Stochastic Domination of Gaussian Maxima: A Resolution of the Weak Simplex Conjecture
Abhijeet Mulgund
math.PR
Jul 15, 2026 · v2
cs.IT math.MG
TL;DR
A Lean formalization of the adaptive tilting and Gaussian product inequality argument proving the stochastic domination is provided.
Abstract
We prove a stochastic comparison for Gaussian maxima. Let $R$ be an $m\times m$ correlation matrix satisfying $R-\mathbf{1} \mathbf{1}^{\mathsf T}/m\succeq0$, let $X\sim\mathcal{N}(0,R)$, and let $Z_1,\ldots,Z_m$ be independent standard Gaussian random variables. Then $\max_{1\leq i\leq m}X_i \leq_{\mathrm{st}} \max_{1\leq i\leq m}Z_i$, or equivalently, $\mathbb{P}\{X_i\leq c\text{ for every }i\}\geqΦ(c)^m$ for every $c\in\mathbb{R}$. This comparison resolves the Weak Simplex Conjecture: among $d+1$ equiprobable equal-energy signals in $\mathbb{R}^d$ transmitted over an additive white Gaussian noise channel, the regular simplex maximizes the probability of correct maximum-likelihood decoding at every signal-to-noise ratio. It also proves the inequality asserted by the Simplex Mean Width Conjecture and gives an exact formula for the largest number of equiprobable messages that can be sent at prescribed energy and error probability by a deterministic no-feedback AWGN code under a per-codeword energy constraint. The proof combines a Gaussian product inequality for log-concave functions with an adaptive tilting argument that makes the inequality applicable to the one-sided threshold events defining the maximum. A lean formalization of this argument is available at
https://github.com/abhmul/weak-simplex-conjecture-lean.
Problem
The Weak Simplex Conjecture asks whether, among d+1 equiprobable equal-energy signals over an AWGN channel, the regular simplex maximizes correct maximum-likelihood decoding probability at every signal-to-noise ratio. The core is a stochastic comparison for Gaussian maxima under a semidefinite condition on the correlation matrix.
Approach
For a correlation matrix R with R - J/m ⪰ 0 and X N(0,R), the authors prove max_i X_i is stochastically dominated by the maximum of independent standard Gaussians. The proof combines a Gaussian product inequality for log-concave functions (with a vanishing first-moment condition) with an adaptive tilting argument that reduces one-sided equal-threshold events to the product inequality. A Lean formalization of the argument is provided.
Results
The stochastic domination is proven, resolving the Weak Simplex Conjecture and implying the Simplex Mean Width inequality, and yielding an exact finite-energy AWGN coding formula under per-codeword energy constraints. Equality cases remain open.
