The Average Singular Value of a Real Square Gaussian Random Matrix Strictly Increases with Dimension

Abstract

We settle the real half of a conjecture of Bandeira, Kennedy and Singer on the dimension dependence of the Gaussian constant governing the little Grothendieck problem over the orthogonal group. For an $N\times N$ standard real Gaussian matrix $G_N$, the average singular value $α_{\mathbb R}(N)=N^{-3/2}\,\mathbb E\|G_N\|_*$ satisfies the quantitative estimate \[ α_{\mathbb R}(N+1)-α_{\mathbb R}(N)>\frac{1}{1000N^2}, \qquad N\ge1. \] Thus, the real constants increase strictly to the Marchenko--Pastur limit $8/(3π)$. The proof is finite-dimensional and exposes a mechanism not visible in the limiting spectral law. We decompose the Laguerre-orthogonal mean into its Laguerre-unitary counterpart and an explicit correction, then complete the resulting finite Laguerre sums to infinite diagonal tails. A bivariate generating function yields a positive diagonal kernel with a dimension-monotone remainder. This puts consecutive orthogonal corrections in common positive coordinates, where the nearest diagonal alone supplies an $N^{-2}$ reserve that dominates the unitary one-step term. The required unitary estimate is derived directly from Abreu's recurrence, and the first five dimensions are handled by exact closed forms.

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“28 ONDREJ HUTNÍK Acknowledgments During the preparation of this work, the author used ChatGPT (OpenAI) as an auxiliary tool for language and stylistic editing, organization of the presentation, and secondary consistency checks. The conception of the work, mathematical results, arguments, and proofs are the author’s own. The author takes fu”

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2026-08-23T05:12:25.266899Z

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