We show Poisson statistics for random band matrices which diagonal entries have Gaussian components. These components are possibly as small as $n^{ -\varepsilon}$. Particularly, our result is applicable for a band matrix cut from the GUE with the band width satisfying $w^{ 3.5}<<n$. A uniform upper bound of the averaged density of states (DOS) is obtained for complex deformed Gaussian band matrices with arbitrary $w$. A lower estimate of the DOS is also proven for arbitrary $w$ in a certain class of band matrices.
Categories: math-ph math.MP math.PR