by P. Vivo (2 credits)
- Simple classification of random matrix models. Gaussian and Wishart ensembles. Warmup calculations: semicircle and Marcenko–Pastur laws.
- Level spacing statistics: Poisson vs Wigner–Dyson.
- Coulomb gas method.
- Orthogonal polynomial technique and numerical checks.
- Largest eigenvalue of a random matrix. Comparison with Extreme Value Statistics for i.i.d. random variables. Tracy–Widom distribution and third-order phase transitions.
- The Replica method. Edwards–Jones formalism. Applications to full and sparse matrices (random graphs).
- Free probability. Sum of random matrices. Blue’s function.