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arXiv:2609.00164

Exact joint eigenvalue densities of non-Hermitian random matrices are Calogero scattering states

Zhenyu Xiao, Ze Chen, Shinsei Ryu

cond-mat.stat-mechcond-mat.mes-hallquant-phSubmitted Aug 31, 2026• Announced Sep 2, 2026• Revised Sep 25, 2026 (v2)
v1 · Aug 31, 2026v2 · Sep 25, 2026
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Abstract

Determining exact joint eigenvalue densities is central to random matrix theory. We solve this long-standing problem for non-Hermitian matrices with transposition symmetry (complex symmetric and complex self-dual) at arbitrary matrix size. Up to a Vandermonde factor, they are scattering-state wave functions of the Calogero model, a line of particles interacting through an inverse-square potential, with the coupling strength set by the symmetry. In contrast to many previously known joint densities, the densities cannot be written as a gas of eigenvalues with pairwise interactions. We further compute the complex level spacing distributions and two-point spectral correlation functions, which carry power-law tails, absent in a Coulomb gas. Our results shed light on the interplay among random matrices, integrability, and symmetry.

Comments
31 pages (10 pages main text and 21 pages Supplemental Material), 3 figures; revised version with updated author list

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Determining exact joint eigenvalue densities is central to random matrix theory. We solve this long-standing problem for non-Hermitian matrices with transposition symmetry (complex symmetric and complex self-dual) at arbitrary matrix size. Up to a Vandermonde factor, they are scattering-state wave functions of the Calogero model, a line of particles interacting through an inverse-square potential, with the coupling strength set by the symmetry. In contrast to many previously known joint densities, the densities cannot be written as a gas of eigenvalues with pairwise interactions. We further compute the complex level spacing distributions and two-point spectral correlation functions, which carry power-law tails, absent in a Coulomb gas. Our results shed light on the interplay among random matrices, integrability, and symmetry.
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-NNN 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 NNN and explicitly evaluate the distributions and representative moments for N=3,4,5,6N=3, 4, 5, 6N=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=3N=3N=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.
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†^\dagger†) and complex self-dual matrices (class AII†^\dagger†), 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.
Motivated by the surge of interest in statistics of non-Hermitian random matrices as a framework for description of universal characteristics of dissipative chaotic quantum many-body systems, we address the problem of characterizing the parametric correlations of spectral densities. Considering parameter-dependent ensemble of complex Ginibre matrices we derive an explicit, closed-form expression for the parametric number covariance in the systems of symmetry class A\mathbf{A}A for eigenvalues in a circular domain containing on average a finite number of eigenvalues in the spectral bulk. This behavior is expected to be universal, as further supported by numerical evidence for the real Ginibre ensemble, non-Hermitian Bernoulli Wigner matrices and bi-unitarily invariant ensembles. We also discuss a relation between parametric correlations of spectral densities and the distribution of the so-called eigenvector non-orthogonality factor, which attracted considerable interest in recent years.
We explore the connections between dissipative quantum phase transitions and non-Hermitian random matrix theory. For this, we work in the framework of the dissipative Dicke model which is archetypal of symmetry-breaking phase transitions in open quantum systems. We establish that the Liouvillian describing the quantum dynamics exhibits distinct spectral features of integrable and chaotic character on the two sides of the critical point. We follow the distribution of the spacings of the complex Liouvillian eigenvalues across the critical point. In the normal and superradiant phases, the distributions are two-dimensional Poisson and that of the Ginibre unitary random matrix ensemble, respectively. Our results are corroborated by computing a recently introduced complex-plane generalization of the consecutive level-spacing ratio distribution. Our approach can be readily adapted for classifying the nature of quantum dynamics across dissipative critical points in other open quantum systems.
We review the ideas of how random matrix theory has to be properly applied to quantum physics; particularly we focus on how the spectrum has to be properly prepared and the random matrix correctly identified before the random matrix and the physical eigenvalue spectrum can be compared. We explain the ideas of the symmetry classification of symmetric matrix spaces and how that yields Dyson's threefold and Altland-Zirnbauer's tenfold way. We also outline how the joint probability density function of the eigenvalues can be calculated from a given probability density function on the matrix space. Furthermore, we dive into the subtleties of the unfolding procedure. For this purpose, we explain the ideas of the local mean level spacing, the local level spacing distribution and the kkk-point correlation functions. We outline the techniques of orthogonal polynomials, determinantal and Pfaffian point processes and their related Fredholm determinants and Pfaffians as well as the supersymmetry method. Moreover, we relate the local spectral statistics to effective Lagrangians that give the relation to non-linear σσσ-models. In all these discussions, we also make brief excursions to non-Hermitian random matrix theory which are useful when studying open quantum systems, for instance.
Abstract In this paper, we present a general framework to solve a fundamental problem in random matrix theory (RMT), i.e. the problem of describing the joint distribution of eigenvalues of the sum of two independent random Hermitian matrices and . Some considerations about the mixture of quantum states are basically subsumed into the above mathematical problem. Instead, we focus on deriving the spectral density of the mixture of adjoint orbits of quantum states in terms of the Duistermaat–Heckman measure, originated from the theory of symplectic geometry. Based on this method, we can obtain the spectral density of the mixture of independent random states. In particular, we obtain explicit formulas for the mixture of random qubits. We also find that, in the two-level quantum system, the average entropy of the equiprobable mixture of n random density matrices chosen from a random state ensemble (specified in the text) increases with the number n . Hence, as a physical application, our results quantitatively explain that the quantum coherence of the mixture monotonously decreases statistically as the number of components n in the mixture. Besides, our method may be used to investigate some statistical properties of a special subclass of unital qubit channels.