Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2011, Issue 4, Pages 60–79
This article is cited in 2 scientific papers (total in 2 papers)
Hodge's idea in percolation: percolation threshold estimation by the unit cell
S. R. Gallyamov, S. A. Mel'chukov
Udmurt State University, Izhevsk, Russia
We consider a percolation problem of knots. The percolation threshold of triangular lattice $x_c=1/2$ was confirmed by the two lattices method and percolation threshold of quadratic 1,2 lattice $x_c=0.40725616$ was obtained.
We propose the method based on Hodge's idea from algebraic geometry to estimate the percolation threshold $x_c$ of the infinite lattice by percolation properties of its unit sell. The model of unit cell of Bete lattice was studied and in the following it was applied for estimation of percolation thresholds of body-centered and face-centered cubic lattices in the three-dimensional case and of hexagonal lattice in the planar case. As a result of estimation the values of $x_c(bcc)=0.24595716$ for BCC, $x_c(fcc)=0.19925370$ for FCC and $x_c=0.69700003$ for hexagonal lattices were obtained.
percolation, percolation threshold, unit cell, Hodge's idea, site problem.
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MSC: 60K35, 82B43
S. R. Gallyamov, S. A. Mel'chukov, “Hodge's idea in percolation: percolation threshold estimation by the unit cell”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2011, no. 4, 60–79
Citation in format AMSBIB
\by S.~R.~Gallyamov, S.~A.~Mel'chukov
\paper Hodge's idea in percolation: percolation threshold estimation by the unit cell
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
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