Spectral Density and Eigenvector Nonorthogonality in Complex Symmetric Random Matrices
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| Format: | Preprint |
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2025
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| author | Akemann, Gernot Fyodorov, Yan V. Savin, Dmitry V. |
| author_facet | Akemann, Gernot Fyodorov, Yan V. Savin, Dmitry V. |
| contents | Non-Hermitian random matrices with statistical spectral characteristics beyond the standard Ginibre ensembles have recently emerged in the description of dissipative quantum many-body systems as well as in non-ergodic wave transport in complex media. We investigate the class AI$^†$ of complex symmetric random matrices, for which available analytic results remain scarce. Using a recently proposed framework by one of the authors, we analyze this class for Gaussian entries and derive an explicit, closed-form expression for the joint distribution of a complex eigenvalue and its right eigenvector for arbitrary matrix size $N\ge 2$ in the entire complex plane. From this, we obtain the distribution of the eigenvector non-orthogonality overlap and the mean eigenvalue density, both for finite $N$ and in the large-$N$ limit. Notably, at the spectral edge both the eigenvalue density and eigenvector statistics exhibits a limiting behavior that differs from the Ginibre universtality class. This behavior is expected to be universal, as further supported by numerical evidence for Bernoulli random matrices. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_21643 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral Density and Eigenvector Nonorthogonality in Complex Symmetric Random Matrices Akemann, Gernot Fyodorov, Yan V. Savin, Dmitry V. Mathematical Physics Disordered Systems and Neural Networks Statistical Mechanics Probability Non-Hermitian random matrices with statistical spectral characteristics beyond the standard Ginibre ensembles have recently emerged in the description of dissipative quantum many-body systems as well as in non-ergodic wave transport in complex media. We investigate the class AI$^†$ of complex symmetric random matrices, for which available analytic results remain scarce. Using a recently proposed framework by one of the authors, we analyze this class for Gaussian entries and derive an explicit, closed-form expression for the joint distribution of a complex eigenvalue and its right eigenvector for arbitrary matrix size $N\ge 2$ in the entire complex plane. From this, we obtain the distribution of the eigenvector non-orthogonality overlap and the mean eigenvalue density, both for finite $N$ and in the large-$N$ limit. Notably, at the spectral edge both the eigenvalue density and eigenvector statistics exhibits a limiting behavior that differs from the Ginibre universtality class. This behavior is expected to be universal, as further supported by numerical evidence for Bernoulli random matrices. |
| title | Spectral Density and Eigenvector Nonorthogonality in Complex Symmetric Random Matrices |
| topic | Mathematical Physics Disordered Systems and Neural Networks Statistical Mechanics Probability |
| url | https://arxiv.org/abs/2511.21643 |