Abstract: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.
| Comments: | 7 pages, 6 figures, accepted at Phys. Rev. E |
| Subjects: | Disordered Systems and Neural Networks (cond-mat.dis-nn); Statistical Mechanics (cond-mat.stat-mech); Probability (math.PR) |
| Cite as: | arXiv:2605.25736 [cond-mat.dis-nn] |
| (or arXiv:2605.25736v1 [cond-mat.dis-nn] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25736 arXiv-issued DOI via DataCite (pending registration) |
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| Related DOI: | https://doi.org/10.1103/xtk6-6c34
DOI(s) linking to related resources |
From: Ulysse Marquis [view email]
[v1]
Mon, 25 May 2026 11:44:47 UTC (14,433 KB)
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