Infinite-Particle Determinantal Dynamics from Brownian Motion on $\mathrm{GL}_N(\mathbb{C})$

Abstract: 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.