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
该研究证明归一化方形复高斯随机矩阵的平均奇异值随维度严格递减，解决了相关猜想，还推广到固定矩形度的复高斯矩阵情形。
AI 中文摘要
我们解决了Bandeira、Kennedy和Singer提出的一个猜想，该猜想源于他们对酉群上小格罗滕迪克问题近似比的分析，方法是证明归一化方形复高斯随机矩阵的平均奇异值随维度严格递减。研究的起点是Abreu推导的递推关系，该关系源自Christoffel-Darboux公式和拉盖尔多项式的Turán行列式。此递推将问题简化为对混合拉盖尔积分的估计。我们通过将相关积分在与权函数$x^{1/2}\\,\mathrm{e}^{-x}$相关的正交基中展开，将该估计转化为显式有限不等式。随后，一个望远镜恒等式将所得不等式转化为正形式。最终的正性论证结合了中心二项式系数的显式估计、对数下界以及小维度的有限验证，完成了方形复高斯情形下平均奇异值随维度严格递减的证明。作为推论，对于固定矩形度$\lambda=0,1,2,\dots$的$N\times(N+\lambda)$复高斯矩阵，也得到了相同的单调性。
英文摘要
Let $W_{N,N+λ}$ have the Laguerre unitary distribution with size $N$ and real shape $λ\ge0$. For $s>0$, we consider the normalized moment $$C_{s,λ}(N)=N^{-s-1}\mathbb{E}[\operatorname{Tr}W_{N,N+λ}^{s}]$$ and its dimension decrement $Γ_{N,s,λ}=C_{s,λ}(N)-C_{s,λ}(N+1)$. Iterating the Laguerre moment recurrence separates this decrement into a square source and a nonnegative shape source. The square source gives the complete finite-dimensional sign diagram: $C_{s,0}(N)$ decreases for $0<s<1$ and $s>2$, increases for $1<s<2$, and is constant for $s\in\{1,2\}$. For every $s>0$ and every $N$, the decrement is strictly increasing in $λ$. The same decomposition determines the critical shrinking-shape scales: $N^{-2s}$ for $0<s<1/2$, $(\log N)/N$ for $s=1/2$, and $N^{-1}$ for $s>1/2$. In the convex range $1<s<2$, where the two sources have opposite signs, the transition occurs when $Nλ_N$ is of order one, with critical constant $$τ_s^*=\frac{s(s-1)(2-s)}{6(2s-1)}.$$ In the convex range, both crossings are unique for every finite $N$, and we determine their locations to second order. At $s=1/2$ we also obtain a bounded-shape two-term expansion, which supplies the unitary estimates used in the companion orthogonal paper.
Commentsv2: Substantially revised and expanded version with a new title. The original half-moment result from v1 is retained, but the proof is replaced by a recurrence-based two-parameter theory covering general positive moments, strict shape monotonicity, critical shrinking-shape scales, and the convex transition. v1: Original proof in the complex setting, based on Abreu's recurrence
