Effect of dimensionality on the percolation thresholds of various d-dimensional lattices

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 87, Issue 3, 2013, Pages

  Torquato, S ^a   Jiao, Y ^b  

^a PRINCETON UNIVERSITY   (United States)

^b Princeton University   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ANALYTICAL ESTIMATES; ASYMPTOTIC EXPANSION; D-DIMENSIONAL LATTICE; EUCLIDEAN DIMENSIONS; OCCUPATION PROBABILITY; PERCOLATION PROBLEMS; PERCOLATION THRESHOLDS; RADIUS OF CONVERGENCE;

ESTIMATION; PERCOLATION (FLUIDS); PERCOLATION (SOLID STATE); SOLVENTS;

PERCOLATION (COMPUTER STORAGE);

EID: 84875835907     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.87.032149     Document Type: Article

References (54)

1. M. E. Fisher and J. W. Essam, J. Math. Phys. JMAPAQ 0022-2488 10.1063/1.1703745 2, 609 (1961).
2. M. E. Fisher and D. S. Gaunt, Phys. Rev. PHRVAO 0031-899X 10.1103/PhysRev.133.A224 133, A224 (1964).
3. J. W. Essam, in Phase Transitions and Critical Phenomena, edited by, C. Domb, and, M. S. Green, (Academic Press, London, 1972), Vol. 2, pp. 197-270.
4. D. S. Gaunt, M. F. Sykes, and H. Ruskin, J. Phys. A: Math. Gen. 1361-6447 10.1088/0305-4470/9/11/015 9, 1899 (1976).
5. D. S. Gaunt and H. Ruskin, J. Phys. A: Math. Gen. 1361-6447 10.1088/0305-4470/11/7/025 11, 1369 (1978).
6. M. Sahimi, B. D. Hughes, L. E. Scriven, and H. T. Davis, J. Phys. A: Math. Gen. 1361-6447 10.1088/0305-4470/16/2/004 16, L67 (1983).
7. J. Adler, Y. Meir, A. Aharony, and A. B. Harris, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.41.9183 41, 9183 (1990).
8. J. J. Binney, N. J. Dowrick, A. J. Fisher, and M. E. J. Newman, The Theory of Critical Phenomena: An Introduction to the Renormalization Group (Oxford University Press, Oxford, UK, 1992).
9. H. L. Frisch and J. K. Percus, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.60.2942 60, 2942 (1999).
10. G. Parisi and F. Slanina, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.62.6554 62, 6554 (2000).
11. M. Skoge, A. Donev, F. H. Stillinger, and S. Torquato, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.74.041127 74, 041127 (2006).
12. S. Torquato and F. H. Stillinger, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.73.031106 73, 031106 (2006).
13. R. D. Rohrmann and A. Santos, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.76.051202 76, 051202 (2007).
14. J. A. van Meel, D. Frenkel, and P. Charbonneau, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.79.030201 79, 030201 (2009).
15. G. Parisi and F. Zamponi, Rev. Mod. Phys. RMPHAT 0034-6861 10.1103/RevModPhys.82.789 82, 789 (2010).
16. L. Lue, M. Bishop, and P. A. Whitlock, J. Chem. Phys. JCPSA6 0021-9606 10.1063/1.3354115 132, 104509 (2010).
17. S. Torquato and F. H. Stillinger, Rev. Mod. Phys. RMPHAT 0034-6861 10.1103/RevModPhys.82.2633 82, 2633 (2010).
18. S. Torquato and F. H. Stillinger, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.68.041113 68, 041113 (2003).
19. C. E. Zachary and S. Torquato, J. Stat. Mech.: Theory Exp. 1742-5468 10.1088/1742-5468/2009/12/P12015 (2009) P12015.
20. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer-Verlag, New York, 1998).
21. S. Torquato, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.82.056109 82, 056109 (2010).
22. H. Cohn and N. Elkies, Ann. Math. ANMAAH 0003-486X 10.4007/annals.2003. 157.689 157, 689 (2003).
23. S. Torquato and F. H. Stillinger, Experimental Math. 1058-6458 10.1080/10586458.2006.10128964 15, 307 (2006);
