Geodesic
Hodge's idea in percolation: percolation threshold estimation by the unit cell
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 4 (2011), pp. 60-79

Voir la notice de l'article provenant de la source Math-Net.Ru

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.
Keywords: percolation, percolation threshold, unit cell, Hodge's idea, site problem. 
@article{VUU_2011_4_a4,
     author = {S. R. Gallyamov and S. A. Mel'chukov},
     title = {Hodge's idea in percolation: percolation threshold estimation by the unit cell},
     journal = {Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹ\^uternye nauki},
     pages = {60--79},
     publisher = {mathdoc},
     number = {4},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VUU_2011_4_a4/}
}
TY  - JOUR
AU  - S. R. Gallyamov
AU  - S. A. Mel'chukov
TI  - Hodge's idea in percolation: percolation threshold estimation by the unit cell
JO  - Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki
PY  - 2011
SP  - 60
EP  - 79
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/VUU_2011_4_a4/
LA  - ru
ID  - VUU_2011_4_a4
ER  - 
%0 Journal Article
%A S. R. Gallyamov
%A S. A. Mel'chukov
%T Hodge's idea in percolation: percolation threshold estimation by the unit cell
%J Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki
%D 2011
%P 60-79
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/VUU_2011_4_a4/
%G ru
%F VUU_2011_4_a4
S. R. Gallyamov; S. A. Mel'chukov. Hodge's idea in percolation: percolation threshold estimation by the unit cell. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 4 (2011), pp. 60-79. http://geodesic.mathdoc.fr/item/VUU_2011_4_a4/