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.

Disordered Systems and Neural Networks (cond-mat.dis-nn); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

arXiv:2609.35230 [cond-mat.dis-nn]

(or arXiv:2609.35230v1 [cond-mat.dis-nn] for this version)

https://doi.org/10.48550/arXiv.2609.35230 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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