Large deviations and fluctuations of real eigenvalues of elliptic random matrices

Abstract

We study real eigenvalues of $N\times N$ real elliptic Ginibre matrices indexed by a non-Hermiticity parameter $0\le \tau <1$, 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 $p_{N,k}^{(\tau )}$ that exactly $k$ eigenvalues are real. In the strong non-Hermiticity regime, where $\tau$ 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 $\zeta$ is the Riemann zeta function. In the weak non-Hermiticity regime, where $\tau =1-\frac{{\alpha }^2}{N}$, 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.

36 pages