Complex symmetric, self-dual, and Ginibre random matrices: Analytical results for three classes of bulk and edge statistics

TL;DR

The paper analyzes local bulk and edge statistics for three Gaussian non-Hermitian matrix classes—complex Ginibre (A), complex symmetric (AI$^ ext{†}$), and complex self-dual (AII$^ ext{†}$)—by computing exactly the expected value of products of $k$ conjugate characteristic polynomials. Employing Grassmann/supersymmetric methods, it derives finite-$N$ expressions for a single pair and establishes global and local large-$N limits, revealing universal effective Lagrangians that are shared across ensembles but with different Goldstone manifolds, thereby confirming distinct local statistics per class. For $k>1$, the bulk results reduce to Itzykson–Zuber-type group integrals over compact groups (with explicit HCIZ form for A) while the edge limits involve non-compact group integrals and Pfaffian/determinant structures; these results collectively support the conjectured three universality classes for complex eigenvalues. The findings deepen understanding of non-Hermitian symmetry classes, connecting random matrix theory to nonlinear σ-models and Coulomb-gas pictures, and open pathways to universality proofs and extensions to additional symmetry classes.

Abstract

Recently, a conjecture about the local bulk statistics of complex eigenvalues has been made based on numerics. It claims that there are only three universality classes, which have all been observed in open chaotic quantum systems. Motivated by these new insights, we compute and compare the expectation values of $k$ pairs of complex conjugate characteristic polynomials in three ensembles of Gaussian non-Hermitian random matrices representative for the three classes: the well-known complex Ginibre ensemble, complex symmetric and complex self-dual matrices. In the Cartan classification scheme of non-Hermitian random matrices they are labelled as class A, AI$^†$ and AII$^†$, respectively. Using the technique of Grassmann variables, we derive explicit expressions for a single pair of expected characteristic polynomials for finite as well as infinite matrix dimension. For the latter we consider the global limit as well as zoom into the edge and the bulk of the spectrum, providing new analytical results for classes AI$^†$ and AII$^†$. For general $k$, we derive the effective Lagrangians corresponding to the non-linear $σ$-models in the respective physical systems. Interestingly, they agree for all three ensembles, while the corresponding Goldstone manifolds, over which one has to perform the remaining integrations, are different and equal the three classical compact groups in the bulk. In particular, our analytical results show that these three ensembles have indeed different local bulk and edge spectral statistics, corroborating the conjecture further.

Figures (2)

• Figure 1: Plots of the rescaled characteristic polynomials for finite-$N$ as a function of $r=zw^*\geq0$ for $N=10,20,50$ (dotted orange, dashed green, solid red, respectively) and the global limit $N\to\infty$ (thick blue). Top left: rescaled density of the complex Ginibre ensemble \ref{['Gin-NF']} (class A); Top right: rescaled characteristic polynomials for the complex symmetric ensemble \ref{['CS-NF']} (class AI$^\dagger$); Bottom: rescaled characteristic polynomials for the complex self-dual ensemble \ref{['QS-NF']} (class AII$^\dagger$). The position of the edge of the global limit is rescaled to unity.
• Figure 2: Plots of the expectation value of a pair of characteristic polynomials in the edge scaling limit as a function of a real $y=(z_0^*\chi+z_0\eta^*)/\sqrt{2}$, which can be achieved by setting $\chi=y/\sqrt{2}z_0^*$ and $\eta^*=y/\sqrt{2}z_0$. We normalised the expectation value by its value at $\chi=\eta^*=0$ for comparability which is given by \ref{['edge.origin']}. The curve for class A (solid blue) with $D_{\rm edge}^{\rm A}(\chi,\eta^*)$ from \ref{['Dedge-A']} normalised by $D_{\rm edge}^{\rm A}(0,0)$ saturates at $2$. Class AI$^\dag$ (dashed orange) given by \ref{['Dedge-CS']} has a clearly visible linear rise, and class AII$^\dag$ (dotted red) following \ref{['Dedge-QS']} shows the decay $\sim-1/\sqrt{2}y$ which starts to be visible for large negative values of $y$.

Theorems & Definitions (6)

• proposition 2.1: One pair of expected characteristic polynomials for finite-$N$
• proposition 3.1: Global limit for a pair of characteristic polynomials
• theorem 3.2: Local limits of a pair of characteristic polynomials at the spectral edge
• proposition 3.3: Local limits of a pair of characteristic polynomials in the spectral bulk
• theorem 4.1: Bulk limit for $k>1$
• theorem 4.2: Edge limit for $k>1$