>
| |
| | 1.436322(2),<ref name="QuintanillaZiff"/> 1.436289(16) <ref name="QuintanillaTorquatoZiff"/>
| |
| |-
| |
| | Ellipses, aspect ratio ε = 2
| |
| | 0.63 <ref name="XiaThorpe88">{{cite journal
| |
| | last = Xia
| |
| | first = W.
| |
| | authorlink =
| |
| | coauthors = M. F. Thorpe
| |
| | title = Percolation properties of random ellipses
| |
| | journal = Physical Review A
| |
| | volume = 38
| |
| | issue = 5
| |
| | year = 1988
| |
| | pages = 2650–2656
| |
| | doi = 10.1103/PhysRevA.38.2650|bibcode = 1988PhRvA..38.2650X
| |
| | pmid = 9900674 }}
| |
| </ref>
| |
| | 0.76
| |
| | 1.94
| |
| |-
| |
| | Ellipses, ε = 5
| |
| | 0.455 <ref name="YiSastry04">{{cite journal
| |
| | last = Yi
| |
| | first = Y.-B.
| |
| | authorlink =
| |
| | coauthors = A. M. Sastry
| |
| | title = Analytical approximation of the percolation threshold for overlapping ellipsoids of revolution
| |
| | journal = Proceedings of the Royal Society A
| |
| | volume = 460
| |
| | issue = 5
| |
| | year = 2007
| |
| | pages = 2353–2380
| |
| | doi = 10.1098/rspa.2004.1279|bibcode = 2004RSPSA.460.2353Y }}
| |
| </ref>
| |
| | 0.607
| |
| | 3.864
| |
| |-
| |
| | Ellipses, ε = 10
| |
| | 0.301 <ref name="YiSastry04"/>
| |
| | 0.358
| |
| | 4.56
| |
| |-
| |
| | Ellipses, ε = 20
| |
| | 0.178 <ref name="YiSastry04"/>
| |
| | 0.196
| |
| | 4.99
| |
| |-
| |
| | Ellipses, ε = 50
| |
| | 0.081 <ref name="YiSastry04"/>
| |
| | 0.084
| |
| | 5.38
| |
| |-
| |
| | Ellipses, ε = 100
| |
| | 0.0417 <ref name="YiSastry04"/>
| |
| | 0.0426
| |
| | 5.42
| |
| |-
| |
| | Ellipses, ε = 1000
| |
| | 0.0043 <ref name="YiSastry04"/>
| |
| | 0.00431
| |
| | 5.5
| |
| |-
| |
| | Aligned squares of side <math>\ell</math>
| |
| | 0.66674349(3),<ref name="MertensMoore12">{{cite journal
| |
| | last = Mertens
| |
| | first = Stephan
| |
| | authorlink =
| |
| | coauthors = Cristopher Moore
| |
| | title = Continuum percolation thresholds in two dimensions
| |
| | journal = Physical Review E
| |
| | volume = 86
| |
| | year = 2012
| |
| | pages = 061109
| |
| | doi = 10.1103/PhysRevE.86.061109|arxiv = 1209.4936 |bibcode = 2012PhRvE..86f1109M }}
| |
| </ref> 0.66653(1),<ref name="TorquatoJiao12b"/> 0.6666(4)<ref name="BakerPaulSreenifasanStanley02">{{cite journal
| |
| | last = Baker
| |
| | first = Don R.
| |
| | authorlink =
| |
| | coauthors = Gerald Paul, Sameet Sreenivasan, H. Eugene Stanley
| |
| | title = Continuum percolation threshold for interpenetrating squares and cubes
| |
| | journal = Physical Review E
| |
| | volume = 66
| |
| | issue = 4
| |
| | year = 2002
| |
| | pages = 046136 [5 pages]
| |
| | doi = 10.1103/PhysRevE.66.046136|arxiv = cond-mat/0203235 |bibcode = 2002PhRvE..66d6136B }}
| |
| </ref>
| |
| | 1.09884280(9),<ref name="MertensMoore12"/> 1.0982(3),<ref name="TorquatoJiao12">{{cite journal
| |
| | last = Torquato
| |
| | first = S.
| |
| | authorlink =
| |
| | coauthors = Y. Jiao
| |
| | title = Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses
| |
| | journal = J. Chem. Phys.
| |
| | volume = 137
| |
| | issue = 7
| |
| | year = 2012
| |
| | pages = 074106
| |
| | doi = 10.1063/1.4742750 |arxiv = 1208.3720 |bibcode = 2012JChPh.137g4106T }}
| |
| </ref> 1.098(1)<ref name="BakerPaulSreenifasanStanley02"/>
| |
| | 1.09884280(9),<ref name="MertensMoore12" /> 1.0982(3),<ref name="TorquatoJiao12">{{cite journal
| |
| | last = Torquato
| |
| | first = S.
| |
| | authorlink =
| |
| | coauthors = Y. Jiao
| |
| | title = Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses
| |
| | journal = J. Chem. Phys.
| |
| | volume = 137
| |
| | issue = 7
| |
| | year = 2012
| |
| | pages = 074106
| |
| | doi = 10.1063/1.4742750 |arxiv = 1208.3720 |bibcode = 2012JChPh.137g4106T }}
| |
| </ref> 1.098(1)<ref name="BakerPaulSreenifasanStanley02" />
| |
| |-
| |
| | Randomly oriented squares
| |
| | 0.62554075(4),<ref name="MertensMoore12"/> 0.6254(2)<ref name="BakerPaulSreenifasanStanley02"/>
| |
| | 0.9822723(1),<ref name="MertensMoore12"/> 0.9819(6)<ref name="BakerPaulSreenifasanStanley02"/> 0.982278(14) <ref name="LiOstling13">{{cite journal
| |
| | last = Li
| |
| | first = Jiantong
| |
| | authorlink =
| |
| | coauthors = Mikael Östling
| |
| | title = Percolation thresholds of two-dimensional continuum systems of rectangles
| |
| | journal = Physical Review E
| |
| | volume = 88
| |
| | issue = 1
| |
| | year = 2013
| |
| | pages = 012101
| |
| | doi = 10.1103/PhysRevE.88.012101|bibcode = 2013PhRvE..88a2101L }}
| |
| </ref>
| |
| | 0.9822723(1),<ref name="MertensMoore12"/> 0.9819(6)<ref name="BakerPaulSreenifasanStanley02"/> 0.982278(14) <ref name="LiOstling13"/>
| |
| |-
| |
| | Rectangles, ε = 1.1
| |
| | 0.624870(7)
| |
| | 0.980484(19)
| |
| | 1.078532(21) <ref name="LiOstling13"/>
| |
| |-
| |
| | Rectangles, ε = 2
| |
| | 0.590635(5)
| |
| | 0.893147(13)
| |
| | 1.786294(26) <ref name="LiOstling13"/>
| |
| |-
| |
| | Rectangles, ε = 3
| |
| | 0.5405983(34)
| |
| | 0.777830(7)
| |
| | 2.333491(22) <ref name="LiOstling13"/>
| |
| |-
| |
| | Rectangles, ε = 4
| |
| | 0.4948145(38)
| |
| | 0.682830(8)
| |
| | 2.731318(30) <ref name="LiOstling13"/>
| |
| |-
| |
| | Rectangles, ε = 5
| |
| | 0.4551398(31)
| |
| | 0.607226(6)
| |
| | 3.036130(28) <ref name="LiOstling13"/>
| |
| |-
| |
| | Rectangles, ε = 10
| |
| | 0.3233507(25)
| |
| | 0.3906022(37)
| |
| | 3.906022(37) <ref name="LiOstling13"/>
| |
| |-
| |
| | Rectangles, ε = 20
| |
| | 0.2048518(22)
| |
| | 0.2292268(27)
| |
| | 4.584535(54) <ref name="LiOstling13"/>
| |
| |-
| |
| | Rectangles, ε = 50
| |
| | 0.09785513(36)
| |
| | 0.1029802(4)
| |
| | 5.149008(20) <ref name="LiOstling13"/>
| |
| |-
| |
| | Rectangles, ε = 100
| |
| | 0.0523676(6)
| |
| | 0.0537886(6)
| |
| | 5.378856(60) <ref name="LiOstling13"/>
| |
| |-
| |
| | Rectangles, ε = 200
| |
| | 0.02714526(34)
| |
| | 0.02752050(35)
| |
| | 5.504099(69) <ref name="LiOstling13"/>
| |
| |-
| |
| | Rectangles, ε = 1000
| |
| | 0.00559424(6)
| |
| | 0.00560995(6)
| |
| | 5.609947(60) <ref name="LiOstling13"/>
| |
| |-
| |
| | Sticks of length <math>\ell</math>
| |
| |
| |
| |
| |
| | 5.6372858(6),<ref name="MertensMoore12"/> 5.63726(2) <ref name="LiZhang09">{{cite journal
| |
| | last = Li
| |
| | first = Jiantong
| |
| | authorlink =
| |
| | coauthors = Shi-Li Zhang
| |
| | title = Finite-size scaling in stick percolation
| |
| | journal = Physical Review E
| |
| | volume = 80
| |
| | issue = 4
| |
| | year = 2009
| |
| | pages = 040104(R)
| |
| | doi = 10.1103/PhysRevE.80.040104|bibcode = 2009PhRvE..80d0104L }}
| |
| </ref>
| |
| |-
| |
| | Power-law disks, x=2.05
| |
| | 0.993(1) <ref name="Sasidevan13">{{cite journal
| |
| | last = Sasidevan
| |
| | first = V.
