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第1行： − 此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。 − − {{short description|Mathematical theory on behavior of connected clusters in a random graph}} − − {{Network science}} − − In [[statistical physics]] and [[mathematics]], '''percolation theory''' describes the behavior of a network when nodes or links are added. This is a type of phase transition, since at a critical fraction of removal the network breaks into [[Glossary of graph theory|connected]] clusters. The applications of percolation theory to [[materials science]] and in many other disciplines are discussed here and in the articles [[network theory]] and [[percolation]]. − − In statistical physics and mathematics, percolation theory describes the behavior of a network when nodes or links are removed. This is a type of phase transition, since at a critical fraction of removal the network breaks into connected clusters. The applications of percolation theory to materials science and in many other disciplines are discussed here and in the articles network theory and percolation. − − 在统计物理学和数学中，逾渗理论描述了当节点或链路被移除时网络的行为。这是一种相变，因为在移除的关键部分，网络分裂成连接的集群。本文讨论了渗流理论在材料科学和其他许多学科中的应用，以及网络理论和渗流理论的文章。 − − − − ==Introduction==<!-- [[Bond percolation]] and [[Site percolation]] redirect to here --> − − ==Introduction==<!-- Bond percolation and Site percolation redirect to here --> − − 引言! ——债券过滤和网站过滤重定向到这里—— − − [[File:perc-wiki.png|thumb|left|A three-dimensional site percolation graph]] − − A three-dimensional site percolation graph − − 三维场地渗流图 − − The [[Flory–Stockmayer theory]] was the first theory investigating percolation processes.<ref>{{Cite book|url=https://books.google.com/?id=Mw_csu3AcB0C&lpg=PA8&dq=Flory%E2%80%93Stockmayer%20theory%20percolation%20theory&pg=PA8#v=onepage&q=Flory%E2%80%93Stockmayer%20theory%20percolation%20theory&f=false|title=Applications Of Percolation Theory|last=Sahini|first=M.|last2=Sahimi|first2=M.|date=2003-07-13|publisher=CRC Press|isbn=978-0-203-22153-2|language=en}}</ref> − − The Flory–Stockmayer theory was the first theory investigating percolation processes. − − 弗洛里-斯托克迈尔理论是第一个研究渗流过程的理论。 − − − − − − A representative question (and the [[etymology|source]] of the name) is as follows. Assume that some liquid is poured on top of some [[porosity|porous]] material. Will the liquid be able to make its way from hole to hole and reach the bottom? This physical question is [[mathematical model|modelled]] mathematically as a [[Grid graph|three-dimensional network]] of {{math|''n'' × ''n'' × ''n''}} [[graph (discrete mathematics)|vertices]], usually called "sites", in which the [[graph (discrete mathematics)|edge]] or "bonds" between each two neighbors may be open (allowing the liquid through) with probability {{math|''p''}}, or closed with probability {{math|1 – ''p''}}, and they are assumed to be independent. Therefore, for a given {{math|''p''}}, what is the probability that an open path (meaning a path, each of whose links is an "open" bond) exists from the top to the bottom? The behavior for large {{math|''n''}} is of primary interest. This problem, called now '''bond percolation''', was introduced in the mathematics literature by {{harvtxt|Broadbent|Hammersley|1957}},<ref name="BroadbentHammersley2008">{{cite journal|last1=Broadbent|first1=S. R.|last2=Hammersley|first2=J. M.|title=Percolation processes|journal=Mathematical Proceedings of the Cambridge Philosophical Society|volume=53|issue=3|year=2008|pages=629|issn=0305-0041|doi=10.1017/S0305004100032680|bibcode = 1957PCPS...53..629B }}</ref> and has been studied intensively by mathematicians and physicists since then. − − A representative question (and the source of the name) is as follows. Assume that some liquid is poured on top of some porous material. Will the liquid be able to make its way from hole to hole and reach the bottom? This physical question is modelled mathematically as a three-dimensional network of vertices, usually called "sites", in which the edge or "bonds" between each two neighbors may be open (allowing the liquid through) with probability , or closed with probability , and they are assumed to be independent. Therefore, for a given , what is the probability that an open path (meaning a path, each of whose links is an "open" bond) exists from the top to the bottom? The behavior for large is of primary interest. This problem, called now bond percolation, was introduced in the mathematics literature by , and has been studied intensively by mathematicians and physicists since then. − − 一个有代表性的问题(以及名字的来源)如下。假设一些液体被倒在一些多孔材料上。液体能够从一个洞到另一个洞并到达底部吗？这个物理问题在数学上被模拟为一个三维顶点网络，通常被称为“网点” ，在这个网络中，每两个邻居之间的边缘或“键”可能是开放的(允许液体通过) ，也可能是封闭的，并且它们被假定是独立的。因此，对于给定的情况，从上到下存在一条开放路径(意味着一条路径，其中每个链接都是一个“开放”键)的概率是多少？大的行为是主要的兴趣。这个问题被称为现在的键逾渗，年被引入数学文献，此后一直被数学家和物理学家深入研究。 − − − − In a slightly different mathematical model for obtaining a random graph, a site is "occupied" with probability {{math|''p''}} or "empty" (in which case its edges are removed) with probability {{math|1 – ''p''}}; the corresponding problem is called '''site percolation'''. The question is the same: for a given ''p'', what is the probability that a path exists between top and bottom? Similarly, one can ask, given a connected graph at what fraction {{math|1 – ''p''}} of failures the graph will become disconnected (no large component). − − In a slightly different mathematical model for obtaining a random graph, a site is "occupied" with probability or "empty" (in which case its edges are removed) with probability ; the corresponding problem is called site percolation. The question is the same: for a given p, what is the probability that a path exists between top and bottom? Similarly, one can ask, given a connected graph at what fraction of failures the graph will become disconnected (no large component). − − 在一个略有不同的获得随机图的数学模型中，一个场地被概率“占据”或者被概率“空”(在这种情况下，它的边被去掉) ; 相应的问题称为场地渗流。问题是一样的: 对于给定的 p，在顶部和底部之间存在一条路径的概率是多少？类似地，我们可以问，给定一个连通图，在失效的部分，图会变成不连通(没有大的组件)。 − − [[File:Tube Network Percolation.gif|thumb|A 3D tube network percolation determination]] − − A 3D tube network percolation determination − − 一种三维管网渗流测定方法 − − The same questions can be asked for any lattice dimension. As is quite typical, it is actually easier to examine [[Infinite graph|infinite]] networks than just large ones. In this case the corresponding question is: does an infinite open cluster exist? That is, is there a path of connected points of infinite length "through" the network? By [[Kolmogorov's zero–one law]], for any given {{math|''p''}}, the probability that an infinite cluster exists is either zero or one. Since this probability is an increasing function of {{math|''p''}} (proof via [[Coupling (probability)|coupling]] argument), there must be a '''critical''' {{math|''p''}} (denoted by {{math|''p''<sub>c</sub>}}) below which the probability is always 0 and above which the probability is always 1. In practice, this criticality is very easy to observe. Even for {{math|''n''}} as small as 100, the probability of an open path from the top to the bottom increases sharply from very close to zero to very close to one in a short span of values of {{math|''p''}}. − − The same questions can be asked for any lattice dimension. As is quite typical, it is actually easier to examine infinite networks than just large ones. In this case the corresponding question is: does an infinite open cluster exist? That is, is there a path of connected points of infinite length "through" the network? By Kolmogorov's zero–one law, for any given , the probability that an infinite cluster exists is either zero or one. Since this probability is an increasing function of (proof via coupling argument), there must be a critical (denoted by ) below which the probability is always 0 and above which the probability is always 1. In practice, this criticality is very easy to observe. Even for as small as 100, the probability of an open path from the top to the bottom increases sharply from very close to zero to very close to one in a short span of values of . − − 对于任何格维数都可以提出同样的问题。这是很典型的，实际上检查无限网络比检查大型网络更容易。在这种情况下，相应的问题是: 是否存在一个无限的疏散集群？也就是说，是否存在一条无限长度的连通点“通过”网络的路径？根据柯尔莫哥罗夫的零一定律，对于任何给定的情况，无限大星系团存在的概率要么是零，要么是一。由于这个概率是一个递增函数(通过耦合参数进行证明) ，因此必须存在一个临界值(表示) ，其中低于该临界值的概率总是0，高于该临界值的概率总是1。在实践中，这种临界性是很容易观察到的。即使对于小到100，从顶部到底部的开放路径的概率急剧增加，从非常接近零到非常接近一在短时间内的值。 − − − − [[Image:Bond percolation p 51.png|thumb|Detail of a bond percolation on the square lattice in two dimensions with percolation probability {{math|''p'' {{=}} 0.51}}]] − − Detail of a bond percolation on the square lattice in two dimensions with percolation probability 0.51}} − − 详细研究了二维正方格子上的键逾渗，逾渗概率为0.51} − − − − For most infinite lattice graphs, {{math|''p''<sub>c</sub>}} cannot be calculated exactly, though in some cases {{math|''p''<sub>c</sub>}} there is an exact value. For example: − − For most infinite lattice graphs, cannot be calculated exactly, though in some cases there is an exact value. For example: − − 对于大多数无限格图，不能精确计算，尽管在某些情况下有一个精确的值。例如: − − − − *for the [[square lattice]] {{math|'''ℤ'''<sup>2</sup>}} in two dimensions, {{math|''p''<sub>c</sub> {{=}} {{sfrac|1|2}}}} for bond percolation, a fact which was an open question for more than 20 years and was finally resolved by [[Harry Kesten]] in the early 1980s,<ref name="BollobásRiordan2006">{{cite journal|last1=Bollobás|first1=Béla|last2=Riordan|first2=Oliver|title=Sharp thresholds and percolation in the plane|journal=Random Structures and Algorithms|volume=29|issue=4|year=2006|pages=524–548|issn=1042-9832|doi=10.1002/rsa.20134|arxiv=math/0412510}}</ref> see {{harvtxt|Kesten|1982}}. For site percolation, the value of <math>pc</math> is not known analyticity but only via simulations of large lattices.<ref>{{cite journal |author=MEJ Newman, RM Ziff|year=2000|title=Efficient Monte Carlo algorithm and high-precision results for percolation |journal=Physical Review Letters |issue=85|volume=19|page=4104 }}</ref> − − *A limit case for lattices in many dimensions{{clarify|date=July 2019}} is given by the [[Bethe lattice]], whose threshold is at {{math|''p''<sub>c</sub> {{=}} {{sfrac|1|''z'' − 1}}}} for a [[coordination number]] {{math|''z''}}. In other words: for the regular [[Tree (graph theory)|tree]] of degree <math>z</math>, <math>p_c</math> is equal to <math>1/(z-1)</math>. − − [[File:Front de percolation.png|thumb]] − − thumb − − 拇指 − − *For random Erdős–Rényi networks of average degree <math><k></math>, {{math|''p''<sub>c</sub> {{=}} {{sfrac|1|<k>}}}}.<ref>{{cite journal |author=Erdős, P. & Rényi, A.