On the Correlation of Increasing Families

TL;DR

This paper improves correlation lower bounds for increasing families on the discrete cube under certain conditions, introduces new tight examples, and simplifies existing proofs using a noise-decrease lemma.

Contribution

It provides stronger correlation bounds under regularity or symmetry conditions and introduces new classes of tight examples, simplifying proofs with a noise-decrease lemma.

Findings

Stronger correlation lower bounds under natural conditions

New classes of examples where bounds are tight

Simplified proof of correlation inequalities using noise lemma

Abstract

The classical correlation inequality of Harris asserts that any two monotone increasing families on the discrete cube are nonnegatively correlated. In 1996, Talagrand 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.