Expansion in n-1 for percolation critical values on the n-cube and ℤn: The first three terms

Combinatorics Probability and Computing

Volumn 15, Issue 5, 2006, Pages 695-713

  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

MATHEMATICAL MODELS;

LACE EXPANSION; PERCOLATION CRITICAL VALUES;

PERCOLATION (COMPUTER STORAGE);

EID: 33746595810     PISSN: 09635483     EISSN: 14692163     Source Type: Journal    
DOI: 10.1017/S0963548306007498     Document Type: Article

References (20)

1. Aizenman, M. and Barsky, D. J. (1987) Sharpness of the phase transition in percolation models. Commun. Math. Phys. 108 489-526.
2. Aizenman, M. and Newman, C. M. (1984) Tree graph inequalities and critical behavior in percolation models. J. Statist. Phys. 36 107-143.
3. Ajtai, M., Komlós, J. and Szemerédi, E. (1982) Largest random component of a k-cube. Combinatorica 2 1-7.
4. Alon, N., Benjamini, I. and Stacey, A. (2004) Percolation on finite graphs and isoperimetric inequalities. Ann. Probab. 32 1727-1745.
5. Alon, N. and Spencer, J. H. (2000) The Probabilistic Method, 2nd edn, Wiley, New York.
6. Bollobás, B. and Kohayakawa, Y. (1994) Percolation in high dimensions. Europ. J. Combin. 15 113-125.
7. Bollobás, B., Kohayakawa, Y. and Łuczak, T. (1992) The evolution of random subgraphs of the cube. Random Struct. Alg. 3 55-90.
8. Borgs, C., Chayes, J. T., van der Hofstad, R., Slade, G. and Spencer, J. (2005) Random subgraphs of finite graphs I: The scaling window under the triangle condition. Random Struct. Alg. 27 137-184.
9. Borgs, C., Chayes, J. T., van der Hofstad, R., Slade, G. and Spencer, J. (2005) Random subgraphs of finite graphs II: The lace expansion and the triangle condition. Ann. Probab. 33 1886-1944.
10. Borgs, C., Chayes, J. T., van der Hofstad, R., Slade, G. and Spencer, J. Random subgraphs of finite graphs III: The phase transition for the n-cube. Combinatorica, to appear.
11. Cox, J. T. and Durrett, R. (1983) Oriented percolation in dimensions d ≥ 4: Bounds and asymptotic formulas. Math. Proc. Cambridge Philos. Soc. 93 151-162.
12. Gaunt, D. S. and Ruskin, H. (1978) Bond percolation processes in d dimensions. J. Phys. A: Math. Gen. 11 1369-1380.
13. Gordon, D. M. (1991) Percolation in high dimensions. J. London Math. Soc. (2) 44 373-384.
14. Grimmett, G. (1999) Percolation, 2nd edn, Springer, Berlin.
15. Hara, T. and Slade, G. (1990) Mean-field critical behaviour for percolation in high dimensions. Commun. Math. Phys. 128 333-391.
16. Hara, T. and Slade, G. (1993) Unpublished appendix to [17]. Available as paper 93-288 at http ://www.ma.utexas.edu/mp_arc.
17. Hara, T. and Slade, G. (1995) The self-avoiding-walk and percolation critical points in high dimensions. Combin. Probab. Comput. 4 197-215.
18. n. Random Struct. Alg. 27 331-357.
19. Kesten, H. (1990) Asymptotics in high dimensions for percolation. In Disorder in Physical Systems (G. R. Grimmett and D. J. A. Welsh, eds), Clarendon Press, Oxford.
20. Menshikov, M. V. (1986) Coincidence of critical points in percolation problems. Soviet Mathematics, Doklady 33 856-859.