Density of States for Random Band Matrices in two dimensions

TL;DR

This paper rigorously analyzes the density of states for two-dimensional random band matrices, demonstrating smoothness and convergence to Wigner's semicircle law with quantifiable precision as the band width increases.

Contribution

It extends the supersymmetric approach to two dimensions, proving smoothness of the density of states and its convergence to the semicircle law with explicit error bounds.

Findings

Density of states is smooth in the 2D random band matrix model.

The density of states converges to Wigner's semicircle law with error $W^{-2+ ext{small}}$.

Extension of supersymmetric methods from 3D to 2D models.

Abstract

We consider a two dimensional random band matrix ensemble, in the limit of infinite volume and fixed but large band width $W$. For this model we rigorously prove smoothness of the averaged density of states. We also prove that the resulting expression coincides with Wigner's semicircle law with a precision $W^{-2+\delta },$ where $\delta\to 0$ when $W\to \infty.$ The proof uses the supersymmetric approach and extends results by Disertori, Pinson and Spencer from three to two dimensions.