Local Optimality of Dictator Functions with Applications to Courtade--Kumar and Li--Médard Conjectures

TL;DR

This work advances the noise-stability program for Boolean functions by proving that dictator functions are locally optimal for the Φ-stability under the Bonami–Beckner operator and by leveraging majorization and hypercontractivity to obtain global optimality results. It confirms the balanced Courtade–Kumar conjecture for ρ in [0,0.914] and the symmetrized Li–Médard conjecture for q in [1.36,2) across all ρ∈[0,1], with computer-assisted proofs underpinning the bounds. The paper also provides extensions to arbitrary Boolean functions and to Ornstein–Uhlenbeck operators, and it develops a rich set of bounds and conjectures for 0<q<1. Together, these results connect majorization theory, hypercontractivity, and information-theoretic stability to sharpen longstanding conjectures at the intersection of discrete analysis and information theory.

Abstract

Given a convex function $Φ:[0,1]\to\mathbb{R}$, the $Φ$-stability of a Boolean function $f$ is $\mathbb{E}[Φ(T_ρf(\mathbf{X}))]$, where $\mathbf{X}$ is a random vector uniformly distributed on the discrete cube $\{\pm1\}^{n}$ and $T_ρ$ is the Bonami-Beckner operator. In this paper, we prove that dictator functions are locally optimal in maximizing the $Φ$-stability of $f$ over all balanced Boolean functions. When focusing on the symmetric $q$-stability, combining this result with our previous bound, we use computer-assisted methods to prove that dictator functions maximize the symmetric $q$-stability for $q=1$ and $ρ\in[0,0.914]$ or for $q\in[1.36,2)$ and all $ρ\in[0,1]$. In other words, we confirm the (balanced) Courtade--Kumar conjecture with the correlation coefficient $ρ\in[0,0.914]$ and the (symmetrized) Li--Médard conjecture with $q\in[1.36,2)$. We conjecture that dictator functions maximize both the symmetric and asymmetric $\frac{1}{2}$-stability over all balanced Boolean functions. Our proofs are based on the majorization of noise operators and hypercontractivity inequalities.

Figures (4)

• Figure 1: The region of $(q,\rho)$ (the gray region here) for which dictator functions maximize the $q$-stability (i.e., for which the Li--Médard conjecture is true). For $q=1$, the corresponding $\rho$ on the boundary of this region is $\rho=0.914$. For $q\ge1.36$, the corresponding $\rho$ on the boundary of this region is $\rho=1$.
• Figure 2: The function of $\rho\mapsto\epsilon_{1,\rho}^{*}$.
• Figure 3: The function of $\rho\mapsto\epsilon_{q,\rho}^{*}$ with $q=1.36$.
• Figure 4: The function of $g$.

Theorems & Definitions (52)

• Definition 1
• Definition 2
• Conjecture 1: Asymmetric $q$-Stability Conjecture
• Conjecture 2: Symmetric $q$-Stability Conjecture
• Definition 3
• Proposition 1
• Definition 4
• Proposition 2
• Proposition 3
• Proposition 4