Spectral Density and Eigenvector Nonorthogonality in Complex Symmetric Random Matrices

TL;DR

The paper delivers an exact analytical framework for the eigenvalue–eigenvector statistics of complex symmetric Gaussian random matrices in the AI$^ullet$ class by deriving the joint distribution $\mathcal{P}(z,\boldsymbol{v})$ via a supersymmetry–Kac–Rice approach. It provides finite-$N$ closed forms for the JPDF, from which the radial eigenvalue density $\rho(r)$ and the diagonal eigenvector overlap distribution $P_r(t)$ are extracted, and then analyzes large-$N$ universal features, revealing distinct edge behavior from Ginibre ensembles. The authors show that the bulk obeys a Ginibre-like universality while the edge exhibits a different scaling function, with Bernoulli simulations supporting universality beyond Gaussianity. These results have implications for dissipation-driven quantum systems and non-ergodic wave transport, where eigenvector nonorthogonality governs sensitivity and relaxation dynamics.

Abstract

Non-Hermitian random matrices with statistical spectral characteristics beyond the standard Ginibre ensembles have recently emerged in the description of dissipative quantum many-body systems as well as in non-ergodic wave transport in complex media. We investigate the class AI$^†$ of complex symmetric random matrices, for which available analytic results remain scarce. Using a recently proposed framework by one of the authors, we analyze 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 $N\ge 2$ in the entire complex plane. From this, we obtain the distribution of the eigenvector non-orthogonality overlap and the mean eigenvalue density, both for finite $N$ and in the large-$N$ limit. Notably, at the spectral edge both the eigenvalue density and eigenvector statistics exhibits a limiting behavior that differs from the Ginibre universtality class. This behavior is expected to be universal, as further supported by numerical evidence for Bernoulli random matrices.

Figures (4)

• Figure 1: The radial spectral density for increasing matrix size $N$. Histograms represent numerical simulations over $10^6$ realizations of Gaussian random matrices for class AI$^{\dagger}$. Solid lines show the exact result (\ref{['P(r)']}), while the dashed line indicates the large-$N$ approximation (\ref{['rho_N']}) for $N=50$. The dotted line marks the limiting triangular law for $\rho(r)$ as $N\to\infty$, corresponding to the circular law for $\rho(r)/r$.
• Figure 2: The distribution of the (rescaled) nonorthogonality factor at the spectral origin for increasing matrix size $N$. Histograms stand for the numerics (the same as in Fig. \ref{['fig:rho']}), whereas solid lines correspond to the exact result (\ref{['P0(t)']}). As $N$ grows, the distribution converges to the limiting form given by Eq. (\ref{['P(tau)']}), with $\left\langle \tau \right\rangle=\frac{1}{2}$ at $r=0$, which is shown by the dashed line.
• Figure 3: Numerical results for Bernoulli complex symmetric random matrices based on $10^5$ realization of size $N=50\;(\times)$, $100\;(\Box)$ and $200\;(\circ)$. (a) Spectral density $\rho(r)$: the solid lines show our large-$N$ prediction (\ref{['rho_N']}). (b) Nonorthogonality distribution $P_r(t)$ at the spectral origin ($r=0$): the solid line indicates the bulk limiting form (\ref{['P(tau)']}). (c) The behavior of $P_r(t)$ at the spectral edge ($r=\sqrt{2}$). The solid line represents our exact expression (\ref{['P(t)']}) for $N=200$, while the dashed line shows the edge limiting form (\ref{['P(sig)']}). Note that, due to the different asymptotic scaling of $t$ with $N$, the convergence to the asymmtotic regime is slower at the edge than in the bulk.
• Figure A1: Large-$N$ asymptotic forms for the complex-symmetric (this work) and the two Ginibre ensembles: (a) the spectral edge profile; (b) the nonorthogonality distribution at the edge ($s=0$).