The Real Ginibre Ensemble with k=O(n) Real Eigenvalues

We consider the ensemble of real Ginibre matrices conditioned to have positive fraction \(\alpha >0\) of real eigenvalues. We demonstrate a large deviations principle for the joint eigenvalue density of such matrices and introduce a two phase log-gas whose stationary distribution coincides with the spectral measure of the ensemble. Using these tools we provide an asymptotic expansion for the probability \(p^n_{\alpha n}\) that an \(n\times n\) Ginibre matrix has \(k=\alpha n\) real eigenvalues and we characterize the spectral measures of these matrices.