Universal Parametric Correlations of Eigenvalues of Random Matrix Ensemble

TL;DR

This paper derives a universal formula for eigenvalue correlations in random matrix ensembles under external perturbations, especially near hard edge singularities, using Dyson Brownian Motion and hydrodynamical equations.

Contribution

It introduces a universal dependence of eigenvalue density correlations on external fields for matrices with hard edge singularities, solved via a hydrodynamical approach.

Findings

Derived a universal density-density correlator formula.

Obtained a Laplace transform expression for variance of linear statistics.

Extended understanding of eigenvalue behavior under external perturbations.

Abstract

Eigenvalue correlations of random matrix ensembles as a function of an external perturbation are investigated vis the Dyson Brownian Motion Model in the situation where the level density has a hard edge singularity. By solving a linearized hydrodynamical equation, a universal dependence of the density-density correlator on the external field is found. As an application we obtain a formula for the variance of linear statistics with the parametric dependence exhibited as a Laplace transform.