发表机构
Institut Louis Bachelier; Ecole Polytechnique, Institut Polytechnique de Paris; Capital Fund Management（路易·巴舍利耶研究所; 巴黎综合理工学院，巴黎理工学院; 资本基金管理公司）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究非厄米随机矩阵约束球面积分的大N渐近，揭示从离域到局域的相变，并提出单特征值大偏差和边界涨落的猜想。
AI 中文摘要
我们研究了非厄米随机矩阵的约束球面积分的大$N$渐近行为，其中两个向量的范数和相互标量积被固定。在离域区域，渐近行为由非厄米变换$\mathcal R_1$和$\mathcal R_2$控制。在鞍点层面，约束积分表现出向局域区域的转变，该区域由移位矩阵的最大奇异值及其关联的左右奇异向量的重叠控制。受厄米球面积分机制和库仑气体图像的启发，我们对单特征值大偏差和边界涨落提出了猜想。在整个分析过程中，我们以数学物理的精神进行。
英文摘要
We study the large-$N$ asymptotics of a constrained spherical integral for non-Hermitian random matrices, in which the norms and mutual scalar product of two vectors are fixed. In the delocalized regime, the asymptotics are governed by the non-Hermitian transforms $\mathcal R_1$ and $\mathcal R_2$. At saddle-point level, the constrained integral exhibits a transition to a localized regime controlled by the largest singular value of a shifted matrix and by the overlap of its associated left and right singular vectors. Motivated by the Hermitian spherical-integral mechanism and by the Coulomb-gas picture, we formulate conjectures for one-eigenvalue large deviations and boundary fluctuations. Throughout, our analysis is carried out in the spirit of mathematical physics.