|year=1959|title=On random graphs I. |journal=Publ. Math. |issue=6|pages= 290–297}}</ref><ref>{{cite journal |author=Erdős, P. & Rényi, A.|year=1960|title=The evolution of random graphs |journal=Publ. Math. Inst. Hung. Acad. Sci. |issue=5|pages= 17–61}}</ref><ref>{{cite journal |author=Bolloba´s, B.|year=1985|title= Random Graphs |journal=Academic}}</ref> − − − − ==Universality== − − The [[Universality (dynamical systems)|universality principle]] states that the numerical value of {{math|''p''<sub>c</sub>}} is determined by the local structure of the graph, whereas the behavior near the critical threshold, pc, is characterized by universal [[critical exponents]]. For example the distribution of the size of clusters at criticality decays as a power law with the same exponents for all 2d lattices. This universality means that for a given dimension, the various critical exponents, the [[fractal dimension]] of the clusters at {{math|''p''<sub>c</sub>}} is independent of the lattice type and percolation type (e.g., bond or site). However, recently percolation has been performed on a [[weighted planar stochastic lattice (WPSL)]] and found that although the dimension of the WPSL coincides with the dimension of the space where it is embedded, its universality class is different from that of all the known planar lattices.<ref>{{cite journal | last1 = Hassan | first1 = M. K. | last2 = Rahman | first2 = M. M. | year = 2015 | title = Percolation on a multifractal scale-free planar stochastic lattice and its universality class | journal = Phys. Rev. E | volume = 92 | issue = 4| page = 040101 | doi=10.1103/PhysRevE.92.040101| pmid = 26565145 | arxiv = 1504.06389 | bibcode = 2015PhRvE..92d0101H }}</ref><ref>{{cite journal | last1 = Hassan | first1 = M. K. | last2 = Rahman | first2 = M. M. | year = 2016 | title = Universality class of site and bond percolation on multi-multifractal scale-free planar stochastic lattice | journal = Phys. Rev. E | volume = 94 | issue = 4| page = 042109 | doi=10.1103/PhysRevE.94.042109| pmid = 27841467 | arxiv = 1604.08699 | bibcode = 2016PhRvE..94d2109H }}</ref> − − The universality principle states that the numerical value of is determined by the local structure of the graph, whereas the behavior near the critical threshold, pc, is characterized by universal critical exponents. For example the distribution of the size of clusters at criticality decays as a power law with the same exponents for all 2d lattices. This universality means that for a given dimension, the various critical exponents, the fractal dimension of the clusters at is independent of the lattice type and percolation type (e.g., bond or site). However, recently percolation has been performed on a weighted planar stochastic lattice (WPSL) and found that although the dimension of the WPSL coincides with the dimension of the space where it is embedded, its universality class is different from that of all the known planar lattices. − − 普适性原理指出，数值的大小是由图的局部结构决定的，而临界阈值附近的行为是拥有属性的普适临界指数。例如，在临界状态下星系团大小的分布呈幂律衰变，所有2d 晶格的指数相同。这种普遍性意味着，对于给定的维数，各种临界指数，团簇的分形维数与晶格类型和逾渗类型无关(例如，键或位置)。然而，最近在加权平面随机晶格(WPSL)上进行的逾渗实验发现，虽然 WPSL 的维数与其所嵌入空间的维数一致，但它的普适性类与所有已知平面晶格的普适性类不同。 − − − − == Phases == − − − − ===Subcritical and supercritical=== − − − − The main fact in the subcritical phase is "exponential decay". That is, when {{math|''p'' < ''p''<sub>c</sub>}}, the probability that a specific point (for example, the origin) is contained in an open cluster (meaning a maximal connected set of "open" edges of the graph) of size {{math|''r''}} decays to zero [[Big O notation#Orders of common functions|exponentially]] in {{math|''r''}}. This was proved for percolation in three and more dimensions by {{harvtxt|Menshikov|1986}} and independently by {{harvtxt|Aizenman|Barsky|1987}}. In two dimensions, it formed part of Kesten's proof that {{math|''p''<sub>c</sub> {{=}} {{sfrac|1|2}}}}.<ref name="Kesten1982">{{Cite book|last1=Kesten|first1=Harry|title=Percolation Theory for Mathematicians|year=1982|doi=10.1007/978-1-4899-2730-9|isbn=978-0-8176-3107-9}}</ref> − − The main fact in the subcritical phase is "exponential decay". That is, when , the probability that a specific point (for example, the origin) is contained in an open cluster (meaning a maximal connected set of "open" edges of the graph) of size decays to zero exponentially in . This was proved for percolation in three and more dimensions by and independently by . In two dimensions, it formed part of Kesten's proof that }}. − − 亚临界阶段的主要事实是“指数衰减”。也就是说，当一个特定的点(例如，原点)包含在一个疏散簇中(意味着图的一个最大连通的“开”边集)的大小以指数方式衰减为零时。这被证明在三个和更多的维度渗透由和独立的。在两个维度中，它构成了 Kesten 的证明的一部分。 − − − − The [[dual graph]] of the square lattice {{math|'''ℤ'''<sup>2</sup>}} is also the square lattice. It follows that, in two dimensions, the supercritical phase is dual to a subcritical percolation process. This provides essentially full information about the supercritical model with {{math|''d'' {{=}} 2}}. The main result for the supercritical phase in three and more dimensions is that, for sufficiently large {{math|''N''}}, there is{{clarify|reason=With probability one?|date=April 2016}} an infinite open cluster in the two-dimensional slab {{math|'''ℤ'''<sup>2</sup> × [0, ''N'']<sup>''d'' − 2</sup>}}. This was proved by {{harvtxt|Grimmett|Marstrand|1990}}.