Complex Eigenvalues in a pseudo-Hermitian \b{eta}-Laguerre ensemble

TL;DR

The paper analyzes a pseudo-Hermitian $eta$-Laguerre ensemble perturbed by a small non-Hermitian corner term, showing that complex eigenvalues organize into a balloon-like region with a cusp, followed by a real tail. Owing to suspected self-averaging, the authors derive the loci of eigenvalues from the mean characteristic polynomial, leveraging Laguerre polynomial asymptotics to obtain a coupled $(u,v)$ description that predicts a large-$N$ circle limit centered at $(N,0)$ with radius $R o frac{1}{2}N^{1+rac{oldsymbol{ elax extalpha}}{2}}$. They demonstrate that, as $N$ grows, the fluctuations around the mean are strongly suppressed along the balloon's boundary but remain anisotropic, being larger perpendicular to the boundary. Numerical simulations validate the theoretical predictions, including the balloon-cusp structure and its asymptotic circle limit, and the results point to self-averaging as a robust feature of this non-Hermitian ensemble. The work opens avenues for extending pseudo-Hermitian analyses to other classical ensembles and Jacobi-type settings.

Abstract

Non-Hermitian PT-symmetric models have been extensively studied in recent years. Following the seminal work that reduced classical random matrix ensembles to a tridiagonal form, several efforts have aimed to generalize this framework to non-Hermitian extensions of the so-called \b{eta}-ensembles. In particular, while the transition of eigenvalues from the real axis to the complex plane has been well characterized for the \b{eta}-Hermite ensemble under symmetry breaking, the behavior of the \b{eta}-Laguerre ensemble in a similar non-Hermitian setting remains less understood. In this work, we investigate an ensemble of unstable matrices isospectral to the \b{eta}-Laguerre ensemble. Introducing a small non-Hermitian perturbation breaks the symmetry and drives the eigenvalues into the complex plane. We derive analytical expressions for the loci of complex-conjugate eigenvalue pairs, which organize into a balloon-like structure in the complex plane, followed by a discrete finite line of real eigenvalues. The asymptotic behavior of these eigenvalues is analyzed in the large matrix-size limit, and our theoretical predictions are supported by numerical simulations.

Figures (8)

• Figure 1: Eigenvalues of an individual realization of matrix $H_{2}(\alpha,\epsilon)$ with $N=50$, $c=0.75$, and $\beta = 1$ as function of parameter $\epsilon$. Top : Eigenvalues' behavior in the complex ${\rm Re}(\lambda) \times {\rm Im}(\lambda)$ plane. Bottom : Eigenvalues' behavior projected onto the ${\rm Re}(\lambda) \times \epsilon$ plane. As the perturbation parameter increases, the eigenvalues move progressively into the complex domain.
• Figure 2: Eigenvalues in the complex plane for individual realizations of $H_{2}(\alpha,\epsilon)$ as function of matrix dimension $N$, for $\beta = 1$, $c = 0.75$, and $\epsilon = 10^{-10}$. Top : Eigenvalues' behavior in the complex plane. Bottom : Eigenvalues' behavior projected in the ${\rm Im}(\lambda) \times N$ plane. One observed that the larger system is, more sensitive it becomes to small perturbations.
• Figure 3: Eigenvalues of the $H_{2}(\alpha,\epsilon)$ ensemble with the parameters $N=60$, $M=80$, $\beta=1$, $\alpha=0.5$ and $\epsilon=10$. Top: Stokes lines (see eqs (\ref{['Xstokes']}) and (\ref{['Ystokes']})) on the complex plane. Bottom: Balloon-like structure formed by the eigenvalues on the complex plane. This structure has an apparent cusp at its right extremity which is followed by a discrete finite line of real eigenvalues. Red filled circles - analytical solution of the coupled eqs. (\ref{['U']}) and (\ref{['V']}). Filled colored circles - numerically evaluated eigenvalues of three random samples.
• Figure 4: Limiting behavior for $N \to \infty$. Top: Solution of the coupled eqs. (\ref{['U']}) and (\ref{['V']}) is compared against the limiting form in eqs. (\ref{['circle']}) for $N = 30000$, $c = 0.85$, $\beta = 1$, and $\alpha = 0.5$. The agreement is very good except for a narrow vicinity of the cusp at the right extremity of the balloon-like structure. Bottom: The density of eigenvalues on the circumference of the balloon-like structure as function of $N$ for different values of $\alpha$. To facilitate the interpretation of the region delineated by the dashed line, we plotted the graph indicated by the red arrow. The inset graph displays the curves corresponding to $\alpha \geq 0.8$.
• Figure 5: Fluctuations around the roots of the mean characteristic polynomial. The blue filled circles depict the spread of fluctuations of the eigenvalues of $100$ samples of matrices with $N=50$, $M=54$,$\beta=1$,$\alpha=1$, and $\epsilon=10^{-10}$. The black filled circles represent the numerical solution of eqs. (\ref{['U']}) and (\ref{['V']}).