Non-Hermitian diluted banded random matrices: Scaling of eigenfunction and spectral properties

Here we introduce the non-Hermitian diluted banded random matrix (nHdBRM) ensemble as the set of $N\times N$ real non-symmetric matrices whose entries are independent Gaussian random variables with zero mean and variance one if $|i-j|<b$ and zero otherwise, moreover off-diagonal matrix elements within the bandwidth $b$ are randomly set to zero such that the sparsity $\alpha$ is defined as the fraction of the $N(b-1)/2$ independent non-vanishing off-diagonal matrix elements. By means of a detailed numerical study we demonstrate that the eigenfunction and spectral properties of the nHdBRM ensemble scale with the parameter $x=\gamma[(b\alpha)^2/N]^\delta$, where $\gamma,\delta\sim 1$. Moreover, the normalized localization length $\beta$ of the eigenfunctions follows a simple scaling law: $\beta = x/(1 + x)$. For comparison purposes, we also report eigenfunction and spectral properties of the Hermitian diluted banded random matrix ensemble.