On the correlation of increasing families

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:144

Article

Kalai, Gil¹  Keller, Nathan²  Mossel, Elchanan^3,4 

[1] Hebrew Univ Jerusalem, Einstein Inst Math, Jerusalem, Israel

[2] Bar Ilan Univ, Dept Math, Ramat Gan, Israel

[3] Univ Penn, Dept Stat, 3730 Walnut St, Philadelphia, PA 19104 USA

[4] Univ Calif Berkeley, Dept Stat & Comp Sci, 367 Evans Hall, Berkeley, CA 94720 USA

关键词: Correlation inequalities;    Noise sensitivity;    Influences;    FKG inequality;    Discrete Fourier analysis;   

DOI  :  10.1016/j.jcta.2016.06.012

来源: Elsevier

【 摘 要 】

The classical correlation inequality of Harris asserts that any two monotone increasing families on the discrete cube are non negatively correlated. In 1996, Talagrand [19] established a lower bound on the correlation in terms of how much the two families depend simultaneously on the same coordinates. Talagrand's method and results inspired a number of important works in combinatorics and probability theory. In this paper we present stronger correlation lower bounds that hold when the increasing families satisfy natural regularity or symmetry conditions. In addition, we present several new classes of examples for which Talagrand's bound is tight. A central tool in the paper is a simple lemma asserting that for monotone events noise decreases correlation. This lemma gives also a very simple derivation of the classical FKG inequality for product measures, and leads to a simplification of part of Talagrand's proof. (C) 2016 Published by Elsevier Inc.

【 授权许可】

Free