Critical and Disordered Systems at ICTP-SAIFR

Random matrix theory

Half-course, spring 2025 · Course description · Syllabus

Lecture notes

1. Introduction & motivation
2. The GOE and its JPDF of eigenvalues
3. From JPDF to spectral density
4. The resolvent
5. The resolvent several ways: cavity & replicas
6. The resolvent several ways: supersymmetry
7. The resolvent several ways: dynamics
8. Free probability
9. Localization

Homework assignments

1. Problem 1.6 Random matrix theory in Jim Sethna’s textbook
2. Dyson Brownian motion & Jacobian for Hermitian random matrices
3. Coulomb gas for Wishart
4. Cavity method & replicas for Wishart
5. Factorization of free products & R transform for Wishart