Functional limit theorem for the determinantal sine-process

Determinantal (fermion) random point processes arise naturally in physics and different areas of mathematics. In particular, they play a central role in the random matrix theory. Despite that during the last two decades determinantal processes have been an object of intensive study, their dynamical properties are not understood very well. Costin and Lebowitz and then Soshnikov have established that a large class of determinantal processes satisfies the Central Limit Theorem. It is known that for many dynamical systems satisfying the CLT the Donsker Invariance Principle (Functional Central Limit Theorem, FCLT) also takes place. The latter states that trajectories of the system, in some appropriate sense, can be approximated by trajectories of the Brownian motion. However, for determinantal processes nothing is known about behavior of their trajectories.
In the first part of the talk I will explain what are determinantal processes and where do they arise. In the second part I will recall the classical FCLT and present results of my joint work with A. Bufetov, where we obtain its analog for one of the most important determinantal processes: the sine-process. It turns out that nothing resembling the Brownian motion arises, but a Gaussian process with a completely different behavior appears.