Percolation theory
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. The Flory–Stockmayer theory was the first theory investigating percolation processes. 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. 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). 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 . For most infinite lattice graphs, cannot be calculated exactly, though in some cases there is an exact value. For example: formula_9 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. Universality. 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, , 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 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. Phases. Subcritical  and supercritical. 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 . 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 . 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 . 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 . Criticality. Percolation has a singularity at the critical point and many properties behave as of a power-law with formula_10, near formula_2. Scaling theory predicts the existence of critical exponents, depending on the number "d" of dimensions, that determine the class of the singularity. When 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  of dimensions satisfies either or . They include: 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 . 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. Applications. In biology, biochemistry, and physical virology. Percolation theory has been used to successfully predict the fragmentation of biological virus shells (capsids), with the fragmentation 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 the broader study of virus disassembly. Interestingly, more stable viral particles (tilings with greater fragmentation thresholds) are found in greater abundance in nature. In ecology. Percolation theory has been applied to studies of how environment fragmentation impacts animal habitats and models of how the plague bacterium "Yersinia pestis" spreads.