Hodge's idea in percolation: percolation threshold estimation by the unit cell

S. R. Gallyamov   ; S. A. Mel'chukov

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 4 (2011), pp. 60-79

Résumé

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.

Détail

Keywords: percolation, percolation threshold, unit cell, Hodge's idea, site problem.

Bibliographie

[1] Gallyamov S. R., Melchukov S. A., “Ob odnom metode raschëta porogov protekaniya kvadratnoi i almaznoi reshëtok v perkolyatsionnoi zadache uzlov”, Vestnik Udmurtskogo universiteta. Matematika. Mekhanika. Kompyuternye nauki, 2009, no. 4, 33–44

[2] Gallyamov S. R., “Porog protekaniya prostoi kubicheskoi reshëtki v zadache uzlov v modeli reshëtki Bëte”, Vestnik Udmurtskogo universiteta. Matematika. Mekhanika. Kompyuternye nauki, 2008, no. 3, 109–115

[3] Kadanoff L. P., Jotze W., Hamblen D., Hecht R., Lewis E. A. S., Palciauskas V. V., Rayl M., Swift J., Aspres D., Kane J., “Static Phenomena Near Critical Points: Theory and Experiment”, Rev. Mod. Phys., 39:2 (1967), 395–431 | DOI | MR

[4] Wilson K. G., “Renormalization group and critical phenomena”, Phys. Rev. B, 4:9 (1971), 3174–3183 | DOI | Zbl

[5] Feder E., Fraktaly, Mir, M., 1991 | MR

[6] Tarasevich Yu. Yu., Perkolyatsiya: teoriya, prilozheniya, algoritmy, Editorial URSS, M., 2002, 52–58

[7] Gould H., Tobochnik J., An introduction to computer simulation methods application to physical systems, Reading, Addison-Wesley Publication Company, 1988; Guld Kh., Tobochnik Ya., Kompyuternoe modelirovanie v fizike, v 2 ch., Ch. 2, Mir, M., 1990, 400 pp.

[8] Ziff R. M., Sapoval B., “The efficient determination of the percolation threshold by a frontier generating walk in a gradient”, J. Phys. A Math. Gen., 19 (1986), L1169–L1172 | DOI | MR

[9] Hodge W. V. D., The theory and applications of harmonic integrals, Cambridge, 1952 | MR

[10] Shioda T., “What is known about the Hodge conjecture?”, Advanced studies in Pure Math., 1 (1983), 55–68 | MR | Zbl

[11] Steenbrink J. H. M., “Some remarks about the Hodge conjecture”, Lect. Notes Math., 1246, 1987, 165–175 | DOI | MR | Zbl

[12] Krasnov V. A., “Algebraicheskie tsikly na veschestvennom algebraicheskom GM-mnogoobrazii i ikh prilozheniya”, Izv. RAN. Ser. matem., 57:4 (1993), 153–173 | MR | Zbl

[13] Gallyamov S. R., Melchukov S. A., “O kriticheskikh indeksakh v trëkhmernoi perkolyatsii v zadachakh uzlov i tvërdykh sfer”, Vestnik Udmurtskogo universiteta. Matematika. Mekhanika. Kompyuternye nauki, 2010, no. 2, 67–80

[14] Stauffer D., Aharony A., Introduction to percolation theory, Tailor Francis, London, 1992 | MR

[15] Lorenz C. D., Ziff R. M., “Universality of the excess number of clusters and the crossing probability function in three-dimensional percolation”, J. Phys. A Math. Gen., 31 (1998), 8147–8157 | DOI | Zbl

[16] Suding P. N., Ziff R. M., “Site percolation thresholds and universal formulas for the Archimedean lattices”, Phys. Rev. E, 60:1 (1999), 275–283 | DOI

[17] Grassberger P., “On the spreading of two-dimensional percolation”, J. Phys. A, 18 (1985), L215–L219 | DOI | MR