<ref name="GrimmettMarstrand1990">{{cite journal|last1=Grimmett|first1=G. R.|last2=Marstrand|first2=J. M.|title=The Supercritical Phase of Percolation is Well Behaved|journal=Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume=430|issue=1879|year=1990|pages=439–457|issn=1364-5021|doi=10.1098/rspa.1990.0100|bibcode=1990RSPSA.430..439G}}</ref> − − The dual graph of the square lattice is also the square lattice. It follows that, in two dimensions, the supercritical phase is dual to a subcritical percolation process. This provides essentially full information about the supercritical model with 2}}. The main result for the supercritical phase in three and more dimensions is that, for sufficiently large , there is an infinite open cluster in the two-dimensional slab . This was proved by . − − 正方格的对偶图也是正方格。因此，在二维情况下，超临界相是双重的亚临界渗流过程。这提供了关于超临界模型的基本上完整的信息。超临界相在三维和更多维度的主要结果是，对于足够大来说，在二维平板中有一个无限的疏散星团。这一点得到了。 − − − − In two dimensions with {{math|''p'' < {{sfrac|1|2}}}}, there is with probability one a unique infinite closed cluster (a closed cluster is a maximal connected set of "closed" edges of the graph). Thus the subcritical phase may be described as finite open islands in an infinite closed ocean. When {{math|''p'' > {{sfrac|1|2}}}} just the opposite occurs, with finite closed islands in an infinite open ocean. The picture is more complicated when {{math|''d'' ≥ 3}} since {{math|''p''<sub>c</sub> < {{sfrac|1|2}}}}, and there is coexistence of infinite open and closed clusters for {{math|''p''}} between {{math|''p''<sub>c</sub>}} and {{math|1 − ''p''<sub>c</sub>}}. − − In two dimensions with }}, there is with probability one a unique infinite closed cluster (a closed cluster is a maximal connected set of "closed" edges of the graph). Thus the subcritical phase may be described as finite open islands in an infinite closed ocean. When }} just the opposite occurs, with finite closed islands in an infinite open ocean. The picture is more complicated when since }}, and there is coexistence of infinite open and closed clusters for between and . − − 在具有}的二维空间中，可能存在唯一的无限闭簇(闭簇是图的“闭”边的最大连通集)。因此，亚临界阶段可以描述为无限封闭海洋中的有限开放岛屿。当}正好相反，在无限的开阔海洋中有限的封闭岛屿。自从}之后，情况就变得更加复杂了，并且在和之间存在无限的开放和封闭集群的共存。 − − For the phase transition nature of percolation see Stauffer and Aharony<ref name=refo>{{cite journal |author= Stauffer, Dietrich; Aharony, Anthony |year=1944|title= Introduction to Percolation Theory |journal=Publ. Math. |issue=6|pages= 290–297|ISBN=978-0-7484-0253-3}}</ref> and Bunde and Havlin<ref name=reft>{{cite journal |author=Bunde A. and Havlin S.|year=1966|title=Fractals and Disordered Systems|journal=Springer|url=http://havlin.biu.ac.il/Shlomo%20Havlin%20books_fds.php}}</ref> . For percolation of networks see Cohen and Havlin<ref>{{cite journal |author= Cohen R. and Havlin S. |year=2010|title= Complex Networks: Structure, Robustness and Function |journal=Cambridge University Press |url=http://havlin.biu.ac.il/Shlomo%20Havlin%20books_com_net.php}}</ref>. − − For the phase transition nature of percolation see Stauffer and Aharony and Bunde and Havlin . For percolation of networks see Cohen and Havlin. − − 关于渗透的相变性质，请参见 Stauffer 和 Aharony 以及 Bunde 和 Havlin。有关网络的渗透，请参见 Cohen 和 Havlin。 − − − − ===Critical=== − − − − Percolation has a [[mathematical singularity|singularity]] at the critical point {{math|''p'' {{=}} ''p''<sub>c</sub>}} and many properties behave as a of power-law with <math>p-p_c</math>, near <math>p_c</math>. [[Critical scaling|Scaling theory]] predicts the existence of [[critical exponents]], depending on the number ''d'' of dimensions, that determine the class of the singularity. When {{math|''d'' {{=}} 2}} these predictions are backed up by arguments from [[conformal field theory]] and [[Schramm–Loewner evolution]], and include predicted numerical values for the exponents. The values of the exponent are given in <ref name=refo/><ref name=reft/>. Most of these predictions are conjectural except when the number {{math|''d''}} of dimensions satisfies either {{math|''d'' {{=}} 2}} or {{math|''d'' ≥ 19}}. They include: − − Percolation has a singularity at the critical point p<sub>c</sub>}} and many properties behave as a of power-law with <math>p-p_c</math>, near <math>p_c</math>. Scaling theory predicts the existence of critical exponents, depending on the number d of dimensions, that determine the class of the singularity. When 2}} these predictions are backed up by arguments from conformal field theory and Schramm–Loewner evolution, and include predicted numerical values for the exponents. The values of the exponent are given in . Most of these predictions are conjectural except when the number of dimensions satisfies either 2}} or . They include: − − Percolation 在临界点 p sub c / sub }处有一个奇点，许多性质与数学 p-p c / math 相近，表现为幂定律。尺度理论根据维数 d 预测了决定奇点类型的临界指数的存在。当2}这些预测由共形场论和 Schramm-Loewner 进化论的论据支持，并包括预测的指数数值。指数的值在。大多数这些预测都是推测的，除非维数满足2}或者。