Spectral properties of non-Hermitian real random matrices with long-range correlations (opens in new tab)

We investigate the spectral properties of non-Hermitian real random matrices whose entries exhibit long-range correlations decaying as~$|r-r'|^{-\alpha}$. We find a progressive breakdown of the circular law, controlled by the decrease of~$\alpha$. In all cases, the radial eigenvalue density decreases away from the origin. At~$\alpha>1$, an effective radius, reminiscent of the circular law, is retrieved, while instead, for~$\alpha<1$, the eigenva...