Diluted banded random matrices: scaling behavior of eigenfunction and spectral properties

We demonstrate that the normalized localization length β of the eigenfunctions of diluted (sparse) banded random matrices follows the scaling law β = x ∗ / ( 1 + x ∗ ) . The scaling parameter of the model is defined as x ∗ ∝ ( b e f f 2 / N ) δ , where b e f f is the average number of non-zero elements per matrix row, N is the matrix size, and δ ∼ 1 . Additionally, we show that x ∗ also scales the spectral properties of the model (up to certain sparsity) characterized by the spacing distribution of eigenvalues.