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 homogeneousPoisson 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).
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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. B45, 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. B10, 1421-1434, 1974.Roy, R.
"Continuum Percolation." http://cinet.vbi.vt.edu/cinet_new/sites/default/files/presentations/r-roy-lecture.pdf.