非厄米随机矩阵的精确联合本征值密度为Calogero散射态
Exact joint eigenvalue densities of non-Hermitian random matrices are Calogero scattering states
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中文总结 AI 辅助
该研究解决了任意规模下具有转置对称性的非厄米随机矩阵的精确联合本征值密度问题，将其与Calogero模型的散射态关联，还计算了复能级间距分布和两点谱关联函数，揭示了随机矩阵、可积性与对称性的相互作用。
中文摘要 AI 辅助
确定精确联合本征值密度是随机矩阵理论的核心问题。我们针对具有转置对称性（复对称与复自对偶）的非厄米矩阵，在任意矩阵规模下解决了这一长期存在的问题。在范德蒙德因子的作用下，它们是Calogero模型的散射态波函数，该模型描述了通过平方反比势相互作用的粒子线，耦合强度由对称性决定。与许多已知的联合密度不同，这些密度无法表示为具有两两相互作用的本征值气体。我们进一步计算了复能级间距分布和两点谱关联函数，它们具有幂律尾，而库仑气体中不存在此类尾。我们的结果揭示了随机矩阵、可积性与对称性之间的相互作用。
英文摘要
Determining exact joint eigenvalue densities is central to random matrix theory. We solve this long-standing problem for non-Hermitian matrices with transposition symmetry (complex symmetric and complex self-dual) at arbitrary matrix size. Up to a Vandermonde factor, they are scattering-state wave functions of the Calogero model, a line of particles interacting through an inverse-square potential, with the coupling strength set by the symmetry. In contrast to many previously known joint densities, the densities cannot be written as a gas of eigenvalues with pairwise interactions. We further compute the complex level spacing distributions and two-point spectral correlation functions, which carry power-law tails, absent in a Coulomb gas. Our results shed light on the interplay among random matrices, integrability, and symmetry.
发表机构
- Princeton University（普林斯顿大学）
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