Spectral properties of non-Hermitian real random matrices with long-range correlations

We investigate the spectral properties of non-Hermitian real random matrices whose entries exhibit long-range correlations decaying as |r-r^{'}|^{-α}. We find a progressive breakdown of the circular law, controlled by the decrease of α. In all cases, the radial eigenvalue density decreases away from the origin. At α>1, an effective radius, reminiscent of the circular law, is retrieved, while instead, for α<1, the eigenvalue distribution broadens with matrix size, and its spectral radius grows like a power law, with exponents numerically consistent with the predictions of the extended central limit theorem for the magnitude of fluctuations. The case α=1 appears as a case with self-similar eigenvalue density and a slowly growing spectral radius. Long-range correlations also enhance clustering of real eigenvalues and slow the resorption of the Saturn effect. These results reveal a correlation-driven transition and suggest the emergence of a new universality class for correlated non-Hermitian random matrices.