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×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 α 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=γ[(bα)^{2}/N]^{δ}, where γ,δ∼1. Moreover, the normalized localization length β of the eigenfunctions follows a simple scaling law: β=x/(1+x). For comparison purposes, we also report eigenfunction and spectral properties of the Hermitian diluted banded random matrix ensemble.