论文标题
根据riemannian浸没和同一轨道的梯度流动的特征值工艺
Eigenvalue processes in light of Riemannian submersion and gradient flow of isospectral orbits
论文作者
论文摘要
We prove eigenvalue processes from dynamical random matrix theory including Dyson Brownian motion, Wishart process, and Dynkin's Brownian motion of ellipsoids are results of projecting Brownian motion through Riemannian submersions induced by isometric action of compact Lie groups, whose orbits have nonzero mean curvature, which contributes to drift terms and is the log gradient of orbit volume function, showing in another way that eigenvalues collide每当纤维退化时。因此，我们提供了统一的处理和在不同环境中的特征值过程与黎曼几何学语言之间的更好联系。 在这种解释下，我们看到了如何自然恢复特征向量过程并得出$β$过程，例如$β$ -Dyson Brownian Motion for General $β> 0 $。 \ textbf {关键字}：特征值过程，对称空间，平均曲率流，梯度流，等光谱，随机矩阵集合
We prove eigenvalue processes from dynamical random matrix theory including Dyson Brownian motion, Wishart process, and Dynkin's Brownian motion of ellipsoids are results of projecting Brownian motion through Riemannian submersions induced by isometric action of compact Lie groups, whose orbits have nonzero mean curvature, which contributes to drift terms and is the log gradient of orbit volume function, showing in another way that eigenvalues collide whenever the fibre is degenerate. We thus provide a unified treatment and better connection between eigenvalue processes in different settings with the language of Riemannian geometry. Under such interpretation, we see how we can naturally recover eigenvector processes and derive $β$ process such as $β$-Dyson Brownian motion for general $β>0$. \textbf{KEYWORDS}: eigenvalue process, symmetric space, mean curvature flow, gradient flow, isospectral manifold, random matrix ensemble