| |
| | authorlink =
| |
| | coauthors =
| |
| | title = Continuum percolation of overlapping discs with a distribution of radii having a power-law tail
| |
| | journal =
| |
| | volume =
| |
| | issue =
| |
| | year =
| |
| | pages =
| |
| | doi = |arxiv = cond-mat/ 1302.0085v2 }}
| |
| </ref>
| |
| | 4.90(1)
| |
| | 0.0380(6)
| |
| |-
| |
| | Power-law disks, x=2.25
| |
| | 0.8591(5) <ref name="Sasidevan13"/>
| |
| | 1.959(5)
| |
| | 0.06930(12)
| |
| |-
| |
| | Power-law disks, x=2.5
| |
| | 0.7836(4) <ref name="Sasidevan13"/>
| |
| | 1.5307(17)
| |
| | 0.09745(11)
| |
| |-
| |
| | Power-law disks, x=4
| |
| | 0.69543(6) <ref name="Sasidevan13"/>
| |
| | 1.18853(19)
| |
| | 0.18916(3)
| |
| |-
| |
| | Power-law disks, x=5
| |
| | 0.68643(13) <ref name="Sasidevan13"/>
| |
| | 1.1597(3)
| |
| | 0.22149(8)
| |
| |-
| |
| | Power-law disks, x=6
| |
| | 0.68241(8) <ref name="Sasidevan13"/>
| |
| | 1.1470(1)
| |
| | 0.24340(5)
| |
| |-
| |
| | Power-law disks, x=7
| |
| | 0.6803(8) <ref name="Sasidevan13"/>
| |
| | 1.140(6)
| |
| | 0.25933(16)
| |
| |-
| |
| | Power-law disks, x=8
| |
| | 0.67917(9) <ref name="Sasidevan13"/>
| |
| | 1.1368(5)
| |
| | 0.27140(7)
| |
| |-
| |
| | Power-law disks, x=9
| |
| | 0.67856(12) <ref name="Sasidevan13"/>
| |
| | 1.1349(4)
| |
| | 0.28098(9)
| |
| |-
| |
| | Voids around disks of radius r
| |
| | 0.159(2) <ref name="vanderMarck96"/>
| |
| |
| |
| |
| |
| |-
| |
| |}
| |
| | |
| <math>\eta_c = \pi r^2 N / L^2</math> equals critical total area for disks, where N is the number of objects and L is the system size.
| |
| | |
| <math>\eta_c = \pi a b N / L^2</math> for ellipses of semi-major and semi-minor axes of a and b, respectively. Aspect ratio <math>\epsilon = a / b </math> with <math>a > b</math>.
| |
| | |
| <math>\eta_c = \ell m N / L^2</math> for rectangles of dimensions <math>\ell</math> and <math>m</math>. Aspect ratio <math>\epsilon = \ell/m</math> with <math>\ell > m</math>.
| |
| | |
| <math>\eta_c = \pi x N / (4 L^2 (x-2))</math> for power-law distributed disks with <math>\hbox{Prob(radius}\ge R) = R^{-x}</math>, <math> R \ge 1 </math>.
| |
| | |
| <math>\phi_c = 1 - e^{-\eta_c} </math> equals critical area fraction.
| |
| | |
| <math>n_c = \ell^2 N / L^2</math> equals number of objects of maximum length <math>\ell = 2 a </math> per unit area.
| |
| | |
| For ellipses, <math>n_c = (4 \epsilon / \pi)\eta_c </math>
| |
| | |
| For void percolation, <math>\phi_c = e^{-\eta_c} </math> is the critical void fraction.
| |
| | |
| For more ellipse values, see <ref name="XiaThorpe88"/>
| |
| | |
| For more rectangle values, see <ref name="LiOstling13" />
| |
| | |
| == Thresholds on 2D random and quasi-lattices ==
| |
| Left to right: (a) Voronoi diagram (solid lines) and its dual, the Delaunay triangulation (dotted lines), for a [[Poisson distribution]] of points, (b) Delaunay triangulation only, (c) Voronoi diagram (black lines) and the covering or line graph (dotted red lines), (d) the Relative Neighborhood Graph (black lines) <ref name="Melchert13"/> superimposed on the Delaunay triangulation (black plus grey lines) for the same set of 128 uniformly distributed random points.
| |
| [[File:VoronoiDelaunay.pdf|225x600px|left|Example image caption]]
| |
| [[File:voronoi.png|Delauney triangulation|250px]]
| |
| [[File:VoronoiCov12.png|300x600px]]
| |
| [[File:RNGonDelaunayTriangulation128vertices.jpg|250x250px|The Relative Neighborhood Graph superimposed on the underlying Delaunay Triangulation (grey).]]
| |
| {{clr}}
| |
| {| class="wikitable"
| |
| |-
| |
| ! Lattice
| |
| ! z
| |
| ! <math>\overline z</math>
| |
| ! Site Percolation Threshold
| |
| ! Bond Percolation Threshold
| |
| |-
| |
| | Relative neighborhood graph
| |
| |
| |
| | 2.5576
| |
| | 0.796(2) <ref name="Melchert13">{{cite journal
| |
| | last = Melchert
| |
| | first = Oliver
| |
| | authorlink =
| |
| | coauthors =
| |
| | title = Percolation thresholds on planar Euclidean relative-neighborhood graphs
| |
| | journal = Physical Review E
| |
| | volume = 87
| |
| | issue = 4
| |
| | pages = 042106
| |
| | doi = 10.1103/PhysRevE.87.042106
| |
| | year = 2013|bibcode = 2013PhRvE..87d2106M|arxiv = 1301.6967}}
| |
| </ref>
| |
| | 0.771(2) <ref name="Melchert13"/>
| |
| |-
| |
| | [[Voronoi tessellation]]
| |
| |3
| |
| |
| |
| | 0.71410(2),<ref name="BeckerZiff09">{{cite journal
| |
| | last = Becker
| |
| | first = A.
| |
| | authorlink =
| |
| | coauthors = R. M. Ziff
| |
| | title = Percolation thresholds on two-dimensional Voronoi networks and Delaunay triangulations
| |
| | journal = Physical Review E
| |
| | volume = 80
| |
| | issue = 4
| |
| | pages = 041101
| |
| | doi = 10.1103/PhysRevE.80.041101
| |
| | year = 2009|bibcode = 2009PhRvE..80d1101B |arxiv = 0906.4360 }}
| |
| </ref> 0.7151* <ref name = "NeherEtAl08"/>
| |
| | 0.68,<ref name="ShanteKirkpatrick71">{{cite journal
| |
| | last = Shante
| |
| | first = K. S.
| |
| | authorlink =
| |
| | coauthors = S. Kirkpatrick
| |
| | title = An introduction to percolation theory
| |
| | journal = Advances in Physics
| |
| | volume = 20
| |
| | issue = 85
| |
| | pages = 325–357
| |
| | doi = 10.1080/00018737100101261
| |
| | year = 1971|bibcode = 1971AdPhy..20..325S }}
| |
| </ref> 0.666931(5),<ref name="BeckerZiff09"/> 0.6670(1) <ref name = "HsuHuang99"/>
| |
| |-
| |
| | Voronoi covering/medial
| |
| | 4
| |
| |
| |
| | 0.666931(2)<ref name="BeckerZiff09"/><ref name = "HsuHuang99"/>
| |
| | 0.53618(2) <ref name="BeckerZiff09"/>
| |
| |-
| |
| | Penrose rhomb dual
| |
| | 4
| |
| |
| |
| | 0.6381(3)<ref name="SakamotoYonezawaHori89"/>
| |
| | 0.5233(2) <ref name="SakamotoYonezawaHori89"/>
| |
| |-
| |
| | [[Penrose tiling|Penrose rhomb]]
| |
| |
| |
| |4
| |
| | 0.5837(3),<ref name="SakamotoYonezawaHori89"/> 0.58391(1)<ref name="ZiffBablievski99">{{cite journal
| |
| | last = Ziff
| |
| | first = R. M.
| |
| | authorlink =
| |
| | coauthors = F. Babalievski
| |
| | title = Site percolation on the Penrose rhomb lattice
| |
| | journal = Physica A
| |
| | volume = 269
| |
| | issue = 2–4
| |
| | pages = 201–210
| |
| | doi = 10.1016/S0378-4371(99)00166-1
| |
| | year = 1999|bibcode = 1999PhyA..269..201Z }}
| |
| </ref>
| |
| | 0.4770(2) <ref name="SakamotoYonezawaHori89"/>
| |
| |-
| |
| | [[Delaunay triangulation]]
| |
| |
| |
| |6
| |
| | 1/2 <ref name="BollobasRiordan06">{{cite journal
| |
| | last = Bollobás
| |
| | first = Béla
| |
| | authorlink =
| |
| | coauthors = Oliver Riordan
| |
| | title = The critical probability for random Voronoi percolation in the plane is 1/2
| |
| | journal = Probab. Theory Relat. Fields
| |
| | volume = 136
| |
| | issue = 3
| |
| | pages = 417–468
| |
| | doi = 10.1007/s00440-005-0490-z
| |
| | year = 2006}}
| |
| </ref>
| |
| | 0.333069(2) <ref name="BeckerZiff09"/><ref name = "HsuHuang99">{{cite journal
| |
| | last = Hsu
| |
| | first = H. P.
