Higher-Dimensional Stick Percolation
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AbstractWe consider two cases of the so-called stick percolation model with sticks of length L. In the first case, the orientation is chosen independently and uniformly, while in the second all sticks are oriented along the same direction. We study their respective critical values $$\lambda _c(L)$$ λ c ( L ) of the percolation phase transition, and in particular we investigate the asymptotic behavior of $$\lambda _c(L)$$ λ c ( L ) as $$L\rightarrow \infty $$ L → ∞ for both of these cases. In the first case we prove that $$\lambda _c(L)\sim L^{-2}$$ λ c ( L ) ∼ L - 2 for any $$d\ge 2,$$ d ≥ 2 , while in the second we prove that $$\lambda _c(L)\sim L^{-1}$$ λ c ( L ) ∼ L - 1 for any $$d\ge 2.$$ d ≥ 2 .
2018 ◽
Vol 61
(1)
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pp. 211-224
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2020 ◽
Vol 35
(22)
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pp. 2050124
2020 ◽
Vol 31
(09)
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pp. 2050129
1996 ◽
Vol 07
(04)
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pp. 609-612
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2017 ◽
Vol 32
(26)
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pp. 1750160
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