Asymptotic expansions in n-1 for percolation critical values on the n-cube and ℤn

Random Structures and Algorithms

Volumn 27, Issue 3, 2005, Pages 331-357

  Van Der Hofstad, Remco ^a   Slade, Gordon ^b  

^a EINDHOVEN UNIVERSITY OF TECHNOLOGY   (Netherlands)

^b UNIVERSITY OF BRITISH COLUMBIA   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 27644515798     PISSN: 10429832     EISSN: None     Source Type: Journal    
DOI: 10.1002/rsa.20074     Document Type: Article

References (21)

1. M. Aizenman and D. J. Barsky, Sharpness of the phase transition in percolation models, Commun Math Phys 108 (1987), 489-526.
2. M. Aizenman and C. M. Newman, Tree graph inequalities and critical behavior in percolation models, J Statist Phys 36 (1984), 107-143.
3. M. Ajtai, J. Komlós, and E. Szemerédi, Largest random component of a k-cube, Combinatorica 2 (1982), 1-7.
4. N. Alon, I. Benjamini, and A. Stacey, Percolation on finite graphs and isoperimetric inequalities, Ann Probab 32 (2004), 1727-1745.
5. T. H. Berlin and M. Kac, The spherical model of a ferromagnet, Phys Rev 86 (1952), 821-835.
6. B. Bollobás and Y. Kohayakawa, Percolation in high dimensions, European J Combin 15 (1994), 113-125.
7. B. Bollobás, Y. Kohayakawa, and T. Luczak, The evolution of random subgraphs of the cube, Random Structures Algorithms 3 (1992), 55-90.
8. C. Borgs, J. T. Chayes, R. van der Hofstad, G. Slade, and J. Spencer, Random subgraphs of finite graphs: I. The scaling window under the triangle condition, Random Structures Algorithms, to appear.
9. C. Borgs, J. T. Chayes, R. van der Hofstad, G. Slade, and J. Spencer, Random subgraphs of finite graphs: II. The lace expansion and the triangle condition, Ann Probab, to appear.
10. C. Borgs, J. T. Chayes, R. van der Hofstad, G. Slade, and J. Spencer, Random subgraphs of finite graphs: III. The phase transition for the n-cube, Combinatorica, to appear.
11. M. E. Fisher and R. R. P. Singh, "Critical points, large-dimensionality expansions, and the Ising spin glass," Disorder in physical systems, Eds., G. R. Grimmett and D. J. A. Welsh, 87-111. Clarendon Press, Oxford, 1990.
12. D. S. Gaunt and H. Ruskin, Bond percolation processes in d dimensions, J Phys A 11 (1978), 1369-1380.
13. P. R. Gerber and M. E. Fisher, Critical temperatures of classical n-vector models on hypercubic lattices, Phys Rev B 10 (1974), 4697-4703.
14. D. M. Gordon, Percolation in high dimensions, J London Math Soc (2) 44 (1991), 373-384.
15. G. Grimmett, Percolation, 2nd edition, Springer, Berlin, 1999.
16. T. Hara and G. Slade, Mean-field critical behaviour for percolation in high dimensions, Commun Math Phys 128 (1990), 333-391.
17. T. Hara and G. Slade, The self-avoiding-walk and percolation critical points in high dimensions, Combin Probab Comput 4 (1995), 197-215.
18. T. Hara and G. Slade, Unpublished appendix to [17], available as Paper 93-288 at http://www.ma.utexas.edu/mp_arc, 1993.
19. n: The first three terms, Combin Probab Comput, to appear.
20. H. Kesten, "Asymptotics in high dimensions for percolation," Disorder in physical systems, Eds., G. R. Grimmett and D. J. A. Welsh, 219-240. Clarendon Press, Oxford, 1990.
21. M. V. Menshikov, Coincidence of critical points in percolation problems, Sov Math Dokl 33 (1986), 856-859.