Patrik Ferrari (Weierstrass Institute)

Random matrices and related problems (11.5 h)

1. Introduction

From micro to macro: occurrence of universal physical laws. What does it mean that a physical system is well modeled by a random matrix ensemble? A few examples.

2. Gaussian Unitary/Orthogonal Ensembles (GUE) of random matrices

Definition. Distribution of eigenvalues. Macroscopic behavior: Wigner semi-circle law. Correlation functions: determinantal structure for GUE. Two important universal limits: bulk and edge scaling limits.

3. Determinantal point processes

Point processes and their determinantal subclass. A class of measures with determinantal structure. Correlation functions and factorial moments. Gap probability: a Fredholm determinant.

4. Tracy-Widom distributions

Definition and properties. GUE Tracy-Widom distribution and Painlevé II equation.

5. Extension of determinantal processes

Linström-Gessel-Viennot theorem. Dyson's Brownian Motion. Extended determinantal point processes. An universal limit: the Airy2 process.

6. Applications to other processes

Polynuclear growth model. Totally asymmetric simple exclusion process.