Eigenvalue Correlations For Banded Matrices

TL;DR

This paper investigates how the eigenvalue distribution of banded matrices evolves when variances of matrix elements change, using a Fokker-Planck framework similar to that of Gaussian ensembles, aiding understanding of physically relevant models.

Contribution

It introduces a Fokker-Planck equation describing eigenvalue evolution in banded matrices, linking it to Gaussian ensemble dynamics and enabling analysis of physically significant cases.

Findings

Eigenvalue distribution evolution follows a Fokker-Planck equation.

Equivalence with Gaussian ensemble perturbations.

Facilitates analysis of physically relevant banded matrices.

Abstract

We study the evolution of the distribution of eigenvalues of $N\times N$ matrix ensembles subject to a change of variances of its matrix elements. Our results indicate that the evolution of the probability density is governed by a Fokker- Planck equation similar to the one governing the time-evolution of the particle- distribution in Wigner-Dyson gas, with relative variances now playing the role of time. This is also similar to the Fokker-Planck equation for the distribution of eigenvalues of a $N\times N$ matrix subject to a random perturbation taken from the standard Gaussian ensembles with perturbation-strength as the "time" variable. This equivalence alonwith the already known correlations of standard Gaussian ensembles can therefore help us to obtain the same for various physically-significant cases modeled by random banded Gaussian ensembles.