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arXiv · 1501.03120

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

Luis Carlos García del Molino·Khashayar Pakdaman·Jonathan Touboul·Gilles Wainrib
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Abstract

We consider the ensemble of Real Ginibre matrices with a positive fraction $\alpha>0$ 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 $p^n_{\alpha n}$ that an $n\times n$ Ginibre matrix has $k=\alpha n$ real eigenvalues and we characterize the spectral measures of these matrices.

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Luis Carlos García del Molino, Khashayar Pakdaman, Jonathan Touboul, Gilles Wainrib. 2015-01-13. The real Ginibre ensemble with $k = O(n)$ real eigenvalues. https://arxiv.org/abs/1501.03120

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