When Majority Fails: Tight Bounds for Correlation Distillation Conjectures

TL;DR

This paper provides nearly tight bounds for two Boolean function conjectures involving the Majority function, clarifying the regimes where they hold or fail for all n ≥ 5.

Contribution

It characterizes the noise parameter regimes where the conjectures hold or fail, refining the original conjectures for all n ≥ 5.

Findings

Both conjectures hold for n=3 in all regimes.

Nearly tight bounds are established for n ≥ 5.

Refined versions of the conjectures are proposed.

Abstract

We study two conjectures posed in the analysis of Boolean functions $f : \{-1, 1\}^n \to \{-1, 1\}$, in both of which, the Majority function plays a central role: the "Majority is Least Stable" (Benjamini et al., 1999) and the "Non-Interactive Correlation Distillation for Erasures" (Yang, 2004; O'Donnell and Wright, 2012). While both conjectures have been refuted in their originally stated form, we obtain a nearly tight characterization of the noise parameter regime in which each of the conjectures hold, for all $n \ge 5$. Whereas, for $n=3$, both conjectures hold in all noise parameter regimes. We state refined versions of both conjectures that we believe captures the spirit of the original conjectures.