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.