GEOMETRIC INFLUENCES

• 成果类型: Article
• 署名作者: Keller, Nathan; Mossel, Elchanan; Sen, Arnab
• 署名单位: Weizmann Institute of Science; University of California System; University of California Berkeley; University of Cambridge
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/11-AOP643
• 发表日期: 2012
• 页码: 1135-1166
• 关键词: zero-one law; product-spaces; BOOLEAN FUNCTIONS; sharp-threshold; gauss space; INEQUALITY; variables

摘要

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 R-n of Gaussian measure t, there exists a coordinate i such that the ith geometric influence of the set is at least ct (l - t) root log n/n, where c is a universal constant. This result is then used to obtain an isoperimetric inequality for the Gaussian measure on R-n and the class of sets invariant under transitive permutation group of the coordinates.