VIRTUAL SEMINAR SERIES -- INTEGRABLE SYSTEMS: Orthogonal Polynomials and Random Matrices
Walter Van Assche (KU Leuven)
Wednesday 28th October, 2020 13:30-14:30 Zoom seminar hosted by ICMS
I will give a brief survey on the connection between orthogonal polynomials and random matrices. Hermite polynomials appear in a natural way in the Gaussian Unitary Ensemble (GUE), Laguerre polynomials in the Wishart Ensemble and Jacobi polynomials for singular values of truncations of unitary random matrices. Multiple orthogonal polynomials appear in random matrix models with external field and products of random matrices. The expected value of the characteristic polynomial of a random matrix is often an orthogonal or multiple orthogonal polynomial and the correlation functions for the eigenvalues or singular values of various random matrices are in terms of determinants containing the Christoffel-Darboux kernel for orthogonal or multiple orthogonal polynomials. This means that eigenvalues or singular values of random matrices form a determinantal point process. Universality then boils down to proving asymptotic results for the Christoffel-Darboux kernel in the bulk of the spectrum, at the soft or hard edges and at critical points where phase transitions occur. Various special functions are needed in the analysis, such as Bessel functions, the Airy function, the Pearcey function, and Meijer G-functions.
For more information, and how to access the seminar through Zoom, see the webpage of the events here.