Instatistical physics andmathematics,percolation theory describes the behavior of a network when nodes or links are added. This is a geometric type ofphase transition, since at a critical fraction of addition the network of small, disconnected clusters merge into significantly largerconnected, so-called spanning clusters. The applications of percolation theory tomaterials science and in many other disciplines are discussed here and in the articlesNetwork theory andPercolation (cognitive psychology).

A representative question (and thesource of the name) is as follows. Assume that some liquid is poured on top of someporous material. Will the liquid be able to make its way from hole to hole and reach the bottom? This physical question ismodelled mathematically as athree-dimensional network ofn ×n ×nvertices, usually called "sites", in which theedge or "bonds" between each two neighbors may be open (allowing the liquid through) with probabilityp, or closed with probability1 –p, and they are assumed to be independent. Therefore, for a givenp, 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 nowbond percolation, was introduced in the mathematics literature byBroadbent & 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 probabilityp or "empty" (in which case its edges are removed) with probability1 –p; the corresponding problem is calledsite percolation. The question is the same: for a givenp, what is the probability that a path exists between top and bottom? Similarly, one can ask, given a connected graph at what fraction1 –p of failures the graph will become disconnected (no large component).

The same questions can be asked for any lattice dimension. As is quite typical, it is actually easier to examineinfinite 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? ByKolmogorov's zero–one law, for any givenp, the probability that an infinite cluster exists is either zero or one. Since this probability is an increasing function ofp (proof viacoupling argument), there must be acriticalp (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 forn 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.

TheFlory–Stockmayer theory (1941), which studied the transition to gelation in polymerization reactions, was the first theory investigating percolation processes.^[2]

The history of the percolation model as we know it has its root in the coal industry. Since the industrial revolution, the economical importance of this source of energy fostered many scientific studies to understand its composition and optimize its use. During the 1930s and 1940s, the qualitative analysis by organic chemistry left more and more room to more quantitative studies.^[3]

In this context, theBritish Coal Utilisation Research Association (BCURA) was created in 1938. It was a research association funded by the coal mines owners. In 1942,Rosalind Franklin, who then recently graduated in chemistry from the university of Cambridge, joined the BCURA. She started research on the density and porosity of coal. During the Second World War, coal was an important strategic resource. It was used as a source of energy, but also was the main constituent of gas masks.

Coal is a porous medium. To measure its 'real' density, one was to sink it in a liquid or a gas whose molecules are small enough to fill its microscopic pores. While trying to measure the density of coal using several gases (helium, methanol, hexane, benzene), and as she found different values depending on the gas used, Rosalind Franklin showed that the pores of coal are made of microstructures of various lengths that act as a microscopic sieve to discriminate the gases. She also discovered that the size of these structures depends on the temperature of carbonation during the coal production. With this research, she obtained a PhD degree and left the BCURA in 1946.^[4]

In the mid fifties, Simon Broadbent worked in the BCURA as a statistician. Among other interests, he studied the use of coal in gas masks. One question is to understand how a fluid can diffuse in the coal pores, modeled as a random maze of open or closed tunnels. In 1954, during a symposium onMonte Carlo methods, he asks questions toJohn Hammersley on the use of numerical methods to analyze this model.^[5]

Broadbent and Hammersley introduced in their article of 1957 a mathematical model to model this phenomenon, that is percolation.

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

• for thesquare latticeℤ² 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 byHarry Kesten in the early 1980s,^[6] seeKesten (1982). For site percolation on the square lattice, the value ofp_c is not known from analytic derivation but only via simulations of large lattices which provide the estimatep_c = 0.59274621 ± 0.00000013.^[7]  
• A limit case for lattices in high dimensions is given by theBethe lattice, whose threshold is atp_c =⁠1/z − 1⁠ for acoordination number z. In other words: for the regulartree of degree${\displaystyle z}$,${\displaystyle p_c}$ is equal to${\displaystyle 1/(z-1)}$.

