Goal: to provide a deeper understanding of the Riemann-Hilbert problems for orthogonal polynomials and applications in random matrix theory.
Course description: This course is a continuation of the course Riemann-Hilbert problems for orthogonal polynomials with applications. We review the basic notions of the Random Matrix Theory and in particular the Gaussian Unitary Ensemble. We describe the Dyson gas in equilibrium and nonequilibrium that allows one to interpret the statistical information of the eigenvalues of random matrices. Furthermore we show alternative descriptions of this statistical information. We discuss different aspects of orthogonal polynomials. One of
these caracterizations is given by a Riemann Hilbert problem. Riemann-Hilbert problem techniques are an efficient and powerfull tool for Random Matrix Theory which we discuss in more detail. In the final part we use the steepest descent method in the asymptotic analysis of orthogonal polynomials.
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