Oscillating density of states near zero energy for matrices made of blocks with possible application to the random flux problem

TL;DR

This paper investigates the oscillatory behavior of eigenvalue densities near zero energy in random block matrices, revealing universal patterns that could apply to quantum systems with random magnetic flux.

Contribution

It explicitly calculates the oscillatory eigenvalue density near zero energy for various random matrix ensembles and demonstrates the universality of these oscillations across different distributions.

Findings

Eigenvalue density exhibits oscillations near zero energy.

Oscillations are universal, independent of distribution details.

Results extend to real symmetric block matrices.

Abstract

We consider random hermitian matrices made of complex blocks. The symmetries of these matrices force them to have pairs of opposite real eigenvalues, so that the average density of eigenvalues must vanish at the origin. These densities are studied for finite $N\times N$ matrices in the Gaussian ensemble. In the large $N$ limit the density of eigenvalues is given by a semi-circle law. However, near the origin there is a region of size $1\over N$ in which this density rises from zero to the semi-circle, going through an oscillatory behavior. This cross-over is calculated explicitly by various techniques. We then show to first order in the non-Gaussian character of the probability distribution that this oscillatory behavior is universal, i.e. independent of the probability distribution. We conjecture that this universality holds to all orders. We then extend our consideration to the more complicated block matrices which arise from lattices of matrices considered in our previous work. Finally, we study the case of random real symmetric matrices made of blocks. By using a remarkable identity we are able to determine the oscillatory behavior in this case also. The universal oscillations studied here may be applicable to the problem of a particle propagating on a lattice with random magnetic flux.