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~$|r-r'|^{-\alpha}$. 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 eigenvalue distribution broadens with matrix size and its spectral radius grows like a power law, with exponents numerically close to the exponents controlling the magnitude of fluctuations in the extended central limit theorem. The case~$\alpha=1$ appears as a case with self-similar eigenvalue density, and 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.