When Majority Fails: Tight Bounds for Correlation Distillation Conjectures

Pritish Kamath; Ravi Kumar; Pasin Manurangsi

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arXiv · 2604.06590

When Majority Fails: Tight Bounds for Correlation Distillation Conjectures

Pritish Kamath·Ravi Kumar·Pasin Manurangsi

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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.

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Pritish Kamath, Ravi Kumar, Pasin Manurangsi. 2026-04-08. When Majority Fails: Tight Bounds for Correlation Distillation Conjectures. https://arxiv.org/abs/2604.06590

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