Percolation theory
In statistical physics and mathematics, percolation theory describes the behaviour of connected clusters in a random graph. The applications of percolation theory to materials science and other domains are discussed in the article percolation.
Introduction
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),^{[1]} 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?
Of course 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.
In some cases p_{c} may be calculated explicitly. For example, for the square lattice Z^{2} 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,^{[2]} see Kesten (1982). A limit case for lattices in many dimensions is given by the Bethe lattice, whose threshold is at p_{c} = 1/(z − 1) for a coordination number z. For most infinite lattice graphs, p_{c} cannot be calculated exactly.
Universality
The universality principle states that the numerical value of p_{c} is determined by the local structure of the graph, whereas the kind of behavior of clusters that is observed below, at, and above p_{c} is independent of the local structure, and therefore, in some sense these behaviors are more natural to consider than p_{c} itself . This universality also 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 albeit 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.^{[3]}
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.^{[4]}
The dual graph of the square lattice Z^{2} 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 an infinite open cluster in the two-dimensional slab Z^{2} × [0, N]^{d−2}. This was proved by Grimmett & Marstrand (1990).^{[5]}
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}.
Critical
The model has a singularity at the critical point p = p_{c} believed to be of power-law type. 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. Most of these predictions are conjectural except when the number d of dimensions satisfies either d = 2 or d ≥ 19. 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.
- α_{2} = −5/48
- α_{6} = −1.
- The shape of a large cluster in two dimensions is conformally invariant.
See Grimmett (1999).^{[6]} In dimension ≥ 19, 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 Hara & Slade (1990).^{[7]}
In dimension 2, 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)^{[8]} in the special case of site percolation on the triangular lattice.
Different models
- The first model studied was Bernoulli percolation. In this model all bonds are independent. This model is called bond percolation by physicists.
- A generalization next was 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.^{[9]}
- Directed percolation, which has connections with the contact process.
- First passage percolation.
- Invasion percolation.
- Percolation with dependency links was introduced by Parshani et al.^{[10]}
- Opinion Model^{[11]}
See also
- Continuum percolation theory
- Critical exponents
- Directed percolation
- Erdős–Rényi model
- Fractal
- Giant component
- Graph Theory
- Invasion percolation
- Network theory
- Percolation threshold
- Percolation critical exponents
- Scale-free network
- Shortest path
References
- ↑ Broadbent, S. R.; Hammersley, J. M. (2008). "Percolation processes". Mathematical Proceedings of the Cambridge Philosophical Society. 53 (03): 629. Bibcode:1957PCPS...53..629B. doi:10.1017/S0305004100032680. ISSN 0305-0041.
- ↑ Bollobás, Béla; Riordan, Oliver (2006). "Sharp thresholds and percolation in the plane". Random Structures and Algorithms. 29 (4): 524–548. doi:10.1002/rsa.20134. ISSN 1042-9832.
- ↑ M. K. Hassan and M. M. Rahman, “Percolation on a multifractal scale-free planar stochastic lattice and its universality class” Phys. Rev. E (Rapid Communication) 92 040101(R) (2015) https://doi.org/10.1103/PhysRevE.92.040101; M. K. Hassan and M. M. Rahman, “Universality class of site and bond percolation on multi-multifractal scale-free planar stochastic lattice” Phys. Rev. E, 94 042109 (2016) https://doi.org/10.1103/PhysRevE.94.042109
- ↑ Kesten, Harry (1982). doi:10.1007/978-1-4899-2730-9. Missing or empty
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(help) - ↑ Grimmett, G. R.; Marstrand, J. M. (1990). "The Supercritical Phase of Percolation is Well Behaved". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 430 (1879): 439–457. doi:10.1098/rspa.1990.0100. ISSN 1364-5021.
- ↑ Grimmett, Geoffrey (1999). 321. doi:10.1007/978-3-662-03981-6. ISSN 0072-7830. Missing or empty
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(help) - ↑ Hara, Takashi; Slade, Gordon (1990). "Mean-field critical behaviour for percolation in high dimensions". Communications in Mathematical Physics. 128 (2): 333–391. Bibcode:1990CMaPh.128..333H. doi:10.1007/BF02108785. ISSN 0010-3616.
- ↑ Smirnov, Stanislav (2001). "Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits". Comptes Rendus de l'Académie des Sciences - Series I - Mathematics. 333 (3): 239–244. Bibcode:2001CRASM.333..239S. doi:10.1016/S0764-4442(01)01991-7. ISSN 0764-4442.
- ↑ Adler, Joan (1991), "Bootstrap percolation", Physica A: Statistical Mechanics and its Applications, 171 (3): 453–470, doi:10.1016/0378-4371(91)90295-n.
- ↑ Parshani, R.; Buldyrev, S. V.; Havlin, S. (2010). "Critical effect of dependency groups on the function of networks". Proceedings of the National Academy of Sciences. 108 (3): 1007–1010. arXiv:1010.4498. Bibcode:2011PNAS..108.1007P. doi:10.1073/pnas.1008404108. ISSN 0027-8424.
- ↑ Shao, Jia; Havlin, Shlomo; Stanley, H. Eugene (2009). "Dynamic Opinion Model and Invasion Percolation". Physical Review Letters. 103 (1). Bibcode:2009PhRvL.103a8701S. doi:10.1103/PhysRevLett.103.018701. ISSN 0031-9007.
- 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
- Bollobás, Béla; Riordan, Oliver (2006), Percolation, Cambridge University Press, ISBN 0521872324
- Broadbent, Simon; Hammersley, John (1957), "Percolation processes I. Crystals and mazes", Proceedings of the Cambridge Philosophical Society, 53: 629–641, Bibcode:1957PCPS...53..629B, doi:10.1017/S0305004100032680
- Bunde A. and Havlin S. (1996), Fractals and Disordered Systems, Springer
- Cohen R. and Havlin S. (2010), Complex Networks: Structure, Robustness and Function, Cambridge University Press
- Grimmett, Geoffrey (1999), Percolation, Springer
- Grimmett, Geoffrey; Marstrand, John (1990), "The supercritical phase of percolation is well behaved", Proceedings of the Royal Society A, 430 (1879): 439–457, Bibcode:1990RSPSA.430..439G, doi:10.1098/rspa.1990.0100
- Hara, Takashi; Slade, Gordon (1990), "Mean-field critical behaviour for percolation in high dimensions", Communications in Mathematical Physics, 128 (2): 333–391, Bibcode:1990CMaPh.128..333H, doi:10.1007/BF02108785
- Kesten, Harry (1982), Percolation theory for mathematicians, Birkhauser
- Menshikov, Mikhail (1986), "Coincidence of critical points in percolation problems", Soviet Mathematics Doklady, 33: 856–859
- Smirnov, Stanislav (2001), "Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits", Comptes Rendus de l'Academie des Sciences, 333 (3): 239–244, Bibcode:2001CRASM.333..239S, doi:10.1016/S0764-4442(01)01991-7
- Stauffer, Dietrich; Aharony, Anthony (1994), Introduction to Percolation Theory (2nd ed.), CRC Press, ISBN 978-0-7484-0253-3
External links
- PercoVIS: a Mac OS X program to visualize percolation on networks in real time
- Interactive Percolation
- Kesten, Harry (May 2006), "What Is ... Percolation?" (PDF), Notices of the American Mathematical Society, Providence, RI: American Mathematical Society, 53 (5): 572–573, ISSN 1088-9477
- Austin, David (July 2008), Percolation: Slipping through the Cracks, American Mathematical Society
- Nanohub online course on Percolation Theory
- Introduction to Percolation Theory: short course by Shlomo Havlin