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 $\lambda \ge 0$. 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 $\lambda \ge 1$ 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 ${\lambda }_{\ast }=1-\pi /4$, 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 $\lambda \ge 1$, and the transition can then all be read from this same series.

v2: Substantially revised and expanded version with a new title. The square monotonicity theorem from v1 is retained with a simpler proof and a stronger explicit bound. New results include the continuous rectangularity transition, uniqueness of both finite-dimensional crossings, and their two-term asymptotics. v1: Original proof in the real setting, based on Abreu's recurrence