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
Abstract:
1. Brief historical introduction: applications.
• Discussion of basic properties of matrices, different random matrix ensembles, rotationally invariant ensembles such as Gaussian ensembles etc.
2. Gaussian ensembles: derivation of the joint probability distribution of eigenvalues, starting from the joint distribution of matrix entries.
3. 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.
4. 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).
5. 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.
6. Perspectives and summary.

Series:
IPhT Courses
Short course title:
Random matrices
Poster:
Arxiv classes: