1808

The small group will study  chaotic properties of determinantal point processes, emphasizing limit theorems and the Gibbs property. From an analytical point of view, the techniques employed will be closely related to the Riemann-hilbert problem. The continuous sine-process of Dyson and its discrete analogue introduced by Borodin, Okounkov and Olshanski  are the most classical examples of stationary determinantal point processes. Their ergodic properties have been extensively studied in the last two decades. In particular, celebrated results of Costin-Lebowitz  and Soshnikov  give the Central Limit Theorem for a wide class of determinantal processes including the sine-process. Nonetheless many key questions about the dynamics of determinantal point processes remain wide open, and the objectives below are open problems for the sine-process itself. The central aim is to give a precise quantitative description of the chaotic behaviour of the trajectories of determinantal point processes. The main difficulty is that the  variance of the number of particles of the sine-process exhibits an extremely slow logarithmic growth as opposed to  linear growth found in most models.