| |
| | authorlink =
| |
| | coauthors = M. C. Huang
| |
| | title = Percolation thresholds, critical exponents, and scaling functions on planar random lattices and their duals
| |
| | journal = Physical Review E
| |
| | volume = 60
| |
| | issue = 1999
| |
| | pages = 6361–6370
| |
| | doi = 10.1103/PhysRevE.60.6361
| |
| | year = 1999|bibcode = 1999PhRvE..60.6361H }}
| |
| </ref>
| |
| |-
| |
| |}
| |
| <nowiki>*</nowiki>Theoretical estimate
| |
| | |
| == Thresholds on slabs ==
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| ! Lattice
| |
| ! z
| |
| ! <math>\overline z</math>
| |
| ! Site Percolation Threshold
| |
| ! Bond Percolation Threshold
| |
| |-
| |
| | h= 2, SC, open b.c.
| |
| |
| |
| |
| |
| | 0.47424 <ref name="SottaLong03"/>
| |
| |
| |
| |-
| |
| | h = 3, BCC, periodic b.c.
| |
| |
| |
| |
| |
| |
| |
| | 0.21113018(38) <ref name="GliozziEtAl05"/>
| |
| |-
| |
| | h = 4, BCC, periodic b.c.
| |
| |
| |
| |
| |
| |
| |
| | 0.20235168(59) <ref name="GliozziEtAl05"/>
| |
| |-
| |
| | h= 4, SC, open b.c.
| |
| |
| |
| |
| |
| | 0.3997 <ref name="SottaLong03">{{cite journal
| |
| | last = Sotta
| |
| | first = P.
| |
| | authorlink =
| |
| | coauthors = D. Long
| |
| | title = The crossover from 2D to 3D percolation: Theory and numerical simulations
| |
| | journal = Eur. Phys. J. E
| |
| | volume = 11
| |
| | issue = 4
| |
| | pages = 375–388
| |
| | doi = 10.1140/epje/i2002-10161-6
| |
| | year = 2003|bibcode = 2003EPJE...11..375S }}
| |
| </ref>
| |
| |
| |
| |-
| |
| | h = 5, SC, periodic b.c.
| |
| |
| |
| |
| |
| |
| |
| | 0.278102(5) <ref name="GliozziEtAl05"/>
| |
| |-
| |
| | h = 6, SC, periodic b.c.
| |
| |
| |
| |
| |
| |
| |
| | 0.272380(2) <ref name="GliozziEtAl05"/>
| |
| |-
| |
| | h = 7, SC, periodic b.c.
| |
| | 5,6
| |
| | 5,6
| |
| | 0.3459514(12) <ref name="GliozziEtAl05">{{cite journal
| |
| | last = Gliozzi
| |
| | first = F.
| |
| | authorlink =
| |
| | coauthors = S. Lottini; M. Panero; A. Rago
| |
| | title = Random percolation as a gauge theory
| |
| | journal = Nuclear Physics B
| |
| | volume = 719
| |
| | issue = 3
| |
| | year = 2005
| |
| | pages = 255–274
| |
| | doi = 10.1016/j.nuclphysb.2005.04.021|arxiv = cond-mat/0502339 |bibcode = 2005NuPhB.719..255G }}
| |
| </ref>
| |
| | 0.268459(1) <ref name="GliozziEtAl05"/>
| |
| |-
| |
| | h= 8, SC, open b.c.
| |
| |
| |
| |
| |
| |0.3557 <ref name="SottaLong03"/>
| |
| |
| |
| |-
| |
| | h = 8, SC, periodic b.c.
| |
| |
| |
| |
| |
| |
| |
| | 0.265615(5) <ref name="GliozziEtAl05"/>
| |
| |-
| |
| |}
| |
| | |
| More for SC open b.c. in Ref.<ref name="SottaLong03"/>
| |
| | |
| h is the thickness of the slab, h x ∞ x ∞.
| |
| | |
| == Thresholds on 3D lattices ==
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| ! Lattice
| |
| ! z
| |
| ! <math>\overline z</math>
| |
| ! Site Percolation Threshold
| |
| ! Bond Percolation Threshold
| |
| ! Dimer Percolation Threshold
| |
| |-
| |
| | (8,3)-a<ref name=TranEtAl12/>
| |
| | 3
| |
| | 3
| |
| | 0.577962(33)<ref name="TranEtAl12">{{cite journal
| |
| | last = Tran
| |
| | first = Jonathan
| |
| | authorlink =
| |
| | coauthors = Ted Yoo, Shane Stahlheber, Alex Small
| |
| | title = Percolation thresholds on 3-dimensional lattices with 3 nearest neighbors
| |
| | journal =
| |
| | volume =
| |
| | issue =
| |
| | year = 2012
| |
| | pages =
| |
| | arxiv = 1211.6531|bibcode = 2013JSMTE..05..014T |doi = 10.1088/1742-5468/2013/05/P05014 }}
| |
| </ref>
| |
| | 0.555700(22)<ref name=TranEtAl12/>
| |
| |
| |
| |-
| |
| | (10,3)-a<ref name=TranEtAl12/>
| |
| | 3
| |
| | 3
| |
| | 0.571404(40)<ref name=TranEtAl12/>
| |
| | 0.551060(37)<ref name=TranEtAl12/>
| |
| |
| |
| |-
| |
| | (10,3)-b<ref name=TranEtAl12/>
| |
| | 3
| |
| | 3
| |
| | 0.565442(40)<ref name=TranEtAl12/>
| |
| | 0.546694(33)<ref name=TranEtAl12/>
| |
| |
| |
| |-
| |
| | [[Ice crystals|ice]]
| |
| | 4
| |
| | 4
| |
| | 0.433(11)<ref name="VyssotskyetalSite61">{{cite journal
| |
| | last = Vyssotsky
| |
| | first = V. A.
| |
| | authorlink =
| |
| | coauthors = H. L. Frisch; E. Sonnenblick; J. M. Hammersley
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| | title = Critical Percolation Probabilities (Site Problem)
| |
| | journal = Physical Review
| |
| | volume = 124
| |
| | issue = 4
| |
| | year = 1961
| |
| | pages = 1021–1022
| |
| | doi = 10.1103/PhysRev.124.1021|bibcode = 1961PhRv..124.1021F }}
| |
| </ref>
| |
| | 0.388(10)<ref name="Vyssotskyetal61">{{cite journal
| |
| | last = Vyssotsky
| |
| | first = V. A.
| |
| | authorlink =
| |
| | coauthors = S. B. Gordon; H. L. Frisch; J. M. Hammersley
| |
| | title = Critical Percolation Probabilities (Bond Problem)
| |
| | journal = Physical Review
| |
| | volume = 123
| |
| | issue = 5
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| | year = 1961
| |
| | pages = 1566–1567
| |
| | doi = 10.1103/PhysRev.123.1566|bibcode = 1961PhRv..123.1566V }}
| |
| </ref>
| |
| |
| |
| |-
| |
| | [[Diamond cubic|diamond]]
| |
| | 4
| |
| | 4
| |
| |0.4299870(4),<ref name="XuWangLvDengl13">{{cite journal
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| | title = Simultaneous analysis of three-dimensional percolation models
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| | arxiv = 1310.5399v1 }}</ref> 0.426(+0.08,-0.02),<ref name="SilvermanAdler90">{{cite journal
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| |
| | first = Amihal
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| | authorlink =
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| | title = Site-percolation threshold for a diamond lattice with diatomic substitution
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| | journal = Physical Review B
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| | volume = 42
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| | year = 1990
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| | pages = 1369–1373
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| | doi = 10.1103/PhysRevB.42.1369|bibcode = 1990PhRvB..42.1369S }}
| |
| </ref> 0.4301(4)<ref name="vanderMarck98">{{cite journal
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| | pages = 529–540
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| | doi = 10.1142/S0129183198000431|arxiv = cond-mat/9802187 |bibcode = 1998IJMPC...9..529V }}
| |
| </ref>
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| | 0.3895892(5),<ref name=XuWangLvDengl13/> 0.390(11),<ref name=Vyssotskyetal61/> 0.3893(2)<ref name=vanderMarck98/>
| |
| |
| |
| |-
| |
| | [[Cubic honeycomb|simple cubic]]
| |
| | 6
| |
| | 6
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| |
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| |
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| |
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| |-
| |
| | |
| | [[Cubic honeycomb|simple cubic]] with 3NN
| |
| | 8
| |
| | 8
| |
| | 0.2455(1) <ref name="Kurzawski"/>
| |
| |-
| |
| | fcc
| |
| | 12
| |
| | 12
| |
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| |0.1201635(10)<ref name=LorenzZiff98a/>
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| |
| |
| |-
| |
| | [[Hexagonal close packed|hcp]]
| |
| | 12
| |
| | 12
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| |
| |
| |
| |-
| |
| | La<sub>2-x</sub> Sr<sub>x</sub> Cu O<sub>4</sub>
| |
| | 12
| |
| | 12
| |
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| |
| |
| |
| |
| |
| |
| |
| |
| |
| |-
| |
| | [[Cubic honeycomb|simple cubic]] with 2NN
| |
| | 12
| |
| | 12
| |
| | 0.1991(1) <ref name="Kurzawski"/>
| |
| |-
| |
| | |
| | Penrose w/8 diagonals
| |
| |
| |
| | 12.764
| |
| | 0.188<ref name=ZakalyukinChizhikov05/>
| |
| | 0.111 <ref name=ZakalyukinChizhikov05/>
| |
| |
| |
| |-
| |
| | [[Cubic honeycomb|simple cubic]] with NN+3NN
| |
| | 14
| |
| | 14
| |
| | 0.1420(1) <ref name="Kurzawski">{{cite journal
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| | arxiv = 1111.3254v2}}
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| </ref>
| |
| |
| |
| |
| |
| |
| |
| |-
| |
| | [[Cubic honeycomb|simple cubic]] with NN+2NN
| |
| | 18
| |
| | 18
| |
| | 0.1372(1),<ref name="Kurzawski"/> 0.13735(5) <ref name="Ziffnnn">{{cite journal
| |
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| | first = R. M.
