Random matrix theory: introduction and applications
Satya Majumdar (LPTMS Orsay)
2015-11-20 10:00, Salle Itzykson, IPhT
2015-11-27 10:00, Salle Itzykson, IPhT
2015-12-04 10:00, Salle Itzykson, IPhT
2015-12-14 10:00, Salle Itzykson, IPhT
2015-12-18 10:00, Salle Itzykson, IPhT
Approved by the École Doctorale ED PIF
- Brief historical introduction: applications.
- Discussion of basic properties of matrices, different random matrix ensembles, rotationally invariant ensembles such as Gaussian ensembles etc.
- Gaussian ensembles: derivation of the joint probability distribution of eigenvalues, starting from the joint distribution of matrix entries.
- Analysis of the spectral properties of eigenvalues: given the joint distribution of eigenvalues, how to calculate various observables such as:
- Average density of eigenvalues - Wigner semi-circle law,
- Counting statistics, spacings between eigenvalues etc,
- Distribution of the extreme (maximum or minimum) eigenvalues.
- Two complementary approaches to study spectral statistics:
- Large $N$ (for an $N\times N$ matrix) method by the Coulomb gas approach: saddle point method,
- Finite $N$ method: for Gaussian unitary ensemble: orthogonal polynomial method (essentially quantum mechanics of free fermions at zero temperature).
- Tracy-Widom distribution: probability distribution of the top eigenvalue. Its appearance in a large number of problems, universality and an associated third order phase transition.
- Perspectives and summary.