Theory of minimum spanning trees. II. Exact graphical methods and perturbation expansion at the percolation threshold

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

Volumn 81, Issue 2, 2010, Pages

  Jackson, T S ^a   Read, N ^a  

^a Yale University   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

FINITE GRAPHS; FIRST ORDER; GRAPHICAL METHODS; LOW-DENSITY EXPANSION; MINIMUM SPANNING TREES; PERCOLATION CLUSTERS; PERCOLATION THRESHOLDS; PERTURBATION EXPANSIONS; RANDOM MINIMA; RENORMALIZATION GROUP; SPACE DIMENSIONS;

FRACTAL DIMENSION; GRAPHIC METHODS; PERCOLATION (COMPUTER STORAGE); SOLVENTS;

PERTURBATION TECHNIQUES;

EID: 77749333035     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.81.021131     Document Type: Article

References (68)

1. T. S. Jackson and N. Read, preceeding paper, Phys. Rev. E 81, 021130 (2010).
2. E. Lawler, Combinatorital Optimization: Networks and Matroids (Dover, Mineola, NY, 2001), Chap..
3. C. H. Papadimitriou and K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity (Dover, Mineola, NY, 1998), Chap..
4. R. E. Tarjan, Data Structures and Network Algorithms (Society for Industrial and Applied Mathematics, Philadeplphia, PA, 1983), Ch..
5. W. J. Cook, W. H. Cunningham, W. R. Pulleyblank, and A. S. Schrijver, Combinatorial Optimization (Wiley Interscience, New York, NY, 1998), Chaps. and.
6. J. B. Kruskal, Proc. Am. Math. Soc. 7, 48 (1956). 10.2307/2033241
7. D. Stauffer and A. Aharony, Introduction to Percolation Theory, 2nd ed. (Taylor & Francis, London, 1994).
8. C. M. Newman and D. L. Stein, Phys. Rev. Lett. 72, 2286 (1994); 10.1103/PhysRevLett.72.2286
9. C. M. Newman and D. L. Stein, J. Stat. Phys. 82, 1113 (1996). 10.1007/BF02179805
10. E. López, S. V. Buldyrev, L. A. Braunstein, S. Havlin, and H. E. Stanley, Phys. Rev. E 72, 056131 (2005); 10.1103/PhysRevE.72.056131
11. S. V. Buldyrev, S. Havlin, and H. E. Stanley, Phys. Rev. E 73, 036128 (2006); 10.1103/PhysRevE.73.036128
12. M. Porto, N. Schwartz, S. Havlin, and A. Bunde, Phys. Rev. E 60, R2448 (1999). 10.1103/PhysRevE.60.R2448
13. L. A. Braunstein, Z. Wu, Y. Chen, S. V. Buldyrev, T. Kalisky, S. Sreenivasan, R. Cohen, E. López, S. Havlin, and H. E. Stanley, Int. J. Bifurcation Chaos Appl. Sci. Eng. 17, 2215 (2007). 10.1142/S0218127407018361
14. M. Cieplak, A. Maritan, and J. R. Banavar, Phys. Rev. Lett. 72, 2320 (1994); 10.1103/PhysRevLett.72.2320
15. M. Cieplak, A. Maritan, and J. R. Banavar, Phys. Rev. Lett. 76, 3754 (1996). 10.1103/PhysRevLett.76.3754
16. R. Dobrin and P. M. Duxbury, Phys. Rev. Lett. 86, 5076 (2001). 10.1103/PhysRevLett.86.5076
17. A. A. Middleton, Phys. Rev. B 61, 14787 (2000). 10.1103/PhysRevB.61.14787
18. B. Wieland and D. B. Wilson, Phys. Rev. E 68, 056101 (2003); 10.1103/PhysRevE.68.056101
19. D. B. Wilson, Phys. Rev. E 69, 037105 (2004). 10.1103/PhysRevE.69.037105
20. Z. Wu, L. A. Braunstein, S. Havlin, and H. E. Stanley, Phys. Rev. Lett. 96, 148702 (2006). 10.1103/PhysRevLett.96.148702
21. M. Aizenman, Nucl. Phys. B 485, 551 (1997). 10.1016/S0550-3213(96)00626-8
22. M. Aizenman, A. Burchard, C. M. Newman, and D. B. Wilson, Random Struct. Algorithms 15, 319 (1999). 10.1002/(SICI)1098-2418(199910/12)15:3/4<319::AID- RSA8>3.0.CO;2-G
23. N. Read, Phys. Rev. E 72, 036114 (2005). 10.1103/PhysRevE.72.036114
24. T. Hara and G. Slade, Commun. Math. Phys. 128, 333 (1990); 10.1007/BF02108785
25. G. Slade, The Lace Expansion and its Applications (Springer, New York, NY, 2004).
26. P. W. Kasteleyn and C. M. Fortuin, J. Phys. Soc. Jpn. (Suppl.) 26, 11 (1969);
27. C. M. Fortuin and P. W. Kasteleyn, Physica (Amsterdam) 57, 536 (1972). 10.1016/0031-8914(72)90045-6
28. F. Y. Wu, Rev. Mod. Phys. 54, 235 (1982). 10.1103/RevModPhys.54.235
29. R. K. P. Zia and D. J. Wallace, J. Phys. A 8, 1495 (1975). 10.1088/0305-4470/8/9/019
