Influence and sharp-threshold theorems for monotonic measures
B. T. Graham, G. R. Grimmett
Abstract
The influence theorem for product measures on the discrete space 0,1N 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.
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