2013/09/30 by Laszlo Erdos, Antti Knowles · 1 citation
Mathematics · Physics and Astronomy · #math-ph #math.MP #math.PR #msc:15B52 #msc:82B44 #msc:82C44
paper · pdf · doi:10.1007/s00220-014-2119-5
arxiv created 2014/03/20 · arxiv updated 2015/06/17
We consider the spectral statistics of large random band matrices on mesoscopic energy scales. We show that the two-point correlation function of the local eigenvalue density exhibits a universal power law behaviour that differs from the Wigner-Dyson-Mehta statistics. This law had been predicted in the physics literature by Altshuler and Shklovskii [4]; it describes the correlations of the eigenvalue density in general metallic samples with weak disorder. Our result rigorously establishes the Altshuler-Shklovskii formulas for band matrices. In two dimensions, where the leading term vanishes owing to an algebraic cancellation, we identify the first non-vanishing term and show that it differs substantially from the prediction of Kravtsov and Lerner [33].