• For a randomtree-like network without degree-degree correlation, it can be shown that such network can have agiant component, and thepercolation threshold (transmission probability) is given by${\displaystyle p_c=\frac{1}{g_1^{\prime }(1)}}$, where${\displaystyle g_1(z)}$ is thegenerating function corresponding to theexcess degree distribution. So, for randomErdős–Rényi networks of average degree${\displaystyle ⟨k⟩}$,p_c =⁠1/⟨k⟩⁠.^[8][9][10]
• In networks with lowclustering,, the critical point gets scaled by${\displaystyle (1-C{)}^{-1}}$ such that:

${\displaystyle p_c=\frac{1}{1-C}\frac{1}{g_1^{\prime }(1)}.}$^[11]

This indicates that for a given degree distribution, the clustering leads to a larger percolation threshold, mainly because for a fixed number of links, the clustering structure reinforces the core of the network with the price of diluting the global connections. For networks with high clustering, strong clustering could induce the core–periphery structure, in which the core and periphery might percolate at different critical points, and the above approximate treatment is not applicable.^[12]

The main fact in the subcritical phase is "exponential decay". That is, whenp <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 sizer decays to zeroexponentially in r. This was proved for percolation in three and more dimensions byMenshikov (1986) and independently byAizenman & Barsky (1987). In two dimensions, it formed part of Kesten's proof thatp_c =⁠1/2⁠.^[13]

Thedual graph of the square latticeℤ² 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 withd = 2. The main result for the supercritical phase in three and more dimensions is that, for sufficiently large N, there is almost certainly an infinite open cluster in the two-dimensional slabℤ² × [0,N]^{d − 2}. This was proved byGrimmett & Marstrand (1990).^[14]

In two dimensions withp <⁠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. Whenp >⁠1/2⁠ just the opposite occurs, with finite closed islands in an infinite open ocean. The picture is more complicated whend ≥ 3 sincep_c <⁠1/2⁠, and there is coexistence of infinite open and closed clusters forp betweenp_c and 1 −p_c.

Percolation has asingularity at the critical pointp =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 ofcritical exponents, depending on the numberd of dimensions, that determine the class of the singularity. Whend = 2 these predictions are backed up by arguments fromconformal field theory andSchramm–Loewner evolution, and include predicted numerical values for the exponents. Most of these predictions are conjectural except when the numberd of dimensions satisfies eitherd = 2 ord ≥ 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 ofr decreasespolynomially, i.e. ison the order ofr^α for some α
  • α does not depend on the particular lattice chosen, or on other local parameters. It depends only on the dimensiond (this is an instance of theuniversality principle).
  • α_d decreases fromd = 2 untild = 6 and then stays fixed.
  • α₂ = −⁠5/48⁠
  • α₆ = −1.
• The shape of a large cluster in two dimensions isconformally invariant.

SeeGrimmett (1999).^[15] In 11 or more dimensions, these facts are largely proved using a technique known as thelace 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 inHara & 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 ofOded Schramm that thescaling limit of a large cluster may be described in terms of aSchramm–Loewner evolution. This conjecture was proved bySmirnov (2001)^[17] in the special case of site percolation on the triangular lattice.

• Directed percolation that models the effect ofgravitational forces acting on the liquid was also introduced inBroadbent & Hammersley (1957),^[1] and has connections with thecontact 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 theFortuin–Kasteleyn random cluster model, which has many connections with theIsing model and otherPotts models.
• Bernoulli (bond) percolation oncomplete graphs is an example of arandom graph. The critical probability is p =⁠1/N⁠, whereN 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 theory has been used to successfully predict the fragmentation of biological virus shells (capsids),^[19][20] with the fragmentation threshold ofHepatitis B viruscapsid predicted and detected experimentally.^[21] 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 gameJenga, and has relevance to the broader study of virus disassembly. More stable viral particles (tilings with greater fragmentation thresholds) are found in greater abundance in nature.^[19]

Percolation theory has been applied to studies of how environment fragmentation impacts animal habitats^[22] and models of how the plague bacteriumYersinia pestis spreads.^[23]

• 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

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