SISSA, Fall 2026
PhD course, 50 hours
Lectures: Monday, Wednesday and Friday, 16:00–18:00
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 introduces 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. We will then explain the modern Erdős–Yau dynamical approach to universality, based 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.
Introduction: Wigner matrices and the semicircle law
Gaussian ensembles (GUE and GOE)
Determinantal point processes
β-ensembles and one-dimensional log-gases
Tridiagonal matrix models, Gaussian β-ensembles and random operators
Haar-distributed matrices and the CUE
The circular β-ensemble and the Sine-β process
Local laws and universality for Wigner matrices
Lecture notes will be updated as the course progresses. The exercises are included in the lecture notes.
Lecture 1. Introduction, Wigner matrices and Gaussian measures.
Lecture 2. Eigenvector delocalization, semicircle local law for the resolvent and Stieltjes transform of the empirical measure.
Lecture 3.