发表机构
Institute of Mathematics, Faculty of Science, Pavol Jozef Šafárik University in Košice（科希策帕沃尔·约瑟夫·沙法里克大学理学院数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究解决了Bandeira等人关于正交群上小Grothendieck问题高斯常数维度依赖性猜想的实部，证明实方形高斯随机矩阵平均奇异值随维度严格递增至Marchenko–Pastur极限。
AI 中文摘要
我们解决了Bandeira、Kennedy和Singer关于正交群上小Grothendieck问题的高斯常数维度依赖性猜想的实部。对于N×N标准实高斯矩阵G_N，平均奇异值α_ℝ(N)=N^{-3/2}𝔼∥G_N∥_*满足定量估计：当N≥1时，α_ℝ(N+1)-α_ℝ(N)>1/(1000N²)。因此，实常数严格递增至Marchenko–Pastur极限8/(3π)。该证明是有限维的，揭示了在极限谱律中不可见的机制。我们将Laguerre-正交均值分解为其Laguerre-酉对应项和显式修正项，再将所得有限Laguerre和补全为无限对角尾项。双变量生成函数产生具有维度单调余项的正对角核，这使得连续正交修正项处于共同正坐标中，其中仅最近的对角项就提供了N^{-2}的储备，其主导了酉单步项。所需的酉估计直接由Abreu递推导出，且前五个维度由精确闭式处理。
英文摘要
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.
Commentsv2: 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
