On Percolation with Fibers or Layers |
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Authors: | Toom André |
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Institution: | (1) University of São Paulo, IME/USP Cx. Postal, 66281 CEP 05315-970 São Paulo, Brazil |
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Abstract: | We consider site percolation on Z
d, directed edges going from any sZ
d to s+A
1,..., s+A
n, where A
1,..., A
n are the same for all sites and at least two of them are noncollinear. A site is closed if it belongs to p+Block, where p is a point in a Poisson distribution in R
dZ
d with a density and Block={sL: |s|M}+{sR
d: |s|}, where L is a linear subspace of R
d, |·| is the Euclidean norm, =max(|A
1|,..., |A
n|) and M is a parameter. We study the behavior of *, the critical value, and P
closed*, corresponding critical percentage of closed sites, when M. Denote R
d/L the factor space. Call two nonzero vectors U, V codirected if U=kV, where k>0. Theorem. If there are A
i and A
j whose projections to R
d/L are not codirected, then *1/M
dim(L) and P
closed* remains separated both from 0 and 1 when M. If projections of all A
1,..., A
n to R
d/L are codirected, then *1/M
dim(L)+1 and P
closed*1/M when M. |
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Keywords: | oriented percolation critical values destruction of materials |
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