Open Access
September 2006 Influence and sharp-threshold theorems for monotonic measures
B. T. Graham, G. R. Grimmett
Ann. Probab. 34(5): 1726-1745 (September 2006). DOI: 10.1214/009117906000000278


The influence theorem for product measures on the discrete space {0,1}N may be extended to probability measures with the property of monotonicity (which is equivalent to “strong positive association”). Corresponding results are valid for probability measures on the cube [0,1]N that are absolutely continuous with respect to Lebesgue measure. These results lead to a sharp-threshold theorem for measures of random-cluster type, and this may be applied to box crossings in the two-dimensional random-cluster model.


Download Citation

B. T. Graham. G. R. Grimmett. "Influence and sharp-threshold theorems for monotonic measures." Ann. Probab. 34 (5) 1726 - 1745, September 2006.


Published: September 2006
First available in Project Euclid: 14 November 2006

zbMATH: 1115.60099
MathSciNet: MR2271479
Digital Object Identifier: 10.1214/009117906000000278

Primary: 60E15 , 60K35
Secondary: 82B31 , 82B43

Keywords: FKG lattice condition , influence , monotonic measure , percolation , positive association , Random-cluster model , sharp threshold

Rights: Copyright © 2006 Institute of Mathematical Statistics

Vol.34 • No. 5 • September 2006
Back to Top