Germ-Grain Model

In continuum percolation theory, the so-called germ-grain model is an obvious generalization of both the Boolean and Boolean-Poisson models which is driven by an arbitrary stationary point process X and which assigns to the points x_i in X arbitrary compact sets A_i in R^d rather than the standard closed balls.

In this scenario, the points x_i are known as the germs while the sets A_i are known as grains. It is not uncommon to consider the union of all grains in a germ-grain model, a collection which is sometimes referred to as the grain cover (Kuronen and Leskelä 2012). The grain cover is sometimes referred to as the basis of the model in question.

In older literature, it is not uncommon to define a germ-grain model by assuming any of several other conditions, e.g., that the grains fail to be independent, by allowing the subsets A_i subset R^d to be random closed sets (not necessarily compact) in R^d, and by considering as a basis all unions of the form

 A= union _((x_i,A_i) in Phi)(A_i+x_i)

where Phi is a non-Poisson marked point process (MPP) on R^d with mark space F; this is in contrast to, e.g., the Boolean-Poisson model which has a basis of the form

 B= union _(x_i in Phi^')A(x_i)

where Phi^' is a Poisson point process in R^d and where A(x_i) is the compact set in R^d assigned to each x_i (Hanisch 1981). Though dated, this notion of a germ-grain model can be further elaborated upon using more rigorous mathematical formality by explicitly writing the MPP Phi=(Phi,F) as

 Phi=sum_(i in N)delta_([X_i(Phi),Z_i(Phi)]),

assuming it satisfies either

 P(Phi in M^_(F))=1


 P(Phi in M_K(F))=1   for all K in K^',

and by defining as a model basis the almost surely (with respect to P) closed set

 Z(Phi)= union _(i in N)(X_i(Phi)+Z_i(Phi)).

The resulting model is said to be driven by Phi or to have been derived from Phi (Heinrich 1992).

See also

AB Percolation, Bernoulli Percolation Model, Bond Percolation, Boolean Model, Boolean-Poisson Model, Bootstrap Percolation, Cayley Tree, Cluster, Cluster Perimeter, Continuum Percolation Theory, Dependent Percolation, Discrete Percolation Theory, Disk Model, First-Passage Percolation, Inhomogeneous Percolation Model, Lattice Animal, Long-Range Percolation Model, Mixed Percolation Model, Oriented Percolation Model, Percolation, Percolation Theory, Percolation Threshold, Polyomino, Random-Cluster Model, Random-Connection Model, Random Walk, s-Cluster, s-Run, Site Percolation

This entry contributed by Christopher Stover

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Hanisch, K. H. "On Classes of Random Sets and Point Process Models." Serdica Bulgariacae Mathematicae Publicationes 7, 160-166, 1981.Heinrich, L. "On Existence and Mixing Problems of Germ-Grain Models." Statistics 23, 271-286, 1992.Kuronen, M. and Leskelä, L. "Hard-Core Thinnings of Germ-Grain Models with Power-Law Grain Sizes." 5 Apr 2012., R. and Roy, R. Continuum Percolation. New York: Cambridge University Press, 2008.

Cite this as:

Stover, Christopher. "Germ-Grain Model." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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