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arXiv · 2511.21643

Spectral Density and Eigenvector Nonorthogonality in Complex Symmetric Random Matrices

Gernot Akemann·Yan V. Fyodorov·Dmitry V. Savin
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Abstract

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$^\dag$ 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.

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Gernot Akemann, Yan V. Fyodorov, Dmitry V. Savin. 2025-11-26. Spectral Density and Eigenvector Nonorthogonality in Complex Symmetric Random Matrices. https://arxiv.org/abs/2511.21643

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