Threshold for monotone symmetric properties through a logarithmic Sobolev inequality

Details Threshold for monotone symmetric properties through a logarithmic Sobolev inequality Threshold for monotone symmetric properties through a logarithmic Sobolev inequality

2005-11-24

arxiv.org/abs/math/0511607v2

Annals of Probability 2005, Vol. 34, No. 5, 1707-1725

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117906000000287 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

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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