Real eigenvalues of non-symmetric random matrices: Transitions and Universality

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In the past 20 years, the study of real eigenvalues of non-symmetric real random matrices has seen important progress. Notwithstanding, central questions still remain open, such as the characterization of their asymptotic statistics and the universality thereof. In this letter we show that for a wide class of matrices, the number kₙ of real eigenvalues of a matrix of size n is asymptotically Gaussian with mean kₙ=O(√n) and variance kₙ(2-√2). Moreover, we show that the limit distribution of real eigenvalues undergoes a transition between bimodal for kₙ=o(√n) to unimodal for kₙ=O(n), with a uniform distribution at the transition. We predict theoretically these behaviours in the Ginibre ensemble using a log-gas approach, and show numerically that they hold for a wide range of random matrices with independent entries beyond the universality class of the circular law.