Influence and sharp-threshold theorems for monotonic measures

Annals of Probability

Volumn 34, Issue 5, 2006, Pages 1726-1745

  Graham, B T ^a   Grimmett, G R ^a  

^a UNIVERSITY OF CAMBRIDGE   (United Kingdom)

Author keywords

FKG lattice condition; Influence; Monotonic measure; Percolation; Positive association; Random cluster model; Sharp threshold

Indexed keywords

EID: 33845504705     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/009117906000000278     Document Type: Article

References (22)

1. BATTY, C. J. K. and BOLLMANN, H. W. (1980). Generalised Holley-Preston inequalities on measure spaces and their products. Z. Wahrsch. Verw. Gebiete 53 157-173. MR0580910
2. BEN-OR, M. and LINIAL, N. (1990). Collective coin flipping. In Randomness and Computation 91-115. Academic Press, New York.
3. BEZUIDENHOUT, C. E., GRIMMETT, G. R. and KESTEN, H. (1993). Strict inequality for critical values of Potts models and random-cluster processes. Comm. Math. Phys. 158 1-16. MR1243713
4. BOLLOBÁS, B. and RIORDAN, O. (2006). The critical probability for random Voronoi percolation in the plane is 1/2. Probab. Theory Related Fields 136 417-468.
5. BOLLOBÁS, B. and RIORDAN, O. (2006). A short proof of the Harris-Kesten theorem. Bull. London Math. Soc. 38 470-484.
6. BOURGAIN, J., KAHN, J., KALAI, G., KATZNELSON, Y. and LINIAL, N. (1992). The influence of variables in product spaces. Israel J. Math. 77 55-64. MR1194785
7. CHAYES, J. T. and CHAYES, L. (1986). Percolation and random media. In Critical Phenomena, Random Systems and Gauge Theories (K. Osterwalder and R. Stora, eds.) 1001-1142. North-Holland, Amsterdam. MR0880545
8. FORTUIN, C. M. (1972). On the random-cluster model. III. The simple random-cluster process. Physica 59 545-570. MR0432137
9. FORTUIN, C. M., KASTELEYN, P. W. and GINIBRE, J. (1971). Correlation inequalities on some partially ordered sets. Comm. Math. Phys. 22 89-103. MR0309498
10. FRIEDGUT, E. (2004). Influences in product spaces: KKL and BKKKL revisited. Combin. Probab. Comput. 13 17-29. MR2034300
11. FRIEDGUT, E. and KALAI, G. (1996). Every monotone graph property has a sharp threshold. Proc. Amer. Math. Soc. 124 2993-3002. MR1371123
12. GRIMMETT, G. R. (1999). Percolation. Springer, Berlin. MR1707339
13. GRIMMETT, G. R. (2003). The random-cluster model. In Probability on Discrete Structures. Encyclopedia of Mathematical Sciences (H. Kesten, ed.) 110 73-123. Springer, Berlin. MR2023651
14. GRIMMETT, G. R. (2006). The Random-Cluster Model. Springer, Berlin.
15. HÄGGSTRÖM, O., KALAI, G. and MOSSEL, E. (2005). A law of large numbers for weighted majority. Adv. in Appl. Math. 37 112-123. MR2232083
16. KAHN, J., KALAI, G. and LINIAL, N. (1988). The influence of variables on Boolean functions. In Proceedings of 29th Symposium on the Foundations of Computer Science 68-80. Computer Science Press.
17. PRESTON, C. J. (1974). A generalization of the FKG inequalities. Comm. Math. Phys. 36 233-241. MR0341553
18. RUSSO, L. (1978). A note on percolation. Z. Wahrsch. Verw. Gebiete 43 39-48. MR0488383
19. RUSSO, L. (1981). On the critical percolation probabilities. Z. Wahrsch. Verw. Gebiete 56 229-237. MR0618273
20. RUSSO, L. (1982). An approximate zero-one law. Z. Wahrsch. Verw. Gebiete 61 129-139. MR0671248
21. SEYMOUR, P. D. and WELSH, D. J. A. (1978). Percolation probabilities on the square lattice. In Advances in Graph Theory (B. Bollobás, ed.) 227-245. North-Holland, Amsterdam. MR0494572
22. TALAGRAND, M. (1994). On Russo's approximate zero-one law. Ann. Probab. 22 1576-1587. MR1303654