Dyson Trace Flow and Multivariate Dynamic Coupled Semicircle Law

Abstract

Interacting random matrix systems are fundamental to modern theoretical physics and data science, yet a unified framework for their analysis has been lacking. This work introduces such a universal framework, built upon two novel concepts: the Dyson Trace Flow characterizing macroscopic fluctuations, and the Multivariate Dynamic Coupled Semicircle Law describing the collective spectral behavior of multiple interacting matrix processes. We establish the stochastic evolution of eigenvalues under asymmetric coupling and prove the mathematical well-posedness of the theory. A large deviation principle is derived, enabling the calculation of rare event probabilities. The framework is extended to nonlinear and non-reciprocal interactions, revealing universal phenomena including exceptional points, bistability, and novel scaling laws. A striking connection to quantum chaos is unveiled through a holographic correspondence with wormhole geometries. By generalizing classical random matrix theory, this work provides powerful tools for understanding neural networks and complex quantum dynamics.