Abstract:
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.
