Delocalization of One-Dimensional Random Band Matrices

Consider an $N\times N$ Hermitian one-dimensional random band matrix with band width $W>N^{1\mathrm{/}2+\mathfrak{c}}$ for any $\mathfrak{c}>0$. 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 $N^{-1+\epsilon }$ for any $\epsilon >0$. (ii) All $L^2$- normalized eigenvectors are delocalized, meaning their $L^{\mathrm{\infty }}$ norms are simultaneously bounded by $N^{-\frac{1}{2}+\epsilon }$ with overwhelming probability, for any $\epsilon >0$. (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.