On Eigenvectors of Random Band Matrices with Large Band

We study random, symmetric $N\times N$ band matrices with a band of size $W$ and Bernoulli random variables as entries. This interpolates between nearest neighbour interaction $W=1$ and Wigner matrices $W=N$. Eigenvectors are known to be localized for $W\ll N^{1\mathrm{/}8}$, delocalized for $W\gg N^{4\mathrm{/}5}$ and it is conjectured that the transition for the bulk occurs at $W\sim N^{1\mathrm{/}2}$. Eigenvalues in the spectral edge change their behavior at $W\sim N^{5\mathrm{/}6}$ but nothing is known about the associated eigenvectors. We show that up to $W \ll N^{5/7}$ any random matrix has with large probability some eigenvectors in the spectral edge, which either exhibit mass concentration or interact strongly on a small scale.