Abstract
We start from applying the general idea of spectral projection (suggested by Olshanski and Borodin and advocated by Tao) to the complex Wishart model. Combining the ideas of spectral projection with the insights from quantum mechanics, we derive in an effortless way all spectral properties of the complex Wishart model: first, the Marchenko-Pastur distribution interpreted as a Bohr-Sommerfeld quantization condition for the hydrogen atom; second, hard (Bessel), soft (Airy) and bulk (sine) microscopic kernels from properly rescaled radial Schrödinger equation for the hydrogen atom. Then, generalizing the ideas based on Schrödinger equation to the case when Hamiltonian is non-Hermitian, we propose an analogous construction for spectral projections of universal kernels for bi-orthogonal ensembles. In particular, we demonstrate that the Narain transform is a natural extension of the Hankel transform for the products of Wishart matrices, yielding an explicit form of the universal kernel at the hard edge. We also show how the change of variables of the rescaled kernel allows us to make the link to the universal kernel of the Muttalib-Borodin ensemble. The proposed construction offers a simple alternative to standard methods of derivation of microscopic kernels. Finally, we speculate, that a suitable extension of the Bochner theorem for Sturm-Liouville operators may provide an additional insight into the classification of microscopic universality classes in random matrix theory.
Highlights
Determinantal point processes [1] appear in several areas of mathematics, physics and applied sciences, ranging from random matrix theory (RMT) to combinatorics and theory of representations
The unique feature of such processes relies on the fact, that the N -point joint probability distribution function is expressed as a determinant of a matrix built from a single, two-point correlation function known as a kernel
Borodin and Olshanski [18] offered a different point of view at kernels built from orthogonal polynomials in random matrix theory
Summary
Determinantal point processes [1] appear in several areas of mathematics, physics and applied sciences, ranging from random matrix theory (RMT) to combinatorics and theory of representations. Such an operator may not be self-adjoint, still, due the fact that its left and right eigenvectors form a bi-orthogonal basis, it is possible to infer the microscopic limit of the kernels, using the spectral projection method. The power of this approach – easiness of calculation of microscopic kernel without the need of Plancherel-Rotach asymptotics – is demonstrated on two examples: singular values of products of Gaussian matrices [42] and the Muttalib-Borodin ensemble [13,37]. In appendix C we recall some properties of the Meijer-G functions
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