Delocalization of One-Dimensional Random Band Matrices

TL;DR

This paper proves that one-dimensional random band matrices with sufficiently large bandwidth exhibit delocalized eigenvectors, adhere to the semicircle law, and have universal eigenvalue statistics, confirming quantum ergodicity and universality in the large N limit.

Contribution

It establishes delocalization, semicircle law validity, quantum ergodicity, and universality for 1D random band matrices with bandwidth exceeding N^{1/2 + c}.

Findings

Eigenvectors are delocalized with bounded L-infinity norms.

Semicircle law holds up to scale N^{-1+ε}.

Eigenvalue statistics match Gaussian unitary ensemble predictions.

Abstract

Consider an $ N \times N$ Hermitian one-dimensional random band matrix with band width $W > N^{1 / 2 + \frak c} $ for any $ {\frak c} > 0$. In the bulk of the spectrum and in the large $ N $ limit, we obtain the following results: (i) The semicircle law holds up to the scale $ N^{-1 + \varepsilon} $ for any $ \varepsilon > 0 $. (ii) All $ L^2 $- normalized eigenvectors are delocalized, meaning their $ L^\infty$ norms are simultaneously bounded by $ N^{-\frac{1}{2} + \varepsilon} $ with overwhelming probability, for any $ \varepsilon > 0 $. (iii) Quantum unique ergodicity holds in the sense that the local $ L^2 $ mass of eigenvectors becomes equidistributed with high probability. (iv) Universality of eigenvalue statistics holds, i.e., the local eigenvalue statistics of these band matrices are given by those of Gaussian unitary ensembles.