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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 percolation threshold is a mathematical concept in percolation theory ... but in the mathematics and physics worlds it generally refers to ... for example, for an ...
For example, in geology, percolation refers to filtration of water through soil and permeable rocks. The water flows to recharge the groundwater in the water table and aquifers . In places where infiltration basins or septic drain fields are planned to dispose of substantial amounts of water, a percolation test is needed beforehand to determine ...
Outside mathematics, the Eden growth model is used to model bacteria growth and deposition of material. Another example is comparing a minimized cost from the Vickrey–Clarke–Groves auction (VCG-auction) to a minimized path from first passage percolation to gauge how pessimistic the VCG-auction is at its lower limit. Both problems are solved ...
For example, exact at the percolation threshold, the percolating network in the system is a fractal with large voids and a ramified structure. The surface interrupts this structure, so the percolating cluster is less likely to come in contact to the surface.
Examples can be found not only in physical phenomena, but also in biology, neuroscience, ecology (e.g. evolution), and economics (e.g. diffusion of innovation). Percolation can be considered to be a branch of the study of dynamical systems or statistical mechanics. In particular, percolation networks exhibit a phase change around a critical ...
Typically, a family of universality classes will have a lower and upper critical dimension: below the lower critical dimension, the universality class becomes degenerate (this dimension is 2d for the Ising model, or for directed percolation, but 1d for undirected percolation), and above the upper critical dimension the critical exponents ...
Thus the Erdős–Rényi process is the mean-field case of percolation. Some significant work was also done on percolation on random graphs. From a physicist's point of view this would still be a mean-field model, so the justification of the research is often formulated in terms of the robustness of the graph, viewed as a communication network.