Random band matrices in the delocalized phase, I: Quantum unique ergodicity and universality

Understand

Consider $N\times N$ symmetric one-dimensional random band matrices with general distribution of the entries and band width $W \geq N^{3/4+\varepsilon}$ for any $\varepsilon>0$.

• 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) The eigenvalues locally converge to the point process given by the Gaussian orthogonal ensemble at any fixed energy.