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

Ulysse Marquis*

  • *Contact author: ulyssepierre.marquis@unitn.it
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Phys. Rev. E 113, 064126 – Published 12 June, 2026

DOI: https://doi.org/10.1103/xtk6-6c34

Abstract

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.

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Phys. Rev. E 113, 064126– Published 12 June, 2026
  • Received 9 March 2026
  • Revised 4 May 2026
  • Accepted 22 May 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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