Ergodic properties of the sine-process

Dyson’s sine-process, the scaling limit of radial parts of Haar measures on unitary groups of growing dimension, is the most classical point process of random matrix theory. In the survey talk, we shall consider the ergodic properties of the sine-process, including the speed of convergence in the Soshnikov Central Limit Theorem and the convergence of its stochastic Euler products to the Gaussian Multiplicative Chaos.