Introduction to the Random Matrix Theory: Gaussian Unitary Ensemble and Beyond

Details Introduction to the Random Matrix Theory: Gaussian Unitary Ensemble and Beyond Introduction to the Random Matrix Theory: Gaussian Unitary Ensemble and Beyond

2004-12-07

arxiv.org/abs/math-ph/0412017v2

Recent perspectives in random matrix theory and number theory, 31--78, London Math. Soc. Lecture Note Ser., 322, Cambridge Uni

Physics

Mathematical Physics

39 pages, 10 figures. These are lectures at the Isaak Newton Institute Summer School (Cambridge, 2004)

Scientific paper

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.

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