30. T. C. Lubensky, in Ill-Condensed Matter, Les Houches Session XXXI, edited by, R. Balian,,, R. Maynard,, and, G. Toulouse, (North-Holland, New York, 1979).
31. W. T. Tutte, J. Comb. Theory, Ser. A 2, 301 (1967); 10.1016/S0021- 9800(67)80032-2
32. W. T. Tutte, J. Comb. Theory, Ser. A 9, 289 (1970). 10.1016/S0021- 9800(70)80067-9
33. J. W. Essam, Rep. Prog. Phys. 43, 833 (1980). 10.1088/0034-4885/43/7/001
34. J. W. Essam, in Phase Transitions and Critical Phenomena, edited by, C. Domb, and, M. S. Green, (Academic Press, New York, 1972), Vol. 2.
35. A. Coniglio and J. W. Essam, J. Phys. A 10, 1917 (1977). 10.1088/0305-4470/10/11/020
36. C. Itzykson and J.-M. Drouffe, Statistical Field Theory (Cambridge University Press, Cambridge, 1989), Vol. 1.
37. D. J. Amit, J. Phys. A 9, 1441 (1976). 10.1088/0305-4470/9/9/006
38. K. Wilson and J. Kogut, Phys. Rep. 12, 75 (1974); 10.1016/0370-1573(74) 90023-4
39. S.-K. Ma, Modern Theory of Critical Phenomena (Benjamin, Reading, MA, 1976);
40. J. L. Cardy, Scaling and Renormalization in Statistical Physics (Cambridge University Press, Cambridge, 1996).
41. D. Amit and V. Martín-Mayor, Field Theory, the Renormalization Group, and Critical Phenomena, 3rd ed. (World Scientific, Hackensack, NJ, 2005).
42. N. Nakanishi, Graph Theory and Feynman Integrals (Gordon and Breach, New York, NY, 1970).
43. C. Itzykson and J.-B. Zuber, Quantum Field Theory (McGraw-Hill, New York, NY, 1980).
44. J. D. Bjorken and S. D. Drell, Relativistic Quantum Fields (McGraw-Hill, New York, NY, 1965).
45. T. Applequist, Ann. Phys. (N.Y.) 54, 27 (1969). 10.1016/0003-4916(69) 90333-9
46. M. Bergère and J. B. Zuber, Commun. Math. Phys. 35, 113 (1974). 10.1007/BF01646611
47. M. C. Bergère and Y.-M. P. Lam, J. Math. Phys. 17, 1546 (1976). 10.1063/1.523078
48. J. Illiopoulos, C. Itzykson, and A. Martin, Rev. Mod. Phys. 47, 165 (1975). 10.1103/RevModPhys.47.165
49. E. Brézin, J. C. le Gillou, and J. Zinn-Justin, in Phase Transitions and Critical Phenomena, edited by, C. Domb, and, M. S. Green, (Academic Press, New York, 1976), Vol. 6.
50. R. G. Priest and T. C. Lubensky, Phys. Rev. B 13, 4159 (1976); 10.1103/PhysRevB.13.4159
51. R. G. Priest and T. C. Lubensky, Phys. Rev. B 14, 5125 (E) (1976). 10.1103/PhysRevB.14.5125.4
52. O. F. de Alcantara Bonfim, J. E. Kirkham, and A. J. McKane, J. Phys. A 14, 2391 (1981). 10.1088/0305-4470/14/9/034
53. J. Cardy and P. Grassberger, J. Phys. A 18, L267 (1985); 10.1088/0305-4470/18/6/001
54. K. H. Janssen, Z. Phys. B: Condens. Matter 58, 311 (1985); 10.1007/BF01303673
55. A. B. Harris, Phys. Rev. B 35, 5056 (1987); 10.1103/PhysRevB.35.5056
56. R. Blumenfeld, et al. J. Phys. A 19, L791 (1986); 10.1088/0305-4470/19/ 13/007
57. Y. Meir and A. B. Harris, Phys. Rev. Lett. 63, 2819 (1989). 10.1103/PhysRevLett.63.2819
58. A. Houghton, J. S. Reeve, and D. J. Wallace, Phys. Rev. B 17, 2956 (1978). 10.1103/PhysRevB.17.2956
59. J. H. van Lint and R. M. Wilson, A Course in Combinatorics, 2nd Ed. (Cambridge University Press, Cambridge, 2001), Chap..
60. S. Sreenivasan, T. Kalisky, L. A. Braunstein, S. V. Buldyrev, S. Havlin, and H. E. Stanley, Phys. Rev. E 70, 046133 (2004). 10.1103/PhysRevE.70.046133
61. J. Komlós, Combinatorica 5, 57 (1985); 10.1007/BF02579443
62. B. Dixon, M. Rauch, and R. E. Tarjan, SIAM J. Comput. 21, 1184 (1992); 10.1137/0221070
63. V. King, Algorithmica 18, 263 (1997). 10.1007/BF02526037
64. D. R. Karger, N. Klein, and R. E. Tarjan, J. ACM 42, 321 (1995). 10.1145/201019.201022
65. N. Bogoliubov and O. Parasiuk, Acta Math. 97, 227 (1957). 10.1007/BF02392399
66. K. Hepp, Commun. Math. Phys. 2, 301 (1966). 10.1007/BF01773358
67. W. Zimmermann, Ann. Phys. (N.Y.) 77, 536 (1973). 10.1016/0003-4916(73) 90429-6
68. C. G. Callan, Jr., in Methods in Field Theory, Les Houches lectures Session XXVIII, edited, R. Balian, and, J. Zinn-Justin, (North Holland, Amsterdam, 1976), p. 41.