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

Abstract

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 $k_n$ of real eigenvalues of a matrix of size $n$ is asymptotically Gaussian with mean ${\bar{k}}_n=\mathcal{O}(\sqrt{n})$ and variance ${\bar{k}}_n(2-\sqrt{2})$. Moreover, we show that the limit distribution of real eigenvalues undergoes a transition between bimodal for $k_n=o(\sqrt{n})$ to unimodal for $k_n=\mathcal{O}(n)$, 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.

10 pages, 12 figures

Cited by in corpus (4)

• The Probability That All Eigenvalues are Real for Products of Truncated Real Orthogonal Random Matrices
• On the number of real eigenvalues of a product of truncated orthogonal random matrices
• On the real spectrum of a product of Gaussian random matrices
• How many eigenvalues of a product of truncated orthogonal matrices are real?