它们包括: − − − − * There are no infinite clusters (open or closed) − − * The probability that there is an open path from some fixed point (say the origin) to a distance of {{math|''r''}} decreases ''polynomially'', i.e. is [[big O notation|on the order of]] {{math|''r''<sup>''α''</sup>}} for some {{math|''α''}} − − ** {{math|''α''}} does not depend on the particular lattice chosen, or on other local parameters. It depends only on the dimension {{math|''d''}} (this is an instance of the [[Universality (dynamical systems)|universality]] principle). − − ** {{math|''α<sub>d</sub>''}} decreases from {{math|''d'' {{=}} 2}} until {{math|''d'' {{=}} 6}} and then stays fixed. − − ** {{math|''α''<sub>2</sub> {{=}} −{{sfrac|5|48}}}} − − ** {{math|''α''<sub>6</sub> {{=}} −1}}. − − * The shape of a large cluster in two dimensions is [[conformal map|conformally invariant]]. − − − − See {{harvtxt|Grimmett|1999}}.<ref name="Grimmett1999">{{Cite book|last1=Grimmett|first1=Geoffrey|title=Percolation|volume=321|year=1999|issn=0072-7830|doi=10.1007/978-3-662-03981-6|series=Grundlehren der mathematischen Wissenschaften|isbn=978-3-642-08442-3}}</ref> In 11 or more dimensions, these facts are largely proved using a technique known as the [[lace expansion]]. It is believed that a version of the lace expansion should be valid for 7 or more dimensions, perhaps with implications also for the threshold case of 6 dimensions. The connection of percolation to the lace expansion is found in {{harvtxt|Hara|Slade|1990}}.<ref name="HaraSlade1990">{{cite journal|last1=Hara|first1=Takashi|last2=Slade|first2=Gordon|title=Mean-field critical behaviour for percolation in high dimensions|journal=Communications in Mathematical Physics|volume=128|issue=2|year=1990|pages=333–391|issn=0010-3616|doi=10.1007/BF02108785|bibcode = 1990CMaPh.128..333H }}</ref> − − See . In 11 or more dimensions, these facts are largely proved using a technique known as the lace expansion. It is believed that a version of the lace expansion should be valid for 7 or more dimensions, perhaps with implications also for the threshold case of 6 dimensions. The connection of percolation to the lace expansion is found in . − − 看。在11个或更多的维度上，这些事实在很大程度上被一种称为花边展开的技术所证明。人们认为，一个版本的花边扩展应该是有效的7个或更多的维度，也许还牵涉到门槛情况下的6个维度。渗透与花边展开的联系见于。 − − − − In two dimensions, the first fact ("no percolation in the critical phase") is proved for many lattices, using duality. Substantial progress has been made on two-dimensional percolation through the conjecture of [[Oded Schramm]] that the [[scaling limit]] of a large cluster may be described in terms of a [[Schramm–Loewner evolution]]. This conjecture was proved by {{harvtxt|Smirnov|2001}}<ref name="Smirnov2001">{{cite journal|last1=Smirnov|first1=Stanislav|title=Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits|journal=Comptes Rendus de l'Académie des Sciences, Série I|volume=333|issue=3|year=2001|pages=239–244|issn=0764-4442|doi=10.1016/S0764-4442(01)01991-7|bibcode = 2001CRASM.333..239S |arxiv=0909.4499|citeseerx=10.1.1.246.2739}}</ref> in the special case of site percolation on the triangular lattice. − − In two dimensions, the first fact ("no percolation in the critical phase") is proved for many lattices, using duality. Substantial progress has been made on two-dimensional percolation through the conjecture of Oded Schramm that the scaling limit of a large cluster may be described in terms of a Schramm–Loewner evolution. This conjecture was proved by in the special case of site percolation on the triangular lattice. − − 在二维中，利用对偶证明了许多晶格的第一个事实(“临界相中没有渗流”)。二维渗流问题已经取得了实质性的进展，欧德 · 施拉姆推测大星系团的标度极限可以用施拉姆 -- 洛夫纳演化来描述。在三角格点上的场地渗流的特殊情况下，证明了这一猜想。 − − − − ==Different models== − − *[[Directed percolation]] that models the effect of [[gravity|gravitational forces acting on the liquid]] was also introduced in {{harvtxt|Broadbent|Hammersley|1957}},<ref name="BroadbentHammersley2008"/> and has connections with the [[contact process (mathematics)|contact process]]. − − *The first model studied was Bernoulli percolation. In this model all bonds are independent. This model is called bond percolation by physicists. − − *A generalization was next introduced as the [[Fortuin–Kasteleyn random cluster model]], which has many connections with the [[Ising model]] and other [[Potts model]]s. − − *Bernoulli (bond) percolation on [[complete graph]]s is an example of a [[random graph]]. The critical probability is {{math|''p'' {{=}} {{sfrac|1|''N''}}}}, where {{math|''N''}} is the number of vertices (sites) of the graph. − − *[[Bootstrap percolation]] removes active cells from clusters when they have too few active neighbors, and looks at the connectivity of the remaining cells.<ref>{{citation − − | last = Adler | first = Joan | authorlink = Joan Adler − − | last = Adler | first = Joan | authorlink = Joan Adler − − 作者: 琼 · 阿德勒 − − | doi = 10.1016/0378-4371(91)90295-n − − | doi = 10.1016/0378-4371(91)90295-n − − | doi 10.1016 / 0378-4371(91)90295-n − − | issue = 3 − − | issue = 3 − − 第三期 − − | journal = Physica A: Statistical Mechanics and Its Applications − − | journal = Physica A: Statistical Mechanics and Its Applications − − 物理学期刊 a: 统计力学及其应用 − − | pages = 453–470 − − | pages = 453–470 − − 第453-470页 − − | title = Bootstrap percolation − − | title = Bootstrap percolation − − | title Bootstrap percolation − − | volume = 171 − − | volume = 171 − − 第171卷 − − | year = 1991| bibcode = 1991PhyA..171..453A}}.</ref> − − | year = 1991| bibcode = 1991PhyA..171..453A}}.</ref> − − 1991年 | bibcode 1991 phya. . 171. . 453 a }} . / ref − − *[[First passage percolation]]. − − *[[Invasion percolation]]. − − * Percolation with dependency links was introduced by Parshani et al.