Universality for a class of random band matrices
  We prove the universality for the eigenvalue gap statistics in the bulk of the spectrum for band matrices, in the regime where the band width is comparable with the dimension of the matrix, $W\sim N$. All previous results concerning universality of non-Gaussian random matrices are for mean-field models. By relying on a new mean-field reduction technique, we deduce universality from quantum unique ergodicity for band matrices. 
Paul Bourgade, Laszlo Erdos, Horng-Tzer Yau, Jun Yin
math.PR math-ph math.MP