Large deviations and fluctuations of real eigenvalues of elliptic random matrices

We study real eigenvalues of $N\times N$ real elliptic Ginibre matrices indexed by a non-Hermiticity parameter $0\leq \tau<1$, 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}^{(\tau)} = -\sqrt\frac{1+\tau}{1-\tau} \frac{\zeta(3/2)}{\sqrt{2\pi}} \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}^{(\tau)} \leq \frac{2}{\pi} \int_0^1 \log\left(1-e^{-\alpha^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.