Threshold for monotone symmetric properties through a logarithmic Sobolev inequality

Abstract

Threshold phenomena are investigated using a general approach, following Talagrand [Ann. Probab. 22 (1994) 1576–1587] and Friedgut and Kalai [Proc. Amer. Math. Soc. 12 (1999) 1017–1054]. The general upper bound for the threshold width of symmetric monotone properties is improved. This follows from a new lower bound on the maximal influence of a variable on a Boolean function. The method of proof is based on a well-known logarithmic Sobolev inequality on {0,1}ⁿ. This new bound is shown to be asymptotically optimal.