TR26-063 | When Majority Fails: Tight Bounds for Correlation Distillation Conjectures | Pritish Kamath, Ravi Kumar, Pasin Manurangsi

We study two conjectures posed in the analysis of Boolean functions $f : \\{-1, 1\\}^n ? \\{?1, 1\\}$, 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.