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
Arxiv classes: