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 $\beta =x^{\ast }/(1+x^{\ast })$. The scaling parameter of the model is defined as $x^{\ast }\propto \left(b_{eff}^2/N\right){}^{\delta }$, where $b_{eff}$ is the average number of non-zero elements per matrix row, N is the matrix size, and $\delta \sim 1$. Additionally, we show that $x^{\ast }$ also scales the spectral properties of the model (up to certain sparsity) characterized by the spacing distribution of eigenvalues. (paper)