The Local Semicircle Law for Random Matrices with a Fourfold Symmetry

TL;DR

This paper proves that random matrices with a fourfold symmetry exhibit the same global and local semicircle law behavior as classical ensembles, despite their additional symmetry constraints.

Contribution

It establishes the local semicircle law for a new symmetry class of random matrices, extending universality results to matrices with fourfold symmetry.

Findings

Density of states converges to the Wigner semicircle law.

Local semicircle law holds on the optimal scale.

Results apply to matrices arising from Fourier transforms of GOE and the Anderson model.

Abstract

We consider real symmetric and complex Hermitian random matrices with the additional symmetry $h_{xy}=h_{N-x,N-y}$. The matrix elements are independent (up to the fourfold symmetry) and not necessarily identically distributed. This ensemble naturally arises as the Fourier transform of a Gaussian orthogonal ensemble (GOE). It also occurs as the flip matrix model - an approximation of the two-dimensional Anderson model at small disorder. We show that the density of states converges to the Wigner semicircle law despite the new symmetry type. We also prove the local version of the semicircle law on the optimal scale.