Exact probability distributions of complex spacing ratios in non-Hermitian random matrices
Abstract
The complex spacing ratio is the complex displacement from a reference eigenvalue to its nearest neighbor divided by the corresponding displacement to its next-to-nearest neighbor. Its statistics provide a useful diagnostic of spectral correlations and nonintegrability in open quantum systems. Here, starting from the exact joint eigenvalue probability densities, we derive finite-N complex-spacing-ratio distributions for the Gaussian ensembles of non-Hermitian random matrices in classes AI† and AII†, realized by complex symmetric and complex self-dual random matrices, respectively. In class AII†, we obtain an exact algebraic expression for arbitrary N and explicitly evaluate the distributions and representative moments for N=3, 4, 5, 6. In class AI†, although the joint density retains a noncompact integral over nonunitary eigenvector degrees of freedom, we analytically derive a normalized one-dimensional integral representation for N=3 and determine the asymptotic behavior, including a logarithmic correction to the cubic level repulsion and a nonanalytic contribution to the angular density. We further confirm these analytical results through direct numerical diagonalization of non-Hermitian random matrices.
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BibTeX
@misc{kawabata2026exact,
title = {{Exact probability distributions of complex spacing ratios in non-Hermitian random matrices}},
author = {Kohei Kawabata},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35230},
eprint = {2609.35230},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35230},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exact probability distributions of complex spacing ratios in non-Hermitian random matrices
AU - Kohei Kawabata
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35230
UR - https://arxiv.org/abs/2609.35230
AB - The complex spacing ratio is the complex displacement from a reference eigenvalue to its nearest neighbor divided by the corresponding displacement to its next-to-nearest neighbor. Its statistics provide a useful diagnostic of spectral correlations and nonintegrability in open quantum systems. Here, starting from the exact joint eigenvalue probability densities, we derive finite-N complex-spacing-ratio distributions for the Gaussian ensembles of non-Hermitian random matrices in classes AI^† and AII^†, realized by complex symmetric and complex self-dual random matrices, respectively. In class AII^†, we obtain an exact algebraic expression for arbitrary N and explicitly evaluate the distributions and representative moments for N=3, 4, 5, 6. In class AI^†, although the joint density retains a noncompact integral over nonunitary eigenvector degrees of freedom, we analytically derive a normalized one-dimensional integral representation for N=3 and determine the asymptotic behavior, including a logarithmic correction to the cubic level repulsion and a nonanalytic contribution to the angular density. We further confirm these analytical results through direct numerical diagonalization of non-Hermitian random matrices.
ER -