<ref name="ParshaniBuldyrev2010">{{cite journal|last1=Parshani|first1=R.|last2=Buldyrev|first2=S. V.|last3=Havlin|first3=S.|title=Critical effect of dependency groups on the function of networks|journal=Proceedings of the National Academy of Sciences|volume=108|issue=3|year=2010|pages=1007–1010|issn=0027-8424|doi=10.1073/pnas.1008404108|arxiv = 1010.4498 |bibcode = 2011PNAS..108.1007P|pmid=21191103|pmc=3024657}}</ref> − − * Percolation and opinion spreading model.<ref name="ShaoHavlin2009">{{cite journal|last1=Shao|first1=Jia|last2=Havlin|first2=Shlomo|last3=Stanley|first3=H. Eugene|title=Dynamic Opinion Model and Invasion Percolation|journal=Physical Review Letters|volume=103|issue=1|pages=018701|year=2009|issn=0031-9007|doi=10.1103/PhysRevLett.103.018701|pmid=19659181|bibcode = 2009PhRvL.103a8701S }}</ref> − − * Percolation under localized attack was introduced by Berezin et al.<ref name="BerezinBashan2015">{{cite journal|last1=Berezin|first1=Yehiel|last2=Bashan|first2=Amir|last3=Danziger|first3=Michael M.|last4=Li|first4=Daqing|last5=Havlin|first5=Shlomo|title=Localized attacks on spatially embedded networks with dependencies|journal=Scientific Reports|volume=5|issue=1|pages=8934|year=2015|issn=2045-2322|doi=10.1038/srep08934|pmid=25757572|pmc=4355725|bibcode=2015NatSR...5E8934B}}</ref> See also Shao et al.<ref>{{Cite journal|last=Shao|first=S.|last2=Huang|first2=X.|last3=Stanley|first3=H.E.|last4=Havlin|first4=S.|date=2015|title=Percolation of localized attack on complex networks|url=|journal=New J. Phys.|volume=17|issue=2|pages=023049|doi=10.1088/1367-2630/17/2/023049|arxiv=1412.3124}}</ref> − − * Percolation of traffic in cities was introduced by Daqing Li et al.<ref name="LiFu2015">{{cite journal|last1=Li|first1=Daqing|last2=Fu|first2=Bowen|last3=Wang|first3=Yunpeng|last4=Lu|first4=Guangquan|last5=Berezin|first5=Yehiel|last6=Stanley|first6=H. Eugene|last7=Havlin|first7=Shlomo|title=Percolation transition in dynamical traffic network with evolving critical bottlenecks|journal=Proceedings of the National Academy of Sciences|volume=112|issue=3|year=2015|pages=669–672|issn=0027-8424|doi=10.1073/pnas.1419185112|pmid=25552558|pmc=4311803|bibcode=2015PNAS..112..669L}}</ref> − − * Introducing recovery of nodes and links in percolation.<ref name="MajdandzicPodobnik2013">{{cite journal|last1=Majdandzic|first1=Antonio|last2=Podobnik|first2=Boris|last3=Buldyrev|first3=Sergey V.|last4=Kenett|first4=Dror Y.|last5=Havlin|first5=Shlomo|last6=Eugene Stanley|first6=H.|title=Spontaneous recovery in dynamical networks|journal=Nature Physics|volume=10|issue=1|year=2013|pages=34–38|issn=1745-2473|doi=10.1038/nphys2819|bibcode=2014NatPh..10...34M}}</ref> − − * Percolation in 2d with a characteristic link length.<ref>{{Cite journal|last=Danziger|first=Michael M.|last2=Shekhtman|first2=Louis M.|last3=Berezin|first3=Yehiel|last4=Havlin|first4=Shlomo|date=2016|title=The effect of spatiality on multiplex networks|journal=EPL (Europhysics Letters)|language=en|volume=115|issue=3|pages=36002|doi=10.1209/0295-5075/115/36002|issn=0295-5075|bibcode=2016EL....11536002D|arxiv=1505.01688}}</ref> This percolation show a novel phenomena of critical stretching phenomena near critical percolation.<ref>{{cite journal |author=Ivan Bonamassa, Bnaya Gross, Michael M Danziger, Shlomo Havlin|year=2019|title= Critical stretching of mean-field regimes in spatial networks |journal=Phys. Rev. Lett. |issue=123|volume=8|page=088301}}</ref> − − *A generalized percolation model that introduces a fraction of reinforced nodes in a network that can function and support their neighborhood was introduced by Yanqing Hu et al.<ref>{{Cite journal|last=Yuan|first=Xin|last2=Hu|first2=Yanqing|last3=Stanley|first3=H. Eugene|last4=Havlin|first4=Shlomo|date=2017-03-28|title=Eradicating catastrophic collapse in interdependent networks via reinforced nodes|journal=Proceedings of the National Academy of Sciences|language=en|volume=114|issue=13|pages=3311–3315|doi=10.1073/pnas.1621369114|issn=0027-8424|pmid=28289204|pmc=5380073|arxiv=1605.04217|bibcode=2017PNAS..114.3311Y}}</ref> − − <!-- LOTS of todo! Other models of percolation. references. Percolation on graphs.--> − − <!-- LOTS of todo! Other models of percolation. references. Percolation on graphs.--> − − ! ——有很多事情要做！其他渗滤模式。参考。图的渗透。—— − − − − == Applications == − − − − === In biology, biochemistry (physical virology), and nanomedicine === − − Percolation theory has been used to successfully predict the fragmentation of biological virus shells (capsids) <ref>{{cite journal | last1 = Brunk | first1 = N. E. | last2 = Lee | first2 = L. S. | last3 = Glazier | first3 = J. A. | last4 = Butske | first4 = W. | last5 = Zlotnick | first5 = A. | year = 2018 | title = Molecular Jenga: the percolation phase transition (collapse) in virus capsids | journal = Physical Biology | volume = 15 | issue = 5| page = 056005 | doi=10.1088/1478-3975/aac194| pmid = 29714713 | pmc = 6004236 }}</ref>, with the percolation threshold of [[Hepatitis B]] virus [[capsid]] predicted and detected experimentally. <ref>{{cite journal | last1 = Lee | first1 = L. S. | last2 = Brunk | first2 = N. | last3 = Haywood | first3 = D. G. | last4 = Keifer | first4 = D. | last5 = Pierson | first5 = E. | last6 = Kondylis | first6 = P. | last7 = Zlotnick | first7 = A. | year = 2017 | title = A molecular breadboard: Removal and replacement of subunits in a hepatitis B virus capsid | journal = Protein Science | volume = 26 | issue = 11| pages = 2170–2180 | doi=10.1002/pro.3265| pmid = 28795465 | pmc = 5654856 }}</ref> When a critical number of subunits has been randomly removed from the nanoscopic shell, it fragments and this fragmentation may be detected using Charge Detection Mass Spectroscopy (CDMS) among other single-particle techniques. This is a molecular analog to the common board game [[Jenga]], and has relevance to virus disassembly. − − Percolation theory has been used to successfully predict the fragmentation of biological virus shells (capsids) , with the percolation threshold of Hepatitis B virus capsid predicted and detected experimentally. When a critical number of subunits has been randomly removed from the nanoscopic shell, it fragments and this fragmentation may be detected using Charge Detection Mass Spectroscopy (CDMS) among other single-particle techniques. This is a molecular analog to the common board game Jenga, and has relevance to virus disassembly. − − 逾渗理论已经成功地用于预测生物病毒外壳(衣壳)的碎裂，并通过实验预测和检测了乙型肝炎病毒外壳的逾渗阈值。当一个临界数量的亚基被随机从纳米级的壳层中移除时，它会碎裂，这种碎裂可以用电荷检测质谱法(CDMS)和其他单粒子技术检测到。这是一个普通的桌面游戏叠叠乐的分子模拟物，与病毒拆解有关。 − − − − === In ecology === − − Percolation theory has been applied to studies of how environment fragmentation impacts animal habitats<ref>{{Cite journal|last=Boswell|first=G. P.|last2=Britton|first2=N. F.|last3=Franks|first3=N. R.|date=1998-10-22|title=Habitat fragmentation, percolation theory and the conservation of a keystone species|journal=Proceedings of the Royal Society of London B: Biological Sciences|language=en|volume=265|issue=1409|pages=1921–1925|doi=10.1098/rspb.1998.0521|issn=0962-8452|pmc=1689475}}</ref> and models of how the plague bacterium ''[[Yersinia pestis]]'' spreads.<ref>{{Cite journal|last=Davis|first=S.|last2=Trapman|first2=P.|last3=Leirs|first3=H.|last4=Begon|first4=M.|last5=Heesterbeek|first5=J. a. P.|date=2008-07-31|title=The abundance threshold for plague as a critical percolation phenomenon|journal=Nature|volume=454|issue=7204|pages=634–637|doi=10.1038/nature07053|issn=1476-4687|pmid=18668107|hdl=1874/29683|hdl-access=free}}</ref> − − Percolation theory has been applied to studies of how environment fragmentation impacts animal habitats and models of how the plague bacterium Yersinia pestis spreads. − − 逾渗理论已被应用于研究环境破碎化如何影响动物栖息地和鼠疫耶尔森菌如何传播的模型。 − − − − ==See also== − − {{Div col}} − − * {{annotated link|Continuum percolation theory}} − − * {{annotated link|Critical exponent}} − − * {{annotated link|Directed percolation}} − − * {{annotated link|Erdős–Rényi model}} − − * {{annotated link|Fractal}} − − * {{annotated link|Giant component}} − − * {{annotated link|Graph theory}} − − * {{annotated link|Interdependent networks}} − − * {{annotated link|Invasion percolation}} − − * {{annotated link|Network theory}} − − * {{annotated link|Percolation threshold}} − − * {{annotated link|Percolation critical exponents}} − − * {{annotated link|Scale-free network}} − − * {{annotated link|Shortest path problem}} − − {{Div col end}} − − − − ==References== − − {{Reflist|30em}} − − − − ===Further reading=== − − {{Div col|colwidth=30em|small=yes}} − − *{{Citation | last1=Aizenman | first1=Michael | authorlink1=Michael Aizenman | last2=Barsky |first2=David | − − title=Sharpness of the phase transition in percolation models | − − title=Sharpness of the phase transition in percolation models | − − 逾渗模型中相变的锐度 | − − journal=Communications in Mathematical Physics | − − journal=Communications in Mathematical Physics | − − 数学物理学通信杂志 | − − volume=108 | − − volume=108 | − − 第108卷 − − year=1987 | − − year=1987 | − − 1987年 − − pages=489–526 | doi=10.1007/BF01212322|bibcode = 1987CMaPh.108..489A | issue=3 }} − − pages=489–526 | doi=10.1007/BF01212322|bibcode = 1987CMaPh.108..489A | issue=3 }} − − 页489-526 | doi 10.1007 / BF01212322 | bibcode 1987CMaPh. 108.489 a | issue 3} − − *{{Citation | url=http://www.cambridge.org/catalogue/catalogue.asp?isbn=0521872324 | − − title=Percolation | first1=Béla | last1=Bollobás | authorlink1=Béla Bollobás | − − title=Percolation | first1=Béla | last1=Bollobás | authorlink1=Béla Bollobás | − − 1 b la | last1 bollob s | authorlink1 b la bollob s | − − first2=Oliver | last2=Riordan | publisher =Cambridge University Press | year= 2006 | isbn=978-0521872324}} − − first2=Oliver | last2=Riordan | publisher =Cambridge University Press | year= 2006 | isbn=978-0521872324}} − − 出版商剑桥大学出版社2006年 − − *{{Citation | last1=Broadbent | first1=Simon | last2=Hammersley | first2=John | authorlink2=John Hammersley | title=Percolation processes I. Crystals and mazes | − − journal=Proceedings of the Cambridge Philosophical Society | − − journal=Proceedings of the Cambridge Philosophical Society | − − 剑桥哲学学会会报 | − − volume=53 | issue=3 | pages=629–641 | year=1957|bibcode = 1957PCPS...53..629B |doi = 10.1017/S0305004100032680 }} − − volume=53 | issue=3 | pages=629–641 | year=1957|bibcode = 1957PCPS...53..629B |doi = 10.1017/S0305004100032680 }} − − 53卷 | 第3期 | 第629-641页 | 1957年 | bibcode 1957PCPS... 53. . 