Satya Majumdar. LPTMS Orsay University, France

Random Matrix Theory and its Applications (13.5h)

1. Basics of Random Matrix Theory (rotationally invariant ensembles, e.g. the three classical Gaussian ensembles): Joint distribution of eigenvalues.

2. Dyson Coulomb gas: computation of the average density of states (Wigner semi-circle): introduction to saddle point method.

3. Wishart matrices: average density (Marcenko-Pastur law), application to principal component analysis (PCA).

4. Spacing distribution (Wigner's surmise), orthogonal polynomials: basic introduction, few examples.

5. Linear statistics via saddle point method: singular integral equation, Tricomi solution, Resolvent method.

6. Several examples of linear statistics: (i) Index distribution of Gaussian and Wishart matrices, (ii) Distribution of conductance and shot noise for chaotic cavities (iii) Entanglement entropy in the pure state of a random bipartite system.

7. Top eigenvalue of a random matrix: Tracy Widom distribution, large deviations of the top eigenvalue: Gaussian, Wishart, Cauchy distributions; experimental results. Universal third-order phase transition.

8. Nonintersecting Brownian motions: Dyson Brownian motion, applications to fluctuating interfaces/steps, connection to Yang-Mills gauge theory and matrix models.