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$^{\dag}$ and AII$^{\dag}$, realized by complex symmetric and complex self-dual random matrices, respectively. In class AII$^{\dag}$, 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$^{\dag}$, 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.