Narain transform for spectral deformations of random matrix models

TL;DR

This paper introduces a novel spectral projection approach using the Narain transform to analyze spectral properties of complex Wishart models and universal kernels, linking quantum mechanics and random matrix theory.

Contribution

It presents the Narain transform as an extension of the Hankel transform for Wishart matrices and offers a new, simpler method for deriving microscopic kernels in random matrix models.

Findings

Derived spectral properties of Wishart models using quantum mechanics analogies.

Introduced the Narain transform for products of Wishart matrices.

Linked rescaled kernels to Muttalib-Borodin ensemble universality.

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 Marcenko-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 Schr\"odinger equation for the hydrogen atom. Then, generalizing the ideas based on Schr\"odinger equation to the case when Hamiltonian is non-Hermitian, we propose an analogous construction for spectral projections of universal kernels built from 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 {\it rescaled} kernel allows to make the link to universal kernel of Muttalib-Borodin ensemble. The proposed construction offers a simple alternative to standard methods of derivation of microscopic kernels, based e.g. on Plancherel Rotach limit of orthogonal polynomials of asymptotics in the Riemann-Hilbert problem. 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.