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 $W \ll N^{1/8}$, delocalized for $W \gg N^{4/5}$ and it is conjectured that the transition for the bulk occurs at $W \sim N^{1/2}$. Eigenvalues in the spectral edge change their behavior at $W \sim N^{5/6}$ 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.

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