Density of states for Random Band Matrix

TL;DR

This paper rigorously proves the smoothness of the averaged density of states for 3D random band matrices and shows it closely matches the Wigner semicircle law with high precision for large band widths.

Contribution

It provides a rigorous proof of the density of states' smoothness and its convergence to the Wigner semicircle law for 3D random band matrices.

Findings

Density of states is smooth for large volume and fixed band width.

Density of states approximates the Wigner semicircle law with $1/W^2$ accuracy.

Results are obtained using the supersymmetric approach.

Abstract

By applying the supersymmetric approach we rigorously prove smoothness of the averaged density of states for a three dimensional random band matrix ensemble, in the limit of infinite volume and fixed band width. We also prove that the resulting expression for the density of states coincides with the Wigner semicircle with a precision $1/W^2$, for $W$ large but finite.