| |
| | authorlink =
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| | coauthors = S. Torquato
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| | title =
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| | journal = To be published
| |
| | volume =
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| | issue =
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| | year = 2007
| |
| | pages = }}
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| </ref>
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |-
| |
| | [[Cubic honeycomb|simple cubic]] with short-length correlation
| |
| | 6+
| |
| | 6+
| |
| | 0.126(1)<ref name="Harter05">{{cite journal
| |
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| |
| | first = T.
| |
| | authorlink =
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| | title = Finite-size scaling analysis of percolation in three-dimensional correlated binary Markov chain random fields
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| |
| </ref>
| |
| |-
| |
| | [[Cubic honeycomb|simple cubic]] with 2NN+3NN
| |
| | 20
| |
| | 20
| |
| | 0.1036(1) <ref name="Kurzawski"/>
| |
| |
| |
| |
| |
| |-
| |
| | [[Cubic honeycomb|simple cubic]] with NN+2NN+3NN
| |
| | 26
| |
| | 26
| |
| | 0.0976(1),<ref name="Kurzawski"/> 0.0976445(10) <ref name="Ziffnnn"/>
| |
| |
| |
| |
| |
| |-
| |
| |}
| |
| | |
| NN = nearest neighbor, 2NN = next-nearest neighbor, 3NN = next-next-nearest neighbor
| |
| | |
| Question: the bond thresholds for the HCP and FCC lattice
| |
| agree within the small statistical error. Are they identical,
| |
| and if not, how far apart are they? Which threshold is expected to be bigger?
| |
| | |
| == Thresholds for 3D continuum models ==
| |
| All overlapping except for jammed spheres.
| |
| {| class="wikitable"
| |
| |-
| |
| ! System
| |
| ! Φ<sub>c</sub>
| |
| ! η<sub>c</sub>
| |
| |-
| |
| | Spheres of radius r
| |
| | 0.289573(2) <ref name="LorenzZiff00">{{cite journal
| |
| | last = Lorenz
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| | first = C. D.
| |
| | authorlink =
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| | coauthors = R. M. Ziff
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| | title = Precise determination of the critical percolation threshold for the three dimensional ''Swiss cheese'' model using a growth algorithm
| |
| | journal = J. Chem. Phys.
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| | volume = 114
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| | issue = 8
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| | year = 2000
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| | pages = 3659
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| | doi = 10.1063/1.1338506 |bibcode = 2001JChPh.114.3659L }}
| |
| </ref>
| |
| | 0.341889(3) <ref name=LorenzZiff00/>
| |
| |-
| |
| | Aligned cylinders
| |
| | 0.2819(2)<ref name="HVLO12"/>
| |
| | 0.3312(1)<ref name="HVLO12">{{cite journal
| |
| | last = Hyytiä
| |
| | first = E.
| |
| | coauthors = J. Virtamo, P. Lassila and J. Ott
| |
| | title = Continuum Percolation Threshold for Permeable Aligned Cylinders and Opportunistic Networking
| |
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| | volume = 16
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| | year = 2012
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| | pages = 1064–1067
| |
| | doi = 10.1109/LCOMM.2012.051512.120497 }}
| |
| </ref>
| |
| |-
| |
| | Aligned cubes of side <math>\ell = 2 a</math>
| |
| | 0.2773(2) <ref name="BakerPaulSreenifasanStanley02"/>
| |
| | 0.3247(3),<ref name="TorquatoJiao12"/> 0.3248(3)<ref name="BakerPaulSreenifasanStanley02"/>
| |
| |-
| |
| | Randomly oriented icosahedra
| |
| |
| |
| | 0.3030(5) <ref name="TorquatoJiao12b"/>
| |
| |-
| |
| | Randomly oriented dodecahedra
| |
| |
| |
| | 0.2949(5) <ref name="TorquatoJiao12b"/>
| |
| |-
| |
| | Randomly oriented octahedra
| |
| |
| |
| | 0.2514(6) <ref name="TorquatoJiao12b"/>
| |
| |-
| |
| | Randomly oriented cubes of side <math>\ell = 2 a</math>
| |
| | 0.2168(2) <ref name="BakerPaulSreenifasanStanley02"/>
| |
| | 0.2444(3),<ref name="BakerPaulSreenifasanStanley02"/> 0.2443(5)<ref name="TorquatoJiao12b">{{cite journal
| |
| | last = Torquato
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| | first = S.
| |
| | authorlink =
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| | coauthors = Y. Jiao
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| | title = Effect of Dimensionality on the Percolation Threshold of Overlapping Nonspherical Hyperparticles
| |
| | journal =
| |
| | volume =
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| | issue =
| |
| | year = 2012
| |
| | arxiv = 1210.0134
| |
| | pages =
| |
| | doi = 10.1103/PhysRevE.87.022111|bibcode = 2013PhRvE..87b2111T }}
| |
| </ref>
| |
| |-
| |
| | Randomly oriented tetrahedra
| |
| |
| |
| | 0.1701(7) <ref name="TorquatoJiao12b"/>
| |
| |-
| |
| | Randomly oriented disks of radius r (in 3D)
| |
| |
| |
| | 0.9614(5)<ref name="YiTawerghi09">{{cite journal
| |
| | last = Yi
| |
| | first = Y. B.
| |
| | authorlink =
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| | coauthors = E. Tawerghi
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| | title = Geometric percolation thresholds of interpenetrating plates in three-dimensional space
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| | volume = 79
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| | year = 2009
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| | pages = 041134
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| | doi = 10.1103/PhysRevE.79.041134 |bibcode = 2009PhRvE..79d1134Y }}
| |
| </ref>
| |
| |-
| |
| | Randomly oriented square plates of side <math>\sqrt{\pi} r</math>
| |
| |
| |
| | 0.8647(6)<ref name=YiTawerghi09/>
| |
| |-
| |
| | Randomly oriented triangular plates of side <math>\sqrt{2 \pi} /3^{1/4} r</math>
| |
| |
| |
| | 0.7295(6)<ref name=YiTawerghi09/>
| |
| |-
| |
| | Voids around disks of radius r
| |
| |
| |
| | 22.86(2)<ref name="YiEsmail12"/>
| |
| |-
| |
| | Voids around oblate ellipsoids of major radius r and aspect ratio 10
| |
| |
| |
| | 15.42(1)<ref name="YiEsmail12"/>
| |
| |-
| |
| | Voids around oblate ellipsoids of major radius r and aspect ratio 2
| |
| |
| |
| | 6.478(8)<ref name="YiEsmail12"/>
| |
| |-
| |
| | Voids around spheres of radius r
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| | title = Network Approach to Void Percolation in a Pack of Unequal Spheres
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| | year = 1996
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| | pages = 1785–1788
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| | doi = 10.1103/PhysRevLett.77.1785
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| | pmid = 10063171
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| | bibcode=1996PhRvL..77.1785V}}
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| </ref> 0.0301(3),<ref name="Rintoul00">{{cite journal
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| </ref> 0.0300(3) <ref name="HoflingMunkFreyFranosch08">{{cite journal
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| | 3.506(8),<ref name="HoflingMunkFreyFranosch08"/> 3.515(6) <ref name="YiEsmail12">{{cite journal
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| |
| | volume = 111
| |
| | issue =
| |
| | year = 2012
| |
| | pages = 124903
| |
| | doi = 10.1063/1.4730333|bibcode = 2012JAP...111l4903Y }}
| |
| </ref>
| |
| |-
| |
| | Jammed spheres (average z = 6)
| |
| | 0.183(3)<ref name="Powell79">{{cite journal
| |
| | last = Powell
| |
| | first = M. J.
| |
| | authorlink =
| |
| | coauthors =
| |
| | title = Site percolation in randomly packed spheres
| |
| | journal = Physical Review B
| |
| | volume = 20
| |
| | issue =
| |
| | year = 1979
| |
| | pages = 4194
| |
| | doi = 10.1103/PhysRevB.20.4194|bibcode = 1979PhRvB..20.4194P }}
| |
| </ref>
| |
| |
| |
| |-
| |
| |}
| |
| | |
| <math>\eta_c = (4/3) \pi r^3 N / L^3</math> is the total volume, where N is the number of objects and L is the system size.
| |
| | |
| <math>\phi_c = 1 - e^{-\eta_c} </math> is the critical volume fraction.
