The Annals of Probability

Threshold for monotone symmetric properties through a logarithmic Sobolev inequality

Raphaël Rossignol

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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.

Article information

Ann. Probab., Volume 34, Number 5 (2006), 1707-1725.

First available in Project Euclid: 14 November 2006

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Zentralblatt MATH identifier

Primary: 60F20: Zero-one laws
Secondary: 28A35: Measures and integrals in product spaces 60E15: Inequalities; stochastic orderings

Threshold influence of variables zero–one law logarithmic Sobolev inequalities


Rossignol, Raphaël. Threshold for monotone symmetric properties through a logarithmic Sobolev inequality. Ann. Probab. 34 (2006), no. 5, 1707--1725. doi:10.1214/009117906000000287.

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