Scaling properties of localization in random band matrices: A sigma -model approach.
We reduce a random-band-matrix (RBM) problem to a one-dimensional, nonlinear, supersymmetric, \ensuremath{\sigma} model. This reduction becomes exact in the limit b\ensuremath{\rightarrow}\ensuremath{\infty}, b being the effective bandwidth. We prove that ${\mathit{b}}^{2}$/N, N being the matrix size, is the relevant scaling parameter. When the mean value of diagonal elements increases linearly along the diagonal an extra scaling parameter arises. These conclusions are in agreement with recent numerical results.