The Annals of Probability

Geometric influences

Nathan Keller, Elchanan Mossel, and Arnab Sen

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Abstract

We present a new definition of influences in product spaces of continuous distributions. Our definition is geometric, and for monotone sets it is identical with the measure of the boundary with respect to uniform enlargement. We prove analogs of the Kahn–Kalai–Linial (KKL) and Talagrand’s influence sum bounds for the new definition. We further prove an analog of a result of Friedgut showing that sets with small “influence sum” are essentially determined by a small number of coordinates. In particular, we establish the following tight analog of the KKL bound: for any set in ℝn of Gaussian measure t, there exists a coordinate i such that the ith geometric influence of the set is at least $ct(1-t)\sqrt{\log n}/n$, where c is a universal constant. This result is then used to obtain an isoperimetric inequality for the Gaussian measure on ℝn and the class of sets invariant under transitive permutation group of the coordinates.

Article information

Source
Ann. Probab., Volume 40, Number 3 (2012), 1135-1166.

Dates
First available in Project Euclid: 4 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.aop/1336136061

Digital Object Identifier
doi:10.1214/11-AOP643

Mathematical Reviews number (MathSciNet)
MR2962089

Zentralblatt MATH identifier
1255.60015

Subjects
Primary: 60C05: Combinatorial probability 05D40: Probabilistic methods

Keywords
Influences product space Kahn–Kalai–Linial influence bound Gaussian measure isoperimetric inequality

Citation

Keller, Nathan; Mossel, Elchanan; Sen, Arnab. Geometric influences. Ann. Probab. 40 (2012), no. 3, 1135--1166. doi:10.1214/11-AOP643. https://projecteuclid.org/euclid.aop/1336136061


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