Geodesic
On one method of calculating percolation thresholds for square and diamond lattices in the percolation problem of knots
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 4 (2009), pp. 33-44

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A method of calculating the percolation threshold $x_c$ in d-dimensional space is proposed based on the average value of the quantity $x_{cL}$ of small-sized lattices $L$. The condition for applicability of the method has limited the range of $2d$ and $3d$ lattices being considered in the problem of knots to square and diamond lattices. The values of $x_{cL}$ for these lattices have calculated in terms of the vector of the initial state of the lattice and the adjacency matrix of the graph corresponding to the lattice with the fraction of knots $x=1$. Percolation thresholds for the square lattice $x_c=0.592744$ and the diamond lattice $x_c=0.430308$ have been calculated.
Keywords: percolation, lattice, percolation threshold, site problem; graph. 
@article{VUU_2009_4_a3,
     author = {S. R. Gallyamov and S. A. Mel'chukov},
     title = {On one method of calculating percolation thresholds for square and diamond lattices in the percolation problem of knots},
     journal = {Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹ\^uternye nauki},
     pages = {33--44},
     publisher = {mathdoc},
     number = {4},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VUU_2009_4_a3/}
}
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S. R. Gallyamov; S. A. Mel'chukov. On one method of calculating percolation thresholds for square and diamond lattices in the percolation problem of knots. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 4 (2009), pp. 33-44. http://geodesic.mathdoc.fr/item/VUU_2009_4_a3/