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W3191148925 abstract "Percolation models were introduced to model fluid flow in a medium, where fluid and medium may be broadly interpreted. Fluid is imagined to flow through the network of sites (vertices) and bonds (edges) of a graph that represents the medium. In the percolation approach, fluid flow is determined by a randomly structured medium, in contrast to the diffusion approach where fluid flow is viewed as random movement in a structureless medium. In the classical bond and site percolation models, edges and vertices, respectively, are randomly open or closed to the passage of fluid. The primary focus is the extent of flow of fluid, particularly the probability that there exists an infinite connected cluster of passable sites and bonds. There are two principal objects of study in classical percolation models: (1) The percolation threshold or critical probability, a proportion of passable sites or bonds that distinguishes between local and infinite flow, is highly dependent on the detailed structure of a graph representing the medium. (2) The behavior of the model near the percolation threshold is also of interest, and is believed to be described by power laws with critical exponents that depend only on the dimension of the medium (see Power Law Process). Percolation models have been applied to a wide range of phenomena in physics, chemistry, biology, and materials science where connectivity and clustering play an important role: Examples include oil flow in sandstone, thermal phase transitions, spontaneous magnetization, electrical conductivity in alloys, and the spread of epidemics. Percolation theory has provided insight into the behavior of more complicated models exhibiting phase transitions and critical phenomena, such as the Ising model and the random cluster model. A proliferation of variations on the classical percolation model have been invented. Brief descriptions of several other models, such as directed percolation, AB percolation, and continuum percolation, are provided later. An extensive literature on percolation has developed with several different emphases. There is an extremely active physics literature on percolation models, containing numerous conjectures, Monte Carlo studies, plausibility arguments, and applications. (See 2, 8, 16, and 19 for physical science and engineering perspectives and applications.) In mathematics, percolation developed as a branch of probability theory. It contributed subadditive stochastic processes and correlation inequalities to probability theory, and percolation concepts play a role in the theory of interacting particle systems. (See 4, 5, and 6 for probabilistic perspectives.) More recently, graph theory and combinatorics have increasingly been employed in percolation theory. In particular, random graph theory methods have had an important impact. (See 1 for a discrete mathematics perspective.) The two classical percolation models are the bond percolation model and the site percolation model. In a bond percolation model on an infinite graph G, each edge of G is open (passable) with probability p, 0 ≤ p ≤ 1, and closed (impassable) otherwise, independently of all other edges. In the site percolation model on G, sites are open with probability p, 0 ≤ p ≤ 1 and closed otherwise, independently, and an edge is open (closed) if and only if both its endpoints are open (closed). In either model, let Pr p and E p denote the probability measure and expectation operator parameterized by p. The open cluster C v containing vertex v of G is the set of vertices that can be reached from v through a path of open edges. The number of vertices in C v is denoted by |C v |. The percolation probability function, denoted θ v (p) = Pr p [|C v | = ∞], is the probability that vertex v is in an infinite open cluster. It is a non-decreasing right-continuous function. The percolation probability function may depend on the vertex v. However, if the underlying graph G is connected, then for any vertices v and w in G, θ v (p) = 0 if and only if θ w (p) = 0. Thus, the critical probability or percolation threshold p c (G) = sup{p:θ v (p) = 0} is independent of the choice of v ∈ G. In fact, by Kolmogorov's zero-one law, if p > p c (G), there is an infinite open cluster in G with probability one. The percolation threshold separates intervals in the parameter space corresponding to local flow and extensive flow of fluid. Hence, the percolation threshold represents a phase transition point, such that the global behavior of the model is dramatically different when p < p c from when p > p c . Much of the interest in percolation theory in the physical sciences and engineering is attributable to the connection with phase transitions and critical phenomena. Several alternative definitions and interpretations of the percolation threshold exist. Historically, p c as defined above, denoted by p H , was called the cluster size critical probability. The mean cluster size critical probability is p T ( G ) = inf { p : E p ( | C v | ) = ∞ } . For any graph G, p T (G) ≤ p H (G). A third definition, the sponge crossing critical probability p S , in terms of the asymptotic behavior of the probability that an open path crosses an expanding rectangular region, played an important role in rigorously determining exact percolation threshold values. Another interpretation, denoted p E is as a singularity of a function describing the expected number of open clusters per site in the graph, was used in a heuristic derivation of percolation threshold values of some planar two-dimensional lattice graphs. For a large class of graphs, p c = p T = p S . An infinite graph G is periodic if it can be embedded in R d , d ≥ 2, so that the graph is invariant under translations by d linearly independent vectors and the vertex set has no accumulation points. If so, we say that G has dimension d. For periodic graphs, the probability distribution of the cluster size decays exponentially when p < p c , implying equality of the three definitions of percolation threshold for periodic graphs in any