24. A. Scardicchio, F. H. Stillinger, and S. Torquato, J. Math. Phys. JMAPAQ 0022-2488 10.1063/1.2897027 49, 043301 (2008).
25. S. Torquato, A. Scardicchio, and C. E. Zachary, J. Stat. Mech.: Theory Exp. 1742-5468 10.1088/1742-5468/2008/11/P11019 (2008) P11019.
26. M. F. Sykes and J. W. Essam, J. Math. Phys. JMAPAQ 0022-2488 10.1063/1.1704215 5, 1117 (1964).
27. C. Domb and M. F. Sykes, Phys. Rev. PHRVAO 0031-899X 10.1103/PhysRev.122. 77 122, 77 (1961).
28. P. Agrawal, S. Redner, P. J. Reynolds, and H. E. Stanley, J. Phys. A: Math. Gen. 1361-6447 10.1088/0305-4470/12/11/018 12, 2073 (1979).
29. G. Stell and J. S. Hoye, J. Phys. A: Math. Gen. 1361-6447 10.1088/0305-4470/18/15/009 18, L951 (1985).
30. J. A. Given and G. Stell, J. Phys. A: Math. Gen. 1361-6447 10.1088/0305-4470/24/14/024 24, 3369 (1991).
31. D. Stauffer and A. Aharony, Introduction to Percolation Theory (Taylor & Francis, London, 1992).
32. M. Sahimi, Applications of Percolation Theory (Taylor & Francis, London, 1994).
33. C. D. Lorenz and R. M. Ziff, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.57.230 57, 230 (1998).
34. C. D. Lorenz and R. M. Ziff, J. Phys. A: Math. Gen. 1361-6447 10.1088/0305-4470/31/40/009 31, 8147 (1998).
35. S. C. van der Marck, J. Phys. A: Math. Gen. 1361-6447 10.1088/0305-4470/31/15/010 31, 3449 (1998).
36. S. C. van der Marck, Int. J. Mod. Phys. C IJMPEO 0129-1831 10.1142/S0129183198000431 09, 529 (1998).
37. M. E. J. Newman and R. M. Ziff, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.85.4104 85, 4104 (2000).
38. C. D. Lorenz, R. May, and R. M. Ziff, J. Stat. Phys. JSTPBS 0022-4715 10.1023/A:1018648130343 98, 961 (2000).
39. P. Grassberger, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.67.036101 67, 036101 (2003).
40. J. Škvor and I. Nezbeda, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.79.041141 79, 041141 (2009).
41. R. M. Ziff and H. Gu, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.79. 020102 79, 020102 (2009).
42. C. Ding, Z. Fu, W. Guo, and F. Y. Wu, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.81.061111 81, 061111 (2010).
43. S. Torquato, J. Chem. Phys. JCPSA6 0021-9606 10.1063/1.3679861 136, 054106 (2012).
44. S. Torquato and Y. Jiao, J. Chem. Phys. JCPSA6 0021-9606 10.1063/1.4742750 137, 074106 (2012).
45. S. Torquato and Y. Jiao, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.87.022111 87, 022111 (2013).
46. A. Coniglio, U. De Angelis, and A. Forlani, J. Phys. A: Math. Gen. 1361-6447 10.1088/0305-4470/10/7/011 10, 1123 (1977).
47. S. Torquato, Random Heterogeneous Materials: Microstructure and Macroscopic Properties (Springer-Verlag, New York, 2002).
48. M. F. Sykes, D. S. Gaunt, and M. Glen, J. Phys. A: Math. Gen. 1361-6447 10.1088/0305-4470/9/10/021 9, 1705 (1976).
49. T. C. Lubensky and J. Isaacson, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.41.829 41, 829 (1978);
50. T. C. Lubensky and J. Isaacson, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.20.2130 20, 2130 (1979).
51. M. F. Sykes, J. L. Martin, and J. W. Essam, J. Phys. A: Math. Gen. JPMNAX 0301-0015 10.1088/0305-4470/6/3/006 6, L306 (1973).
52. M. Aizenman and C. M. Newman, J. Stat. Phys. JSTPBS 0022-4715 10.1007/BF01015729 36, 107 (1984).
53. H. Cohn, Y. Jiao, A. Kumar, and S. Torquato, Geom. Topology 1465-3060 10.2140/gt.2011.15.2235 15, 2235 (2011).
54. k 's (k = 1, 2, 3, 4) for the 10 percolation problems studied in this paper.