Abstract
We study real eigenvalues of N x N real elliptic Ginibre matrices indexed by a non-Hermiticity parameter 0 <= tau < 1. In both the strong (tau is fixed) and weak (tau = 1 - alpha(2)/N) non-Hermiticity 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, we obtain lim(N ->infinity )1/root N logp(N ,kN )((tau))= -root 1 + tau zeta(3/2)/1 - tau root 2 pi for any k(N )= o(root N/log N) as N -> infinity, where zeta is the Riemann zeta function. In the weak non-Hermiticity regime, we obtain lim(N ->infinity )1/N log p(N ,kN )((tau))<= 2/pi integral(1)(0 )log (1 - e (-alpha 2s2))root 1 - s(2 )ds for any k(N )= o(N/logN) as N -> infinity. This inequality is expected to be an equality.


https://doi.org/10.1214/25-EJP1304