Duality between the level statistics of Hermitian and non-Hermitian random matrices
Abstract
The complex spectral statistics of non-Hermitian random matrices provide a standard diagnostic of dissipative quantum chaos. Yet an analytical understanding of how symmetry enriches these universal correlations remains limited. We establish an exact duality between the spectral statistics of Hermitian and non-Hermitian random matrices in the limit of large matrix size. This duality resolves a longstanding analytical problem by yielding the previously unknown eigenvalue pair-correlation functions for complex symmetric (class AI†) and complex self-dual matrices (class AII†), completing the analytical description of bulk pair correlations in the non-Hermitian threefold way. It further gives closed-form spectral densities near the origin for seven additional symmetry classes. Numerical tests in interacting and dissipative quantum models support the universality of these predictions beyond Gaussian ensembles. The duality suggests a deeper connection between chaos in closed and open quantum systems.
Related
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BibTeX
@misc{chen2026duality,
title = {{Duality between the level statistics of Hermitian and non-Hermitian random matrices}},
author = {Ze Chen and Zhenyu Xiao and Yifei Liu and Shinsei Ryu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00162},
eprint = {2609.00162},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00162},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Duality between the level statistics of Hermitian and non-Hermitian random matrices
AU - Ze Chen
AU - Zhenyu Xiao
AU - Yifei Liu
AU - Shinsei Ryu
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.00162
UR - https://arxiv.org/abs/2609.00162
AB - The complex spectral statistics of non-Hermitian random matrices provide a standard diagnostic of dissipative quantum chaos. Yet an analytical understanding of how symmetry enriches these universal correlations remains limited. We establish an exact duality between the spectral statistics of Hermitian and non-Hermitian random matrices in the limit of large matrix size. This duality resolves a longstanding analytical problem by yielding the previously unknown eigenvalue pair-correlation functions for complex symmetric (class AI^†) and complex self-dual matrices (class AII^†), completing the analytical description of bulk pair correlations in the non-Hermitian threefold way. It further gives closed-form spectral densities near the origin for seven additional symmetry classes. Numerical tests in interacting and dissipative quantum models support the universality of these predictions beyond Gaussian ensembles. The duality suggests a deeper connection between chaos in closed and open quantum systems.
ER -