Universal cutoff for Dyson Ornstein Uhlenbeck process

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