2D percolation cluster

A 2-dimensional percolation cluster is a mathematical model of percolation, which is the formation of long-range connectivity in random systems. In engineering and coffee making it refers to the slow flow of fluids through porous networks, but in the mathematics and physics world it generally refers to simplified regular or random lattice models of random systems, and the connectivity on them. The most common percolation model is to take a regular lattice, like a square lattice, and make it into a random network by randomly "occupying" sites (vertices) or bonds (edges) with a statistically independent probability p. At a critical threshold p_c, long-range connectivity first appears, and this is called the Percolation threshold. More complicated systems have several probabilities p₁, p₂, etc., and the transition is characterized by a critical surface. Once can also consider continuum systems, such as overlapping disks and spheres placed randomly, or the negative space (Swiss-cheese models). So far, it has been assumed that the occupation of a site or bond is completely random -- this is the so-called Bernoulli percolation. For a continuum system, this corresponds to the points being placed down by a Poisson process. Further variations involve correlated percolation, such as percolation clusters related to Ising and Potts models of ferromagnets, in which the bonds are put down by the Fortuin-Kasteleyn method. In "bootstrap" or "k-sat" percolation, sites and/or bonds are first made occupied and then successively culled from a system if a site does not have a sufficient number of neighbors. An important other model of percolation, in a different Universality class altogether, is Directed percolation.

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A tremendous amount of work over the last several decades has gone into finding exact and numerical values of the percolation thresholds for a variety of systems. Exact thresholds are only known for certain two-dimensional lattices that can be broken up into a self-dual array, such that under a triangle-triangle transformation, the system remains the same.