International Workshop on
Operator Theory and its Applications
Let $H_R$ be the integral operator defined on $L^2([R,\infty))$ with kernel $h(x+y)$, where $h$ is the Fourier transform of a certain nicely behaved symbol. While the asymptotics of the operator determinant of $I+H_R$ as $R\to\infty$ is rather trivial, the asymptotics as $R\to-\infty$ is not. One expects an answer analogous to the the theory of Toeplitz or Wiener-Hopf determinants, but the precise relationship is not immediately clear.
In he simplest cases, the analogue of a Szego-Widom/Achiezer-Kac Limit Theorem can be proved. The main interest however is in the case of certain symbols of Fisher-Hartwig type. Then the determinant describes probability distribution function of the largest real eigenvalue of the real Ginibre random matrix ensemble in the $N\to\infty$ limit.
Toeplitz, Hankel, and Toeplitz+Hankel matrices play a central role in random matrix theory and several other fields of mathematics and physics. We will show how the theory of Schur polynomials and symmetric functions can be used to obtain several relations between the determinants and minors of these matrices. We will also discuss some applications of this approach to representation theory and orthogonal polynomials.
We explore the recent advances on the orthogonal polynomials with respect to exponentially varying weights, and apply the results to the boundary universality of random normal matrices as well as the structure of a new process, coined arithmetic jellium.
In the theory of random Hermitian matrices, it is well known that Airy kernel describes spacing at the soft edge of the spectrum, and Bessel kernels at the hard edge. In this talk, I will introduce a family of rescaled 2D-Coulomb gas ensembles which interpolate the free boundary case and the hard edge case. I will discuss the edge universality for these ensembles when the underlying potential is radially symmetric. This is based on joint work with Yacin Ameur and Seong-Mi Seo.
In this talk I will present some results about the convergence, in the Kantorovich-Vaserstein distance, of the empirical measure associated to a beta-ensemble towards its limiting measure. In our setting a beta-ensemble will be a random point process distributed according to the beta power of the determinant of a kernel. Particular examples are the spherical ensemble and its generalizations.
Exponential type of a finite positive measure on the real line is defined as the infimum of the length of an interval such that exponential functions with frequencies from that interval span the $L^2$ space with respect to the measure. Finding the type of a given measure is a classical problem of Fourier analysis and spectral theory. In my talk I will discuss recent progress in the area of the type problem along with a new result, showing that the type of a Frostman measure can only equal zero or infinity.
We give expansions, in terms of Schur and Chebyshev polynomials, of reproducing kernels of Christoffel-Darboux type, using results in random matrix theory, for a number of ensembles. We also explain the relationship with the study of minors and inverses of Toeplitz matrices.
This talk is concerned with the asymptotic behavior of the determinants of Toeplitz matrices generated by symbols with an external parameter as the size of the matrices tends to infinity and the external parameter tends to a critical value. We focus on the applications of these double-scaling limits in random matrix theory and quantum spin chain models, and also compare the operator-theoretic approach to the Riemann-Hilbert analysis.
We consider the planar orthogonal polynomials with multiple logarithmic singularities in the potential and show that these orthogonal polynomials are the multiple orthogonal polynomials of Type II. This equivalence allows us to formulate the matrix Riemann-Hilbert problem for $p_n(z)$. We derive the strong asymptotics of $p_n(z)$ when $n$ goes to infinity and find the limiting locus of zeros of $p_n(z)$. This is a joint work with Seung-Yeop Lee (University of South Florida, U.S.).