Random Matrix Theory

(Math 8823 BED)   Jean Bellissard
Professor of Mathematics and Physics

I)- Random matrices

A historical point of view

Dyson's symmetry and Wishart's distributions

Orthogonal polynomials and the Riemann-Hilbert problem

The unitary ensembles, universality of correlations

The gaussians orthogonal and symplectic ensembles

Statistical estimators, applications: Nuclear Physics, Quantum Chaos, the Riemann ζ-function

II)- Free probabilities

Free groups and random walks

Probability spaces and freeness

Tensor algebra and representation of free random variables

Additive and multiplicative free convolutions

Applications: free central limit theorem, free divisibility

Combinatorial aspects of free probabilities

Free entropy, free brownian motion, Fisher's information, large deviations

III)- Supersymmetric methods

Grassmann variable and integrals

Gaussian integrals

Random matrices and SUSY

Random matrices and mesoscopic transport

The diffusive regime of Anderson's model