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We consider the elliptic Ginibre matrices in the orthogonal symmetry classthat interpolates between the real Ginibre ensemble and the Gaussian orthogonalensemble. We obtain the finite size corrections of the real eigenvaluedensities in both the global and edge scaling regimes, as well as in both thestrong and weak non-Hermiticity regimes. Our results extend and provide therate of convergence to the previous recent findings in the aforementionedlimits. In particular, in the Hermitian limit, our results recover the finitesize corrections of the Gaussian orthogonal ensemble established by Forrester,Frankel and Garoni.

Finite size corrections for real eigenvalues of the elliptic Ginibre matrices