Stochastic Domination of Gaussian Maxima: A Resolution of the Weak Simplex Conjecture

2026/07/15 by Abhijeet Mulgund
#math.PR #cs.IT #math.IT #math.MG

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Abstract

We prove a stochastic comparison for Gaussian maxima. Let R be an m× m correlation matrix satisfying R-1 1^{\mathsf T}/m\succeq0, let X\simN(0,R), and let Z₁,…,Z_m be independent standard Gaussian random variables. Then max_{1≤ i≤ m}X_i ≤_st max_{1≤ i≤ m}Z_i, or equivalently, ℙ\X_i≤ c for every i\≥Φ(c)^m for every c∈ℝ. This comparison resolves the Weak Simplex Conjecture: among d+1 equiprobable equal-energy signals in 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.

Citations

• Equivalent Comparisons of Experiments
• Values of Mills' Ratio of Area to Bounding Ordinate and of the Normal Probability Integral for Large Values of the Argument

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