Delocalization for a class of random block band matrices

Abstract

We consider $N\times N$ Hermitian random matrices $H$ consisting of blocks of size $M\ge N^{6/7}$. The matrix elements are i.i.d. within the blocks, close to a Gaussian in the four moment matching sense, but their distribution varies from block to block to form a block-band structure, with an essential band width $M$. We show that the entries of the Green's function $G(z)=(H-z{)}^{-1}$ satisfy the local semicircle law with spectral parameter $z=E+\mathbf{i}\eta$ down to the real axis for any $\eta \gg N^{-1}$, using a combination of the supersymmetry method inspired by \cite{Sh2014} and the Green's function comparison strategy. Previous estimates were valid only for $\eta \gg M^{-1}$. The new estimate also implies that the eigenvectors in the middle of the spectrum are fully delocalized.

arXiv comment81 pages

References in corpus (4)

• Local semicircle law and complete delocalization for Wigner random matrices
• Eigenvector localization for random band matrices with power law band width
• Universality of the local regime for the block band matrices with a finite number of blocks
• Universality of the second mixed moment of the characteristic polynomials of the 1D band matrices: real symmetric case

Cited by in corpus (4)

• Lectures on the local semicircle law for Wigner matrices
• Characteristic polynomials for 1D random band matrices from the localization side
• Density of States for Random Band Matrices in two dimensions
• Poisson statistics for random deformed band matrices with power law band width