  We study the Dyson-Ornstein-Uhlenbeck diffusion process, an evolving gas of
interacting particles. Its invariant law is the beta Hermite ensemble of random
matrix theory, a non-product log-concave distribution. We explore the
convergence to equilibrium of this process for various distances or
divergences, including total variation, relative entropy, and transportation
cost. When the number of particles is sent to infinity, we show that a cutoff
phenomenon occurs: the distance to equilibrium vanishes abruptly at a critical
time. A remarkable feature is that this critical time is independent of the
parameter beta that controls the strength of the interaction, in particular the
result is identical in the non-interacting case, which is nothing but the
Ornstein-Uhlenbeck process. We also provide a complete analysis of the
non-interacting case that reveals some new phenomena. Our work relies among
other ingredients on convexity and functional inequalities, exact solvability,
exact Gaussian formulas, coupling arguments, stochastic calculus, variational
formulas and contraction properties. This work leads, beyond the specific
process that we study, to questions on the high-dimensional analysis of heat
kernels of curved diffusions.
