A note on correlation inequalities for regular increasing families

TL;DR

This paper proves new quantitative correlation inequalities for increasing families and threshold functions in discrete and Gaussian spaces, confirming conjectures about optimal bounds.

Contribution

It establishes the first sharp correlation bounds for balanced increasing families and their Gaussian analogues, verifying key conjectures in the field.

Findings

Linear threshold functions achieve covariance lower bounds of c(log n)/sqrt(n).

Results confirm conjectures of Kalai, Keller, and Mossel.

Extends correlation inequalities to Gaussian halfspaces.

Abstract

This paper establishes quantitative correlation inequalities between monotone events and structured threshold objects in both the discrete cube and Gaussian space. We prove that for any increasing balanced family, there exists a linear threshold function yielding a covariance lower bound of $c \frac{\log n}{\sqrt{n}}$, and extend this principle to halfspaces in Gaussian space. These results verify the conjectures of Kalai, Keller, and Mossel regarding optimal correlation bounds for linear threshold functions and their Gaussian analogues.