On mesoscopic equilibrium for linear statistics in Dyson's Brownian Motion

Understand

In this paper we study mesoscopic fluctuations for Dyson's Brownian motion with $\beta=2$.

• Dyson showed that the Gaussian Unitary Ensemble (GUE) is the invariant measure for this stochastic evolution and conjectured that, when starting from a generic configuration of initial points, the time that is needed for the GUE statistics to become dominant depends on the scale we look at: The microscopic correlations arrive at the equilibrium regime sooner than the macrosopic correlations.
• In this paper we investigate the transition on the intermediate, i.e.
• mesoscopic, scales.