Dyson Trace Flow and Dynamic Coupled Semicircle Law

Abstract: This work introduces a universal framework for analyzing coupled random matrix models, centered around the newly defined Dyson Trace Flow and the Dynamic Coupled Semicircle Law. We derive stochastic differential equations for eigenvalues under asymmetric coupling, establish well-posedness, and prove a large deviation principle with explicit rate functions. The theory is extended to nonlinear and non-reciprocal interactions, revealing phenomena such as exceptional points, bistability, and novel scaling laws. A holographic correspondence with wormhole geometries is established, connecting to quantum chaos. These results generalize classical random matrix theory to interacting systems, with applications in neural networks and quantum dynamics.