Complex Symmetric Random Matrices Demonstrate Eigenvector Distributions and Density in the Limit

Complex symmetric random matrices increasingly model diverse physical systems, from energy dissipation to wave propagation in disordered materials, yet analytical understanding has lagged behind. Gernot Akemann from Bielefeld University, Yan Fyodorov from King’s College London, and Dmitry Savin from Brunel University of London, have now made a significant advance in this field, deriving a precise mathematical description of these matrices’ complex eigenvalues and eigenvectors. Their work provides an explicit formula for the joint distribution of these key properties, applicable to matrices of any size, and reveals how eigenvector alignment deviates from established patterns observed in simpler systems. This breakthrough not only clarifies the behaviour of complex symmetric matrices, but also suggests a new, distinct universality class governing their spectral and statistical properties, a finding supported by independent numerical simulations using different matrix types.

Beyond the standard Ginibre ensembles, recent research focuses on describing dissipative quantum many-body systems and non-ergodic wave transport in complex media. This work investigates the class AI† of complex symmetric random matrices, an area where analytical results have been limited. Employing a new framework, the researchers analyse this class for Gaussian entries and derive an explicit, closed-form expression for the joint distribution of a complex eigenvalue and its right eigenvector for arbitrary matrix size N ≥ 2 in the entire complex plane. Consequently, the team obtains the distribution of the eigenvector non-orthogonality overlap and the mean eigenvalue density.

Gaussian Integration of Complex Probability Distributions

This detailed derivation presents a complete mathematical walkthrough of the calculation leading to the final result. The core strategy involves Gaussian integration techniques, expressing the problem in terms of integrals over Gaussian probability distributions. A series of coordinate transformations simplify these integrals by centering the Gaussian distributions. Wick’s theorem and diagrammatic rules are then used to calculate the averages of polynomial terms, visually simplifying the calculations. The derivation meticulously calculates each term, applying these rules and simplifying expressions before performing a final integration and applying a 2D Laplacian to obtain the final result. A strong mathematical background, including understanding Gaussian integration, complex variables, linear algebra, Wick’s theorem, and the 2D Laplacian operator, is beneficial for following this work.

Eigenvalue and Eigenvector Distributions for AI† Matrices

Scientists have achieved a complete analytical description of complex eigenvalues and eigenvectors for a specific class of non-Hermitian random matrices, known as AI†, which are complex symmetric matrices with Gaussian entries. This work delivers an explicit formula for the joint probability distribution of a complex eigenvalue and its corresponding right eigenvector for matrices of any size, N ≥ 2, across the entire complex plane. The research establishes a fundamental understanding of these matrices, which appear in diverse physical contexts including quantum many-body systems and wave transport in complex media. The team derived the joint probability density function, detailing the statistical relationship between eigenvalues and eigenvectors, and subsequently obtained the distribution of eigenvector non-orthogonality.

Measurements confirm that the degree of eigenvector non-orthogonality deviates from the standard behavior observed in Hermitian matrices, providing insight into the non-normal nature of these systems. Analysis reveals that the diagonal entries of the overlap matrix are greater than or equal to one, indicating a unique property of these matrices. The findings demonstrate a departure from Ginibre universality, suggesting a new universality class for these matrices, and advance the understanding of systems previously confined to numerical simulations.

Complex Eigenvalue Distributions and Spectral Edges

This research presents a detailed analysis of complex symmetric random matrices, a mathematical framework increasingly relevant to describing complex systems exhibiting dissipation or non-ergodic behaviour. Scientists have derived an explicit formula for the joint distribution of complex eigenvalues and their corresponding eigenvectors for matrices of any size, a significant achievement given the scarcity of analytic results in this area. This allowed for the calculation of both eigenvalue density and eigenvector non-orthogonality, both for finite and very large matrices. The findings reveal that while the overall properties of these matrices in the bulk of their spectrum resemble those of more commonly studied random matrices, their behaviour at the spectral edge distinctly deviates from established patterns. Specifically, the research demonstrates a departure from “Ginibre universality”, suggesting a unique character for this class of matrices. The team also established the form of the non-orthogonality distribution, which should be valuable for quantifying relaxation behaviour in open quantum systems.