| |
| | |
| For disks and plates, these are effective volumes and volume fractions.
| |
| | |
| For void ("Swiss-Cheese" model), <math>\phi_c = e^{-\eta_c} </math> is the critical void fraction.
| |
| | |
| For more results on void percolation around ellipsoids and elliptical plates, see.<ref name="YiEsmail12"/>
| |
| | |
| ==Thresholds on hypercubic lattices==
| |
| | |
| {| class="wikitable" border="1"
| |
| |-
| |
| ! d
| |
| ! z
| |
| ! Site Thresholds
| |
| ! Bond Thresholds
| |
| |-
| |
| | 4
| |
| | 8
| |
| | 0.1968861(14),<ref name="Grass03">{{cite journal
| |
| | last = Grassberger
| |
| | first = Peter
| |
| | authorlink =
| |
| | coauthors =
| |
| | title = Critical percolation in high dimensions
| |
| | journal = Physical Review E
| |
| | volume = 67
| |
| | issue = 3
| |
| | year = 2003
| |
| | pages = 4
| |
| | doi = 10.1103/PhysRevE.67.036101 |arxiv = cond-mat/0202144 |bibcode = 2003PhRvE..67c6101G }}
| |
| </ref> 0.196889(3),<ref name="PZS01"/> 0.196901(5) <ref name="Ballesteros97">{{cite journal
| |
| | last = Ballesteros
| |
| | first = H. G.
| |
| | authorlink =
| |
| | coauthors = L. A. Fernández, V. Martín-Mayor, A. Muñoz Sudupe, G. Parisi, and J. J. Ruiz-Lorenzo
| |
| | title = Measures of critical exponents in the four dimensional site percolation
| |
| | journal = Phys. Lett. B
| |
| | volume = 400
| |
| | issue = 3–4
| |
| | year = 1997
| |
| | pages = 346–351
| |
| | doi = 10.1016/S0370-2693(97)00337-7 |arxiv = hep-lat/9612024 |bibcode = 1997PhLB..400..346B }}
| |
| </ref>
| |
| | 0.1601314(13),<ref name="Grass03"/> 0.160130(3),<ref name="PZS01"/> 0.1601310(10) <ref name="DammerHinrichsen04">{{cite journal
| |
| | last = Dammer
| |
| | first = Stephan M
| |
| | authorlink =
| |
| | coauthors = Haye Hinrichsen
| |
| | title = Spreading with immunization in high dimensions
| |
| | journal = J. Stat. Mech: Theory Exp.
| |
| | volume = 2004
| |
| | issue = 7
| |
| | year = 2004
| |
| | pages = P07011
| |
| | doi = 10.1088/1742-5468/2004/07/P07011 |arxiv = cond-mat/0405577 |bibcode = 2004JSMTE..07..011D }}
| |
| </ref>
| |
| |-
| |
| | 5
| |
| | 10
| |
| | 0.1407966(15) <ref name="Grass03"/>
| |
| | 0.118172(1),<ref name="Grass03"/> 0.1181718(3) <ref name="DammerHinrichsen04"/>
| |
| |-
| |
| | 6
| |
| | 12
| |
| | 0.109017(2) <ref name="Grass03"/>
| |
| | 0.0942019(6) <ref name="Grass03"/>
| |
| |-
| |
| | 7
| |
| | 14
| |
| | 0.0889511(9),<ref name="Grass03"/> 0.088939(20) <ref name="SZ99">{{cite journal
| |
| | last = Stauffer
| |
| | first = Dietrich
| |
| | authorlink =
| |
| | coauthors = Robert M. Ziff
| |
| | title = Reexamination of Seven-Dimensional Site Percolation Thresholds
| |
| | journal = International Journal of Modern Physics C
| |
| | volume = 11
| |
| | issue = 1
| |
| | year = 1999
| |
| | pages = 205–209
| |
| | doi = 10.1142/S0129183100000183 |arxiv = cond-mat/9911090 |bibcode = 2000IJMPC..11..205S }}
| |
| </ref>
| |
| | 0.0786752(3) <ref name="Grass03"/>
| |
| |-
| |
| | 8
| |
| | 16
| |
| | 0.0752101(5) <ref name="Grass03"/>
| |
| | 0.06770839(7) <ref name="Grass03"/>
| |
| |-
| |
| | 9
| |
| | 18
| |
| | 0.0652095(3) <ref name="Grass03"/>
| |
| | 0.05949601(5) <ref name="Grass03"/>
| |
| |-
| |
| | 10
| |
| | 20
| |
| | 0.0575930(1) <ref name="Grass03"/>
| |
| | 0.05309258(4) <ref name="Grass03"/>
| |
| |-
| |
| | 11
| |
| | 22
| |
| | 0.05158971(8) <ref name="Grass03"/>
| |
| | 0.04794969(1) <ref name="Grass03"/>
| |
| |-
| |
| | 12
| |
| | 24
| |
| | 0.04673099(6) <ref name="Grass03"/>
| |
| | 0.04372386(1) <ref name="Grass03"/>
| |
| |-
| |
| | 13
| |
| | 26
| |
| | 0.04271508(8) <ref name="Grass03"/>
| |
| | 0.04018762(1) <ref name="Grass03"/>
| |
| |-
| |
| |}
| |
| | |
| {| class="wikitable" border="1"
| |
| |-
| |
| ! d
| |
| ! z
| |
| ! Site Thresholds
| |
| ! Bond Thresholds
| |
| ! τ
| |
| |-
| |
| | 4
| |
| | 8
| |
| | 0.196889(3) <ref name="PZS01"/>
| |
| | 0.160130(3) <ref name="PZS01">{{cite journal
| |
| | last = Paul
| |
| | first = Gerald
| |
| | authorlink =
| |
| | coauthors = Robert M. Ziff, H. Eugene Stanley
| |
| | title = Percolation threshold, Fisher exponent, and shortest path exponent for four and five dimensions
| |
| | journal = Physical Review E
| |
| | volume = 64
| |
| | issue = 2
| |
| | year = 2001
| |
| | pages = 8
| |
| | doi = 10.1103/PhysRevE.64.026115 |arxiv = cond-mat/0101136 |bibcode = 2001PhRvE..64b6115P }}
| |
| </ref>
| |
| | 2.313(3) <ref name="PZS01"/>
| |
| |-
| |
| | 5
| |
| | 10
| |
| | 0.14081(1) <ref name="PZS01"/>
| |
| | 0.118174(4) <ref name="PZS01"/>
| |
| | 2.412(4) <ref name="PZS01"/>
| |
| |-
| |
| |}
| |
| | |
| Simulation parameters and results for p<sub>c</sub> and the Fisher exponent τ.
| |
| | |
| {| class="wikitable" border="1"
| |
| |-
| |
| ! d
| |
| ! z
| |
| ! Site Thresholds
| |
| ! Bond Thresholds
| |
| ! z<sub>spread</sub>
| |
| ! d<sub>min</sub>
| |
| |-
| |
| | 4
| |
| | 8
| |
| | 0.196889 <ref name="PZS01"/>
| |
| | 0.160130 <ref name="PZS01"/>
| |
| | 0.622(2) <ref name="PZS01"/>
| |
| | 1.607(5) <ref name="PZS01"/>
| |
| |-
| |
| | 5
| |
| | 10
| |
| | 0.14081 <ref name="PZS01"/>
| |
| | 0.118174 <ref name="PZS01"/>
| |
| | 0.552(2) <ref name="PZS01"/>
| |
| | 1.812(6) <ref name="PZS01"/>
| |
| |-
| |
| |}
| |
| | |
| Simulation parameters and results for the spreading exponent z<sub>spread</sub> and shortest path exponent.
| |
| | |
| ==Thresholds on kagome lattices in higher dimensions==
| |
| | |
| {| class="wikitable" border="1"
| |
| |-
| |
| ! d
| |
| ! z
| |
| ! Site Thresholds
| |
| ! Bond Thresholds
| |
| ! rw
| |
| |-
| |
| |3
| |
| |6
| |
| |0.3895(2) <ref name="Marck97"/>
| |
| |
| |
| |0.417(1) <ref name="Marck97"/>
| |
| |-
| |
| |4
| |
| |8
| |
| |0.2715(3) <ref name="Marck97">{{cite journal
| |
| | last = van der Marck
| |
| | first = Steven C.
| |
| | authorlink =
| |
| | coauthors =
| |
| | title = Site percolation and random walks on d-dimensional Kagome lattices
| |
| | journal = Journal of Physics A
| |
| | volume = 31
| |
| | issue = 15
| |
| | year = 1998
| |
| | pages = 3449–3460
| |
| | doi = 10.1088/0305-4470/31/15/010 |arxiv =cond-mat/9801112v1 |bibcode = 1998JPhA...31.3449V }}
| |
| </ref>
| |
| |
| |
| |0.274(1) <ref name="Marck97"/>
| |
| |-
| |
| |5
| |
| |10
| |
| |0.2084(4) <ref name="Marck97"/>
| |
| |
| |
| |0.208(1) <ref name="Marck97"/>
| |
| |-
| |
| |6
| |
| |12
| |
| |0.1677(7) <ref name="Marck97"/>
| |
| |
| |
| |0.170(1) <ref name="Marck97"/>
| |
| |-
| |
| |}
| |
| | |
| == Thresholds on hyperbolic, hierarchical, and tree lattices ==
| |
| Visualization of a triangular hyperbolic lattice {3,7} projected on the Poincaré disk <ref name="BaekMinnhagenKim09">{{cite journal
| |
| | last = Baek
| |
| | first = S.K.
