Solvable random matrix ensemble with a logarithmic weakly confining potential

TL;DR

This paper introduces a new solvable random matrix ensemble with a logarithmic weakly confining potential, characterized by Lorentzian eigenvalue density and expressible spectral correlations, with potential applications in quantum physics.

Contribution

It presents a novel solvable ensemble with a logarithmic potential, linking spectral correlations to nonclassical Gegenbauer polynomials and providing a sampling procedure.

Findings

Spectral correlation functions expressed via Gegenbauer polynomials

Eigenvalue density characterized as Lorentzian in the thermodynamic limit

Numerical verification confirms analytical results

Abstract

This work identifies a solvable (in the sense that spectral correlation functions can be expressed in terms of orthogonal polynomials), rotationally invariant random matrix ensemble with a logarithmic weakly confining potential. The ensemble, which can be interpreted as a transformed Jacobi ensemble, is in the thermodynamic limit characterized by a Lorentzian eigenvalue density. It is shown that spectral correlation functions can be expressed in terms of the nonclassical Gegenbauer polynomials $C_n^{(-1/2)}(x)$ with $n \ge 2$, which have been proven to form a complete orthogonal set with respect to the proper weight function. A procedure to sample matrices from the ensemble is outlined and used to provide a numerical verification for some of the analytical results. This ensemble is pointed out to potentially have applications in quantum many-body physics.