When Majority Fails: Tight Bounds for Correlation Distillation Conjectures

2026-04-07

AI Summary

This work investigates the central role of the majority function in the noise stability of Boolean functions and non-interactive correlation extraction, focusing on two classical conjectures: “majority is least stable” and “majority is optimal for correlation extraction under the erasure model.” Although both conjectures have been disproved by counterexamples, this paper provides the first nearly tight characterizations of the conditions under which they hold, using tools from Boolean analysis, noise stability theory, and combinatorial probability. Moreover, it proposes refined formulations that better capture the original intuition. The results establish that both conjectures are valid for all noise parameters when the dimension $ n = 3 $, but for $ n \geq 5 $, they hold only within specific noise regimes, thereby precisely delineating their boundaries of validity.

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.

Problem

• Boolean functions
• Majority function
• Correlation distillation
• Noise stability
• Conjectures

Innovation

• Boolean functions
• Majority function
• Correlation distillation
• Noise stability
• Conjecture refinement