Large deviations and fluctuations of real eigenvalues of elliptic random matrices

We study real eigenvalues of N×N real elliptic Ginibre matrices indexed by a non-Hermiticity parameter 0≤τ<1. In both the strong (τ is fixed) and weak (τ=1−α2N) non-Hermiticity regimes, we prove a central limit theorem for the number of real eigenvalues. We also find the asymptotic behaviour of the probability pN,k(τ) that exactly k eigenvalues are real. In the strong non-Hermiticity regime, we obtain limN→∞ 1 NlogpN,kN(τ)=− 1+τ 1−τζ(3∕2) 2π for any kN=o( N logN) as N→∞, where ζ is the Riemann zeta function. In the weak non-Hermiticity regime, we obtain limN→∞ 1 NlogpN,kN(τ)≤2 π∫01log1−e−α2s2 1−s2ds for any kN=o(N logN) as N→∞. This inequality is expected to be an equality.