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In statistical physics and mathematics, percolation theory describes the behavior of a network when nodes or links are added. This is a geometric type of phase transition, since at a critical fraction of addition the network of small, disconnected clusters merge into significantly larger connected, so-called spanning cluster. The applications of percolation theory to materials science and in many other disciplines are discussed here and in the articles network theory and percolation.

Introduction[]

A three-dimensional site percolation graph

Bond percolation in a square lattice from p=0.3 to p=0.52

The Flory–Stockmayer theory was the first theory investigating percolation processes.^[1]

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 n × n × n vertices, usually called "sites", in which the edge or "bonds" between each two neighbors may be open (allowing the liquid through) with probability p, or closed with probability 1 – p, and they are assumed to be independent. Therefore, for a given 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 n is of primary interest. This problem, called now bond percolation, was introduced in the mathematics literature by Broadbent & Hammersley (1957),^[2] and has been studied intensively by mathematicians and physicists since then.

In a slightly different mathematical model for obtaining a random graph, a site is "occupied" with probability p or "empty" (in which case its edges are removed) with probability 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 1 – p of failures the graph will become disconnected (no large component).

A 3D tube network percolation determination

The same questions can be asked for any lattice dimension. As is quite typical, it is actually easier to examine infinite 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 p, the probability that an infinite cluster exists is either zero or one. Since this probability is an increasing function of p (proof via coupling argument), there must be a critical p (denoted by p_c) 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 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 p.

Detail of a bond percolation on the square lattice in two dimensions with percolation probability p = 0.51

For most infinite lattice graphs, p_c cannot be calculated exactly, though in some cases p_c there is an exact value. For example:

• for the square lattice ℤ² in two dimensions, p_c = 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,^[3] see Kesten (1982). For site percolation, the value of p_c is not known from analytic derivation but only via simulations of large lattices.^[4]  
• A limit case for lattices in high dimensions is given by the Bethe lattice, whose threshold is at p_c = 1/z − 1 for a coordination number z. In other words: for the regular tree of degree ${\displaystyle z}$, ${\displaystyle p_{c}}$ is equal to ${\displaystyle 1/(z-1)}$.

• For random Erdős–Rényi networks of average degree ${\displaystyle \langle k\rangle }$, p_c = 1/⟨k⟩.^[5][6][7]

Universality[]

The universality principle states that the numerical value of p_c is determined by the local structure of the graph, whereas the behavior near the critical threshold, p_c, 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 exponent for all 2d lattices. This universality means that for a given dimension, the various critical exponents, the fractal dimension of the clusters at p_c 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.^[8][9]

Phases[]

Subcritical and supercritical[]

The main fact in the subcritical phase is "exponential decay". That is, when p < p_c, 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 r decays to zero exponentially in r. This was proved for percolation in three and more dimensions by Menshikov (1986) and independently by Aizenman & Barsky (1987). In two dimensions, it formed part of Kesten's proof that p_c = 1/2.^[10]

The dual graph of the square lattice ℤ² 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 d = 2. The main result for the supercritical phase in three and more dimensions is that, for sufficiently large N, there is^{[clarification needed]} an infinite open cluster in the two-dimensional slab ℤ² × [0, N]^{d − 2}. This was proved by Grimmett & Marstrand (1990).^[11]

In two dimensions with p < 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 p > 1/2 just the opposite occurs, with finite closed islands in an infinite open ocean. The picture is more complicated when d ≥ 3 since p_c < 1/2, and there is coexistence of infinite open and closed clusters for p between p_c and 1 − p_c. For the phase transition nature of percolation see Stauffer and Aharony^[12] and Bunde and Havlin^[13] . For percolation of networks see Cohen and Havlin.^[14]

Criticality[]

Zoom in a critical percolation cluster (Click to animate)

Percolation has a singularity at the critical point p = p_c and many properties behave as of a power-law with ${\displaystyle p-p_{c}}$, near ${\displaystyle p_{c}}$. Scaling theory predicts the existence of critical exponents, depending on the number d of dimensions, that determine the class of the singularity. When 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 exponents are given in ^[12] and.^[13] Most of these predictions are conjectural except when the number d of dimensions satisfies either d = 2 or d ≥ 6. They include:

• There are no infinite clusters (open or closed)
• The probability that there is an open path from some fixed point (say the origin) to a distance of r decreases polynomially, i.e. is on the order of r^α for some α
  • α does not depend on the particular lattice chosen, or on other local parameters. It depends only on the dimension d (this is an instance of the universality principle).
  • α_d decreases from d = 2 until d = 6 and then stays fixed.
  • α₂ = −5/48
  • α₆ = −1.
• The shape of a large cluster in two dimensions is conformally invariant.