| |
| | authorlink =
| |
| | coauthors = Petter Minnhagen and Beom Jun Kim
| |
| | title = Comment on 'Monte Carlo simulation study of the two-stage percolation transition in enhanced binary trees'
| |
| | journal = Physica A
| |
| | volume = 42
| |
| | issue = 47
| |
| | pages = 478001
| |
| | doi = 10.1088/1751-8113/42/47/478001
| |
| | year = 2009|bibcode = 2009JPhA...42U8001B |arxiv = 0910.4340 }}
| |
| </ref>
| |
| [[File:TriangularHyperbolic.jpg|250x600px|left|Example image caption]]
| |
| {{clr}}
| |
| | |
| Depiction of the non-planar Hanoi network HN-NP <ref name="ZiffBoetcherCook09"/>
| |
| [[File:Hn-np.jpg|250x600px|left|Example image caption]]
| |
| | |
| {{clr}}
| |
| {| class="wikitable"
| |
| |-
| |
| ! Lattice
| |
| ! z
| |
| |<math>\overline z</math>
| |
| ! Site Percolation Threshold
| |
| !colspan="2"|Bond Percolation Threshold
| |
| |-
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |Lower
| |
| |Upper
| |
| |-
| |
| |{4,5} hyperbolic
| |
| | 5
| |
| |5
| |
| |
| |
| |0.27<ref name="BaekMinnhagenKim09b">{{cite journal
| |
| | last = Baek
| |
| | first = S.K.
| |
| | authorlink =
| |
| | coauthors = Petter Minnhagen and Beom Jun Kim
| |
| | title = Percolation on hyperbolic lattices
| |
| | journal = Physical review
| |
| | volume = 79
| |
| | issue = 1 Pt 1
| |
| | pages = 011124
| |
| | doi = 10.1103/PhysRevE.79.011124
| |
| | year = 2009
| |
| | pmid = 19257018|bibcode = 2009PhRvE..79a1124B |arxiv = 0901.0483 }}
| |
| </ref>
| |
| |0.52<ref name="BaekMinnhagenKim09b"/>
| |
| |-
| |
| |{7,3} hyperbolic
| |
| | 3
| |
| |3
| |
| |
| |
| |0.72<ref name="BaekMinnhagenKim09b"/>
| |
| |0.53<ref name="BaekMinnhagenKim09b"/>
| |
| |-
| |
| |{3,7} hyperbolic
| |
| | 7
| |
| |7
| |
| |
| |
| |0.20<ref name="BaekMinnhagenKim09b"/>
| |
| |0.37<ref name="BaekMinnhagenKim09b"/>
| |
| |-
| |
| |{∞,3} Cayley tree
| |
| | 3
| |
| |3
| |
| | 1/2
| |
| |1/2<ref name="BaekMinnhagenKim09b"/>
| |
| |1<ref name="BaekMinnhagenKim09b"/>
| |
| |-
| |
| |Enhanced binary tree (EBT)
| |
| |
| |
| |
| |
| |
| |
| |0.304(1)<ref name="BaekMinnhagenKim09b"/>
| |
| |0.48,<ref name="BaekMinnhagenKim09b"/> 0.564(1)<ref name="NogawaHasegawa09a">{{cite journal
| |
| | last = Nogawa
| |
| | first = Tomoaki
| |
| | authorlink =
| |
| | coauthors = Takehisa Hasegawa
| |
| | title = Monte Carlo simulation study of the two-stage percolation transition in enhanced binary trees
| |
| | journal = Physica A
| |
| | volume = 42
| |
| | issue = 14
| |
| | pages = 145001
| |
| | doi = 10.1088/1751-8113/42/14/145001
| |
| | year = 2009|bibcode = 2009JPhA...42n5001N |arxiv = 0810.1602 }}
| |
| </ref>
| |
| |-
| |
| |Enhanced binary tree dual
| |
| |
| |
| |
| |
| |
| |
| |0.436(1)<ref name="NogawaHasegawa09a"/>
| |
| |0.696(1)<ref name="NogawaHasegawa09a"/>
| |
| |-
| |
| |Non-Planar Hanoi Network (HN-NP)
| |
| |
| |
| |
| |
| |
| |
| |0.319445<ref name="ZiffBoetcherCook09">{{cite journal
| |
| | last = Ziff
| |
| | first = Robert M.
| |
| | authorlink =
| |
| | coauthors = Stephan Boetcher and Jessica L. Cook
| |
| | title = Monte Carlo simulation study of the two-stage percolation transition in enhanced binary trees
| |
| | journal = Physical Review E
| |
| | volume = 80
| |
| | issue = 4
| |
| | pages = 041115
| |
| | doi = 10.1103/PhysRevE.80.041115
| |
| | year = 2009|bibcode = 2009PhRvE..80d1115B }}
| |
| </ref>
| |
| |0.381996<ref name="ZiffBoetcherCook09"/>
| |
| |-
| |
| | Cayley tree with grandparents
| |
| |
| |
| | 8
| |
| |
| |
| |0.158656326<ref name="Kozakova10">{{cite journal
| |
| | last = Kozáková
| |
| | first = Iva
| |
| | authorlink =
| |
| | coauthors =
| |
| | title = Critical percolation of virtually free groups and other tree-like graphs
| |
| | journal = Annals of Probability
| |
| | volume = 37
| |
| | issue = 6
| |
| | pages = 2262–2296
| |
| | doi = 10.1214/09-AOP458
| |
| | year = 2010}}
| |
| </ref>
| |
| |
| |
| |-
| |
| |}
| |
| Note: {m,n} is the Shläfli symbol, signifying a hyperbolic lattice in which n regular m-gons meet at every vertex
| |
| | |
| Cayley tree (Bethe latttice) with coordination number ''z'': ''p''<sub>c</sub>= 1 / (''z'' - 1)
| |
| | |
| Cayley tree with a distribution of ''z'' with mean <math> \overline z </math>, mean-square <math> \overline{z^2}: </math> ''p''<sub>c</sub>= <math> \overline z / (\overline{z^2} - \overline z) </math> <ref name="CohenEtAl00">{{cite journal
| |
| | last = Cohen
| |
| | first = R
| |
| | authorlink =
| |
| | coauthors = K. Erez, D. Ben-Avraham, S. Havlin
| |
| | title = Resilience of the Internet to random breakdowns
| |
| | journal = Physical Review Letters
| |
| | volume = 85
| |
| | issue =
| |
| | pages = 4626
| |
| | doi = 10.1103/PhysRevLett.85.4626
| |
| | year = 2000|arxiv = cond-mat/0007048 |bibcode = 2000PhRvL..85.4626C }}
| |
| </ref>
| |
| (site or bond threshold)
| |
| | |
| == Thresholds for directed percolation ==
| |
| {| class="wikitable"
| |
| |-
| |
| ! Lattice
| |
| ! ''z''
| |
| ! Site Percolation Threshold
| |
| ! Bond Percolation Threshold
| |
| |-
| |
| | (1+1)-d honeycomb
| |
| | 1.5
| |
| | 0.8399316(2),<ref name="WangEtAl13"/> 0.839933(5),<ref name="JensenGuttmann95">{{cite journal
| |
| | last = Jensen
| |
| | first = Iwan
| |
| | authorlink =
| |
| | coauthors = Anthony J. Guttmann
| |
| | title = Series expansions of the percolation probability for directed square and honeycomb lattices
| |
| | journal = J. Phys. A: Math. Gen.
| |
| | volume = 28
| |
| | issue = 17
| |
| | year = 1995
| |
| | pages = 4813–4833
| |
| | doi = 10.1088/0305-4470/28/17/015 |arxiv = cond-mat/9509121 |bibcode = 1995JPhA...28.4813J }}</ref>
| |
| | 0.8228569(2),<ref name="WangEtAl13"/> 0.82285680(6)<ref name="WangEtAl13"/>
| |
| |-
| |
| | (1+1)-d kagome
| |
| | 2
| |
| | 0.7369317(2),<ref name="WangEtAl13"/> 0.73693182(4)<ref name="Jensen04"/>
| |
| | 0.6589689(2),<ref name="WangEtAl13"/> 0.65896910(8)<ref name="WangEtAl13"/>
| |
| |-
| |
| | (1+1)-d square, diagonal direction
| |
| | 2
| |
| | 0.705489(4),<ref name="LubeckWillmann02">{{cite journal
| |
| | last = Lübeck
| |
| | first = S.
