Pierpaolo Vivo, King's College, London, UK
Random Matrices - Theory and Practice (12h)
1. Simple classification of random matrix models.
Gaussian and Wishart ensembles. Warmup calculations: semicircle and Marcenko-Pastur in three lines (maybe four).
2. Dyson Coulomb gas
Semicircle and Marcenko-Pastur from a saddle-point-of-view.
3. Orthogonal polynomials and level statistics
A hands-on approach.
4. Largest eigenvalue of a random matrix
Tracy Widom distribution and large deviations. Condensation transition.
5. The Replica approach. Edwards-Jones formalism. Applications to full and sparse matrices (random graphs).
6. Correlated Wishart model. Character expansion method. Role of the eigenvectors.