Transfer matrix approach to 1d random band matrices: density of states

TL;DR

This paper analyzes 1D Gaussian Hermitian random band matrices, demonstrating that their average density of states closely follows the Wigner semicircle law with small corrections, under specific size conditions.

Contribution

It provides a transfer matrix approach to establish the density of states for 1D random band matrices, confirming the semicircle law in a new setting.

Findings

Density of states matches Wigner semicircle law up to order W^{-1}

Valid for matrices with size n ≥ C W log W

Uses transfer matrix method for analysis

Abstract

We study the special case of $n\times n$ 1D Gaussian Hermitian random band matrices, when the covariance of the elements is determined by the matrix $J=(-W^2\triangle+1)^{-1}$. Assuming that $n\ge CW\log W\gg 1$, we prove that the averaged density of states coincides with the Wigner semicircle law up to the correction of order $W^{-1}$.