See Grimmett (1999).^[15] In 11 or more dimensions, these facts are largely proved using a technique known as the . 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 Hara & Slade (1990).^[16]

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 Smirnov (2001)^[17] in the special case of site percolation on the triangular lattice.

Different models[]

• Directed percolation that models the effect of gravitational forces acting on the liquid was also introduced in Broadbent & Hammersley (1957),^[2] and has connections with the 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 models.
• Bernoulli (bond) percolation on complete graphs is an example of a random graph. The critical probability is p = 1/N, where 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.^[18]
• First passage percolation.
• Invasion percolation.
• Percolation with dependency links was introduced by Parshani et al.^[19]
• Percolation and opinion spreading model.^[20]
• Percolation under localized attack was introduced by Berezin et al.^[21] See also Shao et al.^[22]
• Percolation of modular networks has been studied by Shay et al.^[23] and Dong et al.^[24]
• Percolation on spatial modular structures has been studied by Gross et al.^[25]
• Percolation of traffic in cities was introduced by Daqing Li et al.^[26] and by Zeng et al.^[27]
• Introducing recovery of nodes and links in percolation.^[28]
• Percolation in 2d with a characteristic link length.^[29] The model displays a new phenomenon, called critical stretching, in the structure and functioning of the network near its critical percolation threshold.^[30] 
• A generalized and decentralized percolation model that introduces a fraction of reinforced nodes in a network that can function and support their neighborhood has been introduced by Yuan et al.^[31]

Applications[]

In biology, biochemistry, and physical virology[]

Percolation theory has been used to successfully predict the fragmentation of biological virus shells (capsids),^[32][33] with the fragmentation threshold of Hepatitis B virus capsid predicted and detected experimentally.^[34] 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 the broader study of virus disassembly. Interestingly, more stable viral particles (tilings with greater fragmentation thresholds) are found in greater abundance in nature.^[32]

In ecology[]

Percolation theory has been applied to studies of how environment fragmentation impacts animal habitats^[35] and models of how the plague bacterium Yersinia pestis spreads.^[36]

Percolation of multilayer interdependent networks[]

Buldyrev et al.^[37] developed a framework to study percolation in multilayer networks with dependency links between the layers. New physical phenomena has been found, including abrupt transitions and cascading failures.^[38] When the networks are embedded in space they become extremely vulnerable even for a very small fraction of dependency links^[39] and for localized attacks on a zero fraction of nodes.^[40][41] When recovery of nodes is introduced, a rich phase diagram is found that include multicritical points, hysteresis and metastable regimes.^[42][43]

In traffic[]

In recent articles, percolation theory has been applied to study traffic in a city. The quality of the global traffic in a city at a given time can be characterized by a single parameter, the percolation critical threshold. The critical threshold represents the velocity below which one can travel in a large fraction of the city network. Above this threshold the city network break into clusters of many sizes and one can travel within relatively small neighborhoods. This novel method is also able to identify repetitive traffic bottlenecks.^[44] Critical exponents characterizing the cluster size distribution of good traffic are similar to those of percolation theory.^[45] It is also found that during rush hours the traffic network can have several metastable states of different network sizes and the alternate between these states.^[46] An empirical study regarding the spatial-temporal size distribution of traffic jams has been performed by Zhang et al.^[47] They found an approximate universal power law for the jam sizes distribution in different cities. A method to identify functional clusters of spatial-temporal streets that represent fluent traffic flow in a city has been developed by Serok et al.^[48]

See also[]

References[]

• Aizenman, Michael; Barsky, David (1987), "Sharpness of the phase transition in percolation models", Communications in Mathematical Physics, 108 (3): 489–526, Bibcode:1987CMaPh.108..489A, doi:10.1007/BF01212322, S2CID 35592821
• Menshikov, Mikhail (1986), "Coincidence of critical points in percolation problems", Soviet Mathematics - Doklady, 33: 856–859

Further reading[]

External links[]

• PercoVIS: a Mac OS X program to visualize percolation on networks in real time
• Interactive Percolation
• Nanohub online course on Percolation Theory
• Introduction to Percolation Theory: short course by Shlomo Havlin