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September 2006 Threshold for monotone symmetric properties through a logarithmic Sobolev inequality
Raphaël Rossignol
Ann. Probab. 34(5): 1707-1725 (September 2006). DOI: 10.1214/009117906000000287


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.


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Raphaël Rossignol. "Threshold for monotone symmetric properties through a logarithmic Sobolev inequality." Ann. Probab. 34 (5) 1707 - 1725, September 2006.


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

zbMATH: 1115.60021
MathSciNet: MR2271478
Digital Object Identifier: 10.1214/009117906000000287

Primary: 60F20
Secondary: 28A35 , 60E15

Keywords: influence of variables , logarithmic Sobolev inequalities , threshold , Zero–one law

Rights: Copyright © 2006 Institute of Mathematical Statistics

Vol.34 • No. 5 • September 2006
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