Dimension Monotonicity in Laguerre Ensembles II: Average Singular Values and the Rectangularity Transition in the Orthogonal Case
Abstract
We study the normalized half moment αℝ(λ)(N) of the size-N Laguerre orthogonal ensemble for real shape λ≥0. At integer shape this is the expected average singular value of an N×(N+λ) real Gaussian matrix. The square mean increases with the dimension, whereas every real shape λ≥1 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≥1, and the transition can then all be read from this same series.
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BibTeX
@misc{hutnik2026dimension,
title = {{Dimension Monotonicity in Laguerre Ensembles II: Average Singular Values and the Rectangularity Transition in the Orthogonal Case}},
author = {Ondrej Hutník},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.12151},
eprint = {2608.12151},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.12151},
}
RIS
TY - UNPB
TI - Dimension Monotonicity in Laguerre Ensembles II: Average Singular Values and the Rectangularity Transition in the Orthogonal Case
AU - Ondrej Hutník
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2608.12151
UR - https://arxiv.org/abs/2608.12151
AB - We study the normalized half moment α_ℝ^(λ)(N) of the size-N Laguerre orthogonal ensemble for real shape λ≥0. At integer shape this is the expected average singular value of an N×(N+λ) real Gaussian matrix. The square mean increases with the dimension, whereas every real shape λ≥1 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≥1, and the transition can then all be read from this same series.
ER -