We investigate real eigenvalues of real elliptic Ginibre matrices of size n, indexed by the parameter of asymmetry . In both the strongly and weakly non-Hermitian regimes, where  is fixed or , respectively, we derive the asymptotic expansion of the probability  that all but a finite number 2l of eigenvalues are real. In particular, we show that the expansion is of the form

and we determine all coefficients explicitly. Furthermore, in the special case where l = 1, we derive the full-order expansions. For the proofs, we employ distinct methods for the strongly and weakly non-Hermitian regimes. In the former case, we utilise potential-theoretic techniques to analyse the free energy of elliptic Ginibre matrices conditioned to have  real eigenvalues, together with the strong Szegö limit theorems. In the latter case, we utilise the skew-orthogonal polynomial formalism and the asymptotic behaviour of the Hermite polynomials.

https://doi.org/10.1088/1361-6544/ae11ef