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The real Ginibre ensemble with \(k=O(n)\) real eigenvalues. (English) Zbl 1343.15024
The authors consider the ensemble of real Ginibre matrices conditioned to have positive fraction \(\alpha >0\) of real eigenvalues. They 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 they provide an asymptotic expansion for the probability \(p_{\alpha n}^n\) that an \(n\times n\) Ginibre matrix has \(k=\alpha n\) real eigenvalues and then they characterize the spectral measures of these matrices.

MSC:
15B52 Random matrices (algebraic aspects)
60F10 Large deviations
60B20 Random matrices (probabilistic aspects)
15A18 Eigenvalues, singular values, and eigenvectors
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