Percolation

Percolation, the fundamental notion at the heart of percolation theory, is a difficult idea to define precisely though it is quite easy to describe qualitatively.

From the narrowest perspective, the term percolation can be defined as a model of a porous medium; indeed, it is from this perspective that the study of percolation theory blossomed, and is typically agreed to be the fundamental physical situation that modern percolation theory attempts to address.

Likewise, it is not uncommon for the term percolation to describe the actual fluid flow within the random media or as the theoretical simulation of such a flow for a given simulated medium.

One of the difficulties underlying the formulation of a precise definition is that some authors choose to define the term relative to the machinery used in its study. For instance, some authors choose to define percolation to be the result of independently removing vertices or edges from some sort of graph (van der Hofstad 2010), a definition which is intimately connected to the fact that graph theory provides one piece of the basis framework for several models of today's discrete percolation theory.

In much the same way, it is not uncommon for the term to be defined as a probabilistic model which exhibits certain behavior, most notably a phase transition (Kesten 2006) or a critical phenomenon (Steif 2009); these two terms most often describe the more ubiquitous term percolation threshold, one of the fundamental probabilistic components of modern percolation theory. As such, this definition is similar to the graph-theoretic definition above in that it addresses another of the main tools-namely, probability theory-used to formalize both discrete and continuum percolation theory. Contrarily, however, some authors use this same terminology to define percolation theory rather than percolation itself (Christensen 2002).

To add to the ambiguity, the term percolation is often used to describe a system which percolates, i.e.,a system within which there exists an infinite connected component (van der Hofstad 2010). In instances of Bernoulli bond and/or site percolation, this notion can be made more precise by defining a probability  on the set of subgraphs of a connected, finite, locally finite multigraph  designating the independent probability with which vertices/edges are considered "open." Under this construction, one says that a percolation occurs within  if  where here,  is an event characterized by the existence of an infinite open cluster within  (Bollobás and Riordan 2006). Analogues of this definition exist in branches of percolation theory other than bond/site theory as well, e.g.,the infinite clump terminology of the Boolean model and the graph-theoretic machinery underlying the random-connection model of continuum percolation theory. Clearly, this definition borrows from both the graph theoretic and probabilistic definitions of percolation.

The term percolation is often prefaced by any of a number of context-specific qualifiers, e.g., AB, bond, bootstrap, continuum, dependent, first-passage, inhomogeneous, long-range, mixed, oriented, site, etc., to indicate additional assumptions made on the underlying system.

REFERENCES

Bollobás, B. and Riordan, O. Percolation. New York: Cambridge University Press, 2006.

Christensen, K. "Percolation Theory." 2002. http://www.mit.edu/~levitov/8.334/notes/percol_notes.pdf.

Grimmett, G. Percolation, 2nd ed. Berlin: Springer-Verlag, 1999.Kesten, H. "What Is Percolation?" Notices Amer. Math. Soc. 53, 572-573, 2006.

Steif, J. E. "A Mini Course on Percolation Theory." 2009. http://www.math.chalmers.se/~steif/perc.pdf.

Van der Hofstad, R. "Percolation and Random Graphs." Ch. 6 in New Perspectives in Stochastic Geometry. Oxford, England: Oxford University Press, pp. 173-247, 2010.