Eigenvalue correlations for banded matrices

We study the evolution of the distribution of eigenvalues of a N×N matrix ensemble subject to a change of the distribution parameters 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 a function of distribution parameters now playing the role of time. This equivalence alongwith the already known particle density correlations of Wigner–Dyson gas can therefore help us to obtain the eigenvalue correlations for various physically significant cases modeled by random banded and sparse Gaussian ensembles.