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 x N real nonsymmetric 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 nonvanishing 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 similar to 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.