Constrained Eigenvalues Density of Invariant Random Matrices Ensembles

TL;DR

This paper derives the exact asymptotic eigenvalue density for invariant random matrix ensembles with eigenvalues constrained within specific intervals, revealing a universal inverse square-root singularity at the barriers.

Contribution

It generalizes the eigenvalue density analysis of Gaussian ensembles to orthogonal, unitary, and symplectic ensembles with boundary constraints.

Findings

Eigenvalue density exhibits inverse square-root singularity at barriers

Results apply to orthogonal, unitary, and symplectic ensembles

Generalizes Dean-Majumdar's Gaussian ensemble findings

Abstract

We compute exact asymptotic of the statistical density of random matrices belonging to invariant random matrices ensemble (RMT) orthogonal, unitary and symplectic ensembles, where all its eigenvalues lie within the interval $[\sigma, +\infty[$ or $]-\infty,\tau]$ or $[\sigma,\tau]$. It is found that the density of eigenvalues generically exhibits an inverse square-root singularity at the location of the barriers. These results generalized the case of Gaussian random matrices ensemble studied by Dean-Majumdar.