Eigenvector localization for random band matrices with power law band   width
  It is shown that certain ensembles of random matrices with entries that vanish outside a band around the diagonal satisfy a localization condition on the resolvent which guarantees that eigenvectors have strong overlap with a vanishing fraction of standard basis vectors, provided the band width $W$ raised to a power $\mu$ remains smaller than the matrix size $N$. For a Gaussian band ensemble, with matrix elements given by i.i.d. centered Gaussians within a band of width $W$, the estimate $\mu \le 8$ holds. 
Jeffrey Schenker