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Universality in Random Matrix Theory

  • Date : 2023/12/20 (Wed.) 15:30~16:30
  • Location : Auditorium, 6 Floor, Institute of Mathematics (NTU Campus)
  • Speaker : Peter Miller (University of Michigan)
  • Abstract : The famous Gaussian Unitary Ensemble (GUE) consists of random Hermitian matrices with elements identically and independently (up to symmetry) distributed as Gaussian random variables. It also has the property that its probability measure is invariant under unitary conjugations. In general unitary invariant ensembles, one maintains the latter property but generalizes the measure by including a potential. We explain how the correlation functions of eigenvalues for such an ensemble can be compactly represented in terms of orthogonal polynomials for a weight involving the potential. Then we demonstrate how Deift, Kriecherbauer, McLaughlin, Venakides, and Zhou proved universality of the correlation functions both in the bulk of the spectrum and at the edge for general convex and analytic potentials, vastly generalizing known results for GUE. Their method uses a characterization of orthogonal polynomials in terms of the solution of a matrix Riemann-Hilbert problem, which applies for completely general weights. Finally, we discuss how the methodology was further extended by McLaughlin and the speaker to extend the universality results to potentials having just two Lipschitz continuous derivatives.