Universal behavior of the corners of Orbital Beta Processes

TL;DR

This paper establishes the universal limiting behavior of eigenvalues at the corners of orbital beta processes, extending known results to a broad class of matrix ensembles and connecting to Gaussian beta corners.

Contribution

It introduces the orbital beta processes and proves their eigenvalue corner universality, generalizing previous results to new matrix ensembles and beta parameters.

Findings

Eigenvalues of small principal submatrices converge to the Gaussian beta corners process.

Established asymptotics for multivariate Bessel functions.

Demonstrated universality across different matrix ensembles and beta values.

Abstract

There is a unique unitarily-invariant ensemble of $N\times N$ Hermitian matrices with a fixed set of real eigenvalues $a_1 > \dots > a_N$. The joint eigenvalue distribution of the $(N - 1)$ top-left principal submatrices of a random matrix from this ensemble is called the orbital unitary process. There are analogous matrix ensembles of symmetric and quaternionic Hermitian matrices that lead to the orbital orthogonal and symplectic processes, respectively. By extrapolation, on the dimension of the base field, of the explicit density formulas, we define the orbital beta processes. We prove the universal behavior of the virtual eigenvalues of the smallest $m$ principal submatrices, when $m$ is independent of $N$ and in such a way that the rescaled empirical measures converge weakly. The limiting object is the Gaussian beta corners process. As a byproduct of our approach, we prove a theorem on the asymptotics of multivariate Bessel functions.