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

Abstract

We study the normalized half moment $α_{\mathbb R}^{(λ)}(N)$ of the size-$N$ Laguerre orthogonal ensemble for real shape $λ\ge0$. At integer shape this is the expected average singular value of an $N\times(N+λ)$ real Gaussian matrix. The square mean increases with the dimension, whereas every real shape $λ\ge1$ decreases. Between these two regimes the decrement is strictly increasing in $λ$, 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 $λ_*=1-π/4$, 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 $λ\ge1$, and the transition can then all be read from this same series.