AI 中文总结
本研究证明了 $\mathrm{GL}_N(\mathbb{C})$ 上乘法布朗运动平方奇异值对数的边缘标度极限为行列式过程，并给出相关核的显式公式及对任意初始条件的扩展。
AI 中文摘要
我们研究了在一般线性群 $\mathrm{GL}_N(\mathbb{C})$ 上乘法布朗运动的平方奇异值对数的边缘标度极限，并考虑了一般初始条件。基于先前的工作 [ arXiv:2605.06429 ]，我们证明了对于增强状态空间中的每一个初始条件，极限动力学都是行列式的，并推导出其扩展多时相关核的显式双轮廓公式。对初始状态的依赖，包括不由粒子坐标确定的信息，由相关的 Laguerre--Pólya 整函数编码。我们进一步将相应的无穷维随机微分方程描述扩展到任意初始配置，包括具有重合坐标的配置。
英文摘要
We study the edge scaling limit of the logarithms of the squared singular values of multiplicative Brownian motion on the general linear group $\mathrm{GL}_N(\mathbb{C})$ with general initial conditions. Building on earlier work [arXiv:2605.06429], we prove that, for every initial condition in the enhanced state space, the limiting dynamics are determinantal and derive an explicit double-contour formula for their extended multitime correlation kernel. The dependence on the initial state, including information not determined by the particle coordinates, is encoded by the associated Laguerre--Pólya entire function. We further extend the corresponding infinite-dimensional stochastic differential equation description to arbitrary initial configurations, including those with coinciding coordinates.
Comments45 pages, 5 figures