| |
| | authorlink =
| |
| | coauthors = R. D. Willmann
| |
| | title = Universal scaling behaviour of directed percolation and the pair contact process in an external field
| |
| | journal = J. Phys. A
| |
| | volume = 35
| |
| | issue =48
| |
| | year = 2002
| |
| | pages = 10205
| |
| | url =
| |
| | doi = 10.1088/0305-4470/35/48/301|arxiv = cond-mat/0210403 |bibcode = 2002JPhA...3510205L }}
| |
| </ref> 0.70548522(4),<ref name="Jensen99">{{cite journal
| |
| | last = Jensen
| |
| | first = Iwan
| |
| | authorlink =
| |
| | coauthors =
| |
| | title = Low-density series expansions for directed percolation: I. A new efficient algorithm with applications to the square lattice
| |
| | journal = J. Phys. A
| |
| | volume = 32
| |
| | issue = 28
| |
| | year = 1999
| |
| | pages = 5233–5249
| |
| | doi = 10.1088/0305-4470/32/28/304 |arxiv = cond-mat/9906036 |bibcode = 1999JPhA...32.5233J }}</ref> 0.70548515(20),<ref name="Jensen04">{{cite journal
| |
| | last = Jensen
| |
| | first = Iwan
| |
| | authorlink =
| |
| | coauthors =
| |
| | title = Low-density series expansions for directed percolation: III. Some two-dimensional lattices
| |
| | journal = J. Phys. A: Math. Gen.
| |
| | volume = 37
| |
| | issue = 04
| |
| | year = 2004
| |
| | pages = 6899–6915
| |
| | doi = 10.1088/0305-4470/37/27/003 |arxiv = cond-mat/0405504|bibcode = 2004JPhA...37.6899J }}</ref>
| |
| 0.7054852(3),<ref name="WangEtAl13">{{cite journal
| |
| | last = Wang
| |
| | first = Junfeng
| |
| | authorlink =
| |
| | coauthors = Zongzheng Zhou, Qingquan Liu, Timothy M. Garoni, Youjin Deng
| |
| | title = A high-precision Monte Carlo study of directed percolation in (d + 1) dimensions
| |
| | journal =
| |
| | volume =
| |
| | issue =
| |
| | year =
| |
| | pages =
| |
| | arxiv = 1201.3006v2 }}</ref>
| |
| | 0.644701(2),<ref name="EssamDeBellAdlerBhatti">{{cite journal
| |
| | last = Essam
| |
| | first = John
| |
| | authorlink =
| |
| | coauthors = K. De'Bell, J. Adler, F. M. Bhatti
| |
| | title = Analysis of extended series for bond percolation on the directed square lattice
| |
| | journal = Physical Review B
| |
| | volume = 33
| |
| | issue = 2
| |
| | year = 1986
| |
| | pages = 1982–1986
| |
| | doi = 10.1103/PhysRevB.33.1982|bibcode = 1986PhRvB..33.1982E }}</ref> 0.644701(1),<ref name="BaxterGuttmann88">{{cite journal
| |
| | last = Baxter
| |
| | first = R. J.
| |
| | coauthors = A. J. Guttmann
| |
| | title = Series expansion of the percolation probability for the directed square lattice
| |
| | journal = J. Phys. A
| |
| | volume = 21
| |
| | issue = 15
| |
| | year = 1988
| |
| | pages = 3193–3204
| |
| | doi = 10.1088/0305-4470/21/15/008|bibcode = 1988JPhA...21.3193B }}</ref> 0.64470015(5),<ref name="Jensen96">{{cite journal
| |
| | last = Jensen
| |
| | first = Iwan
| |
| | authorlink =
| |
| | coauthors =
| |
| | title = Low-density series expansions for directed percolation on square and triangular lattices
| |
| | journal = J. Phys. A
| |
| | volume = 29
| |
| | issue = 22
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| | year = 1996
| |
| | pages = 7013–7040
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| | doi = 10.1088/0305-4470/29/22/007|bibcode = 1996JPhA...29.7013J }}</ref> 0.644700185(5),<ref name="Jensen99"/> 0.6447001(2),<ref name="WangEtAl13"/>
| |
| |-
| |
| | (1+1)-d triangular
| |
| | 3
| |
| | 0.5956468(5),<ref name="Jensen96"/> 0.5956470(3) <ref name="WangEtAl13"/>
| |
| | 0.478025(1),<ref name="Jensen96"/> 0.4780250(4) <ref name="WangEtAl13"/>
| |
| |-
| |
| | (2+1)-d simple cubic, diagonal planes
| |
| | 3
| |
| | 0.43531(1) <ref name="GrassbergerZhang96"/>
| |
| | 0.382223(7) <ref name="GrassbergerZhang96"/>
| |
| |-
| |
| | (2+1)-d square nn (= bcc)
| |
| | 4
| |
| | 0.3445736(3),<ref name="Grassberger09b">{{cite journal
| |
| | last = Grassberger
| |
| | first = P.
| |
| | authorlink =
| |
| | coauthors =
| |
| | title = Local persistence in directed percolation
| |
| | journal = J. Stat. Mech. Th. Exp.
| |
| | volume = 2009
| |
| | issue =8
| |
| | year = 2009
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| | pages = P08021
| |
| | url =
| |
| | doi = 10.1088/1742-5468/2009/08/P08021 |bibcode = 2009JSMTE..08..021G |arxiv = 0907.4021 }}</ref> 0.344575(15) <ref name="LubeckWillmann04"/>
| |
| | 0.2873383(1),<ref name="PerslmanHavlin02">{{cite journal
| |
| | last = Perlsman
| |
| | first = E.
| |
| | authorlink =
| |
| | coauthors = S. Havlin
| |
| | title = Method to estimate critical exponents using numerical studies
| |
| | journal = Europhys. Lett.
| |
| | volume = 58
| |
| | issue = 2
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| | year = 2002
| |
| | pages = 176–181
| |
| | url =
| |
| | doi = 10.1209/epl/i2002-00621-7|bibcode = 2002EL.....58..176P }}
| |
| </ref> 0.287338(3)<ref name="GrassbergerZhang96">{{cite journal
| |
| | last = Grassberger
| |
| | first = P.
| |
| | authorlink =
| |
| | coauthors = Y.-C. Zhang
| |
| | title = "Self-organized" formulation of standard percolation phenomena
| |
| | journal = Physica A
| |
| | volume = 224
| |
| | issue =
| |
| | year = 1996
| |
| | pages = 169–179
| |
| | url =
| |
| | doi = 10.1016/0378-4371(95)00321-5}}
| |
| </ref>
| |
| |-
| |
| | (3+1)-d hypercubic, diagonal planes
| |
| | 4
| |
| |
| |
| | 0.3025(10) <ref name="AdlerBergerDuarteMeir">{{cite journal
| |
| | last = Adler
| |
| | first = Joan
| |
| | authorlink =
| |
| | coauthors = J. Berger, M. A. M. S. Duarte, Y. Meir
| |
| | title = Directed percolation in 3+1 dimensions
| |
| | journal = Physical Review B
| |
| | volume = 37
| |
| | issue = 13
| |
| | year = 1988
| |
| | pages = 7529–7533
| |
| | url =
| |
| | doi = 10.1103/PhysRevB.37.7529|bibcode = 1988PhRvB..37.7529A }}
| |
| </ref>
| |
| |-
| |
| | (3+1)-d cubic, nn
| |
| | 6
| |
| |0.2081040(4) <ref name="Grassberger09b"/>
| |
| | 0.1774970(5) <ref name="DammerHinrichsen04"/>
| |
| |-
| |
| | (3+1)-d body-centered hypercubic
| |
| | 8
| |
| |0.160950(30) <ref name="LubeckWillmann04"/>
| |
| |
| |
| |-
| |
| | (4+1)-d hypercubic, nn
| |
| | 8
| |
| | 0.1461593(2),<ref name="Grassberger09b"/> 0.1461582(3) <ref name="Grassberger09">{{cite journal
| |
| | last = Grassberger
| |
| | first = Peter
| |
| | authorlink =
| |
| | coauthors =
| |
| | title = Logarithmic corrections in (4+1)-dimensional directed percolation
| |
| | journal = Physical Review E
| |
| | volume = 79
| |
| | issue = 5
| |
| | year = 2009
| |
| | pages = 052104
| |
| | doi = 10.1103/PhysRevE.79.052104|bibcode = 2009PhRvE..79e2104G |arxiv = 0904.0804 }}
| |
| </ref>
| |
| | 0.1288557(5) <ref name="DammerHinrichsen04"/>
| |
| |-
| |
| | (4+1)-d body-centered hypercubic
| |
| | 16
| |
| | 0.075582(17) <ref name="LubeckWillmann04">{{cite journal
| |
| | last = Lübeck
| |
| | first = S.
| |
| | authorlink =
| |
| | coauthors = R. D. Willmann
| |
| | title = Universal scaling behavior of directed percolation around the upper critical dimension
| |
| | journal = J. Stat. Phys.
| |
| | volume = 115
| |
| | issue = 5–6
| |
| | year = 2004
| |
| | pages = 1231–1250
| |
| | url =
| |
| | doi = 10.1023/B:JOSS.0000028059.24904.3b|arxiv = cond-mat/0401395 |bibcode = 2004JSP...115.1231L }}
| |
| </ref>
| |
| 0.0755850(3) <ref name="Grassberger09"/>
| |
| |
| |
| |-
| |
| | (5+1)-d hypercubic, nn
| |
| | 10
| |
| | 0.1123373(2) <ref name="Grassberger09b"/>
| |
| | 0.1016796(5) <ref name="DammerHinrichsen04"/>
| |
| |-
| |
| | (5+1)-d body-centered hypercubic
| |
| | 32
| |
| |0.035967(23) <ref name="LubeckWillmann04"/>
| |
| |
| |
| |-
| |
| | (6+1)-d hypercubic, nn
| |
| | 12
| |
| | 0.0913087(2) <ref name="Grassberger09b"/>
| |
| | 0.0841997(14) <ref name="DammerHinrichsen04"/>
| |
| |-
| |
| | (7+1)-d hypercubic,nn
| |
| | 14
| |
| | 0.07699336(7) <ref name="Grassberger09b"/>
| |
| | 0.07195(5) <ref name="DammerHinrichsen04"/>
| |
| |}
| |
| nn = nearest neighbors. For a (d+1)-dimensional hypercubic system, the hypercube is in d dimensions and the time direction points to the 2D nearest neighbors.