629 b | doi 10.1017 / S0305004100032680} − − *{{Citation | author= Bunde A. and [[Shlomo Havlin|Havlin S.]] | title=Fractals and Disordered Systems | − − publisher= Springer| year=1996| url= http://havlin.biu.ac.il/Shlomo%20Havlin%20books_fds.php}} − − publisher= Springer| year=1996| url= http://havlin.biu.ac.il/Shlomo%20Havlin%20books_fds.php}} − − 出版商 Springer | 1996 | url / http://havlin.biu.ac.il/shlomo%20havlin%20books_fds.php } − − *{{Citation | author= Cohen R. and [[Shlomo Havlin|Havlin S.]] | title=Complex Networks: Structure, Robustness and Function | publisher= Cambridge University Press | year=2010| url= http://havlin.biu.ac.il/Shlomo%20Havlin%20books_com_net.php}} − − *{{Citation | last1=Grimmett | first1=Geoffrey |authorlink1=Geoffrey Grimmett| − − title=Percolation | publisher=Springer | year=1999 | − − title=Percolation | publisher=Springer | year=1999 | − − 标题: Percolation | 出版商 Springer | 1999年 | − − url=http://www.statslab.cam.ac.uk/~grg/papers/perc/perc.html}} − − url=http://www.statslab.cam.ac.uk/~grg/papers/perc/perc.html}} − − Http://www.statslab.cam.ac.uk/~grg/papers/perc/perc.html − − *{{Citation | last1=Grimmett | first1=Geoffrey | authorlink1=Geoffrey Grimmett | − − last2=Marstrand | first2=John | − − last2=Marstrand | first2=John | − − 马斯特兰德2 | first2 John | − − title=The supercritical phase of percolation is well behaved | − − title=The supercritical phase of percolation is well behaved | − − 超临界渗滤相表现良好 − − journal=[[Proceedings of the Royal Society A]] | − − journal=Proceedings of the Royal Society A | − − 皇家学会报告杂志 a | − − volume=430 | − − volume=430 | − − 第430卷 − − year=1990 | − − year=1990 | − − 1990年 − − pages=439–457 | doi=10.1098/rspa.1990.0100|bibcode = 1990RSPSA.430..439G | issue=1879 }} − − pages=439–457 | doi=10.1098/rspa.1990.0100|bibcode = 1990RSPSA.430..439G | issue=1879 }} − − 页439-457 | doi 10.1098 / rspa. 1990.0100 | bibcode 1990RSPSA. 430.439 g | issue 1879} − − *{{Citation | last1=Hara | first1=Takashi | last2=Slade | first2=Gordon | year=1990 | − − title=Mean-field critical behaviour for percolation in high dimensions | − − title=Mean-field critical behaviour for percolation in high dimensions | − − 高维渗流的平均场临界行为 | − − journal=Communications in Mathematical Physics | volume=128 | pages=333–391 | doi=10.1007/BF02108785|bibcode = 1990CMaPh.128..333H | issue=2 }} − − journal=Communications in Mathematical Physics | volume=128 | pages=333–391 | doi=10.1007/BF02108785|bibcode = 1990CMaPh.128..333H | issue=2 }} − − 数学物理通讯杂志 | 第128卷 | 第333-391页 | doi 10.1007 / BF02108785 | bibcode 1990CMaPh. 128.333 h | issue 2} − − *{{Citation | title=Percolation theory for mathematicians | last1=Kesten | − − first1=Harry | publisher=Birkhauser | year=1982 | authorlink1=Harry Kesten}} − − first1=Harry | publisher=Birkhauser | year=1982 | authorlink1=Harry Kesten}} − − 1 Harry | publisher Birkhauser | year 1982 | authorlink1 Harry Kesten } − − *{{Citation | last1=Menshikov | first1=Mikhail | authorlink1=Mikhail Vasiliyevich Menshikov| − − title=Coincidence of critical points in percolation problems | − − title=Coincidence of critical points in percolation problems | − − 渗流问题中临界点的巧合 | − − journal=Soviet Mathematics - Doklady | − − journal=Soviet Mathematics - Doklady | − − 苏联数学杂志 | − − volume=33 | − − volume=33 | − − 第33卷 − − year=1986 | − − year=1986 | − − 1986年 − − pages=856–859}} − − pages=856–859}} − − 第856-859页} − − *{{Citation | last1=Smirnov | first1=Stanislav | authorlink1=Stanislav Smirnov | − − title=Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits | year=2001 | journal=Comptes Rendus de l'Académie des Sciences | volume=333 | pages=239–244 | doi=10.1016/S0764-4442(01)01991-7|bibcode = 2001CRASM.333..239S | issue=3 | arxiv=0909.4499 | citeseerx=10.1.1.246.2739 }} − − title=Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits | year=2001 | journal=Comptes Rendus de l'Académie des Sciences | volume=333 | pages=239–244 | doi=10.1016/S0764-4442(01)01991-7|bibcode = 2001CRASM.333..239S | issue=3 | arxiv=0909.4499 | citeseerx=10.1.1.246.2739 }} − − 标题平面上的临界渗透: 共形不变性，Cardy 公式，比例极限 | 2001年 | 杂志 Comptes Rendus de l’ acad mie des Sciences | 卷333 | 页239-244 | doi 10.1016 / S0764-4442(01)01991-7 | bibcode 2001CRASM. 333。 239 s | issue 3 | arxiv 0909.4499 | citeserx 10.1.1.246.2739} − − *{{citation|last1= Stauffer|first1= Dietrich |last2= Aharony|first2= Anthony|title= Introduction to Percolation Theory|edition=2nd|publisher=CRC Press|year=1994|isbn=978-0-7484-0253-3}} − − {{Div col end}} − − {{#seo: − |keywords=渗流模型,临界现象,标度律 − |description=渗流模型介绍 − }}
无编辑摘要
==标度行为==
==标度行为==
标度行为（Scaling）是现在复杂系统研究中的一个非常典型的现象，它体现为系统的若干宏观指标或者某个变量的分布函数满足具有不同幂指数的幂律行为。例如，社会上的收入分布就满足著名的Pareto律，也就是收入分布的密度函数<math>f(x)\sim x^{-1.75}</math>，这是一个幂律分布；再如英语单词中，按其出入频率从大到小排序，则排序为r的单词的出现频率为<math>f(r)\sim r^{-1}</math>，这是著名的Zipf律。另外还有两变量的关系也是幂律的，如生物体新陈代谢和它的Body Size之间满足3/4幂律关系，即<math>F\sim M^{3/4}</math>，这被称为[[Kleiber律]]。再如大家熟知的无标度网络，真实世界很多复杂网络的度分布都满足幂律分布，即<math>p(x)\sim x^{-3}</math>.
标度行为（Scaling）是现在复杂系统研究中的一个非常典型的现象，它体现为系统的若干宏观指标或者某个变量的分布函数满足具有不同幂指数的幂律行为。例如，社会上的收入分布就满足著名的Pareto律，也就是收入分布的密度函数<math>f(x)\sim x^{-1.75}</math>，这是一个幂律分布；再如英语单词中，按其出入频率从大到小排序，则排序为r的单词的出现频率为<math>f(r)\sim r^{-1}</math>，这是著名的Zipf律。另外还有两变量的关系也是幂律的，如生物体新陈代谢和它的Body Size之间满足3/4幂律关系，即<math>F\sim M^{3/4}</math>，这被称为[[Kleiber律]]。再如大家熟知的无标度网络，真实世界很多复杂网络的度分布都满足幂律分布，即<math>p(x)\sim x^{-3}</math>.