Christophe Charlier: Asymptotics of Hankel determinants with a one-cut regular potential and Fisher-Hartwig singularities

Abstract : We will briefly review the notions of Hankel determinants, Fisher-Hartwig singularities and one-cut regular potentials, and show several applications of these determinants in random matrix theory. These Hankel determinants are related to certain orthogonal polynomials, which can in turn be expressed as the solution of a 2x2 Riemann-Hilbert problem. In the second part of the talk, we will show the typical strategy to compute large n asymptotics for n x n Hankel determinants, and in particular how to perform a classical steepest descent analysis. This is intended to be an informal seminar and should be accessible to graduate students.