标题： 非厄米矩阵值布朗运动的特征值、特征向量重叠和正则化弗格勒德-卡迪森行列式 显示英文标题

标题： Eigenvalues, eigenvector-overlaps, and regularized Fuglede-Kadison determinant of the non-Hermitian matrix-valued Brownian motion

摘要： 非厄米矩阵值布朗运动是随机矩阵的随机过程，其元素由独立的复数布朗运动给出。 右特征向量过程和左特征向量过程之间施加了双正交关系，这允许它们进行尺度变换，同时保持特征值过程不变。 特征向量重叠过程是一个厄米矩阵值过程，其中每个元素由右特征向量的重叠和左特征向量的重叠的乘积给出。 我们推导了特征值过程和特征向量重叠过程耦合系统的随机微分方程（SDEs），并证明了所得SDE系统的尺度变换不变性。 与当前矩阵值过程相关的弗格莱德-卡迪森（FK）行列式通过引入一个辅助复变量进行正则化。 这个变量对于给出由正则化FK行列式定义的时间依赖随机场及其平方和对数变体的随机偏微分方程（SPDEs）是必要的。 时间依赖的特征值点过程及其由特征向量重叠过程的对角元素加权的变体与对数正则化FK行列式随机场的导数相关。 我们还讨论了通过对SPDEs进行平均得到的偏微分方程（PDEs）。 显示英文摘要

摘要： The non-Hermitian matrix-valued Brownian motion is the stochastic process of a random matrix whose entries are given by independent complex Brownian motions. The bi-orthogonality relation is imposed between the right and the left eigenvector processes, which allows for their scale transformations with an invariant eigenvalue process. The eigenvector-overlap process is a Hermitian matrix-valued process, each element of which is given by a product of an overlap of right eigenvectors and that of left eigenvectors. We derive a set of stochastic differential equations (SDEs) for the coupled system of the eigenvalue process and the eigenvector-overlap process and prove the scale-transformation invariance of the obtained SDE system. The Fuglede--Kadison (FK) determinant associated with the present matrix-valued process is regularized by introducing an auxiliary complex variable. This variable is necessary to give the stochastic partial differential equations (SPDEs) for the time-dependent random field defined by the regularized FK determinant and for its squared and logarithmic variations. Time-dependent point process of eigenvalues and its variation weighted by the diagonal elements of the eigenvector-overlap process are related to the derivatives of the logarithmic regularized FK-determinant random-field. We also discuss the PDEs obtained by averaging the SPDEs.