# Advanced Topics in Probability Theory, WS 2016/17

Time and Room | Tuesday 10-12 and Thursday, 8-10, LWK 0.011 |

Exam times | 20-21th of February (first trial), 29-30th of March (second trial) |

## Content

In the first part of the lecture, we will consider the Gaussian Unitary Ensemble of random matrices, whose eigenvalues serves as an example of determinantal point processes. We will then discuss this class of point process in a more general framework and develop some mathematical structure related to it.

In the second part of the lecture, we will consider the exclusion process, discuss its construction and properties like stationary measures. Then we will focus on a particular choice of the parameters, in which particles jumps only to one direction. In that case, much more can be computed explicitly as they fits in the framework that we build up in the context of the GUE matrices.

For large time, we will derive the analogue of the central limit theorem (as well as the limit process). The main two differences are that the fluctuations of particle positions growth only like time^{1/3} and the limit law is not Gaussian. They are given, depending on classes of initial conditions, by some distribution laws that appeared before in the domain of random matrix theory.