| |
| | |
| == General formulas for exact results ==
| |
| | |
| Inhomogeneous triangular lattice bond percolation<ref name="SykesEssam"/>
| |
| | |
| <math>
| |
| 1 - p_1 - p_2 - p_3 + p_1 p_2 p_3 = 0
| |
| </math>
| |
| | |
| Inhomogeneous honeycomb lattice bond percolation = kagome lattice site percolation<ref name="SykesEssam"/>
| |
| | |
| <math>
| |
| 1 - p_1 p_2 - p_1 p_3 - p_2 p_3+ p_1 p_2 p_3 = 0
| |
| </math>
| |
| | |
| Inhomogeneous (3,12^2) lattice, site percolation<ref name="SudingZiff99"/>
| |
| <ref name="Wu09">{{cite journal
| |
| | last = Wu
| |
| | first = F. Y.
| |
| | authorlink =
| |
| | coauthors =
| |
| | title = Critical frontier of the Potts and percolation models on triangular-type and kagome-type lattices I: Closed-form expressions
| |
| | journal = Physical Review E
| |
| | volume = 81
| |
| | issue = 6
| |
| | year = 2010
| |
| | pages = 061110
| |
| | doi = 10.1103/PhysRevE.81.061110 | arxiv = 0911.2514|bibcode = 2010PhRvE..81f1110W }}
| |
| </ref>
| |
| | |
| <math>
| |
| 1 - 3(s_1s_2)^2 + (s_1s_2)^3 = 0,
| |
| </math>
| |
| or <math>
| |
| s_1 s_2 = 1 - 2 \sin(\pi/18)
| |
| </math>
| |
| | |
| Inhomogeneous martini lattice, bond percolation
| |
| <ref name="ScullardZiff06">{{cite journal
| |
| | last = Scullard
| |
| | first = Christian R.
| |
| | authorlink =
| |
| | coauthors = R. M. Ziff
| |
| | title = Predictions of bond percolation thresholds for the kagome and Archimedean (3,12<sup>2</sup>) lattices
| |
| | journal = Physical Review E
| |
| | volume = 73
| |
| | issue = 4
| |
| | year = 2006
| |
| | pages = 045102R
| |
| | doi = 10.1103/PhysRevE.73.045102|arxiv = cond-mat/0602431 |bibcode = 2006PhRvE..73d5102S }}
| |
| </ref>
| |
| <math>
| |
| 1 - (p_1 p_2 r_3 + p_2 p_3 r_1 + p_1 p_3 r_2) - (p_1 p_2 r_1 r_2
| |
| + p_1 p_3 r_1 r_3 + p_2 p_3 r_2 r_3) + p_1 p_2 p_3 ( r_1 r_2
| |
| + r_1 r_3 + r_2 r_3) +
| |
| </math>
| |
| <math>
| |
| r_1 r_2 r_3 (p_1 p_2
| |
| + p_1 p_3 + p_2 p_3) - 2 p_1 p_2 p_3 r_1 r_2 r_3 = 0
| |
| </math>
| |
| | |
| Inhomogeneous martini lattice, site percolation). ''r'' = site in the star
| |
| | |
| <math>
| |
| 1 - r (p_1 p_2 + p_1 p_3 + p_2 p_3 - p_1 p_2 p_3) = 0
| |
| </math>
| |
| | |
| Inhomogeneous martini-A (3–7) lattice, bond percolation. Left side (top of "A" to bottom): <math>r_2,\ p_1</math>. Right side: <math>r_1, \ p_2</math>. Cross bond: <math>\ r_3</math>.
| |
| | |
| <math>
| |
| 1 - p_1 r_2 - p_2 r_1 - p_1 p_2 r_3 - p_1 r_1 r_3
| |
| - p_2 r_2 r_3 + p_1 p_2 r_1 r_3 + p_1 p_2 r_2 r_3
| |
| + p_1 r_1 r_2 r_3+ p_2 r_1 r_2 r_3 - p_1 p_2 r_1 r_2 r_3 = 0
| |
| </math>
| |
| | |
| Inhomogeneous martini-B (3–5) lattice, bond percolation
| |
| | |
| Inhomogeneous checkerboard lattice, bond percolation
| |
| <ref name="Wu79">{{cite journal
| |
| | last = Wu
| |
| | first = F. Y.
| |
| | authorlink =
| |
| | coauthors =
| |
| | title = Critical point of planar Potts models
| |
| | journal = Journal of Physics C
| |
| | volume = 12
| |
| | issue = 17
| |
| | year = 1979
| |
| | pages = L645–L650
| |
| | doi = 10.1088/0022-3719/12/17/002|bibcode = 1979JPhC...12L.645W }}
| |
| </ref><ref name="ZiffEtAl12">{{cite journal
| |
| | last = Ziff
| |
| | first = R. M.
| |
| | authorlink =
| |
| | coauthors = C. R. Scullard, J. C. Wierman, M. R. A. Sedlock
| |
| | title = The critical manifolds of inhomogeneous bond percolation on bow-tie and checkerboard lattices
| |
| | journal = Journal of Physics A: Mathematical and Theoretical
| |
| | volume = 45
| |
| | issue = 49
| |
| | year = 2012
| |
| | pages = 494005
| |
| | doi = 10.1088/1751-8113/45/49/494005 |arxiv = 1210.6609 |bibcode = 2012JPhA...45W4005Z }}
| |
| </ref>
| |
| | |
| <math>
| |
| 1 - (p_1 p_2 + p_1 p_3 + p_1 p_4 + p_2 p_3 + p_2 p_4 + p_3 p_4)
| |
| + p_1 p_2 p_3 + p_1 p_2 p_4 + p_1 p_3 p_4 + p_2 p_3 p_4 = 0
| |
| </math>
| |
| | |
| Inhomogeneous bow-tie lattice, bond percolation <ref name="ScullardZiff10"/><ref name="ZiffEtAl12"/>
| |
| | |
| <math>
| |
| 1 - (p_1 p_2 + p_1 p_3 + p_1 p_4 + p_2 p_3 + p_2 p_4 + p_3 p_4)
| |
| + p_1 p_2 p_3 + p_1 p_2 p_4 + p_1 p_3 p_4 + p_2 p_3 p_4 +
| |
| </math>
| |
| <math>
| |
| u(1 - p_1 p_2 - p_3 p_4 + p_1 p_2 p_3 p_4) = 0
| |
| </math>
| |
| | |
| where <math>p_1, p_2, p_3, p_4</math> are the four bonds around the square and <math>u</math> is the diagonal bond connecting the vertex between bonds <math>p_4, p_1</math> and <math>p_2, p_3</math>.
| |
| | |
| == Percolation thresholds of graphs ==
| |
| For random graphs not embedded in space the percolation threshold can be calculated exactly. For example for random regular graphs where all nodes have the same degree k, p<sub>c</sub>=1/k. For [[Erdős–Rényi model|Erdős–Rényi]] (ER) graphs with Poissonian degree distribution, p<sub>c</sub>=1/<k>.<ref>{{cite book |
| |
| title= Complex Networks: Structure, Robustness and Function |
| |
| author= Reuven Cohen, Shlomo Havlin |
| |
| year= 2010 |
| |
| publisher= Cambridge University Press|
| |
| url= http://havlin.biu.ac.il/Shlomo%20Havlin%20books_com_net.php}}</ref> The critical threshold was calculated exactly also for interdependent ER networks.<ref>{{cite journal |
| |
| author= S. V. Buldyrev, R. Parshani, G. Paul, H. E. Stanley, S. Havlin |
| |
| title= Catastrophic cascade of failures in interdependent networks |
| |
| journal= Nature|
| |
| volume= 464 |
| |
| pages= 1025–28 |
| |
| year= 2010
| |
| | doi =10.1038/nature08932 |
| |
| url= http://havlin.biu.ac.il/Publications.php?keyword=Catastrophic+cascade+of+failures+in+interdependent+networks++&year=*&match=all |
| |
| issue= 7291 |bibcode = 2010Natur.464.1025B |arxiv = 0907.1182 }}</ref>
| |
| | |
| ==See also==
| |
| * [[Percolation]]
| |
| * [[Percolation theory]]
| |
| * [[Graph theory]]
| |
| * [[Percolation critical exponents]]
| |
| * [[2D percolation cluster]]
| |
| * [[Directed percolation]]
| |
| * [[Effective Medium Approximations]]
| |
| * [[Epidemic models on lattices]]
| |
| * [[List of uniform tilings|Uniform Tilings]]
| |
| | |
| == References ==
| |
| | |
| <references/>
| |
| <br />
| |
| | |
| {{DEFAULTSORT:Percolation Threshold}}
| |
| [[Category:Critical phenomena]]
| |
| [[Category:Random graphs]]
| |