2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/75081The 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.Probability60E15; 60K35text