Non-Hermitian random matrices with long-range correlations show α-dependent breakdown of the circular law, with spectral radius growing as a power law for α<1 and self-similar density at α=1.
Correlation Functions of Asymmetric Real Matrices
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abstract
We give a closed form for the correlation functions of ensembles of asymmetric real matrices in terms of the Pfaffian of an antisymmetric matrix formed from a $2 \times 2$ matrix kernel associated to the ensemble. We also derive closed forms for the matrix kernel and correlation functions for Ginibre's real ensemble.
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cond-mat.dis-nn 1years
2026 1verdicts
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Spectral properties of non-Hermitian real random matrices with long-range correlations
Non-Hermitian random matrices with long-range correlations show α-dependent breakdown of the circular law, with spectral radius growing as a power law for α<1 and self-similar density at α=1.