dimension. Knowledge of the values of the percolation thresholds of various lattices has been obtained by a variety of methods. The only exact percolation threshold solutions are for infinite trees and some two-dimensional periodic graphs. It is a major open problem to determine exact percolation thresholds for additional graphs, and particularly for any lattice in three or higher dimensions. Since exact solutions do not exist for bond or site percolation on most lattice graphs of interest to physical scientists, considerable knowledge of the percolation threshold value has been gained from inequalities, mathematically rigorous bounds, simulation estimates, confidence intervals, and approximation formulas. The first exact solutions were for regular trees of degree d, obtaining the value p c = 1/(d − 1) for both bond and site models by relating percolation to extinction in a Galton–Watson branching process. Predictions of percolation thresholds were derived in 1964 for the square, triangular, and hexagonal bond models and the triangular site model by an insightful heuristic approach by Sykes and Essam 20. Their predictions were later confirmed in 1980–1982, showing p c = 1/2 for the square lattice bond model, p c = 2 sin (π/18) ≈ 0.347296 for the triangular lattice bond model and p c = 1 − 2 sin (π/18) ≈ 0.652704 for the hexagonal lattice bond model, and p c = 1/2 for the triangular lattice site model and the site model on any full-triangulated periodic lattice. More recent research produced an approach for constructing an infinite class of graphs for which bond percolation thresholds may be exactly determined 22, 23, and 24. The solutions for bond models depend crucially on graph duality. Under rather general conditions, the percolation thresholds of a dual pair of planar lattices sum to one. Since dual graphs exist only for planar graphs, there are currently only exact solutions for two-dimensional lattices. Similarly, exact site model solutions rely on a matching property, with the percolation thresholds of a pair of matching lattices summing to one. The concept of a matching pair of lattices corresponds to the concept of duality for planar graphs; the line graphs of a dual pair of planar graphs is a pair of matching graphs. Thus, exact solutions for site models are known only for certain two-dimensional lattices. No exact critical probabilities are known for models in ℝ d , d ≥ 3. Knowledge of such models is based on numerical evidence. Two simple relationships establish inequalities between percolation thresholds. If H is a subgraph of G, then p c (H) ≥ p c (G), for both site and bond models. If H is obtained by contracting a set of edges in G, then p c (H) ≤ p c (G) for bond models. Under broad conditions, these are strict inequalities. Over the years, many nonrigorous numerical estimates of percolation thresholds have been generated by simulation of configurations on finite portions of the lattice, then extrapolating results from these to obtain the estimate. A variety of simulation approaches and algorithms for extrapolation are used, leading to some disagreement regarding error bounds. One efficient method is due to Newman and Ziff 14. Riordan and Walters 15 developed a method for computing very narrow confidence interval estimates with extremely high confidence levels, such as error probabilities of one-millionth. The confidence intervals are based on simulations, but do not use any extrapolation. Approximation formulas, called “universal formulas” in the physics literature, provide an estimate for the percolation threshold based on a small number of features of the graph, such as average degree and dimension, in order to understand the determinants of the threshold in terms of the characteristics of the medium. (See, e.g., 21.) In inhomogeneous percolation models, edges in different directions or vertices in different locations have different probabilities of being open. For such models, there is a critical surface in the parameter space that corresponds to the percolation threshold in the homogeneous model. Only a few mathematically rigorous results have been established. For the square lattice, with each horizontal edge open with probability p h and each vertical edge open with probability p v , the critical surface is p h + p v = 1. For the triangular lattice, with edges in the three directions open with probability parameters p h , p v , and p d , the critical surface is p h + p v + p d − p h p v p d = 1. (See 5, Section 11.9.) Although the value of the percolation threshold is strongly determined by the local structure of the graph, the nature and behavior of open clusters below, at, and above the threshold are generally believed to be invariant with respect to the local structure. The anticipated behaviors of various percolation functions are described in terms of power laws with the exponents referred to as critical exponents. The universality principle claims that the critical exponents depend only on the dimension of the lattice, and are independent of the detailed structure of the graph. To describe the power law behavior, the notation f(p) ≈ |p − p c |λ typically means that lim p ↓ p c log f ( p ) log ( | p − p c | ) = λ . Similar notation is used as p → p c or for functions of n as n → ∞. The lattices are assumed to be imbedded periodically in dimension d. The most common percolation functions, and the expected form of their power laws, are: Heuristic derivations and simulation evidence suggests, that a number of scaling relations between the critical exponents are satisfied for graphs in any dimension: 2 − α = γ + 2β = β(δ + 1), Δ = δβ, and γ = ν(2 − η). Furthermore, in dimensions 2 ≤ d ≤ 6 there are anticipated hyperscaling relations: dρ = δ + 1 and dν = 2 − α. The scaling relations hold for ℤ d and related spread-out models for sufficiently high dimensions. If the limits defining the critical exponents exist, the scaling and hyperscaling relations (except that involving α) hold for a class of models in d = 2. Physicists have predicted values for the critical exponents when d = 2 or d is large. Conjectured