Random matrix theory was introduced into physics in the 1950s to describe the universal statistical properties of the energy levels of nuclear systems. It was later discovered that the theory also describes the properties of chaotic quantum systems. The central idea of random matrix theory is to replace the Hamiltonian of a chaotic system with a matrix whose entries are random variables and to characterize the statistics of its eigenvalues. The theory has been remarkably successful, with numerous applications in both physics and mathematics.
Dyson introduced a dynamical version of random matrix theory in which the random matrix evolves in time, with its elements following Ornstein–Uhlenbeck stochastic processes, while its eigenvalues evolve according to the dynamics of an overdamped Coulomb gas. This process is known as Dyson Brownian motion. The evolution of this model has been extensively studied, in particular how it evolves from a given initial configuration toward the equilibrium distribution predicted by the time-independent theory.
Although Dyson Brownian motion makes it possible to study the individual dynamics of the eigenvalues, we have recently approached the problem using macroscopic fluctuation theory, which describes the evolution of collective quantities such as the eigenvalue density and its associated current. This approach allowed us to determine the dynamical correlations between the eigenvalues and to uncover how a universal behavior known from static correlations is partially preserved in the dynamical setting. The success of this preliminary study opens the way to addressing more ambitious questions aimed at developing a deeper understanding of the dynamics of Dyson Brownian motion.
