Continuum Percolation Theory

Continuum percolation can be thought of as a continuous, uncountable version of percolation theory-a theory which, in its most studied form, takes place on a discrete, countable point lattice like . Unlike discrete percolation theory, continuum percolation theory involves notions of percolation for  and for various non-discrete subsets thereof.

There are a number of models used to study continuum percolation including but not limited to the disk model, the germ-grain model, and the random-connection model. Perhaps the most well-studied of these methods is the so-called Boolean-Poisson model which roughly consists of centering an independent copy of a random -dimensional shape  at each point of a homogeneous Poisson process  in -dimensional Euclidean space , the result of which is a collection of overlapping shapes spanning a subset of . Using this construction, one devises a percolation theory by considering whether a percolation occurs, i.e.,whether a given random shape is with positive probability part of an infinite clump of random shapes. Percolation is defined similarly in the other models.

The continuum brand of percolation theory was proposed in the early 1960s in order to study bicoastal signal transmission (Gilbert 1961). Experts note that continuum percolation lacks much of the orderly mathematical structure of its discrete counterpart due to the fact that those methods, based largely on enumeration, lose much of their power in the continuum case. Even so, much work has been done to advance the field which is now considered tantamount in a number of areas including condensed matter and material physics (Hall 1985).

REFERENCES

Gawlinski, E. T. and Stanley, H. E. "Continuum Percolation in Two Dimensions: Monte Carlo Tests of Scaling and Universality for Non-Interacting Discs." J. Phys. A: Math. Gen. 14, L291-L299, 1981.

Gilbert, E. N. "Random Plane Networks." Journal of the Society for Industrial and Applied Mathematics 9, 533-543, 1961.

Grimmett, G. Percolation, 2nd ed. Berlin: Springer-Verlag, 1999.

Haan, S. W. and Zwanzig, R. "Series Expansions in a Continuum Percolation Problem." J. Phys. A 10, 1547-1555, 1977.

Hall, P. "On Continuum Percolation." Ann. Probab. 13, 1250-1266, 1985.

Kertesz, J. and Vicsek, T. "Monte Carlo Renormalization Group Study of the Percolation Problem of Discs with a Distribution of Radii." Z. Phys. B 45, 345-350, 1982.

Meester, R. and Roy, R. Continuum Percolation. New York: Cambridge University Press, 2008.

Pike, G. E. and Seager, C. H. "Percolation and Conductivity: A Computer Study I." Phys. Rev. B 10, 1421-1434, 1974.

Roy, R. "Continuum Percolation." http://cinet.vbi.vt.edu/cinet_new/sites/default/files/presentations/r-roy-lecture.pdf.