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 ${\alpha }_{\mathbb{R}}(N)=N^{-3/2}\mathbb{E}\Vert G_N{\Vert }_{\ast }$ 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 $8/(3\pi )$. 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.