I will talk about the construction of the full edge scaling limit of singular values of multiplicative Brownian motion on GL_N(C) starting from general initial conditions. Here the finite-dimensional dynamics are closely related to Heckman-Opdam processes. This construction includes a description of the limiting dynamics in terms of an infinite system of SDE with logarithmic interaction, Brownian Gibbs resampling properties for the paths and the fact that the limiting characteristic polynomial gives rise to a Markov evolution on entire functions which satisfies a certain stochastic PDE. The main driving force that starts the argument is a hidden consistency of the dynamics as the matrix size varies which allows to use techniques from integrable probability. This talk is based on recent joint work with Zahra Sadat Mirsajjadi.
