The real Ginibre ensemble with $k = O(n)$ real eigenvalues

We consider the ensemble of Real Ginibre matrices with a positive fraction $α>0$ of real eigenvalues. We demonstrate a large deviations principle for the joint eigenvalue density of such matrices and we 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 $p^n_{αn}$ that an $n\times n$ Ginibre matrix has $k=αn$ real eigenvalues and we characterize the spectral measures of these matrices.

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