Template:Short description Template:Network science 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 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 (cognitive psychology).
IntroductionEdit
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 Template:Math vertices, usually called "sites", in which the edge or "bonds" between each two neighbors may be open (allowing the liquid through) with probability Template:Math, or closed with probability Template:Math, and they are assumed to be independent. Therefore, for a given Template:Math, 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 Template:Math is of primary interest. This problem, called now bond percolation, was introduced in the mathematics literature by Template:Harvtxt,<ref name="BroadbentHammersley1957">Template:Cite journal</ref> 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 Template:Math or "empty" (in which case its edges are removed) with probability Template:Math; 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 Template:Math 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 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 Template:Math, the probability that an infinite cluster exists is either zero or one. Since this probability is an increasing function of Template:Math (proof via coupling argument), there must be a critical Template:Math (denoted by Template:Math) 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 Template:Math 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 Template:Math.
HistoryEdit
The Flory–Stockmayer theory was the first theory investigating percolation processes.<ref>Template:Cite book</ref>
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. <ref>Template:Cite journal</ref>
In this context, the British 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. <ref>The rosalind franklin papers - the holes in coal: Research at BCURA and in Paris, 1942-1951. https://profiles.nlm.nih.gov/spotlight/kr/feature/coal Template:Webarchive. Accessed: 2022-01-17.</ref>
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 on Monte Carlo methods, he asks questions to John Hammersley on the use of numerical methods to analyze this model. <ref>Template:Cite journal</ref>
Broadbent and Hammersley introduced in their article of 1957 a mathematical model to model this phenomenon, that is percolation.
Computation of the critical parameterEdit
For most infinite lattice graphs, Template:Math cannot be calculated exactly, though in some cases Template:Math there is an exact value. For example:
- for the square lattice Template:Math in two dimensions, Template:Math 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">Template:Cite journal</ref> see Template:Harvtxt. For site percolation on the square lattice, the value of Template:Math is not known from analytic derivation but only via simulations of large lattices which provide the estimate Template:Math 0.59274621 ± 0.00000013.<ref>Template:Cite journal</ref>
- A limit case for lattices in high dimensions is given by the Bethe lattice, whose threshold is at Template:Math for a coordination number Template:Math. In other words: for the regular tree of degree <math>z</math>, <math>p_c</math> is equal to <math>1/(z-1)</math>.
- For a random tree-like network without degree-degree correlation, it can be shown that such network can have a giant component, and the percolation threshold (transmission probability) is given by <math>p_c = \frac{1}{g_1'(1)}</math>, where <math>g_1(z)</math> is the generating function corresponding to the excess degree distribution. So, for random Erdős–Rényi networks of average degree <math>\langle k\rangle</math>, Template:Math.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref><ref>Template:Cite journal</ref>
- In networks with low clustering, <math>
0 < C \ll 1 </math>, the critical point gets scaled by <math> (1-C)^{-1} </math> such that:
<math>p_c = \frac{1}{1-C}\frac{1}{g_1'(1)}.</math><ref>Template:Cite journal</ref>
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.<ref>Template:Cite journal</ref>
PhasesEdit
Subcritical and supercriticalEdit
The main fact in the subcritical phase is "exponential decay". That is, when Template:Math, 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 Template:Math decays to zero exponentially in Template:Math. This was proved for percolation in three and more dimensions by Template:Harvtxt and independently by Template:Harvtxt. In two dimensions, it formed part of Kesten's proof that Template:Math.<ref name="Kesten1982">Template:Cite book</ref>
The dual graph of the square lattice Template:Math 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 Template:Math. The main result for the supercritical phase in three and more dimensions is that, for sufficiently large Template:Math, there is almost certainly an infinite open cluster in the two-dimensional slab Template:Math. This was proved by Template:Harvtxt.<ref name="GrimmettMarstrand1990">Template:Cite journal</ref>
In two dimensions with Template:Math, 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 Template:Math just the opposite occurs, with finite closed islands in an infinite open ocean. The picture is more complicated when Template:Math since Template:Math, and there is coexistence of infinite open and closed clusters for Template:Math between Template:Math and Template:Math.
CriticalityEdit
Percolation has a singularity at the critical point Template:Math and many properties behave as of a 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 Template:Math 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 Template:Math of dimensions satisfies either Template:Math or Template:Math. 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 Template:Math decreases polynomially, i.e. is on the order of Template:Math for some Template:Math
- Template:Math does not depend on the particular lattice chosen, or on other local parameters. It depends only on the dimension Template:Math (this is an instance of the universality principle).
- Template:Math decreases from Template:Math until Template:Math and then stays fixed.
- Template:Math
- Template:Math.
- The shape of a large cluster in two dimensions is conformally invariant.
See Template:Harvtxt.<ref name="Grimmett1999">Template:Cite book</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 Template:Harvtxt.<ref name="HaraSlade1990">Template:Cite journal</ref>
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 Template:Harvtxt<ref name="Smirnov2001">Template:Cite journal</ref> in the special case of site percolation on the triangular lattice.
Different modelsEdit
- Directed percolation that models the effect of gravitational forces acting on the liquid was also introduced in Template:Harvtxt,<ref name="BroadbentHammersley1957"/> 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 Template:Math, where Template:Math 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>Template:Citation.</ref>
- First passage percolation.
- Invasion percolation.
ApplicationsEdit
In biology, biochemistry, and physical virologyEdit
Percolation theory has been used to successfully predict the fragmentation of biological virus shells (capsids),<ref name="Brunk Twarock p. ">Template:Cite journal</ref><ref>Template:Cite journal</ref> with the fragmentation threshold of Hepatitis B virus capsid predicted and detected experimentally.<ref>Template:Cite journal</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 the broader study of virus disassembly. More stable viral particles (tilings with greater fragmentation thresholds) are found in greater abundance in nature.<ref name="Brunk Twarock p. "/>
In ecologyEdit
Percolation theory has been applied to studies of how environment fragmentation impacts animal habitats<ref>Template:Cite journal</ref> and models of how the plague bacterium Yersinia pestis spreads.<ref>Template:Cite journal</ref>
See alsoEdit
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ReferencesEdit
Further readingEdit
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