Open Access
Translator Disclaimer
May 2012 Geometric influences
Nathan Keller, Elchanan Mossel, Arnab Sen
Ann. Probab. 40(3): 1135-1166 (May 2012). DOI: 10.1214/11-AOP643

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.

Citation

Download Citation

Nathan Keller. Elchanan Mossel. Arnab Sen. "Geometric influences." Ann. Probab. 40 (3) 1135 - 1166, May 2012. https://doi.org/10.1214/11-AOP643

Information

Published: May 2012
First available in Project Euclid: 4 May 2012

zbMATH: 1255.60015
MathSciNet: MR2962089
Digital Object Identifier: 10.1214/11-AOP643

Subjects:
Primary: 05D40 , 60C05

Keywords: Gaussian measure , Influences , Isoperimetric inequality , Kahn–Kalai–Linial influence bound , product space

Rights: Copyright © 2012 Institute of Mathematical Statistics

JOURNAL ARTICLE
32 PAGES


SHARE
Vol.40 • No. 3 • May 2012
Back to Top