Program:
Random matrix theory describes the spectral behaviour of many different types of large complex systems and has applications throughout mathematical physics, statistics, data science, finance, etc. One of its central principles is universality: many spectral statistics depend only on a small number of structural features and not on the precise distribution of the matrix entries. Recent mathematical progress has established universality for large classes of random matrices, while the same phenomenon is expected to occur much more broadly in physical systems.
This course will introduce some of the main models in random matrix theory, focusing on the Gaussian Unitary Ensemble (GUE), the Circular Unitary Ensemble (CUE), and their generalizations. The main goal is to study both microscopic eigenvalue fluctuations and macroscopic fluctuations of spectral observables of these models, and to introduce the concept of universality. First, we will derive the fundamental limiting laws of random matrix theory through integrable models; then, we will explain the modern Erdős–Yau dynamical approach to universality, which relies on local laws and the relaxation of Dyson Brownian motion. The subject will be presented as a meeting point of probability theory, analysis and statistical mechanics.
The preliminary plan is to address the following subjects:
- Introduction: matrix models and the semicircle law
- Gaussian ensembles (GUE and GOE)
- Determinantal point processes
- (\beta)-ensembles and one-dimensional log-gases
- Tridiagonal matrix models, Gaussian (\beta)-ensembles and random operators
- Haar-distributed matrices and the CUE
- The circular (\beta)-ensemble and the Sine-(\beta) process
- Local laws and universality for Wigner matrices
The rooms are 136 for all dates, except October 16th, in 134.