values for d = 2 are α = − 2 5 , β = 5 36 , γ = 43 18 , δ = 91 5 , and ν = 4 3 . These values have been rigorously verified only for the triangular lattice site model 18. It is widely expected that the critical exponents for lattices with dimension d ≥ 6 have the corresponding values for a regular tree, which are α = −1, β = 1, γ = 1, δ = 2, δ = 2, ρ = 1 2 , η = 0, and ν = 1 2 . The lace expansion 7 has been valuable for understanding percolation in higher dimensions, including proving many of the physicists' conjectures for bond percolation on the hypercubic lattices Z d with d ≥ 19. Percolation models on two-dimensional lattices are conjectured to have conformal invariance, in the sense that, if the lattice is rescaled so the spacing tends to zero, then certain limiting probabilities of crossings of regions are invariant under conformal mappings of the plane when the model is at criticality. Conformal invariance has been proved for the triangular site percolation model 17. Stochastic Loewner Evolution 12 is a stochastic process of random planar curves generated by a differential equation with Brownian motion as input. It is conjectured to be the scaling limit of various critical percolation models and other stochastic processes in the plane, such as uniform spanning trees and planar loop-erased random walk. A parameter κ determines the distribution of the process, with κ = 6 believed to correspond to the scaling limit of the boundary of percolation clusters at criticality in two-dimensional models. This connection would yield proofs of the conjectured critical exponent values for two-dimensional models, and has been proven to be correct for site percolation on the triangular lattice. A proliferation of variations of the classical percolation models have been invented and investigated, which provide new insights and allow a wide variety of phenomena to be modeled. Some major variations are mentioned briefly or discussed in the following, to suggest the range of applications and behaviors that are possible to model using percolation processes. Several of these models are discussed in 5, Chapter 12. A mixed percolation model allows both vertices and edges, and perhaps also faces, of the graph to be open or closed at random. Directed percolation, sometimes called oriented percolation, randomly retains edges of a directed graph (i.e., a graph in which each edge can be traversed in only one direction). Applications are to model movement in a random medium under the influence of a force such as gravity, a strong wind, or an electric field. Methods and results are surveyed in 3. It is important to model phenomena which do not involve a lattice structure, and a variety of models of continuum percolation have been introduced to do so. Common approaches are to place random objects such as discs or balls in the space, or construct a random irregular graph from a set of random vertices located in the space. Bollobás and Riordan recently made a major advance, proving that the percolation threshold is 1/2 for the Delaunay triangulation constructed from the points of a homogeneous Poisson point process in the plane. Models and results are described in 13, 1, Chapter 8; and 6. In the first-passage percolation model, each edge of the underlying lattice G is assigned a nonnegative random variable that represents its travel time. Interest focuses on the shortest travel time between two vertices over the infinite set of paths available. Ergodic theory for subadditive processes (see Ergodic Theorems) is used to prove that a limiting velocity exists in each direction from the origin, and that a limiting shape exists for the set of vertices that are reached from the origin by time t, as t → ∞. (See 11.) Applications include traffic flow, tumor growth, and spread of epidemics or rumors. The AB percolation or anti-percolation model was introduced to model gelation processes and anti-ferromagnetism. Each vertex is labeled A with probability p and B otherwise, representing two possible chemicals or species that could occupy a site. An edge connects two vertices only if they have opposite labels. If sufficient bonding occurs, the substance represents a gel rather than a liquid. Research has established conditions for the existence and nonexistence of infinite AB percolation clusters on certain lattices, with some surprising results. For example, infinite AB clusters do not occur on the square or hexagonal lattices for any proportions of A and B vertices, but can exist on the triangular lattice for some proportions. Invasion percolation is a dynamic process that models fluid being pumped into a random medium under pressure, displacing other material with the least resistance. Each edge of the graph is independently assigned a random variable that is uniformly distributed on (0, 1). A connected cluster is created by starting from a single vertex and successively adding the edge on the cluster's boundary that has the smallest value. For any p > p c , the set of edges with a value less than p will contain an infinite component of the graph, so once the cluster reaches this component, it will never invade another edge with value greater than p. A central result is that the empirical distribution function of the values of edges in the cluster converges to the uniform distribution on (0, p c (G)). The proliferation of variations has been extensive, with other, better studied models being last passage percolation, fractal percolation, bootstrap percolation, and dynamical percolation. Furthermore, different types of models may be used in various combinations, such as directed first-passage percolation. Lattice Path Combinatorics; Turbulent diffusion; Diffusion Processes; Multivariate Directed Graphs; Poisson Processes; Superadditive and Subadditive Ordering; Point processes, dynamic." @default.
W3191148925 created "2021-08-16" @default.
W3191148925 creator A5064543263 @default.
W3191148925 date "2014-11-18" @default.
W3191148925 modified "2023-09-23" @default.
W3191148925 title "Percolation Theory" @default.
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