Introduction to the random matrix theory: Gaussian unitary ensemble and beyond.

*(English)* Zbl 1204.11151
Mezzadri, Francesco (ed.) et al., Recent perspectives in random matrix theory and number theory. Proceedings of a school that was part of the programme ‘Random matrix approaches in number theory’, Cambridge, UK, January 26–July 16, 2004. Cambridge: Cambridge University Press (ISBN 0-521-62058-9/pbk). London Mathematical Society Lecture Note Series 322, 31-78 (2005).

Summary: These lectures provide an informal introduction into the notions and tools used to analyze statistical properties of eigenvalues of large random Hermitian matrices. After developing the general machinery of orthogonal polynomial method, we study in most detail Gaussian Unitary Ensemble (GUE) as a paradigmatic example. In particular, we discuss Plancherel-Rotach asymptotics of Hermite polynomials in various regimes and employ it in spectral analysis of the GUE. In the last part of the course we discuss general relations between orthogonal polynomials and characteristic polynomials of random matrices which is an active area of current research.

##### MSC:

11M50 | Relations with random matrices |

11M41 | Other Dirichlet series and zeta functions |

15B52 | Random matrices |

33C45 | Orthogonal polynomials and functions of hypergeometric type |

60J65 | Brownian motion |

62H20 | Statistical measures of associations |