Dimension Monotonicity in Laguerre Ensembles II: Average Singular Values and the Rectangularity Transition in the Orthogonal Case

Abstract

We study the normalized half moment $\alpha_{\mathbb R}^{(\lambda)}(N)$ of the size-$N$ Laguerre orthogonal ensemble for real shape $\lambda\ge0$. At integer shape this is the expected average singular value of an $N\times(N+\lambda)$ real Gaussian matrix. The square mean increases with the dimension, whereas every real shape $\lambda\ge1$ decreases. Between these two regimes the decrement is strictly increasing in $\lambda$, and hence has a unique zero in $(0,1)$ for every $N$. There is also a unique crossing of the Marchenko--Pastur limit. Both thresholds converge to $\lambda_*=1-\pi/4$, and their first corrections show that they separate on the scale $(\log N)/N$. The proof starts from an exact decomposition of the real half moment into its complex counterpart and a positive orthogonal correction. Recent unitary estimates take care of the complex term. An Abel completion, together with a Laguerre connection formula, turns the orthogonal correction into a positive diagonal series; the square case, the regime $\lambda\ge1$, and the transition can then all be read from this same series.