Large deviations and fluctuations of real eigenvalues of elliptic random matrices
Sung-Soo Byun, Leslie Molag, Nick Simm
TL;DR
This work analyzes real eigenvalues of real elliptic Ginibre matrices under strong and weak non-Hermiticity, establishing a central limit theorem for their number and detailing large-deviation probabilities for rare real-eigenvalue configurations. The authors develop a generating-function approach built on a finite generating matrix $M_n^{(\tau)}$, whose trace powers encode cumulants and are tractable through a Pfaffian/skew-orthogonal framework. In the strong regime, they show $\frac{1}{\sqrt{2n}}\operatorname{Tr}(M_n^{(\tau)})^{m}$ converges to a universal bound proportional to $\sqrt{\frac{1+\tau}{1-\tau}}$, enabling a CLT with explicit variance depending on $\tau$; in the weak regime, the traces converge to expressions involving $c(\alpha)$, producing an interpolating behavior between GOE and GinOE. Additionally, they derive a sharp generating-function identity, expressible as a determinant, and obtain explicit large-deviation bounds involving the Riemann zeta function and polylogarithms, with a conjectured equality in the weak regime. The results connect cumulants, kernel asymptotics, and generating functions to quantify fluctuations and rare-event probabilities for real eigenvalues in elliptic random matrices.
Abstract
We study real eigenvalues of $N\times N$ real elliptic Ginibre matrices indexed by a non-Hermiticity parameter $0\leq τ<1$, in both the strong and weak non-Hermiticity regime. Here $N$ is assumed to be an even number. In both regimes, we prove a central limit theorem for the number of real eigenvalues. We also find the asymptotic behaviour of the probability $p_{N,k}^{(τ)}$ that exactly $k$ eigenvalues are real. In the strong non-Hermiticity regime, where $τ$ is fixed, we find \begin{align*} \lim_{N\to\infty} \frac{1}{\sqrt{N}} \log p_{N,k_N}^{(τ)} = -\sqrt\frac{1+τ}{1-τ} \frac{ζ(3/2)}{\sqrt{2π}} \end{align*} for any sequence $(k_N)_N$ of even numbers such that $k_N = o(\frac{\sqrt N}{\log N})$ as $N\to\infty$, where $ζ$ is the Riemann zeta function. In the weak non-Hermiticity regime, where $τ=1-\frac{α^2}{N}$, we obtain \begin{align*} \lim_{N\to\infty} \frac{1}{N} \log p_{N,k_N}^{(τ)} \leq \frac{2}π \int_0^1 \log\left(1-e^{-α^2 s^2}\right) \sqrt{1-s^2} \, ds \end{align*} for any sequence $(k_N)_N$ of even numbers such that $k_N=o(\frac{N}{\log N})$ as $n\to\infty$. This inequality is expected to be an equality.