The Real Ginibre Ensemble with $$k=O(n)$$ 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.